Progress in industrial mathematics at ECMI 2010

MATHEMATICS IN INDUSTRY Editors Hans-Georg Bock Frank de Hoog Avner Friedman Arvind Gupta Andr´e Nachbin Helmut Neunzert William R. Pulleyblank Torgeir Rusten Fadil Santosa Anna-Karin Tornberg

THE EUROPEAN CONSORTIUM FOR MATHEMATICS IN INDUSTRY SUBSERIES Managing Editor Vincenzo Capasso Editors Luis L. Bonilla Robert Mattheij Helmut Neunzert Otmar Scherzer

For further volumes: http://www.springer.com/series/4650

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Michael G¨unther Andreas Bartel Markus Brunk Sebastian Sch¨ops Michael Striebel Editors

Progress in Industrial Mathematics at ECMI 2010 With 194 Figures, 71 in color and 26 Tables

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Editors Michael G¨unther University of Wuppertal Wuppertal Germany

Sebastian Sch¨ops University of Wuppertal Wuppertal Germany

Andreas Bartel University of Wuppertal Wuppertal Germany

Michael Striebel University of Wuppertal Wuppertal Germany

Markus Brunk Robert Bosch GmbH Stuttgart Germany

ISBN 978-3-642-25099-6 e-ISBN 978-3-642-25100-9 DOI 10.1007/978-3-642-25100-9 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2012933374 Mathematics Subject Classification (2010): 34, 35, 49, 60, 62, 65, 68, 70, 74, 76, 87, 80, 81, 86, 91, 92, 94 c Springer-Verlag Berlin Heidelberg 2012  This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

The 16th conference of the European Consortium for Mathematics in Industry took place in the Historische Stadthalle Wuppertal, the historical city hall of Wuppertal, Germany, from July 26 to July 30, 2010. This venue, a member of the Historic Conference Centers of Europe, is one of the most remarkable concert halls and conference centers in Europe. The organizers welcomed nearly 250 registered participants from 30 different countries. Most of the participants contributed actively to the scientific program of the conference, which included—besides a poster session accompanied by an extended coffee break sponsored by ST Microelectronics—10 plenary talks, 132 talks within 27 minisymposia and 57 contributed talks. In this proceedings, industrial mathematics appears in a wide range of applications and methods, reflecting the topics addressed at ECMI 2010: from Circuit and Electromagnetic Device Simulation, Model Order Reduction for Chip Design, Uncertainties and Stochastics, Production, Fluids, Life and Environmental Sciences to Dedicated and Versatile Methods. We should mention that of the 106 contributions that were submitted to these proceedings, 76 have been selected for publication after a peer review process. We would like to recall some of the highlights of our conference: On Monday evening, the public lecture Modern Mathematics for Better Technologies was given by Volker Mehrmann, president of Matheon. He emphasized mathematics as an innovation enabler for industry and business, and as an absolutely essential prerequisite for Europe on its way to becoming the leading knowledge-based economy in the world. The same day, Volker Mehrmann chaired a panel discussion to promote and facilitate this process based on the outcome and recommendations of the Forward Look Project Mathematics and Industry of the European Science Foundation. On Wednesday, the Associazione Angelo Marcello Anile and ECMI together awarded the first Anile-ECMI Prize for Mathematics in Industry, which is dedicated to young researchers for excellent PhD theses in industrial mathematics. It was initiated to honor the memory of the former ECMI council member Angelo Marcello Anile (1948–2007), Professor of Applied Mathematics at the University of Catania,

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Photograph: Sebastian Jarych.

Italy. This prize was awarded to Andriy Vasyliovich Hlod for his PhD thesis at TU Eindhoven, The Netherlands. It included an invited talk, which he gave on Jets of Viscous Fluid. Continuing a long tradition of the ECMI conferences and following the suggestion of the Local Organizing Committee, honorary membership of ECMI was awarded to Willi J¨ager, Professor of Applied Mathematics and founder of the IWR at the University of Heidelberg, for his pioneering work in Applied and Industrial Mathematics. The award ceremony took place during the conference dinner in the large ballroom of the historical city hall, a magnificent example of the art nouveau style of architecture. ECMI 2010 would never have been the success it was without the help of many. Among them, first of all, the participants, the speakers and the Program Committee. We thank Lambert T. Koch, Rector of the Bergische Universit¨at Wuppertal, whose negotiation skills—together with the gratefully acknowledged financial support of all our sponsors and partners (see: www.ecmi2010.eu)—allowed us to select for the conference the unique venue of ECMI 2010, the Historische Stadthalle Wuppertal for nearly 1 week. Special thanks go to our team assistants Elvira Mertens, Lisa Hartmann and Eva Winnem¨oller for their excellent administrative support. We are very grateful to our colleagues Matthias Ehrhardt and Roland Pulch for their invaluable work within the local organizing committee. Finally we would like to acknowledge the great support of the many students who helped us during the conference, e.g., by setting up the equipment and answering technical and practical questions. Wuppertal and Stuttgart Germany

Michael G¨unther Andreas Bartel Markus Brunk Sebastian Sch¨ops Michael Striebel

Contents

Part I

Circuit and Electromagnetic Device Simulation

Wavelet Algorithm for Circuit Simulation . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Kai Bittner and Emira Dautbegovic Structural Characterization of Circuit Configurations with Undamped Oscillations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Ricardo Riaza and Caren Tischendorf Entrainment Phenomena in Nonlinear Oscillations . . . . . .. . . . . . . . . . . . . . . . . . . . Hans Georg Brachtendorf and Rainer Laur Initial Conditions and Robust Newton-Raphson for Harmonic Balance Analysis of Free-Running Oscillators .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Jarmo Virtanen, E. Jan W. ter Maten, Theo G.J. Beelen, Mikko Honkala, and Mikko Hulkkonen Rational Modeling Algorithm for Passive Microwave Structures and Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Dirk Deschrijver, Tom Dhaene, Oliver Salazar Celis, and Annie Cuyt An Existence Result for Index-2 PDAE System Arising in Semiconductor Modeling .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Giuseppe Al`ı, Andreas Bartel, and Nella Rotundo Deterministic Numerical Solution of the Boltzmann Transport Equation. Karl Rupp, Tibor Grasser, and Ansgar J¨ungel Analysis of Self-Heating Effects in Sub-Micron Silicon Devices with Electrothermal Monte Carlo Simulations . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Orazio Muscato and Vincenza Di Stefano

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Diffusive Limit of a MEP Hydrodynamical Model Obtained from the Bloch-Boltzmann-Peierls Equations for Semiconductors .. . . . . . . . Giuseppe Al`ı, Vittorio Romano, and Nella Rotundo

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Automatic Layout Optimization of Power Discrete Devices Using Innovative Distributed Model Techniques . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Giuseppe Greco and Salvatore Rinaudo

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3D Stress Simulations of Nano Transistors . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Abderrazzak El Boukili Hybrid Formulations and Discretisations for Magnetoquasistatic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Herbert De Gersem, Stephan Koch, and Thomas Weiland

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A 3D Meshless Approach for Transient Electromagnetic PDEs . . . . . . . . . . . . 107 Guido Ala, Elisa Francomano, and Antonino Spagnuolo Modelling and Analysis of the Nonlinear Dynamics of the Transrapid and Its Guideway . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 113 Michael Dellnitz, Florian Dignath, Kathrin Flaßkamp, Mirko Hessel-von Molo, Martin Kr¨uger, Robert Timmermann, and Qinghua Zheng Part II

Model Order Reduction for Chip Design

A Structure Preserving FGMRES Method for Solving Large Lyapunov Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 131 Matthias Bollh¨ofer and Andr´e K. Eppler Model Reduction for Linear Descriptor Systems with Many Ports . . . . . . . . 137 Peter Benner and Andr´e Schneider Model Order Reduction of Nonlinear Systems By Interpolating Input-Output Behavior .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 145 Michael Striebel and Joost Rommes Krylov-Subspace Based Model Reduction of Nonlinear Circuit Models Using Bilinear and Quadratic-Linear Approximations . . . . . . . . . . . . 153 Peter Benner and Tobias Breiten POD Model Order Reduction of Electrical Networks with Semiconductors Modeled by the Transient Drift–Diffusion Equations . . . . 161 Michael Hinze, Martin Kunkel, and Ulrich Matthes Model Order Reduction of Electrical Circuits with Nonlinear Elements. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 169 Andreas Steinbrecher and Tatjana Stykel

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Structure-Exploiting Symbolic-Numerical Model Reduction of Nonlinear Electrical Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 179 Oliver Schmidt Using Sensitivities for Symbolic Analysis and Model Order Reduction of Systems with Parameter Variation . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 187 Christian Salzig, Matthias Hauser, and Alberto Venturi Multi-Objective Optimization of RF Circuit Blocks via Surrogate Models and NBI and SPEA2 Methods . . . .. . . . . . . . . . . . . . . . . . . . 195 Luciano De Tommasi, Theo G.J. Beelen, Marcel F. Sevat, Joost Rommes, and E. Jan W. ter Maten Part III

Uncertainties and Stochastics

On a Numerical Approximation Scheme for Construction of the Early Exercise Boundary for a Class of Nonlinear Black–Scholes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 207 ˇ coviˇc Daniel Sevˇ On the Stability of a Compact Finite Difference Scheme for Option Pricing .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 215 Bertram D¨uring and Michel Fourni´e Stationary Solutions of Some Nonlinear Black–Scholes Type Equations Arising in Option Pricing .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 223 Maria de F´atima Fabi˜ao, Maria do Ros´ario Grossinho, Eva Morais, and Onofre Alves Sim˜oes Variants of the Combination Technique for Multi-Dimensional Option Pricing .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 231 Janos Benk, Hans-Joachim Bungartz, Aliz-Eva Nagy, and Stefanie Schraufstetter Stochastic PDAE-Model and Associated Monte-Carlo Simulations for Elastic Threads in Turbulent Flows . . . . .. . . . . . . . . . . . . . . . . . . . 239 Nicole Marheineke and Raimund Wegener Production Networks with Stochastic Machinery Default .. . . . . . . . . . . . . . . . . . 247 Simone G¨ottlich, Stephan Martin, and Thorsten Sickenberger Verified Simulation for Robustness Evaluation of Tracking Controllers . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 255 Marco Kletting and Felix Antritter Structural Analysis for the Design of Reliable Controllers and State Estimators for Uncertain Dynamical Systems . . . . . . . . . . . . . . . . . . . . 263 Andreas Rauh and Harald Aschemann

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Wiener Calculus for Differential Equations with Uncertainties . . . . . . . . . . . . 271 Florian Augustin, Peter Rentrop, and Utz Wever Polynomial Chaos and Its Application to Delay Differential Equations with Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 283 Manuel Villegas Caballero Part IV

Production

Nonlinear Metamodeling of Bulky Data and Applications in Automotive Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 295 Igor Nikitin, Lialia Nikitina, and Tanja Clees Heat Transfer During Annealing of Steel Coils . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 303 Winston L. Sweatman, Steven I. Barry, and Mark McGuinness Deformations Arising During Air-Knife Stripping in the Galvanisation of Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 311 Graeme C. Hocking, Winston L. Sweatman, Alistair D. Fitt, and Chris Breward Modelling Preform and Mould Shapes in Blow Moulding . . . . . . . . . . . . . . . . . . 319 J.A.W.M. Groot, R.M.M. Mattheij, and C.G. Giannopapa Asymptotic Analysis of a Multi-Component Wet Chemical Etching Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 327 Jonathan Ward Numerical Treatment of Fiber–Fiber and Fiber-Obstacle Contacts in Technical Textile Manufacturing .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 335 Ferdinand Olawsky, Martin Hering-Bertram, Andre Schmeißer, and Nicole Marheineke A Model of Rotary Spinning Process . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 341 Andriy Hlod, Alphons A.F. van de Ven, and Mark A. Peletier Asymptotic Models of Different Complexity for Viscous Jets and Their Applicability Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 349 Walter Arne, Nicole Marheineke, and Raimund Wegener Wavelet Methods for the Representation, Analysis and Simulation of Optical Surfaces . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 357 Philipp Jester, Christoph Menke, and Karsten Urban Modelling Two-Dimensional Photopolymer Patterns Produced with Multiple-Beam Holography .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 365 Dana Mackey, Tsvetanka Babeva, Izabela Naydenova, and Vincent Toal

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Modeling Berry’s Phase in Graphene by Using a Quantum Kinetic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 373 Omar Morandi and Ferdinand Sch¨urrer Multi Scale Random Sets: From Morphology to Effective Behaviour . . . . . 381 Dominique Jeulin Part V

Modeling, Analysis and Computation of Fluid Dynamics Problems

Mathematical Modelling of Phase Change with a Flowing Thin Film . . . . . 399 Tim G. Myers and Sarah L. Mitchell On Modeling of Curved Jets of Viscous Fluid Hitting a Moving Surface . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 407 Andriy Hlod Air Elimination in Milk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 421 Michael Devereux and William Lee Quantum Navier–Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 427 Ansgar J¨ungel and Josipa-Pina Miliˇsi´c Travelling-Wave Similarity Solutions for an Unsteady Gravity-Driven Dry Patch.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 441 Y. Mohd Yatim, Brian R. Duffy, and Stephen K. Wilson Asymptotics of a Small Liquid Drop on a Cone and Plate Rheometer .. . . . 449 Vincent Cregan, Stephen B.G. O’Brien, and Sean McKee Finger Formation and Non-newtonian Fluids . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 457 Jean P.F. Charpin Numerical Modelling and Simulation of Ship Hull Geometries . . . . . . . . . . . . 465 Michael Hopfensitz, Juan C. Matutat, and Karsten Urban Analysis of Combustion and Turbulence Models in a Cylindrical Combustion Chamber . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 473 Alexandru Dumitrache, Florin Frunzulica, and Horia Dumitrescu Numerical Method for a Stefan-Type Problem with Interior Layers . . . . . . 479 Martin Viscor and Martin Stynes Part VI

Life and Environmental Sciences

The Post-Newtonian Geolocation Problem by TDOA . . . .. . . . . . . . . . . . . . . . . . . . 489 J.M. Gambi, M.C. Rodriguez-Teijeiro, and M.L. Garcia del Pino

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Analytical Method for Inverse Problems of Deep Magneto Telluric Sounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 497 Sharif E. Guseynov Modelling the Mechanical Behaviour of a Pharmaceutical Tablet Using PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 505 Norhayati Ahmat, Hassan Ugail, and Gabriela Gonz´alez Castro On Causality of Thermoacoustic Tomography of Dissipative Tissue. . . . . . . 513 Richard Kowar A Mathematical Model for Drug Delivery . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 521 Vo Thi Ngoc Tuoi, Rongbing Yang, Yury Rochev, and Martin Meere Estimators of the Intensity of Fibre Processes and Applications . . . . . . . . . . . 529 Paola M.V. Rancoita and Alessandra Micheletti Optimal Control Strategies for Stochastic/Deterministic Bioeconomic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 537 Darya Filatova Fishway Optimization Revisited .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 545 Lino J. Alvarez-V´azquez, Aurea Mart´ınez, Joaquim J. J´udice, Carmen Rodr´ıguez, Miguel E. V´azquez-M´endez, and Miguel A. Vilar On a Wind Farm Aggregate Model Based on the Output Rescaling of a Single Turbine Model . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 553 Luciano De Tommasi and Madeleine Gibescu Adaptive Modelling of Coupled Hydrological Processes with Application in Water Management.. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 561 Peter Bastian, Heiko Berninger, Andreas Dedner, Christian Engwer, Patrick Henning, Ralf Kornhuber, Dietmar Kr¨oner, Mario Ohlberger, Oliver Sander, Gerd Schiffler, Nina Shokina, and Kathrin Smetana Part VII

Dedicated and Versatile Methods

FEINS: Finite Element Solver for Shape Optimization with Adjoint Equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 573 Ren´e Schneider Invariant Loading for Full Vehicle Simulation . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 581 Michael Burger, Klaus Dreßler, Albert Marquardt, Michael Morr, and Lothar Witte Structure Preserving Spatial Discretization of a Piezoelectric Beam. . . . . . . 587 Thomas Voß

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Error Estimates for Finite-Dimensional Approximations in Control of Distributed Parameter Systems . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 595 Andreas Rauh, J¨oran Ritzke, and Harald Aschemann Solving Non-smooth Delay Differential Equations with Multiquadrics. . . . 603 Francisco Bernal Higher-Order Matrix Splines for Systems of Second-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 611 Emilio Defez, Michael M. Tung, Jacinto Javier Ib´an˜ ez, and Luna Soler Multiscale Methods in Time and Space .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 619 Konstantin Fackeldey A Cache-Oblivious Sparse Matrix–Vector Multiplication Scheme Based on the Hilbert Curve . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 627 Albert-Jan N. Yzelman and Rob H. Bisseling Absorbing Boundary Conditions for Solving Stationary Schr¨odinger Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 635 Pauline Klein, Xavier Antoine, Christophe Besse, and Matthias Ehrhardt Diffusion on Surfaces of Revolution.. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 643 Michael M. Tung Verified Analysis of a Biomechanics-Related System . . . .. . . . . . . . . . . . . . . . . . . . 651 Ekaterina Auer Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 659 Authors Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 665

Part I

Circuit and Electromagnetic Device Simulation

Overview Today’s electric and electronic industries rely heavily on computer aided engineering tools. The high complexity of devices and the increasing speed of innovation cycles necessitate virtual prototyping. This allows such production at a competitive time to market because virtual experiments are faster and cheaper than their physical ancestors. Thus numerical tools for those simulations play a key role in the electrical engineering industry. Commonly, the underlying principles from physics are well established but the simulation techniques are still a topic of ongoing research, e.g. due to new computational possibilities stemming from advances in computer architectures. The research focuses in particular on improving the general efficiency and robustness of simulations (more accurate results in less time) and the interaction/coupling of multiphysical problems (secondary effects cannot be disregarded anymore). In particular electronic industry relies on efficient numerical experiments of their new designs, e.g., layouts for printed circuit boards. The corresponding circuits result from combining smaller subcircuits. One of the most common configurations is the oscillator, i.e., an electronic device that creates a repetitive signal. Owing to the large number of such devices and their nonlinearity often computational expensive time-domain simulations are necessary. New efficient methods are needed to validate the designs in a reasonable time. To this end K. Bittner and E. Dautbegovic propose in ‘Wavelet Algorithm for Circuit Simulation’ the usage of wavelets in circuit simulation. The authors present a new algorithm based on spline wavelets. The unknowns are expanded into a wavelet representation, which is determined as a solution to nonlinear equations derived from the circuit equations by a Galerkin discretization. Applications, e.g. an oscillator, show that for the same accuracy fewer grid points are needed compared to standard transient analysis. The structural aspects of circuit configurations with oscillations are analyzed by R. Riaza and C. Tischendorf in ‘Structural Characterization of Circuit Configurations with Undamped Oscillations’. Undamped oscillations in linear circuits arise

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from the existence of purely imaginary eigenvalues in the matrix pencil spectrum. The authors investigate the corresponding circuit configurations based on branchoriented circuit models and on several results from digraph theory. H.G. Brachtendorf and R. Laur study in ‘Entrainment Phenomena in Nonlinear Oscillations’ entrainment phenomena in nonlinear oscillations in particular for devices that mix down a radio frequency signal to an intermediate frequency (IF). Circuit designs have been developed using injection locking for the design, but unwanted temporary entrainment known as “pulling” can be a severe cause of performance degradation for zero-IF or low-IF transceivers. The corresponding entrainment effects have been studied for decades. The authors develop a new theory based on a perturbation technique employing Floquet’s theory. On the other hand, when analyzing oscillators in the frequency domain, the simulation often suffers from poor initial conditions. Hence J. Virtanen et al. propose in ‘Initial Conditions and Robust Newton-Raphson for Harmonic Balance Analysis of Free-Running Oscillators’ to use time integration to obtain estimates for the oscillation frequency and for the oscillator solution. They also describe new techniques from bordered matrices and eigenvalue methods to improve Newton methods for finite difference techniques in the time domain as well as for Harmonic Balance. The classical network approach is not sufficient if complex devices must be taken into account and lumped device models are not available. Then the extraction of macromodels by using tabulated S-parameter frequency responses allows us to synthesize SPICE compatible models. In ‘Rational Modeling Algorithm for Passive Microwave Structures and Systems’ D. Deschrijver et al. discuss a method that constructs such macromodels which are passive by construction. The authors apply a new passivity enforcement technique that is able to guarantee passivity by means of an overdetermined least-squares fitting algorithm. In contrast, if the simulation engineer is interested in both the macroscopic behavior of the circuit and additionally in the microscopic phenomena inside of a particular device, then only a full coupled simulation is the way out. G. Al`ı et al. discuss in ‘An Existence Result for Index-2 PDAE System Arising in Semiconductor Modeling’ the existence of solutions for the case of an electric network containing semiconductor devices. The coupled system consists of the Modified Nodal Analysis equations with multi-dimensional elliptic partial differential equations modeling the devices (drift diffusion model). The microscopic behavior of charge carriers in semiconductors is described more accurately by the solution of the Boltzmann transport equation (BTE). However, the Monte-Carlo method, preferably used for solving the BTE suffers from intense numerical complexity. A numerically less expensive approach is presented by K. Rupp et al. in ‘Deterministic Numerical Solution of the Boltzmann Transport Equation’. They consider the deterministic numerical solution of the Boltzmann transport equation and present the deterministic Spherical Harmonic Expansion (SHE) method for the determination of the solution of the BTE. The solution of the SHE-model faces the problem of intense memory consumption. However, the

I Circuit and Electromagnetic Device Simulation

3

authors present a new lossless system matrix compression scheme which allows for a significant reduction of memory consumption. Due to increasing complexity and ongoing miniaturization in the layout and production of semiconductor devices, forecasting of thermal effects and prediction of hotspots has become more and more important within the last years and compels us to develop more reliable models. This subject has been addressed by several authors during the ECMI 2010. In ‘Analysis of Self-Heating Effects in Sub-Micron Silicon Devices with Electrothermal Monte Carlo Simulations’ O. Muscato and V. Di Stefano solve the BTE with a thermal model by a Monte-Carlo method, coupled to a Cattaneo-like equation for the lattice temperature, which is obtained in the framework of extended irreversible thermodynamics. G. Al`ı et al. on the other hand present a new macroscopic energy-transport model incorporating thermal effects in the crystal lattice of the semiconductors. This is done by considering the diffusive limit of MEP hydrodynamical model obtained from the Bloch-Boltzmann-Peierls equation for semiconductors in ‘Diffusive Limit of a MEP Hydrodynamical Model Obtained from the Bloch-Boltzmann-Peierls Equations for Semiconductors’. Thereby, the authors introduce a smallness parameter related to the transition probabilities in the collision operators and a diffuse scaling at the level of the Lagrangian multipliers that appear in the closure relations. Based on this procedure, the authors obtain a system of model equations, which include energy-transport equations that are consistent with the linear irreversible thermodynamics. G. Greco and S. Rinaudo give us some insight into the application of semiconductor device simulation at ST Microelectronics in Catania and present automatic layout optimization of power discrete devices using innovative distributed model techniques, ‘Automatic Layout Optimization of Power Discrete Devices Using Innovative Distributed Model Techniques’. The increasing usage of power MOSFETs in application enforces an optimization of the geometry. This is essential for the reduction of hotspots. Since microscopic models are too costly to simulate in an optimization loop, a distributed model technique is described where the device is divided into several cells connected e.g. by transmission lines. This distributed approach is embedded into an optimization algorithm in order to optimize geometry. A. El Boukili presents 3D stress simulations of nano transistors, ‘3D Stress Simulations of Nano Transistors’. Mechanical stress is intentionally used by semiconductor manufacturers to optimize the performance of devices. This requires semiconductor models to incorporate these effects. The author extends three dimensional semiconductor model equations by coupling them with a mechanical stress model from linear elasticity theory. Based on his simulation results he proposes the modification of existing mobility results and gives interesting ideas in this direction. On the macroscopic level, many devices can be represented by Maxwell’s equations. They describe the spatial distribution of the electromagnetic field by partial differential equations. Simulations follow typically the method of lines: the equations are restated in a more compact formulation (e.g. using the magnetic vector potential as its unknown) and then discretized in space. Finally the resulting system is solved in the time domain.

4

I Circuit and Electromagnetic Device Simulation

In ‘Hybrid Formulations and Discretisations for Magnetoquasistatic Models’ H. De Gersem et al. propose the usage of hybrid formulations and discretizations for the magnetoquasistatic approximation of Maxwell’s equations. This increases the modeling flexibility because distinct model regions can be tackled by different approaches. The authors explain in detail which algebraic solution techniques are necessary for solving the coupled systems of equations and demonstrate their approach using numerical examples. G. Ala et al. diverge from the standard discretization procedures and propose in ‘A 3D Meshless Approach for Transient Electromagnetic PDE’ a new meshless approach for the Maxwell’s 3D full wave equation. They employ the smoothed particle hydrodynamic method by considering the particles as interpolation points, arbitrarily placed in the computational domain. Simulations validate the method and allow a comparison with standard approaches, i.e., the finite difference time domain method. The final contribution to this chapter, ‘Modelling and Analysis of the Nonlinear Dynamics of the Transrapid and Its Guideway’, focusses on a coupled simulation of a real-world industrial example: the Transrapid and its guideway. M. Dellnitz et al. couple the mechanical and electromagnetic parts, i.e., the control subsystems, magnet subsystems, a lateral cross-section and a vertical dynamics model as a multibody system. It is verified using simulations, eigenmode analysis and displacement measurements from train passages on a test track. They show that ground vibrations caused by the vehicle can be significantly reduced by a flexible spring-mass system as a support for the girders. Wuppertal and Stuttgart, Germany

Sebastian Sch¨ops Markus Brunk

Wavelet Algorithm for Circuit Simulation Kai Bittner and Emira Dautbegovic

Abstract We present a new adaptive circuit simulation algorithm based on spline wavelets. The unknown voltages and currents are expanded into a wavelet representation, which is determined as solution of nonlinear equations derived from the circuit equations by a Galerkin discretization. The spline wavelet representation is adaptively refined during the Newton iteration. The resulting approximation requires an almost minimal number of degrees of freedom, and in addition the grid refinement approach enables very efficient numerical computations. Initial numerical tests on various types of electronic circuits show promising results when compared to the standard transient analysis.

1 Introduction Wavelet theory emerged during the twentieth century from the study of CalderonZygmund operators in mathematics, the study of the theory of subband coding in engineering and the study of renormalization group theory in physics. Recent approaches [1, 4, 7–9] to the problem of multirate envelope simulation indicate that wavelets could also be used to address the qualitative simulation challenge by a development of novel wavelet-based circuit simulation techniques capable of an efficient simulation of a mixed analog-digital circuit [6].

K. Bittner () University of Wuppertal, Gaußstraße 20, 42119 Wuppertal, Germany e-mail: [email protected] E. Dautbegovic Infineon Technologies AG, 81726 Munich, Germany e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 1, © Springer-Verlag Berlin Heidelberg 2012

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K. Bittner and E. Dautbegovic

The wavelet expansion of a function f is given as f D

X k2I

ck k C

1 X X

dj k

jk:

(1)

j D0 k2Kj

Here, j refers to a level of resolution, while k describes the localization in time or space, i.e., j k is essentially supported in the neighborhood of a point xj k of the domain, where the wavelet expansion P is defined. The wavelet expansion can be seen as coarse scale approximation k2I ck k by the scaling functions k complemented by information on details of increasing resolution j in terms of the wavelets j k . Since a wavelet basis consist of an infinite number of wavelets one has to consider approximations of f by partial sums of the wavelet expansion (1), which can, e.g., be obtained by ignoring small coefficients.

2 Wavelet-Based Galerkin Method We consider circuit equations in the charge/flux oriented modified nodal analysis (MNA) formulation, which yields a mathematical model in the form of an initialvalue problem of differential-algebraic equations (DAEs):    d  q x.t/ C f x.t/ D s.t/: dt

(2)

Here x is the vector of node potentials and specific branch voltages and q is the vector of charges and fluxes. Vector f comprises static contributions, while s contains the contributions of independent sources. In our adaptive wavelet approach we first discretize the MNA equation (2) in terms of the wavelet basis functions, by expanding P x as a linear combination of wavelets or related basis functions 'i , i.e., x D niD 0 ci 'i . Furthermore, we integrate the circuit equations against test functions ` and obtain the equations Z T     d  q x.t/ C f x.t/  s.t/ ` dt D 0; (3) dt 0 for ` D 1; : : : ; n. Together with the initial conditions x.0/ D x0 , we have now n C 1 vector valued equations, which determine the coefficients ci provided that the test functions ` are chosen suitably to the basis functions 'i . The nonlinear system (3) is solved by Newton’s method. With a good initial guess, Newton’s method is known to converge quadratically. Unfortunately, a good initial guess is usually not available, and convergence can often only be obtained after a large number of (possibly damped) Newton steps. On the other hand, to get a good approximation of the solution of (2), the space X D spanf'k W k D 0; : : : ; ng has to be sufficiently large and the computational cost of each step depends on n D dim X .

Wavelet Algorithm for Circuit Simulation

7

Here, we take advantage from the use of wavelets. The Newton iteration is started on a coarse subspace X0 of small dimension, which provides us with a first coarse approximation x .0/ of the solution. Then x .0/ is used as initial guess for a Newton iteration in a finer space X1  X0 , leading to an improved approximation x .1/ . One positive effect, which can be observed in numerical tests, is that a single Newton step in the beginning of the algorithm is relatively cheap, i.e., having only a poor initial guess for x .i / with i small has only a negligible effect on the performance of the algorithm. On the other hand, due to the excellent initial guess in the higher dimensional spaces Xi with i large, we need only a few of the costly Newton steps, which are necessary in order to achieve a required accuracy. The embedding Xi  Xi C1 is ensured by the use of wavelet subsets, i.e.,  Xi D span fk W k 2 I g [ f

jk

 W .j; k/ 2 i g ;

i  i C1 ;

i.e., we add adaptively more and more wavelets to the expansion. Due to the intrinsic properties of wavelets [6] an adaptive wavelet approximation can provide an efficient representation of functions with steep transients, which often appear in a mixed analog/digital electronic circuit. However, for an efficient circuit simulation we have to take further properties of a wavelet system into account. We consider spline wavelets to be the optimal choice since spline wavelets are the only wavelets with an explicit formulation. This permits the fast computation of function values, derivatives and integrals, which is essential for an efficient solution of a nonlinear problem as given in (2) (see also [3,5]). Spline wavelets have already been used for circuit simulation [10]. However, here we use a completely new approach based on spline wavelets from [2].

3 Numerical Tests A prototype of the proposed wavelet algorithm is implemented within the framework of a productively used circuit simulator and tested on a variety of circuits. For all examples we have compared the CPU time and the grid size (i.e., the number of spline knots or time steps) with the corresponding results from transient analysis of the same circuit simulator. The error is estimated by comparison with well-established and highly-accurate transient analysis. The estimate shown in the signal is the maximal absolute difference over all grid points of the transient analysis, which gives a good approximation of the maximal error. That is, if we can obtain a small error for the wavelet analysis, which proves good agreement with the standard method. In particular, since we compare the solutions of two independent methods we have very good evidence that we approximate the solution of the underlying DAE’s with the estimated error.

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3.1 Amplifier The amplifier is simulated with a pulse signal of period 1 ns, which is modulated by a piecewise smooth amplitude (see Fig. 1). The wavelet method runs over 100 ns. The results show a satisfying performance also for digital-like input signals (Figs. 2 and 3).

3.2 Oscillator The oscillator is an autonomous circuit without an external input signal. The simulation runs over 20 ns. As can be seen from Fig. 4, an excellent agreement with highly-accurate transient analysis is achieved. It should be noted that after the oscillator has reached its periodic steady state the wavelet method works very fast, since the solution from one interval is an excellent initial guess for the next interval.

Fig. 1 Detail of input and output signal for the amplifier

CPU time

Grid size

Wavelet

20 10 5

Transient

Wavelet Transient

20000 10000 5000

2 1 0.5

2000 –4

10

–3

10

–2

10

–1

10

Error 10

–4

–3

10

–2

10

–1

Error

10

Fig. 2 Simulation statistics for the amplifier. Computation time versus error (left), and grid size versus error (right) for transient analysis and adaptive wavelet analysis

Wavelet Algorithm for Circuit Simulation

9

Fig. 3 Output signal for the oscillator CPU time

Wavelet

20

Transient

Grid size

Wavelet Transient

20000

10 10000

5

5000 2 2000

1 –3

10

10

–2

10

—1

Error

10–3

10–2

Error

Fig. 4 Simulation statistics for the oscillator. Computation time versus error (left), and grid size versus error (right) for transient analysis and adaptive wavelet analysis

Fig. 5 Time domain input and output signal for the mixer

3.3 Mixer The mixer is simulated with input frequencies 950 MHz and 1 GHz. The simulation runs over 30 ns. In particular, for high accuracies the number of degrees of freedom is essentially reduced, while the computation time is at least of the same order (Fig. 6).

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K. Bittner and E. Dautbegovic

CPU time

Wavelet

100 50 20 10 5 5 2 1 0.5 0.2

Transient

Grid size

Wavelet Transient

20000 10000 5000 2000 1000

–5

10

–4

10

10

–3

10

–2

10

–1

Error 10

–5

10

–4

10

–3

–2

10

–1

Error

10

Fig. 6 Simulation statistics for the mixer. Computation time versus error (left), and grid size versus error (right) for transient analysis and adaptive wavelet analysis

4 Conclusion The results of the simulations indicate that the wavelet based method may achieve and in some cases surpass performance of the standard transient analysis. Apparently, the number of degrees of freedom can be smaller than for the transient analysis for comparable accuracy. However, this advantage of the wavelet algorithm does not always result (yet) in a smaller computation time. On the other hand it can be expected that the productive implementation of the wavelet algorithm can be further optimized. Therefore our activities on optimization and further development of the wavelet-based algorithm are continuing. Acknowledgements This work has been supported within the EU Seventh Research Framework Project (FP7) ICESTARS with the grant number 214911.

References 1. Barthel, A., Knorr, S., Pulch, R.: Wavelet based methods for multirate partial differentialalgebraic equations. Appl. Numer. Math. 59, 495–506 (2008) 2. Bittner, K.: Biorthogonal spline wavelets on the interval. In: Chen, G., Lai, M.J. (eds.) Wavelets and Splines: Athens 2005, pp. 93–104. Nashboro Press, Brentwood (2006) 3. Bittner, K., Urban, K.: Adaptive wavelet methods using semiorthogonal spline wavelets: Sparse evaluation of nonlinear functions. Appl. Comput. Harmon. Anal. 24, 94–119 (2008) 4. Christoffersen, C., Steer, M.: State-variable-based transient circuit simulation using wavelets. IEEE Microw. Wireless Compon. Lett. 11, 161–163 (2001) 5. Dahmen, W., Schneider, R., Xu, Y.: Nonlinear functionals of wavelet expansions — adaptive reconstruction and fast evaluation. Numer. Math. 86, 49–101 (2000) 6. Dautbegovic, E.: Wavelets in circuit simulation. In: Roos, J., Costa, L. (eds.) Scientific Computing in Electrical Engineering, Mathematics in Industry, vol. 14, pp. 131–142. Springer, Berlin (2010) 7. Dautbegovic, E., Condon, M., Brennan, C.: An efficient nonlinear circuit simulation technique. IEEE Trans. Microw. Theor. Tech. 53, 548–555 (2005)

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8. Soveiko, N., Gad, E., Nakhla, M.: A wavelet-based approach for steady-state analysis of nonlinear circuits with widely separated time scales. IEEE Microw. Wireless Compon. Lett. 17, 451–453 (2007) 9. Zhou, D., Cai, W.: A fast wavelet collocation method for high-speed circuit simulation. IEEE Trans. Circ. Syst. 46, 920–930 (1999) 10. Zhou, D., Cai, W.: A fast wavelet collocation method for high-speed circuit simulation. IEEE Trans. Circ. Syst. I: Fund. Theor. Appl. 46, 920–930 (1999)

Structural Characterization of Circuit Configurations with Undamped Oscillations Ricardo Riaza and Caren Tischendorf

Abstract Undamped oscillations in linear circuits arise from the existence of purely imaginary eigenvalues (PIEs) in the matrix pencil spectrum which characterizes the circuit dynamics. We investigate here the circuit configurations which yield purely imaginary eigenvalues for all values of the capacitances and inductances in strictly passive problems. Our analysis is based on the use of branch-oriented circuit models and on several results from digraph theory.

1 Introduction This communication extends our previous research on qualitative aspects of electrical and electronic circuits [7, 8] by elaborating on the so-called hyperbolicity problem. A circuit composed of independent voltage and current sources, and linear time-invariant resistors, inductors, and capacitors is said to be hyperbolic if all the eigenvalues in the spectrum are away from the imaginary axis [8]; this is a standard concept in dynamical systems theory. Linear time-invariant circuits displaying purely imaginary eigenvalues (PIEs) are important for two main reasons: on the one hand, they are responsible for undamped oscillations in linear cases and, on the other hand, when a linear circuit describes the linearization of a nonlinear one, the existence of PIEs may be responsible for Hopf bifurcations in the nonlinear problem, which in turn generate nonlinear oscillations. Our present goal is to discuss

R. Riaza () Depto. Matem´atica Aplicada TTI, ETSI Telecomunicaci´on, Universidad Polit´ecnica de Madrid, Ciudad Universitaria s/n, 28040 Madrid, Spain e-mail: [email protected] C. Tischendorf Mathematisches Institut, Universit¨at zu K¨oln, Weyertal 86-90, 50931 K¨oln, Germany e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 2, © Springer-Verlag Berlin Heidelberg 2012

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R. Riaza and C. Tischendorf

a full characterization of the configurations which lead to purely imaginary eigenvalues for all positive values of the capacitances and inductances involved in the circuit. Current approaches to circuit analysis in the time-domain make systematic use of models based on differential-algebraic equations (DAEs) (see e.g. [1–3, 5, 6, 9, 10]). Background material on the DAE circuit models to be used is given in Sect. 2. The attention will be focused on so-called branch-oriented models. In Sect. 3 we characterize the circuit configurations which yield PIEs for all reactive values, the graph-theoretic notion of a P-structure being the key element in our analysis. Finally, Sect. 4 comprises some concluding remarks.

2 Circuit Model A linear electrical circuit defined by resistors, capacitors, inductors, and independent voltage and current sources can be modelled by means of the DAE C vc0 D ic Lil0

(1a)

D vl

(1b)

0 D Bc vc C Bl vl C Br vr C Bj vj C Bu vs .t/

(1c)

0 D Qc ic C Ql il C Qr ir C Qj is .t/ C Qu iu

(1d)

0 D ir  Gvr :

(1e)

We split the vectors of (capacitor, inductor, resistor, current and voltage sources) branch voltages and currents as v D .vc ; vl ; vr ; vj ; vs .t// and i D .ic ; il ; ir ; is .t/; iu /. Equations (1c) and (1d) express in matrix form Kirchhoff’s voltage and current laws, making use of the reduced loop and cutset matrices B D .Bc Bl Br Bj Bu /, Q D .Qc Ql Qr Qj Qu /. Further details on these models can be found in [4, 6]. We will assume throughout the paper that the capacitance and inductance matrices C and L are diagonal with positive entries, and that the conductance matrix G is positive definite. Hence, all devices are linear and strictly passive, and capacitors and inductors are uncoupled. Additionally, the circuits will have neither IC-cutsets (that is, cutsets formed just by current sources and/or capacitors) nor VL-loops (namely, loops defined by voltage sources and/or inductors only). The eigenvalue analysis can be simplified by working with the so-called reduced circuit obtained after open-circuiting current sources and short-circuiting voltage sources (cf. [8]). We p will hence look for values of  of the form ˙˛j , with ˛ 2 R  f0g (and j D 1), yielding non-trivial solutions for the linear system C vc D ic

(2a)

Lil D vl

(2b)

Structural Characterization of Circuit Configurations with Undamped Oscillations

15

Bc vc C Bl vl C Br vr D 0

(2c)

Qc ic C Ql il C Qr ir D 0

(2d)

ir D Gvr :

(2e)

Besides standard properties of digraphs [6,8] we will need the concept of a block. Definition 1. A node is said to be an articulation if the removal of it and its incident branches increases the number of connected components of the digraph. A digraph is said to be non-separable if it is connected and has no articulations. A block is a maximal non-separable subgraph. For our purposes, the main property of blocks is the fact that the branches of a block do not belong to any loop or cutset including branches from outside the block. Given a distinguished set of branches K, we will call a loop or cutset including elements from both K and G K a hybrid loop or cutset, respectively. The branches of a block K do not take part either in hybrid loops or in hybrid cutsets.

3 Purely Imaginary Eigenvalues We will make use of some previous results detailed in [8]. Specifically, the circuit pencil is known to have a zero eigenvalue if and only if the circuit has at least one IC-cutset or one VL-loop; we preclude these configurations in order to focus the hyperbolicity analysis on the existence of non-zero, purely imaginary eigenvalues. We also know from [8] that eigenvectors associated with purely imaginary eigenvalues must necessarily have vanishing voltage and current in the resistor branches. Additionally, a well-known property in circuit theory states that all eigenvalues of an LC-circuit are purely imaginary. Hence, if after open-circuiting current sources and short-circuiting voltage sources in a VIRLC circuit there exists an LC-block, then the spectrum includes a PIE. The converse is not true, however; counterexamples can be found in [8], where certain strictly passive RLC circuits without LC-blocks are shown to exhibit PIEs for certain values of the reactances. However, a natural conjecture says that if a circuit has PIEs for all positive values of the reactances, it must be because an LC-block shows up after open-circuiting current sources and short-circuiting voltage sources. This is actually true, as stated below. Theorem 1. A linear, time-invariant circuit has a pair of purely imaginary eigenvalues for all positive values of capacitances and inductances if and only if there exists an LC-block in the circuit obtained after open-circuiting current sources and short-circuiting voltage sources. The proof proceeds via the notion of a P-structure as introduced below. In what follows we work with the reduced RLC circuit without further explicit mention.

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In view of the identities vr D ir D 0 holding true for PIEs, the eigenvalueeigenvector equations (2) read C vc D ic

(3a)

Lil D vl

(3b)

Bc vc C Bl vl D 0

(3c)

Qc ic C Ql il D 0;

(3d)

for which a solution  D ˛j ¤ 0 is assumed to exist for all positive values of C and L. Needless to say, the actual values of  and ˛ will depend on C , L. Fix a set of values for C and L, and focus on the non-vanishing entries of vc , vl , ic and il within an associated eigenvector. Note that, from (3a) and (3b), exactly the same entries vanish in the voltage and the current vector. Additionally, not all vl ’s (hence not all il ’s) can vanish since, otherwise, the equation Bc vc D 0 resulting from (3c) would indicate the existence of a C-cutset. Analogously, not all ic ’s (hence not all vc ’s) may vanish since Ql il D 0 from (3d) would signal an L-loop. Denote by K the set of capacitive and inductive branches with non-vanishing voltage and current in the above-referred eigenvector. We will use the subscript k to denote the corresponding (non-vanishing) entries of vc , vl , ic and il , and also to specify the submatrices of Bc , Bl , Qc and Ql defined by the columns which correspond to K-branches, as well as the capacitances and inductances of the K-branches. This makes it possible to recast (3) as Ck vck D ick

(4a)

Lk ilk D vlk

(4b)

Bck vck C Blk vlk D 0

(4c)

Qck ick C Qlk ilk D 0:

(4d)

Note that every K-branch forms at least one cutset just with other K-branches, as a consequence of the fact that all vck and vlk in (4c) do not vanish. Indeed, since .vck ; vlk / 2 ker .Bck Blk /, this vector can be written as a linear combination of vectors describing K-cutsets; additionally, every K-branch must have a nonvanishing entry in at least one of these vectors since, otherwise, the corresponding entry in vck or vlk would vanish. Proceeding analogously, (4d) indicates that every K-branch forms at least one loop just with other K-branches. This motivates the following definition. Definition 2. A set K of capacitive and inductive branches, together with their incident nodes, is said to form a P-structure if every branch in K forms at least one cutset and at least one loop just with other branches from K. Here the cutset and the loop need not include all the branches in K; nor it must happen that the cutset and the loop involve the same branches. For the sake of

Structural Characterization of Circuit Configurations with Undamped Oscillations

17

terminological simplicity we will use K also to mean the subgraph defined by the K-branches and their incident nodes. P-structures are the candidates which may (but not necessarily do) support the existence of a PIE. In the light of Definition 2, the discussion above indicates that the branches corresponding to the non-vanishing entries of an eigenvector associated with a PIE form a P-structure. Now, an LC-block which does not amount to a single branch can be checked to be a P-structure (note, incidentally, that the P-structures from which a PIE-eigenvector arises include at least two branches, namely a capacitor and an inductor). Certainly, the converse is not true. The proof of Theorem 1 will be closely related to this fact. Fix an eigenvector associated with a PIE, to be denoted by .vc ; vl ; ic ; il /, and consider the associated P-structure K signaled by its non-vanishing entries. Let bk , nk and ck stand for the number of branches, nodes and connected components of K. Lemma 1. If the P-structure K is not a block, then the rank of Bk D .Bck Blk / is greater than bk  nk C ck ; if it is a block then rk Bk D bk  nk C ck . Indeed, K defines a subgraph and therefore its cycle space has dimension bk  nk C ck . This implies that there exist bk  nk C ck linearly independent K-loops, which are loops of the original digraph. If K is a block, then the absence of hybrid loops and cutsets easily yields rk Bk D bk  nk C ck . By contrast, if it is not a block, it can be shown that there must exist a hybrid loop; its K-entries cannot be expressed in terms of those corresponding to the bk  nk C ck K-loops mentioned above, meaning that in this case the rank of Bk must be greater than bk  nk C ck . According to Lemma 1, the first bk  nk C ck rows of (4c) read BQ k vk D 0, where vk stands for .vck ; vlk /. In turn, regarding the cutset matrix Q, notice that ker Qk is spanned by linearly independent K-loops. Since the K-loops are the same in the original circuit and in the K-subcircuit, this means that ker Qk equals ker QQ k , where QQ k is any (reduced) cutset matrix of the K-circuit. This means that (4d) can be recast as QQ k ik D 0, ik standing for .ick ; ilk /. The next result follows from these remarks. Lemma 2. A PIE of the original circuit is also a PIE of the corresponding K-subcircuit, the non-vanishing entries of the original eigenvector defining an eigenvector of the K-subcircuit. Proof of Theorem 1. Our reasoning is supported on the fact that all PIE-eigenvectors must arise from some P-structure, according to Lemma 2. Consider a P-structure K, and choose the values of capacitances and inductances of the K-subcircuit in such a way that all eigenvalues of that subcircuit are simple. This can be done for parameter values lying on an open dense subset in RbCk , where bk is the number of branches in K and RC is the set of positive real numbers, as a consequence of the fact that eigenvalues are given by the roots of the pencil determinantal polynomial, which has the form p.; C; L/ D am .C; L/m C : : : C a0 .C; L/: Note that a0 .C; L/ ¤ 0 because the absence of C-cutsets and L-loops rules out null eigenvalues. Multiple eigenvalues are defined by the intersection of p.; C; L/ D 0 and @p=@.; C; L/ D 0 and, therefore, occur only on a lower dimensional set of the parameter space. This means that the set of values of Ck , Lk for which all

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R. Riaza and C. Tischendorf

eigenvalues are simple is open and dense in RbCk , and implies that the corresponding branch equations have generically corank one. Equivalently, given the system Ck vck D ick

(5a)

Lk ilk D vlk

(5b)

BQck vck C BQlk vlk D 0

(5c)

QQ ck ick C QQ lk ilk D 0;

(5d)

its coefficient matrix has generically corank one when  D ˛j is an eigenvalue of the K-subcircuit. For an eigenvalue of the K-subcircuit to be an eigenvalue of the original circuit, not only (5c) has to be satisfied, but also the additional conditions coming from (4c). This means that the linear system 0

Ck 0 I

0

10

vck

1

B 0 I 0 L C B v C k C B lk C B B CB CD0 @ Bck Blk 0 0 A @ ick A 0 0 QQ ck QQ lk ilk

(6)

must have a non-trivial solution for the same value of . The coefficient matrix of (6) is a row-enlargement of the one corresponding to (5), which as indicated above has generically corank one. If (6) has a non-trivial solution, the additional rows in (6) must be linearly dependent on those within the matrix coming from (5), always for generic values of C , L. Provided that the P-structure K is not a block, there is at least one additional row in the Bk rows of (6) coming from a hybrid loop. It follows from Lemma 1 that this row cannot be expressed as a linear combination of the rows coming from BQ k only. Obviously, it cannot be written just in terms of the BQk and QQ k rows, either. Therefore, this linear dependence relation involves (some of) the Ck , Lk rows. Hence, if it is at all possible that system (6) has a non-vanishing solution, at least one algebraic restriction on the values of Ck and Lk must necessarily be met. We conclude that reactive values leading to PIEs, if any, must lie on a lower dimensional set of the parameter space. This shows that, for a PIE associated with a given P-structure K which is not a block, either it must happen that the PIE is a multiple eigenvalue of the K-subcircuit (and this may happen only for values of Lk , Ck lying on a lower dimensional set), or at least one restriction on the values of these reactances imposed by the existence of a hybrid loop must be met. Since this holds for all P-structures, the fact that none of them is a block restricts the possible values of inductances and capacitances to lie on a finite union of lower dimensional sets, which certainly defines a proper subset of RbCc Cbl . The existence of an LC-block then follows as a necessary condition for a PIE to exist for all values of the reactive parameters. 

Structural Characterization of Circuit Configurations with Undamped Oscillations

19

4 Concluding Remarks We have characterized in this communication the circuit configurations yielding purely imaginary eigenvalues for all values of capacitances and inductances in strictly passive RLC circuits. The hyperbolicity problem in general is however still open and a full structural study is in the scope of future research.

References 1. Est´evez-Schwarz, D., Tischendorf, C.: Structural analysis of electric circuits and consequences for MNA. Int. J. Circ. Theor. Appl. 28, 131–162 (2000) 2. G¨unther, M., Feldmann, U.: CAD-based electric-circuit modeling in industry. I: Mathematical structure and index of network equations. Surv. Math. Ind. 8, 97–129 (1999) 3. G¨unther, M., Feldmann, U.: CAD-based electric-circuit modeling in industry. II: Impact of circuit configurations and parameters. Surv. Math. Ind. 8, 131–157 (1999) 4. Hasler, M., Neirynck, J.: Nonlinear Circuits. Artech House, London (1986) 5. Reis, T.: Circuit synthesis of passive descriptor systems - a modified nodal approach. Int. J. Circ. Theor. Appl. 38, 44–68 (2010) 6. Riaza, R.: Differential-Algebraic Systems. Analytical Aspects and Circuit Applications. World Scientific, Singapore (2008) 7. Riaza, R., Tischendorf, C.: Qualitative features of matrix pencils and DAEs arising in circuit dynamics. Dyn. Syst. 22, 107–131 (2007) 8. Riaza, R., Tischendorf, C.: The hyperbolicity problem in electrical circuit theory. Math. Meth. Appl. Sci. 33, 2037–2049 (2010) 9. Takamatsu, M., Iwata, S.: Index characterization of differential-algebraic equations in hybrid analysis for circuit simulation. Int. J. Circ. Theor. Appl. 38, 419–440 (2010) 10. Tischendorf, C.: Topological index calculation of DAEs in circuit simulation. Surv. Math. Ind. 8, 187–199 (1999)

Entrainment Phenomena in Nonlinear Oscillations Hans Georg Brachtendorf and Rainer Laur

Abstract Entrainment or injection locking is the underlying effect of synchronization. It can therefore be observed in a variety of fields including physics, biology and electronic engineering. In recent years various circuit designs have been developed using injection locking for the design of i.e. quadrature oscillators, frequency dividers and circuits exhibiting low phase noise. On the other hand, unwanted temporary entrainment known as pulling can be a severe cause of performance degradation for zero-IF or low-IF transceivers. Therefore entrainment effects have been studied since decades (i.e. Andronov and Witt, Adler, Kurokawa). A general theory is still missing. In this paper, we give a theory of injection phenomena based on a perturbation technique employing Floquet’s theory. The theory is valid as long as the injected signal power is sufficiently small.

1 Introduction The study of injection or entrainment phenomena goes back to the work of Andronov and Witt [2], Adler [1] and Kurokawa [9]. The method has been used in a variety of circuits designs, i.e. [8,11] in recent years. This demand has spurred the interest in CAD engines for simulating injection phenomena [5, 6]. The technique here is based on previous work of the authors [3, 10].

H.G. Brachtendorf () University of Upper Austria, Softwarepark 11, 4232 Hagenberg, Austria e-mail: [email protected] R. Laur Institute for Electromagnetic Theory and Microelectronics, University of Bremen, Bibliothekstraße 1 – 28359 Bremen, Germany e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 3, © Springer-Verlag Berlin Heidelberg 2012

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H.G. Brachtendorf and R. Laur

In the sequel the system of nonlinear ordinary differential equations of dimension N : x.t/ P D f .x.t// C b.t/ wherein t 2 R is time and x W R ! RN is the vector of the unknown node voltages and branch currents. f W RN ! RN is the vector of sums of currents entering each node and branch voltages. Furthermore b.t/ W R ! RN is the vector of input sources. Moreover, we assume that for b.t/  0 8 t the circuit behaves as a free running oscillator, i.e. xss .t/ D xss .t C T / with period T and fundamental steady state frequency fss D T1 . If energy is periodically injected into the oscillator at frequencies in the vicinity of fss or at multiples (superharmonics) or rational fractions (subharmonics) thereof, i.e. b.t/ D b.t C TO /, it has been observed that the beat frequency f vanishes completely within a specific range fig. 1. The range !  ; ! C is referred to as the locking range. It is well known that the region of entrainment depends on the amplitude of the injection signal and the detuning, referred to as Arnold tongues. From fig. 2 one can see that besides harmonic also super- and subharmonic entrainment can occur. A first understanding of the effect can be obtained by considering the phase portrait depicted in fig. 3. A periodic impulse train stimulus is injected into the oscillator circuit. This impulse train leads to both an orbital and an amplitude deviation. Stability of the oscillator presupposed, the amplitude deviation vanishes |Δw|

Fig. 1 Range of entrainment with bifurcation points ! C and ! 

w–

w

w+

wss

A

Fig. 2 Arnold tongues. The shaded areas depict the locking range as function of the amplitude A

1/2

2/3

1/1

3/2

2

w / w0

Entrainment Phenomena in Nonlinear Oscillations

T

x2

23

T

x2

ϕ

x1

x1 ϕ

Fig. 3 Phase portrait of locking by an impuls train with period TO

whereas the phase deviation remains. Depending on the frequency of the impulse train, the relative phases between the impuls train and the orbits differs. As one can see in fig. 3 (left), the phase shift ' is positive in the first quadrant originating from a stimulus with a frequency larger than fss . On the other hand, the figure on the left corresponds to a stimulus frequency lower than fss , therefore ' is negative in the fourth quadrant. These relative phases between the limit cycle and the injected signal is generally true (i.e. the phase discriminator of a PLL) and later on very important for the numerical technique. It is also clear from the simple example that the locking range depends in first order from the amplitude of the system. In [3] and later in [6] entrainment phenomena have been studied based on a first order perturbation along the limit cycle.

2 Summary of Floquet’s Theory If the orbital deviation from the limit cycle of the free running oscillator is sufficiently small, it is sufficient to consider the variational equation xP D A.t/  x;

A.t/ WD

ˇ d f .x/ˇx.t /Dxss .t / dx

with periodic capacitance matrix, i.e. A.t/ D A.t C T /. Theorem 1. (Floquet) The state transition matrix ˚.t; / of a linear T -periodic system of ODEs can be written as ˚.t; / D U.t/ diag.exp .1 .t  //; : : : ; exp .n .t  // V ./ where U.t/ is a T -periodic matrix and V .t/ D U 1 .t/.

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H.G. Brachtendorf and R. Laur

Because V .t/ D U 1 .t/, the matrix-valued functions V .t/ and U.t/ fulfill a biorthogonality condition which is required below. Lemma 1. (i.e. K¨artner [7]) u1 .t/ D xP ss .t/ is a T -periodic solution of the oscillator variational equation. Without loss of generality u1 .t/ WD xP ss .t/ D f .xss .t//;

1 D 0

is a characteristic Floquet eigenvector (eigenmode) of the corresponding Floquet multiplier 1 D exp .1 T / D 1 Moreover, the Floquet eigenvector u1 .t/ is tangent to the limit cycle. A general solution of the perturbed system can therefore be given by Z '.t; x.t0 // D ˚.t; t0 /  x.t0 / C

t t0

˚.t; /  b./ d

3 A Theory of Entrainment In what follows, the procedure is similar to the phase noise analysis developed by K¨artner [7] and Demir [4]. The derivation follows [3, 10]. We are interested in that contribution of the periodic stimulus b.t/ which leads to a phase deviation whereas the phase deviations are of minor interest. To this end, the stimulus is expanded in the basis U.t C ˛.t// b.t/ D c1 .t/  u1 .t C ˛.t// C    C cn .t/  un .t C ˛.t// with a priori unknown ˛.t/ 2 R. Due to the biorthogonality it follows that c1 .t/ D v1T .t C ˛.t//  b.t/. It can be proven that xss .t C ˛.t// is a solution of the differential equation wherein ˛.t/ is obtained as the solution of the scalar initial value problem (IVP) d ˛.t/ D v1T .t C ˛.t//  b.t/ D c1 .t/; dt

˛.0/ D ˛0

We now use the fact that within a locking range the relative phase between the stimulus and the periodic orbit depends on the detuning what has already been discussed in the introduction. We therefore introduce a family of phase shifted stimuli b .t/ WD b.t C   TO /; 0   < 1 with a priori unknown period TO .

Entrainment Phenomena in Nonlinear Oscillations

25

TO is a function of . The following theorem gives the relation between the two variables. Theorem 2. Let b .t/ WD b.t C   TO /;

0<1

be a family of phase shifted stimuli and ˛0 ./ D

1 TO

Z

TO 0

v1T .t C ˛ .t//  b.t C   TO / dt;

0<1

then TO D .1 C ˛0 .//1  T The proof of the theorem can be found in [3]. The locking range can be calculated by maximizing and minimizing TO . To this end, the normalized locking range is introduced. Definition 1. The normalized locking range is defined by ı WD

!C  ! !ss

The following relation holds by first order ı  ˛0max  ˛0min

4 Simulation Results Van der Pol oscillator. The fig. 4a shows the derivative ˛.t/ P for the maximal detuning frequency. The mean ˛0 is proportional to the detuning according to the theorem. On the other hand, fig. 4b depicts the detuning as a function of the relative phase. This graph is well known from PLL theory, where the relative phase of the input signals at the discriminator depends on the relative frequencies of the signals. Colpitts oscillator. The phase function ˛.t/ P of the Colpitts oscillator are depicted in fig. 5a and the detuning as function of the relative phase in fig. 5b. The behavior is similar to the van der Pol oscillator fig. 4.

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H.G. Brachtendorf and R. Laur

. 1.2 α

b

x 10–5

dw wss (Q)

a 1

6 4

0.8

2

0.6

0

0.4

–2

0.2

–4

0

0

1

2

3

4

5 t/s

da max(t)

x 10–6

–6 –1

6 –5 x 10

0

1

2

3

4

5

dw wss (Q)

dt

6 2pΘ

7

3 2pΘ

4

Fig. 4 Phase deviation as a function of time (left) and detuning as a function of  (right)

a

b dw (Θ) wss

. 0.02 a 0.018

1 0.8

0.016

0.6

0.014

0.4

0.012

0.2

0.01

0

0.008

−0.2

0.006

−0.4

0.004

−0.6

0.002

−0.8

0

0

1

2

3

4

5

6

damax(t)

7

8 t/s

9 x 10–8

x 10–3

−1 –4

–3

–2

–1

0

1

2

dw (Q) wss

dt

Fig. 5 Phase deviation as a function of time (left) and detuning as a function of  (right)

5 Conclusion This paper presents a theory of entrainment based on a first order expansion of the differential equation along the limit cycle employing Floquet’s theory. It is able to predict the locking range for harmonic as well as superharmonic entrainment and is in good agreement with a purely numerical technique as long as the injected power is sufficiently small such that the first order expansion is valid. Acknowledgements This work has been partly supported by the ICESTARS project of the European Community under grant 214911. The authors recognize the work of F. K¨artner, A. Demir

Entrainment Phenomena in Nonlinear Oscillations

27

and J. Roychowdhury on phase noise which had a tremendous influence on this work. They also thank R. Melville for posing the problem.

References 1. Adler, R.: A study of locking phenomena in oscillators. Proc. IRE 34, 351–357 (1946) 2. Andronov, A., Witt, A.: Zur Theorie des Mitnehmens von van der Pol. Archiv f¨ur Elektrotechnik 24, 99–110 (1930) 3. Brachtendorf, H.G.: Theorie und Analyse von autonomen und quasiperiodisch angeregten elektrischen Netzwerken. Eine algorithmisch orientierte Betrachtung. Universit¨at Bremen, Bremen (2001). Habilitationsschrift 4. Demir, A.: Floquet Theory and Nonlinear Perturbation Analysis for Oscillators with Differential-Algebraic Equations. Tech. Rep. ITD-98-33478N, Bell-Laboratories (1998) 5. Demir, A., Roychowdhury, J.: A reliable and efficient procedure for oscillator ppv computation, with phase noise macromodeling applications. IEEE Trans. Comp. Aided Des. Integrated Circ. Syst. 22(2), 188–197 (2003). doi:10.1109/TCAD.2002.806599 6. Harutyunyan, D., Rommes, J., ter Maten, J., Schilders, W.: Simulation of mutually coupled oscillators using nonlinear phase macromodels. IEEE Trans. Comp. Aided Des. Integrated Circ. Syst. 28(10), 1456–1466 (2009). doi:10.1109/TCAD.2009.2026359 7. Kaertner, F.X.: Analysis of white and f ˛ noise in oscillators. Int. J. Circ. Theor. Appl. 18, 485–519 (1990) 8. Kinget, P., Melville, R., Long, D., Gopinathan, V.: An injection-locking scheme for precision quadrature generation. IEEE J. Solid-State Circ. 37(7), 845 –851 (2002). doi:10.1109/ JSSC.2002.1015681 9. Kurokawa, K.: Injection locking of microwave solid-state oscillators. Proc. IEEE 61, 1386– 1410 (1973) 10. Laur, R., Brachtendorf, H.G.: Computerized method for determination and optimization of the synchronization region of a circuit or system (2000). Patent DE10062414 11. Tiebout, M.: A cmos direct injection-locked oscillator topology as high-frequency lowpower frequency divider. IEEE J. Solid-State Circ. 39(7), 1170–1174 (2004). doi:10.1109/ JSSC.2004.829937

Initial Conditions and Robust Newton-Raphson for Harmonic Balance Analysis of Free-Running Oscillators Jarmo Virtanen, E. Jan W. ter Maten, Theo G.J. Beelen, Mikko Honkala, and Mikko Hulkkonen

Abstract Poor initial conditions for Harmonic Balance (HB) analysis of freerunning oscillators may lead to divergence of the direct Newton-Raphson method or may prevent to find the solution within an optimization approach. We exploit time integration to obtain estimates for the oscillation frequency and for the oscillator solution. It also provides an initialization of the probe voltage. Next we describe new techniques from bordered matrices and eigenvalue methods to improve Newton methods for Finite Difference techniques in the time domain as well as for Harmonic Balance. The method gauges the phase shift automatically. No assumption about the range of values of the Periodic Steady State solution is needed.

1 Introduction A free-running oscillator is an autonomous circuit, which has only DC bias sources connected to the circuit and, thus, no periodic excitation. During the time-domain J. Virtanen ()  M. Honkala  M. Hulkkonen Department of Radio Science and Engineering, Aalto University School of Electrical Engineering, P.O. Box 13000, 00076 Aalto, Finland e-mail: [email protected]; [email protected]; [email protected] T.G.J. Beelen NXP Semiconductors, High Tech Campus 46, 5656 AE Eindhoven, The Netherlands e-mail: [email protected] E.J.W. ter Maten Department of Mathematics and Computer Science, Eindhoven University of Technology, CASA, P.O. Box 513, 5600 MB Eindhoven, The Netherlands e-mail: [email protected] Bergische Universit¨at Wuppertal, FB C, Applied Mathematics/Numerical Analysis, Bendahler Str. 29, Zi-503, 42285 Wuppertal, Germany e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 4, © Springer-Verlag Berlin Heidelberg 2012

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J. Virtanen et al.

transient analysis of an oscillator, the oscillation starts by itself due to noise or instability. Long start-up time implies long simulation time to get the Periodic Steady State (PSS) solution. Harmonic balance (HB) analysis is a frequency-domain PSS method. HB is needed for (phase) noise simulations and is more suitable for frequency-dependent linear devices. It may converge faster to the PSS solution of a free oscillator than the transient analysis. To enhance convergence one either modifies the HB equations or one applies artificial excitation. In addition, the oscillation frequency (the fundamental HB frequency), is unknown and one needs a gauge equation and an initial estimate. Frequency domain methods to estimate these can be found in [2, 3, 6–8] (and their references). We present two algorithms for oscillation frequency detection from transient data and improve by (vector) extrapolation [10]. The initialization of the probe voltage amplitude and of the HB solution are considered. Finally we describe new techniques from bordered matrices and eigenvalue methods to improve the Newton method for HB analysis.

2 Initializing HB Oscillator Analysis The oscillator analysis in the APLAC simulator [1] utilizes a probe element and optimization techniques. Inside an optimization loop HB analysis is performed with new values of the optimization variables, being the oscillation frequency, fosc , and the oscillation amplitude, vosc . An artificial excitation probe, being a voltage source in series with a non-zero resistor (to prevent an increase of the DAE-index), is connected to the circuit. The goal of the optimization is to have a zero current through the probe element. For a related procedure see [7]. The initial conditions for the optimization of fosc and vosc are obtained from transient analysis as described next. Initially a (limited) transient analysis is run, followed by a Fourier transform (FFT) to get an impression of the spectrum of the oscillator and of the solution. A spectral line having the largest magnitude indicates the oscillation frequency.

1.42

1.5 FFT parabola fosc

fapprox = 9.880MHz

amplitude / V

1.90 U/V

0.95

–0.1 –0.9

0.47 0.00 6.703M

0.7

8.250M

9.797M freq / Hz

11.344M

12.891M

–1.7 400n

442.5n

485n time/s

527.5n

570n

Fig. 1 Left: Quadratic interpolation of the frequency from the spectrum. Right: Zero-crossing: the x-markers connected with lines show the points used for interpolation

Initial Conditions and Robust Newton-Raphson for Harmonic Balance Analysis

31

Algorithm 1 Algorithmic background for the accelerated Poincar´e map method [4]. 1: Denote the limit of the recursion xnC1 D F.xn / (n  0) by xQ D limn xn . 2: Let en D xn  xQ , then enC1 D ˚en C O .jjen jj2 /. Thus jjenC1 jj2 D O .jjen jj2 / D O .jje0 jj2 /. 3: For the dn mentioned above we have dnC1 D ˚dn C O .jjen  en1 jj2 / D ˚dn C O .jjen1 jj2 / D ˚dn C O .jje0 jj2 / (for n  1). This last 2-terms recursion makes the next steps a bit easier to formulate than the more precise intermediate 3-terms recursion. For n D 0 we have d1 D x1 x0 D e1 e0 D .˚ I/e0 C O .jje0 jj2 /, hence e0 D .˚ I/1 d1 C O .jje0 jj2 /. PkC1 PkC1 PkC1 4: We have 0 D p D 1 p dp D p D 2 p ˚ p1 d1 C 1 d1 C O .jje0 jj2 /, hence pD1 p ˚ p1 d1 D O .jje0 jj2 /. PkC1 PkC1 PkC1 5: Let  D . p D 1 p /Qx. We obtain x C ep / D  C p D 1 p xp D p D 1 p .Q PpC1 PpC1 PkC1 p1 2 p p ˚ e1 C O .jje1 jj / D  C 1 p ˚ e0 C O .jje0 jj2 / D  C 1 p D 1 p ep D  C PpC1 PpC1 p 1 p ˚ .˚  I/ d1 C O .jje0 jj2 / D  C ˚.˚  I/1 1 p ˚ p1 d1 C 1 O .jje0 jj2 / D  C O .jje0 jj2 /. PkC1 PkC1 6: Finally y D p D 1 p xp =. p D 1 p / is a higher order accurate approximation, with error 2 O .jje0 jj /, or, taking the effect " into account, with error O .jje0 jj2 / C O ."/.

Depending on the sampling rate, the actual oscillation frequency may be situated between the sampled frequency points. Therefore, quadratic interpolation with three frequency points around the maximum is used to determine a more accurate estimate for the oscillation frequency, see Fig. 1 (left). An alternative, zero-crossing, method also applies to transient simulation results. The period is determined from the zero crossings in the waveform with the DC-value. Accuracy of the zero crossings is improved by using linear inverse interpolation, see Fig. 1 (right). This can be generalized to a Poincar´e method [4, 5] that determines the next root of a (scalar) phase condition s.x.t// D .x.t/; n/  ˛ D 0 (i.e. solve both x D x˛ [with constraint d.x.t/; n/=dt > 0] and t D t˛ ) and restarts the time integration at t D t˛ with initial value x˛ .1 The values xkC1 D F.xk / (for some function F and k  0), determined in this way, approximate the boundary value used by the ultimate PSS solution. We accelerate by vector extrapolation. Define recursively DkC1 D ŒDk dkC1 , with D0 D ; (empty) and dkC1 D xkC1 xk . Clearly Dk D Xk k for Xk D Œx0 x1 : : : xk  and a difference matrix k . By a QR-decomposition we determine the rank of DkC1 D QkC1 RkC1 , with QkC1 D ŒQk qkC1  and  Rk rkC1 RkC1 D . If j˛kC1 j  " we assume that dkC1 2 Span.Qk / D Span.Dk /, 0 ˛kC1 Pk i.e. we we can write dkC1 D Qk rkC1 D Dk R1 p D 1 p dp where k rkC1 D PkC1 T 1  D .1 ; : : : ; k / D Rk rkC1 . Setting kC1 D  1 we thus have p D 1 p dp D 0. This linear combination has a crucial application. If we express the dn in terms of lower and higher order effects we observe that the sum of the lower order effects nearly cancels. We assume that 1 ¤ 1, that ˚ D @F=@x is uniformly

1

This method is used in Pstar, the in-house analog circuit simulator of NXP Semiconductors.

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bounded and that also .I  ˚/1 exists and is uniformly bounded. We summarize some basic steps in Algorithm 1. Starting with this value y one generates iterands yk with as next extrapolation z. The row x; y; z; : : : converges super-linearly. Storing the tk D t˛ easily provides the period T D tkC1  tk . A final integration over one period gives a time-domain solution from which an initial HB solution is obtained. With n D ek one traces a particular unknown (this choice requires knowledge by the designer about the location where the oscillation occurs). The phase condition can be for a difference of a voltage or a zero condition for a current.

3 VCO Oscillator The methods implemented in APLAC were tested with an industrial VCO circuit that consists of 40 MOSFETS (modelled by BSIM3) and 80 (Juncap) diodes. The expected results based on transient simulation are fosc D 3:25 GHz and vosc D 1:25 V, while the initial values for the analysis were: fosc D 3:0 GHz and vosc D 1:0 V. The circuit has been simulated using HB oscillator analysis without initialization (old), and with FFT and zero crossing (ZeroC) initialization. Table 1 summarizes the CPU times and the number of HB iterations obtained to reach the oscillator solution (fosc D 3:25 GHz and vosc D 1:25 V) as well as initial values of fosc and vosc for the HB based optimization—with the old method user-specified values are used directly, and improved values are obtained using either FFT or ZeroC method. A typical result of the Poincar´e method for a Colpitts oscillator gives 3 outer iterations (extrapolations) with 4, 3, 2 inner iterations, repectively, to build each time a subspace in which extrapolation leads to an improved initial value for solution and period (final error < 1016 ). The zero crossing and the Poincar´e method assume that two successive crossings determine the period. This excludes situations in which four or more crossings really determine the overall period.

4 Newton Raphson The Newton-Raphson method to solve the Harmonic Balance system becomes

Table 1 Number of HB iterations and CPU times, and user-specified (“old”) or improved initial values (“FFT”, “ZeroC”) of fosc and vosc of the VCO circuit Method HBITER CPU/s Initial value of fosc /GHz Initial value of vosc /V Old FFT ZeroC

2,259 47 31

110.6 6.2 1.4

3.0 2.7 3.1

1.00 1.09 1.17

Initial Conditions and Robust Newton-Raphson for Harmonic Balance Analysis

 k

M

XkC1  Xk f kC1  f k



  k k F.Xk ; f k / A b k ; M D : D T k cT ı c X c

33



(1)

Here Ak D

ˇ ˇ @F ˇˇ @F ˇˇ k k k k D ˝  C C G ; b D ; @x ˇXk ;f k @f ˇXk ;f k

(2)

for suitable matrices C and G, that are composed by the local Jacobians and (discrete) Fourier Transforms; ˝ D diag.: : : ; i !k ; : : :/, with !k D k .f k / (for some function k ). The last row in (1) corresponds with the phase equation; usually ı D 0. The matrix A becomes badly conditioned when the Newton iterands converge. This is due to the fact that the time derivative of the PSS solution solves the linearized homogeneous circuit equations when linearized at the PSS solution. Hence when the discretization is exact this time derivative of the ultimate PSS is in the kernel of A. This has led to study more carefully bordered matrices [2, 9] and generalized eigenvalue methods. In [9] the eigentriple .V; W; / is determined such that Œf C C GV D 0 and WT Œf C C G D 0 for the  closest to 1. We approximate the bi-orthogonality relation between V and W by WT C˝ X1 D 0, i.e. in (1) we take cT D WT  C  ˝ and c D 1. We may even consider cT D VT .

5 LC Oscillator We consider an LC tank with a nonlinear resistor that is governed by the following differential equations for the unknowns (v; i ) [v being the nodal voltage; i being the inductor current] 

C 0 0 L



d dt

v.0/ D v0 ;



 1     /  v.t/ 1 v.t/ / S tanh. Gv.t R S C C D0 i.t/ 0 1 i.t/ 0

i.0/ D i0 :

(3) (4)

where C , L and R are the capacitance, inductance and resistance, respectively. The voltage controlled nonlinear resistor is defined by the S and G parameters. The values L D 0:53 nH, C D 1:33 pF, R D 250 ˝, S D 1=R , and G D 1:1=R correspond with an oscillation frequency 6 GHz. Starting with initial conditions T0 D 1:1  2, v0 .t/ D sin.t/, i0 .t/ D 0:2 sin.t/, and N D 101 (100 actual grid points), the PSS solutions are obtained using the old phase-shift condition method and with the new eigenvector gauge method. For both methods we determine the maximum of the normalized correction of the solution and the normalized frequency correction

34

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X: 7 Y: 0.0004908

10

LC OSC.

10

0 LC OSC.

−3

−5

10

Normalized Δf

Max. Normalized Correction

100

X: 7 Y: 1.738e−011

10−10

10

10

X: 7 Y: 1.187e−007

−6

−9

X: 7 Y: 3.698e−011

10−12 OLD Method: Using Phase shift condition New Method: Using Eigenvector

10−15 1

2

3

4

5 6 7 8 Number of Iteration

9

10

11

10−15

OLD Method: Using Phase shift condition New Method: Using Eigenvector

0

2

4 6 8 Number of Iteration

10

12

Fig. 2 LC oscillator: normalized corrections for solutions and frequency

ˇ Xk ˇNormalized D jjXkC1  Xk jj1 =jjXk jj1 ;

ˇ f k ˇNormalized D jf kC1  f k j=jf k j

during each kth Newton-Raphson iteration, which are presented in Fig. 2. A wrong value in the old phase shift condition even prevents convergence for this method [9].

6 Conclusion Time domain initializations have been presented that enhance the convergence options for Harmonic Balance within an outer optimization approach and within a direct Newton-Raphson procedure. One method exploits FFT techniques. A zerocrossing technique was generalized to a Poincar´e method. Here speed up by vector extrapolation was based on Minimal Polynomial Extrapolation. Finally, a new technique for the Newton-Raphson simulation of a free-running oscillator was presented. The generalized eigenvectors for the eigenvalue closest to 1 and the time derivative of the solution provide a robust gauge equation that is dynamically updated within each Newton-Raphson iteration. It was verified that the new method has better convergence properties compared to the popular phase-shift condition method and does not need additional information about the solution. Acknowledgements This work was funded by the EU FP7/2008/ICT/214911 project ICESTARS.

References 1. APLAC Version 8.50 Manual. http://www.awrcorp.com 2. Brambilla, A., Gruosso, G., Storti Gajani, G.: Robust harmonic-probe method for the simulation of oscillators. IEEE Trans. Circ. Syst. I 57(9), 2531–2541 (2010) 3. Duan, X., Mayaram, K.: Frequency-domain simulation of ring oscillators with a multiple-probe method. IEEE Trans. Comp. Aided Des. Integr. Circ. Syst. (TCAD) 25(12), 2833–2842 (2006)

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4. Houben, S.H.M.J.: Circuits in motion – the simulation of electrical oscillators. PhD-Thesis TU Eindhoven, 2003 5. Houben, S.H.M.J., Maubach, J.M.: Periodic Steady-State analysis of free-running oscillators. In: van Rienen, U., G¨unther, M., Hecht, D. (eds.) Scientific Computing in Electrical Engineering, Lect. Notes in Comp. Science and Engineering, vol. 18, pp. 217–224. Springer, Berlin (2001) 6. Kurokawa, K.: Some basic characteristics of broadband negative resistance oscillator circuits. The Bell Syst. Techn. J. 48(6), 1937–1955 (1969) 7. Lampe, S.: Effiziente Verfahren zur Berechnung s¨amtliche Grenzzyklen autonomer Systeme, PhD-Thesis Univ. Bremen, Logos Verlag Berlin, 2003 8. Lampe, S., Brachtendorf, H.G., ter Maten, E.J.W., Onneweer, S.P.: Robust limit cycle calculations of oscillators. In: van Rienen, U., G¨unther, M., Hecht, D. (eds.) Scientific Computing in Electrical Engineering, Lect. Notes in Comp. Science and Engineering, vol. 18, pp. 233–240. Springer, Berlin (2001) 9. Mirzavand Boroujeni, R., ter Maten, J., Beelen, T., Schilders, W., Abdipour, A.: Robust periodic steady state analysis of autonomous oscillators based on generalized eigenvalues. To appear in: Michielsen, B., Poirier, J.-R. (eds.): Scientific Computing in Electrical Engineering SCEE 2010, Series Mathematics in Industry, Vol. 16, pp. 293–302, Springer, Berlin (2012) 10. Skelboe, S.: Computation of the periodic steady-state response of nonlinear networks by extrapolation methods. IEEE Trans. Circ. Syst. 27(3), 161–175 (1980)

Rational Modeling Algorithm for Passive Microwave Structures and Systems Dirk Deschrijver, Tom Dhaene, Oliver Salazar Celis, and Annie Cuyt

Abstract An efficient identification method is proposed for passive rational approximation of frequency domain responses. The method is applied to compute a transfer function from tabulated S-parameter data of a multiport microwave filter. Numerical results validate the robustness and efficacy of the modeling approach.

1 Introduction Broadband characterization of passive linear systems, based on measured and simulated data has become a topic of intense research, mainly due to the increasing speed and decreasing size of electronic circuits. While the operating frequency is increasing well above the multi-GHz frequency range, parasitic effects that were previously ignored cannot be overlooked anymore for accurate system level analysis [1]. This complicates the modeling and design of RF, microwave and millimeterwave components and systems. The number of design failures caused by signal and power integrity problems has become problematic because the existing design tools and modeling methodologies cannot address these issues in an appropriate way [2, 3]. Common approaches to tackle these problems are based on the approximation of tabulated S-parameter frequency responses using rational functions and subsequently synthesizing a SPICE compatible macromodel [4, 5]. These macromodels approximate the complex electromagnetic (EM) behavior of high-speed multi-port

D. Deschrijver ()  T. Dhaene Ghent University – IBBT, Sint Pietersnieuwstraat 41, 9000 Ghent, Belgium e-mail: [email protected]; [email protected] O.S. Celis  A. Cuyt University of Antwerp, Middelheimlaan 1, 2020 Antwerp, Belgium e-mail: [email protected]; [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 5, © Springer-Verlag Berlin Heidelberg 2012

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systems at the input-output ports in the frequency domain by rational functions [6]. The advantage of this approach is that a circuit can be modeled as a black-box, and that no knowledge of its internal logic is required for the modeling process [7]. In 1999, the iterative Vector Fitting (VF) technique was introduced [8–10] to compute such macromodels in a robust and efficient way. It is applicable to both smooth and resonant responses with high orders and wide frequency bands. An additional advantage is that stability of the poles is easily enforced by a simple poleflipping scheme. Since VF is now adopted in many societies of applied engineering, including power systems and microwave systems, it has drawn a lot of attention from researchers, and it has become the de-facto standard for rational approximation. A general overview of the VF methodology was recently presented in [11]. A known restriction of the technique is that the computed macromodels are not guaranteed to be passive by construction. Nevertheless, passivity of the macromodel is of crucial importance since a non-passive macromodel may lead to unstable transient simulations in an unpredictable manner [12, 13]. This paper applies a new passivity enforcement technique that is able to enforce passivity to a nonpassive rational macromodel by means of an overdetermined least-squares fitting algorithm [14]. Numerical results show that the presented approach achieves an excellent tradeoff between computation time and accuracy preservation of the macromodel.

2 Vector Fitting Algorithm Given a set of S-parameter data samples fsk ; Hmn .sk /gK k D 1 , the VF algorithm [8] computes a rational macromodel that matches the frequency response by solving several least squares problems in successive iteration steps (t D 0; : : : ; T ), where 1  m; n  Q denotes the mth row and the nth column of a Q-port transfer matrix H . ˇ ˇ2 arg min ˇ.Hmn /t .s/   t .s/Hmn .s/ˇ

(1)

Both .Hmn /t .s/ and  t .s/ in (1) are expanded as a linear combination of partial fractions (or orthonormal rational functions [15]) that share common poles fapt gPpD1 . .Hmn /t .s/ D

 t .s/ D

P t X cp;mn

s  apt pD1 P X

cQpt

s  apt pD1

t C c0;mn

(2)

C cQ0t

(3)

In the first iteration step (t D 0), the initial starting poles ap0 are selected according to a heuristical scheme [8]. It is trivial to transform (1) into an overdetermined

Rational Modeling Algorithm for Passive Microwave Structures and Systems

39

set of least-squares equations At x t D b t , where the solution vector x t contains the t unknown coefficients fcp;mn ; cQpt gPp D 0 . Based on the coefficients fcQpt g, it is possible to obtain a new set of relocated poles by solving the zeros of  t .s/ as an eigenvalue problem. These relocated poles replace the initial set of poles, and the process of solving (1) is repeated iteratively until the poles are converged to some quasioptimal position. The trivial null solution of (1) is avoided by setting one coefficient cQ0t D 1 or by adding an additional relaxation constraint to (1), as in [9]. It was shown in a previous report that this process is related to the Sanathanan-Koerner iteration [16]. In the final iteration step (t D T ), the transfer function .Hmn /T .s/ in partial fraction form is obtained by solving (1) with  T .s/ D 1. All the details about this Vector Fitting procedure are extensively reported in literature, see [8, 17].

3 Passivity Enforcement Algorithm The passivity constraints for stable and causal macromodels in the scattering case require that the singular values of the transfer matrix H are unitary bounded [18,19] H .j!/Hmn .j!/  0 8! 2 R I  Hmn

(4)

By computing the eigenvalues of an associated Hamiltonian matrix, it is possible to assess algebraically the passivity of the model [20, 21]. In addition, the eigenvalues can be used to exactly pinpoint the boundaries of possible passivity violations. To compensate any occuring violations, the following procedure is used. First a dense set of frequencies !k is used to evaluate the singular value decomposition Hmn .j!k / D Uk ˙k VkH

(5)

where ˙k D diag.1;k ; : : : ; Q;k / is a diagonal matrix that contains the singular values on its main diagonal. The set contains all the samples of the original frequency response, combined with some additional samples in the vicinity of passivity violations [22]. These samples are included to ensure a good resolution of the singular value trajectories. Then, a set of violation parameters is constructed as follows Hmn .j!k / D Uk ˙k VkH

(6)

with ˙k D diag.1;k ; : : : ; Q;k /, where q;k D 0 if q;k  

(7)

q;k D q;k   if q;k > 

(8)

The threshold parameter  is chosen close to, but less than 1. In general, larger values of  (closer to 1) are able to better preserve the accuracy of the model, but

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may require additional iterations to achieve passivity. In order to make the model T passive, a new set of offset residues cp;mn is computed by fitting the violation parameters Hmn .j!k / using the same set of poles of the original model. ˇ ˇ2 arg min ˇ.Hmn /T .j!/  Hmn .j!/ˇ (9) T A new model with reduced passivity violations is obtained by substituting cp;mn T T with cp;mn  cp;mn . The process is repeated iteratively until passivity is reached.

4 Example: Quarter Wavelength Filter In this example, the presented approach is used to compute a passive macromodel of a 2-port quarter wavelength filter. The scattering parameters of the structure are simulated in the frequency domain with a planar full-wave electromagnetic simulator over the frequency range 1–12 GHz. Then, the vector fitting algorithm is used to approximate the response by a 28-pole strictly proper transfer function using 1,000 data samples. The desired model accuracy of the S-parameters is -60 dB or better, which corresponds approximately to 3 significant digits. Even though the simulated data samples are passive, it is seen from Fig. 1 that the resulting macromodel has several small in-band passivity violations. Therefore, 1.002 Original Passive

1.0015 1.001 1.0005

σ(s)

1 0.9995 0.999 0.9985 0.998 0.9975 0.997

0

2

4

6 8 Frequency [GHz]

Fig. 1 Quarter wavelength: singular values of scattering matrix

10

12

Rational Modeling Algorithm for Passive Microwave Structures and Systems

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101 Original Deviation 100

10–1

|Smn|

10–2

10–3

10–4

10–5

10–6

2

4

6

8

10

12

Frequency [GHz]

Fig. 2 Quarter wavelength: magnitude of matrix elements

the proposed passivity enforcement procedure is applied to compensate them, and the algorithm converges to a passive macromodel in a few iteration steps. It is found that the maximum singular value of the scattering matrix decreases monotonically in each iteration step of the algorithm. Figure 2 shows the magnitude of the original frequency response, and the deviation that is caused by the passivity perturbation. It is clear that the overall accuracy of the macromodel is well preserved.

5 Conclusions An efficient algorithm for passive macromodeling of microwave components is proposed. First, a robust macromodeling technique is used to approximate the frequency response with a rational transfer function. Then, the passivity of the model is assessed by using an algebraic passivity test. If the model is nonpassive, an iterative passivity enforcement scheme is applied to perturb the model coefficients using a standard least-squares procedure. Numerical results confirm that the approach is effective, and that the perturbation does not compromise the accuracy of the model.

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Acknowledgements This work was supported by the Fund for Scientific Research Flanders (FWO-Vlaanderen). Dirk Deschrijver is a post-doctoral research fellow of FWO-Vlaanderen.

References 1. Min, S.-H.: “Automated Construction of Macromodels from Frequency Data for Simulation of Distributed Interconnect Networks”, PhD thesis, Georgia Institute of Technology, USA, 2004 2. Choi, K.-L.: “Embedded Passives using Rational Functions in Multi-Layered Substrates”, PhD thesis, Georgia Institute of Technology, USA, 1999 3. Li, E.P., Wei, X.C., Cangellaris, A.C., Liu, E.X., Zhang, Y.J., D’Amore, M., Kim, J., Sudo, T.: “Progress review of electromagnetic compatibility analysis technologies for packages, printed circuit boards, and novel interconnects”. IEEE Trans. Electromagn. C. 52(2), 248–265 (2010) 4. Saraswat, D., Achar, R., Nakhla, M.: “Enforcing Passivity for Rational Function Based Macromodels of Tabulated Data ”. IEEE Conference on Electrical Performance of Electronic Packaging (EPEP 2003), pp. 295–298. Princeton, NJ (2003) 5. Lefteriu, S., Antoulas, A.C.: “A new approach to modeling multiport systems from frequencydomain data”. IEEE Trans. Comp. Aided Des. Integrated Circ. Syst. 29(1), 14–27 (2010) 6. Cuyt, A.: “Nonlinear Numerical Methods and Rational Approximation”. D. Reidel Publishing Company, Dordrecht (1988) 7. Mutnury, B.: “Macromodeling of Nonlinear Driver and Receiver Circuits ”, PhD thesis, Georgia Institute of Technology, GA, USA, 2005 8. Gustavsen, B., Semlyen, A.: “Rational approximation of frequency domain responses by vector fitting”. IEEE Trans. Power Deliv. 14(3), 1052–1061 (1999) 9. Gustavsen, B.: “Improving the pole relocating properties of vector fitting”. IEEE Trans. Power Deliv. 21(3), 1587–1592 (2006) 10. Deschrijver, D., Mrozowski, M., Dhaene, T., De Zutter, D.: “Macromodeling of multiport systems using a fast implementation of the vector fitting method”. IEEE Microw. Wireless Compon. Lett. 18(6), 383–385 (2008) 11. Deschrijver, D., Gustavsen, B., Dhaene, T.: “Advancements in iterative methods for rational approximation in the frequency domain”. IEEE Trans. Power Deliv. 22(3), 1633–1642 (2007) 12. Saraswat, D., Achar, R., Nakhla, M.S.: “Global passivity enforcement algorithm for macromodels of interconnect subnetworks characterized by tabulated data”. IEEE Trans. Very Large Scale Integration (VLSI) Syst. 13(7), 819–832 (2005) 13. Grivet-Talocia, S., Ubolli, A.: “A comparative study of passivity enforcement schemes for linear lumped macromodels”. IEEE Trans. Adv. Packag. 31(4), 673–683 (2008) 14. Dhaene, T., Deschrijver, D., Stevens, N.: “Efficient algorithm for passivity enforcement of s-parameter based macromodels”. IEEE Trans. Microw. Theor. Tech. 57(2), 415–420 (2009) 15. Deschrijver, D., Haegeman, B., Dhaene, T.: “Orthonormal vector fitting: A robust macromodeling tool for rational approximation of frequency domain responses”. IEEE Trans. Adv. Packag. 30(2), 216–225 (2007) 16. Hendrickx, W., Deschrijver, D., Dhaene, T.: “Some remarks on the vector fitting iteration”. Post-Conference Proceedings of EMCI 2004, Mathematics in Industry, pp. 134–138. Springer, Berlin (2006) 17. Deschrijver, D., Dhaene, T.: “A note on the multiplicity of poles in the vector fitting macromodeling method”. IEEE Trans. Microw. Theor. Tech. 55(4), 736–741 (2007 18. Grivet-Talocia, S.: “Passivity enforcement via perturbation of Hamiltonian matrices”. IEEE Trans. Circ. Syst. I, 51(9), 1755–1769 (2004) 19. Deschrijver, D., Dhaene, T.: “Fast passivity enforcement of s-parameter macromodels by pole perturbation”. IEEE Trans. Microw. Theor. Tech. 57(3), 620–626 (2009) 20. Boyd, S., Balakrishnan, V., Kabamba, P.: “A bisection method for computing the Hinf norm of a transfer matrix and related problems”. Math. Contr. Signals Syst. 2, 207–219 (1989)

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21. Zhang, Z., Wong, N.: “Passivity check of S-parameter descriptor systems via s-parameter generalized hamiltonian methods”. IEEE Trans. Adv. Packag. 33(4), 1034–1042 (2010) 22. Chinea, A., Grivet-Talocia, S., Deschrijver, D., Dhaene, T., Knockaert, L.: “On the construction of guaranteed passive macromodels for high-speed channels”. Design, Automation and Test in Europe (2010), Dresden, Germany, pp. 1142–1147, March 2010

An Existence Result for Index-2 PDAE System Arising in Semiconductor Modeling Giuseppe Al`ı, Andreas Bartel, and Nella Rotundo

Abstract We consider an electric network modeled via Modified Nodal Analysis, which is refined by a nonlinear multi-dimensional elliptic partial differential equations as semiconductor device model. This coupled system is a partial differentialalgebraic equation. Starting from an analysis of the pure network equations (without semiconductor) and its index-2 conditions, we propose additional conditions which assure that coupled system is index-2. Then we proof an existence result for the coupled system.

1 Introduction Due to downscaling, distributed models play an important role in circuit simulation. In fact, several phenomena arising in electric networks, negligible at first approximation, are adequately described by systems of partial-differential equations. Several results for coupling and existence of solutions exist already. e.g., for networks connected by lossy transmission lines [5], for coupling with thermal effects [3, 4], for coupling of index-1 circuits with devices [1, 2].

G. Al`ı () Dipartimento di Matematica, Universit`a della Calabria, Arcavacata di Rende 87036, Cosenza, Italy INFN, Gruppo collegato di Cosenza, Arcavacata di Rende 87036, Cosenza, Italy e-mail: [email protected] A. Bartel Bergische Universit¨at Wuppertal, 42097 Wuppertal, Germany e-mail: [email protected] N. Rotundo Universit`a di Catania, Catania, Italy INFN, Gruppo collegato di Cosenza, Arcavacata di Rende 87036, Cosenza, Italy e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 6, © Springer-Verlag Berlin Heidelberg 2012

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Here we consider a system of partial differential-algebraic equations which model an electric network containing semiconductor devices. The system consists of the Modified Nodal Analysis equations, coupled with multi-dimensional elliptic partial differential equations which model the devices. We first study the network equations (using the Modified Nodal Analysis) without the nonlinear coupling term describing the device currents, and we assume these equations to be of tractability index 2. We propose some additional topological conditions that make the full coupled system of index 2. For this coupled system we prove an existence result. Our paper is organized as follows: the next section states the coupled model; Section 3 gives the index-2 decomposition and an example for our conditions developed; Section 4 contains the existence result with a sketched proof.

2 Modeling of Diodes and Networks In this section we summarize the electric network model and its coupling to the drift diffusion equation for spatial distributed semiconductor devices.

2.1 Network Model for Electric Circuits We consider an RLC network, i.e., an electric network containing only resistances, inductances, capacitances, independent current and voltage sources u.t/ D ŒI.t/ V .t/> 2 RnI CnV . To this system we add a nonlinear semiconductor device. Combining Kirchhoff’s laws and characteristic equations for the basic circuit elements, the Modified Nodal Analysis (MNA) yields a system of differentialalgebraic equations [7]. This reads compactly: E xP D Ax C Bu.t/ C .x/;

x.t0 / D x0 ;

t 2 Œt0 ; t1 

(1)

for the unknown x D Œe iL iV > , composed by node potentials e.t/ 2 Rn , currents through inductances and voltage sources iL .t/ 2 RnL , iV .t/ 2 RnV . Matrix E is singular and positive semidefinite and assumed to be symmetric; matrix A is negative semidefinite in the sense x> Ax  0, thus passivity is given. The independent inputs u are stamped by matrix B, which resembles a topology matrix. The term .x/ represents the stamped semiconductor currents; it will be precisely defined below. After this, in Sect. 3, the consistency of the initial data x0 will be discussed.

An Existence Result for Index-2 PDAE System Arising in Semiconductor Modeling

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2.2 Distributed Semiconductor Device Model For simplicity of notation, we consider a circuit containing one distributed-modeled device, only. We assume that the device has nD C 1 terminals, occupies the domain ˝  Rd (d 2 f1; 2; 3g), and is characterized by a doping profile N.x/ (x 2 ˝). We neglect thermal effects and assume electrons (charge q), and holes (positive charge q) to be responsible for the device current, where the device behavior is described by the drift diffusion system [6]. In terms of the number densities of electrons n.x; t/ and holes p.x; t/, respective quasi-Fermi potentials n .x; t/, p .x; t/, respective current densities jn .x; t/, jp .x; t/, and the electrostatic potential .x; t/, this reads: 8 ˆ < r  ."r/ D q.N C p  n/; r  jn D qR; jn D qn nrn ; ˆ : r  jp D qR; jp D qp prp ;

(2) .x; t/ 2 ˝  Œt0 ; t1 :

Here n , p are the mobilities, " the dielectricity and R the recombinationgeneration term. Employing the Maxwell-Boltzmann relations n D ni exp



n UT

 ;

   p D ni exp  UT p ;

with intrinsic concentration ni and thermal voltage UT , we can choose , n , p as unknowns of the problem (2). The mobilities are bounded, strictly positive functions depending on x, n, p and on the electric field r. For the generationrecombination term R D R.n; p/ we assume R.n; p/ D F .n; p/



np n2i

        1 D F .n; p/ exp  nUT p  1 ;

with F  0 (comprising Shockley-Read-Hall and Auger recombination-generation). Furthermore, we assume that the boundary @˝ is the union of two disjoint parts, Dirichlet D D D;0 [ D;1 [    [ D;nD (union of Ohmic device contacts) and Neumann parts N (insulation boundary) with usual boundary conditions:   bi D n D p D eD;k ;

on D;k ; k D 0; : : : ; nD ;

@=@ D @n =@ D @p =@ D 0;

on N D @˝ n D ;

(3)

using built-in potential bi .x/, applied potential eD;k .t/ (at the contact D;k ), and external unit normal to the boundary .

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2.3 Coupling Conditions The applied potentials eD;k are related to e by eD WD ŒeD;0

:::

eD;nD > D S> x;

(4)

where S 2 f0; 1g.nCnLCnV /.nD C1/ is a selection matrix containing exactly one nonzero entry in each column in the first n rows, i.e., eD is in fact selected from e. The electric current through the k-th terminal D;k is Z



jD;k .eD .t// WD 

 jn .x; t/ C jp .x; t/ dD;k :

D;k

It is possible to give a more suitable definition of the currents, in term of volume integrals and appropriate auxiliary functions, which require less regularity of n , p . Notice, jn , jp are time dependent only via eD . The coupling term .x/ in (1) is now given by the stamping of the vector jD D ŒjD;0 : : : ; jD;nD > : .x/ D SjD .S> x/:

(5)

3 Index Concept and Decoupling A matrix chain [8] can be used to decompose the circuit’s unknowns into differential and algebraic components. To recapitulate this for the network model (1) without the nonlinear term .x/, we denote by Q0 a projector onto the kernel of E0 WD E with complementary projector P0 WD I  Q0 . We also define the matrices E1 WD E0  A0 Q0 and A1 WD A0 P0 , with A0 WD A. If E1 is nonsingular, we can write (1) as: E1 .P0 xP C Q0 x/ D A1 x C Bu (6) and obtain the projected equations (index-1) yP D P0 E11 ŒA1 y C Bu.t/ ;

z D Q0 E11 ŒA1 y C Bu.t/ ;

(7)

for the differential component, y D P0 x, and the algebraic component z D Q0 x of x. If E1 is singular, let Q1 denote a projector onto the kernel of E1 . It is always possible to choose Q1 so that Q1 Q0 D O (8) where O denotes the zero matrix. Introducing the new matrices E2 WD E1  A1 Q1 and A2 WD A1 P1 , where P1 WD I  Q1 , the equation (1) without the nonlinear term, can be written as E2 .P1 P0 xP C Q1 x C Q0 x/ D A2 x C Bu.t/:

An Existence Result for Index-2 PDAE System Arising in Semiconductor Modeling

49

If the matrix E2 is nonsingular, by using the properties of the projectors, we can derive equations for the differential component and algebraic components of x D y C w C z: y D P0 P1 x;

w D P0 Q1 x;

z D Q0 x:

(9)

Thus initial data can be assigned for y only: y.t0 / WD y0 , from which w0 ; z0 are computed using the projected algebraic equations. Now, we include also the nonlinear term .x/ defined in (5). We can still introduce the projectors Q0 , Q1 , and as before rewrite the system (1) in the form yP D P0 P1 E21 ŒA2 y C Bu.t/ C .x/ ; w D P0 Q1 E21 ŒA2 y C Bu.t/ C .x/ ; z D Q0 Q1 .wP  w/ C Q0 E21 ŒA2 y C Bu.t/ C .x/ ; where the differential equation contains algebraic variables via .x/ D .yCwCz/. To decouple the equations we assume that  does not depend on z, i.e., we assume S> Q0 D O:

(10)

This assumption is equivalent to the additional topological condition used for the index-1 case in [1], and it amounts to saying that the device’s terminals are connected by paths of capacitors. Moreover, we assume that it is possible to choose Q1 such that, in addition to (8), it holds also P0 Q1 E21 S D O

and Q0 E21 S D O:

(11)

Notice that E2 depends on Q1 . We are not able to characterize topologically these additional conditions. But we will show with an example that there are indeed systems compatible with this assumption. If (10), (11) holds, then solution components w and z do not depend on : yP D P0 P1 E21 ŒA2 y C Bu.t/ C .y C w/ ; w D P0 Q1 E21 ŒA2 y C Bu.t/ ;

z D Q0 Q1 wP C Q0 P1 E21 ŒA2 y C Bu.t/:

This decomposition reveals that the system is still index-2 under the above conditions: replacing w in the differential equation, we obtain an ODE for y, only; then z is obtained from the last algebraic equation after y is known. Example. To show that it is possible to choose Q1 to satisfying also the additional conditions, we investigate a simple example Fig. 1 with incidence matrices:  AC D

 1 0 ; 01

 AR D

 1 ; 1

AV D

  1 ; 0

 AD D

 0 : 1

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MNA equations read:

R

iD

C1

V

C1 eP1 D 1=R.e2  e1 /  iV ;

C2

C2 eP2 D 1=R.e2  e1 /  iD ; 0 D V  e1 :

Fig. 1 Simple circuit schematic For C1 ; C2 ; R ¤ 0, s D iD .e2 /, u.t/ D V .t/ and x D Œe1 e2 iV > , we can recast these MNA equations in the above matrix form (1) using 2

2 3 3 2 3 2 3 C1 0 0 1=R 1=R 1 0 0 E0 D 4 0 C2 0 5; A0 D 4 1=R 1=R 0 5; B D 4 0 5; S D 4 1 5 : 0 0 0 1 0 0 1 0 For this example we find for the matrix chain: 2

3 000 Q0 D 4 0 0 0 5 ; 001

2

3 C1 0 1 E1 D 4 0 C 2 0 5 ; 0 0 0

2

3 1=R 1=R 0 A1 D 4 1=R 1=R 0 5 : 1 0 0

(12)

From this we deduce the general form of the projector Q1 (characterized by Q12 D Q1 , E1 Q1 D O) which satisfies condition (8), with free parameter ˛ 2 R, and consequently the matrix E2 : 3 1 ˛ 0 Q1 D 4 0 0 05; C1 ˛C1 0 2

3 ˛=R 1 C1 C 1=R E2 D E1  A1 Q1 D 4 1=R C2  ˛=R 0 5 : 1 ˛ 0 2

If we check the additional conditions (10) and (11) we will see that  > S> Q0 D 0 0 0 ;

h Q0 E21 S D 0 0 

i>

 > P0 Q1 E21 S D 0 0 0 : (13) Choosing ˛ D 0, which is allowed, the additional conditions are satisfied. ˛C1 C2

;

4 Existence Result The main result of this work is the following existence theorem. Theorem 1. The coupled problem (1)–(5) with u 2 C 1 .Œt0 ; t1 /, index-2 network, additional conditions (10-11) and consistent initial data admits a solution .x; ˚/ 2 C 0 .Œt0 ; t1 /C 0 .Œt0 ; t1 I H 1 .˝/\L1 .˝//, with y D P0 P1 x 2 C 1 .Œt0 ; t1 /.

An Existence Result for Index-2 PDAE System Arising in Semiconductor Modeling

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Moreover, any solution satisfies the estimates:   jy.t/j2  cy K.t/; jw.t/j2  cw K.t/; K.t/ WD e k.t t0 / jy0 j2 C ju0 j2 C kuk2H 1 .Œt0 ;t1 / ; jz.t/j2  cz .jy.t/j C jy.t/j P C ju.t/j C jPu.t/j/ ; inf bi C min eD;i    sup bi C max eD;i ; D

i

D

i

min eD;i  n ; p  max eD;i ; i

i

for some positive constants cy , cw , cz and k depending on E0 , A0 . We sketch the proof of this result: The basis of the result is the passivity condition for the device current .x/ (5), i.e., it holds x> .x/  0: Using the symmetry of E and the passivity of A, one can prove the a priori estimates in Theorem 1 (network variables) as in [1, 2] under the hypothesis that a solution exists. Now the existence prove involves the following steps (see [2] for details): First step. To solve the device equation (2), we define an iteration map ˚  (no time dependence) by freezing all nonlinear coefficient, but keeping the nonlinearity of the recombination-generation term, such that the device stays passive. Thus the a priori estimates do still apply and also the estimates of the potentials are recovered. Second step. The map ˚  is then extended to a map T, which includes the applied potential eD and thus time dependence. By the a priori estimates a suitable subset in function space can be defined, s.t. T is operating only on this subset (self-mapping). Third step. Using Schauder’s fixed-point theorem, we can show that T is a fixedpoint map on the set M. Thus, by construction, any fixed point .eD ; ˚/ D T.eD ; ˚/ leads to a solution of the original coupled problem.

References 1. Al`ı, G., Bartel, A., G¨unther, M., Tischendorf, C.: Elliptic partial differential-algebraic multiphysics models in electrical network design. M3AS 13(9), 1261–1278 (2003) 2. Al`ı, G., Rotundo, N.: An existence result for elliptic partial differential-algebraic equations arising in semiconductor modeling. Nonlinear Anal. 72, 4666–4681 (2010) 3. Al`ı, G., Bartel, A., Culpo, M., de Falco, C.: Analysis of a PDE Thermal Element Model for Electrothermal Circuit Simulation, in: Scientific Computing in Electrical Engineering SCEE 2008, pp. 273–280, Luis R.J.C., Janne, R. (eds.) Springer, Berlin (2010) 4. Bartel A., Partial Differential-Algebraic Models in Chip Design - Thermal and Semiconductor Problems, Fartschrift-Berichte VDI, Reihe 20, Nr. 391, Duesseldorf, VDI Verlag (2004) 5. G¨unther, M.: A PDAE model for interconnected linear RLC networks. Math. Comp. Model. Dyn. Syst. 7(2), 189–203 (2001) 6. Markowich, P.A., Ringhofer, C.A., Schmeiser, C.: Semiconductor Equations (Springer, Berlin, 1990) 7. McCalla, W.J.: Fundamentals of Computer Aided Circuit Simulation. Kluwer, Dordrecht (1988) 8. Tischendorf, C.: Topological index calculation of differential-algebraic equations in circuit simulation. Surv. Math. Ind. 8, 187–199 (1999)

Deterministic Numerical Solution of the Boltzmann Transport Equation Karl Rupp, Tibor Grasser, and Ansgar Jungel ¨

Abstract Due to its deterministic nature, the spherical harmonics expansion method is an attractive alternative to the Monte Carlo method for the solution of the Boltzmann Transport Equation for the purpose of electronic device simulation. However, since the problem is posed in a six-dimensional problem space emerging from the three-dimensional space variable and the three-dimensional momentum variable, deterministic approaches typically suffer from huge memory requirements, which have prohibited their application to two and three-dimensional simulations. To reduce these high memory requirements, we first show that the coupling of the resulting system of partial differential equations is only weak and then propose a new scheme for the lossless compression of the resulting system of linear equations after discretization. This reduces the overall memory requirements significantly and paves the way for deterministic three-dimensional device simulations. Numerical experiments demonstrate the applicability of our method and confirm our theoretical results.

K. Rupp () Christian Doppler Laboratory for Reliability Issues in Microelectronics at the Institute for Microelectronics, TU Wien, Gußhausstraße 27–29/E360, 1040 Wien, Austria e-mail: [email protected] T. Grasser Institute for Microelectronics, TU Wien, Gußhausstraße 27–29/E360, 1040 Wien, Austria e-mail: [email protected] A. J¨ungel Institute for Analysis and Scientific Computing, TU Wien, Wiedner Hauptstraße 8-10/E101, 1040 Wien, Austria e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 7, © Springer-Verlag Berlin Heidelberg 2012

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1 Introduction As long as quantum mechanical effects are negligible, the microscopic behavior of charge carriers in semiconductors is very well described by a distribution function f .x; k; t/ that depends on the spatial coordinate x, the momentum „k and time t, and fulfills the Boltzmann transport equation (BTE). The most commonly used method to solve the BTE is the Monte Carlo method, with the main disadvantage of its computational expense, especially when attempting to reduce the statistical noise in the low density tails of the distribution function. The most prominent alternative to the stochastic Monte Carlo (MC) method is the deterministic spherical harmonics expansion (SHE) method [2]. Recent results demonstrate that higher order expansions, e.g. of order nine, result in excellent agreement with MC simulations, while maintaining the performance benefit [3, 4]. The major challenge of the SHE method is the huge memory consumption reported even for two-dimensional devices at moderate expansion orders [3], which has so far prohibited an application of the SHE method to three-dimensional simulations. To overcome these limitations, we present a new system matrix compression scheme that reduces the memory requirements by up to two orders of magnitude and paves the way for three-dimensional device simulations using the SHE method.

2 The Projected Equations After a truncated expansion of the distribution function into real-valued, orthonormal spherical harmonics Yl;m .; '/ up to order L, f .x; "; ; '; t/ 

L X l X

fl;m .x; "; t/Yl;m .; '/;

(1)

lD0 mDl

a spherical projection of the BTE and application of the H -transform [1], one obtains with Einstein’s summation convention a system of coupled partial differential equations with shifted arguments [3] 0 0 0 ;m0 @fl;m Z C rx  vll;m;m fl 0 ;m0 Z  F   ll;m fl 0 ;m0 Z @t X l 0 ;m0 Iin l 0 ;m0 Iout sl;m fl 0 ;m0 .x; "  „! ; t/Z.x; "  „! ; t/  sl;m fl 0 ;m0 Z D



for all l 2 f0; : : : ; Lg, m 2 fl; : : : ; lg. The generalized density of states Z depends l 0 ;m0 Iin on the band structure and is assumed to be isotropic, F is the force and sl;m

Deterministic Numerical Solution of the Boltzmann Transport Equation 0

55

0

l ;m Iout and sl;m denote the in- and out-scattering coefficients. Function arguments are suppressed wherever appropriate to increase readability. 0 0 0 ;m0 l 0 ;m0 Iin l 0 ;m0 Iout , sl;mI and sl;m were multiples If all coupling coefficients vll;m;m ,  ll;m of the Kronecker delta ıl;l 0 ım;m0 , all equations would be decoupled and could be solved individually. Conversely, nonzero coupling coefficients for all quadruples .l; m; l 0 ; m0 / indicate a tight coupling, which usually complicates the solution process. This is in analogy to systems of linear equations: If the system matrix is diagonal, the solution is found immediately, but if the matrix is dense, typically a lot of computational effort is required to solve the system. l 0 ;m0 Iin l 0 ;m0 Iout It has been shown in [4] that the scattering terms sl;m and sl;m do not couple different expansion coefficients in the case of spherical energy bands. Moreover, the symmetry of the underlying processes yields for general band structures that 0

0

0

0

;m 2i C1;m v2i 2i;m D v2i C1;m D 0;

0

0

;m  2i 2i;m D 

2i 0 C1;m0 2i C1;m

D0

for all permissible integers i , i 0 and m, m0 holds [4]. Therefore, all nonzero coupling coefficients possess different parities in the leading indices. This structural information about the coupling has already been used in a preprocessing step for the solution of the discretized equations in [4]. Under the assumption of spherical energy bands, i.e. ".k/ D "Q.jkj/, the velocity v, the modulus of the wave vector jkj and the generalized density of states only depend on the energy ", but not on the angles ; '. With this it can now be shown 0 0 0 ;m0 that the coupling induced by vll;m;m and  ll;m is weak: Theorem 1 Under the assumption of spherical energy bands, the following holds true for indices l; l 0 2 f0; : : : ; Lg, m 2 fl; : : : ; lg and m0 2 fl 0 ; : : : ; l 0 g: 0

0

1. If vll;m;m is nonzero, then l 2 fl 0 ˙ 1g and m 2 f˙jm0 j ˙ 1; m0 g. 0

0

2. If  ll;m;m is nonzero, then l 2 fl 0 ˙ 1g and m 2 f˙jm0 j ˙ 1; m0 g. A proof is given in [6]. The theorem allows one to better eliminate those 0 0 0 ;m0 , which may not vanish in simulations due to numerical coefficients vll;m;m and  ll;m noise, even though they are analytically zero.

3 Discretization and System Matrix Compression In steady state, a discretization of the expansion coefficients on a staggered grid (cf. [5]) with N grid points leads to a system of linear equations represented by a system matrix S of size N.L C 1/2  N.L C 1/2 . The sparsity of S is a direct consequence of the finite difference or finite volume schemes used. Using the results of Theorem 1, it can be shown [6] that the resulting system matrix S can be written for the case of only elastic scattering mechanisms as

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S D

8 X

Qi ˝ R i

(2)

i D1

where ˝ denotes the Kronecker product. The Qi only depend on the discretization in the .x; H /-space, while the R i are determined by the coupling among spherical harmonics up to order L only. This allows for a representation of S using only O..L C 1/2 C Csparse N / numbers. Since N is typically much larger than .L C 1/2 , the full system matrix can be stored for Csparse D 10 with roughly 80N numbers, which means a reduction of two orders of magnitude compared to the uncompressed case.

4 Solution of the Linear System The matrix compression scheme is of use only if the resulting system of linear equations can be solved without the need to recover the full matrix again. Such a reconstruction is, in principle, necessary if direct solvers are used, because during the solution process the matrix structure is altered in a way that destroys the block structure. For many popular iterative solvers from the family of Krylov methods, it is usually sufficient to provide matrix-vector multiplications [7]. Matrix-vector products for a matrix given as a Kronecker product of two smaller matrices can be carried out in a straightforward manner by decomposing the vector into blocks of suitable size. This allows for the realization of a very memory efficient matrix-vector multiplication if the system matrix is given in the form (2). However, numerical experiments indicate that the full system matrix for the even and odd order expansions coefficients is ill-conditioned. A substantial improvement of the system matrix condition number can be obtained if the unknowns for the odd order expansion coefficients are eliminated in a preprocessing step. However, a direct elimination by altering the system matrix is not possible without destroying the Kronecker product structure. This can be avoided by using the Schur complement, for which we refer the reader to [6]. Thus, the system matrix compression scheme can also benefit from the improved system matrix condition number after elimination of the odd order unknowns.

5 Results We have compared memory requirements for the storage of the system matrix at several expansion orders in a two-dimensional device simulation. The results in Fig. 1 clearly demonstrate the asymptotic advantage of our approach: Already at an expansion order of L D 5, memory savings by a factor of 18 are observed, which increases to 146 at L D 13. With the compressed scheme, the memory required

Deterministic Numerical Solution of the Boltzmann Transport Equation

57

1000

Memory (MegaBytes)

Standard Scheme Compressed Scheme

100

10

1

0

2

4

6

8

10

12

14

Expansion Order L

Fig. 1 Memory used for the uncompressed and the compressed system matrix for different expansion orders L on a three-dimensional grid with 12;500 nodes 100

Relative Memory for System Matrix (Percent)

90 80 70 60 50 40 30 20 Standard Scheme Compressed Scheme

10 0

0

2

4

6 8 10 Expansion Order L

12

14

Fig. 2 Memory used for the system matrix in relation to the total amount of memory used (i.e. system matrix, unknowns and right hand side)

for the system matrix increases only by a few kilobytes as L increases, which is negligible. Since the memory required by the system matrix is of order O.N C L2 / and the memory for the unknowns is of order O.NL2 /, the memory required for the unknowns is much larger than the memory required for the representation of the system matrix for large values of L, cf. Fig. 2. Therefore, the asymptotic

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Execution time (Milliseconds)

1000

100

10

1 Standard Scheme Compressed Scheme 0.1

0

2

4 6 Expansion Order L

8

10

Fig. 3 Execution times for matrix-vector multiplication with the Schur complement on a single core of an Intel Core 2 Quad Q9550 CPU

memory requirements for the full simulation is still O.NL2 /, but the constant of proportionality is of order one, while for the full system matrix it is around 11Csparse  100, so a reduction of memory requirements by two orders of magnitude is obtained. On a single CPU core, the minor price to pay for the dramatic reduction in memory consumption is that the execution times of matrix-vector products with the Schur complement increase by a factor of about two, cf. Fig. 3. However, the proposed matrix compression scheme is very well suited for parallel architectures, because the data required for the system matrix may even fit into the CPU caches, allowing for a very high performance. Moreover, since the system matrix can be written as a sum of Kronecker products, each summand can be computed on a separate core.

6 Conclusion The matrix compression scheme presented in this work reduces the memory requirements for the system matrix arising from a SHE of the BTE from order O.NL2 / to O.N CL2 /, which results in total memory savings for the full simulation run by up to two orders of magnitude. Therefore, our scheme paves the way for three-dimensional device simulations especially for larger expansion orders L. On a single CPU core, the small price to pay is a runtime penalty on matrix-vector multiplication of about a factor of two. However, the proposed method is expected to outperform the traditional storage scheme on parallel architectures.

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59

Acknowledgements Support by the Graduate School PDEtech at the TU Wien is gratefully acknowledged. A. J¨ungel acknowledges partial support from the Austrian Science Fund (FWF), ¨ grants P20214, I395, and WK “Differential Equations”, and the Austrian Exchange Service (OAD), grants ES 08/2010, FR 07/2010, HR 10/2010.

References 1. Gnudi, A., et. al.: Two-dimensional MOSFET simulation by means of a multidimensional spherical harmonics expansion of the boltzmann transport equation. Solid State Electron. 36(4), 575–581 (1993) 2. Goldsman, N., et. al.: A physics-based analytical/numerical solution to the boltzmann transport equation for the use in device simulation. Solid State Electron. 34, 389–396 (1991) 3. Hong, S.M., Jungemann, C.: A fully coupled scheme for a boltzmann-poisson equation solver based on a spherical harmonics expansion. J. Comp. Elec. 8, 225–241 (2009) 4. Jungemann, C., et. al.: Stable discretization of the boltzmann equation based on spherical harmonics, box integration, and a maximum entropy dissipation principle. J. Appl. Phys. 100(2), 024502 (2006) 5. Ringhofer, C.: Numerical methods for the semiconductor boltzmann equation based on spherical harmonics expansions and entropy discretizations. Transport Theor. Stat. Phys. 31, 431–452 (2002) 6. Rupp, K., et. al.: Matrix compression for spherical harmonics expansions of the boltzmann transport equation for semiconductors. J. Comput. Phys. 229, 8750–8765 (2010) 7. Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)

Analysis of Self-Heating Effects in Sub-Micron Silicon Devices with Electrothermal Monte Carlo Simulations Orazio Muscato and Vincenza Di Stefano

Abstract In order to investigate the role of self-heating effects on the electrical characteristics of sub-micron devices, we have implemented a Monte Carlo device simulator that includes the self-consistent solution of the heat transport equation, obtained in the framework of Extended Irreversible Thermodynamics. The lattice temperature is fed back into the electron transport solver through temperaturedependent scattering tables. Simulation results for a nC  n  nC diode are shown.

1 Introduction The increasing of the device density on microelectronics chips could be constrained by the ability to dissipate the extremely high rates of heat generation. For the packaging designers, it is necessary to understand the mechanisms of the heat generation and dissipation inside the device as well as their effects on the electrical performances and reliability. The volumetric heat generation rate inside submicrometric semiconductor devices can be expected to be very high since the device operating power cannot be reduced below a certain level. From the microscopic view point, heat conduction in crystalline materials occurs by lattice vibrations, which produce quanta of energy called phonons. Self-heating starts when the nearly-free conduction band electrons in a semiconductor are accelerated by the electric field. The electrons gain energy from the field, then lose it by inelastical scattering with the boundaries and phonons, heating up the lattice through the mechanism known as Joule heating. The lattice absorbs the extra electron energy, warms to a higher temperature, and in return affects the electronic

O. Muscato ()  V. Di Stefano Dipartimento di Matematica e Informatica, Universit´a di Catania, viale A. Doria 6, 95125 Catania, Italy e-mail: [email protected]; [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 8, © Springer-Verlag Berlin Heidelberg 2012

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transport properties of the material (e.g. the mobility). Phonon-phonon scattering helps to restore thermodynamic equilibrium. Hence one challenge for the device simulation of the next future is the modeling of electric and thermal effects. If the charge carriers are in thermal equilibrium with the lattice, there are small temperature gradients, and the device length is much larger than the phonon mean free path (approximately 200 nm in bulk silicon at room temperature), then this regime (diffusion-like) can be described accurately using the non-isothermal driftdiffusion model [1]. Here the drift-diffusion equations, which deal with the charge transport, are coupled with the thermal transport via a heat flow balance equation, closed by means of the Fourier law, i.e. Cv

@T C rx Q D Hg @t

;

Q D rx T

where Cv is the specific heat,  the thermal conductivity, T the lattice temperature, and Hg the Heat Generation Rate, which is taken as Joule heat, e.g. Hg D J  E

(1)

where E is the electric field and J the current density. When these equations are applied to smaller systems, the deviations with respect to the observations become more obvious [2]. Experimental results prove that, when a temperature difference is applied in a nanometric-scale silicon sample, the heat flux obtained is significantly lower than that predicted by the Fourier law [3]. Some authors suggested that a reduction in the conductivity of the material may be the cause of this behaviour. In order to describe this non-local effect, several authors have tried to model the heat transport in a different way. One of such models is provided by the Extended Irreversible Thermodynamics [4], in which the Fourier law is modified in order that non-local and memory effects are taken into consideration. This theory considers the higher order fluxes as independent variables, and the consecutive transport equations form a hierarchy of coupled evolution equations. In a linear approximation, the equation describing the heat transport is a Cattaneo-like equation [5]  @2 T @T  rx2 T D Hg C 4 @t 2 @t

D

0 2 2 Kn2

q

(2) 

1 C 4 2 Kn2  1

where  is the relaxation time for the higher order fluxes, Kn is the Knudsen number, and 0 the diffusive conductivity. Equation (2) must be coupled to transport equations describing the charge transport in off-equilibrium regimes. One possibility is to couple to the (2) a hydrodynamic-like model as in [6], but in this paper we shall pursue another route.

Analysis of Self-Heating Effects in Sub-Micron Silicon Devices

63

2 The Electrothermal Monte Carlo Simulator Charge transport in semiconductors can be described, from the kinetic viewpoint, using the semiclassical Boltzmann Transport equation (BTE). The Monte Carlo (MC) is a well known algorithm which solves the BTE, using a stochastic particle system, evolving according to Newton’s law of mechanics, subject to the electric field (external and self-consistent), and scattering. In silicon, at room temperature, the main lattice vibrations are due to the acoustic and (six) optical branches [7]. The total scattering rate w.k0 ; k; T /, according to the Fermi rule, writes 0

w.k; k ; T / D Kac .k; k0 /ı.".k0 /  ".k// C 

6 X

Ki .k; k0 / 

i D1

ı.".k0/  ".k/ C „!i /.nqi C 1/ C ı.".k0/  ".k/  „!i /nqi



(3)

where „!i is the i th phonon energy, ".k/ the electron energy dispersion relation which we take in the quasi parabolic band approximation, and T the lattice temperature. This algorithm has been recently extended for solving the kinetic transport equations for the coupled system formed by the electrons and phonons [8], whose main drawback is the huge computational effort. An important issue that arises from the coupling of a MC electronic transport algorithm to any thermal model is the significant difference in the characteristic time scale of electronic and thermal transport. Electronic transients in silicon systems are of the order of tenth of picoseconds, whereas thermal transients may be of the order of nanoseconds, microseconds, or even longer. For this reason the Electrothermal Monte Carlo (ETMC) model has been introduced [9,10], coupling the MC algorithm for the BTE with a steady-state solution of the heat equation. The ETMC is an iterative approach: the initial iteration is a standard isothermal MC (i.e. solving the BTE for the electrons) at room temperature (300 K) until the steady-state is reached ('10 ps). As the steady state is approached, electronic parameters are sampled for typically 10100 ps to generate the results from this iteration. In particular the Heat Generation rate can be computed, in each position x of the device, as a sum over all the phonon emission minus phonon absorption events per unit time, i.e. Hg .x/ D

n X .„!ems  „!abs / Np tsi m

(4)

where n is the electron density, Np ('105 ) the total particle number used in the simulation, tsi m ('100 ps) the total simulation time after the transient is assumed to die out, and „!ems ; „!abs the energy of the phonon emitted or absorbed respectively. This method is a more fundamental approach with respect to (1), because it calculates power dissipation directly from the number of phonon emissions and absorptions, and it allows for the investigation of the relative contribution of different phonon types to heat dissipation, which is not possible with most other

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methods. Then the spatially varying lattice temperature distribution is obtained by solving the steady-state heat diffusion equation (2). In the next iteration the MC algorithm is rerun with the new update lattice temperature distribution (which enters in the scattering probability (3)) and the previous procedure repeated. This iterative procedure is performed until the terminal currents converge to the electrothermal steady-state value.

3 Simulation Results and Conclusions The nC  n  nC silicon diode is the simplest one-dimensional inhomogeneous structure. This diode consists of two highly doped regions (nC called cathode and anode) connected by a less doped region n called channel. In our simulations, the nC regions are 40 nm-long doped to a density ND D 1019 cm3 , while the channel is 20 nm-long doped to a density ND D 1016 cm3 , and the applied bias Vb D 3:2 V. In the Fig. 1 we plot the Heat Generation rate versus the position in the device for some thermal iterations, evaluated by means of (4) in the steady-state regime. We observe that most Joule heat is dissipated in the anode region, and this suggests that the design of the anode region and contacts will play a significant role for heat dissipation in ultrashort devices. In the Fig. 2 we plot the Lattice temperature T versus the position in the device for some thermal iterations, evaluated by means of (2) in the steady-state regime, showing a typical asymmetric behaviour which has been observed in the experiments [6]. On the left of the Fig. 3, we plot the terminal

Heat generation rate ( 1018 W / cm3)

6 Iter = 1 Iter = 4

5

4

3

2

1

0

0

20

40

60

80

100

x (nm)

Fig. 1 Stationary heat generation rate versus position, evaluated by means of (4), for some thermal iterations

Analysis of Self-Heating Effects in Sub-Micron Silicon Devices

65

340 Iter = 1 Iter = 4

Lattice temperature (K)

335 330 325 320 315 310 305 300

0

20

40

60

80

100

x (nm)

Fig. 2 Stationary lattice temperature versus position, evaluated by means of (2), for some thermal iterations 101

92 91.5 100

91

error (%)

current density (mA / (µ m)2)

92.5

90.5 90

10

–1

89.5 89 10–2

88.5 1

1.5

2

2.5

3

3.5

4

0 10

Iteration

Iteration

Fig. 3 (Left) Stationary current density versus thermal iterations. (Right) The error (5) of the current density versus thermal iteration

current density in the device Ji , versus the i th thermal iteration of the ETMC model: we can observe that only four thermal iterations are necessary to obtain the steadystate solution of the current. On the right of the Fig. 3, we plot the relative error of the current density Ji with respect to its final value J4 , i.e. ˇ ˇ ˇ Ji ˇˇ ˇ Eri D ˇ1  ˇ  100 ; J4

i D 1; 2; 3 :

(5)

The log-log scale used in the Fig. 3 shows an exponential convergence rate to the steady-state solution. Finally in the Fig. 4 we plot the I/V characteristics obtained

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current (mA / (µ m)2)

100

80

60 T = 300 K ETMC

40

20

0

0

1

2

3 4 Voltage (V)

5

6

7

Fig. 4 The I/V characteristics obtained with the ETMC model and the isothermal MC at T D 300 K

with the ETMC model, and with an isothermal MC simulator: we can observe that the discrepancy between the two curves become more pronounced for voltage higher than 1 V. The CPU consumed for these simulations was 26 h. The ETMC is a promising and useful tool for understanding energy dissipation and transport in structures whose dimensions are bigger than the phonon and electron wave lengths (i.e. few nanometers). Recent directions regarding the determination of the electrical and thermal conductivity of small devices, thermal rectification, and the role of ubiquitous material interfaces have been undertaken. Acknowledgements The authors have been supported by “Progetti di Ricerca di Ateneo”, Universit´a degli Studi di Catania, and acknowledge CINECA Award N. HP10CHSS93, 2010 for the availability of high performance computing resources and support.

References 1. Wachutka, G.K.: Rigorous thermodynamic treatment of heat generation and conduction in semiconductor device modeling. IEEE Trans. CAD 9(11), 1141–1149 (1990) 2. Pop, E., Sinha, S., Goodson, K.: Heat generation and transport in nanometer scale transistors. Proc. IEEE 94(8), 1587–1601 (2006) 3. Sverdrup, P.G., Sinha, S., Asheghi, M., Uma, S., Goodson, K.E.: Measurement of ballistic phonon conduction near hotspots in silicon. Appl. Phys. Lett. 78(21), 3331–3333 (2001) 4. Jou, D., Casas-V´azquez, J., Lebon, G.: Extended Irreversible Thermodynamics. Springer, Berlin (2001) 5. Alvarez, F.X., Jou, D.: Memory and nonlocal effects in heat transport: from diffusive to ballistic regimes. Appl. Phys. Lett. 90, 083109 (2007)

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6. Lai, J., Majumdar, A.: Concurrent thermal and electrical modeling of sub-micrometric silicon devices. J. Appl. Phys 79(9), 7353–7361 (1996) 7. Jacoboni, C., Reggiani, L.: The Monte Carlo method for the solution of charge transport in semiconductors with applications to covalent materials. Rev. Mod. Phys. 55(3), 645–705 (1983) 8. Rowlette, J.A., Goodson, K.E.: Fully coupled nonequilibrium electronphonon transport in nanometer-scale silicon FETs. IEEE Trans. Elec. Dev. 55, 220–232 (2008) 9. Sadi, T, Kelsall, R, Pilgrim, N.: Simulation of electron transport in InGaAs/AlGaAs HEMTs using an electrothermal Monte Carlo Method. IEEE Trans. Elec. Dev. 53, 1768–1774 (2006) 10. Raleva, K., Vasileska, D., Goodnick, S.M., Dzekov, T.: Modeling thermal effects in nanodevices., J. Comp. Electr. 7(3), 226–230 (2007)

Diffusive Limit of a MEP Hydrodynamical Model Obtained from the Bloch-Boltzmann-Peierls Equations for Semiconductors Giuseppe Al`ı, Vittorio Romano, and Nella Rotundo

Abstract We consider a MEP hydrodynamical model obtained from a set of transport equations for the distribution functions of electrons in conduction band and phonon. We assume that the MEP model contains equations for the electron density fluxes and energy fluxes, and for the phonons energy fluxes. For this system we introduce a small parameter, related to the transition probabilities in the collision terms, and a diffusive scaling at the level of the Lagrangian multipliers appearing in the closure relations. In the diffusive limit, as the small parameter tends to zero, we obtain a model that can be physically interpreted in the framework of linear irreversible thermodynamics.

1 The Bloch-Boltzmann-Peierls Equations Several macroscopic models have been presented for dealing with thermal effects in the crystal lattice of a semiconductor (see [7] for a critical review). For describing adequately these effects from a microscopic point of view, it is necessary to take G. Al`ı () Departimento di Matematica, Universit`a della Calabria, ponte Pietro Bucci 30/B, Arcavacata di Rende, 87036 Cosenza, Italy INFN, Gruppo collegato di Cosenza, Arcavacata di Rende 87036, Cosenza, Italy e-mail: [email protected] V. Romano Dipartimento di Matematica e Informatica, Universit`a di Catania, viale Andrea Doria 6, 95125 Catania, Italy e-mail: [email protected] N. Rotundo Dipartimento di Matematica e Informatica, Universit`a di Catania, viale Andrea Doria 6, 95125 Catania, Italy INFN, Gruppo collegato di Cosenza, Arcavacata di Rende 87036, Cosenza, Italy e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 9, © Springer-Verlag Berlin Heidelberg 2012

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into account nonequilibrium populations of phonons, as done in [3] (see also [4]). Here, we follow the approach used in [5, 6], presenting it in a systematic way. We consider an ensemble of electrons, with charge e, in a semiconductor with nph relevant families of phonons. We denote by f .x; k; t/ the electron distribution function, and by gi .x; q; t/, i D 1; : : : ; nph , the phonon distribution functions, with x 2 ˝  R3 , k; q 2 B  R3 , t 2 RC , where ˝ is the domain of the semiconductor, and B is the first Brillouin zone of the inverse lattice. These distribution functions satisfy the Bloch-Boltzmann-Peierls (BBP) equations: 8 nph X @f e ˆ ˆ < C v.k/  rx f  E  rk f D C0el .f / C Ciel .f; gi /; @t „ i D1 ˆ ˆ : @gi C c .q/  r g D C ph .f; g /; i D 1; : : : ; n ; i x i i ph i @t

(1)

where v.k/ D „1 rk E.k/ is the electron group velocity, and ci .q/ D rq !i .q/, i D 1; : : : ; nph , are the phonon group velocities, with E.k/ denoting the electron dispersion relation (conduction energy band), and !i .q/ the phonon dispersion relations. ph The collision operators C0el .f /, Ciel .f; gi /, Ci .f; gi /, i D 1; : : : ; nph , model the electron-impurity and electron-phonon scattering mechanisms. They are given by Z C0el .f / D Z Ciel .f; gi / D

B

B

ŒP0 .k0 ; k/f 0 .1  f /  P0 .k; k0 /f .1  f 0 / dk0 ; ŒPi .k0 ; kI gi /f 0 .1  f /  Pi .k; k0 I gi /f .1  f 0 / dk0 ;

Z

ph

Ci .f; gi / D 2

wC .k? ; k0 ; q/Œ.g C 1/f 0 .1  f ? /  gf ? .1  f 0 / dk0 ;

B

where the transition rates have the following structure 0 C C  0   Pi .k; k0 I gi / D wC i .k; k ; q /gi C wi .k; k ; q /.gi C 1/; 0 0 w˙ i .k; k ; q/ D si .q/ıŒE.k/  E.k / ˙ „!i .q/;

with q˙ D ˙.k0  k/, k? D k0  q, si depending on the tips of scattering. In the following sections, we present a MEP hydrodynamical model obtained from the BBP equations. For this system we introduce a small parameter, related to the transition probabilities in the collision terms, and a diffusive scaling at the level of the Lagrangian multipliers appearing in the closure relations. In the diffusive limit, we obtain a model that can be physically interpreted in the framework of linear irreversible thermodynamics (see [1] for a similar discussion).

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71

2 MEP-Based Macroscopic Models Macroscopic models for semiconductors can be obtained from a suitable finite subset of the infinite hierarchy of moment equations of the Boltzmann equation. To this aim we consider the sets of weight functions for electrons and for phonons: ˛

W el D f

.k/ W ˛ 2 I g;

Wi

ph

˛ i .q/

Df

W ˛ 2 Ii g;

i D 1; : : : ; nph ;

with index sets I , Ii , and introduce the electron and phonon moments: Z

˛

˛

M D

f dk;

˛ 2 I;

B

Mi˛

Z D B

˛ i gi

dq;

˛ 2 Ii ; i D 1; : : : ; nph :

This moments satisfy the coupled electron-phonon moment equations 8 @M ˛ ˆ ˆ C r  M˛ C eE  N˛ D C0˛ C C ˛ ; < @t ˛ ˆ ˆ : @Mi C r  M˛ D C ˛ ; ˛ 2 Ii ; i i @t

˛ 2 I; (2)

with M˛ .f / D

Z

˛

vf dk;

N˛ .f / D

Z

1 rk „

˛

f dk;

M˛i .gi / D

Z

˛ i ci gi

dq; (3)

C0˛ D

Z

˛

C0el .f / dk; C ˛ D

nph Z X

˛

Ciel .f; gi / dk; Ci˛ D

Z

˛ ph i Ci .f; gi / dq:

i D1

(4) System (2) is not closed, since higher moments appear, and the collision terms in (4) are not expressed in terms of moments of f or gi . To solve this closure problem we can use an extension of the Maximum Entropy Principle (MEP). We introduce the entropy functional (with units such that kB D 1) H.f; g1 ; : : : ; gnph / D H el .f / C H ph .g1 / C    C H ph .gnph /; Z Z el C ph H .f / D h .f / dk; H .g/ D h .g/ dq; B

B

with h˙ .f / D  Œf log f ˙ .1  f / log.1  f /. We choose .f; gi / are estimated by

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nph n X  X   ME ME   X o ˛ ˛ ˛ f ; gi D argmin H C  M .f /M C ˛i Mi˛ .gi /Mi˛ ; ˛2I

i D1 ˛2Ii

where ˛ and ˛i are Lagrangian multipliers. If f ME , giME exist, they are given by  P ˛ f ME D exp ˛2I 

˛



1 C1 ;

 P ˛ giME D exp ˛2Ii i

˛ i



1 1 :

The Lagrangian multipliers are implicitly defined as functions of the moments fM ˛ ; ˛ 2 I g, fMi˛ ; ˛ 2 Ii g by the constraints Z

˛

f ME dk D M ˛ ; ˛ 2 I;

Z

˛ ME i gi

dq D Mi˛ ; ˛ 2 Ii :

(5) ph

We choose the sets of weight functions W el D f1; v; E; Evg, Wi D f„!i ; „!i ci g, with corresponding moments fn; j; W; jW g, fWi ; jW g, and Lagrangian i multipliers f; ƒ; W ; ƒW g, fi ; ƒW i g. Then the resulting moment equations are: 8 @n ˆ ˆ C r  j D 0; ˆ ˆ ˆ @t ˆ ˆ ˆ < @W C r  jW CeE  j D C0W CC W ; ˆ @t ˆ ˆ ˆ ˆ ˆ ˆ @Wi ˆ W : C r  jW i D Ci ; @t

@j C r J C eE  R D C0 C C; @t @jW W C r J W CeE  R W D CW 0 CC ; @t @jW W i C r J W i D Ci ; @t

i D 1; : : : ; nph ; (6)

where J , R, J W , R W , are the moments of f ME with respects to the weight functions v ˝ v, „12 rk ˝ rk E, Ev ˝ v, v ˝ v C E „12 rk ˝ rk E, while J W i is the moment of giME with respects to the weight function „!i ci ˝ ci . System (6) must be ph supplemented with the constraints (5), written for W el , Wi .

3 Diffusive Limit In this section we consider the diffusive limit of system (6). To explain the main idea, we consider a simple isentropic model: @n C r  j D 0; @t

@j C r  J C eE  R D C; @t

(7)

Diffusive Limit of a MEP Hydrodynamical Model Obtained

with the closure J D scaling

1 n

73

j ˝ j C p.n/I, R D nI, C D  1 j. Using the diffusive t D tO=;

j D  Oj;

(8)

formally, we can pass from the moment equations to the drift-diffusion equations: 2

@Oj C r  J  C eEn D Oj @tO

!

rp.n/ C eEn D Oj:

The diffusive scaling (8) depends on a detailed knowledge of the closure relations: the small parameter  arises from the momentum relaxation time, which is a characterizing parameter of C, and the limit of J  as  tends to zero depends on the specific form of the closure. Now, let us go back to the MEP-based models (6). Following the suggestions of the starting example, first we need to identify the smallness parameter from the collision term. We assume the following scaling for the transition probabilities: Scaling 1: P0 .k; k0 / D

1 O P0 .k; k0 /; 

„!i D „!O i ;

si .q/ D

 sOi .q/; 

giME D  gO iME :

(9) (10)

Later, we will see that the small parameter  is related to . Next, since the closure relations are given in terms of the Lagrange multipliers, we propose the following scaling for the Lagrange multipliers corresponding to the fluxes: O Scaling 2: ƒ D  ƒ;

OW; ƒW D  ƒ

OW ƒW i D  ƒi :

We will show that (9), (11) lead to a scaling for the fluxes similar to the one used in the previous example. Introducing the notation O  v C ƒ O W  vE DW .0/ C .1/ ;  D  C W E C  ƒ   O W  ci „!O i DW  .0/ C .1/ ; i D W O i C  ƒ i „! i i i we can express the distribution functions as follows: P ` .`/  ` p`C .f .0/ /..1/ /` DW 1 ; `D0  f  P .0/ P .1/ ` `  .0/ ` .`/ D  gO i; 1 O i; / .i /` DW 1 `D0   p` .g `D0  gi; ;

f ME D f .0/ giME

P1

`D0

(11)

 1 .0/  1 .0/ in which we have set f .0/ D exp..0/ / C 1 , gO i; D exp.i /  1 , and the functions p`˙ are defined iteratively by p0˙ .x/ D 1;

p`˙ .x/ D

d 1 ˙ .1  x/ Œxp`1 .x/; ` dx

`  1:

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.0/

Since  gO i; tends to 1=i as  tends to zero, and the leading order coefficient of the polynomial p` is 1, we obtain  .`/ .0/  .1/ .0/ ` lim gi; D 1=i  i =i :

(12)

 !0

In the expansions (11), we have that .`/

f .`/ .k/ D .1/` f .`/ .k/;

.`/

gi; .q/ D .1/` gi; .q/:

Using this property, and the fact that E.k/, !i .q/ are even functions, expansions (11) lead to analogous expansions for each term in (6): aD

1 X

 2` a.2`/ ;

aD

1 X

 2mC1 a.2mC1/ dk;

(13)

mD0

`D0

W W with a 2 fn; J ; R; W; J W ; R W ; Wi ; J W i g, and a 2 fj; j ; ji g. This leads to the third part of our scaling:

Scaling 3: t D tO=

j D  Oj;

jW D  OjW ;

OW jW i D  ji :

(14)

Applying (9), (10), (11), (14) to system (6), the system of the moment equations becomes: 8 @n 1O @Oj  O ˆ ˆ C r  Oj D 0;  2 C r J C eE  R D C ˆ ; 0C C ˆ ˆ O O   @ t @ t ˆ ˆ ˆ < @OjW @W 1 1 1 OW  OW C CO W ; 2 C C ; C r  OjW C eE  Oj D 2 CO W C r J W C eE  R W D C  0 ˆ O O   0    @ t @ t ˆ ˆ ˆ ˆ ˆ ˆ @OjW  OW ˆ 2 i; : @Wi; C r  OjW D 1 CO W ; C r J W  i; i; D  Ci; ; i D 1; : : : ; nph :  i; @tO @tO

The O on the collision terms refers to the use of the scaled quantities introduced in (9) and (14). The subscript  in the collision terms and in the phonon moments refer to the use of the distribution function  gO iME , as in (10). The latter moments are evaluated by using the scaled dispersion relation !O i . The formal limit of the left-hand sides in the previous equations can be obtained by using (13). In particular, at leading order we find linear relations of the form Oj OjW

! D A

O ƒ OW ƒ

! ;

OjW D ai ƒ OW: i i

For the collision terms, it is difficult to get an explicit expansion analogous to (13). Nevertheless, by using (11), assuming  D O. 2 /, it is possible to show that

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75

O 0, 1 C OW  O  OW  OW 1 OW 1 OW 1 OW the flux collision terms 1 C  0 ,  C ,  C ,  Ci; , 2 C0 ,  C ,  Ci; are of order O.1/ around  D 0. Denoting by .0/ the leading order terms on the right-hand side of the previous scaled equations, and by using (11), we find O .0/ C O W .0/ C

!

O ƒ DB OW ƒ

! D BA 1

Oj

OjW

! ;

O W D bi a1 OjW : O W .0/ D bi  C i i i i

In conclusion, taking the formal limit of (6), after the scaling (9), (11), (14), we obtain the following limit system: 8 @n ˆ O .0/ ; ˆ C r  Oj D 0 r  J .0/ C eE  R .0/ D C ˆ ˆ ˆ @tO ˆ ˆ < @W O W .0/ ; (15) C r  jOW C eE  Oj D CO W .0/ ; r  J W .0/ C eE  R W .0/ D C ˆ @tO ˆ ˆ ˆ ˆ ˆ @Wi W .0/ ˆ O W .0/ ; i D 1; : : : ; nph : O W .0/ ; : r Ji DC C r  OjW i D Ci i @tO The equations on the left column constitute the energy transport system, while the equations on the right column are the constitutive relations for the flux, which can be written in the compact form 0 @

Oj OjW

1 A D A B

r  J .0/ C eE  R .0/

1

r  J W .0/ C eE  .J .0/ C R W .0/ /

OjW D ai b 1 r  J W .0/ ; i i i

! (16)

i D 1; : : : ; nph :

All the coefficients in the above expressions are given in closed form. We do not report them here for lack of space. The limit model (15), (16) is consistent with linear irreversible thermodynamics.

References 1. Anile, A.M., Marrocco, A., Romano, V., Sellier, J.M.: Numerical simulation of 2D Silicon MESFET and MOSFET described by the MEP based energy-transport model with a mixed finite elements scheme, Rapport de Recherche INRIA No. 5095 (2004) 2. Degond, P., G´enieys, S., J¨ungel, A., A steady-state system in non-equilibrium thermodynamics including thermal and electrical effects. Math. Meth. Appl. Sci. 21(15), 1399–1413 (1998) 3. Dreyer, W., Struchtrup, H.: Heat pulse experiment revisited. Continuum Mech. Thermodyn. 5, 3–50 (1993) 4. Galler, M., Sch¨urrer, F.: A deterministic solution method for the coupled system of transport equations for the electrons and phonons in polar semiconductors. J. Phys. A Math Gen. 37, 1479–1497 (2004)

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5. Romano, V., Rusakov, A.: 2D numerical simulations of an electron-phonon hydrodynamical model based on the maximum entropy principle, Comput. Meth. Appl. Mech. Eng. 199, 2741–2751 (2010) 6. Romano, V., Zwierz, M.: Electron-phonon hydrodynamical model for semiconductors, Z. Angew. Math. Phys. 61 1111–1131 (2010) 7. Selberherr, S.: Analysis and Simulation of Semiconductor Devices. Springer, Berlin (1984)

Automatic Layout Optimization of Power Discrete Devices Using Innovative Distributed Model Techniques Giuseppe Greco and Salvatore Rinaudo

Abstract The use of discrete power MOSFET devices has increased in several fields of applications. They are used in all power applications where more and more severe constraints have to be met, forcing designers to use SPICE-like models able to simulate all physical phenomena taking place within the power device. During commutations, current focusing phenomenon could compromise the integrity of the entire device. The optimization of a power MOSFET layout aims at reducing the current density overshoots, referred to as hotspots [IEEE Trans. Power Electr, 15, 575–581, 2000]; changing the geometry within the layout of the device has proven reductions in hotspot areas. A EDA framework using an optimization evolutionary algorithm for this problem analysis will be shown.

1 Introduction The need to analyze the internal electrical behavior of several square millimeters discrete power devices lead us to search new cost efficient solutions in terms of computational efforts. Classical FEM approaches show several limits since the huge 3D structure to be simulated implies very dense meshes whose minimum feature is often aligned, for MOS device, at least to the minimum channel length. So it has been necessary to develop new modelling techniques aimed at reducing the computational resources though guaranteeing good accuracy. PAN (Power Mos Analyzer) [2, 3] is a custom tool developed by STMicroelectronics to analyze the internal behavior of discrete power devices. It represents the core of an innovative optimization framework, available in the industrial context to increase the robustness of the power device [4, 5] through its layout optimization. The

G. Greco ()  S. Rinaudo STMicroelectronics, Catania, Italy e-mail: [email protected]; [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 10, © Springer-Verlag Berlin Heidelberg 2012

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PAN model technique represents power device as an array of elementary cells, each of them representing a portion of silicon area. The elementary cell will contain all information necessary to replicate its electrical and thermal behavior opportunely scaled by area. A lumped-element distributed model will be produced; it takes into account the effects of layout parasites exploiting the high-frequency modelling approach of microstrips to describe both the passive parts of the device and elementary transistor cells.

2 Distributed Model Technique The basic idea is to represent the power device as an array of elementary cells. All cells belonging to the active area will be interconnected through metal and polysilicon paths, which can be modelled as transmission lines, in order to take into account signal time delays due to distances between pilot gate pad and elementary array elements [6, 7]. In a mono-dimensional transmission line the voltage u(x,t) satisfies the following equation [8]: @2 u @u Dk 2 @t @ x

(1)

An interesting case where it is possible to obtain a finite terms solution of (1) is the problem of a line of semi-infinite length, without initial conditions (i.e. by neglecting the transient) and with oscillating boundary condition. More precisely, we will analyze the problem of the electric conduction in an instant enough distant from the starting one, so that the influence of the initial conditions practically do not incise on the distribution of the voltage at the instant of the observation. In the case of a line of semi-infinite length, we will search for a solution of (1) in the domain x > 0 that satisfies the condition: u.0; t/ D Acos.!t/

(2)

In this case it is possible to demonstrate [9] that the solution of (1) satisfying the boundary condition (2) has the following form:  r   r  ! ! u.x; t/ D A exp  x cos  x C !t 2k 2k

(3)

If in this equation we substitute the quantity 1=RC to k we obtain a solution which represents the voltage across the transmission line. Since (3) does not include the contribute of the reflected wave it could be electrically reproduced by a RC network of infinite length as the one reported in Fig. 1. If we want to consider a network with a finite number of RC cells, however, we could eliminate the reflected

Automatic Layout Optimization of Power Discrete Devices

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Fig. 1 Network of elementary RC cells used to model the impact of pilot signal delays across the device layout

Fig. 2 (a) Layout abstraction of the power MOSFET (b) Mapping between matrix and schematic

wave contribute by electrically adapting it through the insertion of Radp and Cadp whose values could be expressed by (4) and (5).

Cadp

p r 2 R Radp D 2 !C !1 p r 2 R ! D 2 C

(4)

(5)

This adapted network will be used to model the impact of gate paths delays along the whole device. The full model is extracted starting from a CAD layout view (Fig. 2a) of the discrete power device by using the STMicroelectronics PAN flow. Through it, it will be possible to translate the layout into a numeric matrix where each value represents a codification of the layout structures so that through a consequent mapping it will be possible to associate the matrix indexes to equivalent schematic parts (Fig. 2b) that will be then synthesized as a spice-like netlist.

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3 Optimization Algorithm The optimization problem to be analyzed implies the use of a non-derivable objective (fitness) function, implemented through the Minimum Square Method, whose analytical expression could not be known being it associated to results coming out from a commercial pspice simulator. Any optimization algorithm which is able to work with discrete variables is suitable for this problem, because it can be tackled as a black box optimization problem (Independence of the concrete application). In order to implement the optimization framework based on the PAN modelling flow we have tested several optimization algorithms, all able to minimize a fitness function without the use of derivatives. In all examined cases, we searched a robust optimization methodology to reduce computational efforts, finding the best solution by limiting the number of fitness evaluation calls (Global optimization at a tolerable computational cost). After the testing phase has been completed, OPT-IA algorithm [10] showed better performances. OPT-IA is a population based evolutionary algorithm (EA) inspired by the human’s clonal selection principle to produce effective methods for search and optimization. Each individual of the population is a candidate solution. Using the cloning operator, an immune algorithm produces individuals with higher objective function values, introducing blind perturbation and selecting their improved mature progenies.

4 Optimization Framework The PAN tool together with the optimization algorithm have been integrated in the EDA flow reported in Fig. 3. The parameters set to be modified by optimization algorithm are the metal fingers distance and the poly-silicon gate area positions (see Fig. 2a). The flow starts from the non-optimized basic layout supplied to the PAN module able to translates it into a numeric layout matrix and finally into a spice-like netlist. It is then simulated and results given to the EA; if end condition is met the flow finishes, otherwise it continues through the generation of other layout solutions. As soon as max number of cycles is reached the final layout matrix is proposed.

5 Result Comparison In Figs. 4 and 5 it is possible to see the simulation results coming from the basic layout performances and the optimized one. In the basic layout we have a central current hotspot with a normalized current peak value during an Unclamped Inductive Switching (UIS) turn-off transient of 1.0.

Automatic Layout Optimization of Power Discrete Devices

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Fig. 3 Optimization framework

Fig. 4 Ipeak current 2D map and waveforms in a unclamped inductive switching turn-off for the starting reference layout

After the optimization loop lasted about 300 cycles, we obtained the best layout shown in Fig. 4 where the current peak has been reduced to 0.73 so obtaining a reduction of about 30%. By observing the optimized layout, for example, wecould see how the W distance has been remodulated.

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Fig. 5 Ipeak current 2D map and waveforms in a unclamped inductive switching turn-off for the optimized layout

6 Conclusion An innovative methodology for discrete device design in pre-silicon phase has been shown. It is aimed at improving robustness and performance in terms of peak current values. This methodology gives designers a new and innovative EDA framework which allows the investigation of issues in previously unaddressed area because of the lack of means. Acknowledgements This research is supported by the European Commission’s, under THERMINATOR project, contract No. ICT-248603 and Marie Curie RTN Project COMSON Grant No. MRTN-CT-2005-019417

References 1. Consoli, A., Gennaro, F., Testa, A., Consentino, G., Frisina, F., Letor, R., Magri, A.: Thermal instability of low voltage power-mosfets. IEEE Trans. Power Electron. 15, 575–581 (2000) 2. Biondi, T., Greco, G., Bazzano, G., Rinaudo, S.: Effect of layout parasitics on the current distribution of power mosfets operated at high switching frequency. J. Comput. Electron. 5(2–3), 149–153 (2006) 3. Biondi, T., Greco, G., Bazzano, G., Rinaudo, S.: Method for modeling large-area transistor devices, and computer program product therefore. U.S.Patent N.11/770,578 deposited in June 2007 4. Chenming, H., Min-Hwa, C., Patel, V.M.: Optimum design of power mosfets. IEEE Trans. Electron. Dev. 31(12), 1693–1700 (1984) 5. Budihardjo, I.K., Lauritzen, P.O., Mantooth, H.A.: Performance requirements for power mosfet models. IEEE Trans. Power Electron. 12(1), 36–45 (1997)

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6. Biondi, T., Greco, G., Bazzano, G., Rinaudo, S.: Analysis of the internal current distribution in power mosfets operated at high switching frequency. Proceedings of MSED,Workshop on Modeling and Simulation of Electron Devices, Pisa, Italy, vol. 15, pp. 4–5, July 2005 7. Biondi, T., Greco, G., Bazzano, G., Rinaudo, S., Allia, M.C., Liotta, S.F. Distributed modeling of layout parasites in large-area high-speed silicon power devices. IEEE Trans. Power Electron. 22(5), 1847–1856 (2007) 8. Pipes, L.A., Harvil, L.R.: Applied Mathematics for Engineers and Physicists. McGraw-Hill International Book Company, (1971) 9. Tikhonov, A.N., Samarskij, A.A.: Equazioni della fisica matematica, Moscow Mir 1981 10. Nicosia, G., Rinaudo, S.: Optimization Population-Based Algorithms for High-Performance Analog Circuits, 2008. SIAM Conference on Optimization – OP08, Boston, MA, USA, 10–13 May 2008

3D Stress Simulations of Nano Transistors Abderrazzak El Boukili

Abstract Mechanical Stress is intentionally used by many semiconductor device manufacturers as Intel, IBM, and TSMC to dramatically increase the performance of the new nano NMOS and PMOS transistors. Deposition induced stress is used to increase carrier mobility through the channel. This paper presents the three dimensional simulations and mathematical modeling of stress distribution after deposition of SiGe pockets in nano PMOS transistors. The originality of this work is to use Finite Volume discretization to solve stress equations in three dimensions and to use Multifrontal Method to solve the resulting large linear systems. Numerical results showing the 3D effects will be presented and analyzed for an Intel 45 nm strained PMOS transistor.

1 Introduction Most of nano semiconductor device manufacturers as Intel, IBM, and TSMC are using mechanical stress to dramatically enhance the performance of nano CMOS transistors at 45nm and below. Stress is used to increase carrier mobility in the channel [1,2]. The electrical measurement show that stress enhanced performance of both NMOS and PMOS by up to 30% [3]. The need of the hour is to use simulation tools and mathematical models to understand the physics of stress and how to attain the desired stress in the channel. Simulations can also explain complex stress issues. This paper is organized as follows. Section 2 outlines the different sources of stress and describes the main techniques used to introduce stress into the channel.

A. El Boukili () Al Akhawayn University, Ifrane 53000, Morocco e-mail: [email protected]

M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 11, © Springer-Verlag Berlin Heidelberg 2012

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Section 3 presents our contribution in the mathematical and physical models of stress and in the numerical methods used to solve the 3D stress equations. Section 4 presents new simulation results and analysis showing significant 3D effects for a 45 nm Intel strained PMOS transistor [1]. These results focus only on stress induced by deposition of S ix Ge1x pockets in source and drain areas. An important point brought by these 3D results is that all the three stress components in x, y, and z directions are significant. The existing theoretical models for stress-enhanced mobility as piezoresistance model assumes that only one stress component is significant (uniaxial stress) and the others are negligible [4]. This means that the existing mobility models should be modified to take into account that the stress could be triaxial and not only uniaxial.

2 Sources of Stress Manufacturers of today nano transistors are using intentional and unintentional stress to boost device performance and reduce size and power. Intensional stress can be global or local. Global stress is based on substrate engineering [5]. Local stress is based on channel engineering. It has significant advantages over global stress and it is the one adopted in mass production of high-performance logic devices [6]. It is introduced into the channel using mainly Intel’s technology or IBM’s technology. Intel’s technology uses the deposition of tensile capping layer for NMOS devices. And for PMOS devices it uses the deposition of S ix Ge1x pockets in source and drain regions. The lattice mismatch in the heterolayer compresses the silicon lattice, which consequently compresses the device channel. For 17% Ge concentration, a compressive stress of 1.4 GPa is generated inside the SiGe layer [7]. The unintentional stress is process induced. Its main sources are: deposition, doping, shallow trench isolation, etching, diffusion, oxidation, salicidation, thermal and material mismatch, proximity, packaging, and layout. The unintentional and intentional stresses should be calculated together to determine the overall stress field in a device.

3 Mathematical Models for Stress in Three Dimension Stress is the distribution of internal body forces of varying intensity due to externally applied forces and/or heat. At mechanical equilibrium, the stress tensor  and the strain tensor  are given by:

3D Stress Simulations of Nano Transistors

1 xx B C B yy C C B B C  D Œij  D B zz C I B xy C C B @ yz A zx 0

87

0

1 xx B C B yy C B C B C  D B zz C B xy C B C @ yz A zx

(1)

The application of stress to a body in equilibrium causes it to undergo deformation and/or motion. A measure of deformation is strain. The relationship between stress and strain depends strongly on the materials. It is the most important relation in modeling stress. Materials used in transistor’s fabrication may behave as elastic, viscous, visco-elastic or plastic depending on temperature and pressure. In this paper, we are assuming that silicon and silicon germanium act as an anisotropic elastic material over a wide temperature range frequently encountered in silicon IC fabrication [5]. For elastic materials the stress-strain relationship is given by Hooke’s law:  D D.  0 / C 0

(2)

where D is the tensor of elastic constants c11 , c12 , and c44 and 0 is the intrinsic strain, and 0 is the intrinsic stress. For SiGe, we are assuming that D has the same form as for Si. For simplicity, we are also assuming that silicon and silicon germanium are isotropic materials. In this case, the elastic constants reduce to the following: c11 D

E.1  / E E ; c12 D ; c44 D .1 C /.1  2/ .1 C /.1  2/ .1 C /

(3)

where E and  represent the Young’s modulus and Poisson’s ratio respectively. In this paper, we are considering the crystal orientation dependence of E and . In three dimension, the strain is related to the displacements by the following strain components [5]: xx D

@u ; @x

yz D zy

  @v @v @w 1 @u ; zz D ; xy D yx D C @y @z 2 @y @x     @w @u 1 @v 1 @w C ; zx D xz D C D 2 @z @y 2 @x @z

yy D

(4)

where u, v, and w are the displacements in x, y, and z directions. We solve in terms of u, v, and w the following system of second order partial differential equations:

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  2   2 @2 w @ v @2 w @2 u @ u @2 u @2 v C C C D a0 ˛ 2 Cˇ C 2 C @x @y 2 @z @x@y @z@x @x@y @z@x   2   2 @2 w @u @2 w @2 v @v @2 v @2 u C C C D b0 ˛ 2 Cˇ C 2 C @y @x 2 @z @x@y @y@z @x@y @y@z   2   2 @2 v @u @2 v @ w @2 w @2 u @2 w C C C D c0 C 2 C ˛ 2 Cˇ @z @x 2 @y @z@x @y@z @z@x @y@z

(5)

where a0 D 

@0;xy @0;xy @0;yz @0;yy @0;zx @0;xx   ; b0 D    @x @y @z @y @x @z @0;zx @0;yz @0;zz   c0 D  @x @y @z

(6)

The coefficients ˛, ˇ, and  depend on the elastic constants c11 , c12 , and c44 . For simplicity, the term D0 is omitted from the right hand side in (6). But, it is included in the implementation. The model in (5) is based on the Newton’s second law of motion governing deformation and the Hooke’s law for elastic materials. We are using homogeneous Dirichlet boundary conditions for u, v, and w on front, end, left, and right sides of the 3D structure. We are solving the system (5) to get the displacements u, v, and w. Then use u, v, and w to get the strain components using (4). And finally, we get stress components using (2). The stress-strain equations (2) can not be solved directly because they have 12 unknowns and only 6 equations. This is why we solve for displacements u, v, and w the system (5) that has 3 equations and 3 unknowns. The intrinsic stress 0 which is an input to (5) is the existing stress in SiGe before deposition. The originality in this paper is to use Finite Volume Method for the discretization of (5) and to use Multifrontal Method to solve the resulting large linear systems [8]. Most of advanced commercial software in the market today for process simulation are based on the Suprem IV software developed by Standforad University in California. Suprem IV uses Finite Elements (fes) for stress and oxidation simulation and uses Finite Volumes (fvs) for diffusion equations. Then, it is switching between two types of meshes (a mesh for (fes) and a mesh for (fvs)) which makes the program too heavy and time consuming. In this work we are showing to the semiconductor community that we could use (fvs) method to solve both diffusion and oxidation equations involving stress. And there is no need to use two types of meshes which will cost a lot especially in 3D. I do understand that Suprem IV uses (fes) for second order oxidation equations to get better accuracy. In particular, a better accuracy on shear stress components. But, there is a method based on Least Squares fitting that can be used with (fvs) method to get also a good accuracy on shear stress components. I am currently working on this issue. An other numerical difficulty is to solve the linear equations. We are using here for the first

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time Multifrontal Method to solve 3D stress equations. I do not know any one using this method in the context of semiconductor equations. This method is faster than the standard LU factorization. And it is more accurate than iterative methods that depend strongly on the preconditioning.

4 3D Numerical Results and Analysis The results in Figs. 1 and 3 show that the stress components xx and zz along the channel and across the channel respectively are both significant. This is one of the main outcomes of this paper. A similar stress distribution has been reported in [9] for a different geometry. The existing theoretical models for stress-enhanced mobility as piezoresistance model assumes that only one stress component is significant (uniaxial stress) and the others are negligible [4]. This means that the existing mobility models should be modified to take into account that the stress could be triaxial and not only uniaxial. Figure 2 shows that the distribution of x stress component is compressive along channel as expected. Figure 3 also shows that the distribution of the z stress component is really nonuniform in the channel. It is varying from 1:5e C 9 dynes cm2 compressive to 1:5e C 9 dynes cm2 tensile stress. A similar result was reported in [9]. On the other hand, these results confirm that our finite volume based implementation of stress equations gives valid and correct results.

Fig. 1 3D Distribution of x stress component along channel in 45 nm PMOS with SiGe source and drain

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Fig. 2 Cut along the channel shows that the x stress component is compressive along channel

Fig. 3 3D Distribution of z stress component across channel in 45 nm PMOS with SiGe source and drain

References 1. Ghani, T., et al.: A 90nm High Volume Manufacturing Logic Technology Featuring Novel 45nm Gate Length Strained Silicon CMOS Transistors. In: IEDM Technical Digest, Washington, DC, USA, pp. 978–980, December 2003 2. Rim, K., Hoyt, J.L., Gibbons, J.F.: Fabrication and analysis of deep submicron strained-Si N-MOSFET’s. IEEE Trans. Electron Dev. 47(7), 1406–1415 (2000)

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3. Krivokapic, Z., et al.: Locally strained ultra-thin channel 25 nm narrow FDSOI devices with metal gate and mesa isolation. In: Proceedings of the 49th Annual IEEE International Electron Device Meeting (IEDM), Washington, DC, USA, pp. 445–448, December (2003) 4. Smith, C.S.K.: Piezoresistance effects in germanium and silicon. Phys. Rev. 94(1), 42–49 (1954) 5. Shah, N.: Stress Modeling of Nanoscale MOSFET. Master thesis, University of Florida (2005) 6. Thompson, S.E., et al.: A logic nanotechnology featuring strained-silicon. IEEE Electron Dev. Lett. 25(4), 191–193 (2004) 7. Al-Bayati, A., et al.: Production processes for inducing strain in CMOS channels. In: Fabtech, Online Information Source for Semiconductor Professionals, White Papers, 26th Edition, Materials and Gases, June (2005) 8. Duff, I.S., Reid, J.K.: The multifrontal solution of indefinite sparse symmetric linear equations. Trans. Math. Software 9(3), 302–325 (1983) 9. Moroz, V., et al.: Analyzing strained-silicon options for stress-engineering transistors. In: Solid State Tech., 49–52 (2004)

Hybrid Formulations and Discretisations for Magnetoquasistatic Models Herbert De Gersem, Stephan Koch, and Thomas Weiland

Abstract This paper aims at increasing the modelling flexibility for magnetoquasistatic finite element simulations by allowing different formulations and different discretisation techniques in distinct model regions. Special care is necessary when conceiving algebraic solution techniques for the coupled systems of equations.

1 Introduction The simulation of electromagnetic fields in electrotechnical devices has become a standard tool in the design thereof. Simulation techniques combine an appropriate formulation of the Maxwell equations with a suitable discretisation in space and time. Many relevant models motivate the application of different formulations and/or different discretisations in different geometric parts of the model. Such hybridisations seem to be straightforward but cause a non-negligible burden on the overall simulation approach. The pre- and post-processing routines have to be adapted accordingly. The system solver has to deal with hybrid matrices. The accuracy and stability of the combined simulation technique has to be proven or verified in practice. Offering the possibility of hybrid modelling, however, significantly increases the applicability and flexibility of electromagnetic field simulation for relevant technical models.

H. De Gersem () Wave Propagation and Signal Processing Research Group, Katholieke Universiteit Leuven, Etienne Sabbelaan 53, 8500 Kortrijk, Belgium e-mail: [email protected] S. Koch  T. Weiland Institut f¨ur Theorie Elektromagnetischer Felder, Technische Universit¨at Darmstadt, Schlossgartenstrasse 8, 64289 Darmstadt, Germany e-mail: [email protected]; [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 12, © Springer-Verlag Berlin Heidelberg 2012

93

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H. De Gersem et al. φ

0

grad

ϕ E J ψ H B A σ = 1/ ρ ρ μ = 1/ ν ν

div

E, A electric scalar potential electric field strength electric current density magnetic scalar potential magnetic field strength magnetic flux density magnetic vector potential conductivity resistivity permeability reluctivity

σ, ρ

J

curl

curl

B

μ, ν

H, T

grad

div

0

ψ

Fig. 1 Electromagnetic field quantities and Tonti’s diagram (time derivative and minus signs are not written)

2 Magneto(quasi)static Formulations We consider the magnetoquasistatic subset of the Maxwell equations, also called the eddy-current problem: r  H D JI rED

(1) dB I dt

(2)

r  J D 0I

(3)

r  B D 0;

(4)

where (1) is Amp`ere’s law, (2) is Faraday-Lenz’s law, (3) is the continuity law for the current and (4) is the magnetic law of Gauss. (3) and (4) can be found from (1) and (2) by left and right application of the divergence operator. The relevant field quantities are defined and ordered graphically in Fig. 1. The continuity law and the magnetic Gauss law can be enforced by expressing the electric current density J and the magnetic flux density B in terms of the electric vector potential T and the magnetic vector potential A respectively: J D r  TI

(5)

B D r  A:

(6)

Hybrid Formulations and Discretisations for Magnetoquasistatic Models

95

A similar integration in space of Amp`ere’s and Faraday-Lenz’s laws leads to the definition of the magnetic scalar potential and the electric scalar potential ' serving as integration constants: H D Tr I ED

dA  r': dt

(7) (8)

The material behaviour is here represented by the relations J D E and B D H where the definition of the material properties can be found in Fig. 1. After some rearrangement, one finds that the magnetoquasistatic field problem can be solved by any of the parabolic partial differential equations @A D r'I @t @ @T D r r  .r  T/ C  @t @t

r  .r  A/ C 

(9) (10)

in terms of the magnetic vector potential A and the electric vector potential T respectively. When the time variation of the magnetic field is negligible, (9) reduces to the stationary-current formulation in terms of the electric scalar potential ':  r  .r'/ D 0:

(11)

When the time variation of the electric current is negligible, (10) leads to the magnetostatic formulation in terms of the magnetic scalar potential :  r  .r / D r  .Hs /

(12)

where the source magnetic field strength Hs is an arbitrary vector field satisfying r  Hs D J. Since we will only apply (12) to current-free regions, Hs D 0.

3 Hybrid Formulations All four formulations (9)–(12) have advantages and disadvantages. The vector potential formulations (9) and (10) solve the full eddy-current problem and account for all time-varying effects, but need the discretisation of a vector field. The scalar potential formulations (11) and (12) are only applicable in the static case (when the electric and magnetic behaviour is decoupled) or in cases where the electric effects are negligible compared to the magnetic effects. The scalar formulations require the discretisation of a scalar field and are computionally less expensive. In practice, eddy-current problems have regions where the full eddy-current effect has to be accounted for, whereas in other regions, only the stationary

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Fig. 2 Magnet model: FE mesh for the outer part and tensor-product grid for the aperture

Γ12

Ω1

Ω2

current effect or the magnetic effect is relevant. Then, it is obvious that a hybrid formulation would be the most efficient simulation approach. In conducting regions, we prefer formulation (9) in terms of the magnetic vector potential over formulation (10) in terms of the electric vector potential, because (10) is known to cause particular topological problems in non-trivial computation domains (e.g. donutshaped domains) [14]. In non-conductive regions, the vector formulation (9) needs three times as much degrees of freedom (DoFs) compared to the scalar stationarycurrent formulation (11) or the scalar magnetostatic formulation (12). It is a straightforward suggestion to replace (9) and to replace (9) by (11) in regions with a negligible magnetic field (12) in non-conductive regions. We consider both replacements separately.

3.1 Magnetoquasistatic-Magnetostatic Hybrid Formulation As a first example, consider the magnet geometry of Fig. 2. The outer part ˝1 consists of the windings and the iron yoke, whereas the inner part ˝2 only covers the magnet aperture. The aperture region is non-conductive and free of currents. In ˝2 , the magnetic field can be described more efficiently by the magnetostatic formulation (12). This leads to the hybrid formulation 8 < :

@A D r' @t r  .r / D 0

r  .r  A/ C 

in ˝1 in ˝2

:

(13)

At the interface 12 D ˝1 \ ˝2 , the normal continuity of the magnetic flux density B D r  A D r and the tangential continuity of the magnetic field strength H D r  A D r should be enforced: 

.r  A C r /  n D 0 ..r  A C r /  n/  n D 0

at 12 ; at 12

(14)

Hybrid Formulations and Discretisations for Magnetoquasistatic Models

97

where n is a vector normal to 12 pointing from ˝1 to ˝2 . This formulation was the standard approach in the eighties, when direct solvers and factorisations where in use and, because of that, the number of DoFs was of primordial importance [15]. Nowadays, such hybrid formulations have been abandoned in favour of monolithic formulations in terms of only the magnetic vector potential. Nevertheless, we will see that the use of (12) may be beneficial when large non-conductive and currentfree regions are present in the model.

3.2 Magnetoquasistatic-Stationary-Current Hybrid Formulation As a second example, consider the accelerator magnet with iron yoke of Fig. 4. The magnetic flux mainly flows in the xy-plane such that for a long magnet like this, a 2D simulation is commonly sufficient. However, when the magnet is ramped from zero to the rated field value, eddy currents are generated in the beam-tube wall. Their simulation necessitates a transient approach. The eddy currents flow along the z-direction in the beam-tube part inside the magnet and turn around in the beam-tube parts at the front and rear sides of the magnet. The shape of the closing paths substantially determines the current distribution in the inside beamtube part. To avoid a transient 3D calculation, it makes sense to coupled the xy-2D magnetoquasistatic model of the magnetically active part with an sz-2D stationarycurrent model of the beam-tube end parts. The magnetically active model part is denoted by ˝1 D 1 ˝ Œ0; za , whereas the stationary current region is denoted by ˝3 D 3 ˝ Œ ı2 ; C 2ı  (Fig. 4). 3 D 13 ˝ Œza ; zb  is the beam-tube surface outside the active magnet part. The intersection of both domain parts is ˝1 \ ˝3 D 13 ˝ Œ ı2 ; C 2ı . The combination of the magnetoquasistatic formulation (9) with the stationary current formulation (11) reads  @A r  .r  A/ C in ˝1   @t C r' D Js ; (15) @A in ˝3 r   @t  r  .r'/ D 0 where the currents in the winding are represented by Js and the currents in the beam tube are represented by r'.

4 Discretisation 4.1 Finite-Element Discretisation Commonly, A is discretized by edge elements wj , whereas by nodal elements Nq :

and ' are discretized

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AD

X

u j wj I

D

X

vq Nq I

D

X

q

j

wq Nq ;

(16)

q

where the DoFs for A, and ' are collected in the vectors u, v and w. The application of the Ritz-Galerkin approach to (9), (12) and (11) leads to du D fI dt

(17)

G v D 0I

(18)

G w D 0:

(19)

K u C M

The matrix coefficients read R K;ij D ˝ r  wj  r  wi d˝ R G;pq D ˝ rNq  rNp d˝

R M;ij D ˝ wj  wi d˝ R fi D ˝ Js  wi d˝

The time discretisation of the semi-discrete system (17) can be done by any implicit time integration technique, e.g., by an implicit Runge-Kutta method. Then, a sparse, symmetric and positive definite system matrix K C ˛M with a parameter ˛ depending on the time-discretisation approach, is obtained. The singular kernel of K C ˛M is spanned by vectors u that are discrete counterparts of gradient fields in the non-conductive parts of ˝1 . For 2D examples, this singularity vanishes. Appropriate curl-curl multigrid techniques for solving (17), possibly accelerated by a Krylov subspace solver, are reported in literature [13]. The system matrices G and G are sparse, symmetric and positive definite. Hence, systems (18) and (19) can be addressed by standard multigrid techniques. Magneto(quasi)static and stationary-current field simulation by (17), (18) and (19) is well understood, is commercially available and is daily applied to engineering examples.

4.2 Discretisation with Magnetostatic Part When (9) is discretised on ˝1 and (12) is discretised on ˝2 , one obtains the mixed system of equations 

K C ˛M B G BT

    u f D ; v 0

(20)

where the integrals at 12 are found to be Z Bi q D 

12

  rNq  wi  d:

(21)

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99

System (20) is further also denoted by Ax D b. The combination of both systems preserves the sparsity and the symmetry, but turns out to be indefinite [1]. Suppose that a cheap inverse of G is available, the mixed system can be turned into the Schur-complement system   T uDf: (22) K C ˛M C BG1  B T The Schur-complement matrix S D K C˛M CBG1  B is symmetric and positive semi-definite but contains large dense blocks due to G1  . The matrix S has the same sparsity pattern as for a FE, boundary-element (BE) coupled formulation where the normal derivatives are eliminated [10]. G is symmetric and positive semi-definite.

4.3 Discretisation with Stationary-Current Part With (11) discretised on ˝3 and (9) discretised on ˝1 , one obtains 

K C ˛M C ˛CT G



   u f D : w 0

(23)

The coefficients of C are related to boundary integrals at ˝1 \ ˝3 : Z Ci q D

˝1 \˝3

Nq wi  d:

(24)

When G can be provided at an acceptable computational cost, one can eliminate the DoFs in w and come up with the Schur-complement system    T u D f: K C ˛ M  CG1  C

(25)

T The matrix Mschur D M  CG1  C can be seen as a modified FE conductance matrix accounting for the closing paths of the currents in the stationary-current model region. Mschur is symmetric and positive semi-definite.

5 Iterative Solution The solution of (22) is typically carried out by an iterative solver without explicitly constructing S. An appropriate solver is the Conjugate Gradient (CG) method. The construction of an efficient preconditioner SQ 1 for S is a tedious task for which problem tailored approaches are found in literature [3]. The mixed system (20) can be solved by the Minimal Residual (MINRES) method [12] or the Quasi-Minimal Residual (QMR) method [6]. MINRES requires

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a symmetric positive definite preconditioner, whereas any symmetric preconditioner will work for QMR. A possible preconditioner for MINRES and QMR is " Q 1 D A 1

# LQ 1 0 Q 1 ; 0 G 

(26)

Q 1 is an Q 1 is a positive semi-definite approximation to .K C˛M /1 and G where L  1 approximation to G . This preconditioner, however, improves the convergence of MINRES or QMR only marginally [4], even if optimal preconditioners are selected Q 1 . A more promising choice is for LQ 1 and G  " Q 1 D A 2

# SQ 1 0 Q 1 : 0 G 

(27)

Q D A and SQ D S, this preconditioner makes MINRES and QMR to converge When A in no more than 2 iteration steps. The construction of an appropriate SQ 1 , however, remains cumbersome. A simple and acceptably efficient choice is an additiveSchwarz-type method [16] 1 C BG B1 : SQ 1 2 D .K C ˛M /

(28)

System (25) can be solved by a preconditioned CG method. Again, this may T become computationally expensive due to the dense blocks CG1  C . The coupled problem (23) is positive semi-definite and can be solved by CG.

6 Hybrid Discretisation In some models, it is beneficial to combine different discretisation techniques, possibly in combination with a hybrid formulation. An example is a particleaccelerator component where the surrounding magnets and/or cavities are designed to provide a specified electromagnetic field at the beam axis. The increasing demand on high precision aperture fields comes along with an increasing need for high accuracies in that model region. Especially when the aperture region is described by the scalar potential formulation (12), it is straightforward to replace the lowest order nodal shape functions by any higher order shape functions. Accelerator components typically have a cylindrical aperture region, here denoted by ˝2 D ŒR; R  Œ0; Œ  ŒZ; Z with radius R and length 2Z. This allows to apply a spectral-element (SE) technique with basis functions Mq .r; ; z/ D Tq1

r  R

e j  Tq2

z Z

(29)

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101

defined at a cylindrical tensor-product grid (Fig. 2). The shape functions vary as Chebyshev functions in r- and z-direction, and as harmonic functions in -direction. Clustering of collocation points at the r D 0 axis is avoided by ranging the r-coordinate between R to R, restricting the -coordinate to Œ0; Œ and using an O even number of collocation points along r [5]. The coupling matrix B D ZT QT D H and the discrete Laplacian G D D D can be provided in factorised form. The operator Z maps the fictitious interface currents from the spectral collocation points of the tensor-product grid onto the edges of the FE mesh. Q selects the spectral degrees of freedom at the collocation points at the interface. D D Œ Dr D Dz H where Dr , D and Dz are differentiation matrices along r,  and z according to the spectral discretisation, which can be invoked by Fast Fourier Transforms (FFTs): Dr D D D Dz D F1 UF, where U is a diagonal matrix [17]. Dr , D and Dz are fully populated, whereas Q can be represented by an index set and B can be stored in a sparse matrix format. The structure of B is sparse with respect to the rows and dense with respect to the columns, which corresponds to the FE and SE DoFs respectively. The computation time for the mixed system (20) is primordially determined by the repetitive application of the system matrix and the preconditioner within the iterative solver. The matrices K C ˛M, G and B resulting from FE discretisation are efficiently stored in a compressed matrix format. When an SE discretisation is applied, G and B are dense and may cause the matrix-vector product to become unacceptably expensive. Then, it is better to make A available by operations on the fly. The SE differentiation matrix D is invoked by FFTs. Q is applied as an index set selecting a subset of the SE DoFs. The FE related matrix parts K C ˛M and Z are made available as sparse matrix-vector products. Standard algebraic preconditioning techniques such as e.g. the Incomplete Cholesky (IC) method are Q 1 and A Q 1 remain available as long as no longer available. The preconditioners A 1 2 1 1 appropriate operators for LQ and SQ 2 are provided. When the Schur-complement system (22) is solved, S is also provided as an operator. Every Krylov-subspace iteration step incorporates the solution by G of the SE model part, which can be done by FFTs.

7 First Example and Numerical Experiments We compare the proposed solution approaches for a 2D model of a superconductive dipole magnet (Fig. 3) in Table 1. The ferromagnetic saturation of the yoke necessitates the application of a fine FE mesh there.  The  aperture region is discretized by spectral basis functions Mq .r; / D Tq1 Rr e j  . As an additional benefit, these basis functions allow an easy calculation of the harmonic field coefficients that decide about the field quality of the magnet. As preconditioner for the FE part, an IC factorisation without fill-in is applied. One series of tests is carried out with systems that are explicitly assembled in a compressed row storage (CRS) format.

102 Fig. 3 Superconductive cos -type dipole magnet

H. De Gersem et al. superconductive coils ferromagnetic yoke aperture

Table 1 Calculation times (in seconds) for the proposed iterative solution techniques. Comparison between the mixed FE-SE system and the Schur-complement system, comparison between the compressed row storage (CRS) form and the matrix-free techniques; comparison to a homogeneous FE model used as a reference (indicated by the subscript “ref”); the FE model part is preconditioned by IC without fill-in Model System Solver Precond. CRS (s) Matrix-free (s) Q 1 FE Lref CG L 27.23 – ref Q 1 9.06 11.02 FE-SE A MINRES A 1 Q 1 FE-SE A MINRES A 8.55 7.56 2 78.34 5.58 FE-SE S CG SQ 1 1 55.34 3.25 FE-SE S CG SQ 1 2

A second series of tests makes use of on-the-fly techniques and FFTs. The numerical experiments show that matrix-free techniques are absolutely necessary. Moreover, solving the Schur-complement system is more efficient than solving the mixed system. The additive-Schwarz preconditioner is more reliable than a preconditioner that only considers the FE part. The FFTs embedded in the on-the-fly techniques only take 10% of the overall simulation time. The increasing modelling flexibility comes together with an improved solver performance, as the hybrid FE-SE model outperforms the pure FE model for a comparable spatial resolution.

8 Second Example The hybrid formulation is applied for the transient simulation of the SIS-100 superconductive magnet that is part of the Facility for Antiproton and Ion Research (FAIR) project of the Helmholtzzentrum f¨ur Schwerionenforschung (GSI) in Darmstadt, Germany [8]. 3D simulations for the beam-tube losses have been reported in [7]. The computation times are rather high. The hybrid formulation proposed here, allows to carry out 2D simulations that provide a comparable accuracy at a significant lower computational costs [2]. The beam tube has an elliptical cross-section and a very thin wall thickness ı. The skin depth

Hybrid Formulations and Discretisations for Magnetoquasistatic Models

s ıskin D

103

2

(30)

2 f tube tube

for a beam-tube conductivity tube and permeability tube is much larger than ı for the expected relevant frequencies f . Therefore, the eddy-current density can be assumed to be homogeneously distributed along the thickness of the beam tube. The stationary-current region ˝3 is a shell of an elliptical cylinder and is parametrised by . ; ; z/, i.e., .x; y; z/ D . cosh cos ;  sinh sin ; z/. The partial differential equation for the stationary-current formulation is written in terms of the LaplaceBeltrami operator 2 r˝ D 3

@2 ' 1 @2 '   C : @z2 2 cosh2  cos2  @2

(31)

In the magnetically active model part ˝1 , we consider the beam tube by a thin sheet model [9, 11], which avoids to mesh the thickness of the beam-tube wall. The conductivity of the beam tube is then considered by completing M with the coefficients Z Mthin D tube wj  wi ı`z ds; (32) ;ij 13

with tube the beam-tube conductivity of the beam-tube material and `z the magnet length. The cross-section of the beam tube and 1 can be represented by 13  Œ 2ı ; 2ı  where 13 is a contour in 1 (Fig. 4). y yoke Ω1

winding

Y13

Ω3

za

magnetic symmetry plane

s

Γ1

aperture

Γ3

Fig. 4 Accelerator magnet with beam tube: computational domain and magnetic flux lines

x electric symmetry plane

zb δ z

beam tube

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a

elliptical beam tube 0.3 mm elliptical beam tube 0.5 mm

beam tube

0.1

4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

b

0.08 z (m)

loss power (W)

5

0.06 0.04 0.02 0

0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 time (s)

0

0.02 0.04 0.06 0.08 rθ (m)

0.1

0.12

Fig. 5 Transient simulation results for the SIS-100 magnet: (a) eddy-current power loss and (b) distribution of the current on the beam-tube end surface

In the magnetoquasistatic region ˝1 , shape functions for the magnetic vector potential A as well as the electric scalar potential ' have to be provided. In the stationary-current problem, only ' is discretised. The 2D translational symmetry of ˝1 and the negligible thickness of ˝3 allow to use specific FE shape functions: N .x;y/

wj .x; y; z/ D j za ez PqQ .s; z/ D MqQ .s; za / zz1 Pq .s; z/ D Mq .s; z/

for A for ' for '

in ˝1 I in ˝1 I in ˝2 ;

(33)

where Nj .x; y/ and Mq .s; z/ are nodal shape functions defined on triangulations of 1 and 3 , respectively (Fig. 4). The simulated eddy-current losses and the current distribution in the beam-tube end region are shown in Fig. 5.

9 Conclusions Hybrid formulations and hybrid discretisations offer an additional modelling flexibility which turns out to be valuable for a relevant class of electromagnetic field models. The efficiency of hybrid approaches, however, requires particular care, especially concerning the algebraic solution techniques applied to the coupled systems of equations.

References 1. Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods. Springer, New York (1996) 2. De Gersem, H., Koch, S., Weiland, T.: Accounting for end effects when calculating eddy currents in thin conductive beam tubes. IEEE Trans. Magn. 45(3), 1040–1043 (2009)

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3. Elman, H., Silvester, D., Wathen, A.: Iterative methods for problems in computational fluid dynamics. Tech. Rep. NA-96/19, Oxford University Computing Laboratory (1996) 4. Fischer, B., Ramage, A., Silvester, D., Wathen, A.: Minimum residual methods for augmented systems. BIT 38(3), 527–543 (1998) 5. Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge (1996) 6. Freund, R., Nachtigal, N.: A new Krylov-subspace method for symmetric indefinite linear systems. In: Ames, W. (ed.) Proceedings of the 14th IMACS World Congress on Computational and Applied Mathematics, pp. 1253–1256 (1994) 7. Koch, S., Trommler, J., De Gersem, H., Weiland, T.: Modeling thin conductive sheets using shell elements in magnetoquasistatic field simulations. IEEE Trans. Magn. 45(3), 1292–1295 (2009) 8. Kovalenko, A., Kalimov, A., Khodzhibagiyan, H., Moritz, G., M¨uhle, C.: Optimization of a superferric nuclotron type dipole for the GSI fast pulsed synchrotron. IEEE Trans. Appl. Superconduct. 12(1), 161–165 (2002) 9. Kr¨ahenb¨uhl, L., Muller, D.: Thin layers in electrical engineering. Example of shell models in analyzing eddy-currents by boundary and finite element methods. IEEE Trans. Magn. 29(5), 1450–1455 (1993) 10. Kurz, S., Russenschuck, S., Siegel, N.: Accurate calculation of fringe fields in the LHC main dipoles. IEEE Trans. Appl. Superconduct. 10(1), 85–88 (2000) 11. Nakata, T., Takahashi, N., Fujiwara, K., Shiraki, Y.: 3-D magnetic field analysis using special elements. IEEE Trans. Magn. 26(5), 2379–2381 (1990) 12. Paige, C., Saunders, M.: Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12(4), 617–629 (1975) 13. Reitzinger, S., Sch¨oberl, J.: An algebraic multigrid method for finite element discretizations with edge elements. Numer. Math. 9(3), 223–238 (2002) 14. Ren, Z.: T-! formulation for eddy-current problems in multiply connected regions. IEEE Trans. Magn. 38(2), 557–560 (2002) 15. Silvester, P., Ferrari, R.: Finite Elements for Electrical Engineers, 2nd edn. Cambridge University Press, Cambridge (1996) 16. Smith, B., Bjørstad, P., Gropp, W.: Domain Decomposition: Parallel multilevel methods for elliptic partial differential equations. Cambridge University Press, Cambridge (1996) 17. Trefethen, L.: Spectral Methods in Matlab. SIAM, Philadelphia (2000)

A 3D Meshless Approach for Transient Electromagnetic PDEs Guido Ala, Elisa Francomano, and Antonino Spagnuolo

Abstract A full wave three dimensional meshless approach for electromagnetic transient simulations is presented. The smoothed particle hydrodynamic (SPH) method is used by considering the particles as interpolation points, arbitrarily placed in the computational domain. Maxwell’s equations in time domain with the assigned boundary and initial conditions are numerically solved by means of the proposed method. The computational tool is assessed and, for the first time, a 3D test problem is simulated in order to validate the proposed approach.

1 Introduction The meshless methods have been successfully used in the simulation of a wide range of applications in many different areas. These methods are a good choice especially when irregular problems geometry with diffused non-homogeneous media have to be modelled. In fact, the main attractiveness of the meshless methods lie in avoiding mesh generation and in using a set of arbitrarily distributed nodes within the problem domain. Moreover, meshless methods can also perform adaptive schemes by adding nodes where fine geometry description is needed. Among meshless methods, smoothed particle hydrodynamics (SPH) [1–9] has been recently re-formulated by the authors, and implemented in the so-called smoothed particle

G. Ala  A. Spagnuolo Dipartimento di Ingegneria Elettrica, Elettronica e delle Telecomunicazioni, di Tecnologie Chimiche, Automatica e Modelli Matematici, Universit´a degli Studi di Palermo, viale delle Scienze 90128 Palermo, Italy e-mail: [email protected]; [email protected] E. Francomano () Dipartimento di Ingegneria Chimica, Gestionale, Informatica e Meccanica, Universit´a degli Studi di Palermo, viale delle Scienze 90128 Palermo, Italy e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 13, © Springer-Verlag Berlin Heidelberg 2012

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electromagnetics (SPEM) method: this in order to solve transient electromagnetic problems directly in time domain [10–15]. In SPEM two set of electric and magnetic nodes (particles) have to be generated keeping information about electric and magnetic field components, respectively. The particles are the points in which the field components are computed at each time step, by using the information belonging to the neighbouring ones. In the paper, the proposed computational tool is assessed and, for the first time, a 3D test problem is simulated and validated. Moreover, in simulating EM fields it is often necessary to control regions with high localized field gradient or reproduce irregular geometries of the problem domain. In these cases uneven particles distribution has to be considered, and consistency restoring techniques have to be introduced. This problem [16], already addressed in 2D simulations [14, 15], will be treated in a forthcoming work for 3D structures. A brief outline of SPEM general is presented in Sect. 2. In Sect. 3 3D test problem is simulated in order to validate and generalize the SPEM method.

2 Fundamental of SPEM Method Following the main features of SPH method, the first step in SPEM is the approximation of a field function by means of the integral representation named as kernel approximation. To this aim, a smoothing kernel function W .r  r 0 ; h/ depending only by spatial parameters (i.e. h, the so-called smoothing length) is introduced; it is multiplied for the field function f .r/ and integrated over the problem domain: Z f .r 0 /W .r  r 0 ; h/dr 0

f h .r/ D

(1)

˝ 0

The W .r  r ; h/ function should satisfy the following conditions: lim W .r  r 0 ; h/ D ı.r  r 0 /

h!0

Z

W .r  r 0 ; h/dr 0 D 1

(2)

(3)

˝

W .r  r 0 ; h/  0 ˇ ˇ W .r  r 0 ; h/ D 0 for ˇr  r 0 ˇ > ˛h

(4) (5)

where ˛ is a constant related to the smoothing kernel for the particle at r. The spatial derivatives of field functions can be easily approximated in SPH. It is interesting to underline that spatial differential operator is transmitted only to the smoothing kernel [1]. By introducing a number of particles covering the problem domain, the kernel approximation can be discretized. The compact condition (5) imposed on the smoothing kernel involves that only a finite number of particles referred as nearest neighboring particles (NNP) have to be considered for a satisfactory approximation.

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Therefore, the kernel approximation is expressed by summing the contribution over all the NNP. In this way the particle approximation is obtained. For a field function, located over a particle i , particle approximation is obtained by averaging function values involving all NNP belonging to the support domain of particle i : f h .ri / D

Ni X

f .rj /W .ri  rj ; h/ 4 Vj

(6)

j D1

where ri and rj are position vectors related to the i th and j th particles, Ni is the number of NNP of particle i and 4Vj is the measure of the domain surrounding j th particle. In order to perform the particle approximation the smoothing kernel is of primary importance. In fact, the kernel defines the extension of the support domain of each particle and determines the consistency and the accuracy of the SPH method. The kernel has to be sufficiently smooth in order to be less sensitive to particles disorder, so reducing the errors in approximating the integral interpolants, provided the particle disorder is not too extreme. The unity condition (3) gives the 0th consistency order; moreover, if the kernel is an even function the consistency increases [4]. When the particle approximation is performed, the consistency condition could not be ensured since, for the particles near the boundary of the problem domain the kernel has to be truncated. In this case ghost particles can be successfully used [1,5,6]. Moreover, when the particles distribution is irregular, lack of consistency can be approached by adopting suitable numerical strategies [14,15]. One of the most popular smoothing kernel is the Gaussian one [7, 8]: W .r  r 0 ; h/ D  exp.R2 /;

RD

jr  r 0 j h

(7)

In 3D formulation,  is expressed as:  D  3=21 h3 . The Gaussian kernel is smooth enough even for high order derivatives. Moreover, the smoothing length h can vary in time and space and has to be opportunely chosen [15]. Every electromagnetic phenomenon, can be described by means of the Maxwell’s curl equations in time domain with proper initial and boundary conditions. In a linear, homogeneous, isotropic, conductive medium the following well-known notation holds: rotE D 

@H ; @t

rotH D J C "

@E @t

(8)

where E and H are the electric and magnetic vector fields, ; ";  are the medium parameters, and J D E. By using the particle approximation, and by introducing an explicit time integration scheme, the previous equations (8) are discretized. For example, for the Ex and Hx components, the following discretized equations hold:

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ExnC =2 .riE / DExn =2 .riE / C 1

1



NNP h X

 4 t Ex .riE / 4t C  " "

i E;H E;H n H E Hzn .rjH /Dy W .ri;j ; hE /  H .r /D W .r ; h / 4 Vj z i y j i i;j

j D1

(9) HxnC1 .riH / D Hxn .riH /  

NNP h X

4t  

i 1 H;E H;E nC1=2 E EznC =2 .rjE /Dy W .ri;j ; hH .rj /Dz W .ri;j ; hH i /  Ey i / 4 Vj

j D1

(10) E;H H;E where ri;j and ri;j are the mutual distances between a E-field particle and a H-field particle, and viceversa: E;H ri;j D riE  rjH I

H;E ri;j D riH  rjE

(11)

and, 4t is the time step. The other four equations for the remaining fields components, can be obtained with indexes permutations. The Courant-FriedrichsLevy stability condition [17–19] requires the time step to be proportional to the smallest spatial point resolution which in SPEM formulation is represented by the smallest smoothing length h of the kernel function W.

3 3D SPEM Simulation The proposed particle method has been validated by comparing the SPEM simulation with finite difference time domain (FDTD) results. A cavity 1 m  1 m  0.5 m with a stationary TE10 wave at 212 MHz is simulated in air: 10,927 E-field particles and 10,926 H-field particles are used (Fig. 1). The walls are considered as perfectly conductors, and ghost particles [1]are introduced in order to treat the lack of consistency due to the truncation of the kernel function related to the boundary particles. In Fig. 2 the Hy component space profile along the middle xz plane is reported: the comparison between SPEM and FDTD simulations agree very well.

4 Conclusions In this paper a full wave three dimensional meshless particle approach for electromagnetic transient simulation is presented. The smoothed particle hydrodynamic

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Fig. 1 Simulation of a cavity 1 m  1 m  0.5 m with a stationary TE10 wave at 212 MHz in: 10,927 E-field particles and 10,926 H-field particles have been used

2

× 10-3 SPEM FDTD

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Hy [A / m] 0

-1

-2 0.1

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0.3

0.4

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0.8

0.9

Fig. 2 Hy space profile along the middle xz plane

method is re-formulated by considering the particles as interpolation points, arbitrarily placed in the computational domain. Maxwell’s equations in time domain with the assigned boundary and initial conditions are numerically solved. The computational tool is assessed and, for the first time, a 3D test problem is simulated. The only even particles distribution has been addressed even if, in simulating EM fields it is often necessary to control regions with high localized field gradient or reproduce irregular geometries of the problem domain. In these cases uneven particles distribution has to be considered, and consistency restoring techniques

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have to be introduced. This problem, already addressed by the authors in 2D simulations, will be treated in a forthcoming work for 3D structures. Acknowledgements This work has been supported by Universit´a degli Studi di Palermo, under project.

References 1. Liu, G.R., Liu, M.B.: Smoothed Particle Hydrodynamics – A Mesh-Free Particle Method. World Scientific Publishing, Singapore (2003) 2. Laguna, P.: Smoothed particle interpolation. Astrophys. J. 439, 814–821 (1994) 3. Monaghan, J.J.: An introduction to SPH. Comp. Phys. Commun. 48, 89–96 (1988) 4. Liu, M.B., Liu, G.R., Lam, K.Y.: Constructing smoothing functions in smoothed particle hydrodynamics with applications. J. Comput. Appl. Math. 155, 263–284 (2003) 5. Belytschko, T., Krongauz, Y., Organ, D., Fleming, M., Krysl, P.: Meshless methods: An overview and recent developments. Comput. Meth. Appl. Mech. Eng. 139, 3–47 (1996) 6. Mirzaei, D., Dehghan, M.: A meshless based method for solution of integral equations. Appl. Numer. Math. 60(3), 245–262 (2010) 7. Monaghan, J.J., Lattanzio, J.C.: A refined particle method for astrophysical problems. Astron. Astrophys. 149, 135–143 (1985) 8. Monaghan, J.J.: Smoothed particle hydrodynamics. Annu. Rev. Astronom. Astrophys. 30, 543–574 (1992) 9. Liu, M.B., Liu, G.R.: Smoothed particle hydrodynamics (SPH): An overview and recent developments. Arch. Comput. Meth. Eng. 17(1), 25–76 (2010) 10. Ala, G., Spagnuolo, A., Viola, F.: An advanced gridless method for electromagnetic transient simulation. In: Proceedings of EMC Europe 2004, pp. 54–59 (2004) 11. Ala, G., Francomano, E., Tortorici, A., Toscano, E., Viola, F.: A smoothed particle interpolation scheme for transient electromagnetic simulation. IEEE Trans. Magn. 42(4), 647–650 (2006) 12. Ala, G., Francomano, E., Tortorici, A., Toscano E., Viola, F.: Smoothed particle electromagnetics: A mesh-free solver for transients. J. Comput. Appl. Math. 191(2), 194–205 (2006) 13. Ala, G., Francomano, E., Tortorici, A., Toscano, E., Viola, F.: A mesh-free particle method for transient full-wave simulation. IEEE Trans. Magn. 43(4), 1333–1336 (2007) 14. Ala, G., Francomano, E., Tortorici, A., Toscano, E., Viola, F.: Corrective meshless particle formulations for time domain Maxwell’s equations. J. Comput. Appl. Math. 210(1), 34–46 (2007) 15. Ala, G., Francomano, E., Tortorici, A., Toscano, E., Viola, F.: On the use of a meshless solver for PDEs governing electromagnetic transients. Appl. Math. Comput. 209(1), 42–51 (2009) 16. Liu, M.B., Liu, G.R.: Restoring particle consistency in smoothed particle hydrodynamics. Appl. Numer. Math. 56, 19–36 (2006) 17. Taflove, A., Hagness, S.: Computational Electrodynamics: The Finite-Difference TimeDomain Method. Artech House, Boston (2000) 18. Sullivan, D.M.: Electromagnetic Simulation Using the FDTD Method. IEEE press, New York (2000) 19. Elsherbeni, A., Demir, V.: The Finite-Difference Time-Domain Method for Electromagnetics with Matlab Simulations. SciTech Publishing, Raleigh (2009)

Modelling and Analysis of the Nonlinear Dynamics of the Transrapid and Its Guideway Michael Dellnitz, Florian Dignath, Kathrin Flaßkamp, Mirko Hessel-von Molo, Martin Kruger, ¨ Robert Timmermann, and Qinghua Zheng

Abstract In the development and optimization of magnetic levitation trains, realistic simulation models of the mechanic, electromagnetic and electronic subsystems both onboard and in the guideway are crucial factors. In this contribution we present coupled mechanical-electromagnetic models of the control subsystems, magnet subsystems, a lateral cross-section and a vertical dynamics model, modeled by the multibody systems method. The models are verified using simulations, eigenmode analysis and displacement measurements from train passages on a test track. The models are suitable e.g. for simulating the effects of train passages on the ground and they are applied to the analysis of a novel guideway support. It is shown that ground vibrations caused by the vehicle can be significantly reduced by a flexible spring-mass system as support for the girders.

1 Introduction The Maglev vehicle Transrapid as described in [7] and [9] is levitated by magnetic forces which pull the vehicle’s levitation chassis towards the guideway from below, as shown in Fig. 1. The levitation magnets are distributed equally along the vehicle

M. Dellnitz  K. Flaßkamp  M. Hessel-von Molo  R. Timmermann () Chair of Applied Mathematics, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany e-mail: [email protected]; [email protected]; [email protected]; [email protected] M. Kr¨uger Heinz Nixdorf Institute, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany e-mail: [email protected] F. Dignath  Q. Zheng ThyssenKrupp Transrapid, Munich, Germany e-mail: [email protected]; [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 14, © Springer-Verlag Berlin Heidelberg 2012

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Fig. 1 Vehicle TR09 on the TVE test facility (picture: Fritz Stoiber Productions, 2008)

and possess poles with alternating fluxes which are part of the synchronous long stator linear motor. Furthermore, guidance magnets lead the vehicle along the guideway. The guideway usually consists of a series of individual concrete girders equipped with stator packs as reaction surface for the levitation magnets and steel rails as reaction surface for the guidance magnets. Within the stator packs the long stator motor winding is placed for the propulsion and braking of the vehicle. An overview of the design of magnets and guideway is given in [10] and a detailed comparison between the system properties of the Transrapid and the system properties of conventional high-speed trains can be found in [11]. Dynamically, the Transrapid represents a typical mechatronic system comprising mechanical, electrical, magnetic, electronic and control subsystems. As the magnetic levitation is inherently instable a well performing and reliable control system is of paramount importance. In order to analyze the dynamic behavior and to optimize the control parameters systematically simulation models can be applied. Such simulations have long been described by various authors, e.g. [2, 6, 8]. While only small models could be handled by computer systems in the early development phase of the Transrapid, today state-of-the-art IT-systems permit the analysis of detailed models, containing the interaction between the subsystems which requires the consideration of many state variables. Lately, a multibody model of a complete Transrapid vehicle consisting of three sections (coaches) has been described [3]. In this model, the controlled forces include effects from the geometric shape of the magnetic field but the nominal magnet forces are calculated by a simplified PD control law for each magnet. The work presented here augments this model by a detailed representation of the electro-magnetic subsystem and the control system yielding the full mechatronic model. In order to validate and analyze the stability of the controlled system in detail, further mechanical representations of the vehicle and of a magnet test bench are generated.

2 Modelling The models for analyzing the lateral and vertical dynamics are constructed using the same basic structure: The main model is divided into submodels for the mechanical subsystem, the electromagnetic subsystem and the electronic subsystem. Figure 2

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disturbances

air gap & mechanical states

air gap desired air gap

Control

voltage

Magnet

magn. force

Mechanics

current

Fig. 2 Schematic representation of the basic model structure guideway coil core

f1

f2

f3

f4

f5

f6

connector

Fig. 3 Image of the left half of a levitation magnet, consisting of six interconnected poles

shows a schematic representation of the submodels’ input-output-structure. In Sect. 2.1 the model of the electronic and electromagnetic subsystem is described. Sections 2.2 and 2.3 explain the mechanical models of the vertical and lateral cross section and how these are coupled with the magnetic model. For more details on the magnetic model see [12].

2.1 Guidance and Levitation Magnets The guidance and levitation magnets (see Fig. 6, left) are very similar in structure and design, thus it is sufficient to construct one model and to adjust its parameters to account for guidance or levitation magnets. Each levitation magnet consists of six poles which are wired in series. The poles are designed such that the fluxes flow mostly through material with high permeability. See Fig. 3 for an illustration. The guidance magnets do not need alternating fluxes because they do not interact with the motor and can be modeled as one pole systems along the x-direction. Maxwell’s equations for quasi-stationary electric and magnetic fields are used to set up the differential equations for the flux computation. We obtain the following equation from the magnetic system (see Fig. 4, left) Rmag  D  ; Rmag 2 R66 ;  D .1 : : : 6 / ;  D .1 : : : 6 / 2 R6

(1)

with Rmag representing the resistances in the magnetic network,  the fluxes and  the magnetic voltages. For each pole its magnetic voltage can be computed as k D nk Ik , k D 1; : : : ; 6 (nk : number of windings, and Ik : electric current through the coil). Similarly, using Kirchhoff’s laws, the governing equations P of the electric substitute system (Fig. 4, right) can be computed to be UM D 6iD1 .Uind;i C Ri IM /.

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RFe1

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f5 + f6

Uind,1

q1

RL3

RL2

RL1

f2 + f3

IM

f2 f1 + f2

RL6

f1 f1

UM

Fig. 4 Substitute systems for the magnetic network (left) and electric network (right)

Rair

qS

fS

qL RL

qS fS

U I

Rel

UM RFe

L

qFe

Uind

Fig. 5 Smaller magnet model (left) and its substitute magnetic (center) and electric structure (right)

This equation can be coupled with (1) using Uind D nP which describes the timedependent behavior of the coil, with Uind being the induced voltage, n the number of coil windings and  the magnetic flux generated by the coil, resulting in a differential equation for the flux. For a given flux, the magnetic force is proportional to the flux squared: Fmag D Kmag  2 . The factor Kmag as well as all other resistances were calculated taking into account material properties and exact geometries. This leads to a model which is suitable for very exact computations but at the cost of higher simulation effort. To overcome this drawback, a substitute model consisting only of a single pole is constructed (see Fig. 5). A parameter fitting is performed such that its input-output-behavior, i.e. electronic and magnetic voltage and current, resembles the behavior of the original model. The magnet control units regulate the size of the air gaps by adjusting the magnet’s input voltages. In each unit the sensor values for air gap, magnet body acceleration and current are filtered, scaled and processed by the actual controller. Afterwards the controller output is transformed into an input voltage for the magnet (cf. Fig. 2).

2.2 Lateral Cross Section Dynamics The lateral dynamics, i.e. vertical, horizontal and rolling motions of the magnetic levitation train are modeled by a multibody system of a lateral cross section of the vehicle. The reference length is one eighth of a Transrapid’s section such that two levitation frames and two secondary suspension units are taken into account. We

Modelling and Analysis of the Nonlinear Dynamics of the Transrapid and Its Guideway

fCB

yCB

zCB aL aR fLC

yLC

zLC zLML

zLMR

car body (CB) elastomer elements air spring rocker air spring (secondary suspension) pendulum levitation chassis (LC) guidance magnet reaction rail stator pack levitation magnet (LM) elastomer spring(primary suspension) guideway girder

117

fCB aL

SCB

yCB

zCB

fLC

SLC

yLC

zLC

Fig. 6 Left: Generalized coordinates .yLC ; zCB ; LC ; yCB ; zCB ; CB ; zLML ; zLMR / and depending auxiliary variables .˛L ; ˛R / for the lateral dynamics model. Right: The angles of the air spring rockers can be computed by the triangle spanned by body fixed points on rocker, car body and lev. chassis

consider the levitation frames to be equipped with guidance magnets and combine the pair of levitation magnets and the pair of primary suspension units to one object respectively (see Fig. 6). The car body, levitation chassis, levitation magnets and the two air spring rockers are idealized as rigid bodies whereas the primary and the secondary suspensions are modeled as massless connections. The suspensions are reduced to linear spring damper systems acting one-dimensionally. Since deviations of the train from the reference position are known to be small, a linear model of the lateral dynamics is appropriate. The cross section model exhibits eight degrees of freedom (as depicted in Fig. 6) assuming the levitation magnets translate only vertically. The rotation about the x axis and the translation in y and z direction of the levitation chassis is described with respect to an inertial frame. The coordinates describing the rotational and translational motion of the car body as well as those of the levitation magnets are defined relative to the chassis. At first it is not obvious that this is an admissible choice of generalized coordinates because the topology of the model contains a kinematic chain consisting of the chassis, car body, both air spring rockers and both pendulums. In principle this would lead to a system of differential algebraic equations. However, we obtain a system of ordinary differential equations by substituting the dependent variables by an explicit formulation of the algebraic equations in terms of the independent variables. The dependent auxiliary variables are the rotation angles of the air spring rockers which can be calculated geometrically as sketched in Fig. 6, right. For more details see [1]. The modelling is realized using the software package Neweul [4] which derives analytically the equations of motion by the Newton-Euler formalism. The explicitly formulated algebraic equations for the air spring rocker angles are implemented in the computer algebra system Maple such that the partial derivatives for the linearized equations of motion can be derived analytically as well.

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2.3 Vertical Dynamics To examine the interaction of the Transrapid with its guideway in detail, the vertical dynamics have been modeled as well. Our aim was to create a suitable model for studying the ground vibrations induced during train passages. The guideway and the vehicle mechanics are modeled separately such that the resulting entire system has a modular structure and any components can be exchanged comfortably. We model a maglev train consisting of three sections as shown in Fig. 7. In order to reduce the complexity, only one section is modeled as a multibody system comprising 28 mechanical degrees of freedom. The other sections are considered by their nominal magnet forces. To merge the three-dimensional vehicle into a two-dimensional model, the effects of the left and right side are summed up. The resulting translational and rotational degrees of freedom are depicted in Fig. 8, left. The equations of motion are derived analytically using Neweul. The simplified version of the electromagnetic subsystem, see Sect. 2.1, is used to model the levitation magnets. The geometry of the stator packs induces a slight dependence on position of the magnet force. This is accounted for by a position dependent scaling as shown in Fig. 8. Although this position dependency is small compared to the nominal load it is responsible for a significant part of the vibrational load. It is possible to switch between a dynamic middle and a dynamic front section by simply using a different parameterization as the topologies of both models are equal. For more details of the vertical model, see [5]. The general approach of coupling the vehicle and the guideway model goes back to [3].

end section

end section

middle section

half-infinite guideway

half-infinite guideway

Fig. 7 Maglev train consisting of three sections and a guideway composed of three girders

Force F / F0

1 0.8 0.6 0.4 0.2 0

0

2

4

6

8

10

Position TM-center of mass

Fig. 8 Left: Generalized coord. of a dynamic middle section. Right: Magnet force modulation

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3 Verification and Analysis As described in the previous section, the lateral and vertical dynamics models are complex, nonlinear, multiphysics simulation models which have to be verified properly. Subsequently, these models can be used to uncover inherent dynamical properties of the systems.

3.1 Verification of the Magnet Models ThyssenKrupp uses a test bench to test the Transrapid’s magnets in various situations. We use a model of the test bench’s mechanics to validate the combined magnet-controller models. At first, linearizations of the full and the reduced magnetcontroller model are compared. Bode-plots show a similar behavior of the two models over a wide range of frequencies. To compare our model to the test stand (see Fig. 9, left), we generated a Bode diagram for the test stand (see Fig. 9, right). The test stand consists of two levitation magnets that follow a forced sinusoidal vertical motion. The frequency of the input motion has been chosen as input and the frequency of the following motion as output for the Bode diagram. The plots show similar but not the same behavior. The differences between the diagrams can be explained by several damping effects that have not been taken into account in the model.

3.2 Linear Analysis of the Lateral and Vertical Dynamics

zF

x

jL zL

jR zR

z magnet poles (following the guideway)

magnet carrier air gap zSR

support structure

phase (deg.)

“guideway” (moving up and down)

magnitude (db)

As a first step we perform an eigenmode analysis of the undamped mechanical subsystems to characterize the shape of the free oscillations. The lateral dynamics

0 –20 –40 –60

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Fig. 9 Left: A schematic representation of the magnet test bench. The guideway can be moved up and down and the magnets and the attached carriers and support structure follow its motion. Right: This Bode diagram compares measurements of the test bench with simulation results of its model (dashed line). As input we chose the up- and downward movement of the guideway and as output the movement of the magnets (left/right; solid and dotted lines)

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model has eight eigenfrequencies. Three are zero and belong to a translation or rotation of the entire system because no reset forces have yet been considered. The remaining five eigenmodes are sketched in Fig. 10 and categorized according to the body that is primarily oscillating. The eigenmodes of the vertical dynamics model can be divided into vertical eigenmodes and those resembling rope oscillations in the levitation magnets caused by their chain-like kinematics. When the damping effects of the primary and secondary suspensions are added, natural oscillations similar to the undamped case can be identified. Next, the mechanical subsystems are connected with the magnetic and controller subsystems. Eigenvalue analyses of the linearized complete systems prove stability for both the lateral and the vertical dynamics systems. We further examine the behavior of the mechanical eigenmodes when adding the other subsystems. A simulation-based analysis shows that some eigenvalues of the mechanical subsystems can still be identified with the former mechanical eigenmodes, e.g. a selection of four eigenmodes of the vertical model can be found in Fig. 11. For a more detailed analysis, we refer to [1] and [5]. These eigenvalues of the complete lateral model can be analyzed with respect to their sensitivity to parameter variations. Among others, we vary the air gap between train and guideway which significantly influences the operating point. The individual eigenvalues react differently to air gap disturbances of the guidance and the levitation magnets. Once more, this verifies the high quality of the model as it reproduces actual behavior realistically.

1st vertical eigenmode (scaled to w0 = 1)

1st rolling eigenmode (1.99 w0)

1st horizontal eigenmode (2.46 w0)

1st eigenm. of 2nd eigen mode of lev.mag. (20.23 w0) lev.mag. (22.20 w0)

Fig. 10 Eigenmodes of the mechanical subsystem of the lateral cross section

1st vertical eigen mode (0.64 w0)

1st rolling eigen mode (0.64 w0)

1st rope oscillation (1.2 w0)

2nd rope oscillation (1.2w0)

Fig. 11 Sketched eigenmodes of the entire vertical model

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driving direction girder

half-infinite guideway

spring-mass system

half-infinite guideway

Fig. 12 Setting for simulation of vehicle passages

4 Novel Guideway As a Transrapid passes a single girder, the interaction of the mechanical and the electromagnetic components induces ground vibrations. New guideway concepts— to be used in sensitive city areas—aim to reduce these vibrations as much as possible. The simulation model of the vertical dynamics is used to approximate these ground vibrations and to study the effects of a new guideway with flexible support. The guideway consists of individual girders with varying length. They are also modeled as rigid bodies with two degrees of freedom. The displacement of a girder at its support is assumed to be proportional to the joint force. Hence, the displacement relative to the static deviation can be used to determine the ground vibrations and it can be computed by simulations of vehicle passages with constant velocity. The model of the vertical dynamics is suitable for this task as it fits quite well with real measurements, see Fig. 13, left. In the novel guideway concept, the girders have a more flexible support due to an additional body that is located between the girder and its foundation and a soft spring between them. Above this additional mass there are the same elastomer springs as in the non flexible model. This support is modeled as an additional spring-mass system as shown in Fig. 12. It is a simple mechanical system with one degree of freedom for the vertical movement. We used the setting that is shown in Fig. 12 to investigate the flexible support. There are three dynamic girders and two additional spring-mass systems. We also used two half-infinite guideways to obtain well-defined initial and final conditions. The initial conditions of the guideway model are chosen such that the girders are leveled out for the static load of the vehicle. This leads to a deviation in the unloaded, initial state, as can be seen from Fig. 12. We computed several simulations with varying stiffness values of the flexible support. Even for very small stiffness values, i.e. for a great flexibility, train passages are still possible, although this leads to high displacements of the girders. The ground vibrations are significantly reduced in particular for high frequencies as it can be seen in the frequency spectra in Fig. 13, right. These spectra again result from simulations of passages with constant velocity. The spectrum with the original support is shown as reference (white line) and three simulations were run with different stiffness values

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amplitude

1.5 force

1 0.5 Measurement Simulation

0 –0.5

0

1

2 time (s)

3

10−3 10−5 0

20

40 60 frequency (w / w0)

80

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Fig. 13 Left: comparison of simulated (white) and measured (black) girder displacement. Right: frequency spectra of ground vibrations: without flexible support (white) and three different stiffness values of the additional spring (black)

of the flexible support. We conclude that it is possible and very effective to use this kind of flexible support for the maglev train Transrapid in order to reduce ground vibrations. Using the lateral dynamics model (cf. Sect. 2.2) and coupling it to a guideway cross section model with varying stiffnesses of the guideway bearing, we also verified the stability of the Transrapid’s levitation at zero velocity.

5 Conclusion Two multidisciplinary models for the Transrapid’s dynamics have been introduced, including detailed submodels of the magnets and the controller units. The lateral and vertical dynamics model have been verified using linear analysis tools, simulations and measurements. As an application of our models, we studied two typical scenarios for magnetic levitation trains: the levitation at zero velocity and the interaction of train and guideway during passages at constant speed. The models can contribute to the further development of the system, e.g. by predicting the system’s behavior on novel guideways.

References 1. Flaßkamp, K.: Analyse der nichtlinear gekoppelten Lateraldynamik der Magnetschwebebahn Transrapid (Analysis of the nonlinearly coupled lateral dynamics of the maglev train Transrapid, in German). Diploma thesis, University of Paderborn (2008) 2. Gottzein, E.: Das Magnetische Rad als autonome Funktionseinheit modularer Trag- und F¨uhrsysteme f¨ur Magnetbahnen. No. 35 in VDI–Fortschritt-Berichte, Reihe 8. VDI, D¨usseldorf (1984) 3. H¨agele, N., Dignath, F.: Vertical dynamics of the Maglev vehicle Transrapid. Multibody Syst. Dyn. 21, 213–231 (2009) 4. Kreuzer, E., Leister, G.: Programmsystem NEWEUL’90. Institute B of Mechanics, University of Stuttgart (1991). AN–24

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5. Kr¨uger, M.: Vertikaldynamik der Magnetschwebebahn Transrapid unter Ber¨ucksichtigung der nichtlinearen Magnetcharakteristik (Vertical dynamics of the maglev train Transrapid under consideration of the nonlinear magnet characteristics, in German). Diploma thesis, University of Paderborn (2008) 6. Meisinger, R.: Beitr¨age zur Regelung einer Magnetschwebebahn auf elastischem Fahrweg. Ph.D. thesis, Fachbereich f¨ur Maschinenwesen, Technical University of Munich (1977) 7. Miller, L.: Transrapid, Innovation f¨ur den Hochgeschwindigkeitsverkehr. Bayerischer Monatsspiegel (4), 34–45 (1998) 8. Popp, K.: Beitr¨age zur Dynamik von Magnetschwebefahrzeugen auf gest¨anderten Fahrwegen. No. 35 in VDI–Fortschritt-Berichte, Reihe 12. VDI Verlag, D¨usseldorf (1978) 9. Raschbichler, H.G.: Entwicklungslinie Magnetschnellbahn Transrapid. In: Rausch, K.F., Rießberger, K., Schaber, H. (eds.) Sonderheft Transrapid, ZEVrail, Glasers Annalen, vol. 127, pp. 10–16. Georg Siemens Verlag, Berlin (2003) 10. Rausch, K.F., Rießberger, K., Schaber, H. (eds.): Sonderheft Transrapid, ZEVrail, Glasers Annalen, vol. 127. Georg Siemens Verlag, Berlin (2003) 11. Schach, R., Jehle, P., Naumann, R.: Transrapid und Rad–Schiene–Hochgeschwindigkeitsbahn. Springer, Berlin (2006) 12. Timmermann, R.: Analytische und experimentelle Untersuchung der Tragmagnetregelung des Transrapid im nichtlinearen Arbeitsbereich (Analytical and experimental analysis of the Transrapid’s levitation magnet controller in its nonlinear working range, in German). Diploma thesis, University of Paderborn (2008)

Part II

Model Order Reduction for Chip Design

Overview Model Order Reduction (MOR) stands for a broad class of methods to reduce the size of a system while for the same inputs it remains able to provide nearly the same outputs at specific ports as the original system. There are several reasons why the systems become ever larger: there is an ongoing trend to include more physical details in the modeling, which leads to more and coupled equations. The increase of frequencies in communication products leads to additional coupling between components. Because of minituarization more components can be placed on the same die in circuit design. And finally, after making a layout, the system of the original design is enlarged with components that model the parasitic couplings. Verifying that such a big system still fulfills the specifications of the overall global design can be beyond the limit of the numerical procedures. Hence the electronics industries have been and still are a driving force in developing mathematical tools to reduce the size of the systems. In several cases one is not even interested in accurate results for all unknowns. On the contrary, the subsystems are intended to provide good results at special interfaces (output ports or terminals) where they communicate to other subsystems. The most simple way to achieve this for a purely resistive network is by eliminating some selected internal unknowns from the system. In this case this can even be done accurately. The remaining system is smaller and (hopefully) enough sparsity has been retained to allow for sparse matrix techniques. This path in research for MOR has led to interest in graph theory and one of the new trends is structured MOR. Structure-preserving MOR opens a new way for studying more coupled systems: electromagnetics coupled to electric circuits and heat, interaction with multibody and hydronamical systems, couplings with systems in biology and in chemistry. Traditionally, MOR starts by considering a linear state space equation, x0 .t/ D Ax C Bu and output is defined as y D Cx C Du. For the same input u one likes to solve a smaller, system of ODEs for x, leading to an approximation xQ

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of x from which an output yQ can be calculated that approximates y. This has led to several approaches: balanced truncation techniques, Krylov-methods, methods based on singular value decomposition. Good introductions on MOR are found in [7, p. 3, 139]. Books dedicated to MOR are [1, 2, 18]. More recent work is found in [3, 10, 12, 17] (all available at Springer), or will appear soon [13]. Balanced truncation provides error estimates, but linear algebra poses restrictions. Krylov methods still do not provide error bounds but fully exploit sparsity techniques that are well-known from linear algebra. In both areas research is being done to reduce the limitations [17, p. 469]. In industrial design processes frequently systems have to be simulated under varying conditions, and often several systems are coupled to form a large one. Hence, industry has inspired research to also consider DAEs rather than ODEs as state space equation (then called descriptor form) [3, p. 53]; to consider the special problem when there are many ports (this pops up when inductive couplings have to be taken into account, or in power grids) [17, p. 523], [3, p. 277]; to study parameterized problems [4, 5]; to take nonlinearity into account [3, pp. 177, 289, 303], [6], [10, p. 441], [12, pp. 293, 307, 476], [16, 22]; and to reconstruct models based on measured input-output behaviour [3, p. 85]. Special topics are synthesis of a reduced model into the input language of a circuit simulator [3, p. 207] and dealing with symbolic methods [3, p. 261], [10, pp. 429,435]. In the field of optimization and statistics techniques like response surface modeling, surrogate modeling, space or manifold mapping are strongly MOR-related; neat adaptive examples of these last are found in [18, p. 157] and in [7, p. 325]. Several conference websites still offer presentations [8, 9, 19, 23, 24]. At ECMI 2010 progress was reported from several national and European projects [11,14,15,20,21] on research topics that were mentioned above: extensions to DAE, structure preservation, exploitation of sparse matrix techniques, treatment of many ports, nonlinearity, techniques for symbolic simulation and finally surrogate models for use in optimization. M. Bollh¨ofer and A.K. Eppler [21] present “A structure preserving FGMRES method for solving large Lyapunov equations” in which a structure-preserving Krylov subspace method for solving large-scale Lyapunov equations is proposed in which the (approximate) solution is of low rank. This problem arises in the balanced truncation technique. The method presented here uses a low-rank approach based on the FGMRES method. For preconditioning the Low Rank Cholesky FactorAlternating Direct Implicit is applied and it preserves the low-rank structures and allows for the use of inner approximate factorizations. P. Benner and A. Schneider [21] present “Model reduction for linear descriptor systems with many ports” containing several numerical improvements for ESVDMOR to enable the application to sparse and very large-scale systems. They prove that ESVDMOR preserves stability, passivity, and reciprocity. This paper provides a survey of these developments and outlines an error estimation for ESVDMOR. M. Striebel and J. Rommes [15] consider “Model order reduction of nonlinear systems by interpolating input-output behavior” with a new approach for model order reduction of parameterized nonlinear systems. Instead of projecting onto the

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dominant state space, an analog macromodel is constructed for the dominant inputoutput behavior. This macromodel is suitable for (re)use in analog circuit simulators. P. Benner and T. Breiten [14,21] highlight “Krylov-subspace based model reduction of nonlinear circuit models using bilinear and quadratic-linear approximations”. Since reduction procedures of existing approaches like TPWL and POD methods are input dependent, models that are subject to variable excitations might not be sufficiently approximated. They generalize Krylov-subspace methods known from linear systems to a more general class of bilinear and quadratic-bilinear systems, respectively. As has recently been shown, a lot of nonlinear dynamics can be represented by the latter systems. It exploits the observation that several nonlinear functions like the exponential have neat simple differential equations that generate them. Advantages and disadvantages of the different approaches are discussed together with the choice of reasonable interpolation points with regard to optimal approximation results. M. Hinze et al. [21] consider “POD Model Order Reduction of electrical networks with semiconductors modeled by the transient Drift-Diffusion equations” based on snapshot collection in the Proper Orthogonal Decomposition (POD) approach. It is applied to integrated circuits with semiconductors modeled by the transient drift-diffusion equations (DDEs). Discretization of the DDEs with mixed finite elements in space yields a high dimensional DAE. POD, and POD combined with discrete empirical interpolation (DEIM, [6]), can be used to reduce the dimension of the model. A. Steinbrecher and T. Stykel [21] split a circuit into a linear and a nonlinear part in “Model order reduction of electrical circuits with nonlinear elements”. Model reduction of the linear part is done with a passivity-preserving balanced truncation technique. By this, error control is possible for the whole problem. O. Schmidt [21] presents work on “Structure-exploiting symbolic-numerical model reduction of nonlinear electrical circuits” in which he exploits the hierarchical structure of analog electronical circuits. Thereby, the entire circuit is considered as a system of interconnected subcircuits. A newly developed algorithm uses a novel concept of subsystem sensitivities and reduces the subcircuits separately from each other. Thus, a coupling of different reduction techniques is permitted. His PhD Thesis is mentioned in his references. Ch. Salzig et al.[11, 20, 21] describe in “Using sensitivities for symbolic analysis and model order reduction of systems with parameter variation” symbolic model order reduction techniques and an adaptation of the analysis of nominal systems to design circuits, which are robust with respect to parameter variation. Therefore, new sensitivity based methods are introduced to estimate the output of statistical systems and to improve the performance of the statistical MOR methods. L. De Tommasi et al. [15] present work on “Multi-objective optimization of RF circuit blocks via surrogate models and NBI and SPEA2 methods”. Multiobjective optimization techniques can be categorized globally into deterministic and evolutionary methods. Examples of such methods are the Normal Boundary Intersection (NBI) method and the Strength Pareto Evolutionary Algorithm (SPEA2), respectively. With both methods one explores trade-offs between conflicting perfor-

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mances. Surrogate models can replace expensive circuit simulations so enabling faster computation of circuit performances. As surrogate models of behavioral parameters and performance outcomes look-up tables with interpolation and Neural Network models are considered. Wuppertal, Germany

E. Jan W. ter Maten Michael Striebel

References 1. Antoulas, A.C.: Approximation of Large-Scale Dynamical Systems. SIAM Publications, Philadelphia (2005) 2. Benner, P., Mehrmann, V., Sorensen, D. (eds.): Dimension reduction of large-scale systems. In: Lecture Notes in Computational Science and Engineering, vol. 45. Springer, Berlin (2005) 3. Benner, P., Hinze, M., ter Maten, E.J.W. (eds.): Model reduction for circuit simulation. In: Lecture Notes in Electrical Engineering, vol. 74. Springer, Berlin (2011) 4. Bond, B.N., Daniel, L.: A piecewise-linear moment-matching approach to parameterized model-order reduction for highly nonlinear systems. IEEE Trans. Comp.-Aided Des. Integr. Circ. Syst. (TCAD) 26-12, 2116–2129 (2007) 5. Bond, B.N., Daniel, L.: Stable reduced models for nonlinear descriptor systems through piecewise-linear approximation and projection. IEEE Trans. Comp.-Aided Des. Integr. Circ. Syst. (TCAD) 28-10, 1467–1480 (2009) 6. Chaturantabut, S., Sorensen, D.C.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32-5, 2737–2764 (2010) 7. Ciuprina, C., Ioan, D. (eds.): Scientific Computing in Electrical Engineering SCEE 2006. In: Mathematics in Industry, vol. 11, Springer, Berlin (2008) 8. Cluster Symposium on Model Order Reduction, TU Eindhoven, 6 and 13 Dec 2006. http:// www.win.tue.nl/casa/meetings/special/cluster/ (2006) 9. COMSON Autumn School on Future Developments in Model Order Reduction, Terschelling, 21–25 Sep 2009, org. by TU Eindhoven and NXP Semiconductors, http://www.win.tue.nl/casa/ meetings/special/mor09/ (2009) 10. Fitt, A.D., Norbury, J., Ockendon, H., Wilson, E. (eds.): Progress in industrial mathematics at ECMI 2008. In: Mathematics in Industry, vol. 15, Springer, Berlin (2010) 11. Hiespana, Hierarchical simulation of nanoelectronic systems for controlling process variations, Fraunhofer Internal Programs under Grant No. MAVO 817759, http://www.hiespana. fraunhofer.de/ (2008–2011) 12. Korytowski, A., Malanowski, K., Mitkowski, W., Szymkat, M. (eds.): System Modeling and Optimization, IFIP AICT 312. IFIP Advances in Information and Communication Technology, vol. 312. Springer, Berlin (2009) 13. Michielsen, B., Poirer, J.-R. (eds.): Scientific Computing in Electrical Engineering SCEE 2010, vol. 16 of Mathematics in Industry, The European Consortium of Mathematics in Industry, Springer-Verlag Berlin Heidelberg 2012 14. MoreSim4Nano, Model Reduction for Fast Simulation of new Semiconductor Structures in Nano- and Microsystems-Technology, BMBF-Programm Mathematik f¨ur Innovationen in Industrie und Dienstleistungen, funded by the German Federal Ministry of Education and Research, http://www.moresim4nano.org/index.php?lang=en (2010–2012) 15. O-MOORE-NICE, Operational MOdel Order REduction for Nanoscale IC Electronics, FP6 Marie Curie Transfer of Knowledge, MTKI-CT-2006-042477, http://www.tu-chemnitz.de/ mathematik/industrie technik/projekte/omoorenice/index.php?lang=en (2007–2010)

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16. Rewie´nski, M., White, J.: A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices IEEE Trans. Comp.-Aided Des. Integr. Circ. Syst. (TCAD) 22-2, 155–170 (2003) 17. Roos, J., Costa, L.R.J. (eds.): Scientific computing in electrical engineering SCEE 2008. In: Mathematics in Industry, vol. 14. Springer, Berlin (2010) 18. Schilders, W.H.A., van der Vorst, H.A., Rommes, J. (eds.): Model order reduction: Theory, research aspects and applications. In: Mathematics in Industry, vol. 13. Springer, Berlin (2008) 19. Symposium on Recent Advances in Model Order Reduction, TU Eindhoven, 23 Nov 2007. http://www.win.tue.nl/casa/meetings/special/mor07/ (2007) 20. SymTeCO, Symbolic Techniques for Circuit Optimization, FP6 Marie Curie Transfer of Knowledge (2007–2010) 21. SyreNe, Systemreduktion f¨ur IC Design in der Nanoelektronik, BMBF-Programm Mathematik f¨ur Innovationen in Industrie und Dienstleistungen, funded by the German Federal Ministry of Education and Research. http://www.tu-chemnitz.de/mathematik/syrene/ (2007–2010) 22. Volkwein, S.: Model reduction using proper orthogonal decomposition. http://www.uni-graz. at/imawww/volkwein/POD.pdf (2008) 23. Workshop Model Reduction for Circuit Simulation, Univ. Hamburg, 30–31 Oct 2008. http:// www.math.uni-hamburg.de/spag/zms/syrene/ (2008) 24. Workshop Model Reduction for Complex Dynamical Systems, TU Berlin, 2–4 Dec 2010. http://www3.math.tu-berlin.de/modred2010/ (2010)

A Structure Preserving FGMRES Method for Solving Large Lyapunov Equations Matthias Bollh¨ofer and Andr´e K. Eppler

Abstract We present a structure-preserving Krylov subspace method for solving large-scale Lyapunov equations where the (approximate) solution is of low rank. This problem arises, e.g., from model order reduction techniques based on Balanced Truncation for large-scale descriptor systems such as those in the simulation of large electrical circuits. The method presented here uses a low-rank approach based on the FGMRES method. For preconditioning the Low Rank Cholesky Factor-Alternating Direct Implicit is applied which turns out to preserve the low-rank structures and allows for the use of inner approximate factorizations.

1 Introduction In very large system integrated (VLSI) technology advances in size and speed leads to differential-algebraic equations (DAE) with several hundred million elements. For verification of the model a full simulation is often impossible. To do this efficiently model order reduction (MOR) methods have been recognized as a key technology in constructing reduced order models, in particular those methods that inherit the central properties of the underlying circuit like stability and passivity. The original linear circuit can be modelled by a DAE of type E x.t/ P D Ax.t/ C Bu.t/

(1)

y.t/ D Cx.t/ C Du.t/

M. Bollh¨ofer  A.K. Eppler () TU Braunschweig, Institute Computational Mathematics, Pockelsstr. 14, 38106 Braunschweig, Germany e-mail: [email protected]; [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 15, © Springer-Verlag Berlin Heidelberg 2012

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where A; E 2 Rn;n ; B 2 Rn;m ; C 2 Rp;n ; D 2 Rp;m with m; p  n. The Q A; Q B; Q CQ ; DQ such reduced model approximates the system by smaller matrices E; that the dimension n is replaced by some l  n. For circuit equations Positive Real Balanced Truncation (BT) is a suitable method to reduce the dimension and preserve passivity at the same time as shown in [6, 7]. The main idea behind this method consists of balancing the solutions of associated projected Lur’e equations. For special cases these simplify to projected algebraic Riccati equations. For solving these second order matrix equations with Newton’s method (see [1]) in every step projected, generalized Lyapunov equations of the following type need to be solved Ak Xk ET C EXk ATk D Pl Bk BTk PTl ; where Xk D Pr Xk PTr ;

(2)

AT Yk E C ET Yk Ak D PTr CkT Ck Pr ; where Yk D PTl Yk Pl :

(3)

Here Pl ; Pr refer to the left and right projection of the matrix pencil E  Ak to the subspace of finite eigenvalues. This is necessary because the matrix E is not invertible when dealing with electrical circuits. The matrix Ak and the matrix of the right hand side Bk may change in every outer Newton step, while the matrix E remains unchanged from the descriptor system. In the case of MOR we are dealing with a small number of input and output signals which in turn yields low rank matrices B (resp. C ). Moreover, assuming that the system is passive, we obtain symmetric and positive semidefinite solutions Xk ; Yk of (2), (3) which are approximately of low rank, i.e., Xk D Lk LTk , Yk D Rk RkT . It should be noted that for linear RCI networks the descriptor system can symmetrized, meaning that it is necessary only to solve one Riccati equation and therefore only one Lyapunov equation (2) is required at each Newton step, see [6]. In Sects. 2 and 3 we are going to explain how we combine the two methods to a new solver. First we comment on the LRCF-ADI method and then we state how this can be used within Krylov subspace methods. In Sect. 4 we compare our numerical approach with the LRCF-ADI method for one example circuit and also investigate the use of inner approximate factorizations.

2 Low Rank Cholesky Factor-ADI (LRCF-ADI) We now discuss solving generalized, projected Lyapunov equations of the form AXET C EXAT D Pl BBT PTl ; where X D Pr XPTr :

(4)

One of the most popular methods for solving Lyapunov equations is the alternating direction implicit (ADI) method which was first demonstrated in [10]. The transfer to the generalized case as part of BT can be found in [9]. Formally, for ADI one has to solve a sequence of pairs of equations

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.E C j A/Xj  1 D Pl BB T PlT  Xj 1 .E  j A/T 2

Xj .E C j A/T D Pl BB T PlT  .E  j A/Xj  1 2

for j D 1; 2; 3; : : : ; t. Xj can be computed more efficiently using its representation as low-rank Cholesky factors Xj D Lj LTj , see [5]. The most efficient way, usually referred to as LRCF-ADI [4] uses elegant way of computing Lj from Lj 1 such that the computational work grows at most linearly with the number of shifts t. In this simplest case Lj can be computed from Lj 1 via p p   Rej = Rej 1 Fj 1  .j C Nj 1 /.E C j A/1 AFj 1 ;   Lj D Lj 1 ; Fj ; Fj D

p j D 2; : : : ; t, where F1 D Re1 .E C 1 A/1 B, L1 D F1 . In practice it is by far too expensive to compute the residual kALj Lj E T C ELj LTj AT C Pl BB T PlT kF . One can bypass this problem via the updated QR-factorization as shown in [5]. and  Here one computes  updates QR factorizations of the form Q R D AL ; EL ; P B , Q R D 1 1 j j l 2 2   ELj ; ALj ; Pl B . Then the norm reduces to kR1 R2T kF . For the performance of the ADI method it is essential to solve systems .E C j A/ efficiently. In case of circuit equations these matrices become very large and sparse so the LU-decomposition can be used to solve the systems. The other advantage is, that once these factorizations are computed they can be reused in the next cycle of the ADI method.

3 Low Rank Krylov Subspace Methods For the solution of generalized Lyapunov we discuss the use of Krylov subspace methods. Here we will concentrate on the FGMRES [8] method. In contrast to the standard preconditioned GMRES procedure this allows for changing preconditioners in every iteration step. The elementary operations needed in Krylov subspace methods are matrix vector multiplications, scalar products as well as linear combinations of vectors. These templates will be reformulated in a low-rank pseudo-arithmetic that uses a representation V ZV T similar to the Cholesky-factor representation. It is easy to see that a linear combination of symmetric low rank matrices leads again to a symmetric matrix of lower rank. Furthermore the “matrixvector” multiplication by the Lyapunov operator applied to V ZV T  AVZV T ET C EVZV T AT D ŒAVEV

 0 Z ŒAVEVT ; Z 0

can be rewritten as a symmetric matrix of lower rank as well. As for the linear combination and the “matrix-vector” product after the concatenation of columns

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ŒAVEV  the rank may increase. Suppose we have a low rank matrix X D V ZV T . The two main steps in our rank truncation strategy are using an rank revealing QR and an eigenvalue decomposition. First we compute V D QR˘ where ˘ is just column pivoting (see [3]) and cut off the rank according to R based on O R. O After that we calculate a relative tolerance. We denote the remainders by Q; O RO T D U˙U T and discard small a symmetric eigenvalue decomposition of RZ O Here we use the same tolerance again. As a result values of ˙ to get UO and ˙. we have X D VO ˙O VO T with a unitary matrix VO D QO UO and a diagonal matrix ˙O . Beside the symmetric low-rank pseudo-arithmetic we need to use preconditioning that is consistent with this representation. E.g., diagonal preconditioning can be implemented fairly easy but would violate the symmetric low-rank representation. We therefore propose to use the LRCF-ADI method instead. Suppose that at step i , the FGMRES method computes Arnoldi vectors Vi D Vi Zi ViT represented as symmetric low rank matrices. Then one preconditioning step using LRCF-ADI consists of one cycle j D 1; : : : ; t starting with right hand side B D Vi (compare (4)) in the symmetric low-rank format. LRCF-ADI returns the Cholesky factor Lt which induces a symmetric low rank matrix Wi D Lt Dt LTt for i D 1; : : : ; m, where Dt D It ˝ Zi . As mentioned earlier, computing the residual during ADI iteration is quite costly and is usually replaced by the updated QR-factorization [5]. For our algorithm this can be avoided to save time and we can work with the residual norm computed by the FGMRES algorithm.

4 Numerical Experiments We have tested our method on several examples from NEC Europe and compared the solution with the state of the art method LRCF-ADI as stand-alone solver for the associated generalized, projected Lyapunov equations [9]. Here we only consider a RC-chain with 2002 capacitors, 2003 resistors and three voltage sources yielding state dimension n D 2007 and rank 3 of the right hand side. Since we concentrate on developing a new Lyapunov solver we set Ak D A and Bk D B. We want to solve the Lyapunov equation up to a estimated residual norm of 1010 . We start the FGMRES with initial guess zero. As stated before it is essential for the performance of the LRCF-ADI method to solve the shifted systems E C j A fast and reliable. We investigate the influence of approximative solves of these linear systems to both methods. For this purpose we use the software package ILUPACK, see [2]. A preconditioner for each of the systems E Cj A is computed and we solve for three different tolerances for the inner iterative solver, 1012 , 108 and 104 respectively. To allow for such variable preconditioning techniques it is necessary to use the FGMRES method instead of regular GMRES. Fifteen different real shifts are chosen as explained in [5]. All computations were done on a workstation with c Intel(R) Xeon(TM) MP CPU 3.66 GHz and 16 GB RAM using MatlabVersion 7.7.0.471 (R2008b).

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In Fig. 1 the convergence history with respect to the number of ADI steps is plotted. The black lines refer to the proposed LR-FGMRES while the grey lines belong to the LRCF-ADI. It can be seen that the LRCF-ADI method stagnates for the tolerances 108 and 104 at a residual norm of about the same order of magnitude. The values in Fig. 1 are in fact 6:6e  5 and 4:3e  9. We stop the LRCF-ADI iteration after a maximum of 100 steps but only the first 60 ones are plotted here. With our LR-FGMRES we still get convergence even if the systems are only approximately solved. So our method is almost not affected by this. One can not distinguish the three different curves for the LR-FGMRES in Fig. 1. With 3 cycles of FGMRES each of them using 15 ADI iterations we are at the same number of steps as the LRCF-ADI method. In Table 1 one can see the cpu-time needed to solve the Lyapunov equations according to the related tolerance for solving the shifted systems inside the LRCFADI iteration. Table 2 shows the rank development of the right hand sides B D Vi 1 from (4) during the preconditioning step. Initially the rank is always 3. We observe for our method that the ranks of the right hand side increase faster as we raise the tolerance. This explains the longer computational time.

residual norm

100

LRCF−ADI LR−FGMRES

tol. 1e−4 10–5

tol 1e−8 10–10

tol. 1e−12 0

10

20

30 # ADI steps

40

50

60

Fig. 1 Convergence history of residual norms Table 1 Running time [s] dependent on tolerance

Tolerance

Table 2 Rank development dependent on tolerance

FGMRES step i 1 2 3

12

10 108 104

LRCF-ADI

LR-FGMRES

2.7 – –

6.4 15.7 33.8 1012 3 8 25

108 3 19 52

104 3 21 93

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While the LRCF-ADI method is fast it only converges about the same tolerance as one solves the shifted linear systems. The FGMRES turned out to be more robust with respect to this tolerance, this effect was demonstrated by the use of the Multilevel-ILU.

5 Conclusions In this paper we proposed a new method for solving large generalized, projected Lyapunov equations. Using Krylov subspace methods as outer iteration we were able to improve robustness of the LRCF-ADI method when using incomplete factorizations. This is intended to use where direct solvers can not be applied. For applications where larger rank does not contribute too much to the total computation time both methods are suitable. The increase of the ranks in our method will be subject of future work. Another advantage of our solver is the cheaply computable convergence criterion delivered by FGMRES. Other preconditioned Krylov subspace methods like CG, QMR or BiCGStab can be considered within this framework. This work can easily be transfered and applied to the case of standard Lyapunov equations. Acknowledgements The authors would like to thank Carsten Neff and Achim Basermann formerly from NEC Europe for providing the circuit examples and especially Tatjana Stykel from TU Berlin for providing the Matlab codes of the LRCF-ADI method. This work is supported by the German Federal Ministry of Education and Research (BMBF) grant no. 03BOPAE4.

References 1. Benner, P., Li, J.-R., Penzl, T.: Numerical solution of large-scale Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems. Numer. Lin. Algebra Appl. 15, 755– 777 (2008) 2. Bollh¨ofer, M., Saad, Y.: Multilevel preconditioners constructed from inverse-based ILUs. SIAM J. Sci. Comput. 27, 1627–1650 (2006) 3. Golub, G.H., Van Loan, C.F.: Matrix Computations (Johns Hopkins Studies in Mathematical Sciences), 3rd edn. The Johns Hopkins University Press, Baltimore (1996) 4. Li, J.R., White, J.: Low rank solution of Lyapunov equations. SIAM J. Matrix Anal. Appl. 24(1), 260–280 (2002) 5. Penzl, T.: A cyclic low-rank smith method for large sparse Lyapunov equations. SIAM J. Sci. Comput. 21(4), 1401–1418 (2000) 6. Reis, T., Stykel, T.: PABTEC: Passivity-preserving balanced truncation for electrical circuits. IEEE Trans. Comp. Aided Des. Integr. Circ. Syst. 29(9), 1354–1367 (2010) 7. Reis, T., Stykel, T.: Positive real and bounded real balancing for model reduction of descriptor systems. Int. J. Contr. 83(1), 74–88 (2010) 8. Saad, Y.: A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14(2), 461–469 (1993) 9. Stykel, T.: Low-rank iterative methods for projected generalized Lyapunov equations. Electron. Trans. Numer. Anal. 30, 187–202 (2008) 10. Wachspress, E.L.: Iterative solution of the Lyapunov matrix equation. Appl. Math. Lett. 1(1), 87–90 (1988)

Model Reduction for Linear Descriptor Systems with Many Ports Peter Benner and Andr´e Schneider

Abstract Circuit simulation for power grid models leads to the challenge of model order reduction for linear descriptor systems with many ports. Based on the ESVDMOR idea of Feldmann and Liu [Proceedings of the 2004 IEEE/ACM International Conference on Computer-aided design (2004)], we have proposed several numerical improvements for ESVDMOR to enable the application to sparse and very largescale systems. In further investigations we have proven that ESVDMOR is, under a few assumptions, stability, passivity, and reciprocity preserving. This paper provides a survey of these developments and outlines error estimation for ESVDMOR.

1 Introduction Research in circuit simulation deals among others with linear parasitic systems which are, if they are in usual form, very suitable for model order reduction (MOR). Consequently, MOR became a standard tool over the last decades. Unfortunately, many known approaches are not able to handle a very special structure of today’s systems, namely a large number of I/O-terminals. In recent years, this problem became a focus of numerous investigations. Especially from the industrial point of view, this problem needs to be solved as fast as possible to avoid a deadlock in process development due to a lack of simulation know-how. There is a basic idea of Feldmann and Liu [4], on which our work is based on. We modify the algorithm in a way such that it does not need expensive computational steps anymore, e.g. we replace a full SVD by a truncated one. Consequently, it becomes applicable for very large-scale linear continuous time-invariant systems up to order n D 106 , or even larger. Beyond that, we discuss questions about passivity, stability and reciprocity

P. Benner  A. Schneider () Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany e-mail: [email protected]; [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 16, © Springer-Verlag Berlin Heidelberg 2012

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preservation, which are again very important for real world applications. Especially reciprocity, i.e. the possibility of synthesizing the reduced model as a circuit in applications of circuit simulation, is a very important question. Often reduced order models are dense and not physically interpretable. Error analysis and industrial implementation are the last stages on the way to provide a useful and powerful tool to handle these special structures mentioned above. This paper gives an overview about the theoretical results. Due to space limitations, extensive numerical studies will be presented in a succeeding publication.

2 Theoretical Properties of ESVDMOR Modeling of dynamical processes from various application areas, e.g. circuit simulation, mechanical constructs, and biological or biochemical reactions, leads to linear time-invariant continuous-time descriptor systems C x.t/ P D Gx.t/ C Bu.t/;

y.t/ D Lx.t/;

x.0/ D x0 ;

(1)

with C; G 2 Rnn , B 2 Rnmin , L 2 Rmout n , x.t/ 2 Rn containing internal state variables, u.t/ 2 Rmin the vector of inputs, y.t/ 2 Rmout being the output vector, x0 2 Rn the initial value, n the number of state variables, and the number of inputs min , which is not necessarily equal to the number of outputs mout . We assume the transfer function of (1) in the frequency domain to be H.s/ D L.sC C G/1 B:

(2)

In this section we briefly discuss the basics of (E)SVDMOR for systems with O.n/  O.mi n=out /. In [4, 6] it is shown that it is possible to make use of inner system correlations regarding input and output terminals. Consider the i th block moment of (2) as mi D L.G 1 C /i G 1 B, i D 0; 1; : : :, where mi is an mout  min matrix. These moments are equalPto the coefficients of the Taylor series expansion 1 i of (2) about s0 D 0, H.s/ D i D0 mi .s/ : The expansion in s D s0 leads to frequency-shifted moments mi .s0 / D L..s0 C C G/1 C /i .s0 C C G/1 B;

i D 0; 1; : : :

(3)

P Thus, the Taylor series expansion including these moments is H.s/ D 1 i D0 mi .s  s0 /i : We use r different (frequency shifted) block moments to create the input response matrix MI and the output response matrix MO , which are defined as: T  MI D m0 T ; m1 T ; : : : ; mr1 T ;

MO D Œm0 ; m1 ; : : : ; mr1 T :

(4)

Note that if the number of rows in each matrix of (4) is not larger than the number of columns, then r has to be increased. SVDMOR can be seen as a special case

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of ESVDMOR with r D 1, i.e., only m0 is used. Next, we apply the SVD to (4) in order to obtain a low rank approximation MI D UI ˙I VIT  UIri ˙Iri VITr ; and i MO D UO ˙O VOT  UOro ˙Oro VOTro , where ˙Iri and ˙Oro are ri  ri and ro  ro diagonal matrices, VIri and VOro are min  ri and mout  ro isometric matrices that contain the dominant column subspaces of MI and MO , and UIri and UOro are rmout  ri and rmin  ro isometric matrices. They are not used any further. The values ri  min and ro  mout denote the number of the virtual input and output terminals of the terminal reduced order model that are equal to the number of significant, i.e., not neglected singular values. The approximations of B and L using the matrices VITr and VOTro lead to B  Br VITr and L  VOro Lr , where i i Br 2 Rnri and Lr 2 Rro n are consequences of applying the Moore-Penrose pseudoinverse (denoted by ./C ) of VITr and VOro (which are isometric) to B and i

L, respectively. In detail, we have Br D BVIri .VITr VIri /1 D BVITr C D BVIri and i

i

Lr D .VOTro VOro /1 VOTro L D VOCro L D VOTro L; where Br 2 Rnri and Lr 2 Rro n . This leads to the desired decomposition of the transfer function b .s/ D VOr Lr .G C sC /1 Br VIT ; H.s/  H o „ ƒ‚ … ri WDHr .s/

which is equivalent to a terminal reduction step. VOro and VITr can be understood i as operators mapping the information from the original terminals to the virtual ones and back. The new inner transfer function Hr .s/, which has just a few virtual inputs and outputs, can be further reduced by means of any established MOR method, such that HQ r .s/ D LQ r .GQ C s CQ /1 BQ r  Hr .s/: (5) Equation (5) is of essential matter for property preservation of the whole method, see Sect. 3. We end up with a very compact terminal reduced and reduced-order model HQ r .s/ that approximates the original transfer function, i. e. b .s/ D VOr Hr .s/VIT  H b r .s/ D VOr HQ r .s/VIT : H.s/  H o o r r i

i

(6)

3 Numerical Algorithm, Properties, and Error Estimation In this section, we briefly describe the numerical improvements we have implemented. Then we show preservation properties of the method and at the end we discuss error estimation for ESVDMOR. The SVD is one of the crucial ingredients of the original idea. We forbear to perform a full SVD and neglect some of the singular triples simply because it is too expensive. Instead, we perform an efficient truncated SVD to calculate just the needed singular values (SV), i.e., SVs that are kept as well as SVs needed for error estimation. Additionally, we do not compute the moments in (3) explicitly but use

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Algorithm 2 Computation of the components y i a D Bx rC1 ; a D G 1 a; for i D 1 to r do y i D La; a D Ca; a D G 1 a; end for

an iterative application of matrix vector multiplication to factors of the moments. For illustration, we choose MI consisting of r different moments. All presented approaches apply similarly to MO . The singular triples of MI are computed  by 0 MI using solving an eigenvalue computation of the augmented matrix A D MIT 0 Krylov subspaces and matrix vector multiplication. The output of this multiplication is a vector y 2 Rrmout Cmin of the same structure as the input vector x, such that T  Ax DW y D ..y 1 /T ; .y 2 /T ; : : : ; .y rC1 /T /T ; where y i D y.i 1/mout C1 ; : : : ; yi mout and y rC1 D .yrmout C1 ; : : : ; yrmout Cmin /T ; for i D 1; : : : ; r. After matrix multiplication we get the components y i and y rC1 of vector y as y i D mi1 x rC1

and

y rC1 D m0 T x 1 C    C mr1 T x r :

(7)

For efficiency reasons, we replace the block moments with their factors. We compute the r C 1 parts of y by repeatedly applying the same factors to parts of x, depending on which part of (7) it is. The computation of y i follows Algorithm 2. The computation of y rC1 is more involved, but follows the same recursive principle. For large r, there is a chance that numerical stability problems accrue, but in practice, r is often small. In the following, we summarize several facts on preservation of stability, passivity, and reciprocity in ESVDMOR reduced-order models. For detailed proofs, see [2]. Defining the descriptor system (1) as asymptotically stable if lim x.t/ D 0 t !1

for all solutions x.t/ of C x.t/ P D Gx.t/, we have the following theorem: Theorem 1. Consider an asymptotically stable system (1) with its transfer function (2). The ESVDMOR reduced-order system corresponding to (6) is asymptotically stable iff the inner reduction (5) is stability preserving. A possible stability preserving model reduction method that can be applied along the lines of Theorem 1 is balanced truncation for regular descriptor systems, see [7]. Regarding passivity we note that a system is passive iff its transfer function is positive real [1]. The definition of positive realness can be found, e.g., in [5]. This definition requires min D mout D m. If we assume L D B T , such that H.s/ D B T .sC C G/1 B and 

  B1 G1 G2 C1 0 x D u; xP C G2T 0 0 0 C2

  y D B1 0 x;

(8)

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where G1 , C1 , C2 are symmetric, G1 ; C1  0 (i.e., both matrices are positive semidefinite), and C2 > 0 (i.e., C2 is positive definite), then H.s/ is positive real and thus the system is passive. This is a common structure among linear circuit models, see [5]. Theorem 2. Consider a passive system of the form (8). The ESVDMOR reduced system (6) is passive iff the inner reduction (5) is passivity preserving. Definition 1. A transfer function (2) is reciprocal if there exists m1 ; m2 2 N with m1 C m2 D m, such that for ˙e D diag.Im1 ; Im2 / and all s 2 C, where H.s/ has no pole, it holds H.s/˙e D ˙e H T .s/: The matrix ˙e is called external signature of the system. A descriptor system is called reciprocal if its transfer function is reciprocal. As a consequence, a transfer function of a reciprocal system has the form H.s/ D  H11 .s/ H12 .s/ T T ; where H11 .s/ D H11 .s/ 2 Rm1 ;m1 and H22 .s/ D H22 .s/ 2 T .s/ H22 .s/ H12 Rm2 ;m2 . Theorem 3. Consider a reciprocal system of the form (8). The ESVDMOR reduced system (6) is reciprocal iff the inner reduction (5) is reciprocity preserving. Next, we discuss some ideas on how to get a global error bound for ESVDMOR. For details

see [3]. The error caused by a SVD of MI is eMI D





MI .r/  UIri ˙rIi VITr D rIi C1 ; where ˙rIi D diag.1I  : : :  rIi  rIi C1  i

2

: : :  mI in  0/  ˙rIi D diag.1I  : : :  rIi /: The error for the square root



P variant of balanced truncation is bounded by Hr  HQ r H1  2 nkD`C1 Ok D ı; in case we keep the ` largest Oi . Due to (6) and the triangle inequality, the total ESVDMOR error in spectral norm on the imaginary axis can be expressed locally as















etot D H.i!/  HO r .i!/  H.i!/  HO .i!/ C HO .i!/  HO r .i!/ : (9) 2 2 „ ƒ‚ … „ ƒ‚ …2 Deout

ein

The error caused by the inner reduction follows from (6) and (9) as







ein D VOro Hr .s/VITr  VOro HQ r .s/VITr D Hr .s/  HQ r .s/ 2  ı; i i 2

due to the fact that the spectral norm is invariant under orthogonal transformations. The outer reduction error eout in the SVDMOR case is based on MI D MOT D m0 D B T .s0 C C G/1 B D U˙V T D U˙U T  Ur ˙r UrT : The local terminal reduction error eout then is

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eout D H.s0 /  HO .s0 /

2



(SVD) I=O D B T .s0 C C G/1 B  Ur UrT B T .s0 C C G/1 BUr UrT 2 D kC1 ;

(U=V)

if we keep k singular values or terminals. Then the total error in the SVDMOR case in spectral norm is n X I=O etot  kC1 C 2 O j : (10) j DlC1

For the ESVDMOR case with r  1 (r times mi within the ansatz matrices), see [3].

4 Conclusions This work gives an overview of the ESVDMOR approach with which, in combination with the right choice of the method in (5), it is possible to preserve stability, passivity, and reciprocity. Additionally, the possibility of a global error bound is given in (10). Despite the industrial need for such algorithms, very large-scale real world examples are hard to come by due to confidentiality. We have successfully reduced an academic state space example of order 105 with originally circa half as much I/O-terminals and we have investigated an industrial circuit model of order 103 with a few hundred pins. In any case, just as in standard MOR methods, the approaches are very dependent on the decay of the SVs. Furthermore, the reduced order model should be evaluated iteratively and in factorized form. Otherwise, ESVDMOR would be inefficient and we might end up with a very large-scale dense model due to the mapping back to the original terminals. With respect to the given hints, ESVDMOR is a powerful tool to reduce linear descriptor systems with many terminals. Acknowledgements The work reported in this paper was supported by the German Federal Ministry of Education and Research (BMBF), grant no. 03BEPAE1. Responsibility for the contents of this publication rests with the authors.

References 1. Anderson, B., Vongpanitlerd, S.: Network Analysis and Synthesis. Prentice Hall, New Jersey (1973) 2. Benner, P., Schneider, A.: On stability, passivity, and reciprocity preservation of ESVDMOR. In: Benner, P., Hinze, M., ter Maten, J. (eds.) Model Reduction for Circuit Simulation, Lecture Notes in Electrical Engineering, vol. 74, pp. 267–278. Springer, Berlin (2011) 3. Benner, P., Schneider, A.: Some Remarks on A Priori Error Estimation for ESVDMOR, Scientific Computing in Electrical Engineering SCEE 2010. In: Michielsen, B., Poirier, J.-R. (eds.), Mathematics in Industry, vol. 16, pp. 15–24. Springer, Berlin/Heidelberg, Germany (2012)

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4. Feldmann, P., Liu, F.: Sparse and efficient reduced order modeling of linear subcircuits with large number of terminals. In: ICCAD ’04: Proceedings of the 2004 IEEE/ACM International Conference on Computer-aided design, pp. 88–92. IEEE Computer Society, Washington, DC (2004) 5. Freund, R.W.: On Pad´e-type model order reduction of J-Hermitian linear dynamical systems. Lin. Algebra Appl. 429(10), 2451–2464 (2008) 6. Liu, P., Tan, S.X.D., Yan, B., McGaughy, B.: An efficient terminal and model order reduction algorithm. Integr. VLSI J. 41(2), 210–218 (2008) 7. Mehrmann, V., Stykel, T.: Balanced truncation model reduction for large-scale systems in descriptor form. In: Benner, P., Mehrmann, V., Sorensen, D. (eds.) Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol. 45, pp. 83–115. Springer, Berlin (2005)

Model Order Reduction of Nonlinear Systems By Interpolating Input-Output Behavior Michael Striebel and Joost Rommes

Abstract In this paper we propose a new approach for model order reduction of parameterized nonlinear systems. Instead of projecting onto the dominant state space, an analog macromodel is constructed for the dominant input-output behavior. This macromodel is suitable for (re)use in analog circuit simulators. The performance of the approach is illustrated for a benchmark nonlinear system.

1 Introduction Simulation of VLSI chips is becoming CPU and memory intensive, or even infeasible, due to the increasing amount of layout parasitics and devices in analog designs. A popular method for speeding up and/or enabling simulation of largescale dynamical systems is Model Order Reduction (MOR) [1]. For linear systems (large parasitic networks), several methods have been developed that are now used in industrial circuit simulators. Well-known methods for nonlinear systems in circuit simulation are Proper Orthogonal Decomposition (POD) based methods [3] and piecewise-linearization (PWL) methods [5]. Both approaches try to obtain reduction by projection on the dominant dynamics. However, both approaches may suffer from difficulties that may limit their practical use [7]. We present a new method for the reduction of large nonlinear systems. The most significant difference with respect to existing methods is that instead of focusing

M. Striebel () Bergische Universit¨at Wuppertal, 42119 Wuppertal, Germany e-mail: [email protected] J. Rommes NXP Semiconductors, 5656 AE Eindhoven, The Netherlands e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 17, © Springer-Verlag Berlin Heidelberg 2012

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on the dominant state dynamics, the proposed method captures the dominant inputoutput behavior.

2 Circuit Modeling and Simulation In general, nonlinear circuit blocks connected to other blocks at nP pins are modeled using modified nodal analysis (MNA) yielding network equations 0 D AC 0D

d qC .ATC e/ C AR r.ATR e/ C AL jL C AV jV C AI i.t/  Apin jpin ; dt

d ˚ L .jL /  ATL e; dt

(1a) (1b)

0 D v.t/  ATV e;

(1c)

0 D vpin  ATpin e;

(1d)

where e.t/ 2 Rne , jL .t/ 2 RnL , jV .t/ 2 RnV denote the unknown node voltages and currents through inductors and voltage sources, determined by the equations (1a), (1b) and(1c), respectively. The incidence matrices A˝ 2 f0; ˙1gne n˝ , describe the placement of the basic network elements resistor (˝ = R), capacitor (C), inductor (L), voltage (V) and current (I) source, respectively. The, in general nonlinear, characteristics of the network elements are represented by qC ./, ˚./, r./, i.t/, v.t/. The incidence matrix Apin 2 f0; ˙1gne nP selects the pins from the nodes. At the pins communication with the outside is done in terms of pin voltages and pin currents, vpin ; jpin 2 RnP , respectively. Injecting, i.e., prescribing the pin voltages vpin , the pin currents jpin become additional unknowns, determined by (1d), which will be passed back to system the block is embedded in, or vice versa. By this, a circuit unit turns into an input-output system, represented in the compact form 0D

d q.x/ C j.x/ C s.t/ C BuI dt

y D BT x;

(2)

where u.t/; y.t/ 2 RnP represent the input and output of the system and x.t/ 2 Rn (n D ne C nV C nL C nP ) denote the internal states. Note, that in the following we will omit the excitation s.t/. Systems of type (2) are solved by numerical time integration. The backward Euler, used here for didactical reasons only, discretizes the system, yielding 0D

1 Œq.xn /  q.xn1 / C j.xn / C Bun I h

yn D BT xn :

(3)

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147

Given un and xn1 , (3) defines xn and yn , i.e., approximations to x.tn / and y.tn / at tn D tn1 C h. Applying a Newton-Raphson technique to solve this problem, a series of linear equations has to be solved, made up of 1 C.Nx/ C G.Nx/I h

1 q.Nx/ C j.Nx/I h

q.xn1 /;

(4)

d d i.e., the Jacobians, C./ D dx q./, G./ D dx j./, and element functions evaluated at intermediate Newton iterands xN , and a term q.xn1 / reflecting the history of the dynamic elements.

3 MOR via Input-Output Behavior Macromodeling Often one is interested only in the input-output behaviour of (2), i.e., in the way u.t/ is translated to y.t/. Classically, MOR replaces (2) by a dynamical system of reduced dimension r  n. One demands that given the same input u.t/, the substitute dynamical system with internal states z.t/ 2 Rr produces (almost) the same output y.t/ as the full system. MOR for linear systems reached a high level of maturity. Based on considerations in the frequency domain, (2) is usually projected onto a lower dimensional subspace. For nonlinear problems the situation is different. Here, one can not change to the frequency domain and a projection to a lower dimensional subspace may reduce the dimension of the system but not the computational costs evaluating the system. We propose an approach to reproduce the input-output mapping, starting from timedomain considerations. Being interested in the translation of input to output reads, in terms of the discretised problem (3): we are interested in yn and xn is an auxiliary quantity only. Hence, ideally we are able to replace the system (3) by an input-output mapping  W R n P ! Rn P ;

un 7! yn D .un /:

(5)

At first glance it becomes clear that this is not realisable, as not only a combined evaluation of fq; jg and fC; Gg is needed but also the dynamics’ history q.xn1 /. However, homogeneous structures, comprising only resistive (R), capacitive (C) or inductive (L) elements, the mapping  ˝ (˝ D R; C; L), can be replaced by a compact model. In this way a macroelement with nP ports is created that responds to u (almost) in the same way as the combination of the invidual elements. The basic concept is that it does not have internal states, but the response .u/ and the @ corresponding Jacobian T D @u .u/ is realized by interpolation from tabulated data (see Table 1), gathered in some training phase. For device modelling , table models have been used before [2, 4]; we, however use table models for complete circuit blocks.

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Capacitive Structures The distribution of charges and voltages in a network of capacitors is described by 0 D AC q.ATC e/  Apin qpin ;

0 D vpin  ATpin e;

(6a)

where qpin are point charges at the structure’s pins. In other words: we map a large number of charges q./ to nP point charges qpin . Here the voltage vpin 2 RnP is prescribed at the pins. This formulation reflects the charge conservation and is the integral form of Kirchhoff’s current law (1a) for purely capacitive structures. As a capacitor defines a voltage-charge relation, we choose the pin voltages vpin as input parameters. Assuming sufficient regularity of the capacitance matrix @ qC .w/, (6a) implicitly defines the node voltages and pin charges as Cq .w/ WD @w functions of the pin voltages, i.e., e D e.vpin / and q D q.vpin /, respectively. We differentiate (6a) with respect to vpin to get (with the nP  nP identity matrix InP ) 0 D AC Cq .ATC e/ATC 0 D InP  ATpin

@qpin @e  Apin ; @vpin @vpin

(6b)

@e @vpin

The column in Table 1 reflecting the charge replies is made up of fqp;1 ; : : : ; qp;k g. Items for the column in Table 1 describing the Jacobians TC are found by solving @qpin .i / jvp;i DW Qp;i DW TC , i.e., from the Schur complement for @v@epin jvp;i and @vpin  1  1 .i / Apin Qp;i D TC D ATpin AC Cq .ATC ei /ATC

2 RnP nP :

(6c)

We construct Table 1, describing the mapping pin-voltages to -charges: 1. Choose a discrete set of k 2 N terminal voltages vp;1 ; : : : ; vp;k with vp;i 2 RnP 2. For each i 2 f1; : : : ; kg a. Compute ei D e.vp;i / and qp;i D qpin .vp;i / by solving (6a) for vpin D vp;i @qpin b. Solve the linear system (6b) for @v@epin jvp;i and @vpin jvp;i DW Qp;i . Here, Cq ./ is evaluated at ATC ei . This amounts to computing the Schur complement (6c). 3. The parameters for a capacitive macromodel from Table 1 are Table 1 Macromodel – tabulated data

u

u.1/



u.k/

˝ T˝

.1/ ˝ .1/ T˝

 

˝ .k/ T˝

.k/

Model Order Reduction of Nonlinear Systems By Interpolating

u.i / D vp;i ;

.i /

 C D qp;i ;

149 .i /

TC D Qp;i

for i D 1; : : : ; k where vp;i 2 RnP , qp;i 2 RnP , Qp;i 2 RnP nP Resistive, Inductive and Parameterized Systems Also for purely resistive and purely inductive blocks, table models can be extracted in a similar case. The systems (6a), (6b) used in the training phase are adapted here to both cases: for the former we sweep voltages to get currents; in the latter we sweep currents to get magnetic fluxes. In addition, we can also treat parameterized, homogeneous blocks, by sweeping the corresponding parameters.

3.1 Using the Macromodels A system containing subblocks, of purely static (i.e., resistive) and purely dynamic (i.e., capacitive or inductive) nature can basically be modelled by 0D

d d q.x/ C j.x/ C s.t/ C BR  R .BTR x/ C BC  C .BTC x/; dt dt

(7)

with incidence matrices BR ; BC describing the interfaces. In this way, we accomodate the characteristics of a subblock being reactive or nonreactive. Applying any numerical time integration technique to (7), we see, that the basic ingredients for the systems to be solved in this process are (cf. (4))    1 C Ce TC .Nx/ C G C e TR .Nx/I h

1 Œq C e  C  .Nx/ C Œj C e  R  .Nx/I h

(8)

q.xn1 / and e  C .xn1 /; e˝ ./ D B˝ T ˝ .BT / for ˝ 2 fR; C g. where e  ˝ ./ D B˝  ˝ .BT˝ / and T ˝ Recall, that evaluation of the macromodel-functions and the corresponding Jacobians are realized by interpolation from the corresponding Table 1. See [6] for more details on how piecewise linear models can help convergence of DC analysis.

4 Numerical Experiments The presented approach has been implemented in MATLAB, where for interpolation the available functions interpf1,2,3,ng have been used. The circuit shown in Fig. 1 contains N D 10 purely capacitive blocks. Each block is made up of 100 pairs of a linear capacitor and a varactor. Hence the

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2

3

4

2N−1

5

2N

2N+1

Fig. 1 Test circuit 1 original table

voltage [V]

0.5 node 21 0 node 11 −0.5 node 3 −1

0

2

4

6

8

10 12 time [µs]

0

2

full system

reduced system extraction time: 1:59 sec

3 h 19 min 16 sec

1 min 59 sec

4

6

Fig. 2 Varactor circuit: voltages at node 3,11,21

dimension of the full model is 2;021. The varactor is modelled as a nonlinear p capacitor with qd .u/ D 1:8  1  1  0:5 u . For the capacitive block a compact model is derived by sweeping vpin D f0:0; ˙0:3; ˙0:6g. For testing, the block was instantiated N D 10 times and a voltage source v.t/ D sin.2  105  t/ C 0:4  sin.2  107  t/ was chosen. From Fig. 2 a speedup of about 100, with a dimension reduced to 31, and an almost perfect matching, replacing each of the blocks with a tablemodel becomes obvious.

5 Conclusion We have presented a method that directly approximates the input-output behaviour of nonlinear circuits by interpolating precomputed contributions to the network equations. The method is suitable for large problems as well. Numerical results confirm that significant speedups can be obtained while maintaining accuracy. Future work focuses on mixed static/dynamic circuits and advanced interpolation methods.

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References 1. Antoulas, A.C.: Approximation of Large-Scale Dynamical Systems. SIAM, Philadelphia (2005) 2. Bourenkov, V., McCarthy, K., Mathewson, A.: MOS table models for circuit simulation. IEEE Trans. Comp.-Aided Des. Integrated Circ. Syst. 24(3), 352–362 (2005). doi:10.1109/ TCAD.2004.842818 3. Chaturantabut, S., Sorensen, D.C.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010) 4. Meijer, P.: Fast and smooth highly nonlinear multidimensional table models for device modelling. IEEE Trans. Circ. Syst. 37(3), 335–346 (1990) 5. Rewie´nski, M.J., White, J.: A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices. IEEE Trans. CAD Int. Circ. Syst. 22(2), 155–170 (2003) 6. Roos, J., Valtonen, M.: An efficient piecewise-linear dc analysis method for general non-linear circuits. Int. J. Circ. Theor. Appl. 27, 311–330 (1999) 7. Striebel, M., Rommes, J.: Model order reduction of nonlinear systems: Status, open issues, and applications. Tech. Rep. CSC/08-07, Technische Universit¨at Chemnitz (2008)

Krylov-Subspace Based Model Reduction of Nonlinear Circuit Models Using Bilinear and Quadratic-Linear Approximations Peter Benner and Tobias Breiten

Abstract We discuss Krylov-subspace based model reduction techniques for nonlinear control systems. Since reduction procedures of existent approaches like TPWL and POD methods are input dependent, models that are subject to variable excitations might not be sufficiently approximated. We will overcome this problem by generalizing Krylov-subspace methods known from linear systems to a more general class of bilinear and quadratic-bilinear systems, respectively. As has recently been shown, a lot of nonlinear dynamics can be represented by the latter systems. We will explain advantages and disadvantages of the different approaches and discuss the choice of reasonable interpolation points with regard to optimal approximation results. A nonlinear RC circuit will serve as a numerical test example.

1 Introduction An important tool in the analysis of complex physical phenomena is the simulation of the underlying mathematical models which are often given by systems of ordinary and/or partial differential equations. As one is interested in models as accurate as possible, linear models are often insufficient such that one is faced with large-scale nonlinear systems. Frequently, these cannot be handled efficiently, necessitating model order reduction, i.e., the construction of a smaller system approximating the original one. Here, we consider nonlinear control-affine systems ˙W

x.t/ P D f .x.t// C Bu.t/;

y.t/ D C x.t/;

x.0/ D x0 ;

(1)

P. Benner ()  T. Breiten Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany e-mail: [email protected]; [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 18, © Springer-Verlag Berlin Heidelberg 2012

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with f W Rn ! Rn nonlinear, B 2 Rnm ; C 2 Rpn ; x 2 Rn ; u 2 Rm ; y 2 Rp : We now want to construct a reduced system ˙O W

PO O x.t/ D fO.x.t// O C Bu.t/;

y.t/ O D CO x.t/; O

x.0/ O D xO 0 ;

O ; C 2 RpnO ; x 2 RnO ; u 2 Rm ; yO 2 Rp ; nO  n with fO W RnO ! RnO , B 2 Rnm s.t. yO  y for all admissible u. While most existing nonlinear reduction methods like POD and TPWL, see [1, 2], require specific training inputs and thus are input dependent, we will discuss two Krylov-based techniques which overcome this drawback. These are based on the approximation of ˙ by a corresponding bilinear system or on representing ˙ as quadratic-bilinear system.

2 Bilinear Control Systems A common approach for the reduction of nonlinear systems of the form (1) is to first approximate them by systems with a simpler structure. Due to reasons of accuracy, instead of linearizing f around an operating point, we use Carleman bilinerization. The basic idea is to incorporate higher order terms of the Taylor expansion of f; leading to more accurate approximations, see [3], using a bilinear control system ˙W

x.t/ P D Ax.t/ C

m X

Nj x.t/uj .t/ C Bu.t/;

y.t/ D C x.t/;

x.0/ D x0 ;

j D1

where A; Nj 2 Rnn ; B 2 Rnm ; C 2 Rpn ; x 2 Rn ; u 2 Rm ; y 2 Rp : Though these systems have been studied throughout several decades now, recently an increased interest in the generalization of successful linear reduction techniques to bilinear systems has emerged, e.g. [4–6]. For simplicity, in the following we want to stick to the SISO case, i.e. m D p D 1; u.t/ D u1 .t/; N1 D N: As is well-known from [3], the output of a bilinear system can be described by a Volterra series, generalizing the impulse response of a linear system, as y.t/D

1 Z tZ X j D1 0

Z

t1

tj 1

::: 0

h.t1 ; : : : ; tj /u.t  t1  : : :  tj /    u.t  tj /dtj    dt1 ;

0

(2) with kernels h.t1 ; : : : ; tj / D C e Atj N    e At2 Ne At1 B: Passing to frequency domain by multivariable Laplace transform yields generalized transfer functions H.s1 ; : : : ; sj / D C.sj I  A/1 N    .s2 I  A/1 N.s1 I  A/1 B: The following statement now generalizes the rational interpolation problem known for linear system theory (see [7] and references therein).

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Theorem 1. Let a bilinear system ˙ and f1 ; : : : ; q g 2 Cq ; r 2 N; be given. Assume that V and W are given as biorthogonal bases of the unions of the column spaces  V1 D .1 I  Vj D .1 I  W1 D .1 I  Wj D .1 I

  A/1 B; : : : ; .q I  A/1 B ;

  A/1 N Vj 1 ; : : : ; .q I  A/1 N Vj 1 ;   AT /1 C; : : : ; .q I  AT /1 C ;

j  r;

  AT /1 N T Wj 1 ; : : : ; .q I  AT /1 N T Wj 1 ;

(3)

j  r:

(4)

If ˙O is constructed by projection, i.e., AO D W T AV; nO D W T N V; BO D W T B; CO D C V , then the following Hermite interpolation conditions hold for si 2 f1 ; : : : ; q g: Hj .s1 ; : : : ; sj / D HO j .s1 ; : : : ; sj /; @ O @ Hj .s1 ; : : : ; sj /; Hj .s1 ; : : : ; sj / D @sk @sk

j  2r; j D 1; : : : ; r;

k D 1; : : : ; j:

As for linear systems, a crucial fact obviously is the choice of reasonable interpolation points. For this, let us take a look at the H2 -norm introduced in [8], which can alternatively be computed via transfer functions. Lemma 1. Let a bilinear system ˙ be given and let .A/ D f1 ; : : : ; n g denote the spectrum of A: Then the H2 -norm of ˙ is given as jj˙jj2H2 D

n 1 X X j D1 `j D1



n X

  ˚`1 ;:::;`j Hj `1 ; : : : ; `j ;

`1 D1

where ˚`1 ;:::;`j D lim Hj .s1 ; : : : ; sj /.s1 `1 /    .sj `j / denotes a generalized sk !`k

residue associated with the j th transfer function. A careful analysis of the above expression and the derivation of the H2 -norm of the error system now lead to a slight change of the successful iterative rational Krylov algorithm (IRKA) proposed in [7]—due to space limitation, we only provide the resulting method as Algorithm 1.

3 Quadratic-Bilinear Control Systems Let us now switch to quadratic-bilinear (QB) control systems, given as follows x.t/DA P 1 x.t/CA2 x.t/˝x.t/CN x.t/u.t/CBu.t/;

y.t/ D C x.t/; x.0/ D x0 ;

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Algorithm 1 Bilinear Iterative Rational Krylov Algorithm (Bilinear-IRKA) INPUT: A, N , B, C , r, q O NO , B, O CO OUTPUT: A, 1: Make an initial selection S WD f1 ; : : : ; q g: 2: while (change in S > ) do r 3: Compute V D ŒV1 ; : : : ; Vr  and W D ŒW1 ; : : : ; Wr  2 Rn.qCCq / as in (3,4). 4: Compute truncated SVD Vq and Wq of V and W: O 5: AO D .WqT Vq /1 WqT AVq ; S .A/ 6: end while 7: NO D .WqT Vq /1 WqT N Vq ; BO D .WqT Vq /1 WqT B; CO D C Vq :

2

where A1 ; N 2 Rnn ; A2 2 Rnn ; B; C T 2 Rn : The advantage of QB systems is that they allow the exact representation of a large class of nonlinear systems. In contrast to this, Carleman bilinearization employing the quadratic term of the Taylor expansion of the nonlinear system leads to a bilinear system which is only an approximation of the original system. Moreover, the quadratic-bilinearization is often achieved with an acceptable increase of the state dimension (e.g., doubling or tripling the original system dimension) in contrast to Carleman bilinearization leading to a system of . C 1/-times the size  of the original model. The idea using quadratic-bilinear systems emerged quite recently and can be found in [9]. Due to space limitations, we only provide an example illustrating the basic idea. Example 1. Let us consider a two-dimensional nonlinear control system. xP 1 D exp.x2 / 

q

x12 C 1;

xP 2 D sin x2 C u

Next, we introduce new state variables in order to get rid of the exponential, root and trigonometric functions: z1 WD exp.x2 /;

z2 WD

q

x12 C 1;

z3 WD sin x2 ;

z4 WD cos x2 :

This allows to construct a six-dimensional quadratic-bilinear system: xP 1 D z1  z2 ; zP2 D

2  x1  z1  z2 2  z2

z2 .t />0 8t

D

xP 2 D z3 C u; x1  z1 ;

zP1 D z1  .z3 C u/;

zP3 D z4  .z3 C u/;

zP4 D z3  .z3 C u/:

Note that in the above example, we have used successive differentiations in order to perform the transformation. In general, adding algebraic equations might be necessary as well, see [9]. An efficient tool for understanding nonlinear systems is variational analysis, see [3]. For this, we consider an input of the form ˛u.t/ and further assume the system to be given by a series of homogeneous subsystems, i.e. the response should be of

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Algorithm 2 Quadratic-Bilinear Model Order Reduction INPUT: A1 , A2 , N , B, C , q1 , q2 ,  O CO OUTPUT: AO1 , AO2 , NO , B,  1: U D Kq1 .A1  I /1 ; .A1  I /1 B 2: for i D 1 to q2 do   fWi denoting i -th column of W g 3: Wi D Kq2 iC1 .A1  2I /1 ; .A1  2I /1 N Ui 4: for j D 1 to min.q2   i C 1; i / do1  5: Zi D Kq2 ij C2 .A1  2I / ; .A1  2I /1 A2 .Ui ˝ Uj C Uj ˝ Ui / 6: end for 7: end for 8: V D orth.ŒU; W; Z/ 9: AO1 D V T A1 V; AO2 D V T A2 V ˝ V; NO D V T N V; BO D V T B; CO T D V T C:

the form x.t/ D ˛x1 .t/ C ˛ 2 x2 .t/ C ˛ 3 x3 .t/ C : : : . Finally, a comparison of the coefficients leads to a series of coupled linear systems xP 1 D A1 x1 C Bu;

xP 2 D A1 x2 C A2 x1 ˝ x1 C N x1 u;

xP 3 D A1 x3 C A2 .x1 ˝ x2 C x2 ˝ x1 / C N x2 u;

:::

Although an explicit solution formula similar to (2) has not been proposed so far, the growing exponential approach allows a characterization in the frequency domain via generalized transfer functions as well. Here, we will only state the first two transfer functions: H1 .s1 / D C.s1 I  A1 /1 B DW C G1 .s1 / and H2 .s1 ; s2 / D

1 C ..s1 C s2 /I  A1 /1 ŒN .G1 .s1 / C G1 .s2 // 2Š CA2 .G1 .s1 / ˝ G1 .s2 / C G1 .s2 / ˝ G1 .s1 // :

Analogous to the bilinear case, multivariable expansions about a specific interpolation point allow the characterization via multimoments which can be matched by the reduced system if the Krylov subspaces are incorporated in the projection subspace as in Algorithm 2. Note the changes in the shift of the Krylov matrix for the second transfer function, as well as the symmetric starting vectors which are missing in [9].

4 Numerical Example We will now judge the quality of the two different approaches by means of a standard numerical test example. For a detailed model description of the investigated nonlinear RC circuit, we refer to [4]. The circuit consists of  resistors with voltage dependency g.v/ D exp.40v/ C v  1: While a Carleman linearization up to second order leads to a bilinearized system of dimension  C  2 ; there exists a clever quadratic-bilinearization which only yields a state dimension of 2  ; see [9].

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10−2 10− 3

IRKA points, ˆn = 4 + 4·4 B-IRKA points, ˆn = 21

10−3

Relative error

Relative error

10−2

−4

10

10− 4 10− 5 10−6

10−5 0

0.2 0.4 0.6 0.8

1 Time t

1.2 1.4 1.6 1.8

2

10−7

0

0.2 0.4

0.6 0.8

1

1.2

1.4 1.6 1.8

2

Time t

Fig. 1 Left. Relative errors for bilinearized system with  D 100. Right. Quadratic-bilinear system with  D 1; 000 and interpolation point  D 1

Figure 1 shows the relative error between the reduced bilinear system and the bilinearized system of order C 2 : On the other hand, the reduced quadratic-bilinear system is directly compared to the original nonlinear one. For the bilinear system, we compare the choice of 21 interpolation points computed by Algorithm 1 with 4 interpolation points obtained by IRKA which were used to construct a reduced bilinear system interpolating 4 values of the first transfer function and 4  4 values of the second transfer function leading to a reduced system of similar size. For implementation details see, e.g., [5]. At least in our example, the new approach leads to better results. On the other hand, for the QB system, we compare the approximations around a specific interpolation point . D 1/ for a varying number of matched derivatives of the first and second transfer functions, respectively. Note that the moderate increase of the state dimension allows to reduce a circuit consisting of 1; 000 resistors for which a bilinearization up to second order would have lead to a bilinearized system of unmanageable size.

5 Conclusions We have discussed two Krylov-based model reduction techniques applicable to a class of general nonlinear systems. While the Carleman-based approach might easily extend manageable system dimensions, the transformation to a quadraticbilinear system seems to be an interesting alternative to TPWL and POD. However, the choice of the interpolation points and the possibility of using two-sided projection methods seem to be tricky and thus should be further investigated.

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References 1. Chaturantabut, S., Sorensen, D.C.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010) 2. Rewienski, M.: A trajectory piecewise-linear approach to model order reduction of nonlinear dynamical systems. Ph.D. thesis, Massachusetts Institute of Technology (2003) 3. Rugh, W.: Nonlinear System Theory – The Volterra/Wiener Approach. The Johns Hopkins University Press, Baltimore (1981) 4. Bai, Z., Skoogh, D.: A projection method for model reduction of bilinear dynamical systems. Lin. Algebra Appl. 415(2–3), 406–425 (2006) 5. Breiten, T., Damm, T.: Krylov subspace methods for model order reduction of bilinear control systems. Syst. Contr. Lett. 59(8), 443–450 (2010) 6. Benner, P., Damm, T.: Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems. SIAM J. Contr. Optim. 49(2), 686–711 (2011) DOI: 10.1137/09075041X. 7. Gugercin, S., Antoulas, A., Beattie, C.: H2 model reduction for large-scale dynamical systems. SIAM J. Matrix Anal. Appl. 30(2), 609–638 (2008) 8. Zhang, L., Lam, J.: On H2 model reduction of bilinear systems. Automatica 38(2), 205–216 (2002) 9. Gu, C.: QLMOR: A new projection-based approach for nonlinear model order reduction. In: ICCAD ’09: Proc. Intl. Conf. CAD, pp. 389–396. ACM, New York (2009)

POD Model Order Reduction of Electrical Networks with Semiconductors Modeled by the Transient Drift–Diffusion Equations Michael Hinze, Martin Kunkel, and Ulrich Matthes

Abstract We consider POD model order reduction (MOR) of integrated circuits with semiconductors modeled by the transient drift-diffusion equations (DDEs). Discretization of the DDEs with mixed finite elements in space yields a high dimensional DAE. We sketch how POD, and POD combined with discrete empirical interpolation (DEIM) can be used to reduce the dimension of the model.

AMS subject classifications: 93A30, 65B99, 65M60, 65M20

1 Introduction In this article we investigate a POD-based MOR for semiconductors in electrical networks. In [9] POD-MOR is proposed to obtain a reduced surrogate model conserving as much of the DDEs structure as possible in the reduced order model. This approach in [8] is extended to parametrized electrical networks using the greedy sampling proposed in [11]. Advantage of the POD approach are the higher accuracy of the model and fewer model parameters. On the other hand, numerical simulations are more expensive. For a comprehensive overview of the drift-diffusion equations we refer to [4, 10, 13].

M. Hinze ()  U. Matthes Department of Mathematics, University of Hamburg, Bundesstr. 55, 20146 Hamburg, Germany e-mail: [email protected]; [email protected] M. Kunkel Institute of Mathematics, University of W¨urzburg, Am Hubland, 97074 W¨urzburg, Germany e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 19, © Springer-Verlag Berlin Heidelberg 2012

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The scaled DDEs are given by 

D n  p  C;

(1)

@t n C n div Jn D R.n; p/;

(2)

@t p C p div Jp D R.n; p/;

(3)

Jn D rn  nr ;

(4)

Jp D rp  pr ;

(5) U 

"UT with constants  WD L2 qkC , n WD UTL2 n and p WD TL2 p , where L denotes k1 a specific length of the semiconductor, see e.g. [13]. Semiconductors in electrical networks obtained by a modified nodal analysis are now modeled by the time– discrete version of (1)–(5), which results in a partial DAE of the form

Problem 1 (full model). AC

d > qC .A> C e.t/; t/ C AR g.AR e.t/; t/ C AL jL .t/ C AV jV .t/ dt C AS jS .t/ C AI is .t/ D 0;

0

0

(6)

d L .jL .t/; t/  A> L e.t/ D 0; dt

(7)

A> V e.t/  vs .t/ D 0;

(8)

jS .t/  C1 Jn .t/  C2 Jp .t/  C3 gP .t/ D 0;

(9)

0

1

B M n.t/ C LP B C B C P C B ML p.t/ B C C AFEM B C 0 B C @ A 0 0

1 .t/ B n.t/ C B C B C B p.t/ C B C C F .nh ; p h ; g h /  b.e.t// D 0; Bg .t/C B C @ Jn .t/ A Jp .t/

(10)

where (9) represents the discretized linear coupling condition (11) Z jS;k D

.Jn C Jp  "@t r /   d: O;k

(11)

POD Model Order Reduction of Elctrical Networks with Semiconductors e1(t)

Fig. 1 Basic test circuit with one diode. The network is described by

163 e2(t)

jV (t)

y (t,x) n (t,x)

ns(t)

AV D

p(t,x)



1;

 AS D 1;  AR D 0; g.A> R e; t/ D

0 1 1

> > >

R

; ; ;

1 e2 .t/ R

Here, e denotes the node potentials, and jL and jV the currents of inductive and voltage source branches, respectively. The electrostatic potential is denoted by .t; x/, the electron and hole concentrations by n.t; x/ and p.t; x/, and the current densities by Jn .t; x/ and Jp .t; x/. q is the elementary charge, " the dielectricity, n and p are the mobilities of electrons and holes. The temperature is assumed to be constant which leads to a constant thermal voltage UT . The function C is the time independent doping profile. We focus on the Shockley-ReadHall recombination. Furthermore, the incidence matrix A D ŒAR ; AC ; AL ; AV ; AI represents the network topology, e.g. at each non mass node i , aij D 1 if the branch j leaves node i and aij D  1 if the branch j enters node i and aij D 0 elsewhere. qC , g and L are continuously differentiable functions defining the voltage-current relations of the network components. The continuous functions vs and is are the voltage and current sources. For a basic example consider the network in Fig. 1. Further details are given in [8]. The analytical and numerical analysis of systems of this form is subject to current research, see [3, 7, 14, 16].

2 Model Reduction We use POD-MOR applied to the DD part (10) to construct a dimension-reduced surrogate model for (6)–(10). For this purpose we run a simulation of the unreduced system and collect l snapshots h .tk ; /, nh .tk ; /, p h .tk ; /, g h .tk ; /, Jnh .tk ; /, Jph .tk ; / at time instances tk 2 ft1 ; : : : ; tl g  Œ0; T . We use the time instances

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delivered by the DAE integrator. The snapshot variant of POD introduced in [15] finds a best approximation of the space spanned by the snapshots w.r.t. to the considered scalar product. Since every component of the state vector z WD . ; n; p; g ; Jn ; Jp / has its own physical meaning we apply POD MOR to each component separately. The time-snapshot POD procedure now delivers Galerkin ansatz spaces for , n, p, g , Jn , and Jp and we set POD .t/ WD U .t/, nPOD .t/ WD Un n .t/; : : :. The injection matrices U 2 RN s , Un 2 RN sn ; : : :, contain the (time independent) POD basis functions, and the vectors ./ the corresponding time-variant coefficients. The numbers s./ denote the respective number of POD basis functions included. Assembling the POD system yields a reduced model with similar structure as (6)–(10), see [8] for details. All matrix-matrix multiplications are calculated in an offline-phase. The nonlinear functional F has to be evaluated online, the arguments have to be interpreted as functions in space. For the reduction of the nonlinearity we use DEIM proposed in [5], see also [2].

3 Numerical Implementation and Results The FEM is implemented in C++ based on the finite element library deal.II [1]. The high dimensional DAE is integrated using the DASPK software package [12]. We assume that the differentiation index of the network is 1. Otherwise one should switch to alternative integrators. The derivative of the nonlinear functional F with respect to nj .t/, pj .t/, g ;j .t/ is difficult to compute and thus we calculate the Jacobians by automatic differentiation with the package ADOL-C [17]. We implement the preconditioning subroutine of DASPK using SuperLU [6]. A basic test circuit with a single 1-dimensional diode is depicted in Fig. 1. The parameters of the diode are summarized in [8]. The input vs .t/ is chosen to be sinusoidal with amplitude 5 ŒV . In the sequel the frequency of the voltage source will be considered as a model parameter. Figure 2 validates the POD reduced and the POD-DEIM reduced model at the reference frequency of 5109 ŒH z w.r.t. the lack of information . It shows that both reduction techniques perform equally well. The number of POD and DEIM-POD basis functions s./ for each variable is chosen such that the indicated approximation quality is reached, i.e.  WD  ' n ' p ' g ' Jn ' Jp : In Fig. 3 the simulation times are plotted versus the lack of information . The POD reduced order model does not reduce the simulation times significantly for the chosen parameters. The reason for this is the dependency on the number of variables of the unreduced system. Here, the unreduced system contains 1,000 finite elements which yields 12,012 unknowns. The POD-DEIM reduced order model behaves very well and leads to a reduction in simulation time of about 90% without reducing the accuracy of the reduced model. However, we have to report a minor drawback; not all tested reduced models converge for large .s/  3  105 . This is indicated in

rel. L2−error of output jV

POD Model Order Reduction of Elctrical Networks with Semiconductors

165

100

POD 10 – 5 10 –7

DEIM 10 –6

10 –5

10 –4

10 –3

lack of information Δ(s)

Fig. 2 Relative error between reduced and unreduced problem at the fixed frequency 5  109 ŒH z 60 POD simulation time [sec]

50

DEIM unreduced

40 30 20 10 0 10 –7

10– 6

10– 5

10–4

10–3

lack of information Δ(s)

Fig. 3 Time consumption for simulation runs of Fig. 2. The horizontal line indicates the time consumption for the simulation of the original full system

the figures by missing squares. This effect is even more pronounced for spatially two-dimensional semiconductors. In Fig. 4 we plot the corresponding total number of required POD basis functions. It can be seen that with the number of POD basis functions increasing linearly, the lack of information tends to zero exponentially. Furthermore, the number of DEIM interpolation indices behaves in the same way. In Fig. 5 we investigate the dependence of the reduced models on the number of finite elements N . One sees that the simulation times of the unreduced model depends linearly on N . The POD reduced order model still depends on N linearly with a smaller constant. The dependence on N of our DEIM-POD implementation is negligible.

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number of basis functions

250 POD 200

DEIM

150 100 50 0 10–7

10–6

10 –5 10 –4 lack of information Δ(s)

10 –3

Fig. 4 The number of required POD basis function and DEIM interpolation indices grows only logarithmically with the requested information content 150

simulation time [sec]

POD DEIM unreduced

100

50

0

0

500

1000

1500

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number of finite elements

Fig. 5 Computation times of the unreduced and the reduced order models plotted versus the number of finite elements

Finally, we in Fig. 6 analyze the behaviour of the models with respect to parameter changes. We consider the frequency of the sinusoidal input voltage as model parameter. The reduced order models are created based on snapshots gathered in a full simulation at a frequency of 5  109 ŒH z . We see that the POD model and the POD-DEIM model behave very similar. The adaptive enlargement of the POD basis using the residual greedy approach of [11] is discussed in [8]. Summarizing all numerical results we conclude that the significantly faster PODDEIM reduction method yields a reduced order model with the same qualitative behaviour as the reduced model obtained by classical POD-MOR.

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10–2 rel. L2− error of output jV

POD DEIM 10–3

10–4

108

1010

1012

frequency [Hz]

Fig. 6 The reduced models are compared with the unreduced model at various input frequencies Acknowledgements The work reported in this paper was supported by the German Federal Ministry of Education and Research (BMBF), grant no. 03HIPAE5. Responsibility for the contents of this publication rests with the authors.

References 1. Bangerth, W., Hartmann, R., Kanschat, G.: deal.II – a general-purpose object-oriented finite element library. ACM Trans. Math. Softw. 33(4) (2007) 2. Barrault, M., Maday, Y., Nguyen, N.C., Patera, A.T.: An empirical interpolation method: Application to efficient reduced-basis discretization of partial differential equations. C.R. Acad. Sci. Paris, Serie I 339, 667–672 (2004) 3. Bodestedt, M., Tischendorf, C.: PDAE models of integrated circuits and index analysis. Math. Comput. Model. Dyn. Syst. 13(1), 1–17 (2007) 4. Brezzi, F., Marini, L., Micheletti, S., Pietra, P., Sacco, R., Wang, S.: Discretization of semiconductor device problems. I. In: Schilders, W.H.A., et al. (eds.) Handbook of Numerical Analysis, vol. XIII, pp. 317–441. Special Volume: Numerical Methods in Electromagnetics. Elsevier, Amsterdam. (2005) 5. Chaturantabut, S., Sorensen, D.C.: Discrete empirical interpolation for nonlinear model reduction. Tech. Rep. 09-05, Department of Computational and Applied Mathematics, Rice University (2009) 6. Demmel, J.W., Eisenstat, S.C., Gilbert, J.R., Li, X.S., Liu, J.W.H.: A supernodal approach to sparse partial pivoting. SIAM J. Matrix Anal. Appl. 20(3), 720–755 (1999) 7. G¨unther, M.: Partielle differential-algebraische Systeme in der numerischen Zeitbereichsanalyse elektrischer Schaltungen. VDI Fortschritts-Berichte, Reihe 20, Nr. 343 (2001) 8. Hinze, M., Kunkel, M.: Residual based sampling in POD model order reduction of driftdiffusion equations in parameterized electrical networks. Z. Angew. Math. Mech. 92(2), 91–104 (2012) 9. Hinze, M., Kunkel, M., Vierling, M.: POD model order reduction of drift-diffusion equations in electrical networks. In: Benner, P., Hinze, M., Ter Maten, E.J.W. (eds.) Lecture Notes in Electrical Engineering, vol. 74, pp. 177-192. Springer, New York (2011) 10. Markowich, P.: The Stationary Semiconductor Device Equations. Computational Microelectronics. Springer, New York (1986)

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11. Patera, A., Rozza, G.: Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations. version 1.0. Copyright MIT 2006–2007, to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering (2007) 12. Petzold, L.R.: A description of DASSL: A differential/algebraic system solver. IMACS Trans. Sci. Comput. 1, 65–68 (1993) 13. Selberherr, S.: Analysis and Simulation of Semiconductor Devices. Springer, New York (1984) 14. Selva Soto, M., Tischendorf, C.: Numerical analysis of DAEs from coupled circuit and semiconductor simulation. Appl. Numer. Math. 53(2–4), 471–488 (2005) 15. Sirovich, L.: Turbulence and the dynamics of coherent structures I: Coherent structures. II: Symmetries and transformations. III: Dynamics and scaling. Q. Appl. Math. 45, 561–590 (1987) 16. Tischendorf, C.: Coupled Systems of Differential Algebraic and Partial Differential Equations in Circuit and Device Simulation. Habilitation thesis, Humboldt-University of Berlin (2003) 17. Walther, A., Griewank, A.: ADOL-C–Manual, ‘ADOL-C: a package for the automatic differentiation of algorithms written in C/C++’. https://projects.coin-or.org/ADOL-C

Model Order Reduction of Electrical Circuits with Nonlinear Elements Andreas Steinbrecher and Tatjana Stykel

Abstract In this paper, we present a model order reduction approach for nonlinear circuit equations. The approach is based on decoupling linear and nonlinear subcircuits of the electrical circuit, followed by model reduction of the linear part using a passivity-preserving balanced truncation technique. A numerical example is given.

1 Introduction The efficient and robust numerical simulation of electrical circuits plays a major role in computer aided design of electronic devices. While the structural size of such devices is decreasing, the complexity of the electrical circuits is increasing. This leads to a system of model equations in form of differential-algebraic equations (DAEs) with a huge number of unknowns. Simulation of such models is unacceptably time and storage consuming. Model order reduction presents a way out of this problem. A general idea of model reduction is to replace a large-scale system by a much smaller model which approximates the input-output relation of the original system within a required accuracy and preserves important system properties. There exists a large variety of model reduction techniques for linear circuit equations, e.g., [1, 3, 4]. For nonlinear systems, only a few methods including trajectory piecewise linearization (TPWL) and proper orthogonal decomposition (POD) have been developed, e.g., [5, 6]. A standard approach for model reduction of nonlinear circuits with large linear subnetworks is to extract linear subsystems and replace them by reduced-order models, e.g., [2, 3]. Although this approach is widely used in industrial practice, only little attention has been paid to the extraction

A. Steinbrecher ()  T. Stykel Institut f¨ur Mathematik, TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany e-mail: [email protected]; [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 20, © Springer-Verlag Berlin Heidelberg 2012

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procedure, approximation quality and properties of reduced-order nonlinear models. In this paper, we consider model reduction of general nonlinear circuits. We develop a topology based decoupling technique that avoids the increasing of the index and guarantees the well-posedness of decoupled linear subsystems.

2 Circuit Equations A commonly used tool for modeling electrical circuits is the Modified Nodal Analysis (MNA). An electrical circuit can be modeled as a directed graph whose nodes correspond to the nodes of the circuit and whose branches correspond to the circuit elements. Using Kirchhoff’s laws as well as the branch constitutive relations, the dynamics of an electrical circuit can be described by a DAE system (or descriptor system) of the form E .x/

d x D A x C f .x/ C B u; dt

y D B Tx;

(1a)

with 2 2 3 3 3 0 AL AV  AC C .ATC /ATC 0 0 6 6 7 7 6 0 L.L / 0 7 0 5; x D 4 L 5; E .x/ D 4 5; A D 4 ATL 0 0 0 0 ATV 0 0 V (1b) 2 3 2 3     AR g.ATR / AI 0 uI I 5 4 5 f .x/ D 4 ; y D  : ; B D ; u D 0 0 0 uV V 0 0 I (1c) 2

Here, x, u and y are descriptor vector, input and output, respectively,  is the vector of node potentials, L , V and I are the vectors of currents through inductors, voltage sources and current sources, respectively, uV and uI are the vectors of voltages of voltage sources and current sources, respectively. We will distinguish between linear circuit elements (denoted by a bar) that are characterized by linear current-voltage relations and nonlinear circuit components (denoted by a tilde) that are characterized by nonlinear current-voltage relations. Without loss of generality, we assume that the circuit elements are ordered such that the incidence matrices describing the circuit topology have the form AC D ŒACN ACQ  2 Rn; nCN CnCQ , n; nLN CnLQ n; nRN CnRQ AL D ŒAL , AR D ŒAR , AV 2 Rn; nV N ALQ  2 R N ARQ  2 R n; nI and AI 2 R , where the incidence matrices ACN , AL N and AR N correspond to the linear circuit components, and ACQ, ALQ and ARQ correspond to the nonlinear circuit components. Furthermore, the conductance matrix-valued function C W RnC ! RnC; nC , the inductance matrix-valued function L W RnL ! RnL; nL and the resistor

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relation g W RnR ! RnR given by "

C .ATC / D

CN

0

e.AT / 0 C Q C

"

# ; L.L / D

N L

0

e .LQ / 0 L

"

# ; g.ATR / D

T GNAR N 

#

e g .ATQ / R

(1d) characterize the physical properties of the capacitors, inductors and resistors, respectively. Here LQ 2 RnLQ is the vector of currents through the nonlinear inductors. We will assume that the matrices AV and ŒAC AL AR AV  have full rank, the matrices C .ATC / and L.L / are symmetric, uniformly positive definite and the function g.ATR / is monotonically increasing for all admissible  and L . These assumptions imply that the circuit elements do not generate energy, i.e., the circuit is passive.

3 Model Reduction for Nonlinear Circuits In this section, we present a model reduction technique for nonlinear circuits with a small number of nonlinear elements. This technique is based on decoupling of the linear and nonlinear subcircuits in a suitable way, reduction of the linear part using the PABTEC method [4] followed by an adequate recoupling of the unchanged nonlinear part and the reduced linear part to obtain a nonlinear reduced-order model.

3.1 Decoupling of Linear and Nonlinear Subcircuits Our goal is to extract a linear subcircuit from a nonlinear circuit. For this purpose, we replace nonlinear inductors and nonlinear capacitors by controlled current sources and controlled voltage sources, respectively. The nonlinear resistors are replaced by an equivalent circuit consisting of two serial linear resistors and one controlled current source connected parallel to one of the resistors. Such replacements guarantee that neither additional loops of capacitors and voltage sources nor cutsets of inductors and/or current sources occur in the decoupled linear subcircuit meaning that its index coincides with the index of the original circuit. On the equation level these replacements result in the following coupled system of linear and nonlinear equations. Consider the circuit equations (1). Let ARQ DA1Q CA2Q be decomposed such that A1Q 2 f0; 1gn; nRQ and A2Q 2 f1; 0gn; nRQ , R R R R and let the matrices G1 ; G2 2 RnRQ ;nRQ be given such that G1 and G2 are both symmetric, positive definite. Introduce new functions uCQ D ATCQ;

z D .G1 C G2 /G11e g .ATRQ /  G2 ATRQ :

(2)

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Then system (1) together with the relations z D .G1 C G2 /1 ..G1 .A1RQ /T  G2 .A2RQ /T /  z /; e.uCQ/ CQ D C

d u Q dt C

(3a) (3b)

for the additional unknowns z 2 RnRQ and CQ 2 RnCQ is equivalent to the system d e L .LQ / LQ D ATLQ ; dt

(4)

coupled with the linear system E

d x` D Ax` C Bu` ; dt

y` D B T x` :

(5a)

Here, the system matrices are in the MNA form 3 AC CATC 0 0 7 6 E D4 0 L 0 5; 0 0 0 2

3 3 2 AI 0 AR GATR AL AV 7 7 6 6 AD4 0 5 ATL 0 0 5; B D 4 0 0 I 0 0 ATV (5b) 2

with incidence and element matrices " #       1 2 AR ACN AN AV ACQ N ARQ ARQ ; AR D ; AV D ;(5c) AC D ; AL D L 0 0 0 0 0 I I # " AI A2RQ ALQ N ; G D diag.GN; G1 ; G2 /; AI D ; C D CN; L D L (5d) 0 I 0 i i h h T T T uT` D TI Tz TLQ uTV uTCQ , y`T D and x`T D T Tz L N V CQ ,  T T T T T y1 y2 y3 y4 y5 . With (2), equivalence here means that Œx T Tz TCQT solves (1) and (3) if and only if Œx`T T Q T solves (5) and (4). More details and the proof can L be found in [7].

3.2 Model-Order Reduction of the Linear Subsystem We now apply the PABTEC method [4] to the linear system (5) with a transfer function G .s/ D B T .sE A/1 B. The assumptions above on the nonlinear system

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(1) guarantee that the following projected Lur’e equations E X.A  BBT /T C .A  BBT /XET C 2Pl BBTPlT D 2Kc KcT ; EXB  Pl BMT0 D Kc JcT ;

X D Pr XPTr

Jc JcT D I  M0 M0T ;

(6)

and E T Y .A  BBT / C .A  BBT /T YE C 2PrTBBTPr D 2KoT Ko ;  E T YB C PrTBM0 D KoT Jo ;

JoT Jo D I  M0T M0 ;

Y D PlT YPl

(7)

are solvable for X , Kc , Jc and Y , Ko , Jo , respectively, Here Pr and Pl are the spectral projectors onto the right and left deflating subspaces of E  .A  BBT / corresponding to the finite eigenvalues and M0 D I 2 lims!1 B T .sE ACBBT /1 B. The minimal solutions Xmin and Ymin of (6) and (7) that satisfy 0  Xmin  X and 0  Ymin  Y for all symmetric solutions X and Y of (6) and (7), are called the controllability and observability Gramians of system (5). Using the block structure of the system matrices (5b), we can show that Pl D Sint Pr Sint and Ymin DSint Xmin Sint with Sint D diag.In CnRQ ; InLN ; InV CnCQ /. Model reduction consists in approximating the large-scale DAE system (5) of order n` D n C nRQ C nL N C nV C nCQ by a reduced-order model d O `; EO xO ` D AOxO ` C Bu dt

yO D CO xO ` ;

(8)

where xO ` 2 Rr and r  n` . It is required that the approximate system (8) captures the input-output behavior of (5) to a required accuracy and preserves passivity. The PABTEC model reduction method is based on transforming system (5) into a balanced form whose controllability and observability Gramians are equal and diagonal. Then a reduced-order model is computed by truncating the states corresponding to the small diagonal elements of the balanced Gramians. We summarize the PABTEC method in Algorithm 1. One can show that the reduced model computed by the PABTEC method preserves passivity and we have the error bound e  G kH1  2kI C G k2H .rf C1 C : : : C q /; kG 1 provided 2kI C G kH1 .rf C1 C : : : C q / < 1, see [4] for details. If Dc D I  M0 M0T is nonsingular, then Do D I  M0T M0 is also nonsingular and the projected Lur’e equation (6) is equivalent to the projected Riccati equation Pl HPTl C FXFT C EXET C EXQXET D 0;

X D Pr XPTr

(9)

1 T 1 T T T with F D A  BBT  2Pl BD1 o M0 B Pr , H D 2BDo B and Q D 2BDc B . Such an equation can be solved via Newton’s method [4]. Note that the matrix M0 and the projectors Pl and Pr required in (9) can be constructed in explicit form

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Algorithm 1 Passivity-preserving balanced truncation for electrical circuits (PABTEC) 1. Compute the Cholesky factor R of the minimal solution Xmin D RRT of (6). 2. Compute the eigenvalue decompositions RTSint ER D ŒU1 U2 diag.1 ; 2 /ŒV1 V2 T and .I  M0 /Sext D U0 0 U0T with Sext D diag.InI CnRQ CnLQ ; InV CnCQ /, where ŒU1 U2 , ŒV1 V2  and U0 are orthogonal, 1 D diag.1 ; : : : ; rf /, 2 D diag.rf C1 ; : : : ; q / and O 1; : : : ;  O m /. 0 D diag. 3. Compute the reduced-order system (8) with  I 0 EO D ; 0 0 

# " p   T T 1 W T Bp A T 2 W BC 2 W 1 p OD AO D ; B ; B1 = 2 2  2 B1 B T T 2 I  B1 C1   TTB p ; CO T D T C1 = 2

where B1 D S0 j0 j1=2 U0T Sext ; C1 D U0 j0 j1=2 ; O 1 /; : : : ; sign. O m //; j0 j D diag.j O 1 j; : : : ; j O m j/; W D Sint R U1 j1 j1=2 ; S0 D diag.sign. T D R U1 S1 j1 j1=2 ; S1 D diag.sign.1 /; : : : ; sign.rf //; j1 j D diag.j1 j; : : : ; jrf j/:

exploiting the topological structure of the circuit equations (5), see [4]. For largescale problems, the numerical solution of projected Lur’e equations is currently under investigation.

3.3 Recoupling the Nonlinear and Reduced Linear Subsystems     Let BO D BO1 BO 2 BO 3 BO 4 BO 5 and CO T D CO 1T CO 2T CO 3T CO 4T CO 5T in (8) be partitioned in blocks according to u` and y` , respectively. Since the vector yO` D CO xO ` is an approximation to the output vector y` of system (5), we have .A2RQ /T   z  CO 2 xO ` ;

ATLQ   CO 3 xO ` ;

 CQ  CO 5 xO ` :

(10)

Then (4) and (3b) are approximated by d e L .OLQ / OLQ D CO 3 xO ` ; dt

d dt

Ce.OuCQ/ uO CQ D CO 5 xO ` ;

(11)

respectively, where OLQ and uO CQ are approximations to LQ and uCQ, respectively. Furthermore, for z defined in (2), z defined in (3a) and uRQ D ATRQ  2 RnRQ , we have g .uRQ /  G2 uRQ ; z D .G1 C G2 /G11e .A2RQ /T   z D ATRQ  C G11e g .ATRQ / D uRQ C G11e g .uRQ /:

(12)

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Then the first approximation in (10) together with (12) imply the relation g .OuRQ /; 0 D G1 CO 2 xO `  G1 uO RQ C e

(13)

where uO RQ approximates uRQ . Combining (8), (11) and (13) and adding to xO ` also the approximations OLQ , uO CQ, and uO RQ as state variables, we get the reduced-order nonlinear DAE system d O EO .x/ O xO D AOxO C fO.x/ O C Bu;

yO D COxO

dt

with 2O 3 2O A C BO 2 .G1 C G2 /CO 2 E 0 0 0 6 7 60 L e .OLQ / 0 07 CO 3 6 6 EO .x/ O D6 7; AO D 6 4 40 0 CO 5 Ce.OuCQ/ 0 5 0 0 0 0 G1 CO 2 2

2 O 3 0 B1 6 0 7 6 0 7 O 6 fO.x/ O D6 4 0 5; B D 4 0 e g .OuRQ / 0

2 OT 3 C1 BO 4 6 0 7 7; COT D 6 0 4 0 0 5 0 0

CO 4T 0 0 0

3 BO3 BO 5 BO 2 G1 0 0 0 7 7 7; 0 0 0 5 0

0 G1

2

3 xO `   6 7 7 7; xO D 6OLQ 7; yO D yO1 4uO Q 5 5 yO4 C uO RQ 3

that approximates the original nonlinear system (1). This reduced model can now be used instead of (1) in the analysis of the dynamical behavior of the circuit.

4 Numerical Experiments Consider an electrical circuit with 1990 linear resistors, 10 diodes, 991 linear inductors, 10 nonlinear inductors, 2000 linear capacitors and 1 voltage source which is modeled by a DAE system (1) of dimension 4003. This system was approximated by a reduced model of dimension 203. The numerical simulation of both models is done for t 2 Œ0; 0:05 s using the BDF method of order 2 with fixed stepsize of length 5105 . Table 1 shows the numerical results. In the upper plot, in Figure 1 we present the output y.t/ D V .t/ of the original system and the output y.t/ O D OV .t/ of the reduced system. In the lower plot, the error V .t/ D jy.t/ O  y.t/j is shown.

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Table 1 Numerical results dimension of the original nonlinear system simulation time for the original system (to )

4003 4557s

tolerance for model reduction of the linear subsystem time for model reduction

1e-05 822s

dimension of the reduced system simulation time for the reduced nonlinear system (tr )

203 67s

absolute error in the output speedup (to =tr )

4.4e-06 68.5

× 10−3 Output: negative current through voltage source orig. system red. system

−iV

15 10 5 0 0

0.01

0.02

0.03

0.04

0.05

0.04

0.05

t Error in the output

× 10−5

ΔiV

4

2

0

0

0.01

0.02

0.03 t

Fig. 1 Simulation results for the original and the reduced systems

References 1. Freund, R.: SPRIM: structure-preserving reduced-order interconnect macromodeling. In: Technical Digest of the 2004 IEEE/ACM International Conference on Computer-Aided Design, pp. 80–87. Los Alamos, CA (2004) 2. Heinkenschloss, M., Reis, T.: Model reduction for a class of nonlinear electrical circuits by reduction of linear subcircuits. Technical Report 702-2010, DFG Research Center M ATHEON , Technische Universit¨at Berlin (2010) 3. Odabasioglu, A., Celik, M., Pileggi, L.: PRIMA: Passive reduced-order interconnect macromodeling algorithm. IEEE Trans. Circ. Syst. 17(8), 645–654 (1998) 4. Reis, T., Stykel, T.: PABTEC: Passivity-preserving balanced truncation for electrical circuits. IEEE Trans. Comp.-Aided Des. Integrated Circ. Syst. 29(9), 1354–1367 (2010) 5. Rewie´nski, M.J.: A trajectory piecewise-linear approach to model order reduction of nonlinear dynamical systems. Ph.D. thesis, Massachusetts Institute of Technology (2003)

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6. Sirovich, L.: Turbulence and the dynamics of coherent structures. I: Coherent structures. II: Symmetries and transformations. III: Dynamics and scaling. Q. Appl. Math. 45, 561–590 (1987) 7. Steinbrecher, A., Stykel, T.: Model order reduction of nonlinear circuit equations. Preprint 2011/02, Technische Universit¨at Berlin (2011)

Structure-Exploiting Symbolic-Numerical Model Reduction of Nonlinear Electrical Circuits Oliver Schmidt

Abstract This paper presents new strategies for the analysis and reduction of systems of ever-growing size and complexity by exploiting the hierarchical structure of analog electronical circuits. Thereby, the entire circuit is considered as a system of interconnected subcircuits. A newly developed algorithm uses a novel concept of subsystem sensitivities and reduces the subcircuits separately from each other. Thus, a coupling of different reduction techniques is permitted. Finally, the practicability of the new techniques is demonstrated on an amplifier circuit example typically used in industrial applications.

1 Introduction In order to avoid immense time and financial effort for the production of deficiently designed prototypes of integrated circuits (ICs), industrial circuit design uses mathematical models and simulations in order to predict and analyse the physical behavior of electronical systems. Thereby, analog circuits are modelled by systems of DAEs, i.e. sets of differential equations with additional algebraic constraints composed of component characteristics and Kirchhoff laws. Nowadays, models of modern ICs contain up to several millions of equations. Since design verification requires a high number of simulation runs with different input excitations, model reduction has become inevitable. Besides lots of numerical reduction techniques, there also exist symbolic methods [2, 3, 6, 7]. They indeed are costly to compute, but allow deeper analytical insights into the behavior of the system by maintaining the system parameters in their symbolic form. The basic idea behind these methods is a stepwise reduction of the original system by comparing its

O. Schmidt () Fraunhofer ITWM, Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany e-mail: [email protected]; [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 21, © Springer-Verlag Berlin Heidelberg 2012

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reference solution to the solution of the so far reduced system. These comparisons are made by using error functions to measure the error between the two solutions. In order to couple different reduction techniques for the processing of systems of ever-growing size and complexity, new strategies, which exploit the hierarchical structure of the entire system, are presented in the following.

2 Hierarchical Modelling and Model Reduction In general, electronical circuits have a hierarchical structure: While the entire system consists of interconnected blocks such as amplifiers, current mirrors, or polarization circuits, each block itself is an interconnecting network of components like diodes, resistors, transistors, etc. Hence, the entire circuit is a network of coupled or interconnected subcircuits. Moreover, each subcircuit itself might have such a structure, which causes the hierarchy of the entire circuit. The main idea of the new hierarchical model reduction approach in this paper is the exploitation of this hierarchy for a coupling of different reduction techniques suitable for and in accordance with the modelling of corresponding subcircuits. This also allows for a faster processing of smaller subproblems if the administrative cost does not get out of hand. Furthermore, particularly in the case of symbolic model reduction, it makes larger circuits manageable at all. First experiments using this approach involved a differential-amplifier circuit and have been very promising [5]. Moreover, the obtained models proved to be very robust w.r.t. different input excitations and simulation speed-up. However, by reducing the subcircuits separately, only the errors at their terminals can be controlled, but not the error at the output of the entire system. Since in the general nonlinear case there is no relation between these errors available, a new algorithm has been developed that chooses a suitable degree of reduction for each subcircuit. This is achieved by measuring the influence of each subcircuit on the behavior of the entire circuit as described in the following section.

2.1 Subsystem Sensitivities Consider an analog circuit ˙, given as an interconnection S of k subsystems Ti by   ˙ D fTi j i D 1; : : : ; kg; S :

(1)

In order to measure the influence of a single subcircuit T on the behavior of ˙ excited by an input u, we proceed as follows. First of all, m reduced models of T of different accuracies are generated. Then T is successively replaced in ˙ by these reduced models. After each replacement, a simulation of the current configuration is

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performed and the global error at the output of the original system ˙ is measured by a suitable error function. The reduced models together with the corresponding errors are arranged in the subsystem sensitivity vector or simply the subsystem sensitivity: Algorithm 2 Heuristically reasonable order of subsystem replacements.

  Input: segmented electrical circuit ˙ D f Ti j i D 1; : : : ; k g; S , input u, error bound "   ri;j  ri;j  Output: reduced entire system ˙Q D f Ti j i D 1; : : : ; k g; S , where Ti are suitably reduced subsystems, E.y; y˙Q /  ", and where y˙Q is the output of ˙Q forall the subsystems Ti do Li WD order.sTi / w.r.t. E.y; yTi ;ri;j / r Ti i;0 WD Ti end L WD fL1 ; : : : ; Lk g y WD solve.˙; u/ ˙Q WD ˙ repeat compute .rp;q ; E.y; yTp ;rp;q // WD min .min.Li // w.r.t. E.y; yTi ;ri;j / rp;q

rp;q

i;Li 2L

replace current Tp 0 by Tp update.˙Q / y˙Q WD solve.˙Q ; u/ "out WD E.y; y˙Q / if "out  " then Lp WD Lp n fmin.Lp /g if Lp D ; then L WD L n fLp g else rp;q rp;q reset Tp to Tp 0 update.˙Q / L WD L n fLp g end until L D ;

Definition 1. Let an analog circuit ˙ be given by (1). Further, let reduction information rij such as “symbolic reduction, 10% error” or “Arnoldi reduction, iterate 5 steps” be given, that specifies how to reduce the i th subcircuit Ti in order to obtain its j th reduced model. Finally, let E be an error function for measuring the error on the reference solution y, i.e. the output of the original entire system ˙. Then the sensitivity of a subsystem1 Ti is defined by sTi D

    ri;1 ; E.y; yTi ;ri;1 / ; : : : ; ri;mi ; E.y; yTi ;ri;mi / :

(2)

In this notation, yTi ;ri;j is the solution of the configuration of ˙, where Ti is replaced by its j th reduced model obtained by applying the reduction technique encoded in ri;j , while the remaining subsystems are the original ones.

1

We will not further distinguish between “subcircuits” and “subsystems”.

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2.2 Subsystem Ranking In this section, the subsystem sensitivities are used to derive a heuristically reasonable order of subsystem replacements. Basically, the reduced models of all subsystems are ordered increasingly w.r.t. their errors on the global output of ˙. Then, the subsystem replacements are performed in the resulting order and the accumulated global error is checked by simulating the current configuration. If the user-given error bound " is exceeded, the current configuration is reset to the previous one that was best in list and the remaining reduced models of the “failing subcircuit” are removed from the list of candidates. Otherwise, the next model replacement is performed and the procedure is repeated. Algorithm 1 gives a more r detailed description, where Ti i;j denotes the j th reduced model of Ti . Following this procedure, the subsystems of ˙ are successively replaced by appropriate reduced models, while the global error on its output y is controlled.

3 Application Example The algorithms have been implemented in Analog Insydes [1], a symbolic analysis and reduction tool developed at the Fraunhofer ITWM. In this section, they are applied to a circuit example typically used in industry, namely, the operational amplifier op741. As Fig. 1 shows, it is divided into seven subsystems. More details, e.g. about their functionality, can be found in [4, Appendix C]. The aim is a symbolic reduction of the amplifier with a 10% overall error bound in time domain. The results of a direct non-hierarchical approach [2,7] are compared

CM1

Vid

CM3

DP

CM4

V$26

LS

CM2 DAR

PP

Fig. 1 Operational amplifier op741 composed of seven subsystems CM1–3, DP, DAR, LS, and PP, and with input source Vid and output V$26

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voltage [V]

voltage [V]

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5 0.0005 –0.5

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Fig. 2 Input excitation (left) and reference solution (middle) of the operational amplifier op741, as well as the outputs of both the original (right, solid) and the hybrid reduced systems (dotted)

to the new hierarchical reduction algorithm. The computations are performed with a sine wave voltage excitation of 0.8 V and 1 kHz frequency for the input source Vid on the time interval T D Œ0 s; 0:002 s. The voltage potential V$26 at node 26 defines the output of the entire system. The input together with the corresponding reference solution are shown in Fig. 2. Since the reference solution is pulse-shaped [4], a newly designed error function E  is used in order to cope with small delays in “jumps” of the solution (cf. [4, Sect. 4.6]). As a second error function, the L 2 -norm is used simultaneously. Furthermore, in order to obtain reduced models of the seven subcircuits with different accuracies, a sweep of thirteen error bounds is provided2: ˚  sw D 1%; 2%; 5%; 10%; 20%; : : : ; 90%; 100% :

(3)

By using the modified nodal analysis, one obtains a system containing 215 equations with 1;050 terms. Table 1 summarizes the results of the different reduction approaches including the time costs of the derivation of the respective reduced entire models. While “non-hierarchical” corresponds to the direct non-hierarchical method, “hierarchical” denotes a slightly improved version of the new algorithms described in this paper. The “hybrid” approach finally means a subsequent non-hierarchical reduction applied to the “interim solutions” of the hierarchically reduced systems permitting the same 10% error bound. This leads to a more significant reduction and faster simulation times, whereas this is not the case for a “double non-hierarchical” or a “double hierarchical” reduction. See [4] for details on the latter two methods. The graph on the right of Fig. 2 provides a qualitative impression of the results of the hybrid reduced systems. In order to check the robustness of the hierarchically reduced systems, some other input excitations are applied, namely, a sine wave with 3 kHz frequency, a sum of sine waves with frequencies of 250, 500, and 2,000 Hz, and a pulse excitation of 250 Hz. The output curves of the hybrid reduced models coincide almost perfectly, cf. [4]. Moreover, the speed-up in simulation time is up to a factor of 19, see Table 2. It shows selected systems identified by their number of equations and terms.

2

Note that very low tolerances <1% lead to almost no reduction for any of the seven subsystems.

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Table 1 Overview of the results of the reduction of the operational amplifier op741. The computations were performed on a machine with 8 Quad-core AMD Opteron 8384 “Shanghai” (32 cores in total) with 2:7 GHz and 512 GB RAM on a SuSE Linux 10.1 system

Time costs (h)

215 eqns., 1; 050 terms, 26.0 s sim. time Non-hierarchical Hierarchical Hybrid 10:5 2:5 <4

L -norm

Eqns./terms Error (%) Sim. time (s) Time costs (h)

97=593 2:51 16:0 >12

139=362 7:16 11:4 2:5

34=92 5:68 2:2 <4

E

Eqns./terms Error (%) Sim. time (s)

80=405 0:37 9:5

132=336 0:08 13:1

34=93 5:32 2:0

Original system: Error fct. 2

Table 2 Speed-up of simulation of a hybrid reduced entire system w.r.t. the original one System

Voltage pulse (s)

3 kHz sine wave (s)

Sum of sine waves (s)

215=1; 050 34=92

106 6.6

273 14.1

104 10.5

4 Conclusion The hierarchical and hybrid reduction approaches could be applied successfully to the op741 amplifier leading to robust models, accelerated simulations, and significant savings in time costs for the reduction. This shows the large potential of these methods, particularly in the symbolic case, where larger circuits may not be processed directly due to their high level of complexity. Acknowledgements The work reported in this paper has been carried out within the project SyreNe – System Reduction for Nanoscale IC Design (www.syrene.org). It is supported by the German Federal Ministry of Education and Research (BMBF) under grant no. 03LAPAE6.

References 1. Analog Insydes – The Intelligent Symbolic Design System for Analog Circuits. Homepage: http://www.analog-insydes.de 2. Halfmann, T., Wichmann, T.: Overview of Symbolic Methods in Industrial Analog Circuit Design. Berichte des Fraunhofer ITWM, Nr. 44 (2003) 3. Hennig, E.: Symbolic Approximation and Modeling Techniques for Analysis and Design of Analog Circuits. Ph.D. thesis, Universit¨at Kaiserslautern, Shaker, Aachen (2000) 4. Schmidt, O.: Structure-Exploiting Coupled Symbolic-Numerical Model Reduction For Electrical Networks. Ph.D. thesis, Technische Universit¨at Kaiserslautern, Cuvillier, G¨ottingen (2010)

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5. Schmidt, O., Halfmann, T., Lang, P.: Coupling of numerical and symbolic techniques for model order reduction in circuit design. In: Benner, P., Hinze, M., ter Maten, E.J.W. (eds.) Model Reduction for Circuit Simulation, Lecture Notes in Electr. Eng., vol. 74. Springer, Berlin (2011) 6. Sommer, R., Hennig, E., Nitsche, G., Schwarz, P., Broz, J.: Automatic nonlin. behavioral model generation using symb. circ. anal. In: Fakhfakh, M., Tlelo-Cuautle, E., Fern´andez, F.V. (eds.) Design of Analog Circuits through Symbolic Analysis, Bentham Science Publishers, IL (2010) 7. Wichmann, T.: Symbolische Reduktionsverfahren f¨ur nichtlineare DAE-Systeme. Ph.D. thesis, Technische Universit¨at Kaiserslautern, Shaker, Aachen (2004)

Using Sensitivities for Symbolic Analysis and Model Order Reduction of Systems with Parameter Variation Christian Salzig, Matthias Hauser, and Alberto Venturi

Abstract The ongoing trend from micro- to nanoelectronics causes the growth of the relative parameter variation during the integrated electronic circuits production resulting in a consequent reduction of the production yield. Thus, symbolic model order reduction (MOR) techniques which were developed for design and analysis of nominal systems have to be adapted to assist the design of circuits which are robust with respect to parameter variation. Therefore, new sensitivity based methods have to be introduced to estimate the output of statistical systems and to improve the performance of the statistical MOR methods.

1 Introduction To guarantee a high yield of functional devices in the production of nanoelectronic circuits high robustness with respect to disturbed parameters is necessary. Hence the design of these components requires exact knowledge of the system behavior and its dependence on the system parameters. Often, deep insights into the system behavior of modern electronical circuits are missing due to its complexity. One possible solution is the simplification and reduction of the behavioral model of the circuit. The Fraunhofer ITWM considers electronic circuits at system level using its EDA software Analog Insydes [1], for system modeling, analysis and symbolic approximation. In this paper a short description of the statistical MOR is given. Thereafter, it presents a sensitivity-based procedure for estimating the output

C. Salzig ()  M. Hauser  A. Venturi Fraunhofer Institute for Industrial Mathematics ITWM, Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany e-mail: [email protected]

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distribution of statistical systems and improving the performance of statistical MOR methods. Analog Insydes 2011 is unique in reducing nonlinear transient systems symbolically. Thus, the combination of this approximation capability with statistics-based verification, which is presented below, is a significant advancement with respect to the state-of-the-art.

2 Model Order Reduction of Systems with Parameter Variation In this section the MOR method for symbolic systems with parameter variations is presented, which will motivate the subsequent sections. For more details, see [7]. Starting with the netlist, model cards and statistics information of an electronic circuit the corresponding symbolic nonlinear differential-algebraic system of equations (DAE) d f . dt x; x; p; t/ D 0 (1) with parameter variations is generated. Here x 2 Rn denotes the state vector, p 2 Rm a vector of distributed parameters, t the time and f W Rn  Rn  Rm  R ! Rn a model function. Simplifying this nonlinear DAE (1) symbolically means reducing it to its most relevant terms, decreasing its model order and keeping its symbolic form [5, 8]. Note that each equation is formulated as a sum of terms which consists of variables, parameters or nonlinear expressions of them. Before a MOR can be run, a reference data set of the original system has to be calculated. This data will be used for verification of the results of the reduction process. Due to the parameter variation the nominal system simulation is not sufficient as reference data. A Monte Carlo simulation of the original system has to be taken instead. Using the reference a ranking of the influence of each term on the system output behavior can be calculated. Therefore, each term is deleted consecutively and the output of the resulting system is evaluated. Here again, a Monte Carlo simulation is required to estimate the output distribution. This data can be compared to the reference using a statistical error function, as e.g. the Kolmogorov-SmirnovTest [3]. This results in an estimation which terms can be deleted without violating a predefined error bound. In the final reduction step terms with similar rankings are arranged in clusters and deleted iteratively. After each deletion the resulting system is Monte Carlo simulated and the output distribution is validated by the reference as it was done in the ranking step. If the cancelation causes an error beyond the given bound the reduction is undone and appropriate sub-clusters are tested in succession. Otherwise the reduction is kept and the next cluster is taken. After evaluating all clusters a reduced behavioral model is generated, whose statistical behavior approximates the original system and the error due to the reduction is below the pre-defined limit (Fig. 1).

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Fig. 1 Workflow of a symbolic model reduction in Analog Insydes

Considering the introduced method, it is conspicuous that the system output is evaluated very often. Here in the reference, ranking and final reduction step a Monte Carlo simulation is run. Unfortunately, Monte Carlo simulations are very timeconsuming. Thus, alternative and computational cheaper techniques to compute or estimate the output of systems with parameter variation are needed. The sensitivity analysis is an applicable solution to the performance problem, which will be demonstrated in the following sections.

3 Sensitivity Analysis and Its Usage The effects of distributed parameters on the system behavior of a system can be analysed by the methods of the sensitivity analysis [6]. Therefore, a model parameter is perturbed and the difference in the system output is measured. Since Analog Insydes allows the generation of symbolic model equations of a circuit, it is possible to determine the sensitivities in an analytical way. In the static case the model (1) is simplified to f .x; p/ D 0

(2)

with f W Rn  Rm ! Rn , since the time dependency disappears. Then the sensitivities are defined as the derivatives of the output variables with respect to the system parameters, that is SD

ˇ dy ˇˇ 2 Rqm dp ˇfx0 ;p0 g

(3)

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Deriving the system of equations (2) by the model parameters leads to ˇ ˇ ˇ ˇ ˇ ˇ ˇ dx ˇˇ @f df @f ˇ ˇ .x; p/ˇˇ .x; p/ .x; p/  C D D0 ˇ ˇ ˇ @x @p dp fx0 ;p0 g dp fx0 ;p0 g fx0 ;p0 g fx0 ;p0 g

(4)

Here x 2 Rn denotes the state vector, p 2 Rm the vector of parameters, x0 2 Rn the operating point and p0 2 Rm the nominal design point. Note that the sensitivities S are the solution of the system of equations (4) since the system output vector y 2 Rq is a sub-vector of state x. Especially, the system of sensitivities (4) can be decoupled. For each of m parameters a system of equations of size n has to be solved instead of solving one large system of equations of size n  m. This avoids numerical problems and allows the usage of parallel solvers which leads to an additional performance gain.

3.1 Approximation of the Output Distribution of Systems with Parameter Variation Using the sensitivities (3) the system output for a sample pQ 2 Rm from the parameter distribution p can be estimated by yjpQ  yjp0 C S  .pQ  p0 /

(5)

where yjp0 denotes the solution of the nominal system. This allows the propagation of the parameter distributions through the system resulting in an estimation of the output distributions using the following fact: A multivariate normally distributed random variable vector stays normally distributed under affine transformation [4]. More precise, let the vector P  N.; ˙/ (6) be normally distributed with location vector  and covariance matrix ˙ and Y D c C BP

(7)

be an affine transformation with vector c and matrix B of appropriate sizes. Then the random variable vector Y is distributed by Y  N.c C B; B˙B T /

(8)

To handle arbitrary distributions, we approximate the joint distribution of the parameters p by a sum of radial basis functions (RBFs) with gaussian kernels [2]: p

r X i D1

ai N.i ; ˙i /

(9)

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with ai 2 R, i 2 Rm , ˙i 2 Rmm for all i 2 f1; : : : ; rg, r 2 N. Then, the nominal design point is split into the linear combination p0 D

r X

ai i

(10)

i D1

Now, using the first order approximation (5) and the property of affine transformed parameter distributions (8) the distributed output y 2 Rq can be approximated by y

r X i D1

ai N. yjp0 C S.i  p0 /; S˙i S T /

(11)

without any computational expensive Monte Carlo simulation.

4 Example In the following the approximation of the output distribution by sensitivities is shown. Therefore a voltage limiter circuit (Fig. 2) is considered, which is described by 254 differential-algebraic equations (2). For visualization, here only two distributed parameters and one output are chosen. As parameters the two resistors of the voltage divider are distributed normally respectively triangular (Fig. 3). The output of the circuit is the voltage Vout . The two-dimensional distribution is approximated by 400 RBFs, which is shown in Fig. 4. Using the sensitivities (3) of the system, (11) yields the distribution of the voltage Vout as a weighted sum of Gaussian distributions. Figure 5 compares the probability density function of resulting distribution to a histogram of a Monte Carlo simulation of the voltage limiter circuit.

Fig. 2 Voltage limiter circuit

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Fig. 3 Joint distribution of resistors (resistances in Ohm)

8. × 10–10 6. × 10–10 4. × 10–10 2. × 10–10 0

9 × 106 8 × 106 0

7 × 106 R2 6 × 106

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Fig. 4 Approximation of Fig. 3 by 400 RBFs

8. × 10–10 6. × 10–10 4. × 10–10 2. × 10–10 0

9 × 106 8 × 106 7 × 106 R2

0 500 R1

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Fig. 5 Comparison of a Monte Carlo simulation (500 samples) and the approximated output distribution

3.

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output voltage [V]

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Table 1 Computation time Time (s)

Monte Carlo simulation

Approximation by RBFs

Sensitivity based estimation

422

7.9

1.5

The performance gain is presented exemplary in Table 1. The parameter distribution has to be approximated by RBFs once before the MOR is started. Then each Monte Carlo simulation, here needing 422 s, can be replaced by the sensitivity based estimation of the output distribution that needs about 1.5 s (less than 0.36%). Acknowledgements This work has been carried out under grant of the Fraunhofer market-driven prospective research within the project Hierarchical Simulation of Nanoelectronic Systems for Control of Process Variations (HIESPANA).

References 1. Analog Insydes: The intelligent symbolic design system for analog circuits. www.analog-insydes.com 2. Buhmann, M.D.: Radial Basis Functions: Theory and Implementations. Cambridge University Press, Cambridge (2003) 3. Freedman, D., Pisani, R., Purves, R.: Statistics, 4th edn. W.W.Norton and Company, NY (2007) 4. Georgii, H.O., Ortgiese, M., Baake, E.: Stochastics: Introduction to Probability and Statistics. de Gruyter, NY (2008) 5. Halfmann, T., Broz, J., Knoth, C., Platte, D., Rotter, P.: Generation of efficient behavioral models using model compilation and model reduction techniques. In: Proceedings of Xth International Workshop on Symbolic and Numerical Methods, Modeling and Applications to Circuit Design. SMACD, Erfurt, Germany (2008) 6. Saltelli, A., Chan, K., Scott, E.M. (eds.): Sensitivity Analysis. Wiley Series in Probability and Statistics. Wiley, NY (2000) 7. Salzig, C., Hauser, M.: Design of robust electronic circuits for yield optimization. In: Proceedings of XIth International Workshop on Symbolic and Numerical Methods, Modeling and Applications to Circuit Design. SM2ACD, Tunis-Gammarth, Tunisia (2010) 8. Wichmann, T.: Symbolische Reduktionsverfahren f¨ur nichtlineare DAE-Systeme. Ph.D. thesis, Fraunhofer ITWM Kaiserslautern (2004)

Multi-Objective Optimization of RF Circuit Blocks via Surrogate Models and NBI and SPEA2 Methods Luciano De Tommasi, Theo G.J. Beelen, Marcel F. Sevat, Joost Rommes, and E. Jan W. ter Maten

Abstract Multi-objective optimization techniques can be categorized globally into deterministic and evolutionary methods. Examples of such methods are the Normal Boundary Intersection (NBI) method and the Strength Pareto Evolutionary Algorithm (SPEA2), respectively. With both methods one explores trade-offs between conflicting performances. Surrogate models can replace expensive circuit simulations so enabling faster computation of circuit performances. As surrogate models of behavioral parameters and performance outcomes, we consider look-up tables with interpolation and Neural Network models.

L. De Tommasi United Technologies Research Center, Lee Mills House, Prospect Row, Cork, Ireland e-mail: [email protected] T.G.J. Beelen  J. Rommes NXP Semiconductors, Central R&D, High Tech Campus 46, 5656AE Eindhoven, The Netherlands e-mail: [email protected]; [email protected] M.F. Sevat LPD, Nieuwe Eyckholt 292e, 6419 DJ Heerlen, The Netherlands e-mail: [email protected] E.J.W. ter Maten () Department of Mathematics and Computer Science, Eindhoven University of Technology, CASA, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Bergische Universit¨at Wuppertal, FB C, AMNA, Bendahler Str. 29/503, 42285 Wuppertal, Germany e-mail: [email protected]; [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 23, © Springer-Verlag Berlin Heidelberg 2012

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1 Introduction: Multi-Objective Optimization Problem The design parameters (input) x and performances, or performance parameters, (output) f are assumed to be in the Design Space D and the Performance Space P, respectively. We assume that D is feasible, i.e., all x 2 D satisfy the imposed constraints (reflected by inequalities for a function c.x/). Also the f 2 P can be constrained (reflected by inequalities for a function g.f/). We define D and P by D D fx 2 Rm jc.x/  0g; with c.x/ 2 Rq ; P D ff 2 Rn j9x2D f D f.x/; g.f/  0g; with g.f/ 2 Rp : The design problem is a multi-objective optimization problem, i.e. a constrained simultaneous minimization of several performances fk .x/ 1 f1 .x/ C B Minimizex 2 D f.x/ D @ ::: A such that g.f/  0: fn .x/ 0

A simple single-objective optimization can be done by constrained minimizing a weighted sum of performances Minimizex 2 D f .x/ D

X

ki  fi .x/ such that g.f/  0:

i

Here obvious problems arise. The multi-objective problem admits multiple solutions, whereas the single objective problem admits isolated solutions. No rigorous criteria exist to choose the weights fki g. In practice, several optimization runs (with different fki g) are needed to find a suitable solution of the design problem. More basically, in general, there is no single design x 2 D that can minimize all performances fk , k D 1; : : : ; n simultaneously. The set of solutions of the multiobjective optimization problem are Pareto optimal, i.e. it is only possible to improve one performance at the cost of others. This leads to the concept of “dominance”. Let a; b 2 Rn , then a D .a1 ; : : : ; an / dominates b D .b1 ; : : : ; bn / if and only if a  b W, 8i 2f1;:::;ng .ai  bi / ^ 9i 2f1;:::;ng .ai < bi /: A performance vector f? is said to be Pareto-optimal if it is non-dominated within P, i.e. :9f2P Œf  f? . The set of all Pareto-optimal points in P is called the Pareto Front of P. The corresponding set in D is called the Pareto Source.

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2 Surrogate Modeling Recently several techniques emerged to compute the Pareto Front. The most obvious one deals with trade-off analysis from available data. Hence (in principle) no new simulations are needed. The search for Pareto optimal points is done by applying non-dominated sorting. An efficient implementation by Yi Cao [2] is found on the MATLAB central website (mex function). An alternative is to perform Performance Space Exploration. Here one builds one or more surrogate models, each of them derived by a set of circuit simulations (samples), starting from an initial design. With adaptive sampling the models are improved [11], which requires accessibility of a (circuit) simulator. The models can be generated by several techniques [7] (including look-up tables with interpolation and neural network models). The approach can also be applied to derive symbolic models, that may include a new trade-off problem between Fitness (approximation error) and Complexity [5]. In practice, in both cases, the number of parameters is still restricting (up to 6–10). Here interesting progress is derived using a nearly orthogonal and space-filling Latin Hypercube [1, 3]. Writing x D .x.1/ ; x.2/ / one may reduce the parameter dependency in the surrogate modeling and consider behavioral parameters b D b.x.1/ /, followed by performance computations f D f.b; x.2// using algebraic expressions. Error amplification from b to f may occur (see [8] for the IIP2 performance of a Low Noise Amplifier). Clearly, when the surrogate models are available one can use them in the forward modeling in more cheaply generating additional data for improving tradeoff analysis. However, the models can also be used in reverse modeling, i.e. in applying them to dedicated Pareto Front methods like NBI (Normal Boundary Intersection method [4, 10]) and SPEA2 (Strength Pareto Evolutionary Algorithm 2 [12]).

3 NBI: Normal Boundary Intersection Method  Assume f D

f1 .x/ f2 .x/

 W D ! P. The Algorithm [4, 10] looks like

1. Determine a minimizer x?k of each fk .x/. Let f?k D f.x?k /. This is a global optimization problem for each fk .x/ and critical for the next step. MATLABs fmincon.m allows nonlinear constraints. It implements a local optimization procedure: it starts from a user-specified point and may stop in a local minimum. More robust was direct.m [9] which provides global optimization using Lipschitzian optimization. It only allows domain boundary constraints. 2. Determine the straight line L (convex hull of the individual minima) in P between f?1 and f?2 .

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3. Determine the normal n to this line in direction of decreasing f. Next • Original [10]: Select N points fk D k f?1 C .1  k /f?2 , k 2 Œ0; 1 on L . • Modification: Select N points xk D k x?1 C.1k /x?2 , k 2 Œ0; 1 on G , line in D. For convex f we have fxk D f.xk /  fk H) redefine fk D min.fk ; fxk /. 4. For each fk determine pk 2 P that maximizes the distance t along n, starting in fk . Without constraints these pk are on the Pareto Front. We solve max

.t; x/2R.D \ f1 .P//

t;

subject to

p.x/ D F C tn;

where F is a point of the convex hull of the individual minima. Note, that x has to be feasible. Also these are global optimization problems, but less critical. Here the starting point allows fmincon to provide good results. When during the maximization process a constraint in P is encountered this process is stopped, say with performance vector Qf. This does not necessarily mean that Qf is located on the Pareto Front: there may be a vector Of  Qf that also satisfies that constraint. We apply a refinement procedure. Let xQ 2 D with Qf D Qf.Qx/. Next • Determine four neighboring points xQ N ; xQ E ; xQ S ; xQ W at a small distance from xQ and calculate f.QxK / (K D N; E; S; W ). Compare step 6 of the SPEA2 Algorithm in Sect. 4. • Replace Qf by the best performance vector (based on the dominance relation) out of the set fQf; f.QxN /; : : : ; f.QxW /g. This still does not guarantee a point on the Pareto Front, it just gives an improvement. In general a more sophisticated approach is needed.

4 SPEA2: Strength Pareto Evolutionary Algorithm 2 The SPEA2 Algorithm [12] allows constraints both in D and in P. It looks like • Initialize an internal I and and external E set of points in P (last being approximations of Pareto Front). • Iteration loop 1. E c Dcopy(E ). U D I [ E c . 2. Determine fitness of individuals in U [“fitter” when not dominated in P and not too close to each other; impose constraints in P]. 3. Update E with fittest individuals from U . 4. Select individuals from U , randomly based; “fitter” points have a higher probability in being chosen. 5. Recombine selected individuals. This exploits convexity using a randomly chosen weighting. 6. Mutate recombined individuals. By properly defining the probability density function in mutating the result (f.i. after a gradient calculation) one can push the convex hull in P to the Pareto front.

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7. Repopulate I with mutated individuals. 8. Verify iteration termination criterion. • Output E as best approximation found to the source of the Pareto front.

5 Examples A good testing example appeared to be p WD f1 .x; y/ D x 2 C .y  1/2 , q WD f2 .x; y/ D .x  2/2 C y 2 , for .x; y/ 2 ŒxL ; xU   ŒyL ; yU  and .p; q/ 2 ŒpL ; pU ŒqL ; qU . Observe that f is convex. By considering the mapping of vertical and horizontal lines in D into P one can obtain impressions of the Pareto Front to check the outcomes of the algorithms. One can also observe the effect of constraints. A more realistic example is provided by a weakly nonlinear, narrowband Low Noise Amplifier (LNA), Fig. 1. • Design parameters: x.1/ D .W; L; Ls ; Lm ; f; VGS /. • Extra circuit parameters x.2/ D .Zs ; Zl /. • Typical circuit performances f D f.x.1/; x.2/ / D .P; Av ; a ; IIP2; IIP3; NF/. We considered reverse modeling using look-up table models vs. analytic expressions, both with constrained optimization. • Normalized design constraints: 0 < Wn < 1 and 0 < Lmn < 0:6. • Performance constraints: Av > 13 dB, a < 10 dB, min.IIP2; IIP3/ > 0 dBm. • (O1) Minimize P and maximize IIP 3 and (O2) maximize Av and maximize IIP2.

Vdd Vout

W,L

Vgs

Zl

Lm Zs

Ls

Vs LNA

Fig. 1 A weakly nonlinear, narrowband, low noise amplifier (LNA) [6, 8]. Design parameters: W; L are transistor width and length; Ls ; Lm are inductances; VGS is the gate-source bias voltage difference; f is the frequency. Zs ; Zl are the source and load impedances. Performances: power P , voltage gain Av , input reflection a , 2nd order and 3rd order linearity IIP 2; IIP 3, noise figure NF

200 Pareto Source

Pareto Front

0.65

20

0.6

18 16

0.55

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IIP2

0.5

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Fig. 2 Pareto Front determined by SPEA2 for (O2). This involved reverse modeling using look-up table models (100  100 meshpoints). Here NBI (using fmincon) failed in finding a global minimum

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0.45 0.4

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Av

For (O1) NBI and SPEA2 worked successfully using surrogate models based on neural networks. For (O2) we used look-up table models. Here the NBI method (using fmincon) failed in finding a global minimum. Figure 2 shows the SPEA2 result.

6 Conclusions Direct modeling of performances was more robust than modeling of intermediate “behavioral” parameters. We considered look-up tables and applied interpolation. Also the size of tables was investigated. Neural network models were accurate, but expensive in generating. The NBI method was improved in several ways. DIRECT provided a robust global optimizer for the start. Also the start of the directional optimization step was improved. Without constraints it covers the whole Pareto front in nice detail. However, with constraints, as above in (O2), still more work has to be done. SPEA2 is more robust than NBI. Constraints can be applied on both design variables and on performances (including those not involved in the trade-off). The results were confirmed by considering a Low Noise Amplifier. Acknowledgements The work by the first (LDT, Univ. of Antwerp) and fourth (JR, NXP) author was funded by the EU Marie Curie FP7 MTKI-CT-2006-042477 project O-MOORE-NICE!

References 1. Beelen, T.G.J., ter Maten, E.J.W., Sihaloho, H.J., van Eijndhoven, S.J.L.: Behavioral modeling of the dominant dynamics in input-output transfer of linear(ized) circuits. Procedia Comp. Sci. 1(1), 347–355 (2010)

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2. Cao, Y.: http://www.mathworks.com/matlabcentral/fileexchange/17251-pareto-front (2007/ 2008) 3. Cioppa, T.M., Lucas, T.W.: Efficient nearly orthogonal and space-filling Latin Hypercubes. Technometrics 49-1, 45–55 (2007) 4. Das, I., Dennis, J.E.: Normal-Boundary Intersection: A new method for generating Pareto optimal points in multicriteria optimization problems. SIAM J. Optim. 8-3, 631–657 (1998) 5. DataModeler: http://www.evolved-analytics.com 6. De Tommasi, L., Gorissen, D., Croon, J., Dhaene, T.: Surrogate modeling of low noise amplifiers based on transistor level simulations. In: Roos, J., Costa, R.J. (eds.) Scientific Computing in Electrical Engineering SCEE 2008. Mathematics in Industry, vol. 14, pp. 225–232. Springer, Berlin (2010) 7. De Tommasi, L., Gorrisen, D., Croon, J.A., Dhaene, T.: Surrogate modeling of RF circuit blocks. In: Fitt, A.D., Norbury, J., Ockendon, H., Wilsson, E. (eds.) Progress in industrial mathematics at ECMI 2008. Mathematics in industry, vol. 15, pp. 447–452. Springer, Berlin (2010) 8. De Tommasi, L., Rommes, J., Beelen, T., Sevat, M., Croon, J.A., Dhaene, T.: Forward and reverse modeling of low noise amplifiers based on circuit simulations. In: Benner, P., Hinze, M., ter Maten, E.J.W. (eds.) Model Reduction for Circuit Simulation. Lecture Notes in Electrical Engineering, vol. 74, pp. 111–124. Springer, Berlin (2011) 9. Finkel, D.E.: Global optimization with the direct algorithm. PhD-Thesis North Carolina State University (2005). http://pages.cs.wisc.edu/ferris/cs726/direct.m 10. Stehr, G., Gr¨ab, H.E., Antreich, K.J.: Analog performance space exploration by NormalBoundary Intersection and by Fourier-Motzkin elimination. IEEE Trans. Comp.-Aided Des. Integrated Circ. Syst. 26-10, 1733–1745 (2007) 11. SUMO toolbox: http://www.sumo.intec.ugent.be/?q=sumo toolbox 12. Zitzler, E., Laumanns, M., Thiele, L.: SPEA2: Improving the Strength Pareto Evolutionary Algorithm, Techn. Report TIK Report 103, ETH Z¨urich. http://www.tik.ee.ethz.ch/pisa/ selectors/spea2/spea2 c source.html (2001)

Part III

Uncertainties and Stochastics

Overview Mathematical modeling of real life applications often yields systems consisting of ordinary differential equations, partial differential equations or more complex combinations and generalizations. Owing to advancements in the reliability of corresponding numerical methods, other types of problems are considered in connection to existing models now. For example, the influence of uncertainties on the solution of the model will be characterized and quantified appropriately. One can classify the kind of uncertainty into two types: aleatoric uncertainties and epistemic uncertainties. On the one hand, aleatoric situations include inherent uncertainties, which cannot be reduced further. This case is given if a repetition of the same experiment always yields different results, for example. Consequently, a stochastic modeling is required to incorporate these effects. On the other hand, epistemic situations represent uncertainties, which are caused by a lack of knowledge. For example, errors in measurements or a simplification of mathematical models make a precise description of a real life application impossible. Again a stochastic modeling is often reasonable. However, alternative approaches can be applied like interval analysis, see [1], or fuzzy sets, see [4], for example. Both aleatoric uncertainties and epistemic uncertainties can appear in the same technical or economic application. Moreover, the interpretation of uncertainties may differ on the level of the mathematical model in comparison to the underlying real life problem. In dynamic systems, uncertainties are often present in the form of time-dependent noise, which corresponds to the aleatoric type in a broad class of problems. The noise can be modeled mathematically by a Wiener process, also referred to as Brownian motion. It follows the class of stochastic differential equations, see [5], where random effects are included in time-dependent ordinary differential equations. Likewise, stochastic partial differential equations result for problems depending on time as well as space, which represent an active field of research.

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Stochastic ordinary differential equations often appear in problems from financial engineering. For example, the fair price of an option corresponding to some asset or a basket of assets has to be determined. While the price of an asset follows from a stochastic differential equation, the fair price of an option is modeled by a deterministic partial differential equation of parabolic type in some important cases, i.e., the famous class of the Black–Scholes equations, see [3, 6]. In general systems of differential equations, the exact values of some parameters may be unknown in the model, which is something that causes epistemic uncertainties. If the crucial parameters are replaced by random variables, it follows a corresponding stochastic model. Such a random-dependent system can be resolved by a Monte–Carlo simulation, for example. Techniques based on the polynomial chaos, also referred to as the Wiener calculus, allow for alternative approaches. The polynomial chaos involves expansions using orthogonal polynomials in random space, see [2, 7, 8]. Corresponding numerical methods follow from an intrusive approach via a stochastic Galerkin method or a non-intrusive approach via a stochastic collocation. At the ECMI 2010 conference, several contributions have addressed stochastics and uncertainty quantification. Emphasis was placed on problems both from financial engineering and financial mathematics. Further contributions treat problems from mechanical engineering, aerodynamics and other industrial applications. ˇ coviˇc considers the determination of the fair price of American type D. Sevˇ options modeled by nonlinear Black–Scholes equations, where the volatility depends on the unknown option value in his paper “On a Numerical Approximation Scheme for Construction of the Early Exercise Boundary for a Class of Nonlinear Black–Scholes Equations”. Numerical methods for the early exercise boundary of this problem are investigated. B. D¨uring and M. Fourni´e apply the Heston model, i.e., an extension of the Black–Scholes equation including a stochastic volatility. In their paper “On the Stability of a Compact Finite Difference Scheme for Option Pricing” they analyze the von Neumann stability of a high-order finite difference method used to solve the partial differential equations. In “Stationary Solutions of Some Nonlinear Black–Scholes Type Equations Arising in Option Pricing”, F. Fabi˜ao et al. show the existence of specific solutions in a class of nonlinear Black–Scholes equations, which model the price of an option including transaction costs and a stochastic volatility again. The proofs of the existence theorems are outlined. In “Variants of the Combination Technique for Multi-Dimensional Option Pricing”, J. Benk et al. consider partial differential equations of Black–Scholes type for options corresponding to a basket of many assets. To meet the challenge of dealing with the curse of dimensionality, several discretizations using sparse grids are constructed and compared to each other. N. Marheineke and R. Wegener apply a stochastic modeling in turbulent flows corresponding to aerodynamics in their paper “Stochastic PDAE-Model and Associated Monte–Carlo Simulations for Elastic Threads in Turbulent Flows”. It follows a model based on stochastic partial differential equations, where additional

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algebraic constraints appear. The authors introduce two surrogate models to resolve the dynamics with noise. In “Production Networks With Stochastic Machinery Default”, S. G¨ottlich et al. investigate a time-continuous production network, where aleatoric uncertainties are included by a random breakdown of individual processors. The stochastic model used is a coupled system of partial differential equations and ordinary differential equations. The authors present results of numerical simulations for a test example. M. Kletting and F. Antritter consider a magnetic leviation system in “Verified Simulation for Robustness Evaluation of Tracking Controllers”, i.e., time-dependent ordinary differential equations, in connection to tracking controllers. Epistemic uncertainties are assumed in the system’s parameters as well as the initial values. The authors apply an interval analysis for a corresponding uncertainty quantification. Results of numerical simulations are given. In “Structural Analysis for the Design of Reliable Controllers and State Estimators for Uncertain Dynamical Systems”, A. Rauh and H. Aschemann investigate the controllability and observability of semi-explicit systems of differential algebraic equations. Epistemic uncertainties are considered in the system’s parameters and in form of measurement errors and modeling errors. Again an interval analysis yields an uncertainty quantification. An example is presented where a space discretization of a specific heat equation results in time-dependent differential algebraic equations. F. Augustin et al. consider the polynomial chaos to resolve stochastic models, where random parameters are applied to represent epistemic uncertainties in the paper “Wiener Calculus for Differential Equations with Uncertainties”. Both the intrusive and the non-intrusive approach for the determination of the polynomial chaos expansions are outlined. The authors illustrate results of numerical simulations for two examples: a problem from mechanical engineering modeled by a space-dependent partial differential equation of elliptic type and the van-der-Pol oscillator, i.e., a time-dependent system of ordinary differential equations. In “Polynomial Chaos and Its Application to Delay Differential Equations With Uncertainties”, M. Villegas Caballero examines delay differential equations with uncertain delays. The stability of corresponding solutions is analyzed with respect to a random delay, where the polynomial chaos expansion is applied within a nonintrusive approach. Wuppertal, Germany

Roland Pulch Matthias Ehrhardt

References 1. Alt, R., Frommer, A., Baker Kearfott, R., Luther, W. (eds.): Numerical Software with Result Verification. Springer, Berlin (2004) 2. Ghanem, R.G., Spanos, P.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991)

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3. Higham, D.J.: An Introduction to Financial Option Valuation. Cambridge University Press, Cambridge (2004) 4. M¨oller, B., Beer, M.: Fuzzy Randomness. Uncertainty in Civil Engineering and Computational Mechanics. Springer, Berlin (2004) 5. Øksendal, B.: Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin (2002) 6. Seydel, R.: Tools for Computational Finance, 4th edn. Springer, Berlin (2009) 7. Xiu, D.: Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, Princeton (2010) 8. Xiu, D., Karniadakis, G.E.: The Wiener-Askey polynomial Chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)

On a Numerical Approximation Scheme for Construction of the Early Exercise Boundary for a Class of Nonlinear Black–Scholes Equations ˇ coviˇc Daniel Sevˇ

Abstract The purpose of this paper is to construct the early exercise boundary for a class of nonlinear Black–Scholes equations with a volatility function depending on the option price. We review and revisit a method how to transform the problem into a solution of a time depending nonlinear parabolic equation defined on a fixed domain. An example of numerical computation of the early exercise boundary for a nonlinear Black–Scholes equation is also presented.

1 Black–Scholes Equations with a Nonlinear Volatility Function The main purpose of this paper is to review and revisit the fixed domain transformation method adopted for solving a class of nonlinear Black–Scholes equations having the form: @V 1 @2 V @V C .r  q/S C  2 .S 2 @2S V; S; T  t/S 2 2  rV D 0; S > 0; t 2 .0; T /: @t @S 2 @S (1) A solution V D V .S; t/ can be identified with a price V of the option contract in the future maturity time T > 0 (e.g. call or put) where S > 0 is the underlying asset value at the present time t 2 Œ0; T /. Here, r > 0 is the riskless interest rate, q  0 is the dividend yield rate of the underlying asset. For American style of a call option, the free boundary problem consists in construction of the early exercise position Sf D Sf .t/ and the solution V D V .S; t/ to (1) defined on the time dependent

ˇ coviˇc () D. Sevˇ Department of Applied Mathematics and Statistics, Faculty of Mathematics, Physics & Informatics, Comenius University, 842 48 Bratislava, Slovak Republic e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 24, © Springer-Verlag Berlin Heidelberg 2012

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domain 0 < S < Sf .t/; 0 < t < T (cf. Kwok [13]). V is subjected to the boundary conditions yielding C 1 smooth pasting of V .S; t/ and V .S; T / at S D Sf .t/: V .0; t/ D 0; V .Sf .t/; t/ D Sf .t/  E; @S V .Sf .t/; t/ D 1;

(2)

and the terminal pay-off condition at expiry t D T; V .S; T / D .S  E/C ;

(3)

where E > 0 is the exercise price. We briefly mention a motivation for studying the nonlinear Black–Scholes equation having the form of (1). Such equations with a volatility function .S 2 @2S V; S; T  t/ arise from modeling the option prices by taking into account nontrivial transaction costs (cf. Leland [14], Hoggard et al. [10], Avellaneda and Paras [3]), market feedbacks and effects due to large traders choosing given stocktrading strategies (Frey [7], Frey and Patie [8], Frey and Stremme [9], During et al. [6], Sch¨onbucher and Wilmott [15]), the risk adjusted pricing methodology ˇ coviˇc model due to Kratka [12] and its modification developed by Jandaˇcka and Sevˇ [11, 17]). As an example for application of the numerical method, we consider a nonlinear model taking into account imperfect replication and investor’s preferences which has been proposed by Barles and Soner in [4]. If investor’s preferences are characterized by an exponential utility function they derived a nonlinear Black– Scholes equation with the volatility function  given by    2 .S 2 @2S V; S; / D O 2 1 C  .a2 e r S 2 @2S V / :

(4)

Here O 2 > 0 is a constant historical volatility of the asset p price returns,  is the unique solution to the ODE:  0 .x/ D . .x/ C 1/=.2 x .x/  x/;  .0/ D 0 and a  0 is a constant depending transaction costs and investor’s risk aversion 1 parameter (see [4] for details). The function  satisfies:  .x/ D O.x 3 / for x ! 0 and  .x/ D O.x/ for x ! 1. For practical purposes, the solution  .x/ can be constructed from an implicit equation obtained in [5]. We revisit an iterative numerical algorithm for solving the free boundary problem ˇ coviˇc in [17]. The key idea of this method consists in (1)–(2) developed by Sevˇ transformation of the free boundary problem into a semilinear parabolic equation defined on a fixed spatial domain coupled with a nonlocal algebraic constraint equation for the free boundary position. This method has been analyzed and utilized in a series of papers [1, 2, 16–19] by Ehrhardt and Ankudinova and the author. The disadvantage of the original method consists in the necessity of solving an algebraic constraint equation. In this approach, highly accurate evaluation of the derivative of a solution at one point entering the algebraic constraint is needed (cf. [17]). In this note, we present a new efficient way how to overcome this difficulty by considering an equivalent integrated form of the algebraic constraint. We also present results

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of numerical calculation of the free boundary position for the Barles and Soner nonlinear extension of the Black–Scholes model.

2 Fixed Domain Transformation of the Free Boundary Problem We recall method how to transform the free boundary problem (1)–(3) into a form of a nonlinear parabolic equation defined on a fixed domain and satisfying a nonlocal algebraic constraint equation developed by the author in [17]. It is based on the following change of independent variables and the transformed function ˘ D ˘.x; / defined as follows:  D T  t;

x D ln ../=S / ;

˘.x; / D V .S; t/  S @S V .S; t/;

(5)

where ./ D Sf .T  /. Clearly,  2 .0; T / and x 2 .0; 1/ iff S 2 .0; Sf .t//. The boundary value x D 0 corresponds to the free boundary position S D Sf .t/ whereas x D C 1 corresponds to the default value S D 0 of the underlying asset. Under the structural assumption 0
(6)

˘.0; / D  E; ˘.C1; / D 0; x > 0;  2 .0; T /;  E; for x < ln.r=q/; ˘.x; 0/ D 0; otherwise, ./ D

 2 .@x ˘.0; /; ./; / @˘ rE C .0; /; q 2q @x d

. /

with .0/ D

rE ; q

(7)

where  2 D  2 .@x ˘.x; /; ./e x ; /; b./ D dt. / Cr q (cf. [17]). Notice that (7) is not quite appropriate for construction of a robust numerical approximation scheme since any small inaccuracy in approximation of the value @x ˘.0; / is immediately transferred in to the entire computational domain x 2 .0; 1/ through the free boundary function ./ entering (6). Instead of (7), we present a new equivalent integrated equation for the free boundary position ./. Indeed, integrating the governing equation (6) for x 2 .0; 1/ taking into account the boundary conditions

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˘.0; / D  E; ˘.1; / D 0 (and consequently @x ˘.1; / D 0), we obtain the following spatially integrated form of the algebraic constraint: Z 1  d  E ln ./ C ˘.x; /dx C q./  qE d 0  Z 1 1 2 @˘ x C .x; / C r˘.x; / dx D 0: (8)   .@x ˘.x; /; ./e ; / 2 @x 0

3 Numerical Scheme Based on Operator Splitting Technique The idea of the iterative numerical algorithm is based on the original numerical discretization scheme proposed by the author in [17]. We modify this method by considering the alternative integrated form (8) of the constraint between ˘ and . The spatial domain x 2 .0; 1/ is restricted to a finite interval of values x 2 .0; L/ where L > 0 is sufficiently large. For practical purposes one can take L  3 (see [17]). Let us denote by k > 0 the time step, k D T =m and by h > 0 the spatial step, h D L=n where m; n 2 N stand for the number of time and space j discretization steps, respectively. We denote by ˘i an approximation of ˘.xi ; j /; j j   .j /; b  b.j / where xi D ih; j D j k. We furthermore denote by ˘ j the j vector ˘ j D f˘i ; i D 1; : : : ; ng. We approximate the value of the volatility  at the node .xi ; j / by the finite difference approximation as follows: i D ..˘i C1  ˘i /= h; j e xi ; j /: j

j

j

We set ˘i0 .x/ D ˘.xi ; 0/. Next, following the idea of the operator splitting method discussed in [17], we decompose the above problem into two parts—a convection 1 part and a diffusive part by introducing an auxiliary intermediate step ˘ j  2 . Our discretization of (8) and (6) reads as follows: (Integrated form of the algebraic part)   E ln j D E ln j 1 C I0 .˘ j 1 /  I0 .˘ j / C k qE  qj  I1 .j ; ˘ j / ; (9) R1 where I0 .˘ / stands for numerical trapezoid quadrature of the integral 0 ˘./d  whereas I1 .j ; ˘ / is a trapezoid quadrature of the second integral in (8), i.e. Z

1

I1 . ; ˘ /  j

0

  1 2 @˘ j x .x/ C r˘.x/ dx:   .@x ˘.x/;  e ; j / 2 @x

(Convective part) 1

˘ j  2  ˘ j 1 @ 1 C b j ˘ j  2 D 0; k @x

(10)

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(Diffusive part) 1

. j /2 @ j 1 @ ˘j  ˘j2  ˘  k 2 @x 2 @x

  j 2 @ j . / ˘ C r˘ j D 0: @x

(11)

The convective part can be approximated by an explicit solution to the transport j1 equation @ ˘Q C b./@x ˘Q D 0. Thus the spatial approximation ˘i 2 can be constructed from the formula  j 1 ˘ j1 .i /; if i D xi  ln j C ln j 1  .r  q/k > 0; ˘i 2 D (12) E; otherwise, j 1

where a piecewise linear interpolation between discrete values ˘i ; i D 0; 1; :::; n; is being used to compute the value ˘ j 1 .xi  ln j C ln j 1  .r  q/k/. The diffusive part can be solved numerically by means of finite differences. Using a central finite difference approximation of the derivative @x ˘ j we obtain j  12

j

˘i  ˘i k j

.i 1 /2

j

j

C r˘i  j

j

˘i  ˘i 1 h

j

j

j

j ˘  ˘i 1 .i /2 ˘i C1  ˘i 1 j  .i /2 i C1 2 2h 2h h !

D 0:

(13)

Now, (9), (12) and (13) can be rewritten in the operator form: j D F .˘ j ; j /;

1

˘ j  2 D T .˘ j ; j /;

1

A .˘ j ; j /˘ j D ˘ j  2 ;

where F .˘ j ; j / is the right-hand side of the integrated algebraic equation (9). The operator T .˘ j ; j / is the transport equation solver given by the right-hand side of (12) and A D A .˘ j ; j / is a tridiagonal matrix with coefficients given corresponding to (13). At each time level j ; j D 1; : : : ; m, the above system can be solved approximately by means of successive iterations procedure. Given a discrete solution ˘ j 1 , we start up iterations by defining ˘ j;0 D ˘ j 1 ; j;0 D j 1 . Then the .p C 1/-th approximation of ˘ j and j is obtained as a solution to the system: j;pC1 D F .˘ j;p ; j;p /;

1

˘ j  2 ;pC1 D T .˘ j;p ; j;pC1 /; 1

A .˘ j;p ; j;pC1 /˘ j;pC1 D ˘ j  2 ;pC1 :

(14)

We repeat the procedure for p D 0; 1 : : : ; pmax , until the prescribed tolerance is achieved. At the end of this section, we present a numerical example of approximation of the early exercise boundary for the Barles and Soner model by means of a

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Fig. 1 A comparison of . / D Sf .T   / (solid line) for the Barles and Soner model with a D 0:15 and for the Black–Scholes equation, i.e. a D 0

solution to the transformed system of equations. In this model the volatility is given by expression (4). A discrete solution pair .˘; / has been computed by our iterative algorithm for the model parameters: E D 10; T D 1 (one year), r D 0:1 (10% p.a) , q D 0:05 (5% p.a.) and O D 0:2. As for the numerical parameters, we chose n D 750 spatial points and m D 225;000 time discretization steps. The step k D T =m represents 140 s in the real time scale. In order to achieve the precision 107 we used pmax D 6 micro-iterates in (14). A graphical plot of the early exercise boundary ./ D Sf .T  / is shown in Fig. 1. Taking a positive value of the risk aversion coefficient a D 0:15 results in a substantial increase of the free boundary position ./ in comparison to the linear Black–Scholes equation with constant volatility  D O . Notice that the Barles and Soner model for a D 0 coincides with the linear Black–Scholes model with constant volatility. Acknowledgements The author was supported by VEGA 1/0747/12 grant

References 1. Ankudinova, J., Ehrhardt, M.: Fixed domain transformations and highly accurate compact schemes for nonlinear Black-Scholes equations for American options. In: Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing, pp. 243–273. Nova Science Publishers, Inc., Hauppauge (2008) 2. Ankudinova, J., Ehrhardt, M.: On the numerical solution of nonlinear Black-Scholes equations. Comput. Math. Appl. 56(3), 799–812 (2008) 3. Avellaneda, M., Par´as, A.: Dynamic hedging portfolios for derivative securities in the presence of large transaction costs. Appl. Math. Finance 1, 165–193 (1994) 4. Barles, G., Soner, H.M.: Option pricing with transaction costs and a nonlinear Black-Scholes equation. Finance Stoch. 2(4), 369–397 (1998) 5. Company, R., Navarro, E., Pintos, J., Ponsoda, E.: Numerical solution of linear and nonlinear Black-Scholes option pricing equations. Comput. Math. Appl. 56(3), 813–821 (2008)

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6. D¨uring, B., Fourni´e, M., J¨ungel, A.: High order compact finite difference schemes for a nonlinear Black-Scholes equation. Int. J. Theor. Appl. Finance 6(7), 767–789 (2003) 7. Frey, R.: Market illiquidity as a source of model risk in dynamic hedging in model risk. In: RISK Publications. R. Gibson Ed., London (2000) 8. Frey, R., Patie, P.: Risk management for derivatives in illiquid markets: a simulation study. In: Advances in finance and stochastics, pp. 137–159. Springer, Berlin (2002) 9. Frey, R., Stremme, A.: Market volatility and feedback effects from dynamic hedging. Math. Finance 7(4), 351–374 (1997) 10. Hoggard, T., Whalley, A.E., Wilmott, P.: Hedging option portfolios in the presence of transaction costs. Adv. Futures Options Res. 7, 21–35 (1994) ˇ coviˇc, D.: On the risk-adjusted pricing-methodology-based valuation of 11. Jandaˇcka, M., Sevˇ vanilla options and explanation of the volatility smile. J. Appl. Math. 3, 235–258 (2005) 12. Kratka, M.: No mystery behind the smile. Risk 9, 67–71 (1998) 13. Kwok, Y.K.: Mathematical models of financial derivatives. Springer Finance. Springer-Verlag Singapore, Singapore (1998) 14. Leland, H.E.: Option pricing and replication with transaction costs. J. Finance 40, 1283–1301 (1985) 15. Sch¨onbucher, P.J., Wilmott, P.: The feedback effect of hedging in illiquid markets. SIAM J. Appl. Math. 61(1), 232–272 (electronic) (2000) ˇ coviˇc, D.: Analysis of the free boundary for the pricing of an American call option. 16. Sevˇ European J. Appl. Math. 12(1), 25–37 (2001) ˇ coviˇc, D.: An iterative algorithm for evaluating approximations to the optimal exercise 17. Sevˇ boundary for a nonlinear Black-Scholes equation. Can. Appl. Math. Q. 15(1), 77–97 (2007) ˇ coviˇc, D.: Transformation methods for evaluating approximations to the optimal exercise 18. Sevˇ boundary for linear and nonlinear Black–Scholes equations. In: Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing, pp. 153–198. Nova Science Publishers, Inc., Hauppauge (2008) ˇ coviˇc, D., Chadam, J.: The early exercise boundary for the American put near: 19. Stamicar, R., Sevˇ numerical approximation. Canad. Appl. Math. Quart. 7(4), 427–444 (1999)

On the Stability of a Compact Finite Difference Scheme for Option Pricing Bertram During ¨ and Michel Fourni´e

Abstract In this short paper we are concerned with the von Neumann stability analysis of a compact high-order finite difference scheme for option pricing in the Heston stochastic volatility model. We first review stability results in the case of vanishing correlation and then present some new results on the behavior of the amplification factor for non-zero correlation.

1 Introduction The Heston model [8] is a stochastic volatility model for option pricing where the option price V as function of price of the underlying S , volatility  and time t solves   1 1 Vt C S 2 VS S C vS VS C v 2 V  C rS VS C   .   /   V  rV D 0; 2 2 (1) for S;  > 0, 0  t < T and subject to a suitable final condition, e.g. V .S; ; T / D max.K  S; 0/; in case of a European put option with strike price K: In (1),   , v;   ; and  denote the constant mean reversion speed, volatility of volatility, long-run mean of volatility, and market price of volatility risk parameter, respectively. The “boundary” conditions in the case of the put option read as follows

B. D¨uring () Department of Mathematics, University of Sussex, Brighton, BN1 9QH, United Kingdom e-mail: [email protected] M. Fourni´e Institut de Math´ematiques de Toulouse, Universit´e de Toulouse et CNRS (UMR 5219), France e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 25, © Springer-Verlag Berlin Heidelberg 2012

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V .0; ; t/ D Ke r.T t / ; V .S; ; t/ ! 0; V .S; ; t/ ! 0;

T > t  0;  > 0;

T > t  0;  > 0; as S ! 1; T > t  0; S > 0; as  ! 0 and  ! 1:

For constant parameters, one can employ Fourier transform techniques and obtain a system of ordinary differential equations which can be solved analytically [8]. In general, however, when the parameters are not constant, (1) has to be solved numerically. In the mathematical literature, there are many papers on numerical methods for option pricing with a single underlying. Most approaches use standard, second order finite difference methods. Compact high-order finite difference schemes were proposed, e.g. in [6, 7, 12]. For option pricing in the Heston model different second order finite difference methods for solving the American option pricing problem are compared in [10]. In [9] different, low order ADI (alternating direction implicit) schemes are adapted to the Heston model to include the mixed spatial derivative term. Other approaches include finite element-finite volume [16], multigrid [2], sparse wavelet [11], or spectral methods [15]. In [3, 4] we proposed a new high-order compact finite difference scheme for option pricing in the Heston model. It can easily be adapted to other stochastic volatility models (e.g. [5]). In this short paper we focus on the von Neumann stability analysis of the new scheme. We first review stability results that we obtained in [3] in the case of vanishing correlation, i.e.  D 0: Then we present some new results on the behavior of the amplification factor for non-zero correlation.

2 High Order Compact Scheme Let us introduce the modified parameters,  D   C ;  D     =.  C /; which allow us to study the problem with one parameter less. Under the transformation of variables x D ln.S=K/; y D =v; tQ D T  t; u D exp.r tQ/V =K; (we immediately drop the tilde in the following) and using the modified parameters,  and , we then obtain from (1), ut  12 vy.uxx C uyy /  vyuxy C

1 2

 vy  r ux    vy uy D 0; v

(2)

which has to be solved on R  RC  .0; T / with transformed initial and boundary conditions. For the discretization, we replace R by ŒR1 ; R1  and RC by Œ0; R2  with R1 ; R2 > 0: For simplicity, we consider a uniform grid with mesh width h in both the x- and y-direction, Z D fxi 2 ŒR1 ; R1  W xi D ih, i D  N; : : : ; N g  fyj 2 Œ0; R2  W yj D j h, j D 0; : : : ; M g consisting of .2N C 1/  .M C 1/ grid points, with R1 D N h; R2 D M h and time step k. Let uni;j denote the approximate solution of (2) in .xi ; yj / at the time t n D nk and let un D .uni;j /. On the truncated

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numerical domain we impose artificial boundary conditions. We difference at time t D .1  /t n C t nC1 , where 0    1: This yields a class of integrators that include the forward Euler ( D 0), Crank-Nicolson ( D 1=2) and backward Euler ( D 1) schemes. The fourth-order compact finite difference scheme derived in [3] uses a ninepoint computational stencil involving the eight nearest neighboring points of the reference grid point .i; j / 0

1 ui 1;j C1 D u6 ui;j C1 D u2 ui C1;j C1 D u5 @ ui 1;j D u3 ui;j D u0 ui C1;j D u1 A : ui 1;j 1 D u7 ui;j 1 D u4 ui C1;j 1 D u8 The resulting fully discrete difference scheme for node .i; j / at the time level n can be written in the form P8 P nC1 D 8lD0 l unl ; (3) lD0 ˇl ul where the coefficients ˇl ; l are given by (a detailed derivation is presented in [3]) ˇ0 D...2yj 2  8/v 4 C ..8  8r/yj  8r/v 3 C .8 2 yj 2 C 8r 2 /v 2  16 2 vyj C 8 2  2 /k C 16v 3 yj /h2 C .162 C 40/yj 2 v 4 k; ˇ1;3 D ˙ ..v 2  v 4  yj v 3 /k  .yj C 2/v 3 C 2v 2 r/h3 C ...yj 2 C 2/v 4 C ..4r C 2/yj C 4r/v 3  .2 C 4r 2 /v 2 /k C 2v 3 yj /h2 ˙ .4v 4 yj 2 C .8yj 2   8yj r/v 3 C 8yj v 2 /kh C .82  8/yj 2 v 4 k; ˇ2;4 D ˙ ..2 2 v  2 2 v 2 yj  2v 3 /k  2v 2 yj  C 2v  2v 3 /h3 C ..2v 4 C 2yj v 3 C .4 2 yj 2 C 2/v 2 C 8 2 vyj  4 2  2 /k C 2v 3 yj /h2 ˙ ..8yj 2  C 8yj r/v 3  4v 4 yj 2   8v 2 yj /kh C .82  8/yj 2 v 4 k; ˇ5;7 D..v 4  C .y 2  C yj  C r/v 3 C . C 2r/yj v 2  2rv/k C v 3 yj /h2 ˙ ..2 C 1/yj 2 v 4 C ..2 C 4/yj 2 C .4r  2r/yj /v 3 C .2  4/yj v 2 /kh C .2  42  6/yj 2 v 4 k; ˇ6;8 D..v 4  C .yj 2   yj   r/v 3 C .  2r/yj v 2 C 2rv/k  v 3 yj /h2 ˙ ..2  1/yj 2 v 4 C ..2  4/yj 2 C .2r  4r/yj /v 3 C .4  2/yj v 2 /kh C .42 C 6  2/yj 2 v 4 k; 0 D16v 3 yj h2 C .1  /k...8  2yj 2 /v 4 C ..8 C 8r/yj C 8r/v 3 C .8r 2  8 2 yj 2 /v 2 C 16 2vyj  8 2  2 /h2 C .40 C 162 /yj 2 v 4 /;

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1;3 D ˙ .2r  .yj C 2/v/v 2 h3 C 2v 3 yj h2 C .1  /k.˙.vyj C v 2  /v 2 h3 C .v 2 yj 2  .4r C 2/vyj C 4r 2 C 2  2v 2  4vr/v 2 h2 ˙ ..4v C 8/v 3 yj 2 C .8 C 8vr/v 2 yj /h C .8v 2  8v 2 2 /v 2 yj 2 /; 2;4 D ˙ .2v  2v 2 yj   2v 3 /h3 C 2v 3 yj h2 C .1  /k.˙2.v 3    2 v C  2 v 2 yj /h3 C .4 2 v 2 yj 2  .2v 2 C 8/vyj C 2.2  v 2 /  2v 4 /h2 ˙ ..8v 3  C 4v 4 /yj 2 C .8v 2  8v 3 r/yj /h C .8v 4 2 C 8v 4 /yj 2 /; 5;7 Dv 3 yj h2 C .1  /k..v 3 yj 2   v.v C 2rv C v 2 /yj  v.v 2 r  2r C v 3 //h2 ˙ .v.2v 3  C v 3 C 4v 2  C 2v 2 /yj 2 C v.2v C 4v C 4v 2 r C 2v 2 r/yj /h C v.2v 3 C 6v 3  C 4v 3 2 /yj 2 /; 6;8 D  v 3 yj h2 C .1  /k..v 3 yj 2  C v.v C 2rv C v 2 /yj C v.v 2 r  2r C v 3 //h2 ˙ .v.2v 3  C v 3 C 4v 2   2v 2 /yj 2 C v.2v  4v C 4v 2 r  2v 2 r/yj /h C v.2v 3  6v 3  C 4v 3 2 /yj 2 /: When multiple indexes are used with ˙ and  signs, the first and second index corresponds to the upper and lower sign, respectively. In the Crank-Nicolson case  D 1=2; the resulting scheme is of order two in time and of order four in space.

3 Stability Results We study the von Neumann stability of the scheme (for frozen coefficients). Note that our numerical experiments that we reported in [3, 4] did not reveal any stability problems. To reduce the high number of parameters, we assume zero interest rate, r D 0; and choose the parameter  D 1=2: We rewrite uni;j as uni;j D g n e I i z1 CIj z2 ;

(4)

where I is the imaginary unit, g n is the amplitude at time level n, and z1 D 2 h=1 and z2 D 2 h=2 are phase angles with wavelengths 1;2 in the range Œ0; 2 Œ. A (necessary) condition for stability is fulfilled if for all z1;2 the amplification factor G D g nC1 =g n satisfies jGj2  1  0: (5) An expression for G can be found using (4) in (3). We recall the following theorem. Theorem 1 (cf. [3]). For r D  D 0 and  D 1=2 (Crank-Nicolson), scheme (3) satisfies stability condition (5).

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One key ingredient of the proof in [3] is to define new variables c1;2 D cos.z1;2 =2/; s1;2 D sin.z1;2 =2/; W D  2.vy  /s2 =v; V D 2vys1 =; which allow us to express G in terms of h; k; ; V; W and trigonometric functions only. For non-zero correlation the situation becomes more involved. Additional terms appear in the expression for the amplification factor G and we face an additional degree of freedom through . Since we have proven stability condition (5) holds for  D 0 it seems reasonable to assume it holds at least for values of  close to zero. In practical applications, however, correlation can be strongly negative. Few theoretical results can be obtained, some of them are given in the following lemma. Lemma 1. For any , r D 0,  D 1=2 (Crank-Nicolson) it holds: if either c1 D ˙ 1 or c2 D ˙ 1 or y D 0, then the stability condition (5) is satisfied. Proof. We can prove by direct computation • if y D 0, it holds jGj2  1 D 0; and • if c12 D 1 (then V D 0), it holds jGj2  1 D  2; and • if c22 D 1 (then W D 0), it holds jGj2  1 D 64˛kV h2 s1 =.ˇ1 s1 C ˇ2 /; where ˛ D h2 .c12  1/  6c1 2  12, ˇ1 D 32kV .c12  1/h4  192kV .c1 2 C 2/h2 , ˇ2 D 16.c12 1/h6 C. 2 k 2 V 2 .c1 2 1/64c1 4 256.c1 2 C1//h4 12 2 k 2 V 2 .c1 2 C 2/h2 C 144 2 k 2 V 2 .c12  1/, s1 2 Œ0; 1; and V  0. It is simple to prove that ˛  0, ˇ1  0, ˇ2  0 and conclude. t u Since at present a complete analysis for non-zero correlation seems out of reach, we resort to performing numerical studies of the amplification factor G. To this end, we fix some parameters to practical relevant values, v D 0:1;  D 2;  D 0:01: We replace all sinus terms in (5) by equivalent cosinus expressions. Then, condition (5) depends on  and five other parameters: c1 , c2 , y, h, k. We reformulate condition (5) into a constrained optimization problem with constraints induced by typical parameter ranges: c1 ; c2 2 Œ1; 1, y 2 Œ0; 2, h 2 Œ106 ; 101  and k 2 Œ1012 ; 101  (no real restriction on the mesh widths). For different values of  fixed in Œ1; 0, we search maxc1 ;c2 ;y;h;k jG./j2  1

(6)

which has to be less or equal to zero. A line-search global-optimization algorithm based on the Powell’s and Brent’s methods [1, 14] is employed. More precisely, we use the DirectSearch optimization package v.2 for Maple [13] and its derivative-free optimisation method CDOS (Conjugate Direction with Orthogonal Shift). Solving (6) for 50 uniform values of  2 Œ1; 0, we find that the stability condition is always satisfied. The maxima for each  are always negative and very close to 0. This result is in agreement with Lemma 1 (jGj2  1 D 0 for y D 0). We conjecture that the stability condition (5) is satisfied although hard to prove analytically. Moreover, these results give information on the location of the maxima. We observe that extrema are often attained for y close to 0 as already mentioned, and for the extreme values of c1;2 D ˙ 1 which correspond to vanishing V and W; respectively, and h; k seem to be linked. By Lemma 1 the stability condition is

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Fig. 1 Numerical results for (6) for  2 Œ1; 0 and y 2 Œ"; 2=10 (left), y 2 Œ2=10; 2 (right)

satisfied for such values which induce drastic simplification in G: To study the behavior of G according to  away from these values, we solve (6) restricting the range of the parameters to exclude those specific values (where stability is satisfied). We consider c1;2 2 Œ1 C "; 1  " with " D 106 , h 2 Œ106 ; 101  and fix k D h2 as suggested by the above results and the parabolic nature of the PDE. We split the interval for y into Œ"; 2=10 (to exclude 0 and observe significant maximum values) and Œ2=10; 2 (to exclude y D =v which cancels W ). The plots in Figure 1 give the maxima obtained for 50 uniform values of  2 Œ1; 0 and illustrates the influence of : The stability condition is more and more difficult to obtain as  & 1 or y & 0. The stability condition is always satisfied. We refer to [3] for additional numerical experiments where we monitored the error of numerical solutions for vanishing and for non-zero correlation. We observed a similar behavior for both cases and did not observe any stability problems. Acknowledgements Bertram D¨uring acknowledges support from the Isaac Newton Institute for Mathematical Sciences in Cambridge (UK), where a part of this paper has been written, and from ¨ the Austrian-Croatian Project HR 01/2010 of the Austrian Exchange Service (OAD). The authors are grateful to the anonymous referee for his constructive remarks and suggestions.

References 1. Brent, R.P.: Algorithms for Minimization without Derivatives, Prentice-Hall, Englewood Cliffs, NJ (1973) 2. Clarke, N., Parrott, K.: Multigrid for American option pricing with stochastic volatility. Appl. Math. Finance 6(3), 177–195 (1999) 3. D¨uring, B., Fourni´e, M.: High-order compact finite difference scheme for option pricing in stochastic volatility models, preprint, 2010. Available at SSRN: http://ssrn.com/ abstractD1646885 4. D¨uring, B., Fourni´e, M.: Compact finite difference scheme for option pricing in Heston’s model. In: AIP Conference Proceedings 1281, Numerical Analysis and Applied Mathematics,

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Simos, T.E. et al. (eds.), pp. 219–222, American Institute of Physics, Melville, NY (2010) 5. D¨uring, B.: Asset pricing under information with stochastic volatility. Rev. Deriv. Res. 12(2), 141–167 (2009) 6. D¨uring, B., Fourni´e, M., J¨ungel, A.: Convergence of a high-order compact finite difference scheme for a nonlinear Black-Scholes equation. Math. Mod. Num. Anal. 38(2), 359–369 (2004) 7. D¨uring, B., Fourni´e, M., J¨ungel, A.: High-order compact finite difference schemes for a nonlinear Black-Scholes equation. Intern. J. Theor. Appl. Finance 6(7), 767–789 (2003) 8. Heston, S.L.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6(2), 327–343 (1993) 9. in’t Hout, K.J., Foulon, S.: ADI finite difference schemes for option pricing in the Heston model with correlation. Int. J. Numer. Anal. Mod. 7, 303–320 (2010) 10. Ikonen, S., Toivanen, J.: Efficient numerical methods for pricing American options under stochastic volatility. Numer. Meth. Part. Differ. Equat. 24(1), 104–126 (2008) 11. Hilber, N., Matache, A., Schwab, C.: Sparse wavelet methods for option pricing under stochastic volatility. J. Comput. Financ. 8(4), 1–42 (2005) 12. Liao, W., Khaliq, A.Q.M.: High-order compact scheme for solving nonlinear Black-Scholes equation with transaction cost. Int. J. Comput. Math. 86(6), 1009–1023 (2009) 13. Moiseev, S.: Universal derivative-free optimization method with quadratic convergence, http://arxiv.org/abs/1102.1347v1 14. Powell, M.J.D.: An efficient method for finding the minimum of a function of several variables without calculating derivatives. Computer J. 7, 155–162 (1964) 15. Zhu, W., Kopriva, D.A.: A spectral element approximation to price European options with one asset and stochastic volatility. J. Sci. Comput. 42(3), 426–446 (2010) 16. Zvan, R., Forsyth, P.A., Vetzal, K.R.: Penalty methods for American options with stochastic volatility. J. Comp. Appl. Math. 91(2), 199–218 (1998)

Stationary Solutions of Some Nonlinear Black–Scholes Type Equations Arising in Option Pricing Maria de F´atima Fabi˜ao, Maria do Ros´ario Grossinho, Eva Morais, and Onofre Alves Sim˜oes

Abstract Black–Scholes equation has been widely used by academicians and practitioners. In the classical model, transaction costs are not considered and volatility is assumed to be constant, which is not consistent with practice. Having the works of Leland (J. Finance 40:1283–1301, 1985) and Avellaneda et al. (Int. J. Theor. Appl. Finance 1:289–310, 1998) in view, we present two results that contribute to the mathematical study of the above questions. We prove the existence of stationary solutions of nonlinear versions of the standard parabolic Black–Scholes PDE, following the framework of Amster et al. (J. Math. Anal. Appl. 276:231–238, 2002; J. Math. Anal. Appl. 303:688–695, 2005), and using the upper and lower solutions method.

1 Introduction In this paper we study some equations related to option pricing. The classical Black– Scholes model concerns the price of a call or a put option on an underlying asset when the latter is modelled as a geometric Brownian motion. Consider the BS equation, with the final condition, which represents the pay-off function, 1 Vt C rS VS C S 2  2 VS S  rV D 0; 2

V .T I S / D .S /;

M. de F´atima Fabi˜ao  M. do Ros´ario Grossinho ()  O.A. Sim˜oes CEMAPRE, ISEG, Technical University of Lisbon, Rua do Quelhas 6, 1200-781 Lisboa, Portugal e-mail: [email protected]; [email protected]; [email protected] E. Morais Department of Mathematics, UTAD, Apartado 1013, 5000-801 Vila Real, Portugal; and also CEMAPRE, ISEG, Technical University of Lisbon, Rua do Quelhas 6, 1200-781 Lisboa, Portugal e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 26, © Springer-Verlag Berlin Heidelberg 2012

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where V is the option price, S is the stock price at time t, T is the maturity time, r is the interest short rate and  is the stock volatility. In the BS model no transaction costs are considered and the volatility is constant. We will present two results related to these two aspects. We look for stationary solutions and use the upper and lower solutions method applied to nonlinear PDEs. These solutions give the option value as a function of the stock price (which can be relevant in certain options such as perpetual options).

2 Black–Scholes Model with Linear Transaction Costs 2.1 Analytical and Financial Setting In [8], it is considered a model with costs proportional to the value of the transaction. In [2], the cost is assumed to be a percentage of the transaction, given by a linear function f of the number of shares traded, i.e., f ./ D 1  2 ; where  is the number of shares traded and 1 ; 2 > 0: The following nonlinear BS type equation is obtained, where t is the interval between transactions, 1 Vt C  2 S 2 VSS  1 S 2 2

r

2 jVSS j C 2 S 3  2 VSS2 C r.SV S  V / D 0: t

(1)

If VSS > 0 and 1 is assumed to be small, the above equation can be written as 1 2 2  S VSS C 2 S 3  2 VSS2 C r.SV S  V / D 0 Vt C e (2) 2   q 1 2 2 2 where e  D 1  2  t > 0. So, V is a convex solution of (1) in the S -variable if and only if V is a convex solution of (2). We shall look for the existence and localization of convex stationary solutions of (2), without any Lipschitz assumption, which complements the result of [2]. For other insights on the problem see also [7] and [10].

2.2 Problem I Let 0 < c < d and consider the Problem I consisting of the equation 1 2 2 e  S VSS C 2 S 3  2 VSS2 C r.S VS  V / D 0 2

in .c; d /

with the Dirichlet boundary conditions V .c/ D Vc and V .d / D Vd .

(3)

Nonlinear Black–Scholes Type Equations

Theorem 1. If that

Moreover

Vd d

Vc c

<

225

then the Problem I has a unique convex solution V such

Vd  Vc dVc  cVd Vd S  V .S /  SC : d d c d c

V 0 .S /  Vdd

in

Œc; d 

and V 0 .c/ <

Vd Vc d c

< V 0 .d / :

Proof. We sketch briefly the proof. For more details see [6]. Put as notation V 00 D VSS : Then, solving (3) is equivalent to finding V that satisfies 2

00

V D

Q 2 S2 ˙

q

4

Q 4 S4  42 S 3  2 r.V 0 S  V / 22  2 S 3

:

So, we are led to consider the following auxiliary problem   V 00 C g S; V; V 0 D 0; 2

0

where g .S; V; V / D

Q 2 S2 

q

V .c/ D Vc ;

V .d / D Vd

(4)

4

Q 4 S4 C 42 S 3  2 r jV 0 S  V j 22  2 S 3

 0:

It is clear that a convex solution V of (4) satisfying V0 S  V  0 is a solution of Problem I. Moreover, any solution V of (4) is convex since V 00 Dg .S; V; V 0 /  0: So, we study now problem (4). Consider the following functions, defined in Œc; d ; ˛ .S / D

Vd S d

and ˇ .S / D

Vd  Vc dVc  cVd SC : d c d c

Standard computations show that ˛ and ˇ are lower and upper solutions of (4), respectively, and ˛  ˇ. Moreover, the function g satisfies the Nagumo condition in   Vd  Vc dVc  cVd Vd S x SC : E D .S; x; y/ 2 Œc; d   R2 W d d c d c p In fact, easy computations show that, jg .S; x; y/j  k1 C k2 jyj  ' .y/ : R C1 R C1 y y dy D 0 p dy D C1, it is clear that the function As 0 ' .y/ k1 C k2 jyj g satisfies the Nagumo condition in E: So, by the result contained in [4], we can derive that there exists a convex solution V of (4) such that ˛ .S / D 0

Vd dVc  cVd Vd  Vc S  V .S /  SC D ˇ .S / : d d c d c

We show that V S  V  0. As ˛ .d / D V .d / D ˇ .d / and ˛ .S /  V .S /  c ˇ .S / ; it follows that ˇ 0 .d /  V0 .d /  ˛ 0 .d / ; that is, Vdd V  V0 .d /  Vdd : c In particular, we have V0 .d / d  Vd D V .d / : (5)

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 0 Since V00 is nonnegative, then V0 .S / S  V .S / D V00 .S / S C V0 .S /  V0 .S / D V00 .S / S  0 and so V0 .S / S  V .S / is nondecreasing in S . Therefore, by (5), V0 .S / S  V .S /  V0 .d / d  V .d /  0:   Then jV0 S  V j D  V0 S  V and   g S; V ; V0 D

2

Q 2 S2 

q

  4 Q 4 S4  42 S 3  2 r V0 S  V 22  2 S 3

:

So, V is a convex solution of 2

00

V D

Q 2 S2 C

q

4

Q 4 S4  42 S 3  2 r .V 0 S  V / 22  2 S 3

;

and, thus, of 2  2 S 3 .V 00 /2 C 12 Q 2 S 2 V 00 C r .V 0 S  V / D 0 in c; d Œ, satisfying the boundary conditions V .c/ D Vc and V .d / D Vd . 0 Moreover, as seen before, ˇ 0 .d /  V0 .d /  ˛ 0 .d / ; that is, Vd Vd  Vc  V0 .d /  : d c d Vd : By the notions of upper and d lower solutions together, once more, with the convexity of V ; the following strict 0 c inequalities can be easily derived, V0 .c/ < Vdd V c < V .d / : The proof is finished. Since V is convex, V0 is increasing. So, V0 .S / 

3 Black–Scholes Model with Stochastic Volatility

3.1 Analytical and Financial Setting Some generalizations of the classical BS model have been obtained by developing a stochastic volatility model, with a general correlation between the asset price and its volatility. We refer [3], where the following BS type equation is derived 1 1 1 Vt C  2 S 2 VSS C  2 2 V  C  2 S VS   2 V C rS VS D rV: 2 2 2 In this equation, S and  are both stochastic, with a correlation coefficient , and is the volatility of the volatility.

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The nonlinear stationary problem has been studied in [1] by a Newton-type iteration, and the method of upper and lower solutions. The theorem we present concerns a different case and states moreover a localization result.

3.2 Problem II  2 Let ˝, with ˝  RC , be open with boundary @˝ of class C 2 . Consider 8 1 1 1 ˆ ˆ <  2 S 2 VSS C  2 2 V  C  2 S VS   2 V C rS VS D r .V / 2 2 2 ˆ ˆ : V .S; / D h.S; / on @˝: The function  W R ! RC is of class C 1 , the function W RC ! Œa; C1Œ is continuous(a> 0) and bounded above, the function W ˝ !1; 1Œ is continuous and h 2 C ˝ is positive. Some of the hypotheses appear quite naturally, due to the fact that the problem is related to an option pricing model. As notation, S0 ; S1 Œ RC is the projection of ˝ on the S -axis, 0 ; 1 Œ RC is the projection of ˝ on the -axis, 0 D infRC , 1 D supRC and 1 D sup˝ , a1 D 12 02 S02 I a2 D 12 02 02 I b2 D  12 max f0; 1 g 12 1 .  2 We observe that ˝ S0 ; S1 Œ0 ; 1 Œ  RC and ˝  ŒS0 ; S1   Œ0 ; 1 . Define and h0 as   b a1 1 a2 2 a2 .1 0 / 2 > 0; WD max ; e r .S1  S0 /2 .1  0 /2 h0 WD min fh.S; / W .S; / 2 @˝g : Rs Theorem 2. Set .s/ D 0 ./ d  for s 2 R and suppose that inf 2

.h0 /  .s/

s 2 Œ0;h0 

.h0  s/2

< :

(6)

Then Problem II has a solution V , such that 0 < "  V  h1 W D max@˝ h: Proof. We sketch the proof. For more details see [5]. The existence of ordered lower and upper solutions of Problem II will be shown. Step 1: Lower solution ˛ of Problem II. Consider the initial value problem .C u0 /0 D r.u .// u.a/ D 

and u0 .a/ D 0;

(7)

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C > 0. This IVP follows in the framework of the IVP considered in [5]. Moreover, using (6) it can be seen that the condition (7) on the primitive of  of Proposition 2 in [5] is satisfied. So, by this proposition there exists a solution u of (7) such that h0  u.x/   > 0 Put ˛.S; / D u./

and u0 .x/  0:

for .S; / 2 ˝: Then

1 2 2 1 1  S ˛SS C  2 2 ˛  C  2 S˛S   2 ˛ C rS˛S 2 2 2 1 2 2 1 2 D  ˛    ˛ 2 2 1 2 2 00 1  0 0 u  1 12 1 u0 D Au00 ./ C Bu0 ./ D .C u0 /0 D r.u .// 2 2 D r.˛ .S; //: Since u./  h0 D min@˝ h.S; /; then ˛.S; /  h.S; / on @˝: Step 2: Upper solution ˇ of Problem II. Take h1 D max@˝ h.S; /. Set ˇ.S; /  h1 for .S; / 2 ˝. Of course ˇ 2 C 2 .˝/, 1 2 2 1 1  S ˇSS C  2 2 ˇ  C  2 SˇS   2 ˇ C rSˇS D 0  r.ˇ/ in ˝ 2 2 2 and ˇ.S; /  h.S; /

on @˝:

Step 3: Final arguments Since ˛.S; /  h0  h1 D ˇ.S; /; by the classic result that can be found in [9], Problem II has a solution V such that 0 <   V .S; /  max@˝ h.S; /: Acknowledgements The authors are grateful to the anonymous referees for their valuable remarks.

References 1. Amster, P., Averbuj, C.G., Mariani, M.C.: Solutions to a stationary nonlinear Black–Scholes type equation. J. Math. Anal. Appl. 276, 231–238 (2002) 2. Amster, P., Averbuj, C.G., Mariani, M.C., Rial, D.: A Black-Scholes option pricing model with transaction costs. J. Math. Anal. Appl., 303, 688-695 (2005) 3. Avellaneda, M., Zhu, Y.: Risk neutral stochastic volatility model. Internat. J. Theor. Appl. Finance, 1, 289–310 (1998) 4. De Coster, C., Habets, P.: Two-Point Boundary Value Problems Lower and Upper Solutions. Elsevier (2006)

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5. Fabi˜ao, F., Grossinho, M.R., Sim˜oes, O.: Positive solutions of a Dirichlet problem for a stationary nonlinear Black-Scholes equation. Nonlinear Anal. Theor. Meth. Appl. 71, 4624–4631 (2009) 6. Grossinho, M.R., Morais, E.: A note on a stationary problem for a Black-Scholes equation with transaction costs. Int. J. of Pure and Appl. Math. 51, 579–587 (2009) 7. Kim, S.: Hedging option portfolios with transaction costs and bandwidth. J. KSIAM 14(2), 77–84 (2000) 8. Leland, H.E.: Option pricing and replication with transaction costs. J. Finance 40, 1283–1301 (1985) 9. Mawhin, J.: Points fixes, points critiques et probl`emes aux limites. S´eminaire de Math´ematiques Sup´erieures , 92. Presses de l’Universit´e de Montr´eal, Montreal, QC (1985) 10. Wilmott, P., Howison, S., Dewynne, J.: The mathematics of financial derivatives. A student Introduction, Cambridge University press, Cambridge (1995)

Variants of the Combination Technique for Multi-Dimensional Option Pricing Janos Benk, Hans-Joachim Bungartz, Aliz-Eva Nagy, and Stefanie Schraufstetter

Abstract In this paper, we present an approach for pricing different types of multi-dimensional options with partial differential equations. To deal with the curse of dimensionality in higher dimensions, sparse grids are used for discretization. We compare different variants of the combination technique that lead to some sparse grid-like discretizations and show that, for more than two dimensions, only combination techniques with specific properties are able to handle the C 1 discontinuities in the payoff function and, thus, lead to good convergence results.

1 Introduction Sparse grids are often used to tackle multi-dimensional problems since, in contrast to full grids, they can break the curse of dimensionality to some extent by reducing the number of grid points. Thus, in option pricing, the Black–Scholes partial differential equation (BS-PDE) is often solved on sparse grids with the help of a combination technique. Reisinger [9], for example, solves a transformed BS-PDE with such an extrapolation-type approach and uses additionally a non-equidistant grid to improve convergence and accuracy. Leentvaar and Oosterlee [5] extend this idea and introduce another coordinate transformation combined with grid stretching. Mertens[8] transforms the BS-PDE to a simple heat equation and solves it on a so-called “modified sparse grid”. On this non-stretched sparse grid, the BS-PDE

J. Benk  H.-J. Bungartz  S. Schraufstetter () Technische Universit¨at M¨unchen, Boltzmannstr. 3, 85748 Garching, Germany e-mail: [email protected]; [email protected]; [email protected] A.-E. Nagy Universitatea Technicˆa Cluj-Napoca, Romania e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 27, © Springer-Verlag Berlin Heidelberg 2012

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is solved directly in the hierarchical basis without any combination technique. Bungartz et al. [2] presents another direct sparse grid approach that additionally includes local adaptivity. To deal with various types of options in an efficient and general way, we rely on the Theta-notation introduced in [3]. In the previous publication [10], we presented PDE-based pricing with this notation. The main idea is to decompose a complicated financial contract into basic operations. The application of these operators to the given payoff function in the specified order leads to the price function of the option. Provided operators are: • Theta operator: It represents the waiting time between different actions. During this time, the underlyings of the option follow a stochastic path, and we have to solve the corresponding PDE backwards in time. We denote this operator in the operator sequence with the expression fTheta dtg with waiting time dt. • Da operator: It represents a transaction or a cash flow in a financial contract. This might have effect either on the payoff or on one of the underlying values, i.e. a grid axis. In scripts, this operator is denoted with the assignment expression fS D exprg, where expr represents a general expression. An example for this is a dividend payment S D 0:95  S . • Decision operator: During maturity, decisions must be taken in specific options. These are modeled with the decision operator and represented by the construct fif (cond ) body t rue else body f alse endg with an optional false branch. To tackle the C 1 -discontinuities of the payoff function, we investigate different variants of the combination technique which are presented in Sect. 2. We show several numerical examples in Sect. 3 and close with a short conclusion.

2 Combination Technique The combination technique [4] is an efficient method to cope with the curse of dimensionality. Reisinger [9] and Leentvaar and Oosterlee [5] use this technique to price options in higher dimensions. The idea is to combine PDE solutions fl .x/ computed on coarse full grids with mesh widths hi D 2li , i D 1; : : : ; d , for different level vectors l D .l1 ; : : : ; ld / 2 Nd to a sparse grid solution fnc .x/, where n indicates the minimal mesh width 2n . In the following, we consider three different variants of the combination technique: • Standard Combination Technique (S-CT) The standard combination technique fnc .x/

D

d 1 X qD0

  X d 1 .1/ fl .x/ q q

jlj1 Dnq

introduced in [4] results in a common sparse grid of level n.

(1)

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• Truncated Combination Technique (T-CT) The truncated combination technique is a modified version of the S-CT, motivated by the fact that anisotropic basis functions (i.e. long support in only very few dimensions, small support along all other dimensions) lead to problems when representing payoff functions with C 1 -discontinuities on sparse grids. Thus, we again use (1), but add the additional level restriction li > lt r;i 8 i to the terms summed up such that strong anisotropic grids are neglected. In the numerical examples of the next section, we set lt r D . n2 ; : : : ; n2 /, which leads to the same technique as presented in [6]. • Two-Scale Combination Technique (TS-CT) This type of combination technique was introduced in [7] for the twodimensional case and uses only grids with two different scales. For this, we q q define N D n2 and level vectors lq with li D N for i ¤ q and lq D n as well as 0 l D .N; : : : ; N / and combine the corresponding full grids by the formula fnc .x/ D

d X

flq .x/  .d  1/  fl0 .x/:

(2)

qD1

To solve a PDE with one of these combination techniques when the Theta operator is applied, we project the given sparse grid function on all full grids that are needed by the chosen combination technique. On each full grid, we solve the PDE and finally combine the solutions at the end of the waiting time dt with the respective combination formula to fnc .x/ to get a sparse grid solution again.

3 Numerical Examples For the numerical examples, we modeled all underlyings S D .S1 ; S2 ; : : : ; Sd / with a geometric Brownian motion. Hence, the BS-PDE for the price V .S; t/ has the form X @2 V .S; t/ @V .S; t/ @V .S; t/ 1X X C i .S; t/ C i .S; t/j .S; t/i;j @t @Si 2 i D1 j D1 @Si @Sj i D1 d

d

d

rV .S; t/ D 0 with volatilities i .S; t/ D i  Si , correlations i;j , and drifts i .S; t/ D i  Si . Note that computations with other underlying models can be done in a straightforward manner by using different definitions for i .S; t/ and i .S; t/. We discretized the BS-PDE with second-order finite differences and used a geometrical multigrid solver. This solver is able to take into account constraints for the solution, without solving an optimization problem, by applying the constraints after each implicit time step directly. Instead of underlying-specific transformations such as a logarithmic

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transformation we solely rely on grid stretching to achieve better accuracy. Time discretization was done with the second-order scheme of Crank and Nicolson. In the following, we show numerical results for different types of options and different dimensions: • 2-dimensional European basket call option with dividends In our first example, we consider a basket option with a dividend payment at half maturity. The corresponding operator sequence is fTheta 0:3; S1 D S1  0:97; S2 D S2  0:95; Theta 0:3; V D max.S1 C S2  1:90; 0/g. The last operation defines the payoff of the call option which serves as terminal condition for the BS-PDE (3). The error of S-CT has an upper bound only if the mixed second derivatives are bounded [1]. Generally, this does not hold, since the C 1 -discontinuity is usually not parallel to one grid axis. Tabular 1 shows that, in two dimensions, S-CT converges well to the full grid solution, despite the lack of an upper bound for the mixed second derivatives. This convergence is most probably due to grid stretching, and the small number of anisotropic full grids used by S-CT. • 3-dimensional American basket put option Here, the operator sequence is fTheta 1:0; V D P g with P D max.300  S1  S2  S3 ; 0/ and an additional constraint in form of a decision operator fif (V < P ) V D P endg applied at every time step t D 102 . Tabular 2 shows that, for this example, S-CT is not converging exactly to the price all other techniques converge to. In three dimensions, S-CT considers already a large number of strong anisotropic full grids, and the payoff function does not have bounded mixed second derivatives. These factors disturb the extrapolation in case of a C 1 -discontinuous payoff. On the other hand, TS-CT and T-CT do not use these anisotropic grids. For both techniques, the price converges to the Table 1 Results for the 2-dimensional European basket call option for Si D 1, i D r D 0:05, i D 0:4, and  D Œ1:0; 0:5I 0:5; 1:0. The columns contain the sparse grid level n, the option price for the specified stock value S, the relative error at S pointwise as well as on a domain around S in both the L2 and L1 norm. The reference values are given by the solution on the grid with the highest level. In the last line the average order of convergence is shown, which we denote with p Level Full grid S-CT

4 5 6 7 8 9 10 p

Price

Relative error

L2

L1

Price

Relative error

L2

L1

0.242612 0.242139 0.241792 0.241629 0.241593

4.216e-3 2.258e-3 8.224e-4 1.496e-4

4.497e-3 1.741e-3 4.599e-4 6.211e-5

1.198e-2 4.494e-3 1.176e-3 1.477e-4

0.242612 0.242139 0.241792 0.241630 0.241593 0.241583 0.241580

4.269e-3 2.312e-3 8.761e-4 2.043e-4 5.371e-5 1.171e-5

4.520e-3 1.764e-3 4.834e-4 8.571e-5 2.370e-5 5.046e-6

1.204e-2 4.550e-3 1.232e-3 2.047e-4 5.703e-5 1.336e-5

1.6

2.05

2.11

1.7

1.96

1.96

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Table 2 Results for the 3-dimensional American basket put option with Si D 100, i D r D 0:05, i D 0:4, and  D Œ1:0; 0:3; 0:5I 0:3; 1:0; 0:3I 0:5; 0:3; 1:0 Level Full grid S-CT Price 4 5 6 7 8 9

25.7287 26.0752 26.1813 26.2104

p

Relative error 1.838e-2 5.158e-3 1.112e-3

L2

L1

Price

4.682e-3 1.319e-3 2.847e-4

5.243e-3 1.446e-3 3.050e-4

23.8341 23.0534 25.3658 26.7681 26.2613 26.3226

2.02

2.02

2.05

Level T-CT

4 5 6 7 8 p

Relative error 2.854e-1 1.242e-1 3.635e-2 1.693e-2 2.327e-3

L2

L1

6.746e-2 3.153e-2 8.099e-3 4.725e-3 8.021e-4

8.686e-2 4.422e-2 1.088e-2 6.945e-3 1.586e-3

1.73

1.6

1.44

TS-CT

Price

Relative error

L2

L1

Price

Relative error

L2

L1

24.9771 25.8992 26.2047 26.2145 26.2135

4.716e-2 1.199e-2 3.357e-4 4.087e-5

1.212e-2 3.124e-3 8.436e-5 1.422e-5

1.410e-2 3.303e-3 1.477e-4 3.335e-5

22.7946 22.8885 26.1612 26.1933 26.2108

1.301e-1 1.266e-1 1.672e-3 4.492e-4

3.550e-2 3.485e-2 5.185e-4 2.275e-4

5.167e-2 5.189e-2 7.694e-4 4.679e-4

3.4

3.24

2.9

2.71

2.42

2.26

reference solution given by the full grid. For T-CT and TS-CT, there were not any theoretical error bounds derived yet, such that we consider only the empirical results for the convergence. These variants of the combination technique can handle the C 1 -discontinuous payoff in higher dimensions. The costs for this are the partially elimination of the main advantage of the S-CT, given by tackling the curse of dimensionality. However, T-CT and TS-CT are more efficient than solving the PDE on a full grid. • 5- and 6-dimensional European basket call options We finally consider higher dimensional basket options given by the scriptP language expression fTheta 1:0; V D max. diD 1 Si  d; 0/g with d 2 f5; 6g. In 5 dimensions (Table 3) as well as in 6 dimensions (Table 4), divergence can be observed for S-CT, whereas both T-CT and TS-CT again converge to the reference result. Here, additionally, the curse of dimensionality in case of the full grid can be noticed, since computations were only possible up to level 3 or 4, respectively. The measured orders of convergence are for the full grids and T-CT, as expected around 2.0 and higher than [10], where no stretching was used. For TSCT, the measured order is only around 1.5, but there are too few levels for convergence analysis. In theory, as it is the case for Table 2, p should be around 2.0.

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Table 3 Results for the 5-dimensional European basket call option with Si D 1, i D 0:4, i D 0, r D 0, and i;i D 1:0; i;j D 0:1; i ¤ j . For lack of space, we restricted here to the L1 -norm Level Full grid Level S-CT T-CT TS-CT Price

L1

2 3 4

0.3816 1.331e-1 3 0.4181 2.384e-2 4 0.4261 5 6

p

2.48

Price

L1

Price

L1

Price

L1

0.1263 2.984e-1 0.41574 5.551e-2 0.80486 3.766e-1 0.6210 2.122e-1 0.42324 1.608e-2 0.41973 5.013e-2 0.3256 2.101e-1 0.42845 2.805e-3 0.42247 5.095e-2 0.4248 0.42865 0.42822 0.25

2.15

1.43

Table 4 Results for the 6-dimensional European basket call option (settings as specified in Table 3) Level Full grid Level S-CT T-CT TS-CT

1 2 3

Price L1 0.2388 2.550e-1 3 0.4344 1.283e-1 4 0.4745 5 6

p

0.93

Price –0.0494 0.9139 0.1623 0.6237

L1 6.7316e-1 6.4355e-1 5.7402e-1 0.11

Price 0.47105 0.48133 0.48587 0.48577

L1 6.352e-2 4.273e-2 3.472e-3 2.07

Price 1.0082 0.4758 0.4780 0.4855

L1 5.227e-1 5.990e-2 6.470e-2 1.5

4 Conclusion In this paper, we showed different variants of the combination technique to price various types of options in multi-dimensions. It turned out that the standard combination technique does not converge in higher dimensions to the correct result and, thus, can not handle C 1 -discontinuities in the payoff function. This is reasoned in the strong anisotropic grids the technique uses and the lack of an upper bound for the mixed second derivatives of the payoff functions. Omitting the anisotropic grids, as it is done in case of the two-scale technique and the truncated combination technique, this problem is avoided and leads to good convergence results. Acknowledgements We thank for the support from the German Federal Ministry of Education and Research (BMBF).

References 1. Bungartz, H.J., Griebel, M.: Sparse grids. Acta Numerica 13, 1–123 (2004) 2. Bungartz, H.J., Heinecke, A., Pfl¨uger, D., Schraufstetter, S.: Parallelizing a black-scholes solver based on finite elements and sparse grids. In: Proceedings of IEEE International Parallel & Distributed Processing Symposium, Atlanta, USA (2010) 3. Dirnstorfer, S.: An introduction to theta-calculus. SSRN eLibrary (2005)

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4. Griebel, M., Schneider, M., Zenger, C.: A combination technique for the solution of sparse grid problems. In: de Groen, P., Beauwens, R. (eds.) Iterative Methods in Linear Algebra, pp. 263–281. IMACS (1992) 5. Leentvaar, C., Oosterlee, C.: On coordinate transformation and grid stretching for sparse grid pricing of basket options. J. Comput. Appl. Math. 222(1), 193–209 (2008); Special Issue: Numerical PDE Methods in Finance 6. Leentvaar, C.C.W.: Pricing multi-asset options with sparse grids. Ph.D. thesis, Tu Delft (2008) 7. Liu, F., Zhou, A.: Two-scale finite element discretization for partial differential equations. J. Comput. Math. 24(3), 373–392 (2006) 8. Mertens, T.: Option pricing with sparse grids. Computing in Economics and Finance 2005 449, Society for Computational Economics (2005) 9. Reisinger, C.: Numerische methoden f¨ur hochdimensionale parabolische gleichungen am beispiel von optionspreisaufgaben. Ph.D. thesis, Ruprecht-Karls-Universit¨at Heidelberg (2004) 10. Schraufstetter, S., Benk, J.: A general pricing technique based on theta-calculus and sparse grids. In: Proceedings of the ENUMATH Conference. Uppsala (2010)

Stochastic PDAE-Model and Associated Monte-Carlo Simulations for Elastic Threads in Turbulent Flows Nicole Marheineke and Raimund Wegener

Abstract Considering the motion of a long slender elastic thread in turbulent flows, a stochastic aerodynamic drag force concept was derived for a one-way coupling on top of a k- turbulence description in Marheineke and Wegener (SIAM J. Appl. Math. 66:1703–1726, 2006). In this paper we present a generalization of this concept that allows the simulation of practically relevant fluid-solid interactions and yields very convincing results in comparison to experiments. Thereby, it reduces the complex problem to two surrogate models: a universally valid drag model for all Reynolds number regimes and incident flow directions and a turbulence correlation model.

1 Stochastic Elastic Generalized String Model Consider a single elastic thread of slenderness ratio ı D d=l  1 with length l and circular cross-sections of typical diameter d that is immersed in a subsonic highly turbulent air flow with small pressure gradients and Mach number Ma < 1=3. Its dynamics is mainly due to the acting aerodynamic force. The determination of this force requires in principle a two-way coupling of solid structure and fluid flow with no-slip interface conditions. In case of slender threads and turbulent flows, the needed high resolution and adaptive grid refinement make the direct numerical simulation of the coupled fluid-solid-problem not only extremely costly

N. Marheineke () FAU Erlangen-N¨urnberg, Department Mathematik, Cauerstr. 11, 91058 Erlangen, Germany e-mail: [email protected] R. Wegener Fraunhofer-Institut f¨ur Techno- und Wirtschaftsmathematik (ITWM), Fraunhofer Platz 1, 67663 Kaiserslautern, Germany e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 28, © Springer-Verlag Berlin Heidelberg 2012

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and complex, but also still impossible for practically relevant applications. Since the thread’s influence on the turbulent flow is negligibly small due to the slender geometry, it makes sense to associate to the force a stochastic drag that characterizes the turbulent flow effects on the thread and allows a one-way coupling, [5, 6]. We represent the thread as arc-length parameterized time-dependent curve r W 3 Œ0; l  RC 0 ! R with line weight .A/. Then, its dynamics can be asymptotically modeled by a system of stochastic partial differential equations with algebraic constraint of inextensibility, i.e., k@s rk2 D 1

(1a)

.A/ @tt r ds dt D f @s .T @s r  @s .EI @ss r// C .A/ g C a.r; @t r; @s r; s; t/ g ds dt (1b) C A.r; @t r; @s r; s; t/  dws;t supplemented with appropriate initial and boundary conditions, where N t/  w; k.x; t/; .x; t/; .x; t/; d.s//; a.x; w; ; s; t/ D m.; u.x;

(1c)

N t/  w; k.x; t/; .x; t/; .x; t/; d.s// A.x; w; ; s; t/ D L.; u.x;

(1d)

N t/  w; k.x; t/; .x; t/; .x; t//  D.; u.x; and k:k2 is the Euclidean norm. This stochastic elastic generalized string model is deduced from the dynamical Kirchhoff-Love equations [1] for a Cosserat rod being capable of large, geometrically nonlinear deformations, neglecting torsion, [4,5]. In (1b) the change of the momentum is balanced by the acting internal and external forces. The internal line forces stem from bending stiffness indicated by Young’s modulus and the moment of inertia .EI/ as well as from traction. The tractive force T W Œ0; l  RC 0 ! R can be viewed as Lagrangian multiplier to (1a). The external line forces come from gravity g and aerodynamics a, A. The aerodynamic force is derived on basis of a stochastic k- turbulence model. Expressing the instantaneous flow velocity as sum of a mean and a fluctuating part, the Reynolds-averaged Navier-Stokes equations (RANS) yield a deterministic 3 description for the mean velocity uN W R3  RC 0 ! R , whereas two further transport C equations for the kinetic turbulent energy k W R3  RC and dissipation 0 ! R C 0 rate  W R3  RC ! R characterize the random fluctuations u according to 0 k D EŒu0  u0 =2 and  D  EŒru0 W ru0  with kinematic viscosity , density  and expectation EŒ:. Analogously, the aerodynamic force is split into a mean and a fluctuating part. Acting as additive Gaussian noise in (1b), it depends on the N k, , and , , cf. (1c), (1d). Thereby, the deterministic mean flow quantities u, force m W S 2  R3  .RC /4 ! R3 as well as the associated splitting operator L W S 2  R3  .RC /4 ! R33 are determined by the chosen air drag model f which N t/@t r, and is a function of the mean relative velocity between fluid and thread, u.r; the thread tangent @s r. The correlated fluctuations are asymptotically approximated

Stochastic PDAE-Model and Associated Monte-Carlo Simulations for Elastic Threads

241

by Gaussian white noise with turbulence-dependent amplitude, where .ws;t ; .s; t/ 2 3 Œ0; l  RC 0 / denotes a R -valued Wiener process (Brownian motion). The amplitude 2 3 C 3 D W S  R  .R / ! R33 represents the integral effects of the localized centered Gaussian velocity fluctuations on the relevant thread scales by containing the necessary information of the spatial and temporal correlations of the doublevelocity fluctuations  D EŒu0 ˝ u0 . Consequently, the performance of the aerodynamic force mainly relies on two models, i.e. the air drag model f (inducing m and L) and the turbulence correlation approximation  (inducing D). Applying the Global-from-Local Concept of [5] we present here local models that we globalize by superposition, for details see [6]. So, we handle the delicate interaction problem by help of two surrogate models: a drag model for an incompressible flow around an inclined infinitely long circular cylinder and a correlation model for incompressible homogeneous isotropic turbulence. 1.0.1 Air Drag Model In an incompressible flow, the force f acting on a fixed, infinitely long circular cylinder is exclusively caused by friction and inertia, [7–9]. It depends on the material and geometrical properties (fluid density , kinematic viscosity , cylinder diameter d) and the specific inflow situation (inflow velocity v, cylinder orientation , kk2 D 1). Non-dimensionalizing the line force f and flow velocity v with the typical mass d3 , length d and time d2 = yields a reduction of the dependencies,  2 f.; v; ; ; d/ D f d

  d ; v ; 

vD

 v: d

We focus on the dimensionless quantity f .; v/. Assuming v 6k  at first, we introduce a .; v/-induced orthonormal basis .n; b; / by nD

v  v  ; vn

b D   n;

v D v  ;

vn D

q

v 2  v2 :

Because of the rotational invariance of the force, its components depend only on the scalar products v   and v 2 . The binormal force component vanishes in case of a circular cylinder due to symmetry reasons such that f .; v/ D fn .vn ; v /n C f .vn ; v / holds. For the dependencies of normal and tangential component, Hoerner [3] postulated an independence principle which is strictly proved for a stationary flow in [6]. Theorem 1 (Independence Principle) • The normal force fn is independent of the tangential velocity v . • The tangential force f depends linearly on the tangential velocity v .

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The force can be expressed in terms of normal and tangential drag cn , c or resistance rn , r functions fn .vn ; v / D vn2 cn .vn / D vn rn .vn /;

f .vn ; v / D v vn c .vn / D v r .vn /:

Model 2 (Universal Drag) [6] The continuously differentiable drag functions cn , c are composed of Oseen theory, Taylor heuristic and numerical simulations. They are experimentally validated and hold true for all Reynolds number regimes,v 6k    8 v / 1  vn2 .S 2  S=2 C 5=16/=.32S / ˆ < 4=.S P n  3 j cn .vn / D exp j D0 pn;j ln vn ˆ : p 2= vn C 0:5   8  1/vn / 1  vn2 .2S 2  2S C 1/=.16.2S  1// ˆ < 4=..2S P  3 c .vn / D exp p;j lnj vn j D0 ˆ : p = vn

vn < v1 v1  vn  v2 v2 < vn vn < v1 v1  vn  v2 v2 < vn

with S.vn / D 2:0022  ln vn , transition points v1 D 0:1, v2 D 100, amplitude  D 2. The C 1 -regularity involves the parameters pn;0 D 1:6911, pn;1 D 6:7222101, pn;2 D 3:3287  102 , pn;3 D 3:5015  103 and p;0 D 1:1552, p;1 D 6:8479  101 , p;2 D 1:4884  102 , p;3 D 7:4966  104 . To be also applicable in the special case of a transversal incident flow v k  and to allow for a realistic smooth force f , the drag need to be adapted for vn ! 0. Taking into account Stokes theory for finitely long cylinders a ı-based regularization of the associated resistance functions rn , r is proposed in [6]. It matches Stokes resistance coefficients of higher order for vn  1 to those of Model 2, assuming ı < 3:5102. Coming back to the turbulent flow around a moving long flexible thread, we generalize the drag by glueing together the locally valid results for the cylinder. The force acting on the thread at a certain position is then given by f .@s r; u.r; t/  @t r/ D f .@s r; .Nu.r; t/  @t r/ C u0 .r; t//; where r and u D uN C u0 describe the dimensionless thread curve and flow velocity. Analogously to the RANS-averaging ansatz, the force is approximated by an appropriately chosen linear Gaussian process that is split into a mean part m and a fluctuation part. The drag fluctuations inherit the stochastic properties of the turbulence by being modeled linearly in the locally isotropic, centered Gaussian velocity fluctuations with the matrix-valued linearization operator L. In particular, we have ! ! r Z 1  2 2k m.; v; k/ D  exp d f ; v C .2/3=2 R3 3 2

Stochastic PDAE-Model and Associated Monte-Carlo Simulations for Elastic Threads

1 L.; v; k/ D .2/3=2

r

3 2k

r

Z R3

f

; v C

243

! !  2 2k  ˝  exp d; 3 2

depending on thread tangent , mean relative velocity v between mean flow and thread and turbulent kinetic energy k. From the dimensionless quantities we return to the associated dimensional ones in (1) by using      2 d d2 d d2 m.; v; k; v; ; d/ D m ; v; 2 k ; L.; v; k; v; ; d/ D L ; v; 2 k : d    

1.0.2 Turbulence Correlation Model The amplitude D represents the integral effects of the spatial and temporal correlations of the double-velocity fluctuations on the relevant thread scales. In an incompressible, homogeneous and isotropic turbulent flow, D depends on the turbulence properties (turbulent kinetic energy k, dissipation rate , kinematic viscosity ) and the specific thread-flow relation (mean relative velocity v, thread tangent , kk2 D 1). Non-dimensionalizing the correlation representant D, mean velocity v and viscosity  with the typical turbulent length k3=2 = and time k= yields a reduction of the dependencies, D.; v; k; ; / D

  k7=4 1  D ; p v; 2  ;  k k

vD

p kv;

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k2 : 

We proceed with the dimensionless quantity D.; v; /. Considering an advectiondriven flow, the correlations  of the velocity fluctuations can be modeled by help of an initial correlation tensor  0 and a temporal decay function ', i.e. O t C tO/ ˝ u0 .x; O tO/ D  0 .x  t uN / '.t/: .x C x; O t C tO; x; O tO/ D EŒu0 .x C x; The Fourier transform F 0 of the initial correlations is the spectral density which is exclusively determined by the scalar-valued energy spectrum E in case of incompressible isotropic turbulence. Gathering the existing knowledge [2] about E we use Model 3 (Energy Spectrum) [5] The continuously differentiable energy spectrum 8 5=3 P6

j ˆ

ˆ j D4 aj . 1 / < 1 E. ; / D CK 5=3 ˆ ˆ : 5=3 P9

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< 1

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where the -dependent transition wave numbers 1 and 2 are implicitly given by Z

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induces a velocity fluctuation field that agrees with Kolmogorov’s 5/3-Law and the k- turbulence model. The Kolmogorov constant is CK D 1=2. and the regularity parameters are a4 D 230=9, a5 D 391=9, a6 D 170=9, b7 D 209=9, b8 D 352=9, b9 D 152=9. The restriction 0 < < crit  3:86 coming from the condition 0 < 1 < 2 < 1 is practically irrelevant, since the turbulence theory presupposes  1. The temporal decay is assumed to be '.t/ D exp.t 2 =2/ with Fourier transform F' . Then, the tensor-valued amplitude D formulated in the .; v/-induced orthonormal basis is D.; v; / D dn .vn ; / n ˝ n C db .vn ; / b ˝ b C d .vn ; /  ˝  Z 1 E. ; / 2 ln;b; .vn / d ; .vn ; / D 4 dn;b;

0 Z =2 ln;b; . / D fsin2 ˇ; cos2 ˇ; 1g F' . cos ˇ/ dˇ 0

Since dn2 C db2 D d2 holds, the effort for the computation of D reduces to the evaluation of two scalar-valued functions d , db depending on two parameters.

2 Results and Discussion The system (1) of stochastic partial differential equation that models the thread dynamics in turbulent flows is implemented in the software tool FIDYST,1 where it is solved by a method of lines. The use of a spatial finite difference method of higher order ensures the appropriate approximation of the algebraic constraint. The Box-Muller method generates the Gaussian deviates for the stochastic force. Incorporating the force amplitude explicitely, the time integration is realized by a semi-implicit Euler method with step size control. So far, our proposed stochastic force model is successfully applied to the simulation of thread-turbulence interactions in technical textile manufacturing. In [6] for example, we show its performance in a specific industrial melt-spinning process of nonwoven materials where hundreds of threads are computed in parallel by Monte-Carlo simulations. The numerical results turn out to coincide very well

1

FIDYST: Fiber Dynamics Simulation Tool developed at Fraunhofer ITWM, Kaiserslautern, for details see [4].

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with experimental data. For further applications, numerical simulations and figures we refer to the contribution by Olawsky et al. in this book.

References 1. Antman, S.S.: Nonlinear Problems of Elasticity. Springer, New York (2004) 2. Frisch, U.: Turbulence. The Legacy of A. N. Kolmogorov. Cambridge University Press, Cambridge (1995) 3. Hoerner, S.F.: Fluid-dynamic drag. Practical information on aerodynamic drag and hydrodynamic resistance. Published by the author. Obtainable from ISVA (1965) 4. Klar, A., Marheineke, N., Wegener, R.: Hierarchy of mathematical models for production processes of technical details. ZAMM 89, 941–961 (2009) 5. Marheineke, N., Wegener, R.: Fiber dynamics in turbulent flows: General modeling framework. SIAM J. Appl. Math. 66(5), 1703–1726 (2006) 6. Marheineke, N., Wegener, R.: Modeling and application of a stochastic drag for fibers in turbulent flows. Int. J. Multiphas. Flow. 37(2), 136–148 (2011) 7. Schlichting, H.: Grenzschicht-Theorie. Verlag G. Braun, Karlsruhe (1982) 8. Sumer, B.M., Fredsoe, J.: Hydrodynamics around cylindrical structures. World Scientific Publishing, London (2006) 9. Zdravkovich, M.M.: Flow around circular cylinders, Vol 1: Fundamentals. Oxford University Press, New York (1997)

Production Networks with Stochastic Machinery Default Simone G¨ottlich, Stephan Martin, and Thorsten Sickenberger

Abstract We present a model of production networks that includes random breakdowns of individual processors. The defaults of processors are exponentially distributed and the time-continuous formulation of network dynamics yields a coupled PDE-ODE system with Markovian switching. Its solution is a piecewise deterministic process, which allows us to use a modified stochastic simulation algorithm to trace stochastic events and to simulate the deterministic behavior of the network between them. The impact of stochastic default is illustrated with an exemplary Monte-Carlo simulation.

1 Stochastic Network Model Real world production planning in many cases no longer is a deterministic problem, but includes the consideration of stochastic events, such as machine failure (see e.g. [7]). Even though random failures are unavoidable by their very nature, the gathering of statistical data on e.g. mean availability of machines helps planners to

S. G¨ottlich School of Business Informatics and Mathematics, University of Mannheim, 68131 Mannheim, Germany e-mail: [email protected] S. Martin () Department of Mathematics, TU Kaiserslautern, P.O. Box 3049, 67653 Kaiserslautern, Germany e-mail: [email protected] T. Sickenberger Maxwell Institute and Heriot-Watt University, Department of Mathematics, Edinburgh EH14 4AS, UK e-mail: [email protected]

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deal with unexpected failure and to optimize production. In this work, we present a method to include stochastic failure into a mathematical network model for timecontinuous supply chains. We start from the network model originally discussed in [5]. Here, differential equations are used to model and study the dynamics of a production network. Each machine (processor) is represented by an arc e of given length l e, and each distribution knot of the supply chain is represented by a vertex connecting at least two arcs. The topology of the supply chain is thus modeled using a directed, connected graph. The flow of products is modeled using a continuous product density rather than tracing individual items (see [2] for an up-to-date introduction to continuous models). The propagation of product density through a processor can be interpreted as monitoring the level of completion of the particular production process. Properties of processors e 2 f1; : : : ; M g are modeled with (i) a finite processing capacity e, and (ii) a finite processing speed v e . In front of each processor, there is a buffer (queue) q e, where products that temporarily cannot enter a processor are stocked. In order to model machine default, we assume that breakdown times of individual processors as well as their repair times are independent exponentially distributed, e e such that the parameters on and off describe mean time between failures (MTBF) and mean repair time (MRT), respectively. From that, one obtains a stochastic process r e taking values in f0; 1g to model the evolution of operating and inoperating time periods of a processor, which depends on the time t > 0 as well as on the random sample ! 2 ˝, whose argument ! is usually dropped. The choice of exponential distributions is a natural choice motivated in queuing theory and related models for chains of processors (see [3]). A realization or scenario of the multivariate state process r.t/ D .r 1 .t/; : : : ; M r .t// 2 f0; 1gM is given by a set of sampled switching points tNi , where one of the M processors changes its current state from operating to in-operating (1 to 0) or vice versa. These points can be sampled distributed random P viae an exponential e e e e variable with rate parameter sum D M e D 1  where  D 1=on or  D 1=off , respectively. It describes the length of the time period up to the next switching and  the sampled switching points create a set T D ftN0 ; tN1 ; : : : g of strictly increasing times. Between these times r.t/ is a random constant (see [1, 8]). Having the signal of on and off states available, we include it into the network model by setting the capacity e to zero, if the respective processor is currently in off-state. This means that no more products are processed—no matter their current level of completion inside the processor—and that no more products enter the processor. Arriving product inflow is stored in the respective queue. Once the processor gets back to work, production is restarted at the same stage it was stopped and no products are lost. The network model reads as follows. Let t 2 Œt0 ; T . The density e .x; t/ of goods inside processor e satisfies the hyperbolic PDE @t e .x; t/ C @x f e .e .x; t// D 0;

x 2 Œae ; b e ; l e D b e  ae

(1)

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and the transport of goods along the arc e is described by the flux function f e .e .x; t// D min.v e  e .x; t/; e  r e .t//:

(2)

The queue-load q e .t/ of each processor is modeled by the ODE e e @t q e .t/ D gin .t/  gout .t/;

(3)

describing the difference of all incoming and outgoing fluxes, which are designed as e gin .t/ D

8
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f eN .eN .b eN ; t//

if e has predecessors if e has no predecessor

in

(4)

and ( e gout .t/

D

e minfgin .t/ ; e  r e .t/g

if q e .t/ D 0,

e  r e .t/

if q e .t/ > 0,

(5)

respectively. Ginv .t/ is the network inflow, s denotes a map identifying any arc with its vertex of origin, and As.e/;e .t/ are control functions setting distribution rates at vertices. The PDE boundary condition couples the ODE and PDE via the outgoing flux function e e .ae ; t/ D gout .q e .t//=v e

(PDE boundary conditions).

(6)

See [6] for a more detailed description and discussion of the model. In the present work, we restrict ourselves to constant distribution rates As.e/;e .t/ WD ˛ s.e/ , e.g. the naive control, where outgoing products are equally distributed on all successive processors.

2 Numerical Aspects Realizations of the stochastic supply chain model are computed by a subsequent iteration of two algorithms: the simulation of switching points of the multivariate state-process r and the deterministic computations between switchings. It is therefore related to the stochastic simulation algorithm introduced, see [4], to compute a scenario of the Chemical Master Equation, which is a hybrid ODE. However, in our approach the deterministic dynamics is modeled by a coupled ODE-PDE system. A sketch of the simulation algorithm reads as follows: (1) Due to its memoryless property, the sampling of the next switching points w.r.t. the exponential distribution of r.t/ can be done by drawing first an

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exponential distributed random variable to sample the next switching point, and second, a uniform distributed one to sample the next switching processor. (2) Between sampled switching points t 2 ŒtNi ; tNiC1  the solution of the coupled ODE-PDE system is approximated by an interaction of an ODE and PDE solver. For simplicity, the hyperbolic PDE is discretized by an upwind scheme, whereas the ODE is solved by the explicit Euler method. An interesting part of the algorithm is the coupling of both equations through (4), (5): Careful attention has to be taken in sequencing the numerical iteration in a correct order, such that no mass is lost in during a switch. For details, we again refer to [6]. Let us note, that the deterministic algorithm is ideally parallelizable and can treat any network topology, including cycles and loops.

3 A Case Study: The Diamond Network For the scope of this article, we want to illustrate the stochastic network model and the impact of default with a case study of averaged quantities obtained from a Monte-Carlo simulation. To do so, we study the diamond network illustrated in Fig. 1, which consists of 7 processors and two diversion controls ˛1 , ˛2 located in front of processors 2, 3 and 4, 5 respectively. Processing rates are 1 D 40, 2 D 30, 3 D 20, 4 D 20, 5 D 5, 6 D 10, 7 D 10, all processors share common length l e D 1 and velocity v e D 1. The inflow of products is illustrated in Fig. 1. e e Stochastic default is set to on D 0:95, off D 0:05, except for processors 2 and 4, e e where on D off D 0:5. We fix the controls to ˛1 D 0:6 (60% into proc. 2); ˛2 D 0:5. In particular, the chosen setting creates several bottlenecks in the network, where the total capacity behind a network vertex is smaller than the possible maximum inflow. In total, we have a setting where the influence of stochasticity on the networks production performance can be hardly predicted without the use of simulation. One intuitive guess to predict the dynamics of the network could be as follows: Since we

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know the long-run average availability of each processor, the expectational capacity of each processor is given by eguess D EΠr e e  D

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and we can compute the stochastic dynamics using these capacities in a purely deterministic simulation. We include this computation in our example to demonstrate that this intuitive guess clearly does not work. We simulate N D 1;000 R scenarios on a time interval Œ0; 100 to ensure that all incoming product flow of Gin .t/dt D 250 units are completely processed, even in extreme events. In Fig. 2, we show queue loads in front of the processors over time. In orderto illustrate  the dynamics of the e e processors in Fig. 3, we plot the sample mean of  b a ; t for each processor and 2 different times. For both figures, the 95% confidence intervals of the sample means are too small to be well-distinct in the plot. Looking at Fig. 2, we see that stochastic queues can either significantly overtop deterministic ones (q2, q3, q4) or stay below deterministic peaks (q5, q6, q7), and take longer to be emptied. The intuitive guess predicts the emergence of queuing in q2 but fails to do so in q3, q4. In Fig. 3, processor emptying sees delay caused by stochasticity, which is considerably large in p2 and p4. The intuitive guess is never an approximation to the stochastic case and can even be worse than the deterministic setting, c.f. p2 and p4. Finally, the impact of stochastic dynamics can be observed in the total production time. Figure 4 shows the probability distribution of total production time over 1,000 scenarios. We can see that both the deterministic as well as the intuitive guess production time lie outside the stochastic distribution.

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Acknowledgements This work was supported by the BC/DAAD ARC project “Robust simulation of networks with random switching” (1349/50021880), the DFG grant HE 5386/6–1, and Leverhulme Trust grant F/00 276/K.

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References 1. Davis, M.H.A.: Markov Models and Optimisation. Monograph on Statistics and Applied Probability 49, Chapmand & Hall, London (1993) 2. D’Apice, C., G¨ottlich, S., Herty, M., Piccoli, B.: Modeling, Simulation, and Optimization of Supply Chains: A Continuous Approach. SIAM (2010) 3. Degond, P., Ringhofer, C.: Stochastic dynamics of long supply chains with random breakdowns. SIAM J. Appl. Math. 68(1), 59–79 (2007) 4. Gillespie, D.T.: A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Phys. Chem. A 104, 403–434 (1976) 5. G¨ottlich, S, Herty, M., Klar, A.: Network models for supply chains. Comm. Math. Sci. 3(4), 545–559 (2005) 6. G¨ottlich, S., Martin, S., Sickenberger, T.: Time-continuous production networks with random breakdowns. Networks and Heterogeneous Media (NHM) 6(4), 695–714 (2011) DOI: 10.3934/nhm.2011.6.695 7. Kelly, F.P., Zachary, S., Ziedins, I. (eds.): Stochastic Networks: Theory and Applications. Oxford University Press, Oxford (2002) 8. Mao, X., Yuan, C.: Stochastic Differential Equations with Markovian Switching. Imperial College Press, London (2006)

Verified Simulation for Robustness Evaluation of Tracking Controllers Marco Kletting and Felix Antritter

Abstract In this contribution a method for investigating the robustness properties of tracking controllers using verified simulation is suggested. This method allows to compare the controllers with respect to robustness against uncertainties in the parameters of the plant and uncertain initial conditions of measured and unmeasured states. A suitable robustness criterion is formulated which can be evaluated using interval methods. To illustrate the approach, we compare the robustness properties of three conceputally different flatness based tracking controllers with dynamic output feedback, which are applied to a simple example system.

1 Introduction Flatness based controller design [6, 7] is a powerful tool for motion planning and trajectory tracking for linear and nonlinear systems. It has been applied successfully to numerous problems of industrial relevance. However, a major drawback is the lack of techniques that allow to investigate the robustness of flatness based tracking controllers against, e.g., parameter uncertainties in the plant and measurement uncertainties due to non-ideal sensors. It has been shown in [2, 10, 11] that interval methods [4,5] are a suitable tool for analyzing the properties of the resulting closed loop. Using interval methods, the

M. Kletting Multi-Function Airborne Radars (OPES22), Cassidian Electronics, Woerthstr. 85, 89077 Ulm, Germany e-mail: [email protected] F. Antritter () Automatisierungs- und Regelungstechnik, Universit¨at der Bundeswehr M¨unchen, Werner-Heisenberg-Weg 39, 85579 Neubiberg e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 30, © Springer-Verlag Berlin Heidelberg 2012

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maximum admissible range of parameter uncertainties in the plant and in the initial state is determined such that the deviation from a desired trajectory is guaranteed within specified tolerances. In this paper we illustrate the approach for the comparison of the robustness of three different flatness based tracking controllers. The compared approaches are the “classical” flatness based tracking controller with exact linearization of the tracking error dynamics (see e.g. [7]), the tracking controller with exact feedforward linearization [9] and an approach using a nonlinear feedforward together with a linear error feedback [1,3]. The proposed robustness analysis using interval methods cannot be used to obtain general results on the different controllers but it allows to evaluate the performance for a given system. This is an important contribution compared to previous relatively unsharp discussions as in [9]. Here, we chose the magnetic levitation system which is a structurally rather simple single input differentially flat system and hence simplifies the discussion. The system has been chosen in [9] as an example system for the illustration of the feedforward linearizing controller. This paper is organized as follows. In Sect. 2 we introduce the magnetic levitation system and present the three controllers as well as the tracking observer and reduced order output feedback respectively, which are necessary for the implementation of the controllers. Then, in Sect. 3 a suitable robustness criterion is formulated, which can be evaluated with Taylor model based verified integration of nonlinear uncertain systems. Simulation results are shown in Sect. 4 and conclusions are drawn in Sect. 5.

2 Magnetic Levitation System A simplified model of a magnetic levitation system is given by [12] d x dt 1

D x2 ;

d x dt 2

D

u2 k g m .c  x1 /2

(1)

3

with the nominal parameters k0 D 58:041 kgs2 cm , g0 D 981 cm , m0 D 0:0844 kg and A2 s2 c0 D 0:11 cm. Establishing the flatness property of (1) amounts to derive a differential parameterization of the states and the inputs with the flat output. A flat output yf of (1) is given by yf D x1 . For system (1) it is rather simple to derive the differential parameterization with the flat output: .x1 ; x2 / D

D .yf ; dtd yf /; r m u D u .yf ; dtd yf ; yRf / D .c  yf / .yRf C g/: k d x .yf ; dt yf /

(2) (3)

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For the load of the levitation system a set point change is considered, i.e. a trajectory has to be planned such that the following boundary conditions are satisfied: .x1 .0 s/; x2 .0 s// D .0:4 cm; 0

cm cm /; .x1 .0:2 s/; x2 .0:2 s// D .0:2 cm; 0 /: s s (4)

In view of the differential parameterization (2) this yields the boundary cond ditions .yf ;d .0 s/; dtd yf ;d .0 s// D .0:4 cm; 0 cm s /, .yf ;d .0:2 s/; dt yf ;d .0:2 s// D cm .0:2 cm; 0 s / for a corresponding trajectory yf ;d for the flat output. This can be satisfied by assigning for yf ;d a third order polynomial in t. A feedforward controller which, in the case that the initial states x.0/ coincide with the assumption in (4), achieves this trajectory is obtained to r ud D

d u .yf ;d ; dt yf ;d ; yRf;d /

D .c  yf ;d /

m .yRf;d C g/: k

(5)

The “original” flatness based tracking controller (see e.g. [7]) which stabilizes the designed trajectory results to r uel D .c  yf /

m ..yRf;d  aQ 1 dtd e  aQ 0 e/ C g/ kc

(6)

with e D yf  yf ;d (recall also the transformation (2)). For the given system (1) in nonlinear controller form it is easy to verify, by substitution into the system equations, that this controller achieves the linear tracking error dynamics 0 D eR C aQ 1 dtd e C aQ 0 e:

(7)

The linearization matrices of (1) clearly have the structure   @f ˇˇ 0 1 ; D AD ˇ ac0 .t/ 0 @x xd ;ud

  @f ˇˇ 0 BD : D ˇ bc .t/ @u xd ;ud

(8)

Then, the so-called feedforward linearizing controller [9] results to r uf l D .c  yf ;d /

m ..yRf;d  aQ 1 dtd e  .ac0 .t/ C aQ 0 /e/ C g/: kc

(9)

This controller has been parameterized such that it yields the same linearized closed loop dynamics as (6). Finally, we consider a controller with a nonlinear feedforward and a linear feedback ulin D ud C

 1  aQ 1 dtd e  .ac0 .t/ C aQ 0 /e : bc .t/

(10)

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Again, this controller has been parameterized such that the same linearized closed loop dynamics result. All used controllers need full state information for the implementation (recall (2)). However, it is assumed that only the flat output of (1) is available for measurement, i.e. y D h.x/ D x1 :

(11)

Thus, to implement the controllers, a nonlinear tracking observer [8], has to be implemented. It has the form (the observer gains li are used to assign time invariant poles in the observer normal form) d O1 dt x

D xO 2  l1 .t/.x1  xO 1 /;

d O2 dt x

D

u2 k  g  l2 .t/.x1  xO 1 /: m .c  xO 1 /2

(12)

Using the measured output (11) and the observer state xO 2 the feedbacks (6) and (9) can be implemented. On the other hand the linear feedback part of the controller (10) can directly be estimated with a first order linear dynamic output feedback [1], which yields a linear time invariant estimation error between uO lin and ulin : d dt 

D aO .t/ C by .t/y;

uO lin D ud C  C cy .t/y

with  2 R:

(13)

3 Interval Methods and Robustness Analysis It is assumed that the initial conditions x.0/ and the system parameters p of the magnetic levitation system are uncertain but bounded by interval vectors Œx.0/ D Œx.0/I x.0/ and Œp D ŒpI p. It is further assumed that constraints for the at most tolerable deviations from the reference trajectory for the controlled system can be specified in the following manner jxi .t/  xi;d .t/j < ıi ;

i D 1; 2I 8 t 2 Œ0 s; 0:2 s:

(14)

The goal of the robustness analysis is to determine parameter values and initial states Sin D

ˇ    ˇ x.0/ D z.0/ˇˇ jxi .t/  xi;d .t/j  ıi 8t 2 Œ0 s; 0:2 s; i D 1; 2 ; p

(15)

for which it can be guaranteed that the conditions for robustness in equation (14) are fulfilled. Here z is the extended state vector defined by the system states x and parameters p. The determination of the set Sin is done by splitting the extended initial state vector Œz.0/ in subboxes and performing a verified integration [1] over

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the desired time span. For the verified integration a Taylor model based solver as implemented in COSY-VI [5] is used. Then, three cases have to be distinguished: 1. If for some t 2 Œ0 s; 0:2 s, the resulting enclosure of the trajectory is completely outside the specified tolerances the corresponding box is not admissible and can be deleted. 2. If on the other hand the resulting enclosures of the trajectory lie completely inside the tolerance for all t 2 Œ0 s; 0:2 s, the corresponding box is admissible. 3. Subboxes which lead to enclosures of the trajectories which are partially (but not completely) outside for some t 2 Œ0 s; 0:2 s but also not completely inside for all t 2 Œ0 s; 0:2 s have to be split further until a user given maximum number of splitting operations is reached.

4 Simulation Results For the simulation, we assumed that the parameter k is not exactly known but 3 bounded by k 2 Œ54I 62 kgs2 cm . Additionally, we assume that the inital state x2 .0/ A2 is uncertain but bounded by x2 .0/ 2 Œ0:01I 0:01 cm. Figure 1 shows the parameter values in this range for which the robustness criterion (14) with ı1 2 Œ0:2I 0:2  103 m and ı2 2 Œ0:01I 0:01 m/s is satisfied for the controller with exact linearization and with feedforward linearization respectively with tracking observer. Finally, Fig. 2 shows the result for the linear controller, which is estimated by the reduced order output feedback (13). The parameters aQ have been chosen such that the roots of (7) are placed at 70. The eigenvalues of the tracking observer in observer normal form have been placed at 140. Also the root of the first order estimation error dynamics resulting from the linear dynamic output feedback (13) has been placed at 140. Thus, for all controllers a comparable behaviour of the linearization has been

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Fig. 1 Admissible (yellow), undecided (red) and non-admissible (green) parameters for flatness based tracking controller with exact linearization and observer (left) and for the flatness based tracking controller with feedforward linearization and observer (right)

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Fig. 2 Admissible (yellow), undecided (red) and non-admissible (green) parameters for linear dynamic output feedback

used. The feedforward linearizing controller can indeed accept a bigger parameter range than the controller with exact linearization. However, the biggest admissible parameter range is achieved by the linear controller. Additionally, it is much simpler to implement the linear controller with standard commercially available controllers. The analysis yields clear results for the considered system, but as has been mentioned already in the introduction, the result cannot be generalized to robustness properties of the different controllers for other systems.

5 Conclusion In this paper the robustness of three different flatness based tracking controllers which use only output feedback for a magnetic levitation system has been analyzed using interval methods. Verified integration of subsets of the uncertain parameter and initial state interval led to guaranteed enclosures of the admissible sets of the parameter and the initial state which can be kept within the specified tolerances by the investigated controllers. Based on the robustness analysis the most robust controller for the given system could be determined. Let us emphasize once more that such explicit results are a new contribution for flatness based tracking controllers. The method can also be extended to other control strategies. And for the evaluation of the robustness analysis also other validated ODE solvers like VNODE [13] or VALENCIA-IVP [4] can be used.

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References 1. Antritter, F.: Tracking Control for Nonlinear Dynamics using Differential Parameterizations (PhD-Thesis). Shaker, Aachen (2007) 2. Antritter, F., Kletting, M., Hofer, E.P.: Robustness analysis of flatness based tracking controllers using interval methods. Int. J. Control (2007) 3. Antritter, F., M¨uller, B., Deutscher, J.: Tracking control for nonlinear flat systems by linear dynamic output feedback. Proceedings NOLCOS 2004, Stuttgart (2004) 4. Auer, E., Rauh, A., Hofer, E.P., Luther, W.: Validated Modeling of Mechanical Systems with SmartMOBILE: Improvement of Performance by ValEncIA-IVP. In: Proc. Dagstuhl-Seminar 06021: Reliable Implementation of Real Number Algorithms: Theory and Practice, volume 5045 of Lecture Notes on Computer Science. Dagstuhl, Germany (2008) 5. Berz, M., Makino, K.: Verified Integration of ODEs and Flows Using Differential Algebraic Methods on High-Order Taylor Models. Reliable Comput. 4, 361–369 (1998) 6. Fliess, M., L´evine, J., Martin, P., Rouchon, P.: Flatness and defect of nonlinear systems: introductory theory and examples. Int. J. Contr. 61, 1327–1361 (1995) 7. Fliess, M., L´evine, J., Martin, P., Rouchon, P.: A Lie-B¨acklund approach to equivalence and flatness of nonlinear systems. IEEE Trans. Aut. Contr. 44, 922–937 (1999) 8. Fliess, M., Rudolph, J.: Local tracking observers for flat systems. Proceedings of the Symposium on Control, Optimization and Supervision, CESA ’96 IMACS Multiconference, Lille, France pp. 213–217 (1996) 9. Hagenmeyer, V., Delaleau, E.: Exact feedforward linearization based on differential flatness. Int. J. Contr. 76, 537–556 (2003) 10. Kletting, M., Antritter, F., Hofer, E.P.: Guaranteed robust tracking with flatness based controllers applying interval methods. In: 12th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics SCAN 2006, Duisburg, Germany, Book of abstracts (2006) 11. Kletting, M., Antritter, F., Hofer, E.P.: Robust flatness based controller design using interval methods. In: Proceedings NOLCOS 2007 in Pretoria (2007) 12. Levine, J., Lottin, J., Ponsart, J.C.: A nonlinear approach to the control of magnetic bearings. IEEE Transaction on Control Systems Technology pp. 545–552 (1996) 13. Nedialkov, N.S., Jackson, K.R.: Methods for Initial Value Problems for Ordinary Differential Equations. In Kulisch, U., Lohner, R., Facius, A. (eds.), Perspectives on Enclosure Methods. pp. 219–264. Springer, Vienna (2001)

Structural Analysis for the Design of Reliable Controllers and State Estimators for Uncertain Dynamical Systems Andreas Rauh and Harald Aschemann

Abstract The task of designing feedforward control strategies for finitedimensional systems in such a way that the output variables match predefined trajectories can be formulated in terms of an initial value problem (IVP) for a set of differential-algebraic equations (DAEs). The same holds for the reconstruction of internal variables and parameters on the basis of measured data. In this contribution, we discuss criteria for the solvability of both DAE problems and their relations to controllability and observability of dynamical systems. The practical applicability of this type of problem formulation is demonstrated by numerical results.

1 Verified Solution of DAEs in Controller and Observer Design Systems of DAEs are a natural description for the mathematical modeling of many real-life processes consisting of the interconnection of different physical components with their own dynamic behavior. Such interconnected systems can be described by separate subsystem models (for instance related to electric drives and mechanical components in power trains) and interface conditions connecting these components by a description of power flow or, for example, geometric side conditions imposed by links or joints. In this paper, we consider semi-explicit DAEs xP .t/ D f .x .t/ ; y .t/ ; t/ 0 D g .x .t/ ; y .t/ ; t /

with f W D 7! Rnx

(1)

with g W D 7! Rny ; D  Rnx  Rny  R1 ;

(2)

A. Rauh ()  H. Aschemann Chair of Mechatronics, University of Rostock, D-18059 Rostock, Germany e-mail: [email protected]; [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 31, © Springer-Verlag Berlin Heidelberg 2012

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and the consistent initial conditions x .t0 / and y .t0 /. These DAEs may further depend on uncertain parameters p. To simplify the notation, the dependency on p is not explicitly h i denoted. However, all presented results are also applicable to pi 2 p i I p i with p i < p i , i D 1; : : : ; np . Computational procedures determining verified enclosures to the solution of IVPs to DAEs containing the sets of all reachable states in a guaranteed way have been presented in [8]. The corresponding enclosures for the differential and algebraic variables xi .t/ and yj .t/, respectively, are defined by Œxi .t/ WD xapp;i .tk / C .t  tk /  xP app;i .tk / 

C ŒRx;i .tk / C .t  tk /  ŒRP x;i .t/



yj .t/ WD yapp;j .tk / C .t  tk /  yPapp;j

and   .tk / C Ry;j .t/

(3)

with i D 1; : : : ; nx , j D 1; : : : ; ny , and t 2 Œtk I tkC1 , t0  t  tf . In (3), tk and tkC1 are two subsequent points of time between which guaranteed state enclosures are determined. For t D t0 , the conditions Œx .t0 / D xapp .t0 / C ŒRx .t0 /

and

  Œy .t0 / D yapp .t0 / C Ry .t0 /

(4)

have to be fulfilled with approximate solutions xapp .t/ and yapp .t/. They are computed, for example, by the non-verified DAE solver DAETS [5–7]. The following three-stage algorithm implemented in VALENCIA-IVP allows us to determine guaranteed state enclosures of a system of DAEs using the Krawczyk iteration which solves nonlinear algebraic equations in a verified way. Step 1. Compute hidden constraints that have to be fulfilled for the verified enclosures of the initial conditions x .t0 / and y .t0 / as well as for the time responses x .t/ and y .t/, t > t0 , by considering algebraic equations gi .x/ which do not depend explicitly on y. Differentiation with respect to time leads to the hidden constraints !T j 1 @Lf gi .x/ d j gi .x/ j D f .x; y/ D Lf gi .x/ D 0 with L0f gi .x/ D gi .x/: @x dtj (5) j

The Lie derivatives Lf gi .x/ are computed automatically by using FADBAD++ [2] j

up to the smallest j > 0 for which Lf gi .x/ depends on at least one component of y. Step 2. Compute initial conditions for (1) such that the constraints gi .x/ D 0 and (5) are fulfilled using the Krawczyk iteration. Step 3. Substitute the state enclosures (3) for the vectors x .t/ and y .t/ in (1) and solve the resulting equations for ŒRP x .t/ and ŒRy .t/ with the help of the Krawczyk iteration. The constraints (5) are employed to restrict the set of feasible solutions.

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Besides simulation of systems with known control inputs, DAE solvers can be employed for trajectory planning, computation of feedforward control, and estimation of system parameters, disturbances, and non-measured state variables. In these cases, further (usually time-dependent) algebraic constraints are included in the system model (1). These constraints characterize the desired system behavior (in control design) or represent knowledge on measured data (in estimation tasks). In the following, only the tasks of feedforward control and estimation are considered. For feedforward control, the inputs u .t/ are computed as components of the vector y .t/ in the DAEs (1) with the desired system outputs specified by 0 D h .x .t/ ; u .t/ ; q .t/ ; t /  .yd .t/ C yt ol .t//. In these constraints, ytol .t/ 2 Œytol .t/ represents worst-case interval bounds for the tolerances ytol .t/ between the actual and desired outputs y .t/ and yd .t/. Moreover, the function h .x .t/ ; u .t/ ; q .t/ ; t / (or shortly h .x/) relates the states x to the (measured) outputs in dependence on further parameters q. The resulting DAE system is solved by a verified or non-verified solver (VALENCIA-IVP or DAETS) for consistent u .t/ and x .t/. Compared with symbolic approaches applicable to feedforward control of exactly input-to-state linearizable sets of ordinary differential equations (ODEs) (as a special case of differentially flat systems) [3, 4], numerical interval-based approaches are more flexible since uncertain parameters and initial states as well as robustness requirements can be expressed directly in the constraints above. Furthermore, this approach can also handle differentially non-flat systems with stable internal dynamics [9]. Additionally, the DAE-based solution procedure can be used to implement a onestage interval observer. To estimate non-measured quantities, the equation h iT iT h .nx 1/ T T T q.x/ D ym D h.x/T Lf h.x/T : : : Lfnx 1 h.x/T (6) yPm : : : ym .i / describing the measured variables ym .t/ and their i -th derivatives ym .t/ has to .i / nx be solved for the state vector x .t/ 2 R . In (6), y .t/ is expressed as the Lie m  

derivative Lif h.x/ D Lf Lif1 h.x/ , i D 0; : : : ; nx  1, of the output h .x/ along

@ h.x/  f .x/. The the vector field f .x/ with L0f h.x/ D h .x/ and Lf h.x/ D @x equation (6) can be solved (at least locally) for x, if the observability matrix Q.x/ D h iT @ i with Qi .x/ D @x Lf h.x/ has the full rank Q0T .x/ Q1T .x/ : : : QnTx 1 .x/ nx [1, 9]. As the Lie derivatives in (6) correspond to the hidden constraints (5), the DAE solver can be applied directly after including ym .t/ D h .x .t// in the algebraic constraints g.

2 Control of a Distributed Heating System To visualize the DAE-based procedures for control and estimation, we consider the heating system in Fig. 1. Its controlled variable is the rod temperature at a given position. Control and disturbance inputs are provided by four Peltier elements and

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Peltier element

Fig. 1 Experimental setup of a distributed heating system

cooling units. The rod temperature # .z; t/ depends both on the spatial variable z and on the time t according to the parabolic partial differential equation (PDE)  @2 #.z; t/ ˛ ˛ @#.z; t/  C #.z; t/ D #U ; 2 @t cp @z hcp hcp

(7)

which is discretized in its spatial coordinate into finite volume elements for offline simulation as well as online state and disturbance estimation. With the coefficients a11 D 

˛Als C s bh s bh ˛Als C 2s bh ; a12 D ; and a22 D  ls ms cp ls ms cp ls ms cp

(8)

and the heat flow u .t/ into the first segment of the rod, balancing of heat exchange between four volume elements leads to the ODEs 2 3 2 3 3 3 2 2 3 2 xP 1 .t/ 1 e1 .t/ a11 a12 0 0 x1 .t/ 6 7 6 7 7 7 6 6 7 6 6xP 2 .t/7 6a12 a22 a12 0 7 6x2 .t/7 607 6e .t/7 6 7D6 7 C 1 6 7 u .t/ C ˛A 6 2 7 76 6xP .t/7 6 0 a a a 7 6x .t/7 m c 607 7 ms cp 6 s p 4 5 12 22 12 5 4 3 4 3 5 4 5 4e3 .t/5 xP 4 .t/ x4 .t/ e4 .t/ 0 0 a12 a11 0 (9) for the temperatures xi .t/ in the segments i D 1; : : : ; n D 4. The goal of feedforward  control is the computation of an input u .t/ D u1 .t/ D 0:5  u1 .t/ C u1 .t/ in such a way that the temperature y .t/ in an arbitrary segment tracks the desired profile     #f  #0 3;600 s yd .t/ D #0 C 1 C tanh k t  (10) 2 2 with #0 D #U .0/, #f D #0 C 10 K, and k D 0:0015.

Structural Analysis for the Reliable Control of Uncertain Dynamical Systems

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The error terms e1 .t/ ; : : : ; e4 .t/ result from both the discretization of the PDE and unmodeled disturbances estimated by a Luenberger observer and the novel DAE-based approach, see Fig. 2. Without these errors, all ei are equal to the ambient temperature #U . The interval observer detects the point of time from which on the Luenberger observer yields consistent estimates. Both estimators use the measured temperatures y1 D x1 and y2 D x4 . For the implementation of the intervalbased observer, the ODEs (9) are extended by eP D 0 with e D e1 D : : : D e4 . To quantify the influence of measurement errors, the uncertainties xi 2 yj CŒ1 I 1 K, xP i 2 Œ0:5 I 1:5 yPj , i 2 f1; 4g, j 2 f1; 2g are considered. To reduce the influence of model errors and disturbances, the control is extended according to u .t/ D u1 .t/ C u2 .t/ by a PI output feedback u2 .t/ compensating the system’s largest time constant [8]. For specification of the flat output g .x; t / D x4 .t/  yd .t/ D 0, the structural analysis performed in VALENCIA-IVP for the hidden constraints (5) shows that the Lie derivative L4f g corresponds to the smallest order of the derivative of g .x; t / which is influenced explicitly by the input u. Since the number of unknown states and inputs coincides with the number of hidden constraints, L1f g D 0; : : : ; L4f g D 0 can be solved directly using interval Newton techniques for x1 , x2 , x3 , and u. Note that x4 is known a-priori by g D L0f g D 0 for each t. Therefore, no additional initial conditions are required for the synthesis of the feedforward control. However, this means that deviations of the actual initial temperatures from the ones specified by L1f g D 0; : : : ; L4f g D 0 inevitably lead to tracking errors y .t/  yd .t/ ¤ 0. These deviations can be compensated by output feedback controllers as introduced above. For a non-flat output, e.g. g .x; t / D x3 .t/  yd .t/ D 0, the order ı of the derivative of g influenced directly by u is smaller than the number of unknowns. Therefore, L1f g D 0; : : : ; Lıf g D 0 cannot be solved directly for the states x and the input u. Thus, further information about initial conditions is required. Firstly, we identify sets of ODEs or DAEs including the system output which can be solved as an IVP. In the case of ODEs, all initial conditions can be specified arbitrarily. In the case of a DAE system, the initial conditions have to fulfill g D L0f g D 0 and, if necessary, the constraints L1f g D 0; : : : ; Lf g D 0,  < ı. Secondly, this solution

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u (t) in W 50 40 30 20 10 0 0

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is substituted for the corresponding states in LfC1 g D 0; : : : ; Lıf g D 0 which are solved for the remaining states and the input u .t/. For specification of x3 as the desired output, it is at least necessary to know the initial temperature x4 .0/. Then, an IVP for the ODE for x4 .t/ is solved in the first stage with yd .t/ D x3 .t/. This information is substituted for x4 .t/ in L1f g D 0; : : : ; Lıf g D 0, which can now be solved for the remaining unknowns, see also [8]. In Fig. 3, the feedforward control obtained by DAETS is displayed for different variations # D #f #0 of the output temperature.

3 Conclusions and Outlook on Future Research DAE-based approaches for design and verification of control strategies were presented and applied to a finite volume representation of a distributed heating system. Using interval-based solution procedures after a problem reformulation in terms of DAEs, a structural analysis provides verified information concerning solvability. Moreover, an interval-based state estimator allows one to make guaranteed statements whether Luenberger-type observers provide estimates which are consistent with the mathematical model including uncertainties and measurement errors. These procedures are currently being extended towards an automatic computation of feedforward control laws for systems which can only be linearized using dynamic state feedback. This is typical for non-quasi-linear DAEs and, generally, for differentially flat systems with multiple inputs and outputs if the sum of the relative degrees exceeds the dimension of the state vector. In this case, dynamic extensions of the inputs are required [8]. A systematic procedure for control and estimator design is being developed on the basis of the work presented in this paper.

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References 1. Aschemann, H., Minisini, J., Rauh, A.: Interval Arithmetic Techniques for the Design of Controllers for Nonlinear Dynamical Systems with Applications in Mechatronics—Part 1. Izvestiya RAN. Teoriya i sistemy upravleniya. J. Comput. Syst. Sci. Int. (3), 3–14 (2010) 2. Bendsten, C., Stauning, O.: FADBAD++, Version 2.1 (2007); http://www.fadbad.com 3. Fliess, M., L´evine, J., Martin, P., Rouchon, P.: Flatness and Defect of Nonlinear Systems: Introductory Theory and Examples. Int. J. Contr. 61, 1327–1361 (1995) 4. Marquez, H.J.: Nonlinear Control Systems. Wiley, New Jersey (2003) 5. Nedialkov, N.S., Pryce, J.D.: Solving Differential-Algebraic Equations by Taylor Series (I): Computing Taylor Coefficients. BIT 45(3), 561–591 (2005) 6. Nedialkov, N.S., Pryce, J.D.: Solving Differential-Algebraic Equations by Taylor Series (II): Computing the System Jacobian. BIT 47(1), 121–135 (2007) 7. Nedialkov, N.S., Pryce, J.D.: Solving Differential-Algebraic Equations by Taylor Series (III): the DAETS Code. J. Numer. Anal. Ind. Appl. Math. 3, 61–80 (2008) 8. Rauh, A., Auer, E.: Interval Approaches to Reliable Control of Dynamical Systems. In: Proceedings of Dagstuhl Seminar 09471, Computer-Assisted Proofs—Tools, Methods, and Applications. Dagstuhl, Germany (2009); drops.dagstuhl.de/portals/index.php 9. Rauh, A., Minisini, J., Hofer, E.P.: Verification Techniques for Sensitivity Analysis and Design of Controllers for Nonlinear Dynamic Systems with Uncertainties. Special Issue of the Int. J. Appl. Math. Comput. Science AMCS, “Verified Methods: Applications in Medicine and Engineering” 19(3), 425–439 (2009)

Wiener Calculus for Differential Equations with Uncertainties Florian Augustin, Peter Rentrop, and Utz Wever

Abstract In technical applications uncertainties are a topic of increasing interest. During the last years the Polynomial Chaos of N. Wiener was revealed to be a cheap alternative to Monte Carlo simulations. In this paper we apply Polynomial Chaos to stationary and transient problems, both from industry and academics. For each application chances and limits of Polynomial Chaos are discussed. The presented problems show the need for new theoretical results.

1 Introduction A typical field where academic and industrial research go into a fertile connection is the field of electric circuits simulation. The development history of typical simulations codes like TITAN (SIEMENS AG) or PSTAR (Philips AG) shows this interaction. From the viewpoint of mathematics large systems of ordinary differential equations (ODEs) or differential algebraic equations (DAEs) are numerically solved. Their modeling base is a network-approach, going back to the SPICE code from Nagel. Standard electric or electronic elements like MOSFETs, diodes or quartz oscillators are modeled by subcircuits, consisting of the basic elements: voltage sources, current sources, capacitors, resistors and inductors. Their special values are given from fitting procedures. A typical MOSFET subcircuit can include more than 100 measured values. A simple example including the basic elements

F. Augustin  P. Rentrop () Department of Mathematics (M2), Technische Universit¨at M¨unchen, Boltzmannstr. 3, 85748 Garching, Germany e-mail: [email protected]; [email protected] U. Wever SIEMENS AG, Otto-Hahn-Ring 6, 81730, Munich, Germany e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 32, © Springer-Verlag Berlin Heidelberg 2012

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Fig. 1 Diagram of a MOSFET, consisting of subcircuits and basic elements [7]

is given in Fig. 1. Obviously there is a need for estimating the influence of critical measurements. There are at least two possibilities: 1. Solve the sensitivity equations, which is an option in many simulation codes. 2. Perform a Monte Carlo simulation for critical parameters. The latter is an often used but expensive way. Especially in the case that the second moment of the parameter distribution is limited, an idea of N. Wiener—denoted by Polynomial Chaos—may lead to a cheaper approach. In the following we outline the general idea of N. Wiener in Sect. 2 and discuss its application to stationary and transient problems. In Sect. 3 we first apply the method of Polynomial Chaos to a stationary problem, an elastic bracket where Young’s modulus is assumed to be uniformly distributed. Afterwards, in Sect. 4 we take a look at the random van der Pol equation, which is an example of a transient problem. We will close this article by drawing conclusions in Sect. 5.

2 Polynomial Chaos Approach Due to N. Wiener In this section we first discuss the idea of N. Wiener to develop a random variable with finite second moment into a series of orthogonal polynomials, see [18]. Afterwards we will present how this idea can be used to approximate the solution of random differential equations. Assume .˝; F ; P / to be a probability space with F being a -algebra over ˝. P is the associated probability measure. Furthermore we consider independent Gaussian random variables  WD .1 ; : : : ; m / mapping from .˝; F ; P / to .Rm ; B m ; P /. Here, B denotes the Borel -algebra and P denotes the range measure of . For a detailed introduction to probability theory we refer to [12, 15]. If F is generated by , then every square integrable mapping u W .˝; F ; P / ! .Rn ; B n ; Pu / can be decomposed into a series of orthogonal Hermite polynomials u..!// D

1 X i D0

qi i ..!//;

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see [9]. We denote the space of square integrable functions by L2 .˝; F ; P / and abbreviate it with L2 in the forthcoming. The inner product ˝

˛

1

i ./; j ./ WD p .2/m

Z

i ./j ./e 

T  2

d

Rm

is defined by the density function of . It induces, that fi g1 i D0 are the Hermite polynomials. To keep the notation less cumbersome we assume the polynomials to be normalized so that ˝

˛ i ; j D ıij

with the Kronecker function ıij D 1 for i D j and ıi;j D 0 for i ¤ j . Note that in this case it always holds 0  1. The restriction to Hermite polynomials was avoided by Xiu and Karniadakis in [20], where they extended the Polynomial Chaos expansions to polynomials from the Askey scheme. They allowed the random vector  to have distributions with densities that correspond to weighting functions of orthogonal polynomials. I.e. the density function serves as weighting function in the L2 inner product which induces the orthogonality of the respective polynomials. This extension is called generalized Polynomial Chaos (gPC). In fact there is a close connection between popular distributions and polynomials from the Askey scheme. For example the density of a uniformly distributed random variable corresponds to Legendre polynomials and the density of the gamma distribution corresponds to the Laguerre polynomials. A selection of further relations is shown in Table 1. As it can be seen from Table 1 the gPC is not restricted to continuous distributions. The convergence of the generalized series expansions is not covered by the work of N. Wiener. In the case of compact range of , the convergence follows from the theorem of Stone-Weierstraß. For a detailed discussion on convergence of gPC expansions we refer to [6]. We remark, that it is also possible to prescribe different distributions for fi gm i D1 as long as they are independent and satisfy the conditions on convergence of the series expansion. Knowing the coefficients fqi g1 i D0 of the gPC expansion it is easy to compute the moments of u. Due to the orthogonality of the polynomials the expectation value is given by Table 1 Relation between density functions of well-known distributions and orthogonal polynomials from the Askey scheme

Distribution

Orthogonal polynomials

Gauss Gamma Beta Uniform Poisson Hypergeometric

Hermite Laguerre Jacobi Legendre Charlie Hahn

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Z E.u/ D

u..!//dP.!/ D

Z X 1

qi i ..!// 1 dP.!/ D q0

(1)

˝ i D0

˝

and the variance is Z V .u/ D u..!//2 dP.!/  E.u/2 ˝

Z D ˝

D

1 X

1 X

!0 qi i ..!// @

i D0

1 X

1 qj j ..!//A dP.!/  q02

(2)

j D0

qi2 :

i D1

This directly follows from Parseval’s identity. The gPC expansion of a random variable in L2 can be used to represent the solution of a given differential equation. Assume the differential equation operator T and the system T .u.t; x; /; / D 0;

(3)

consisting of n differential equations. Here, the random vector  describes the uncertainty in the parameters of the mathematical model. Consequently, the solution u is not only a function in time t and space x, but it becomes a random field over the probability space .Rm ; B m ; P /. Assuming u 2 L2 for every .t; x/ of its domain it can be decomposed into a gPC expansion u.t; x; .!// D

1 X

qi .t; x/i ..!//:

i D0

The choice of orthogonal polynomials fi g1 i D0 depends on the distribution of the parameters . Truncation of the series at polynomials of degree d results in an approximation uO .t; x; .!// D

p X

qi .t; x/i ..!//:

(4)

i D0 p

Computation of the coefficient functions fqi gi D0 yields an approximation uO of the solution u. We remark, that the length p C 1 of the truncated series increases very fast with increasing polynomial degree d and number m of random parameters:

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pC1D

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.d C m/Š : d Š mŠ

Thus, methods based on the gPC approach may become inefficient in these cases. This holds especially for transient problems as we will see in Sect. 4. The methods for computing the coefficient functions can be divided into two classes, the intrusive methods and the non-intrusive methods. For simplicity of notation we write  instead of .!/ in the following.

2.1 Non-intrusive Methods The non-intrusive methods are sampling based methods to compute the coefficient p p functions fqi gi D0 . From the orthogonality of the normalized polynomials fi gi D0 it holds qi .t; x/ D hu.t; x; /; i ./i for every .t; x/ in the domain of dependence of u. The integral on the right-hand side is the projection of u onto the i -th polynomial. It can be approximated by using q q a cubature rule with nodes f j gj D1 and weights fwj gj D1 : qi .t; x/ 

q X

u.t; x;  j /i . j /wj :

j D1

We see, that the underlying deterministic differential equation has to be solved q times for the realizations of  given by the cubature nodes. For the approximation being a polynomial of maximal degree d the natural choice of cubature rule is a tensor Gauss cubature. Although being exact in the case that a sufficient number q of nodes is chosen, the efficiency is poor. Especially for a large number m of random parameters Gauss cubature becomes unfeasible. In these cases the sparse grid cubature of Smolyak, see [8], should be used. Approaches based on the non-intrusive methods are the Stochastic Collocation method (SCM) [19] and the Stochastic Finite Element method (SFEM) [3, 14]. The advantage of these methods is, that they can be used as a black-box which only needs the well-tested and robust codes for the underlying deterministic problem.

2.2 Intrusive Methods The intrusive methods are based on a Galerkin projection on the probability space .Rm ; B m ; P / in order to obtain a system of differential equations for the coefficient

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functions. Therefore we consider the stochastic weak form of (3): hT .u.t; x; /; /;  ./i D 0

for all  2 L2 .˝; F ; P /:

The stochastic weak form is then restricted to the finite dimensional subspace Cp p spanned by the polynomials fi gi D0 . Thus, we search for an approximation uO 2 Cp of the solution, satisfying hT .Ou.t; x; /; /; i ./i D 0

for all i D 0; : : : ; p:

(5)

The solution of (5), which is a system of deterministic equations for the coefficient p functions fqi .t; x/gi D0 , yields the approximation as in (4). Note, that the dimension of the system is n.p C 1/, which can be very large depending on the length p C 1 of the truncated series expansion. For a deeper insight into this stochastic Galerkin methods (SGM) we refer to Sect. 4, where we will apply it to the van der Pol equation. The projection of the governing equation p onto the polynomials fi gi D0 is performed in a pre-processing step up to a very high accuracy. So at the run time of the code, the deterministic projected system of p equations for the coefficients fqi .t; x/gi D0 is readily available for evaluation.

3 The Elastic Bracket In this section we will consider the bracket as shown in Fig. 2, which consists of two different materials for the cross beam and the bracket itself. The bracket is fixed at one side and a force is applied on the opposite side. We consider the Young’s modules of the bracket itself and the cross beam as random parameters. The interest is about the distribution and moments of the resulting von Mises stress.

Fig. 2 Structure of the bracket, which consists of two different materials for the cross beam and the bracket itself. It is fixed on the bottom side and a force is applied to the top side

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The continuous momentum equation is given by 

@ij @ij @2 ui D C fim D C fiv @t 2 @xj @xj

for i;j 2 f1; 2; 3g;

with  being the density of the material and fui g3iD1 , ffim g3iD1 , ffiv g3iD1 its velocities, forces per unit mass and forces per unit volume, respectively. Further it holds for the strain   @uj 1 @ui ij D C 2 @xj @xi and for an isotropic linear-elastic material we have E ij D 1C

! 3  X kk ; ij C ıij 1  2 kD1

with Young’s modulus E and the Poisson number . Abbreviatory we write the equations above as uR D r .2/ C fm D r .2/ C fv  .2/ D 12 .ru C .ru/T /  .2/ D trace. .2/ / I C 2  .2/ ; with u WD .u1 ; u2 ; u3 /T , the unit matrix I and WD

E ; .1 C /.1  2/

WD

E : 2.1 C /

 .2/ and  .2/ are represented by the matrices 0

 .2/

1 11 12 13 WD @ 21 22 23 A 31 32 33

0

and  .2/

1 11 12 13 WD @ 21 22 23 A : 31 32 33

The von Mises stress is defined by r sMises D

.s1  s2 /2 C .s1  s3 /2 C .s2  s3 /2 ; 2

(6)

where s1 ; s2 ; s3 are the eigenvalues of stress tensor  .2/ . We specify the random parameters as follows. The Young’s modulus of the bracket is uniformly distributed on the interval Œ10:000; 12:000 and the one of

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EXP. MISES

DEV. MISES

6.73 E–01 2.84 E+01 5.60 E+01 8.37 E+01 1.11 E+02 1.39 E+02 1.67 E+02 1.94 E+02 Current View Min = 6.73 E–01 X = 1.01 E–01 Y = 7.01 E+00 Max = 1.94 E+02 X = 4.56 E+00 Y = 1.24 E+01

6.84 E–05 3.89 E–01 7.79 E–01 1.17 E+00 1.56 E+00 1.95 E+00 2.34 E+00 2.72 E+00 Current View Min = 6.84 E–05 X = 0.00 E+00 Y = 1.94 E+01 Max = 2.72 E+00 X = 4.56 E+00 Y = 1.24 E+01

Fig. 3 Expectation value (left) and variance (right) of the von Mises stress

the cross beam on the interval Œ50:000; 60:000. The Poisson ratio of the bracket material is set to 0:25 and the one of the cross beam to 0:3. The FEM computations are performed in 2D. According to the non-intrusive method, see Sect. 2, the von Mises stress is point-wise (in each node) represented as a gPC expansion. The expectation value and the variance are evaluated by the formulas (1) and (2). Thus, it remains to compute the expansion coefficients qi D hsMises ; i i ; where sMises is the von Mises stress. The numerical cubature is performed by the adaptive procedure described in [1]. We will not go into details of this method, but we remark that it is well suited for high dimensional cubature over smooth integrands. Figure 3 shows the expectation value and the variance of the von Mises stress. We used an approximation degree 6 in each dimension for the adaptive Gauss cubature. In comparison to standard multi dimensional Gauss cubature requiring 49 function evaluations, the adaptive Gauss cubature only needs 32 function evaluations for a relative error tolerance of 104 . We refer to [1] for a more detailed discussion on the efficiency of the adaptive Gauss cubature.

4 The van der Pol Equation Within this section we will discuss the application the SGM from Sect. 2.2 by means of the van der Pol equation. This equation is a simplified mathematical model of the electrical circuit shown in Fig. 4, see [11]. The fundamental rules of Kirchhoff and the governing equations for the basic elements yield the differential equation 0 D C UR C



d RC C f .U / L dU

 d dt U

C

 1  Rf .U / C U  Uop : L

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L

N1

N2

C

Uop

tunnel diode

U0 = 0

Fig. 4 Circuit with an inductance L, a capacitance C , a resistance R, a tunnel diode and the grounding U0 as well as an operational voltage Uop . Additionally two nodes N1 and N2 are plotted

Here, U denotes the voltage at node N2 , Uop denotes the operational voltage, L the inductance, C the capacitance, R the resistance. Realistic choices of these technical parameters are C D 107 F , L D 2  103 H , R D 1V =A and Uop D 0:25V . The function f describes the characteristics for the tunnel diode. For the 1N4448 diode it can be fitted by a cubic polynomial f .U / D a1 U C a2 U 2 C a3 U 3 with coefficients a1 D 1:80048

A ; V

a2 D 8:766

A V2

and

In the following we use the van der Pol equation   uR .t; / D  1  u.t; /2 dtd u.t; /  u.t; /;

a3 D 10:8

A : V3

t 20; 15Œ

(7)

with deterministic initial values u.0; /  1, dtd u.0; /  0. The uncertain parameter in (7) is modeled by a uniformly distributed random variable   U .Œ4:75; 5:25/. Following P.E. Zadunaisky who stated “I have a theory that whenever you want to get in trouble with a method, look for the Van-der-Pol equation.”, see [10], we apply the intrusive method to approximate the solution of (7). The expectation value and the variance are plotted in Fig. 5. Therefore we used an approximation of degree d D 8 in the truncated gPC expansion. Again formulas (1) and (2) are used to compute the expectation value and the variance. The coefficient functions fqi g8i D 0 are computed from the projected system of differential equations qR0 .t/ D

1 X

d q .t/ci;0 dt i



i D0

qR1 .t/ D

2 X

qR2 .t/ D

i D0

qi .t/qj .t/ dtd qk .t/di;j;k;0  q0 .t/

i;j;kD0 d q .t/ci;1 dt i



i D0 8 X

8 X

8 X

qi .t/qj .t/ dtd qk .t/di;j;k;1  q1 .t/

i;j;kD0 d q .t/ci;2 dt i



8 X i;j;kD0

qi .t/qj .t/ dtd qk .t/di;j;k;2  q2 .t/

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3

0.9 0.8 0.7

1 variance

expectation value

2

0 −1

0.6 0.5 0.4 0.3 0.2

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−3 0

2.5

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Fig. 5 Expectation value (left) and variance (right) of the solution of the van der Pol oscillator in the interval Œ0; 15 5

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−0.5

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10

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Fig. 6 Coefficients fqi g8iD0 of the truncated generalized Wiener expansion of the solution of the van der Pol equation in the interval Œ0; 15. Ordering: from top left to bottom right with increasing index

:: : qR8 .t/ D

8 X i D0

d dt qi .t/ci;8



8 X

qi .t/qj .t/ dtd qk .t/di;j;k;8  q8 .t/:

i;j;kD0

˛ ˝ The constants c and d are defined by ci;l WD hi ; l i and di;j;k;l WD i j k ; l . The result is plotted in Fig. 6.

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On the one hand the magnitude of the coefficient functions decays with increasing index. But the choice of realization of the uncertain parameter influences the period of the respective solution. Thus, with increasing time, the location of the steep gradients of the solution differ more and more. This deviation is non-linear, so that a higher polynomial approximation is needed for large times. We see, although being small, the magnitude of the latest coefficient functions becomes not negligible after about t D 12. This shows, that the truncation error gains importance with increasing time. Nevertheless, the comparison to the first two moments estimated by Monte-Carlo simulations revealed no significant differences to the moments plotted in Fig. 5. An easy way to reduce the truncation error is to add more terms to the truncated series expansion. This procedure will cause large computational costs to compute the solution of the projected system of differential equations for the coefficient functions. A way out of this dilemma is proposed by Wan and Karniadakis in [17], where they propose the Multi-Element gPC approach. For a detailed discussion about convergence properties of this method we refer to [2].

5 Conclusions As we have seen gPC is a promising approach to treat random differential equations. On the one hand, due to its relation to generalized Fourier series, it is easy to apply. In the case of non-intrusive methods it can even be used as a black-box, which only has to be called by the well-known and problem adapted solvers for the underlying deterministic problem. On the other hand, there are many drawbacks of this method. For example the curse of dimensionality and the convergence rates of the truncated gPC expansion. There is an active field of research dealing with these problems. The problem of the curse of dimensionality can be reduced by using adaptive or sparse cubature rules, see [8]. Furthermore a detailed convergence analysis can be found in [2, 4] for example. Methods based on gPC expansions work well in the case of stationary problems, but exhibit problems in the transient case [1]. Though the method has been applied successfully to various problems [5, 13, 16]. We close this article by quoting H. G. Matthies (IWR TU-Braunschweig) on methods based on gPC: “[...] this is a young topic with old roots covering stochastics, stochastic modelling, multidimensional integration, thin grids, efficient solvers for thin tensor approximations, [...]. [...] To my feeling the actual status in development is comparable to the FEM in the early 1970s—there is much to do and we expect interesting results.” in GAMM Rundbrief 1/2010. Acknowledgements The authors thank Prof. Dr. A. Gilg and Dr. M. Paffrath from SIEMENS AG for their support and many interesting discussions and ideas on the topic of this paper.

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References 1. Augustin, F., Gilg, A., Paffrath, M., Rentrop, P., Wever, U.: Polynomial chaos for the approximation of uncertainties: Chances and limits. Eur. J. Appl. Math. 19(2), 149–190 (2008) 2. Augustin, F., Rentrop, P.: Stochastic Galerkin techniques for random ordinary differential equations. Numerische Mathematik (submitted 2010) 3. Babuska, I.M., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45(3), 1005–1034 (2007) 4. Babuska, I.M., Tempone, R., Zouraris, G.E.: Galerkin finite element approximation of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42(2), 800–825 (2004) 5. Chauvi`ere, C., Hesthaven, J.S., Lurati, L.: Computational modeling of uncertainty in timedomain electromagnetics. SIAM J. Sci. Comput. 28(2), 751–775 (2006) 6. Ernst, O.G., Mugler, A., Starkloff, H.J., Ullmann, E.: On the convergence of generalized polynomial chaos expansions (2010). (Preprint 60) DFG SPP 1324 http://www.dfg-spp1324. de/download/preprints/preprint060.pdf 7. Feldmann, U., Denk, G.: Private communication, Infineon AG (1993) 8. Gerstner, T., Griebel, M.: Numerical integration using sparse grids. Numer. Algorithm. 18(3–4), 209–232 (1998) 9. Ghanem, R.G., Spanos, P.D.: Stochastic finite elements: A spectral approach. Springer, New York (1991) 10. Hairer, E., Nørsett, S.P., Wanner, G.: Solving ordinary differential equations 1. Springer, Berlin, Heidelberg (1993) 11. Kampowski, W., Rentrop, P., Schmidt, W.: Classification and numerical simulation of electric circuits. Surv. Math. Ind. 2, 23–65 (1992) 12. Klenke, A.: Probability theory. Universitext. Springer, London (2008) 13. Le Maˆıtre, O.P., Knio, O.: Spectral methods for uncertainty quantification. Scientific Computation. Springer, Dordrecht, Heidelberg, London, New York (2010) 14. Matthies, H.G.: Stochastic finite elements: computational approaches to stochastic partial differential equations. ZAMM 88(11), 849–873 (2008) 15. Øksendal, B.: Stochastic differential equations, 5th edn. Universitext. Springer, Berlin, Heidelberg, New York (2000) 16. Pulch, R.: Polynomial chaos for multirate partial differential algebraic equations with random parameters. Appl. Numer. Math. 59(10), 2610–2624 (2009) 17. Wan, X., Karniadakis, G.E.: Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J. Sci. Comput. 28(3), 901–928 (2006) 18. Wiener, N.: The homogeneous chaos. Am. J. Math. 60(4), 897–936 (1938) 19. Xiu, D.: Efficient collocation approach for parametric uncertainty analysis. Comm. Comput. Phys. 2(2), 293–309 (2007) 20. Xiu, D., Karniadakis, G.E.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)

Polynomial Chaos and Its Application to Delay Differential Equations with Uncertainties Manuel Villegas Caballero

Abstract Polynomial Chaos (PC) plays an increasingly important role when dealing with uncertainties. PC replaces the random parameters in the system through an expansion in orthogonal polynomials. Its main advantage compared to the standard Monte Carlo (MC) methods is the speed of performance. Systems in several fields in nature and technology can be described by delay differential equations (DDEs). Appearance of time lags in DDEs can influence decisively their behavior, e.g. affecting dynamical properties like the stability. Here the basic aspects of PC will be reviewed and applied to the stability analysis of DDEs with uncertain delays, and the performance of PC and MC methods will be compared.

1 Introduction In a variety of applications in industry or biology, DDEs appear in the modeling step. Sometimes the underlying subprocesses occurring are not well known or it is not convenient to describe them in detail. Though not considered, they are not instantaneous, and thus, its incidence on the system takes place with some delay. Such delays are relevant to dynamic properties like stability. Uncertainties can be found naturally when describing systems. In biology it could be the randomness of a parameter due to biodiversity. In industry, it can come from lack of precise measurements of certain parameters. That makes necessary to apply methods which can deal with these uncertainties. Traditionally, MC methods have

M.V. Caballero () Siemens AG & Technische Universit¨at M¨unchen, Germany e-mail: [email protected]

M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 33, © Springer-Verlag Berlin Heidelberg 2012

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been used, being both easy-to-use and easy-to-interpret. Nonetheless, being brute force techniques, they can become inefficient for some tasks. An alternative procedure to MC can be used, which relies on the PC expansion. Instead of evaluating many realizations with values of the stochastic parameters given by random generated numbers, in the PC techniques these parameters are expanded in orthogonal polynomials. This allows for a spectral decomposition where the coefficients of the PC expansion for the solution can be calculated. Its main advantage with respect to MC is the speed up in performance. In Sect. 2 an introduction to PC will be given. Section 3 exploits the Lambert W function for the stability analysis of DDEs. Finally, in Sect. 4, the difference between deterministic and stochastic solutions in terms of stability is presented. PC and MC are applied to estimate the stability of DDEs with uncertain delays and results are compared.

2 Polynomial Chaos We do not intend to give a comprehensive treatment of PC. The interested reader should consult [7] for an extensive introduction in the subject. Due to Wiener [9], homogeneous chaos is defined as a span of Hermite polynomial functionals of a Gaussian process. According to the theorem of Cameron and Martin [1], the Fourier-Hermite series can approximate any functionals in L2 and converges in the L2 sense. So, second-order random processes can be expanded in terms of orthogonal Hermite polynomials. A general second order random process x.t; /, with t the time and  the random event, can be represented in the form x.t; / D

1 X

qi .t/Hi ..//;

(1)

i D0

where the Hi are the Hermite polynomials in terms of the standard normal distributed variables . The polynomial basis fHi gi 2N0 forms a complete orthogonal basis of the Hilbert space H with the inner product Z

1

hf ./; g./i D p .2/n

1 T

f ./g./e  2   d :

(2)

Rn

So, H D spanfH0 ./; H1 ./; H2 ./; : : :g. Thus, each functional x.t; .// 2 H with finite variance, h x.t; .//; x.t; .// i < 1, may be approximated by a series of Hermite polynomials in the L2 sense x.t; .// 

M X i D0

qi .t/ Hi ..//:

(3)

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Due to the orthogonality of the base functions, the coefficients qi read qi .t/ D

hx.t; /; Hi ./i ; hHi ./; Hi ./i

i D 0; : : : ; M:

(4)

For low number of random variables, a multi-dimensional Gauss-Hermite quadrature can be efficiently used [3]. The expectation value and the variance can be computed as follows E.x/ D q0 hH0 ./; H0 ./i;

 2 .x/ D

N X

qi2 hHi ./; Hi ./i:

(5)

i D1

When n random variables are considered, N C1D

.n C p/Š nŠpŠ

(6)

is the number of basis polynomials up to degree p [7]. For normal distributed random variables the exponential convergence of the homogeneous chaos expansion can be shown, cf. [10]. For random variables distributed other than normal, homogeneous chaos expansion still works, but optimal convergence is only achieved with a particular choice of the orthogonal polynomial basis. This is called generalized polynomial chaos (gPC). In this generalization not only the Hermite polynomials are used, but a series of orthogonal hypergeometric polynomials grouped in the so called Askey scheme can be taken to construct the expansion. In this scheme, f˚i gi 2N0 is the appropriate family of orthogonal polynomials. See [10] for details.

3 Delay Equations Many systems in nature or technology can be very complex, comprising physical, chemical or biological processes in different time scales. For a realistic description, some subprocesses must be neglected, though they are not instantaneous. Thus, the introduction of a delay can help to simplify the model. Several examples ranging from enzyme kinetics to control engineering can be found in the literature [4]. We focus on the linear scalar DDE with one delay y 0 .t/ D ay.t/ C by.t  /:

(7)

There are several routines available for numerically solving (7). An explicit solution is possible y.t/ D

C1 X kD1

ck e k t ;

1 k D a C Wk .bea / ; 

(8)

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where Wk is the k-th branch of the Lambert-W function, see [2]. This function satisfies Wk .c/e Wk .c/ D c. Due to the infinite number of terms, (8) is in general unsuited for computational calculation of the solution. Nevertheless, interesting results about stability can be gained. The appearance of k in the exponent allows to assert y 0 .t/ D ay.t/ C by.t  / asymptotic stable

,

<.k / < 0 for all k: (9)

Though there are still an infinite number of values to be computed, it can be shown that k D 0, called the principal branch, determines the stability, [8]. Hence, we regard 0 as the principal eigenvalue. The same analysis is possible for linear systems of DDEs, y 0 .t/ D Ay.t/ C By.t  /, whenever A and B are simultaneous triangularizable matrices, cf. [6].

4 The Logistic Equation with Random Delay The logistic equation with delay y 0 .t/ D ry.t/.1  y.t  //;

(10)

has been proposed to describe the population growth under finite resources [5]. Equation (10) has been conveniently rescaled, y.t/  N.t/=K, where N is the number of individuals and K the carrying capacity of the medium. r stands for the reproductive rate. It has two equilibrium points, P1 D 0 and P2 D 1. The stability analysis of (10) has been made elsewhere [4]. P1 is easily determined to be unstable, while the dynamics of P2 is more complex. Here, the Lambert W function allows a more compact description of its stability. Linearization around equilibrium, with x.t/  y.t/1, gives x 0 .t/ D rx.t /. Via the Lambert W function, the principal eigenvalue of the characteristic equation reads 0 D

1 W0 .r/: 

(11)

W0 is calculated using the Halley iteration scheme, after an initial guess made with an asymptotic expansion for W0 [2]. We can thus determine the stability of P2 in dependence of the parameters r and . In Fig. 1(a) we can see the evolution of the eigenvalues s0 and s1 with r D 1 and increasing delay . Up to  D 1=e, s0 is real and negative, and the only principal eigenvalue. The equilibrium point P2 is stable and the trajectory of the solution approaches it without oscillations. For  > 1=e, s0 2 C, and s1 becomes a principal eigenvalue, complex conjugate of s0 . Now, though P2 is still attractive, the approach of the trajectory is oscillatory. At  D =2 the principal eigenvalues

Polynomial Chaos and Its Application to Delay Differential Equations

a

2

b 2.5

s0 s−1

1.5 1 0.5

¡(s)

τ=e

287

2

τ = p/2 (Hopf bifurcation)

−1

τ = 0.2 τ = 1.2 τ = 1.7

1.5 y(t)

0

1

−0.5 −1

0.5

−1.5 −2 −3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

0

0

10

¬(s)

20

30

40

50

Time

Fig. 1 (a) Evolution of the eigenvalues s0 and s1 with increasing delay  . The arrows show the direction of evolution of the eigenvalues. (b) Solution of the logistic equation with delay for some values of  Table 1 <.0 / for deterministic and random delay. Elapsed time in seconds Method EŒ<.0 /  Œ0  El. time [s] Deterministic ( D 1:58) 0.00263 0 0.01 Polynomial Chaosa 0.00620 0.0981 1.2 Monte Carlo (4  105 samples)a 0.00621 0.0979 45

cross the imaginary axis to the half-plane < > 0, and P2 becomes unstable. Thus, a supercritical Hopf bifurcation takes place, where P2 loses its stability and a stable limit cycle appears. That accounts for an oscillatory behavior of the solution around P2 . Figure 1(b) shows solutions y.t/ for different values of the delay , agreeing with the prediction through the eigenvalue analysis. Retake (10), with   Beta.2; 2/ and centered at  D 1:58. Following (8) k D

1 Wk ./ ; 

k 2 Z:

(12)

The truncated gPC expansion is used for computing the coefficients k 

7 X i D0

qik ˚i ./;

 D 1:58 C 0:4

)

˝ ˛ k ./; ˚j ./ ˛; qjk D ˝ ˚j ./; ˚j ./

(13)

where the ˚i ’s are the Jacobi polynomials, ˚i D Pi2;2 , and  D 2 0  1;  0  Beta.2; 2/, according to the gPC schema in Sect. 2. The numerical computation of the coefficients qjk are thus made with a Gauss-Jacobi quadrature using 5 nodes. Table 1 indicates the agreement between the values of <.0 / calculated with PC and MC. 4  105 samples showed enough accuracy for MC computation (EŒ<.0 / D 0:00608 for 1  105 samples, EŒ<.0 / D 0:00619 for 2  105 samples). The execution time is lower in the case of PC. Figure 2 displays k of (12) for k D  3; : : : ; 2, with  D 1:58 and the expected value of k , EŒk , for   Beta.2; 2/ centered at 1:58. A general shift of the

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10

Unstable

ℑ (λk)

5 Stable 0 −5

τ~ Beta (2, 2) τ deterministic

−10 −1.5

−1

0.5

0

ℜ (λk)

Fig. 2 k of (12) for k D  3; : : : ; 2, with  D 1:58 and expected value of k , EŒk , for   Bet a.2; 2/ centered at 1:58. The values with the largest real part, k D  1; 0, determine the asymptotic stability

real part to the left is observed, and 1 ; 0 , the values with the largest real part, experience a change of the sign, and thus a change of stability. For the stochastic distributed , the system described by (10) is expected to be asymptotically stable, which is not the case for the deterministic . The variance is also an important parameter to take into account, indicating how likely a realization of the system will be unstable.

5 Conclusions In this work, the effect of delay in the stability of linear DDEs has been analysed. In case of stochastic calculation, the efficiency of PC could be shown, with execution times much lower than MC. For linear systems, where A and B are simultaneous triangularizable, the same procedure as for (7) can be applied, with PC calculations being an efficient alternative to MC in case of uncertainties.

References 1. Cameron, R., Martin, W.: The orthogonal development of nonlinear functionals in series of Fourier-Hermite functionals. Ann. Math. 48(2), 385–392 (1947) 2. Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: On the Lambert W function. Adv. Comput. Math. 5, 329–359 (1996) 3. Davis, P., Rabinowitz, P.: Methods of Numerical Integration. 2nd edn. Academic Press, New York (1984) 4. Erneux, T.: Applied Delay Differential Equations. Springer, New York (2009)

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5. Hutchinson, G.: Circular causal systems in ecology. Ann. New York Acad. Sci. 50, 221–246 (1948) 6. Jarlebring, E., Damm, T.: The Lambert W function and the spectrum of some multidimensional time-delay systems. Automatica 43, 2124–2128 (2007) 7. Le Maˆıtre, O.P., Knio, O.M.: Spectral methods for uncertainty quantification. Springer, New York (2010) 8. Shinozaki, H., Mori, T.: Robust stability analysis of linear time-delay systems by Lambert W function: Some extreme points results. Automatica 42, 1971–1999 (2006) 9. Wiener, N.: The homogeneous chaos. Amer. J. Math. 60, 897–936 (1938) 10. Xiu, D., Karniadakis, G.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)

Part IV

Production

Overview Virtual Product Development Nowadays, virtual product development is divided into three basic stages: the simulation, the data analysis and the product optimization with respect to multiple objectives, e.g. total weight, fuel consumption and production costs in the automotive industry, e.g. ‘Nonlinear Metamodeling of Bulky Data and Applications in Automotive Design’ by I. Nikitin et al. Usually, for the optimization a number of simulations is needed to evaluate appropriately the complex objective function for a particular sample of a virtual product. To reduce the computational effort “Design of Experiments” (DoE) is applied, where a space of design variables is sampled by a limited number of simulations. This approach leads to a mathematical model that describes the input/output behavior between design variables and design objectives including the model’s tolerance limits to control the accuracy. In real industrial applications these simulations cannot be performed in real time, i.e., ideally with a response time of fractions of a second and thus innovative mathematical strategies in this field are a future challenge.

Steel Production For the automotive industry steel is the most important raw material. Here, one major problem to be dealt with is the fact that the manufactured steel becomes brittle during cold rolling to produce sheet metal. This can be circumvented by annealing, i.e., coiled metal strips are heated to a high temperature which is then maintained for several hours. For an optimized heating process one has to determine the coil that is slowest to heat to minimise this very costly heating time. Thus, heating costs

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can be reduced and poorly annealed steel that needs reprocessing can be avoided. This issue is addressed in ‘Heat Transfer During Annealing of Steel Coils’ by W.L. Sweatman et al. A second basic issue in steel sheet production is protecting the steel surface against corrosion which is usually done by coating it with a metal alloy, e.g. zinc/aluminium. Typically, in the continuous hot-dipped galvanising process, the steel strip is passed through a bath of molten alloy and then drawn upward until the coating solidifies. Alongside the rising steel strip, a pair of air knives (high velocity air jets) control coating thickness by forcing surplus molten alloy back downwards. Surface quality issues for Bluescope Steel are analyzed by G.C. Hocking et al. in ‘Deformations Arising During Air-Knife Stripping in the Galvanisation of Steel’. For fixed processing conditions there seems to be a critical air knife pressure below which coating is satisfactory but above which defects may appear. Pocks, the most serious defects, involve substantial local thinning of the coating and a correspondingly severe reduction in corrosion protection. New innovative mathematical models improve earlier models by including air-knife shear.

Glass Manufacturing The important final production step in the forming of hollow containers, e.g. bottles, is the blow moulding: a so-called pre-form of hot viscous material is transferred into a mould. Afterwards, the pre-form is inflated with pressurized air until it takes the mould shape and then it is left to cool down. Eventually, it is removed from the mould, see ‘Modelling Preform and Mould Shapes in Blow Moulding’ by J.A.W.M. Groot et al. There are a couple of relevant mathematical questions on the relation between the pre-form shape and the container shape, e.g. well-posedness and sensitivity, necessary constraints on the mould wall and equilibrium states of the container shape in blow moulding. These issues can be separated into two basic classes. First, given the shape of the pre-form, the forward problem is to determine the shape of the container. Conversely, the inverse problem is to compute the shape of the pre-form, given the shape of the container. Another important problem of glass production is the wet chemical edging being used in a wide range of industrial applications, e.g. integrated circuits or glass micro-fluidic devices. In ‘Asymptotic Analysis of a Multi-Component Wet Chemical Etching Model’ J. Ward considers the case of lead crystal glassware, decorative features cut into the glass leave it optically opaque and consequently polishing is required to restore its transparency. Here, one has to model the dependency of the macroscopic normal velocity on various energy and curvature considerations and moreover, for multi-component systems, it is not known how this velocity depends on the acids present and the composition of the glass. Mathematically the edging process is described by coupled linear ordinary differential equations. The long time

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edging behavior is analysed by Ward with the method of steepest descents and it is applied to calculate the limiting etch rate.

Textile Fabrication The simulation of fibers in production processes requires the consideration of contacts between fibers and machine parts as well as contacts between fibers. For the current fiber model F. Olawsky et al. developed in ‘Numerical Treatment of Fiber– Fiber and Fiber-Obstacle Contacts in Technical Textile Manufacturing’ a contact model which could be integrated into the system and which is able to handle real production processes with complex machine parts. For the numerical simulation of industrial processes in technical textile manufacturing an elastic fiber is modeled by a one-dimensional arc-length parameterized time-dependent curve. A rotary spinning process is used to produce aramide fibers. In this process thin jets of polymer solution emerge from the nozzles of the rotating rotor and flow towards the cylindrical coagulator. At the coagulator the jets hit the water curtain in which they solidify forming fibers. The rotary spinning is described by a steady jet of viscous Newtonian fluid between the rotor and the coagulator. The jet model of A. Hlod et al., ‘A Model of Rotary Spinning Process’, includes the effects of inertia, longitudinal viscosity, and centrifugal and Coriolis forces. For the jet model the specific type of the boundary conditions depends on the balance between the inertia and viscosity in the momentum transfer through the jet cross-section. Hlod et al. found two possible flow regimes in rotary spinning: First, viscous-inertial, where viscosity dominates at the rotor and inertia at the coagulator, Secondly, inertial, where inertia dominates everywhere in the jet. For the simulation of slender viscous jets in spinning processes W. Arne et al. presented in ‘Asymptotic Models of Different Complexity for Viscous Jets and Their Applicability Regimes’ asymptotic models of differing complexity. Here, a jet is a long slender body whose dynamics can be reduced to an one-dimensional description by averaging the underlying balance laws over its cross-sections. In the slenderness limit a viscous Cosserat rod reduces to a string system. Two string models are proposed, i.e., inertial and viscous-inertial string models, that differ in the closure conditions and hence yield a boundary value problem and an interface problem, respectively. The conditions for the jet are also discussed by Hlod et al. in ‘A Model of Rotary Spinning Process’. Arne et al. show that the convergence/applicability regimes, where the respective string solution is the asymptotic limit to the rod, are disjoint and cover nearly the whole parameter space of Reynolds, Froude, Rossby numbers and jet length. The transition hyperplane is explored analytically for the gravitational two-dimensional scenario.

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Material Modelling P. Jester et al. present ‘Wavelet Methods for the Representation, Analysis and Simulation of Optical Surfaces’. Therein they apply wavelet methods and its multiresolution properties for optical surface analysis and simulation. On the one hand they show the usability of a wavelet representation in a ray trace algorithm. On the other hand they present numerical experiments for the detection of local errors. In ‘Modelling Two-Dimensional Photopolymer Patterns Produced with MultipleBeam Holography’, D. Mackey et al. present a new mathematical model for a process that allows the production of spatially periodic variations in the refractive index of a photopolymer material by exposing it to suitable illumination patterns. This is of special interest, since periodic structures as photonic crystals are used in nanotechnology and plasmonics. To date mathematical models for multibeam holography do not reflect experimental data, because they do not capture diffraction at higher frequencies. In ‘Modeling Berry’s Phase in Graphene by Using Quantum Kinetic Approach’, O. Morandi and F. Sch¨urrer investigate the novel material graphene. Certain recently discovered effects seem to justify the investigation of the electron behavior in graphene in the solid state framework and the authors consider the formulation of the multiband particle-hole dynamics in graphene using the Wigner formalism with a first-order „-correction. D. Jeulin presents ‘Multi Scale Random Sets: From Morphology to Effective Behaviour’. Therein he addresses rather generally the modelling of complex material where microstructures and heterogenous multiscale textures are involved. He deals with multiscale random sets modeling real problems arising in different fields of Material Science and proposes several methods in order to predict materials properties. Wuppertal, Germany

Matthias Ehrhardt Sebastian Sch¨ops

Nonlinear Metamodeling of Bulky Data and Applications in Automotive Design Igor Nikitin, Lialia Nikitina, and Tanja Clees

Abstract We describe and discuss methods for nonlinear metamodeling of simulation databases featuring continuous exploration of simulation results, tolerance prediction, sensitivity analysis, and rapid interpolation of bulky FEM data. The methods have been implemented in the design-parameter optimization tool DesParO. Reallife applications from the automotive industry show their efficiency.

1 Introduction Simulation is an integral component of today’s virtual product development. From the viewpoint of product development the real purpose is product optimization, and simulation is “only” a means for the purpose. Optimization is a search for the best possible product with respect to multiple objectives (multiobjective optimization), e.g. total weight, fuel consumption and production costs, while simulation provides an evaluation of objectives for a particular sample of a virtual product. (Engineering) Optimization usually requires a number of simulation runs, the results form a simulation dataset. To keep simulation time as short as possible, “Design of Experiments” (DoE, [7]) is applied, where a space of design variables is sampled by a limited number of simulations. On the basis of these samples, a mathematical model is constructed, e.g. a response surface [2], which describes the dependence between design variables and design objectives. Modern metamodels [3–6] describe not only the value of design objective but also its tolerance limits, which allow to control the precision of the result. Moreover, not only scalar design objectives, but whole simulation results from bulky datasets can be modeled [5]. I. Nikitin  L. Nikitina ()  T. Clees Fraunhofer Institute for Algorithms and Scientific Computing SCAI, 53754 Sankt Augustin, Germany e-mail: [email protected]; [email protected]; [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 34, © Springer-Verlag Berlin Heidelberg 2012

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However, today’s industrial simulations have such a high complexity that they cannot be performed in real time, in spite of the permanently increasing power of computers and parallelization techniques. Real-time performance means support of interactive work, i.e. ideally a response time of a tenth of a second. Hence, innovative solution strategies in simulation, optimization and data analysis are required.

2 Metamodeling of Bulky Data Design parameter optimization considers dependence of the numerical result on variations of a vector p of Npar control parameters pk , k D 1; : : : ; Npar . In automative crash, such control parameters can be e.g. thicknesses of car body components, initial velocity of collision, material properties etc. Metamodeling is an interpolation or approximation method applied to existing simulation results to predict the result of a new simulation for intermediate values of parameters. Often non-linear methods are used to catch non-linear parametric dependencies. The usage of radial basis functions (RBFs) is quite convenient, i.e. a representation of the form X

f .p/ D

ci ˚.jp  p i j/

(1)

i D1;:::;Nexp

where p i are coordinates of the i -th run out of Nexp in the space of design variables; f is the objective value, interpolated to a new point p; ˚ are special functions, depending only on the Euclidean distance between the points p and p i . The coefficients ci can be obtained by solving a linear system fi D

X

cj ˚.jp i  p j j/;

(2)

j

where fi are the user-provided data, which could be single criteria or bulky simulation results, associated with i -th simulation run. The solution can be found by inverting a moderately sized Nexp  Nexp system matrix ˚ij D ˚.jp i  p j j/: Theorems [1] guarantee non-degeneracy of this matrix for arbitrary dimension of the P design space and arbitrary dataset. The result can be written as f .p/ D i wi .p/fi ; with the weights wi .p/ D

X

˚ij1 ˚.jp  p j j/:

(3)

j

This representation makes an interpolation of bulky simulation data possible at a rate (computational complexity) linear with respect to the size of the dataset. p A suitable choice for the RBF is the multi-quadric function ˚.r/ D c 2 C r 2 ; where c is a constant defining smoothness of the function near data point p D p i .

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In between data points, the result of the interpolation is a cumulative effect of many contributions and, except of this smoothness, there is almost no influence of coefficient c to the result. Therefore adjustment procedures (learning) are not necessary, in contrast to other interpolation techniques like Kriging and neural networks [4]. RBF interpolation can also be combined with polynomial detrending, adding a polynomial part P to the right-hand side of (1). This allows reconstructing exactly polynomial dependencies and generally improving precision of interpolation. Formulae for coefficients ci and interpolation weights wi can be updated accordingly. Metamodeling is often used for the solution of engineering optimization problems. Engineers shall simultaneously optimize multiple objectives, coming e.g. from different types of simulations, cost or quality measures. The optimum is (usually) not an isolated point but a hypersurface (Pareto front) composed of points satisfying a tradeoff property, i.e. none of the criteria can be improved without a simultaneous degradation of at least one other criterion. DesParO [5, 6] allows interactive exploration of design problems by a user-driven continuous change of design variables and instant evaluation of design objectives. Constraints to the design objectives can be applied by setting upper/lower boundaries on corresponding axes. Optimization, e.g. minimization of objectives, can be achieved by shifting their upper boundaries towards the minimum, while the availability of solutions is immediately visible as appearance of “green islands” on the axes of design variables and design objectives. The purpose of linear sensitivity analysis is, based on the determination of @fi a Jacobi matrix J D J.p/ with Jik D @p .p/ or, in a different normalization, a k sensitivity matrix S D S.p/ with Si k D Jik rms.pk /; a decision which parameters mostly influence the data, often used to reduce the optimization problem considered. Metamodeling can support sensitivity analysis for non-linear dependencies. Instead of the matrix S other quantities can be used as sensitivity measures. For example, Skmax .p/ D max jS.p/j ; pk

(4)

with the maximum taken over the k-th central cross-section in parameter space: pj D

pjmin C pjmax 2

for all j ¤ k, and pk 2 Œpkmin ; pkmax :

(5)

Also the sign of S.p/ can be tested along the cross section, and a global sign of dependence can be introduced, equal to C1 if all S.p/ are positive on the crosssection, 1 if all S.p/ are negative and 0 for sign-changing S.p/. In DesParO, this information is presented graphically, cf. Fig. 1 (left). For the evaluation of sensitivity measures, one can use not only cross-sections but other subsets, e.g. providing a dense sampling of the parameter space. A further challenge is not only to interpolate data in a new parameter point, but also to estimate the precision (“tolerance”) of this interpolation. In an existing data

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Fig. 1 DesParO’s graphical user interface with a parameter-sensitivity matrix (left) and the geometry viewer for interpolated simulation results (right)

point, this precision can be estimated using a known cross-validation procedure: the data point is removed, data are interpolated to this point and compared with the actually measured value at this point. Formally, for this comparison one needs to construct a new metamodel where the data point is removed. For an RBF metamodel this step can be performed analytically, applying the SMW formula, resulting in the following direct expression for the tolerance ei : ei D f interpol .pi /  f actual .pi / D 

ci ˚ii1

(6)

where ci and ˚ were defined above. This tolerance formula is applicable also for bulky data, since ci D

X

˚ij1 fj for all i D 1; : : : ; Nexp

(7)

j D1;:::;Nexp

is represented as a weighted sum of data items. In between data points, the soestimated tolerance can be interpolated, using e.g. the same RBF weights wi .

3 Specifics of Bulky Data One of the major challenges of bulky data analysis is the huge amount of data items to be processed, typically with Ndata reaching hundreds of millions. Special methods are required to support processing of such amounts of data. The number

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of experiments is usually varied from dozens to hundreds. In order to keep computational cost reasonable, Nexp is required to be as minimal as possible. A dimension-reduction technique applicable here is provided by principal component analysis (PCA), here used in form of singular value decomposition (SVD). As an input, the experimental data matrix X D .xij , i D 1; : : : ; Ndata , j D 1; : : : ; Nexp is used, where every column forms one experiment, every row forms a data item varied in experiments. An average value is row-wise subtracted, forming a centered data matrix dX with dxij D xij  < xi >. This matrix is decomposed as dx D UV T where  is a diagonal matrix of size Nexp  Nexp , U is a column-orthogonal matrix (U T U D 1) of size Ndata Nexp , V an orthogonal square matrix (V T V D V V T D 1) of size Nexp  Nexp . A computationally efficient method to perform this decomposition in our practical case with Ndata  Nexp is to find the Gram matrix G D dx T dx; perform its spectral decomposition G D V2 V T ; and compute the matrix U D dxV1 : The Gram matrix has a moderate size ofNexp  Nexp which allows to load the whole G-matrix to memory and apply O any suitable method for spectral matrix decomposition (e.g. JacobiAt’s method). Practically, the computational complexity of this step is negligible compared with the other steps. Computation of the G-matrix can be performed by sequential reading of data items and incrementing the elements of the G-matrix, e.g. for the n-th data item (n D 1 : : : ; Ndata ) in pseudo-code: Gij D Gij C dxni dxnj ;

i; j D 1; : : : ; Nexp ;

(8)

2 2 requiring O.Ndata Nexp / floating point operations and O.Nexp / memory. Computation of the U -matrix requires to read the database once more, performing floating point operations in O.Ndata Nexp Nmodes / (if the Nmodes first columns of U are taken 2 into account) and storing O.Nexp / data items. The SVD spectrum is represented by elements of the diagonal matrix . These values are non-negative and sorted in non-ascending order. If all these values starting from a certain number Nmodes C 1 are omitted (i.e. set to zero), the resulting reconstruction of data matrix will have a deviation r. L2-norm of this deviation gives

err2 D

X ij

rij2 D

X

2k

(9)

k>Nmodes

i.e. the sum of squares of omitted spectral values. This formula allows controlling precision of the reconstructed data matrix. Usually k rapidly decreases with k, and a few largest values give sufficient precision. In this way, SVD provides an efficient representation of the data matrix as a linear combination of a small number of U -vectors. It is convenient to scale these vectors by entries of the diagonal -matrix, writing  D U. Column vectors of  are called spectral modes or principal components, while column vectors of the V -matrix define distribution of experiments over principal components.

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The advantage of such a representation is that a suitable selection of Nmodes  Nexp leads to a reduction of storage and the possibility to accelerate all processing operations applied to the data matrix. Particularly, RBF metamodeling of bulky data represents an interpolated result as a weighted sum: dxi D

X

dxij wj :

(10)

j D1;:::;Nexp

With SVD representation above this gives dxi D

X

ip gp ;

X

.Vjp wj /

(11)

j D1;:::;Nexp

pD1;:::;Nmodes

providing an acceleration factor of the weighted sum.

gp D

Nexp , Nmodes

compared with the direct computation of

4 Industrial Example The small, but representative demonstration model shown on Fig. 1 (right) contains around 10,000 nodes, 45 timesteps, 101 simulations. Two parameters are varied representing thicknesses of two layers composing a part of a B-pillar. The purpose is to find a (Pareto optimal) combination of parameters simultaneously minimizing the total mass of the part and crash intrusion in a selected middle point. While methods of multiobjective optimization combined with RBF metamodeling are applied to solve this problem, SVD has been used for a compact representation of bulky data. SVD has been recently integrated to our interactive optimization tool DesParO and allowed to accelerate algorithms responsible for interpolation of bulky data by two orders of magnitude to fulfil real time requirements. In DesParO, the user can interactively change parameter values, immediately see variations of optimization criteria and—with SVD—of complete simulation result.

5 Conclusion We have described methods for supporting nonlinear metamodeling of a simulation database, continuous exploration of simulation results, tolerance prediction, sensitivity analysis, and rapid interpolation of bulky FEM data. The effiency of the methods has recently been demonstrated for several real-life application cases from automotive industry. One exemplary demonstration model has been described. Acknowledgements This work was supported by the FhG Internal Programs under Grants No. MAVO 816 450 (CAROD) and MAVO 817759 (HIESPANA).

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References 1. Buhmann, M.: Radial Basis Functions: Theory and Implementations. Cambridge University Press, Cambridge (2003) 2. Donoho, D.: High-Dimensional Data Analysis: The Curses and Blessings of Dimensionality. Lecture on August 8, 2000, to the American Mathematical Society “Math Challenges of the 21st Century” (2000) Available from http://www-stat.stanford.edu/ donoho, last access date: February 1st (2012) 3. Jones, D., Schonlau, M., Welch, W.: Efficient global optimization of expensive black-box functions. J. Global Optim. 13, 455–492 (1998) 4. Keane, A.J., Leary, S., Sobester, A.: On the design of optimization strategies based on global response surface approximation models. J. Global Optim. 33, 31–59 (2005) 5. Nikitina, L., Nikitin, I., Stork, A., Thole, C.A., Klimenko, S., Astakhov, Y.: Towards interactive simulation in automotive design. Vis. Comput. 24, 947–953 (2008) 6. Nikitina, L., Nikitin, I., Thole, C.A., Steffes-lai, D., Kersten, R., Bruns, J.: Constrained optimization with DesParO. In: Proceedings of the Conference on Virtual Product Development in Automotive Engineering, Prien, 21–22 March 2007 7. Tukey, J.: Exploratory Data Analysis. Addison-Wesley, London (1997)

Heat Transfer During Annealing of Steel Coils Winston L. Sweatman, Steven I. Barry, and Mark McGuinness

Abstract Steel becomes brittle during the cold rolling process which is used to produce sheet metal. Heat treatment (annealing) is required to release stresses and reform the crystalline structure. The 2008 Mathematics-and Statistics-inIndustry Study Group in Wollongong (MISG08) modelled the approach used by New Zealand Steel for which steel coils are heated in a batch annealing furnace. Determining the temperature within each coil is complicated by height-dependent gaps within the coils. Deciding on suitable boundary conditions for the outside of the coils provides a further challenge. This is explored with two alternative models. Having made reasonable assumptions, a linear model is found to be sufficient for modelling the heating process and allows the cold point in the steel coil to be established.

1 Introduction Manufactured steel becomes brittle during cold rolling to produce sheet metal. Annealing reforms the crystal structure. Initially, coiled metal strips are heated to a high temperature (1; 000 K). This temperature is then maintained for several hours. The time required for the initial heating is determined by the part of the coil that is slowest to heat. To minimise the heating time this point needs to be found and W.L. Sweatman () Massey University, Auckland, New Zealand e-mail: [email protected] S.I. Barry Australian National University, Canberra, Australia e-mail: [email protected] M. McGuinness Victoria University of Wellington, Wellington, New Zealand e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 35, © Springer-Verlag Berlin Heidelberg 2012

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the time for it to reach the desired temperature obtained. Simple and accurate models that incorporate different coil properties will allow this process to be optimised reducing heating costs and avoiding poorly annealed steel that needs reprocessing. In Auckland, New Zealand Steel use a Uniflow Annealing System (UAS) furnace. Steel coils, typically nine in a horizontal square formation, are placed on their ends with their axes aligned vertically on top of a ventilated steel platform. The platform is moved into the furnace. The coils are heated directly by radiant burners spaced across the ceiling of the furnace. Additional burners along the sides are shrouded and do not directly heat the coils. The gas within the furnace is an inert mixture of nitrogen (93% by volume) and hydrogen. Circulating this gas provides indirect heating. Experimental data for the furnace are limited because there are practical difficulties in taking measurements. A full account of the work of MISG08 is given in [1] with some additional consideration of the problem in [2]. In the following, we provide a concise summary of a model of the coil annealing process, identifying the key features, and further investigate the assumed boundary conditions on the ends of the coils.

2 Modelling the Steel Coils We can model each individual coil as a continuous vertical hollow cylinder with anisotropic (position and direction dependent) thermal conductivity. Vertically, conduction is in the axial z direction, in individual coil layers, with the conductivity kz [J/m/s/K] that of steel ks . Radially, there is effectively a lower conductivity, kr , due to gaps between layers of the metal. Temperature T D T .r; z; t/ [K] is governed by @.cp T / 1 @ D @t r @r

    @ @T @T kr r C kz ; @r @z @z

(1)

where cp [J/kg/K] and  [kg/m3] are the heat capacity and density of steel, respectively. Table 1 lists appropriate dimensions and properties. Initially T D T0 . With the paucity of experimental data, finding suitable boundary conditions for the outside of the coils requires modelling assumptions. The ventilated platform is presumed to reach furnace temperature extremely quickly. The coils heat by radiation, conduction and convection. Heating of each coil’s flat surfaces (the ends z D 0; L) is expected to be very effective due to radiation from the heaters above and conduction from the ventilated platform below. These surfaces are, for the present, taken to rapidly assume furnace temperature Tg . (Alternative boundary conditions are considered further in Sect. 3.) The curved surfaces r D a; b, heated by convection, have the boundary condition kr @T =@r D ˙H.T  Tg / (Newton’s Law of Cooling), where H is the heat transfer coefficient. A number of approaches for estimating the value of H have been considered, however, for the units used here, they all give values in the range 3–5 [1]. The concept of mean action time

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Table 1 Typical steel, coil and furnace properties Steel density  7,854 Steel thermal conductivity ks 60.5–30 Steel thermal capacity cp 434–1,169 Steel strip thickness 0.4–3 Steel strip width L 700–1,500 Coil mass 10–20 Coil inner diameter a 508 Coil outer diameter b 1.5 Gas thermal conductivity kg 0.06 Furnace circulation 800 Platform mass 37 Furnace dimensions 6.5  6.5  4 Furnace temperature Tg 1,000 Initial coil temperature T0 300

kg/m3 at 300 K W/m/K at 300 K–1,000 K J/kg/K at 300 K–1,000 K mm mm tonnes mm m W/m/K m3 /min tonnes m3 K K

[1, 3] can be used to allow for changing thermal capacity cp and conductivity ks with rising steel temperatures but is not considered further here. Coils can be modelled as concentric annular cylinders of metal separated by hot gas [4, 5]. This approximates the different mechanisms of heat transport (direct contact, gaseous diffusion, and radiation) between layers of the coil [6–8]. The effective conductivity across the layers can be taken to be keff 

ds C dg ds ks

C

dg kg

(2)

where ds and dg are the thicknesses of the steel and gas layers. This expression is exact for the steady state and the limit of infinite layers [9]. It is a reasonable model here. A potential complication is vertical variation of the gaps, as a rolled steel strip has a crown: it is thinner at the edges than the middle. Examples at 1,000 K suggest that in the central 3=4 of the coil by height (middle of the strip), the radial conductivity kr is about 1=2 to 3=4 that of steel ks . This rapidly decreases towards the flat ends where thinning due to the crowning occurs [1]. However, as the ends are heated very effectively from above and below this has little effect. Potentially, radial conductivity kr is also radially dependent due to coil tension and differential expansion during heating, although these effects are thought to be small. If the coil is assumed homogeneous, but with constant anisotropic heat conductivities kr , kz , separation and Sturm–Liouville Theory lead to a solution for the temperature T (cf. [1, 2, 10]). First, the heat transfer equation (1) is rewritten as 1 @ @T D Dr @t r @r

    @T @ @T r C Dz : @r @z @z

(3)

The diffusivities Dr D kr =.cp /, Dz D kz =.cp / are assumed constant. The system is rescaled so that r D br  , z D Lz , t D .cp L2 =kz /t  , u D .T  Tg /=.T0  Tg /,

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where r  ; z ; t  ; u are non-dimensional. Relative diffusivity D D kr L2 =kz b 2 , ˛ D a=b, and h D H b=kr . The  notation is dropped for convenience. The new system is   @u 1 @ @u @2 u DD r C 2; @t r @r @r @z     @u @u D D hu; Œut D0 D 1; ŒuzD0;1 D 0: @r rD˛ @r rD1

(4) (5)

This now represents the non-dimensional cooling of a unit cylinder from an initial temperature of unity to a surrounding temperature of zero. Separating variables as u.r; z; t/ D R.r/Z.z/T .t/, a series solution can be found with Bessel functions uD

1 1 X X

Amn e .Dm Cn /t sin.n z/ Cm .r/; 2

2

(6)

nD1 mD1

where the eigenvalues n , m satisfy ˇ ˇ ˇ hJ .˛ / C  J .˛ / hY .˛ / C  Y .˛ / ˇ m m 1 m 0 m m 1 m ˇ ˇ 0 n D n; ˇ ˇ D 0; ˇ hJ0 .m /  m J1 .m / hY0 .m /  m Y1 .m / ˇ

(7)

and m are found numerically. Coefficients Amn and functions Cm .r/ are given by R1

R1

Amn D

0 sin.n z/d z R1 2 0 sin .n z/ d z

a

rCm dr

a

rCm2 dr

R1

4 n .1

D

m



 .1/n / ŒrCm1 1˛   1 ; 2 r 2 Cm2 C Cm1 ˛

Cm  J0 .m r/ C Bm Y0 .m r/ ; Cm1  J1 .m r/ C Bm Y1 .m r/: Bm D

. hm J1 .m /  J0 .m // .Y0 .m / 

m h Y1 .m //

:

(8) (9) (10)

In numerical simulations, only very modest differences occurred when variation in radial conductivity was introduced to allow for height dependence due to crowning[1]. The Newtonian boundary condition is the main restriction on radial heat flow rather than conductivity rates within the coil. If the curved sides were heated more directly then the cold point’s heating time would reduce [2]. The leading eigenfunction associated with eigenvalues 1 and 1 D  dominates the linear model. Consequently, time dependence is governed by the decay of 2

2

2

e.D1 C1 /t D e.D1 C /t : 2

(11)

At the cold point, in scaled units, z D 1=2 and r D rc where rc is given by the extremum of the function C1 .r/. As illustrated in Fig. 1 the coil’s cold point is closer to the inner curved surface due to its smaller surface area (and hence heat flux).

Heat Transfer During Annealing of Steel Coils

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Contour of temperature at latest time

0.9 0.8

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0.8

0.9 0.8

0.7

0.7

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0.7

0.6

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0.6

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z

0.5 0.5

0.4 0.4

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0.2

0.1

0.1 0

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0.8

0.9

1

0.2 0.1 0

0.4

0.5

0.6 0.7 radius r

0.8

0.9

1

Fig. 1 Contour plots of scaled temperature for a coil cross-section during heating. In the left-hand figure, the boundary conditions of Sect. 2 apply at the coil ends (T D Tg ). The cold spot (white) is nearer the inside of the coil (r D 1=3). The curved surfaces are still approaching the surrounding gas temperature (black). In the right-hand figure, the alternative conditions (Sect. 3) are applied

3 Alternative Boundary Conditions In Sect. 2, we have assumed that the ends of the coils heat so rapidly that we may assume that they are at the furnace temperature. This assumption was based upon radiation from the heaters above and conduction from the ventilated steel platform below. Here we consider what will happen if this is not the case. The heating from above comes from a grid of radiant heaters. As well as spaces between the heaters, in places these may be partially obscured by the roof structure. If we suppose that the top of a coil is not heated directly by radiation then a similar boundary condition can be applied here to those used for the sides of the coils. The gaps in the surface due to the layers of coil might have a small effect, however, we may allow for this by modifying this surface’s heat transfer coefficient (HE ) so kz @T =@z D HE .T  Tg /. Non-dimensionalisation proceeds as before, however, in (5), ŒuzD1 D 0 is replaced by Œ@u=@zzD1 D hE u where hE D HE L=kz . The solution is similar except that now the eigenvalues n must satisfy hE tan .n / C n D 0:

(12)

This gives values of n 2 Œn  =2; n. (If heat transfer hE ! 1, we obtain n D n in agreement with our earlier result, (7). If hE ! 0 (insulation on the top surface), then n ! n  =2.)

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Corresponding to the change in the leading eigenvalue 1 , which decreases from  towards =2, the extremum in the eigenfunction sin .1 z/ moves from 1=2 towards 1. This means that the cold point will shift upwards towards the top surface of the coil, as is to be expected with less efficient heating there. (In the limit hE ! 0 (insulation on the top surface) the cold point would be on the top surface.) Using the new value of 1 in (11), the timescale for the heating is observed to increase by a factor of .D21 C  2 /=.D21 C 21 /. In Fig. 1, we compare a coil modelled with these alternative boundary conditions with the previous model. As expected, now that the heating of the top of the coil is less effective, the cold point has shifted upwards. It is again apparent, that the crucial limiting factor is the coil’s exterior heating, rather than internal conduction. Other scenarios may be incorporated into the model in a similar way.

4 Conclusions We have presented a simple model of the heating process for a steel coil during annealing. The model illustrates that the major constriction is the slow transport of heat through sides of coils that are only heated indirectly. This is of more significance than the anisotropic conduction rates of heat within the coils. The linearised model seems adequate for calculations and itself tends to be dominated by a leading eigenvalue term. Alternative boundary conditions for the coils have been considered. Acknowledgements We are grateful to the NZ Steel representatives and thank other participants of the 2008 Mathematics-in-Industry Study Group.

References 1. McGuinness, M., Sweatman, W.L., Baowan, D., Barry, S.I.: Annealing Steel Coils. In: Merchant, T., Edwards, M., Mercer, G. (eds.) Proceedings of the 2008 Mathematics and Statistics in Industry Study Group, pp. 61–80. University of Wollongong, Australia (2009) 2. Hocking, G.C., Sweatman, W.L., Fitt, A.D., Breward, C.: Deformations during jet-stripping in the galvanizing process. J. Eng. Math. 70, 297–306 (2011), doi: 10.1007/s10665-010-9394-8 3. Landman, K., McGuinness, M.: Mean Action Time for Diffusive Processes. J. Appl. Math. Decis. Sci. 4(2), 125–141 (2000) 4. Stikker, U.O.: Numerical simulation of the coil annealing process. In: Mathematical Models in Metallurgical Process Development, Iron and Steel Institute, Special Report, 123, pp. 104–113 (1970) 5. Willms, A.R.: An exact solution of Stikker’s nonlinear heat equation. SIAM J. Appl. Math. 55(4), 1059–1073 (1995) 6. Sridhar, M.R., Yovanovicht, M.M.: Review of elastic and plastic contact conductance models: comparison with experiment. J. Thermophys. Heat Trans. 8, 633–640 (1994) 7. Zuo, Y., Wu, W., Zhang, X., Lin, L., Xiang, S., Liu, T., Niu, L., Huang, X.: A study of heat transfer in high-performance hydrogen Bell-type annealing furnaces. Heat Tran. Asian Res. 30(8), 615–623 (2001)

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8. Zhang, X., Yu, F., Wu, W., Zuo, Y.: Application of radial effective thermal conductivity for heat transfer model of steel coils in HPH furnace Int. J. Thermophys. 24(5), 1395–1405 (2003) 9. Hickson, R., Barry, S., Mercer, G.: Exact and numerical solutions for effective diffusivity and time lag through multiple layers. ANZIAM J. (E) 50, C682–C695 (2009) 10. Budak, B.M., Samarskii, A.A., Tikhonov, A.N.: A collection of problems in mathematical physics. Dover, New York (1964)

Deformations Arising During Air-Knife Stripping in the Galvanisation of Steel Graeme C. Hocking, Winston L. Sweatman, Alistair D. Fitt, and Chris Breward

Abstract During sheet steel production, the steel surface is usually coated with metal alloy for corrosion protection. This can be done by passing the steel through a bath of the molten metal coating, and controlling thickness with a pair of air knives on either side of the ascending steel strip. Surface quality problems have arisen with recent developments in production. The process was considered at the 2009 Mathematics-and-Statistics-in-Industry Study Group in Wollongong (MISG09) and in subsequent investigations. Previous analyses are extended by the addition of shear terms and by exploring the effect of increased air-jet speeds. A first-order partial differential equation governs the system. This may be used to determine the steadystate coating shape and to study the evolution of any defects that may form.

1 Introduction In steel sheet production, the steel surface is usually coated with a metal alloy (e.g. zinc/aluminium) to protect against corrosion. In the continuous hot-dipped galvanising process, the steel strip is passed through a bath of molten alloy and then

G.C. Hocking Murdoch University, Perth, Murdoch WA 6150, Australia e-mail: [email protected] W.L. Sweatman () Massey University, Auckland, New Zealand e-mail: [email protected] A.D. Fitt University of Southampton, Southampton, UK e-mail: [email protected] C. Breward University of Oxford, Oxford, UK e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 36, © Springer-Verlag Berlin Heidelberg 2012

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drawn upward until the coating solidifies. Alongside the rising steel strip, a pair of air knives (high velocity air jets) control coating thickness by forcing surplus molten alloy back downwards. New advanced coatings have led to surface quality issues for Bluescope Steel who brought the problem to the MISG 2009 study group. For fixed processing conditions there seems to be a critical air knife pressure below which coating is satisfactory but above which defects may appear. Pocks, the most serious defects, involve substantial local thinning of the coating and a correspondingly severe reduction in corrosion protection. New models [1, 2] improve upon earlier models [3–7] by including air-knife shear. Here, we summarise model development.

2 Modelling the Coating Process The development of the mathematical model [1, 2] parallels that of Tuck [3], however, an extra term due to the air-knife shear is included. The model assumes a thin coating and almost-uni-directional flow. The air knife is modelled by a surface pressure distribution and a surface shear stress. The flow is taken as twodimensional, incompressible, laminar and unsteady, governed by the Navier–Stokes equations: 1  ut C uux C wuz D  px C .uxx C uzz/  g;  

(1)

1  wt C uwx C wwz D  pz C .wxx C wzz /;  

(2)

ux C wz D 0;

(3)

where t is time, x and z are respectively the vertical and horizontal coordinates, q D .u; w/ is fluid velocity and p pressure. Subscripts indicate differentiation. Other parameters and typical values are listed in Table 1. Air-knife pressure pa .x/ and shear stress a .x/ are assumed specified. Coating fluid boundary conditions are u D U;

wD0

at z D 0 (the substrate);

(4)

uz D a .x/; p  pa .x/ D  ; ht C uhx D w at z D h.x; t/ (the free surface): (5) Curvature   hxx (variation of h with x is small). Set t D .L=U /tN, x D Lx, N N pa .x/ D .U= 2 L/P .x/ z D LNz, u D U uN , w D U w, N p D .U= 2 L/p, N h D Lh, and a .x/ D .U=L/G.x/. Drop overbars for convenience. To leading order (1)–(5) become px D uzz  S; u D 1;

pz D 0; wD0

ux C wz D 0; at z D 0;

(6) (7)

Deformations Arising During Air-Knife Stripping in the Galvanisation of Steel Table 1 Parameters and some typical values Density of the coating Density of steel Dynamic viscosity of the coating Gravitational acceleration Surface tension coefficient of the coating Scale of thickness of coating Vertical length scale—half-width of air jet Half-width of the steel strip Upward speed of the steel strip Maximum centreline speed of the air jet Capillary number (Surface tension) Surface tension quotient Reynolds number Stokes number Length ratio Pressure scaling Shear scaling

uz D G.x/;

3  103 7  103 103 9:8 101 5  106 5  103 103 2:5 30 2:5  102 4  108 37; 500 0:0003 103 5  105 500

 s  g  h0 L d U Ua Ca D U = C D  3 =Ca Re D UL= S D gh20 =U  D h0 =L U = 2 L U =L

p  P .x/ D C hxx ;

ht C uhx D w

313

kg m3 kg m3 kg m1 s1 m s2 N m1 m m m m s1 m s1

kg m1 s2 kg m1 s2

at z D h;

(8)

P .x/ and G.x/ being non-dimensional pressure and shear acting on the coating surface. Based on experimental work [8, 9], realistic functional forms for them are [2]: P .x/ D PMAX .1 C 0:6x 4 /3=2 ; 8 h i <sign.x/G 0:22jxj3 MAX erf.0:41jxj/ C 0:54jxje G.x/ D :sign.x/GMAX Œ1:115  0:24 log jxj

(9) if jxj < 1:73 if jxj  1:73.

(10)

Ignoring terms multiplied by  2 Re D h20 U=.L/  0:04 and solving (6)–(8)  1 2 p D P .x/  C hxx ; z  hz C zG.x/ C 1; u D .S C P .x/  C hxxx / 2 (11)   1 1 1 1 w D z2 hx .S C P 0 .x/  C hxxx /  .P 00 .x/  Chxxxx / z3  hz2  z2 G 0 .x/: 2 6 2 2 (12) 0



These values satisfy (6)–(8) except that the last boundary condition becomes a PDE:   1 1 ht C h C h2 G.x/  h3 .S C P 0 .x/  C hxxx / D 0: 2 3 x

(13)

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3 Steady-State Solutions Surface tension is neglected as C  4  108 (also see [5]). Integrating (13) without time dependence gives the coating fluid flux Q. This is identical for all x. So Q D f .h; x/ D h C

h2 h3 G.x/  .S C P 0 .x//; 2 3

dQ h2 h3 D 0 D h0 Œ1 C hG.x/  h2 .S C P 0 .x// C G 0 .x/  P 00 .x/: dx 2 3

(14) (15)

Coefficients of the cubic in (14) vary continuously with changes in G.x/ and P .x/. Consider the control point .xc ; hc /, where the cubic has a double root. As @f .hc ; xc / D 1 C hc G.xc /  h2c ŒS C P 0 .xc / D 0; @h s   !  G.xc / 1 S C P 0 .xc / 1˙ 1C4 : hc D 2 S C P 0 .xc / G.xc /2

(16)

(17)

The negative sign is appropriate. Evaluating (15) at x D xc , using (16), noting hc ¤ 0, G 0 .xc / D

2 hc P 00 .xc /: 3

(18)

A simple practical approach to finding the control point is to evaluate hc at each x using (17) and find the associated flux Q with (14). The required xc and hc correspond to the minimum Qc . Using PMAX D 0:01, GMAX D 0:1 in (9) and (10), then xc   1:107, hc  6:172, Qc  3:529 and upstream coating thickness h  43 and that downstream hC  3:33. The final coating thickness decreases as the pressure of the air jet increases. Correspondingly, the thickness of the upstream layer increases as more of the coating alloy drains downwards. Also, as expected, the flux decreases as the coating thickness is reduced.

4 Evolution of Coating Deformations Starting with an established coating process in steady state we consider the evolution of a small surface deformation. It is unclear what the origins of such a perturbation might be. In [1], various possibilities are considered including impaction by a particle embedded in the air knife jet. In [2], it is found that a quite large single pressure pulse in the air knife (20%) only creates a small dip, while rapid periodic

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variations in the pressure (shudder of the air knife) have a minimal effect upon the coating. Re-writing (13) ht C c.h; x/hx D A.h; x/;

(19)

we observe the propagation speed c.h; x/ and amplitude A.h; x/ of a disturbance are c.h; x/ D 1 C G.x/h  h2 .S C P 0 .x//;

(20)

A.h; x/ D h2 .hP 00 .x/=3  G 0 .x/=2/:

(21)

Note that c.h; x/ D 0 at .xc ; hc / from (16), and disturbances below the control point propagate downwards towards the bath, while those above propagate upward and are our primary interest. A comparison of c.h; x/ and A.h; x/ for a weaker jet and a stronger one [2], verifies results of linear analysis [1, 3]. The term A.h; x/ is only significant over a very narrow region close to the control point, whereas the term c.h; x/ remains significant well downstream where it approaches 1  h2 S . Furthermore, for the weaker jet, gravity plays a more significant role as h is larger and G.x/ is smaller. Close to the air knife centre, coating behaviour is directed by the interaction between the shear and pressure terms. Beyond this thin band, c.h; x/  1 C G.x/h  h2 S and A.h; x/ remains small for all pressures. This suggests that beyond the immediate influence of the jet, perturbations will be marginally stable (neither growing nor decaying) as found in the analysis [3]. There are circumstances where points near the surface travel faster than those near the substrate and others where the reverse is true [1, 2]. In the higher pressure case, the shear term has a greater effect upon c.h; x/. If c.h; x/ > 1 (the high pressure case), then disturbances travel upward faster than the substrate, and depressions break forward (upward). If c.x; h/ < 1 (lower air-knife pressure) disturbances travel more slowly than the sheet and depressions break backward (downward) into themselves. Numerical explorations initially used the method of characteristics [1], and later [2] the method of lines (cf. [10]). They confirmed the analysis. In particular, a 2 dip ı.x/ D 0:3e .x0:5/ was added to the steady-state coating thickness and its evolution followed for realistic choices of shear and pressure (9,10), S D 0:0015, GMAX D 10PMAX , and a range of values PMAX D 0:0025; 0:005; 0:01 and 0:05 [2]. At lower values of PMAX , the air knife’s influence is very narrow and shear and gravity quickly dominate. The coating surface is slower than the substrate and the dip steepens at the leading edge prior to breaking backwards (downward). At higher values of PMAX , the coating is very thin, and shear terms persist, so that points near the substrate are slower and the back of the dip breaks forwards (upward). For intermediate values of PMAX the depression continues with only minor changes in shape for a long distance upward. After the disturbance passes, the coating near the air knife returns to steady state.

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5 Conclusions The equations derived by Tuck [3] have been modified to include a shear term as in [1, 2]. Results suggest that a disturbance’s evolution depends on the air-knife shear G.x/ at larger values of pressure P .x/, and is dominated by gravity at lower values. Propagation speed depends on the coating thickness. Disturbances of any sort are more persistent for thinner coatings, as then gravity is insufficient to cause a dip to break back into itself and fill the hole. Surface tension and metallurgical effects of solidification may have some effect on these disturbances. However, it would appear unlikely that they will be important with the parameters considered here [2,5]. There seems no indication that the fluid dynamics undergoes any new calamitous change that might be responsible for the pitting of the surface. Dimensional values will depend on the actual parameters. For illustration, a final coating thickness of about 15 m results when using the typical values given here (Table 1) with PMAX  0:01 (associated with air-knife speed 28 ms1 and steel strip speed 2.5 ms1 ) . Increasing the air-knife speed or decreasing the strip speed effectively raises PMAX . In this example, the transition to potential difficulties occurs when the coating thickness drops below h  1 (a thickness of about 5 m) which is associated with PMAX  0:05 or a doubling in air-knife speed combined with slowing of the strip speed. To obtain thinner coats major increases in air-knife speed are needed unless the strip speed is reduced. Slowing the strip speed, although it has the apparent effect of an increase in pressure, allows more time for any disturbances to decay. Application to a real situation depends on the actual values for the other parameters and the numbers here are purely indicative. Acknowledgements We are grateful to Cat Tu and Daniel Yuen from Bluescope Steel, Wollongong, Australia and also thank other participants of the 2009 Mathematics-and-Statisticsin-Industry Study Group.

References 1. Hocking, G.C., Sweatman, W.L., Roberts, M.E., Fitt A.D.: Coating Deformations in the continuous hot-dipped galvanizing process. In: Merchant, T., Edwards, M., Mercer, G. (eds.) Proceedings of the 2009 Mathematics and Statistics in Industry Study Group, pp. 75–89. University of Wollongong, Australia (2010) 2. Hocking, G.C., Sweatman, W.L., Fitt, A.D., Breward, C.: Deformations during jet-stripping in the galvanizing process. J. Eng. Math. 70, 297–306 (2011) doi: 10.1007/s10665-010-9394-8 3. Tuck, E.O.: Continuous coating with gravity and jet stripping. Phys. Fluid. 26(9), 2352–2358 (1983) 4. Tuck, E.O., Bentwich, M., Van der Hoek, J.: The free-boundary problem for gravity-driven unidirectional viscous flows. IMA J. Appl. Math. 30, 191–208 (1983) 5. Tuck, E.O., Vanden-Broeck, J.-M.: Influence of surface tension on jet-stripped continuous coating of sheet materials. Amer. Inst. Chem. Engineers J. 30, 808–811 (1984) 6. Tu, C.V.: Optimisation of lip gap for thin film coating in the jet stripping process. Proceedings of the 5th International Conference on Manufacturing Engineering. University of Wollongong, Australia (1990)

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7. Thornton, J.A., Graff, H.F.: An analytical description of the jet-finishing process for hot-dip metallic coatings on strip. Metall. Mater. Trans. B 7B, 607–618 (1976) 8. Elsaadawy, E.A., Hanumanth, G.S., Balthazaar, A.K.S., McDermid, J.R., Hrymak, A.N., Forbes, J.F.: Coating weight model for the continuous hot-dip galvanizing process. Metall. Mater. Trans. B 38B, 413–424 (2007) 9. Tu, C.V., Wood, D.H.: Wall pressure and shear stress measurements beneath an impinging jet. Exp. Ther. Fluid Sci. 13, 364–373 (1996) 10. Hamdi, S., Schiesser, W.E., Griffiths, G.W.: Method of Lines, Scholarpedia 2(7), 2859 (2007) doi: 10.4249/scholarpedia.2859

Modelling Preform and Mould Shapes in Blow Moulding J.A.W.M. Groot, R.M.M. Mattheij, and C.G. Giannopapa

Abstract Blow moulding is an essential stage of manufacturing glass and polymer containers, i.e. bottles or jars. A preform is brought into a mould and subsequently blown into the mould shape to produce the container. Two different problems regarding blow moulding are considered: the forward problem, which consists of determining the mould shape from the preform shape, and the inverse problem, which consists of determining the optimal preform shape corresponding to the designed container shape. This paper is concerned with the constraints on the mould surface and sensitivity to perturbations in the shape for both problems.

1 Introduction Blow moulding or forming is an important stage in the manufacturing of hollow containers, such as bottles or jars. It is usually the final stage of a container forming process. In the blow moulding stage a so-called pre-from of hot viscous material is transferred into a mould. Subsequently, the preform is inflated with pressurised air until it takes the mould shape. The container is then left to cool down and removed from the mould. Figure 1 illustrates the process. Blow moulding finds its applications in glass or plastic container manufacturing. This paper is focussed on the relation between the pre-form shape and the container shape in blow moulding. Regarding this relation, the following two problems are considered.

J.A.W.M. Groot ()  R.M.M. Mattheij  C.G.Giannopapa Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands e-mail: [email protected]; [email protected]; [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 37, © Springer-Verlag Berlin Heidelberg 2012

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Fig. 1 Schematic drawing of the final blow stage ring

preform

mould

Fig. 2 Blow moulding problem description

1. The forward problem is to determine the shape of the container, given the shape of the pre-form. Thus the forward problem represents the physical process of blowing the pre-form into the mould shape. 2. The inverse problem is to determine the shape of the pre-form, given the shape of the container. Usually, a container with a certain shape is desired, but in order to blow this container first the pre-form needs to be known. This paper investigates for which container shapes a pre-from shape can exist and vice versa. It deals with relevant issues regarding the relation between the pre-form shape and the container shape, such as well-posedness and sensitivity, necessary constraints on the mould wall and equilibrium states of the container shape in blow moulding.

2 Problem Formulation This section gives a mathematical formulation for the forward and inverse problem. Let 1;0 , 2;0 , i and m be the inner and outer pre-form surfaces, the inner container surface and the mould surface, respectively, as depicted in Fig. 2 for a 2D axialsymmetrical jar. The dotted lines represent the variable surfaces in the problem formulation.

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In the mathematical model used for the forward problem, the molten material is modelled as an incompressible, viscous fluid. Since viscous forces dominate, the material flow can be described by a Stokes flow problem: 

 

r .ru/  rp D f; r u D 0;

(1)

where u is the flow velocity,  is the viscosity, p is the pressure and f is a body force. The viscosity is typically highly temperature dependent. Boundary conditions include an inflow pressure at the mould entrance and free-stress conditions for air and no-slip conditions for the material on the mould wall. The propagation of the material surfaces 1 .t/ and 2 .t/ follows the ordinary differential equation dx D u; dt

(2)

The forward problem is formulated as follows: find the location of inner glass surfaces 1 at t D t  , such that 2 .t  / D m and 2 .t/ ¤ m for t < t  , given 1 .0/ D 1;0 and 2 .0/ D 2;0 . It can be solved by means of numerical simulation, e.g. [1, 7] (glass blowing) and [3, 8, 9] (polymer stretch blow moulding). The inverse problem is to find the location of the initial glass surface 2;0 D 2 .0/, given the surfaces 1 and 2 at time t D t  , such that 2 .t  / D m and 2 .t/ ¤ m for t < t  , and given 1 .0/ D 1;0 . It can be solved by numerical optimisation [4, 5].

3 Equilibrium State In order to investigate whether the inner container surface converges towards an equilibrium during blow moulding, a constant air pressure pa is exerted on the inner material surface. The outer material surface is the mould wall. Equations (1), (2) hold in the material domain ˝. If the viscosity of air is negligible compared to the viscosity of the material, the following relation holds on the free surface:



ru n  pn D  n;

(3)

where  is the surface tension,  is the curvature and p WD pm  pa is the pressure difference between air and material. In equilibrium the flow velocity is zero, thus (1) simplifies to: rp D f; p D  ;

in ˝;

(4)

on 1 [ 2 :

(5)

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Fig. 3 Outer glass surface near p268corner in equilibrium

Consider the following cases. If f D 0, the curvature is constant, i.e. the surface is spherical. No equilibrium state can be reached in general, since the geometry of the bottle is not spherical and the material is incompressible. If also  D 0, the system is in equilibrium if pm D pa irrespective the shape. If f D gez , with gravitational acceleration g, the curvature is linear in z. If  D 0, the glass steadily flows to the bottom of the bottle. Because of the no-slip condition on the mould wall, this equilibrium state cannot be reached. Next consider an outer surface approaching a corner (see Fig. 3). Because of surface tension the surface reaches an equilibrium state before it reaches the corner. For the design of the mould the location of the surface in equilibrium is of interest. Let f D gez , then from (4) and (5) it follows that   D p0  gz;

(6)

where the axis z D 0 corresponds to p D p0 . For a two-dimensional curve of the form z.r/ the curvature can be written as   2  32 .r/ D z00 .r/ 1 C z0 .r/ : (7) Substitution in (6) yields a differential equation for z.r/    2  32 ˇz00 D   z 1 C z0 ; with

ˇD

 ; g

D

(8)

p0 : g

(9)

Boundary conditions are z.rc / D H;

z.L/ D zc ;

z0 .rc / D tan ';

z0 .L/ D cot ';

(10)

where ' is the contact angle between the surface and the mould (see Fig. 3), which can be determined experimentally (e.g. [6]). The two extra boundary conditions are

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323

Fig. 4 Typical equilibrium surface

necessary to determine the unknowns rc and zc . Successive multiplication of (8) by z0 and integration results in the first order ordinary differential equation v u z Du t  0

4ˇ 2 2  1: 2 z CE

(11)

Substitution of boundary conditions (10) into (11) yields E D 2ˇ cos ' C .H C /2 ;

zc D  

q

  2ˇ sin '  14  C .H C /2 : (12)

In order to find rc separation of variables is applied to (11) and the differential equation is integrated over the interval .rc ; L/, which leads to Zzc  H

! 12

4ˇ 2 .z/2 CE

2  1

dz D L  rc :

(13)

Finally, differential equation (11) with initial condition z.rc / D H can be solved numerically. Typical values are ˇ D 0:2,  D 20, H D 10, L D 1 and ' D 9 . The corresponding equilibrium surface is shown in Fig. 4. The walls in Fig. 4 are on the left and lower axes. The time scale at which an equilibrium state can be expected can be estimated by dimensional analysis. Near an equilibrium state the velocity is close to zero, hence the pressure difference is of the same order of magnitude as the surface force, p   . Let L be the local length scale and V the typical flow velocity, then V .

 L : 

(14)

Consider typical values   104 Pa s; L  1;   1Pa m. Then the typical velocity is V D 104 m s1 . With L  102 m, the typical time scale can be measured as

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L  100s: V

(15)

Furthermore, for a temperature dependent problem the viscosity will typically be of a larger order of magnitude by the time the equilibrium state is reached, so the time scale will be much larger. Moreover, elastic forces will play a more important role as the material cools down. It can be concluded that the mould shape will deviate somewhat from the equilibrium state, which is something that has to be taken into account when designing the mould.

4 Sensitivity Analysis A sensitivity analysis is applied to the 2D axial symmetrical case. The equations for the propagation of interface i in spherical coordinates .r; '/ around the center of the mould opening are i dri 1 h @ D ur .ri ; 'i / D . sin ' .ri ; '// 'D'i dt ri sin 'i @' h i 1 @ d'i D u' .ri ; 'i / D  .r .r; 'i // : ri rDri dt ri @r

(16) (17)

where is the stream function. The thickness of a perturbation in the shape is of interest. Therefore, the propagation of the perturbation in radial direction is estimated. Assume that d'i  0: (18) dt Then from (17) it follows that .ri ; 'i / 

Q .'i / ri

;

(19)

for some function Q W Œ0; 2 ! R. Subsequently, (16) gives  1   3 3 ri .'i ; t/  3 cot 'i Q .'i / C Q 0 .'i / t C ri;0

(20)

Finally, a small perturbation 0 <   1 in preform radius ri;0 gives a perturbation in the container radius rQi .t/ D ri .t/ C

2 ri;0

ri2 .t/

 C O. 2 /;

and since the mass flow in azimuthal direction is negligible

(21)

Modelling Preform and Mould Shapes in Blow Moulding

a

Original preform

b

Original container

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c

d

Preform with bump

Container with bump

Fig. 5 Blow moulding wine glass with and without bump

rQ1 .t  / D r1 .t  / C

2 ri;0

ri2 .t  /

 C O. 2 /:

(22)

Figure 5 shows a simulation of the blow process for a 2D axial-symmetrical wine glass with perturbation (Fig. 5a, b) and without perturbation (Fig. 5c, d; green circles) in the preform. The simulation model described in [1, 2] is used. The radius of the glass opening is 3.5 cm and the height is 7 cm. The bump has a thickness of 0.175 cm on the symmetry axis (where u' D 0). The preform has outer radius r2;0 D 4:025 cm on the symmetry axis. Then the thickness of the perturbation in the inner container surface is 0.058 cm—a decrease by factor 3 (see (22)). It can be concluded that the inverse problem is sensitive to perturbations in the preform.

References 1. Giannopapa, C.G.: Development of a computer simulation model for blowing glass containers. J. Man. Sci. Eng. 130 (2008) 2. Giannopapa, C.G., Groot, J.A.W.M.: Modeling the blow-blow forming process in glass container manufacturing: a comparison between computations and experiments. J. Fluid. Eng. 133(2), 021,103 (2011) 3. Groot, J.A.W.M., Giannopapa, C.G., Mattheij, R.M.M.: A computer simulation model for the stretch blow moulding process of polymer containers. In: Proceedings of the ASME 2010 Pressure Vessels & Piping Division/K-PVP Conference (July 18–22, 2010) 4. Groot, J.A.W.M., Giannopapa, C.G., Mattheij, R.M.M.: Numerical optimisation of blowing glass parison shapes. In: Proceedings of PVP 2009: ASME Pressure Vessels and Piping Division Conference (July 26–30, 2009) 5. Lochegnies, D., Moreau, P., Guilbaut, R.: A reverse engineering approach to the design of the blank mould for the glass blow and blow process. Glass Tech. 46(2), 116–120 (2005) 6. Passerone, A., Valbusa, G., Biagini, E.: The titanium-molten glass system: interactions and wetting. J. Mat. Sci. 12, 2465–2474 (1977) 7. C´esar de S´a, J.M.A.: Numerical modelling of glass forming processes. Eng. Comput. 3, 266–275 (1986) 8. Schmidt, F.M., Agassant, J.F., Bellet, M., Desoutter, L.: Viscoelastic simulation of pet stretch/blow molding process. J. Non-Newtonian Fluid Mech. 64, 19–42 (1996)

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9. Wang, S., Makinouchi, A., Okamoto, M., Kotaka, T., Maeshima, M., Ibe, N., Nakagawa, T.: 3D FEM simulation of the stretch blow molding process with a two-stage material model. In: Materials, ANTEC 1999 Plastics: Bridging the Millennia, vol. 2: Materials Society of Plastics Engineers, Brookfield, Conn. pp. 977–981 (1999)

Asymptotic Analysis of a Multi-Component Wet Chemical Etching Model Jonathan Ward

Abstract This paper investigates wet chemical etching of lead crystal glass, where it is necessary to use both hydrofluoric and sulphuric acid to dissolve all of the components of the glass. We consider a simple model of multi-component etching, consisting of a set of coupled linear ordinary differential equations. The long time behaviour is analysed using the method of steepest descents and the limiting etch rate is determined.

1 Introduction and Model Set-Up Wet chemical etching is used in a wide variety of industrial applications, from the manufacture of Integrated Circuits to the fabrication of glass micro-fluidic devices [3, 4, 8, 9]. This paper is motivated by multi-component etching used in the production of lead crystal glassware. In this case, decorative features cut into the glass leave it optically opaque and consequently polishing is required to restore its transparency [8]. This process consists of three steps: first the glass is immersed in an etching tank, then it is rinsed in water and finally the reaction products are collected in a settlement tank. The focus of this paper is on the initial wet chemical etching step, where it is necessary to use both HF and H2 SO4 in order to dissolve all of the components of the glass, namely SiO2 , PbO and K2 O. Such multi-component systems pose a non-trivial problem in determining the consequent etching rate. A number of experimental studies of wet chemical etching of glass have been conducted, see [8] for a review. In short, these studies present measurements of etching rates [4] and their dependence on etchant and glass composition [7, 9, 11]. To our knowledge, there has not been an experimental study of multi-component

J. Ward () MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 38, © Springer-Verlag Berlin Heidelberg 2012

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glasses where different types of etchant are required. An observation related to the work in this paper is that etching a smooth surface (e.g. mechanically polished) causes it to develop cusp like features [8,9], thus roughening it slightly. Furthermore, the height of such features is found to be normally distributed [10]. In the mathematics literature, the simplest models of macroscopic surface evolution have been well studied [1]. For a single component system, asymptotic results have been obtained for the diffusion limited case around a resist edge, where the normal velocity is taken to be proportional to the concentration gradient [6]. In general, the normal velocity will depend on various energy and curvature considerations; such effects have been considered in stressed media [5]. For multicomponent systems, it is not obvious how the macroscopic normal velocity of the etched surface depends on the composition of the glass and the acids present. Thus our approach in this paper is to consider a microscopic model that captures the salient features of multi-component etching in a simple way. This model was proposed at the 62nd European Study Group with Industry organised by the Mathematics Applications Consortium for Science and Industry at the University of Limerick in Ireland. As a simple model, we imagine that the glass consists of equallyPspaced layers n D 0; : : : ; N each composed of a fraction fj of species j , where Jj D1 fj D 1. We suppose that each layer is of the order of a few nanometers thick, much smaller than the height of macroscopic features to be etched (which are typically of the order of micrometres). Thus for multi-component systems, our goal is to determine the resulting etching rate. We model the evolution fraction of exposed surface at P of the j j layer n of species j , denoted by n (where j;n n D 1), by the reaction equations d dt d dt

j 0

D Aj

j 0;

j n

D Aj

j n

C fj

(1) X

Ak

k n1 ;

n  1:

(2)

k

The negative terms account for the removal of atoms from the exposed surface and the positive terms model the creation of exposed surface due to the total amount of surface removed in the layer above. We consider an initially flat surface, given by j 0 .0/

D fj ;

j n .0/

D 0:

(3)

A typical numerical solution of (1–3) for a simple caseP with just two species j is illustrated in Fig. 1. The exposed surface (where n D j n is non-zero) is diffuse and propagates downwards with an almost constant velocity. Thus the flat glass is etched and roughened as time progresses.

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329

ψn 0.08 0.07 t = 20 1 0.06 0.05 t2= 60

0.04 0.03

t3= 100

t4= 140

0.02 0.01 0

0

50

100

150

200

250

300

n

Fig. 1 Simulation results of (1–3) for two species plotted against n at the four time instants t1 –t4 . The solid black curve is the total fraction of exposed surface n , the light grey dashed line and j the dark grey solid line are the fraction of exposed surface n of species 1 and 2 respectively. The fractions of each species in the glass are f1 D 0:3 and f2 D 0:7, and the reaction rates are A1 D 1 and A2 D 2

2 Microscopic Model Analysis In this section, we apply an inductive Laplace transform technique to the discrete rate equations, (1) and (2), which yields an integral equation that can be analysed j j with standard asymptotic methods. Defining the Laplace transform of n as n , (1) and (2) become 0 D

fj  C Aj

nj D

fj X k Ak n1 ;  C Aj

j

and n  1;

(4)

k

respectively, where  is the transform variable. Solving the recursion relation prescribed by (4), we find nj D

Solutions

j n

fj g./n ;  C Aj

where

g./ WD

X Ak fk :  C Ak k

are found by taking the inverse Laplace transform, j n

1 D 2i

Z C

fj g./n et  d;  C Aj

(5)

where the contour C is chosen to the right of the J poles of the integrand, located at  D Aj . The integral (5) can be solved explicitly by calculating the residue at the

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poles. This approach can be followed through, although it yields little insight into the nature of the solution. Instead we look to use asymptotic methods to determine the large time behaviour j of n when n  t. Solutions presented in Fig. 1 show that the surface propagates with near p constant velocity whilst diffusing. This suggests the transformation n D vt C  t , where v > 0 and  is a travelling wave variable. We then define ./ WD  C v ln g./;

(6)

such that (5) takes the form j n

D

1 2i

Z C

h i p fj exp t C t ln g d:  C Aj

(7) j

Our next step is to determine the travelling wave velocity v D v at which n is stationary. To achieve this we consider the case where  D 0, i.e. we solve for j n along a ray with velocity v D n=t. We can then analyse (7) using the method of steepest descents (see [2] for details). In short, we look to deform the contour of integration C in such a way that the imaginary part of ./ remains constant along the new contour C 0 . We must not cross any poles of the integrand when deforming the contour. Then in the limit t ! 1, the contribution to the integral is dominated by the global maximum of Re ./ on C 0 . Such points are saddle points of ./ in the complex  plane and hence satisfy 0 ./ D 0. In this case the saddles points are solutions to g 0 ./ D 0; (8) 0 ./ D 1 C v g./ corresponding to the 2J  1 roots of the polynomial J X j D1

 Y Aj fj  C Aj  v . C Ak /2 D 0: k¤j

Thus the saddle points must typically be calculated numerically. In Fig. 2a we plot contours in the complex -plane along which Im  is constant, where arrows are used to indicate increasing Re  and the saddle points are labelled with white markers. It is possible to prove that there is only ever one admissible saddle point, however we will omit this complex argument from this paper and refer the reader to [12] where it is discussed in detail. The correct choice of saddle can however be easily identified. Suppose that the species j are ordered such that A1 < A2 <    < AJ . For real  > A1 , g is monotonically decreasing, thus it follows that Re  has a minimum at  D  say, which is shown in [12] to be the only admissible saddle. The asymptotic solution along a ray with velocity v is then

Asymptotic Analysis of a Multi-Component Wet Chemical Etching Model

a

b

2

331

10−1

σ(t)

1.5

10−2 1

λi 0.5

10−3

0

λ∗

−0.5 −1 −2.5

−2 −A2

−1.5

−1

λ1

−0.5

10−4

0

0

0.5

50

λr

−A1

100

t

150

Fig. 2 Panel (a): Contour plot of Im , see (6), in the complex  plane for the two species example illustrated in Fig. 1 with v D v , given by (9). Poles and saddle points are indicated by grey and white markers respectively; branch cuts are labelled with thick grey lines. Arrows indicate the direction of increasing Re . Panel (b): Root mean square error  .t / between numerical solutions presented in Fig. 1 and asymptotic theory (10)

s j n



2 00 t . /

!

fj exp .t. // :  C Aj

Since solutions cannot grow, we look for cases where . / D 0. We find that this occurs when  D 0 and hence from (8) the critical velocity is given by X fk v D Ak

!1 :

(9)

k

We now return to the integral equation (7) given the velocity v . We know that the dominant contribution to the integral will come from the saddle point at  D 0, so we Taylor expand the integrand about this point for small values of  D    and . The exponent then becomes, p 00 . / 2 1 p  t   t: ./t C g./ t  2 v We then complete the square using the substitution r z WD i

  00 . / p  : t  2 v 00 . /

This results in a Gaussian integral with solution

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j n

  .n  vt/2 :  exp  2 00 p 2v  . /t Aj 200 . /t fj

(10)

Thus the long time behaviour of our microscopic model (1–3) are Gaussian p profiles moving at a constant velocity v , given by (9), which diffuse like t . To quantify the performance of our asymptotic solution, we calculate the root mean squared difference .t/ between the numerical solutions presented in Fig. 1 and the asymptotic prediction (10). These results are presented in Fig. 2b; clearly the agreement is extremely good.

3 Conclusion In summary, we have presented a simple microscopic model of multi-component etching that can be analysed using asymptotic methods. From this we can connect the microscopic dynamics, based on atomic length scales, to the macroscopic etching process that takes place at length scales of the order of micrometres. Our microscopic analysis indicates that the (macroscopic) normal velocity of the surface is related to the reaction rates and glass composition according to (9). Moreover, the spread of the fraction of exposed surface relatespto the microscopic roughness of the surface. Since this increases with time like t, we can be sure that macroscopic features will be smoothed long before the microscopic roughness grows to a macroscopic level. It would be interesting to combine our work with a macroscopic etching model since we could potentially predict optimal etching times that minimise both macroscopic and microscopic roughness. Acknowledgements I would like to thank Andrew Fowler and Stephen O’Brien for bringing this problem to my attention and providing useful discussion. I acknowledge the support of the Mathematics Applications Consortium for Science and Industry (www.macsi.ul.ie) funded by the Science Foundation Ireland mathematics initiative grant 06/MI/005.

References 1. Barab´asi, A.L., Stanley, H.: Fractal Concepts in Surface Growth. Cambridge University Press, Cambridge (1995) 2. Bender, C.M., Orszag, S.A.: Advanced Mathematical Methods for Scientists and Engineers. Springer, Berlin (1999) 3. Harrison, J.D., Fluri, K., Seiler, K., Fan, Z., Effenhauser, C.S., Manz, A.: Micromachining a miniturized capillary electrophoresis-based chemical analysis system on a chip. Science 261, 895–897 (1993) 4. Judge, J.S.: A study of the dissolution of SiO2 in acidic fluoride solutions. J. Electrochem. Soc. 118, 1772–1775 (1971) 5. Kim, K.S., Hurtado, J.A., Tan, H.: Evolution of a surface-roughness spectrum caused by stress in nanometer-scale chemical etching. Phys. Rev. Lett. 83, 3872–3875 (1999)

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6. Kuiken, H.K.: Etching: a two-dimensional mathematical approach. Proc. Roy. Soc. Lond. Series A 392, 199–225 (1984) 7. Spierings, G.A.C.M.: Compositional effects in the dissolution of multicomponent silicate glasses in aqueous HF solutions. J. Mater. Sci. 26, 3329–3336 (1991) 8. Spierings, G.A.C.M.: Wet chemical etching of silicate glasses in hydrofluoric acid based solutions. J. Mater. Sci. 28, 6261–6273 (1993) 9. Spierings, G.A.C.M., Van Dijk, J.: The dissolution of Na2 O-MgO-CaO-SiO2 glass in aqueous HF solutions. J. Mater. Sci. 22, 1869–1874 (1987) 10. Tavassoly, M.T., Dashtdar, M.: Height distribution on a rough plane and specularly diffracted light amplitude are fourier transform pair. Optic. Comm. 281, 2397–2405 (2008) 11. Tenney, A.S., Ghezzo, M.: Etch rates of doped oxides in solutions of buffered HF. J. Electrochem. Soc. 120, 1091–1095 (1973) 12. Ward, J.A., Fowler, A.C., O’Brien, S.B.G.: Acid polishing of lead glass. J. Math. Ind. 1(1) (2011) DOI 10.1186/2190-5983-1-1

Numerical Treatment of Fiber–Fiber and Fiber-Obstacle Contacts in Technical Textile Manufacturing Ferdinand Olawsky, Martin Hering-Bertram, Andre Schmeißer, and Nicole Marheineke

Abstract For the numerical simulation of industrial processes in technical textile manufacturing an elastic fiber is modeled by a one-dimensional arc-length parameterized time-dependent curve. Neglecting extension and shear effects, its dynamics is described in terms of the Kirchhoff–Love equations by accumulating the acting inner and outer forces. They stem from traction and bending as well as from external loads like gravity and aerodynamic drag. Regarding fiber bundles consisting of several hundreds of fibers, the mathematical modeling and numerical treatment of fiber–fiber and fiber-obstacle contacts are presented.

1 Fiber Model The overall work concentrates on the simulation of fiber bundles in industrial processes like nonwoven production (spunbond, meltblown), cotton processing or glass wool production. For the simulation of an elastic, inextensible fiber a generalized string model is used. The fiber with length l is represented as 3 an arc-length parameterized time-dependent curve r W Œ0; l  RC 0 ! R with line F. Olawsky () University of Applied Sciences, Moltkestr. 30, D-76133 Karlsruhe e-mail: [email protected] M. Hering-Bertram Rhein-Waal University of Applied Sciences, Nollenburgerweg 115, D-46446 Emmerich e-mail: [email protected] Andre Schmeißer Fraunhofer Institut f¨ur Techno- und Wirtschaftsmathematik, Fraunhofer Platz 1, D-67663 Kaiserslautern e-mail: [email protected] N. Marheineke University of Erlangen-N¨urnberg, Cauerstr. 11, D-91058 Erlangen e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 39, © Springer-Verlag Berlin Heidelberg 2012

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weight A. Its dynamics can be asymptotically modeled by a system of partial differentail equations with algebraic constraint of inextensibility .A/@t t r D @s .T @s r  @s .EI @ss r// C .A/g C fair C fcontact k@s rk2 D 1

(1) (2)

supplemented by appropriate initial and boundary conditions. In (1) the change of the momentum is balanced by the acting internal and external forces. The internal line forces consist of bending stiffness indicated by Young’s modulus and the moment of inertia EI as well as of traction. The tractive force T can be viewed as Lagrangian multiplier to (2). The external line forces include gravity g and aerodynamic forces fai r . Considering turbulent flow fields, the aerodynamic forces are modeled by stochastic Gaussian processes such that (1) becomes a stochastic partial differential equation. For details about the fiber model and aerodynamics see the contribution of Marheineke and Wegener in this book and [3–5]. For the simulation of fiber contacts with surfaces or other fibers contact forces fcont act are introduced. These contact forces are realized in a discrete way, which is explained in the next sections. Equations (1) and (2) are discretized by a spatial finite volume method. The time integration is realized by a semi-implicit Euler method to ensure the algebraic and geometric constraints. In each time step the resulting nonlinear system is solved by Newton’s method. The linear systems are banded and are solved efficiently by LU decomposition.

2 Collision Modeling The simulation of fibers in production processes requires the consideration of contacts between fibers and machine parts as well as contacts between fibers. For computer animations several collision models and collision simulations are developed, see [1, 2]. For the presented fiber model with the constraint (2) a contact model has to be developed which could be integrated into the system and which is able to handle real production processes with complex machine parts.

2.1 Fiber Collisions with Machine Parts In industrial processes the fibers hit the surface of machine parts like conveyor belts, funnels or pipes. The simulation of the fiber-surface contacts consists of two steps: contact detection and force modeling. At the end of a time step the fiber-surface contacts have to be detected, i.e. a discretization point of the fiber has hit the surface. If a fiber-surface contact is detected the time step has to be repeated with an additional contact force for the considered discretization point fcontact D n:

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Fig. 1 Simulation of fibers inside a pipe. Left: pipe with triangulated surface, right: fibers streaming through the pipe

The vector n is the normal vector of the surface, the Lagrangian parameter  is the strength of the contact force which is determined by a geometrical constraint. For the contact with a plane this geometrical constraint is given by hri  p; ni D 0; with the fibers discretization point ri , a point p in the plane and the normal vector n. For the simulation the surfaces of the machine parts are triangulated, thus there is no limit in the complexity of the considered surfaces. But for a triangulated surface the normal vector n is not continuously differentiable. A smoothed implicit surface S with a continuously differentiable normal vector has to be reconstructed out of the triangulated surface. Our approach uses a linear combination of the planes of the triangles X SW ˛i hx  pi ; ni i D 0: P

i

The weights ˛i with ˛i D 1 are modeled using the distance d.Ti ; x/ of a point x 2 R3 to a triangle Ti . With this distance function we define the weights w.jd.Ti ; x/j/ ˛i D P j w.jd.Tj ; x/j/

with

  d2 w.d / D exp  2 : r

The parameter r controls the accuracy of the smoothed surface with respect to the triangulated surface. For r ! 0 the surface S fits the triangulated surface. In Fig. 1 a simulation of fibers inside a pipe is shown.

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Fig. 2 Discretization points P1, P2 and P3 of a fiber and the reconstructed smooth fiber with Bezier curves in the vertices. Contact force (red arrow) is split to the discretization points (black arrows)

2.2 Fiber–Fiber Collisions For the simulation of fiber–fiber collisions a similar approach is used. At the end of a time step fiber–fiber contacts have to be detected. To determine these contacts the discretized fibers have to be reconstructed as a curve. If a contact is detected, the time step has to be repeated with additional contact forces to avoid the intersection of the fibers. fiber 1 W

fcont act D Cn

fiber 2 W

fcont act D n

(3)

The contact force acts on both fibers with the same strength  but in contrary directions. The vector n is normal to both fibers at the contact points. The Lagrangian parameter  is determined by a geometrical constraint describing that the distance between the fibers equals the diameter of the fibers. The most simple reconstruction of the discretized fiber is a polyline which is not continuously differentiable. Using cubic splines results in smooth curves for which the contact detection is much to complicated. The polyline defined by the discretization points of the fiber is smoothed by Bezier curves in the vertices (see Fig. 2). For this reconstructed fiber the contact detection and the splitting of the contact force can be handled. In Fig. 3 a simulation of two fibers with fiber–fiber contacts is shown.

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Fig. 3 Simulation of two fibers with fiber–fiber contact

3 Parallel Computations The simulation of several fibers should be performed in parallel. While the simulation of many fibers without fiber–fiber contacts results in a trivial parallelization, the simulation with fiber–fiber contacts is much more complex and needs efficient parallel numerical schemes. To solve the discretized nonlinear equations with Newton’s method, a linear system has to be solved for each fiber in each Newton step. In the case of fiber– fiber-contact simulations the dynamics of the fibers are coupled resulting in a large linear system 0 10 1 0 1 B1 A1 x1 b1 B C C B B A2 B2 C B x2 C B b2 C B C B :: C B :: C D B :: C : :: (4) B C C B B : CB : C B : C B C @ AN BN A @ xN A @ bN A C1 C2    CN D

y

d

The matrices Ai are the Jacobians corresponding to (1) and (2). The matrices Bi are the Jacobians to the contact forces (3). The geometrical constraints for the fiber– fiber contact yield the Jacobians Ci and D. The linear system (4) could be computed and solved in parallel. Therefore the Schur complement system " D

N X i D1

# Ci A1 i Bi

yDd

N X

Ci A1 i bi

(5)

i D1

is solved with LU decomposition in the first Newton step and with LU-preconditioned GMRES in further Newton steps, [6]. After solving (5) the linear system (4) can be solved in parallel. In Fig. 4 a simulation with 96 fibers is shown which was performed on a parallel computer.

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Fig. 4 Simulation of a steel brush. Parallel simulation with 96 fibers

4 Remarks All presented models are implemented in the software tool FIDYST (Fiber Dynamics Simulation Tool) which is developed at Fraunhofer ITWM, Kaiserslautern. This software tool is used to simulate industrial production processes for technical textiles and so the applicability of the models is proved in many projects.

References 1. Baraff, D., Witkin, A.: Large Steps in Cloth Simulation. Proceedings of SIGGRAPH ’98, pp. 43–54 (1998) 2. Goldenthal, R., Harmon, D., Fattal, R., Bercovier, M., Grinspun, E.: Efficient Simulation of Inextensible Cloth. ACM Transactions on Graphics, Proceedings of SIGGRAPH 2007, vol. 26(3) (2007) 3. Klar, A., Marheineke, N., Wegener, R.: Hierarchy of mathematical models for production processes of technical details. Z. Angew. Math. Mech. 89, 941–961 (2009) 4. Marheineke, N., Wegener, R.: Fiber dynamics in turbulent flows: General modeling framework. SIAM J. Appl. Math. 66(5), 1703–1726 (2006) 5. Marheineke, N.,Wegener, R.: Modeling and application of a stochastic drag for fibers in turbulent flows. Int. J. Multiphase Flow 37:136–148 (2011) 6. Zhang, F. (ed.): The Schur Complement and Its Applications. Springer, Berlin (2005)

A Model of Rotary Spinning Process Andriy Hlod, Alphons A.F. van de Ven, and Mark A. Peletier

Abstract A rotary spinning process is used to produce aramide fibers. In this process thin jets of polymer solution emerge from the nozzles of the rotating rotor and flow towards the cylindrical coagulator. At the coagulator the jets hit the water curtain in which they solidify forming fibers. The rotary spinning is described by a steady jet of viscous Newtonian fluid between the rotor and the coagulator. The jet model includes the effects of inertia, longitudinal viscosity, and centrifugal and Coriolis forces. For the jet model the specific type of the boundary conditions depends on the balance between the inertia and viscosity in the momentum transfer through the jet cross-section. Based on that we find two possible flow regimes in rotary spinning: (1) viscous-inertial, where viscosity dominates at the rotor and inertia at the coagulator (2) inertial, where inertia dominates everywhere in the jet. Moreover, there are two situations where spinning is not possible, either due to lack of a steady-jet solution or because the jet wraps around the rotor. Finally, we characterize the parameter space.

A. Hlod () Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands e-mail: [email protected] A.A.F. van de Ven Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands e-mail: [email protected] M.A. Peletier Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 40, © Springer-Verlag Berlin Heidelberg 2012

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1 Rotary Spinning Process A rotary spinning process is used to produce aramid fibers. In this paper we propose a model of the jet in the rotary spinning which is aimed to understand the process and obtain a process window. A rotary spinning machine consists of a rotor and a coagulator having a form of vertical cylinders [2, 5]. Along the inner wall of the coagulator the water falls creating a curtain; see Fig. 1. Small holes are in the rotor’s lateral surface. The rotor is placed inside the coagulator so that the symmetry axes of both cylinders coincide. The rotor rotates counterclockwise and hot polymer solution is pumped through the rotor’s nozzles. The solution, in the form of thin jets, flows to the coagulator under the influence of Coriolis and centrifugal forces. After hitting the water curtain at the coagulator wall the jets solidify forming fibers. The problem of modeling rotary spinning was presented to the mathematical community at the 48th European Study Group Mathematics with Industry in Delft (2004) [2]. In the report, the jet was described using the string model. However, at that time the equations could not be solved due to the assumption that the jet always leaves the nozzle radially, which does not reflect the reality. Later, the model equations of the rotary spinning process were solved in [5, 6]. It has been shown there that the jet orientation at the nozzle is determined by the jet itself. However, understanding why the jet orientation at the nozzle cannot be prescribed a priori was missing. In this paper we present the analysis of the rotary spinning model [2, 5, 6] and we employ the boundary conditions derived in [4][3]. From that we obtain the complete characterization of the parameter space. Jets in similar configurations without coagulator are studied in [1, 7–15].

Polymer solution Rotor Coagulator

Water curtain

Fig. 1 Rotary spinning process

Washing & drying

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2 Model In the model of the rotary spinning process we consider one steady jet (see Fig. 2) in two dimensions in a horizontal frame. By doing this we neglect the vertical motion due to gravity, so the jet moves horizontally from the nozzle of the rotor to the contact point with the coagulator. In a fixed reference frame the rotor rotates counterclockwise with angular velocity ˝. The radii of the rotor and the coagulator are Rrot and Rcoag , respectively. We parameterize the jet by its length s with s D 0 at the nozzle and s D send at the contact with the coagulator. Note that send is the length of the jet, and is unknown in advance. The jet leaves the nozzle with the flow velocity vnozzle . At the contact with the coagulator the jet sticks to it having the flow velocity ˝Rcoag . To describe the jet position in the rotating reference frame of the rotor, we use two sets of coordinates: either the polar coordinates R and ˇ, with the origin at the center of the rotor, or the arc length of the jet s, and the angle  the tangent to the jet makes with the radial direction; see Fig. 2. The relation between the two sets are given by (4), (5). Two (inertial) body forces act on the jet in the rotating reference frame, i.e the centrifugal, Fc , and Coriolis, FC , force. The system describing the stationary jet follows from the conservations of mass and momentum. We scale the flow velocity v with respect to ˝Rcoag , both R and s with respect to Rcoag . Then the system is fully described by three dimensionless

ey

W

b (s)

s=0

vnozzle ex

Rcoag

R(s)

R rot

f(s)

s Fc

Fc send WRcoag

Fig. 2 A schematic picture of the rotary spinning process

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2 numbers B D 3=.!Rcoag /, R0 D Rrot =Rcoag , Dr D vnozzle =.˝Rcoag /, where  is the kinematic viscosity of the fluid. The resulting system (see [3, Chapter 5, p. 89]) is

 0 .s/ D cos..s//R.s/=v.s/;

(1)

.s/ 0 .s/ D R.s/ sin..s//=v.s/  sin..s//.s/=R.s/ C 2;

(2)

0

2

v .s/ D .v.s/  .s/v.s//=B;

(3)

R0 .s/ D cos..s//;

(4)

0

ˇ .s/ D  sin..s//=R.s/;

(5)

together with the boundary conditions v.0/ D Dr; v.send / D 1; R.0/ D R0 ; ˇ.0/ D 0; R.send / D 1; if .0/ > 0; then .0/ D 0; if .send / < 0; then .send / D =2: Here, v.s/ is the flow velocity in the jet, .s/ represents the momentum transfer through the jet cross-section. The boundary conditions for the jet orientation .s/ are determined by the sign of .s/; see [3, Chapter 2]. From (1) it follows that .s/ is a strictly increasing function implying that there are three possibilities for the sign of .s/: • The first situation is .s/ > 0 everywhere in the jet, and then inertia dominates everywhere in the momentum transfer through the jet cross-section. We call this jet flow regime inertial. In this case the jet must be aligned with the radial nozzle direction, .0/ D 0. • In the second situation .s/ changes sign from negative to positive, and viscosity dominates near the nozzle and inertia near the coagulator in the momentum transfer through the jet cross-section. We call this jet flow regime viscousinertial. In this case we cannot prescribe any boundary condition for the jet orientation. However, from (2), it follows that at the point s0 where .s0 / D 0 the jet should be aligned with the direction of the resulting external force acting on the jet, yielding .s0 / D arcsin.2v.s0 /=R.s0 //: (6) • In the third situation .s/ < 0 everywhere in the jet, and viscosity dominates everywhere in the momentum transfer through the jet cross-section. We call this jet flow regime viscous. For viscous jet we require tangency with the coagulator .send / D =2. However, the viscous jet situation is not possible in the current setup because of the following argument. The border between the parameter regions for the viscous and viscous-inertia jets should satisfy the condition .send / D 0. Then from (6) and the boundary conditions it follows that sin..send // D 2 leading to a contradiction.

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The following solution strategy is suggested. We multiply (1) by sin..s// and (2) by cos..s//, and add them. In the resulting equation we use (4) which allows us to integrate and obtain the following relation sin..s//.s/ D

R.s/2  C1 ; R.s/

(7)

where C1 D R0 for the inertial jet, C1 D R.s0 / for the viscous-inertial jet. Then system (7), (1), (3)–(6) is solved using the shooting method. The parameter space is described by the three dimensionless numbers R0 , Dr, and B. In Fig. 3 we present the partitioning of the parameter space in the R0 ; Dr-plane for different B. In this plane we observe the regions of inertial jet, viscous-inertial jet, and two regions of nonexistence of a jet solution. For the parameter regions “no solution 1” the cause of nonexistence is that the jet does not reach the coagulator and wraps round the rotor. For the parameter regions “no solution 2” the jet reaches the coagulator, but the flow velocity at the coagulator cannot be matched, indicating the unsteady jet. In the region “no solution” both causes of nonexistence are possible. The borders between the viscous-inertial and inertial regimes are calculated from the condition .0/ D 0, and the borders of the nonexisting regions are obtained using monotonicity properties of the jet solution together with the condition .send / D =2. 1.0

1.0

B = 0.15

B = 0.01 0.8

0.8

inertial jet inertial jet Dr

0.6

Dr

0.6 0.4 0.2

no solution 1

viscous-inertial jet

0.4

viscous-inertial jet no solution 2

0.2

no solution 1 no solution 2

0.0 0.0

0.2

0.4

0.6

0.8

0.0 0.0

1.0

0.2

0.4

R0

0.6

0.8

1.0

0.6

0.8

1.0

R0

1.0

1.0 B = 0.2617

B = 0.04

0.8

0.8

inertial jet

inertial jet 0.6 Dr

Dr

0.6

0.4 0.2 0.0 0.0

0.4

0.2

no solution

0.2

no solution 1

no solution 2 0.4

0.6

0.8

1.0

0.0 0.0

R0

Fig. 3 Parameter regions in the R0 ; Dr-plane for different B

0.2

0.4 R0

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For B close to zero (see the plot for B D 0:01) the region of the inertial jet occupies almost the whole plane except a narrow area near Dr D 0. The viscous-inertial jet region forms a long and narrow strip along the line Dr D 0. The nonexistence regions “no solution 1” and “no solution 2” are small areas concentrated near the points Dr D 0, R0 D 0, and Dr D 0, R0 D 1, respectively. With increasing B (see the plot for B D 0:15) the border of the inertial jet region rises, the viscous-inertial jet region becomes higher and shorter, and the nonexistence regions grow. For B D 0:2617 the viscous-inertial jet regions disappears (shrunk to one curve). For larger B > 0:2617 (see the plot for B D 0:4) the viscous-inertial jet is absent, the inertial jet region becomes smaller and the nonexistence region ”no solution” expands.

3 Conclusions The steady jet model in rotary spinning configuration is studied. The jet is described by a system of ODE’s on an unknown domain. The scaled system describing the jet is characterized by three dimensionless parameters. For the jet in rotary spinning we distinguish three situations: the inertial jet, the viscous-inertial jet, and a steady jet solution does not exist. There are two causes of the nonexistence of the jet between the rotor and the coagulator (1) the jet wraps around the rotor, and (2) the flow velocity at the coagulator cannot be reached, indicating an unsteady jet. Acknowledgements The authors would like to acknowledge Teijin Aramid, a part of the Teijin group of companies.

References 1. Decent, S.P., King, A.C., Wallwork, I.M.: Free jets spun from a prilling tower. J. Eng. Math. 42(3), 265–282 (2002) 2. den Decker, P., Knoester, H., Meerman, H., Dekker K. van Horssen, W., Vuik, C., Wesseling, P., Prokert, G., van ’t Hof, B., van Beckum, F.: The rotor spinning process for fibre production. In: Proceedings of the 48th European Study Group Mathematics with Industry (Delft, 15–19 March 2004), pp. 35–48 (2004) 3. Hlod, A.: Curved jets of viscous fluid: Interactions with a moving wall. Ph.D. thesis, Eindhoven University of Technology (2009) 4. Hlod, A., Aarts, A.C.T., van de Ven, A.A.F., Peletier, M.A.: Three flow regimes of viscous jet falling onto a moving surface. IMA J. Appl. Math. (2011). DOI 10.1093/imamat/hxr017. URL http://imamat.oxfordjournals.org/content/early/2011/03/23/imamat.hxr017.abstract 5. Kolk, E.: Mathematical models for a rotor spinning process. Interim report, TU Delft (2005) 6. Kolk, E.: Mathematical models for a rotor spinning process. Final report, TU Delft (2005) 7. Marheineke, N., Wegener, R.: Asymptotic model for the dynamics of curved viscous fibres with surface tension. J. Fluid. Mech. 622(-1), 345–369 (2009) 8. Panda, S.: The dynamics of viscous fibers. Ph.D. thesis, Technische Universit¨at Kaiserslautern (2006)

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9. Panda, S., Marheineke, N., Wegener, R.: Systematic derivation of an asymptotic model for the dynamics of curved viscous fibers. Math. Meth. Appl. Sci. 31(10), 1153–1173 (2008) 10. Parau, E.I., Decent, S., Simmons, M., Wong, D.C.Y., King, A.C.: Nonlinear viscous liquid jets from a rotating orifice. J. Eng. Math. 57(2), 159–179 (2007) 11. Parau, E.I., Decent, S.P., King, A.C., Simmons, M.J.H., Wong, D.C.: Nonlinear travelling waves on a spiralling liquid jet. Wave Motion 43(7), 599–618 (2006) 12. Partridge, L., Wong, D.C.Y., Simmons, M.J.H., Parau, E.I., Decent, S.P.: Experimental and theoretical description of the break-up of curved liquid jets in the prilling process. Chem. Eng. Res. Des. 83(A11), 1267–1275 (2005) 13. Uddin, J., Decent, S.P., Simmons, M.J.: The instability of shear thinning and shear thickening spiralling liquid jets: linear theory. Trans. ASME J. Fluid. Eng. 128(5), 968–975 (2006) 14. Wallwork, I.M., Decent, S.P., King, A.C., Schulkes, R.M.S.M.: The trajectory and stability of a spiralling liquid jet. Part 1. Inviscid theory. J. Fluid. Mech. 459, 43–66 (2002) 15. Wong, D.C.Y., Simmons, M.J.H., Decent, S.P., Parau, E.I., King, A.C.: Break-up dynamics and drop size distributions created from spiralling liquid jets. Int. J. Multiphas. Flow 30(5), 499–520 (2004)

Asymptotic Models of Different Complexity for Viscous Jets and Their Applicability Regimes Walter Arne, Nicole Marheineke, and Raimund Wegener

Abstract This paper presents asymptotic models of different complexity for the simulation of slender viscous jets in spinning processes. In the slenderness limit a viscous Cosserat rod reduces to a string system. We propose two string models, i.e. inertial and viscous-inertial string models, that differ in the closure conditions and hence yield a boundary value problem and an interface problem, respectively. Their convergence/applicability regimes where the respective string solution is the asymptotic limit to the rod turn out to be disjoint and to cover nearly the whole parameter space of Reynolds, Froude, Rossby numbers and jet length. We explore the transition hyperplane analytically for the gravitational two-dimensional scenario.

Viscous Rod and String Models A jet is a long slender body whose dynamics can be reduced to an one-dimensional description by averaging the underlying balance laws over its cross-sections. This procedure is based on the assumption that the displacement field in each crosssection can be expressed in terms of a finite number of vector- and tensor-valued quantities. The special Cosserat rod theory [1] consists of only two constitutive ele-

W. Arne Universit¨at Kassel, Fachbereich Mathematik und Naturwissenschaften, Heinrich Plett Str. 40, D-34132 Kassel, Germany e-mail: [email protected] N. Marheineke () FAU Erlangen-N¨urnberg, Department Mathematik, Cauerstr. 11, 91058, Erlangen, Germany e-mail: [email protected] R. Wegener Fraunhofer ITWM, Fraunhofer Platz 1, D-67663 Kaiserslautern, Germany e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 41, © Springer-Verlag Berlin Heidelberg 2012

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ments in the three-dimensional Euclidean space E3 , a curve r W Q ! E3 specifying the position and an orthonormal director triad fd1 ; d2 ; d3 g W Q ! E3 characterizing the orientation of the cross-sections. In Q D f.s; t/ 2 R2 j s 2 Œsa .t/; sb .t/; t > 0g, s is the arc-length parameter and t the time. In a spinning process a viscous liquid jet is extruded from a nozzle. It grows and moves due to viscous friction, surface tension, gravity and aerodynamic forces. Since we are interested in the bending behavior of the jet at the nozzle we focus on a spun fiber jet of certain length with stress-free end. Surface tension and temperature effects are neglected. For an easy objective formulation of the material laws [7], the associated stationary viscous Cosserat P rod is stated P in the director basis. Note that to an arbitrary vector field x D 3i D 1 xi di D 3i D 1 xM i ei 2 E3 , we indicate the coordinate tupels corresponding to the director basis and a fixed outer basis fe1 ; e2 ; e3 g by x D .x1 ; x2 ; x3 / and xM D .xM 1 ; xM 2 ; xM 3 / 2 R3 , respectively. The director basis can be transformed into the outer basis by the tensor-valued rotation D, i.e. D D ei ˝ di D Dij ei ˝ ej 2 E3 ˝ E3 with associated orthogonal matrix D D .Dij / D .di  ej / 2 SO.3/. For the coordinates, x D D  xM holds. The cross-product x  D is defined as mapping .x  D/ W R3 ! R3 , y 7! x  .R  y/. Canonical basis vectors in R3 are denoted by ei , i D 1; 2; 3, e.g. e1 D .1; 0; 0/. Then, the dimensionless rod model is [2] D  @s Mr D e3

(1)

@s D D   D 1 4 @s  D  n3 C uP3=2  m 3 3 1 @s u D un3 3   1 M @s n D   n C Re u  e3 C un3 e3  D  f 3   4 1 1 1 uP3  m  n3 P2    .P2  /D  Ml @s m D mC 2 n  e3 CRe  3 12 4 supplemented with geometric and kinematic boundary conditions at the nozzle s D 0 and stress-free dynamic boundary conditions at the jet length s D ` Mr.0/ D r0 ;

R.0/ D R0 ;

.0/ D 0;

u.0/ D 1;

n.`/ D 0;

m.`/ D 0

and diagonal matrix Pk D diag.1; 1; k/, k 2 R. The boundary value problem prescribes the behavior of jet curve Mr, orientation D, curvature , intrinsic velocity u, contact forces n and couples m due to the acting outer forces Mf and couples Ml. It is made dimensionless by help of the following reference values: s0 D r0 D R, 0 D R1 , u0 D U , n0 D Ud 2 =.4R/, m0 D Ud 4 =.16R2 /, f0 D U 2 d 2 =.4R/ and l0 D U 2 d 4 =.64R2/ with jet density , viscosity ,

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velocity U at the nozzle, diameter d and length L as well as typical length of the spinning plant R. Thus, the spinning is characterized by the following dimensionless numbers: Reynolds number Re D UR= as ratio between inertia and viscosity,  D d=R and ` D L=R as length ratios between fiber diameter, length respectively and plant length scale. Moreover, specifying the external load as gravity, we get Mf D eg =.Fr2 u/ in direction eg with Froude number Fr D U=pgR as ratio between inertia and gravity (g gravitational acceleration). Considering rotational spinning processes, the transition to stationarity can be obtained by introducing a rotating outer basis. Then, artificial Coriolis and centrifugal forces and associated couples enter the equations in Mf, Ml and the Rossby number as ratio of inertia and rotation becomes a further important dimensionless number, see e.g. [2, 3, 5]. Theorem 1 (Slenderness Limit). [3] In the asymptotic limit of slenderness  ! 0, the equations for the viscous rod reduce to a string model for jet curve, tangent, intrinsic velocity and tangential stress .Mr; ; M u; N D n3 /, i.e., @s Mr D M   1 M ; M N @s M D q @s M D p D Mf  .Mf  / u Re

(2) kk M 2D1

1 uN 3   1 M uN  f  M : @s N D Re 3 @s u D

The function q that depends parametrically on the characteristic numbers as all other quantities, crucially affects the string system. The term appears explicitely as factor of @s M and requires a special consideration of the limit model in view of boundary conditions, solvability and approximation quality. In [5] q is interpreted as sum of inertial and viscous energies. As consequence of acting forces, q can be assumed to increase monotonically on Œ0; `, moreover q.`/ > 0 since u > 0. Thus, two cases can be distinguished, i.e. q.0/ > 0 and q.0/  0, specifying two different string regimes, the inertial and the viscous-inertial strings, respectively. Note that the monotonicity of q is no property of the rod (1) where N D n3 . Definition 1 (Inertial and Viscous-Inertial Strings). • The inertial string model Si is a boundary value problem where the string equations (2) are supplemented with the rod-associated boundary conditions Mr.0/ D r0 ;

u.0/ D 1;

N.`/ D 0;

.0/ M D 0 :

• The viscous-inertial string model Svi is a interface problem (at the transition point s ? ) where the string equations (2) are supplemented with

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Mr.0/ D r0 ;

u.0/ D 1;

N.`/ D 0;

q.s ? / D 0;

p.s ? / D 0;

s ? 2 Œ0; `Œ:

Si is the classical string model known in literature. Being well-posed in the parameter regime where q.0/ > 0 holds, its solutions turn out to be the asymptotic limit of the rod solutions as  ! 0 (see e.g. [2] for a rotational 2d scenario). But, if q.0/ ! 0, the string solutions in M rise boundary layers at the nozzle that finally cause the break-down of numerics. The reason lies in a non-removable singularity since p.0/ ¤ 0. The inertial string model Si fails analytically. Moreover, it admits no solution for q.0/  0, as a non-existence result in [5] shows. The more complex rod model in contrast allows for angular momentum effects and resolves the strong curvature changes at the nozzle by help of a boundary layer. It is applicable without any restrictions. The ability to handle all parameter ranges of practical interest in simulation and optimization makes the rod model obviously superior to the string approach. These observations correspond to previous studies on fluid-mechanical sewing machines investigating jet lay-down onto a moving belt, [4, 8]. However, by containing the slenderness parameter  explicitely, it is no asymptotic model of zeroth order and requires a careful numerical treatment in case of small . The viscous-inertial string model Svi is an approach that circumvents the introduction of a higher order model but overcomes the thitherto limitations of Si . Following an idea of Hlod et al. [6], we replace the physical boundary condition for the jet direction at the nozzle M .0/ by an interface condition ensuring the continuity of the string quantities in the transition point s ? that is characterized by q.s ? / D 0. Thereby, boundary layers are cut off. But note that to avoid the occurrence of a singularity in s ? , not only p.s ? / has to vanish, but also the ratio .p=q/.s ? / has to be finite for each component, see [3] to this point.

Applicability Regimes for Jets Exposed to Gravity The compatibility of the string models Si , Svi , their applicability and validity in the respective parameter regimes as asymptotic simplification of the rod model are studied here for jets under gravity. Setting Mr.0/ D e2 , M .0/ D e2 , Mf D e1 =.Fr2 u/ and Ml D 0, the jet stays in the e1 -e2 -plane. With d2 ?f and d2 D e3 , the rotation D is prescribed by a single angle ˇ 2 Œ=2; . The contact force acts in the d1 -d3 plane, curvature and contact couple are oriented in d2 -direction. Thus we abbreviate  D 2 , m D m2 and Mr D .Mr1 ; rM2 /. Then, the rod model (1) becomes @s Mr D .cos ˇ; sin ˇ/

Mr.0/ D .0; 1/  2

@s ˇ D 

ˇ.0/ D

1 4 @s  D  n3 C um 3 3

.0/ D 0

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@s u D

1 un3 3

353

u.0/ D 1

Re sin ˇ Fr2 u Re Re cos ˇ un3 C 2 @s n3 D n1 C 3 Fr u 1 4 Re .um  n3 / @s m D 2 n1 C  3 4

@s n1 D n3  Reu C

n1 .`/ D 0 n3 .`/ D 0 m.`/ D 0;

and the string models (2) simplify to @s Mr D .cos ˇ; sin ˇ/   1 1 sin ˇ u N @s ˇ D q @s ˇ D p D 2 ; Re Fr u

Mr.0/ D .0; 1/ ˇ.0/ D

 for Si ; 2

q.s ? / D p.s ? / D 0 for Svi 1 uN 3 Re cos ˇ Re uN C 2 @s N D 3 Fr u @s u D

u.0/ D 1 N.`/ D 0:

In contrast to Si , the string equations equipped with the interface conditions allow for solutions to all parameter tupels .Re; Fr; `/. They describe straight jets in direction of gravity Mr.s/ D .s; 1/, ˇ   with @s u D

1 uN; 3

u.0/ D 1;

@s N D

Re 1 Re uN  2 ; 3 Fr u

N.`/ D 0

independent of s ? . However, note that only the solutions with s ? 2 Œ0; `Œ, where s ? is the root of .u  N=Re/.s ? / D q.s ? / D 0, present the jet behavior corresponding to the viscous-inertial string model Svi . The other solutions are meaningless. Theorem 2 (String-Transition Surface). [3] Let q W Œ0; ` ! RC 0 be a composition of the Airy functions Ai, Bi and their derivatives Ai0 , Bi0 , 

12 q.s/ D  P

1=3

Ai0 .'.s//Bi0 .'.0//  Bi0 .'.s//Ai0 .'.0// Ai.'.s//Bi0 .'.0//  Bi.'.s//Ai0 .'.0//

with '.s/ D .3=.2P//1=3.Re=3 s  1/. Then, the transition surface q.0/ D 0 of Svi is determined by the parameter tupels .Re; Fr; `/ solving   Re 6 6 1 ` q.`/  D 0; q .`/ C P 3 P 3

P D Re Fr2 :

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Its asymptotic limits are 3  1:4; 2 mini j i j3 p Fr D `

Ai0 . i / D 0

P D Re Fr2 D

for Re ! 1 for Re ! 0:

Furthermore, the associated jets that are straight in direction of gravity satisfy u.s/ D .P=6 q 2 .s/  Re=3 s C 1/1 and N.s/ D Re.u  q/.s/. The analytically prescribed transition surface of Svi coincides with the one of Si , as respective numerical computations of Si show. So, the transition surface between the inertial and viscous-inertial jet behavior is coexistently the border surface that separates the existence regimes of the two string models. A numerical analysis on rod-to-string convergence shows further that the existence regimes are also the regimes of convergence where the respective string solutions are the asymptotic limits of the rod solutions in a L 2 -sense, [3]. The string-transition surface is visualized as curves corresponding to different lengths ` in the .Re; Fr/space in Fig. 1. While the inviscid asymptote ispindependent of `, the jet length effects the viscous limit. But this relation Fr D ` is carried into the problem by the scaling with another macroscopic length in view of an underlying 3d spinning process. If only one length scale is relevant for the problem (as it is actually the case in such a pure gravitational scenario), we have L=R D ` D 1 and the viscous limit is certainly Fr D 1. The transition curves corresponding to shorter jets (smaller `) lie below the ones of longer jets (larger `) which implies a bigger inertial jet regime. This effect is also observed for thicker jets (larger ). These investigations on the applicability regimes of the string models are extended to viscous jets in three-dimensional rotational spinning processes in [3].

101

100

Si

Si

10

0

ε→0

l=1

Fr

Fr

l=4 Svi

l=1 / 4 Svi

10−1 10−2

ReFr 2 = c 10−1

100 Re

101

–1

10 102

10–3

10–2

10–1

100

101

Re

Fig. 1 Transition curve in .Re; Fr/-space separating applicability regimes of Svi and Si that lie below and above the curve, left: for different lengths `, right: for ` D 1 in comparison to respective rod quantities for varying thickness  2 f101 ; 102 ; 103 g (dashed lines)

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References 1. Antman, S.S.: Nonlinear Problems of Elasticity. Springer, New York (2006) 2. Arne, W., Marheineke, N., Meister, A., Wegener, R.: Numerical analysis of Cosserat rod and string models for viscous jets in rotational spinning processes. Math. Mod. Meth. Appl. Sci. 20(10), 1941–1965 (2010) 3. Arne, W., Marheineke, N., Wegener, R.: Asymptotic transition of cosserat rod to string models for curved inertial viscous jets. Math. Mod. Meth. Appl. Sci. 21(10), 1987–2018 (2011) 4. Chiu-Webster, S., Lister, J.R.: The fall of a viscous thread onto a moving surface: a ’fluidmechanical sewing machine’. J. Fluid. Mech. 569, 89–111 (2006) 5. G¨otz, T., Klar, A., Unterreiter, A., Wegener, R.: Numerical evidence for the non-existence of solutions to the equations describing rotational fiber spinning. Math. Mod. Meth. Appl. Sci. 18(10), 1829–1844 (2008) 6. Hlod, A., Aarts, A.C.T., van de Ven, A.A.F., Peletier, M.A.: Mathematical model of falling of a viscous jet onto a moving surface. Euro. J. Appl. Math. 18, 659–677 (2007) 7. Ribe, N.M.: Coiling of viscous jets. Proc. Roy. Soc. London, A 2051, 3223–3239 (2004) 8. Ribe, N.M., Lister, J.R., Chiu-Webster, S.: Stability of a dragged viscous thread: onset of ’stitching’ in a fluid-mechanical ’sewing machine’. Phys. Fluid. 18, 124,105 (2006)

Wavelet Methods for the Representation, Analysis and Simulation of Optical Surfaces Philipp Jester, Christoph Menke, and Karsten Urban

Abstract Wavelets are well adapted to describe local perturbations as well as global structures and so lead to an alternative representation of optical surfaces that combines a high approximation accuracy with a fast evaluation. We show that such a representation is usable in a ray trace algorithm to describe aspherical or free-form surfaces and give results for the achieved accuracy. Moreover the representation can be used for wavelet analysis of manufactured surfaces. We present numerical experiments for the detection of local errors, the separation of low and mid spatial frequency errors and the localization of regions with varying quality. A broad field of applications in optics, especially in tolerancing and manufacturing, is expected.

1 Introduction Elements with non-spherical surfaces are frequently used in modern optical systems, e.g. mobile phone lenses. Usually, the aspheric surfaces are described in terms of a power series expansion. This representation turns out to be unsatisfactory in some cases, like free-form surfaces or aspheric surfaces including manufacturing errors. A description of an as-built surface with polynomials will need a high number of terms because of the local structures that are present on the surface. Nevertheless P. Jester () Institute of Numerical Mathematics, Ulm University, 89081 Ulm, Germany e-mail: [email protected] C. Menke Corporate Research and Technology, Carl Zeiss AG, Carl-Zeiss-Str. 22, 73447 Oberkochen, Germany e-mail: [email protected] K. Urban Institute of Numerical Mathematics, Ulm University, 89081 Ulm, Germany e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 42, © Springer-Verlag Berlin Heidelberg 2012

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this description will be afflicted by a large approximation error. Wavelets offer an alternative surface representation that is better adapted to local structures.

2 B-Spline and Wavelet Representation We focus on smooth optical surfaces which may contain local defects. As they are well adapted to represent local and global structures we choose cardinal B-splines as basis functions. Their well-known locality and recursion properties are essential for a fast evaluation. Moreover one can show that a refinement equation holds. This enables B-splines to generate a Multiresolution Analysis (MRA), see [2, 3]. We fix ' a B-spline of order d  2 and use it as scaling function of our MRA. The properties of a MRA imply the existence of a dual scaling function 'Q such that .'./; '. Q  k//L2 .R/ D ı0;k ;

k 2 Z;

holds. For a piecewise continuous function g W R ! R we denote by gj;k .x/ WD 2j=2 g.2j x  k/ a scaled and shifted variant of g. Using this notation we define the biorthogonal projection Pj W L2 .R/ ! Sj of a function f 2 L2 .R/ onto the space Sj WD closL2 .R/ f'j;k W k 2 Zg by Pj f D

X .f; 'Qj;k /L2 .R/ 'j;k :

(1)

k2Z

The dual space SQj is defined in a similar way. Let us now introduce wavelets. Let Wj ; WQ j be the complement spaces such that Sj C1 D Sj ˚ Wj ;

Sj ?WQ j ;

SQj C1 D SQj ˚ WQ j ;

SQj ?Wj ;

holds. A function is called primal wavelet if f j;k W k 2 Zg is a Riesz basis for Wj , i.e. for some coefficients dj;k there exist constants 0 < c‰  C‰ such that  X  2 c‰ jdj;k j   dj;k  j;k2Z j;k2Z X

2    j;k  

 C‰

L2 .R/

X

jdj;k j2 :

j;k2Z

The corresponding dual wavelet is denoted by Q . Now we obtain an equivalent representation to (1) in terms of wavelets Pj f D

j 1 X X .f; Q j;k /L2 .R/ `D1 k2Z

j;k ;

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where we set 1;k WD '0;k . Defining the coefficients cj;k WD .f; 'Qj;k /L2 .R/ , dj;k WD .f; Q j;k /L2 .R/ the Fast Wavelet Transform maps cj WD .cj;k /k to dj WD .dj;k /k , FWT W cj 7! .c0 ; d0 ; : : : ; dj 1 /: The FWT is of linear complexity as long as primal and dual scaling function are compactly supported. Two dimensional scaling functions and wavelets are constructed by the bivariate tensor product. We are now ready to describe and analyze optical surfaces with wavelets.

3 Accuracy First, we have to show that the projection in (1) can be used in a ray trace algorithm with sufficient accuracy. The following error estimate is well-known for a smooth function f 2 H s .R/, kf  Pj f kL2 .R/ . 2js kf kH s .R/ ;

0  s  d:

We need to compute the coefficients cj;k . This cannot be done directly because there is no analytic form for the dual scaling function available. Instead, we use the following quasi-interpolation scheme cj;k  cNj;k WD 2j=2

m X

d;l f .2j .k C `//

`Dm

with m WD b d 1 2 c and weights d;l , see [1, 5]. The quasi-interpolant PNj f WD

X

cNj;k 'j;k

k2Z

is of the same order of approximation as Pj . It holds: kf  PNj f kL2 .R/ D O.2js /:

(2)

We focus on using this representation to describe an optical surface in a ray trace algorithm. Therefore the deviation of every single ray is important. We choose an aspheric test surface and compute the maximal deviation of intersection points .j / calculated with respect to our method xi and the usual representation xi , i.e. .j /

Mj WD max fkxi  xi kg; i D1;:::;n

for n D 106 rays.

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10–2 10–3 10–4

d=2 d=3 d=4 d=5 d=6

10–5 10–6 10–7 10–8 10–9 10–10 10–11

4

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7 Level j

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Figure 1 shows an error plot for different orders d D 2; : : : ; 6. The accuracy of the ray trace algorithm is fixed to " D 1012 . We see that an approximation accuracy up to a sufficient tolerance is reached for orders d D 5; 6 and level j D 8, which corresponds to 256  256 coefficients. Moreover it is evident that the rates of convergence for the L2 -error in (2) almost hold for the maximal deviation. Therefore we plot best-fit lines with corresponding slope d in Fig. 1.

4 Wavelet Analysis Next, we describe the use of wavelets to analyze given real and synthetic optical surfaces, for a detailed report see [4]. To demonstrate the detection of local errors we add an artificial perturbation to a measured aspheric surface. This error and the resulting combined surface are shown in Fig. 2. The combined wavelet details of level j D 5; 6 are shown in Fig. 3, where we define the combined wavelet detail by the element-wise euclidean norm of horizontal, vertical and diagonal wavelet detail. The local perturbation is clearly visible in the wavelet decomposition of level j D 6 at position .0:4; 0:25/. One notices also a true error in the center of the surface. Moreover wavelets are well adapted to separate low and mid spatial frequency errors. In optics these errors are typically modeled by Zernike polynomials and Power Spectral Density (PSD) functions. To show how wavelets perform we simulate these two errors and superposed them, the resulting surface is seen in Fig. 4. The pure ZFR-error and its reconstruction from the wavelet decomposition are shown in Fig. 5. In the inner region the ZFR-error is well reconstructed, where near the boundary the extension of the data impacts the results. Finally we use wavelets to localize mid spatial frequency perturbed domains. Therefore we split our surface in outer and inner domain and add artificial errors,

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modeled by a PSD function, with different intensities for each domain. To obtain a more realistic model we add a ZFR-error as above. An example is shown in the left part of Fig. 6. The combined wavelet detail in the right part of Fig. 6 clearly

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shows the differing domains. The localization in space and frequency of compactly supported wavelets offers an advantage over Fourier methods for the detection of local surface structures. This can be exploited in the manufacturing of optical surfaces. The wavelet analysis helps to restrict expensive processing to regions with insufficient surface accuracy.

5 Summary The presented wavelet method can be used to approximate an optical surface up to an desired accuracy and has been implemented in a ray trace algorithm. We applied wavelets to analyze and detect common errors of optical surfaces. Isolated local errors are detected as well as domains of varying quality. The wavelet representation enables the separation of errors with differing frequencies. This opens a broad field of applications in tolerancing and manufacturing of aspheric optical elements.

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References 1. Bittner, K., Urban, K.: Adaptive wavelet methods using semiorthogonal spline wavelets: sparse evaluation of nonlinear functions. Appl. Comput. Harmon. Anal. 24(1), 94–119 (2008) 2. de Boor, C.: A practical guide to splines. Applied Mathematical Sciences, vol. 27, revised edn. Springer, New York (2001) 3. Cohen, A., Daubechies, I., Feauveau, J.C.: Biorthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math. 45(5), 485–560 (1992) 4. Jester, P., Menke, C., Urban, K.: Wavelet methods for the representation, analysis and simulation of optical surfaces, IMA J. Appl. Math. 76(6) (2011) 5. Zheludev, V.A.: Local quasi-interpolation splines and Fourier transforms. Dokl. Akad. Nauk SSSR 282(6), 1293–1298 (1985)

Modelling Two-Dimensional Photopolymer Patterns Produced with Multiple-Beam Holography Dana Mackey, Tsvetanka Babeva, Izabela Naydenova, and Vincent Toal

Abstract Periodic structures referred to as photonic crystals attract considerable interest due to their potential applications in areas such as nanotechnology, photonics, plasmonics, etc. Among various techniques used for their fabrication, multiple-beam holography is a promising method enabling defect-free structures to be produced in a single step over large areas. In this paper we use a mathematical model describing photopolymerisation to simulate two-dimensional structures produced by the interference pattern of three noncoplanar beams. The holographic recording of different lattices is studied by variation of certain parameters such as beam wave vectors, time and intensity of illumination.

1 Introduction Grating evolution in photopolymers has been studied by several authors ([1–3], etc.). It is known that exposing a photopolymer material to an illumination pattern causes light-induced mass transport of the system components. The recorded holographic D. Mackey () School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland e-mail: [email protected] T. Babeva Institute of Optical Materials and Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria e-mail: [email protected] I. Naydenova Centre for Industrial and Engineering Optics, Dublin Institute of Technology, Ireland e-mail: [email protected] V. Toal Centre for Industrial and Engineering Optics, Dublin Institute of Technology, Ireland e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 43, © Springer-Verlag Berlin Heidelberg 2012

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grating is then due to a spatial variation of the refractive index resulting from changes in the density of the monomer and polymer species. However, the common feature of most theoretical models proposed to date is that they fail to describe the experimental observation of poor diffraction efficiency at high spatial frequencies. A two-way diffusion theory was proposed in [4,5] and states that the counter diffusion of short-chain polymer molecules away from the bright fringes is responsible for the reduction in diffraction efficiency. To verify this assumption, a new mathematical model for the formation of a weak grating after short exposure times was proposed in [6, 7]. This model accounts for both monomer and polymer diffusion and moreover distinguishes between short polymer chains capable of diffusing and long polymer chains that are immobile. The time evolution of refractive index modulation was calculated and compared with experimental results for different values of the diffusion coefficients, polymerization rates, intensity and spatial frequency of recording. It was concluded that the two-way diffusion model successfully predicts the observed grating deterioration at high frequencies and this has been further validated by good agreement within other experimental settings [8]. In this paper, the model is generalised to describe two-dimensional holographic gratings and preliminary results regarding numerical investigations of photopolymerisation patterns produced by the interference pattern of three noncoplanar beams are presented.

2 Three-Beam Holography Procedures for implementing two-dimensional optical interference profiles through holography, together with strategies for optimising the desired lattice characteristics in terms of propagation and polarisation parameters can be found in the literature (see, for example, [9–11]). In this work we restrict our study to patterns produced by three non-coplanar beams with electric field vectors Em .r/ D Em exp.i Km  r C i ım / em ; m D 1; 2; 3; where Em is the real amplitude, ım is the absolute phase and em are unit vectors which define the polarisation directions. The propagation vectors Km D

2 .cos.m / sin.m /; sin.m / sin.m /; cos.m // ; 

are expressed in the spherical coordinate system illustrated in Fig. 1. ( is the recording wavelength.) The illumination intensity is defined as the square of the total electric field and, for the case of linearly polarised waves, given in [10] as

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Fig. 1 The geometry of the three beams

z

K1

K3

K2

θ

θ

θ y φ3

x

I.r/ D

3 X lD1

El2 C

3 X

2El Em cos lm cos..Kl  Km /  r C ıl  ım /

(1)

1Dl<m

where lm is the angle between Kl and Km . Since the interference pattern resulting from three noncollinear propagation vectors is always two-dimensional, we choose the coordinate system so that all polar propagation angles are equal, 1 D 2 D 3  , and one reciprocal wave vector, say K2  K1 , is parallel to the y-axis. The intensity can then be conveniently expressed as " I.x; y/ D I0

1C

3 X

# Vlm cos.Alm x C Blm y/

(2)

1Dl<m

where 2 2 .cos l  cos m / sin ; .sin l  sin m / sin  Blm D     D cos.l  m / cos2  cos l cos m C sin l sin m

Alm D 3 Vlm 2

 cos  sin.l  m / sin.

l



m/

C sin2  cos

l

cos

m:

The maximum intensity points form a two-dimensional lattice with primitive vectors  .1; 0/; a1 D sin./ .cos.1 /  cos.2 // and we let a D ja1 j and b D ja2 j.

  1 C 2  a2 D cot. /; 1 2 sin./ sin.2 / 2

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3 Two-Dimensional Diffusion-Polymerisation Model The model proposed in [6, 7] takes into account monomer and polymer diffusion, creation of short polymer chains and introduces a simple “immobilization” mechanism which mimics the growth of polymer chains to the extent where they cannot diffuse any longer. This diffusion-polymerisation model consists of partial differential equations for three unknown functions which represent the concentrations of monomer, m, short polymer chains, p1 and long polymer chains, p2 , which is easily generalised to two dimensions. @m D Dm m  ˚.t/ F .x; y/ m @t @p1 D r .D.x; y/rp1 / C ˚.t/ ŒF .x; y/ m   mp1  @t @p2 D ˚.t/  mp1 ; @t The spatial domain is chosen as a  x  a, b  y  b and t  0. Here Dm is the monomer diffusion constant and, as justified in [6], we assume that the short polymer diffusion coefficient, as well as the polymerization rate are proportional to the illumination, D.x; y/ D Dp I.x; y/ and F .x; y/ D F0 I.x; y/, where I.x; y/ is given by (2) and F0 is the polymerization constant. We assume the initial conditions are given by m.x; 0/ D m0 and p1 .x; 0/ D p2 .x; 0/ D 0. To account for a finite exposure time, te , we have introduced the step function ˚.t/ D 1 if t  te and ˚.t/ D 0 if t > te . For convenience with the numerical simulations, the system is then non-dimensionalised by choosing xN D

x ; a

yN D

y ; b

m N D

m ; m0

pNi D

pi .i D 1; 2/: m0

The refractive index of a material consisting of a mixture of components can be calculated with the well-known Lorentz–Lorenz equation, X n2  1 n2i  1 D ; ˚ i n2 C 2 n2i C 2 i

(3)

where n is the effective refractive index of the mixture, ni are the refractive indices of the components (monomer, polymer and binder) determined experimentally, and ˚i are the normalized concentrations of the components. (For more details of this calculation see, for example, [6].)

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4 Numerical Results and Conclusions The non-dimensional model was integrated numerically using a finite difference ADI (Alternating Direction Implicit) scheme. We choose a simple interference pattern based on three linearly polarized beams with the propagation and polarization parameters chosen as in [9]:  D 0:384, 1 D 4 , 2 D 3 , 3 D  3 , 4 4 1 D 2:56, 2 D 0:178, 3 D 0:918. This beam configuration gives rise to an illumination pattern characterized by a square lattice (since the primitive vectors a1 , a2 are orthogonal and a D b) and, in this context, the partial differential equations model presented in the previous section can be supplemented by zero-flux boundary conditions. The recording wavelength was kept constant at  D 532  109 m. The numerical values used for the diffusion constants are Dm D 107 cm2 /s, Dp D 109 cm2 /s and the polymerization rate constant was varied between F0 D 0:01  10 s1 . The exposure time is te D 5 s and the system was integrated for a total of 50 s. The spatial distribution of the refractive index was calculated from (3), as a function of the monomer and polymer concentrations, at various times during and after exposure and the resulting grating formation was compared with the illumination pattern. It was found that, in general, the accuracy of copying was good except for parameter regimes where diffusion times were much smaller than polymerisation times. (This effect could be illustrated, for example, by decreasing the polymerisation rate F0 —see Fig. 2). Figure 3 shows spatial concentration profiles of long polymers obtained after exposure for two values of F0 . It is noted

(a) Illumination pattern

(c)

F0 = 0.1

(b)

(d)

F0 = 1

F0 = 0.01

Fig. 2 Comparison of illumination pattern and refractive index patterns after exposure

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(a) F0 = 0.1

(b)

F0 = 0.01

Fig. 3 Spatial distribution of long polymers after exposure

that for low values of F0 (or other parameter combinations which result in low ratios of diffusion to polymerisation times), the concentration profiles of long polymers develop contributions from higher harmonics. (This is in accordance with a similar observation made in [7] for the case of the one-dimensional illumination pattern.) In conclusion, a preliminary theoretical and computational study was presented with the purpose of assessing the suitability of a mathematical model for describing gratings produced with multiple beam holography. More such simulations are needed in order to assess the influence of various parameters (related to both the photopolymer system and the recording process) on the accuracy of copying a twodimensional light pattern into a change of the refractive index and such research is currently being carried out by our team.

References 1. Piazzola, S., Jenkins, B.: First-harmonic diffusion model for holographic grating formation in photopolymers. J. Opt. Soc. Am. B 17, 1147–1157 (2000) 2. Zhao, G., Mouroulis, P.: Diffusion model of hologram formation in dry photopolymer materials. J. Mod. Opt. 41, 1929–1939 (1994) 3. Colvin, V.L., Larson, R.G., Harris, A.L. Schilling, M.L.: Quantitative model of volume hologram formation in photopolymers. J. Appl. Phys. 81, 5913–5923 (1997) 4. Martin, S., Naydenova, I., Jallapuram, R., Howard, R. Toal, V.: Two-way diffusion model for the recording mechanism in a self developing dry acrylamide photopolymer. Proc. SPIE, 6252, 62525–625217 (2006) 5. Naydenova, I., Jallapuram, R., Howard, R., Martin, S., Toal, V.: Investigation of the diffusion processes in a self-processing acrylamide-based photopolymer system. Appl. Opt. 43, 2900– 2905 (2004) 6. Babeva, T., Naydenova, I., Mackey, D., Martin, S. Toal, V.: Two-way diffusion model for short exposure holographic grating, formation in acrylamide based photopolymers. J. Opt. Soc. Am. B 27(2), 197–203 (2010) 7. Mackey, D., Babeva, T., Naydenova, I. Toal, V.: A diffusion model for spatially dependent photopolymerisation. In: Fitt, A.D., Norbury, J., Ockendon, H., Wilson, E. (eds.) Progress in Industrial Mathematics at ECMI 2008, Mathematics in Industry, vol. 15, pp. 253–259. Springer, Berlin (2010)

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8. Babeva, T., Mackey, D., Naydenova, I., Martin, S., Toal, V.: Study of the photoinduced surface relief modulation in photopolymers caused by illumination with a Gaussian beam of light, J. Optic. 12, 124011 (2010) 9. Escuti, M.J., Crawford, G.P.: Holographic photonic crystals. Opt. Eng. 43(9), 1973–1987 (2004) 10. Cai, L.Z., Yang, X.L. Wang, Y.R.: Formation of a microfiber bundle by interference of three noncoplanar beams, Optic. Lett. 26(23), 1858–1860 (2001) 11. Mao, W., Zhong, Y., Dong, J., Wang, H.: Crystallography of two-dimensional photonic lattices formed by holography of three noncoplanar beams. J. Opt. Soc. Am. B 22(5), 1085–1091 (2005)

Modeling Berry’s Phase in Graphene by Using a Quantum Kinetic Approach Omar Morandi and Ferdinand Schurrer ¨

Abstract The influence of Barry’s phase on the particle motion in graphene is analyzed by means of a quantum phase-space approach. Our procedure is based on a reformulation of the Wigner formalism where the multiband particle-hole dynamics is described in terms of the Berry curvature. The emergence of some adiabatic parameters for the description of the quasi-classical trajectories in the presence of an external electric field is also discussed.

1 Introduction Graphene is a novel material consisting of a flat monolayer of carbon atoms packed in a two-dimensional honeycomb lattice. In a quite wide range of energy around the Dirac point, electrons and holes propagate as massless Fermions. In the present work we address the following well established graphene Hamiltonian [1, 3, 6] c D i vF „   rr C z C 0 U.r/ ; H

(1)

which describes the motion of an electron-hole pair in a graphene sheet in the presence of an external potential U.r/, vF is the and  the band gap   Fermi velocity (which is zero for pure graphene). Here  D x ; y ; z is the Pauli vector-matrix, and 0 denotes the identity 2  2 matrix. A novel set of phenomena has been recently discovered in such a material (for example the integer quantum Hall effect and the Aharonov–Bohm effect [5, 7]), opening the intriguing possibility to study the implication of the relativistic-like electron behaviour in the solid state framework. Despite the increasing effort

O. Morandi ()  F. Sch¨urrer Institute of Theoretical and Computational Physics, TU Graz, Petersgasse 16, 8010 Graz, Austria e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 44, © Springer-Verlag Berlin Heidelberg 2012

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devoted to the theoretical study of graphene, the physical explanation of some properties of this material (for example the limit of the mobility and the residual conductivity) is still unclear. In solid state physics, we are typically interested in macroscopic phenomena which are slow in time and smooth in space on atomic scale. Such a simple picture cannot give complete account of effects to first-order in the fields. In particular, in a crystal as graphene, where the velocity operator has off-diagonal elements and the electric field mixes the bands, the usual definition of the group velocity does no longer apply. The theory of Berry phases offers an elegant explanation of this effect in terms of the intrinsic curvature of the perturbed band. The phase-space formulation of quantum mechanics of Wigner offers a framework in which quantum phenomena can be described with a classical language and the question of the quantum-classical correspondence can be directly investigated. In this contribution, we present a procedure based on the Wigner formalism where the multibanddynamics is described in terms of the Berry curvature. For these reasons, an approach where both the kinetic characteristics of the particles and the pseudospin degree of freedom are described in a full quantum framework, seems to be a promising approach for studying graphene.

2 Quasi-diagonal Wigner Model The principal aim of this contribution is to study the multiband dynamics in a graphene sheet and to discuss some interesting connections between our kinetic description of the quantum dynamics and the Berry phase approach, which usually is studied at the Schr¨odinger level. In the general case, the full quantum mechanical description of motion consists of a rather complex set of coupled equations, where a clear interpretation of the dynamics is hampered by the presence of the highly nonlocal operators. For this reason, in order to profit from the close analogy between the classical mechanics and the Wigner formalism, we limit ourselves to consider the first-order „-correction to the classical dynamics. This approximation is justified in the limit where the external electric potential U.r/ can be considered as a sufficiently smooth function, so that only the first-order terms (proportional to the electric field) play a significant role in the dynamics. We derive the equation of motion of the particles in the quantum kinetic b defined on a suitable Hilbert space formalism. Given a differential operator A h i H, we consider the map A .r; p/ D W 1 AO , obtained by applying the Wigner 0 b transformation to the kernel KA b .x; x / of the operator A :

Z A .r; p/ 

     i p r C ; r  e „ d: KA b 2 2

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Let us now fix an orthonormal basis D f i j i D 1; 2; : : :g. A generic mixed state R b Œh D  .x; x0 /h.x0 / dx0 , whose kernel is the is defined by the density operator S P density matrix  .x; x0 / D i;j ij i .x/ tj .x0 /. Here, the superscript t denotes the transposition. The von Neumann equation governs the evolution of the density operator: i„

h i b @S c; S b : D H @t

(3)

Equation (3) expresses the evolution of the system in operational form. By using the Weyl operator W 1 , this equation can be mapped in an evolution equation defined on the phase plane r  p. In particular, the symbol h i associated to the graphene 1 HO D vF   p C  z  C 0 U.r/ Hamiltonian given in (1) is H .r; p/  W (in this simple case the usual quantization rule p ! i „r holds true). We consider c0   b  bS b  where  b .r; rr / is a unitary 2  2 matrix operator (the superscript S c0 is the density matrix of the  denotes hermitian transposition). The operator S 0 b i . A convenient quantum kinetic system, expressed in the new basis set i D  b in terms of description of the hole-electron system is obtained by identifying  the symbol  .r; p/  W 1 Œ, where  .r; p/ diagonalizes the Hamiltonian H .r; p/. We have H   D ;

(4)

where  D z E.p/ q C 0 U.r/ is the diagonal matrix related to the relativistic-like c0 equation of motion is still (3) but with the spectrum E D vF2 jpj2 C 2 . The S

b

c bH b  . By applying the operator W 1 to (3), we obtain the Hamiltonian H 0   h i c0 in the phase space .r; p/: final equation of motion for the symbol S 0 D W 1 S i„

  @S 0 D H 0; S 0 ? ; @t

(5)

where the brackets denote commutation ŒA ; B? D A ? B  B ? A , and the star-Moyal product ? is defined as A ?B A e

i„ 2

  !  ! rr rp rp rr

B:

(6)

After some algebra, up to the first order in „, (5) gives   vF p @f ˙ D ˙p  rr f ˙ C rr U  rp f ˙ ˙ i Bf i  Bf i ; @t 1 C 2 jpj   @f i D i Af i C rr U  rp f i C i B f C  f  ; @t

(7) (8)

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where we defined

S0 .2 „/2



! f C .r; p/ f i .r; p/ , and f i .r; p/ f  .r; p/

2vF .M /2 AD p jpj C .p ^ rr U /z jpj2 „ 1 C 2 BD Here, M˙ . / D

r

MC M px C ipy .p ^ rr U /z : 2 jpj3

1˙ p1

1C

, D 2

vF jpj 

(9) (10)

 @U and .p ^ rr U /z D  @r py C x

@U p @ry x

 .

The equations (7)–(8) describe the quantum mechanical evolution of a two-particle q

system due to the band structure of graphene E.k/ D vF2 „2 jkj2 C 2 . Here C  f (f ) represents the electron (hole) distribution function in the presence of an external electric field that modifies the periodic crystal potential. Since  is a diagonal matrix, the free motion of the electron-hole pairs in the upper and lower conically shaped energy surfaces ˙ ˙ is described in terms of the evolution of two non-interacting populations of particles. We note that our procedure is derived in a full quantum context, without invoking the usual generalization of the semi classical motion to the quantum one, where the substitution k ! i rr in the semiclassical expression of the energy spectrum E.k/ is assumed (See for example [4]). 2.0.1 Analogy with the Adiabatic Berry Phase Approximation

We discuss the analogy of our approach and the well-known Berry phase formalism. The Berry connection plays an explicit role in spin dynamics and in describing spinorbit interaction. The formal analogy between spin and band degree of freedom suggests us to investigate the effects of including the Berry phase in the evolution of a many-band electron system. In particular, we show that in our approach the quantum corrections to the semi-classical equation of motion, describing the interband tunneling, are given in terms of the Berry connection. The Berry theory considers the adiabatic evolution of a physical system described by a Hamiltonian H ..t// whose time dependence is given by a set of parameters  D . 1 ; : : : ; n / that follow a given trajectory in the parameter space (see for example [2] and references therein). By uj ./ and "j ./, respectively, we denote the set of eigenvectors and eigenvalues of H ./. Berry’s adiabatic theory states that if a system is initially described by a certain eigenvector ui ./, the vector state of the system at time t is given by i

.t/ D ui ..t// e i i .t / „

Rt 0

"i Œ.t 0 / dt 0

;

(11)

where the term i is denoted by dynamicalRphase factor and can be obtained as the path integral along the -trajectory, i D˛ Ai i ./  d, of the Berry connection ˝ A./ given by Aij ./ D i ui ./jr uj ./ .

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In order to see the analogy of such a procedure with our approach, we group together all the states belonging to the same energy cone. Instead to consider the c .r; rr / given in (1) (that is a differential operator acting on the full Hamiltonian H wave function), we apply the Berry approach to the Hamiltonian symbol H (which is a simple matrix where p plays the role of the adiabatic variable ). In this way, the analogons of the eigenvectors uj ./ (eigenvalues "j ./) in our procedure are represented by the solutions of (4) and the Berry connection is equal to the matrix X   Aij .p/ D .p/rp .p/ ij D i k rp kj : k

Furthermore, it is well known that the phase factor i does not appear when we consider Floquet projectors (and it is the reason for which the Berry geometry is defined in terms of fiber bundles where the phase information is “attached” to the projector manifold). In fact, from (11) we obtain that j .t/i h .t/j D jui Œ.t/ i h ui Œ.t/ j : In our case, the distribution functions f C and f  , respectively, are the Wigner functions related to the p-dependent eigenspace projectors juC .p/i h uC .p/j and ju .p/i h u .p/j of H (or “diagonal projectors”) for which the Berry phases cancel out. On the contrary, the function f i is related to the “band transition” operator juC .p/i h u .p/j that, for a given trajectory p.t/ (in the following we address the problem how to define such a trajectory), cumulates a Berry phase equal to j

C .t/i h  .t/j

R

t i 0 0 0 D juC Œp.t/ i h u Œp.t/ j e i ŒC .t / .t / „ 0 ."C Œp.t /" Œp.t // dt :

In particular @ j @t

"C .p/  " .p/ dp  .ACC  A /  j C i h  j (12) Ci h j D i dt „ # " 

dp .M /2 2vF p^ Di  p p j C i h  j : (13) jpj2 dt z „ 1 C 2 

holds true. For dp D rr U , the square bracket of (13) becomes equal to the function dt A defined in (9), and the equation of motion agrees with the evolution equation (8) for the interband function f i . Since the Berry connection appears explicitly in the equation of motion of the function f i , our method is particularly suited to highlight the role of the Berry phases in the evolution of the system. One of the characteristics of the Berry connection is the divergence in the proximity of points where the bands intersect. In gapless graphene such a divergence can be found in the neighborhood of the Dirac point p D 0 (where the upper and the lower cone “touch”). For that reason, from (8) we see that the natural oscillation frequency A of f i behaves like

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1=jpj when  D 0. The Wigner–Moyal formalism we adopted, is essentially a projection operation, where at the lower order of the „ expansion, the dynamics of the system is restricted to two single-band energy levels. In view of (13), the term proportional to the Berry curvature can be regarded as the result of the ”residual” interaction of those projected-out energy levels. One of the main differences between our approach and the direct application of the Berry procedure is that in the formalism that we adopted, the equations of motion are expressed in terms of the kinetic variables position and momentum. We can thus profit from the fact that in the quantum phase-space formulation of quantum mechanics, position and momentum are just variables and not operators. This situation eases the definition of adiabatic trajectories. In the standard application of the Berry theory, one recognizes in the Hamiltonian the presence of a timedependent parameter R.t/ (which evolves accordingly to the adiabatic time scale). The trajectory of R.t/ is fixed and independent from the system under study. This limits the application of the Berry theory to situations where an “internal” system, described by a Schr¨odinger equation with a time-dependent Hamiltonian, is coupled to an “environment” which is the source of the modification of the Hamiltonian. A usual trick to circumvent this limit and to use the Berry method in systems where the previous subdivision does not apply (as for example in the study of the Aharonov–Bohm (A-B) interference pattern in the presence of a stationary magnetic field), is to define some “external variables” of the system in terms of a suitable set of quasi-constant observables. For example, in the case of the AB effect, one assumes that the particle is confined in a small box located at a position R0 and that the wave packet moves without spreading along a certain trajectory R0 .t/. On the contrary, in our approach there is a natural identification of the phase-space variables with the adiabatic Berry parameter R. This is the case because, at the leading order in „, the evolution equation for the multicomponent Wigner function becomes the classical Liouville equation. In this way, the simple change of variables .r; p/ ! .r.t/; p.t// where .r.t/; p.t// are the classical trajectory, leads toa system of equations where the˛ constant-in-time Berry  ˝ connection i .p/rp .p/ ij  rr U.r/ D i ui .p/jrp uj .p/  rr U.r/, is substituted by the analogous term evaluated along the particle flux. Acknowledgements This work has been supported by the Austrian Science Fund, Vienna, under the contract No. P 21326 - N 16.

References 1. Beenakker, C.W.J.: Colloquium: Andreev reflection and Klein tunneling in graphene. Rev. Mod. Phys. 80(4), 1337–1354 (2008) 2. Bohm, A., Mostafazadeh, A., Koizumi, H., Niu, Q., Zwanziger, J.: The Geometric Phase in Quantum Systems: Foundations, Mathematical Concepts, and Applications in Molecular and Condensed Matter Physics. Sringer, Berlin (2003)

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3. Castro Neto, A.H., Guinea, F., Peres, N.M.R., Novoselov, K.S., Geim, A.K.: The electronic properties of graphene. Rev. Mod. Phys. 81(1), 109–162 (2009) 4. Markowich, P.A., Ringhofer, C.A., Schmeiser, C.: Semiconductor Equations, vol. 8. Springer, Berlin (2002) 5. Morozov, S.V., Novoselov, K.S., Katsnelson, M.I., Schedin, F., Ponomarenko, L.A., Jiang, D., Geim, A.K.: Strong suppression of weak localization in graphene. Phys. Rev. Lett. 97(1), 016,801 (2006) 6. Novikov, D.S.: Elastic scattering theory and transport in graphene. Phys. Rev. B 76(24), 245–435 (2007) 7. Zhang, Y., Tan, Y., Stormer, H.L., Kim, P.: Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438, 201–204 (2005)

Multi Scale Random Sets: From Morphology to Effective Behaviour Dominique Jeulin

Abstract Complex microstructures in materials and in biology often involve multi-scale heterogeneous textures, that we model by random sets derived from Mathematical Morphology. Our approach starts from 2D or 3D images; a complete morphological characterization is performed, and used for the identification of a model of random structure. Simulations of realistic microstructures are introduced in a numerical solver to compute appropriate fields (electric, elastic, velocity, . . . ) and to estimate the effective properties by numerical homogenization, accounting for scale dependent statistical fluctuations of the fields.

1 Introduction As a result of the non-homogeneous dispersion of a charge in a matrix, like carbon black in [33], many nanocomposite materials show an arrangement of aggregates at different scales. To predict the effective properties of such composites (like the dielectric permittivity or the elastic moduli), it is necessary to know the properties of the two components (charge and matrix), and their spatial distribution. For this purpose, we developed a general methodology, based on the theory of random sets [11, 22, 24, 34]: the morphology is summarized and can be simulated by multi-scale random models accounting for the heterogeneous distribution of aggregates. The identification of the model is made from image analysis. From 3D simulations of the model, overall properties like the percolation threshold or the effective properties can be estimated by numerical homogenization, or sometimes by estimations. After a reminder of the basic tools to characterize random sets and of the Boolean model, we illustrate our approach by an introduction to the multi scale combination of D. Jeulin () Centre de Morphologie Math´ematique, Math´ematiques et Syst`emes, MINES ParisTech, 35, rue Saint-Honor´e, F77300 Fontainebleau, France e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 45, © Springer-Verlag Berlin Heidelberg 2012

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random sets, to the Cox Boolean model, and to some iterations of random media. In the last part, a prediction of effective properties is obtained by numerical simulations on representative volume elements of the microstructure.

2 Principle of Random Structures Modeling, and the Boolean Random Set The heterogeneity of materials can be handled through a probabilistic approach, which enables us to generate models and simulation of the microstructures.

2.1 Random Set Properties When considering two-phase materials (for instance a set of particles A embedded in a matrix Ac ), we use a model of random set A [11, 22, 24, 34], fully characterized from a probabilistic point of view by its Choquet capacity T .K/ defined on the compact sets K, from (1) below, where P denotes a probability: T .K/ D P .K \ A ¤ ;/ D 1  Q.K/ D 1  P .K  Ac /:

(1)

In practice, T .K/ can be estimated by area fraction measurements on 2D images (from true microstructures, or from simulations), after morphological dilation of the set A by the set K [11,22,24,34], noted A ˚ K, or calculated for a given theoretical model. Equation (1) is used for the identification of a model (estimation of its parameters, and test of its validity). Particular cases of morphological properties deduced from (1) are the volume fraction Vv , the covariance (a useful tool to detect the presence of scales or anisotropies), the distribution of distances of a point in Ac to the boundary of A. The access to 3D images of microstructures by means of X-ray microtomography [6] makes it possible to use 3D compact sets K (like balls B.r/with various radii r) to characterize the random set.

2.2 Boolean Model Some materials (like porous media or composite materials) can be simulated by means of a basic random set model, namely by distributions of overlapping spheres. A random set model, the Boolean model, was proposed by G. Matheron [22, 24] to reproduce this situation. First, we consider a location of centres of spheres by means of a Poisson point process with intensity . In this condition, a volume V contains a random number of centres N following a Poisson distribution with

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average value V . Then a random set is obtained by the union of random grains (any L compact set). For this model, the Choquet capacity is given by (2), where V .A0 ˚ K/ 0 is the average volume of the random primary grain A , dilated by K: L T .K/ D 1  exp V .A0 ˚ K/

(2)

As a particular case, the volume fraction VV of the grains, is given by (3): 0

q D 1  VV D e  V .A / :

(3)

Using for K a pair of points, three points, or a ball with radius r gives access to the covariance, the third order moment, and to the spherical contact distribution (namely the distribution of the distance of a random point in Ac to the boundary of A). For a given population of random grains, these morphological functions are theroretically available from (2). In the special case of spherical grains, the distribution function of the speres radii can be estimated from the covariance [3], and therefore the model can be identified from the two points statistics, which is a one-dimensional information. For more general situations, the population of grains requires higher order moments for the identification of the model.

2.3 Percolation of the Boolean Model For materials made of components with a high contrast of properties, like for instance carbon black in a polymeric matrix, there is a strong effect on the macroscopic properties when a given phase percolates through the structure, inducing connected paths in the samples of the medium. Analytical estimations of the percolation of Boolean models of cylinders are available with the excluded volume model [1]. More recent analytical estimates of the percolation threshold of isotropic Boolean models with convex grains are based on the zeroes of the connectivity number [3, 15, 27]. They give an estimate of the percolation threshold of the grains pc1 and of Ac . These two percolation thresholds are different, as a result of the fact that the two sets A and Ac are non-symmetrical. The percolation threshold can also be estimated on simulations of the microstructure. It appears that a Boolean model with anisotropic primary grains (for instance sphero-cylinders) shows a much lower percolation threshold (0:01145 for an aspect ratio l=r D 100 [1]) than for isotropic grains (0:2895 for spheres [16, 32]. This can explain the expected outstanding mechanical, electrical or chemical properties of composites containing carbon nanotubes, mainly due to their shape, giving a low percolation threshold. The percolation of the complementary set of a Boolean model of spheres obtained analytically from the connectivity number and by simulations are given by 0:05698 and 0:0540 ˙ 0:005 respectively [18].

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3 Multiscale Combination of Independent Random Sets Starting from the basic models, more complex structures, such as superposition of scales, or fluctuations of the local volume fraction p of one phase can be generated in a simple way by a combination of various random sets, considering independent realizations. A convenient construction of multiscale models makes use of the union or intersection of independent random sets Ai with different scales. In the case of intersections, A D \i Ai , and it is easy to show that we have: P .K/ D P fK  .\i Ai /g D ˘i P fK  Ai g:

(4)

The result (4) is exact without any approximation, whatever the independent random sets Ai and their scales. For instance the overall volume fraction is the product of the volume fractions of the Ai . Similarly the binary covariance C.h/ D P fx 2 A; x C h 2 Ag, and more generally the n points probabilities are obtained as a product of the corresponding individual n points probabilities. A lower bound of the corresponding percolation threshold pc can be estimated by the products of percolation thresholds pci when the scales are widely separated: pc ' ˘i pci . This model was used to simulate the random distribution of carbon black in composites by means of the intersection of three Boolean models of spheres at different scales, reproducing the carbon black particles, aggregates, and zones of exclusion in the matrix [1]. The identification of the model is made from the measurement of the overall binary covariance on C.h/ images obtained by transmission elecron microscope on thin sections of the material.

4 The Multi Scale Cox Boolean Model 4.1 The Cox Point Process and the Boolean Model In another way to account for a non homogeneous distribution of random grains, it is possible to replace the Poisson point process by a Cox point process [10]. Consider a positive random function (RF) giving a non homogenous intensity .x/. For any realization of this RF, a Poisson point process with intensity .x/ is generated (for any realization, the number of points in a domain D follows a Poisson distribution R with average .D/ D D .dx/. If 'K ./ is the Laplace transform of the positive L where EA0 states for the mathematical expectation random variable EA0 f.A0 ˚ K/g, with respect to the random set A0 , we have: 0

L

T .K/ D 1  E fe EA0 f.A ˚K/g g D 1  'K .1/: An interesting particular case is obtained by means of a constant intensity  inside a first random set A. For instance A is a Boolean model of spheres with a large

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radius R. We then keep the points of a Poisson point process contained in A, as germs for centers of spheres with a smaller radius r [16]. The random measure becomes .dx/ D 1A .x/dx, where 1A .x/ is the indicator function of the set A (1A .x/ D 1 if x 2 A and 1A .x/ D 0 if x 2 Ac ). We have T .K/ D 1  ˚K ./;

(5)

where ˚K ./ is the Laplace transform of the random variable EA0 fV ..AL0 ˚K/\A/g obtained on the realizations of the random set A, after averaging over the realizations of the primary grain A0 . As a particular case for a deterministic grain A0 the Choquet capacity of the Boolean Cox model is deduced from the distribution of the change of support of the set A over the compact set AL0 ˚ K (or probability law of V ..AL0 ˚ K/ \ A/). It is usually difficult to access to this law for any random set A, but it can be easily estimated from simulations. We can therefore estimate the theroretical covariance (or higher order moments) of the model. In the case of a large separation of scales between the two sets A and A0 the covariance and the three points probability are asymptotically equal to the corresponding theoretical values (4) given for the intersection of independent random sets (Jeulin, 2010, “unpublished”). This result is valid for any n points probability, giving an approximation of the corresponding moments for this Cox Boolean model. For thin sections in electron microscopy, we can notice that the available experimental information is a projection of the set A through the thickness e, and therefore we can estimate T .K ˚ e/, from which the model can be identified, after replacing the primary grain A0 by A0 ˚ e. For carbon black nanocomposites, it is usual to model the distribution of carbon black particles by means of a three scale model [7,16,33]: spherical carbon particles (with a possible distribution of radii f1 .r/) are located on Poisson points inside inclusion zones (Boolean model of spheres with a distribution of radii f2 .r/) and outside of exclusion zones (Boolean model of spheres with a distribution of radii f3 .r/). The identification of the parameters of the model is made by means of an iterative optimization process, minimizing the difference between probabilistic properties of simulations and of images of the material: in [7, 9], the multi scale model is identified from measurements of the covariance, the third order moment, and of the area fraction after 2D closings on transmission electron images.

4.2 Percolation of the Cox Boolean Model As expected, the generation of aggregates produces random media with a lower percolation threshold: for a large separation of scales in a two scales model, the lower bound of the overall percolation threshold pc is given by the product pc1 pc2 of the corresponding thresholds, as in the case of the intersection of independent random sets. Models involving iteration of scales generate microstructure with a very low percolating threshold, and therefore with improved macroscopic properties when the percolating component presents the higher property (e.g. conductivity or

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elastic moduli). It shows that a typical homogeneous distribution of grains in space (like for the standard Boolean model) does not produce a medium with optimal properties. The percolation threshold of various multi scale models was estimated in various situations [16–18] . In [18], we get pc D 0:0849 for a scale factor equal to 30 in a two scale Cox Boolean model of spheres, close to 0:28972 D 0:0839. In [17] the percolation threshold of sphero-cylinders (aspect ratio l=r D 100) with centres located in a primary Boolean model of spheres (with Vv D 0:32) percolates for pc D 0:0056. Again, considering the case of aggregates of carbon nanotubes, high performances are expected from such a microstructure for a very low volume fraction of charge, due to the presence of clusters.

5 Change of Scale in Random Media In materials sciences, one is interested by the prediction of the macroscopic behavior of a physical system from its microscopic behavior. Models of random sets as seen before can give a good description of the morphological arrangement of components, and are a crucial step towards the connexion between morhological textures and the overall physical properties. The field of prediction of the effective properties (overall properties of an equivalent homogeneous medium) of random heterogeneous media from their microstructure is usually denominated as Homogenization. Most relevant tools are based on variational principles, from which bounds of the effective properties can be derived. In some specific situations based on multiscale random sets bounds can coincide and therefore provide an estimation of the properties, as illustrated below. In more general cases, an estimation of the effective behaviour is obtained from numerical simulations on realizations of random media on finite domains. This numerical approach is relevant, provided the notion of Representative Volume Element (RVE) can be correctly taken into account for random sets, as seen below.

6 Multi Scale Iterations of Random Media Using multiscale iterations of random media gives processes to generate two components microstructures with “optimal properties”, i.e. with the highest possible effective properties, given the volume fraction and the physical properties of components. The probably most famous case is given by the Hashin coated spheres assemblage [6]. It can be shown that for this isotropic morphology the effective conductivity (or effective dielectric permittivity) is given by the upper (resp. lower) Hashin–Shtrikman (H-S) bound when the material with the highest (resp. lowest) conductivity is put on the outer layer of spheres. We give here two other types of models, based on iterations of random sets.

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6.1 Dilated Poisson Hyperplanes Third order bounds (depending on the three-points statistics of a random medium) of effective properties (like conductivity, elastic moduli) were derived in [2, 26]. from a classical variational principle. It was shown that for an isotropic two component random set, these bounds depend on two positive functions of the volume fraction p, bounded by 1 [28]. When these functions summarizing the three points probability of a random set are equal to 1 (resp. 0), the two third order bounds coincide with the upper (resp. lower) Hashin–Shtrikman (H-S) bound. We have shown [12, 13] that for an infinite union of dilated Poisson flats with widely separated scales, we get a multi scale structure percolating for any volume fraction, and corresponding to this situation: its effective permittivity is given by the upper (resp. lower) HS bound, when attributing the highest permittivity to the union of dilated flats (resp. to their complementary set). This result is valid for a porous medium, where the upper (H-S) bound becomes the effective permittivity. The same result holds when considering the case of the bulk modulus in linear elasticity.

6.2 A Two Scales Hard Core Composite In some situations, a microstructure is made of non overlapping objects (for instance spheres), described by a so-called hard-core process [34]. It turns out that for this model with spheres of a single radius, numerical simulations show that the obtained effective properties are close to the lower H-S bound for a “soft” matrix, and close to the higher H-S bound for “soft” spheres (like pores) [8, 35]. This can be explained by the fact that for this model the morphological parameter involved in the calculation of bounds is close to 1 in the first case [36], and to 0 in the second one. This point can be used to give an approximate estimate of the effective properties of a material containing non overlapping aggregates (similar to the mentioned Cox Boolean model), obtained in two steps, as illustrated in an elastomeric matrix containing carbon black particles CB [8]. Composite spherical aggregates are made of percolating CB spheres (with volume fraction p1 ) and of the elastomeric component (with volume fraction 1  p1 ). The elastic properties of the aggregates are estimated by the third order upper bound of a Boolean model of spheres (or equivalently of a hard core model) with volume fraction p1 . In a second step, these spherical aggregates (with a radius much larger than the radius of CB spheres, involving a separation of scales) are implanted according to a second hard core process with volume fraction p2 , giving an overall volume fraction p D p1 p2 . The overall moduli are now estimated by the H-S lower bound computed for this composite. As shown in [14], the estimates of the shear modulus G, obtained for different values of p1 are consistent with the experimental measurements obtained for various mixes, a more uniform distribution in space giving a lower G. For a given p, G increases when p1 decreases, making easier the percolation in the

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two-scale model. A more accurate estimate is obtained by computer intensive numerical simulations [8].

7 Prediction of Effective Properties by Numerical Simulations An efficient way to solve the problem of homogenization of physical properties, for instance to predict the dielectric permittivity or the effective mechanical properties of heterogeneous media, makes use of numerical solutions of the corresponding partial differential equations (PDE) solutions, before estimating the effective properties by spatial averaging of the solution. This requires the input of 3D images (real images, as obtained by confocal microscopy, or by X-ray microtomography) or simulations, so-called digital materials. In a second step a computational code (Finite Elements, Fast Fourier Transform, PDE numerical solver) is implemented.

7.1 Digital Materials We use a method derived from [5, 29], to estimate the equivalent macroscopic dielectric constant " from the structure of a material and the properties of its constituents [30]. For this, we compute the electric field E.x/ inside the material, knowing the local dielectric permittivity ".x/, by application of a constant electric field E0 , and of periodic boundary conditions. An iterative calculation in the Fourier space is used to estimate E.x/, giving " by means of a spatial average <>. The numerical solution is obtained by means of the Green function of a homogeneous reference medium for the corresponding PDE to the electrical field or for the elastic field problem. Using the FFT approach is very versatile and does not require any meshing of the microstructure, in contrast with other numerical methods such as Finite Elements. In elasticity, stress and strain field maps are obtained on the scale of microstructure, to get a detailed study of the effect of the microstructure on the local fields. With the Morph–Hom code developed by F. Willot [37, 38] images of large sizes (up to 15003) can be handled for any contrast of properties, and therefore the electrostatic or elastic behaviour of porous media or of rigid media was studied by numerical techniques. This was applied to the Boolean model of spheres with any volume fractions [37] and to two-scales and three-scales Cox Boolean models of spheres [38]. The reinforcing effect of the iteration of scales with a large separation is clearly observed in the case of a very high conductivity or of a rigid phase, as a result of a lower percolation threshold. In the case of porous media, the reverse effect is observed, as would appear in the case of a damage at the interface between clusters of rigid particles and the matrix.

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7.2 Statistical Approach of the RVE When working on images of a material or on realizations of a random medium, a natural question arises [12, 20]: what is the representativity of the effective property estimated on a bounded domain of a microstructure? In other words, what is the size of a so-called “Representative Volume Element” RVE. In many industrial applications, mainly in the case of electronic parts such as MEMS, the size of the microstructure cannot be neglected with respect to the size of parts. The parts cannot be considered as an infinite homogeneous medium, and fluctuations of effective properties are observed and must be accounted for. We address this problem by means of a probabilistic approach giving size-dependent intervals of confidence, initially developed in the framework of the homogenization of the elastic moduli of random media [4, 20], and based on the size effect of the variance of the effective properties of simulations of random media. This approach was applied to the elastic properties of rigid and of porous Boolean models of spheres [37, 38].

7.2.1 The Integral Range We consider fluctuations of average values over different realizations of a random medium inside the domain B with the volume V . In Geostatistics [23], it is well known that for an ergodic stationary random function Z.x/, with mathematical N / expectation E.Z/, one can compute the variance DZ2 .V / of its average value Z.V over the volume V as a function of the central covariance function C .h/ of Z.x/ by: DZ2 .V / D

1 V2

Z Z C .x  y/ dxdy; B

(6)

B

where C .h/ D Ef.Z.x/  E.Z// .Z.x C h/  E.Z//g: For a large specimen (with V  A3 ), equation (6) can be expressed to the first order in 1=V as a function of the integral range in the space R3 , A3 , by DZ2 .V / D DZ2 with A3 D

1 DZ2

A3 ; V Z R3

(7) C .h/ dh;

(8)

where DZ2 is the point variance of Z.x/ (here estimated on simulations) and A3 is the integral range of the random function Z.x/, defined when the integral in equations (6) and (8) is finite. The asymptotic scaling law (7) is valid for an additive variable Z over the region of interest B. To estimate the effective elasticity or permittivity tensors from simulations, we have to compute the average stress

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hi and strain h"i (elastic case) or electric displacement hDi and electrical field hEi. For the applied boundary conditions, the local modulus is obtained from the estimations of a scalar, namely the average in the domain B of the stress, strain, electric displacement, or electric field. Therefore the variance of the local effective property follows the equation (7) when the integral range A3 of the relevant field is known. Since the theoretical covariance of the fields ( or ") is not available, the integral range can be estimated according to the procedure proposed by G. Matheron for any random function [25]: working with realizations of Z.x/ on domains B with an increasing volume V (or in the present case considering subdomains of large simulations, with a wide range of sizes), the parameter A3 is estimated by fitting the obtained variance according to the expression (7).

7.2.2 Practical Determination of the Size of the RVE When considering a material as a realization of a random set or of a random function RF, the idea that there exists one single possible minimal RVE size must be left out. Instead, the size of a RVE can be defined for a physical property Z, a contrast, and a given precision in the estimation of the effective properties depending on the number of realizations that one is ready to generate. By means of a standard statistical approach, the absolute error abs and the relative error rela on the mean value obtained with n independent realizations of volume V are deduced from the 95% interval of confidence by: abs D

2DZ .V / 2DZ .V / abs D p ; rela D p : Z n Z n

(9)

The size of the RVE can now be defined as the volume for which for instance n D 1 realization (as a result of an ergodicity assumption on the microstructure) is necessary to estimate the mean property Z with a relative error (for instance rela D 1%), provided we know the variance DZ2 .V /. Alternatively, we can decide to operate on smaller volumes (provided no bias is introduced by the boundary conditions), and consider n realizations to obtain the same relative error. This methodology was applied to the case of the dielectric permittivity of various random media [19], and to the elastic properties and thermal conductivity of a Vorono¨ı mosaic[20], of materials from food industry [21], or of Boolean models of spheres [37]. Further results were obtained for the Boolean model and for the multiscale Cox Boolean model of rigid spheres (with radius 10 voxels, the larger scale being a Boolean model of spheres with radius 100 voxels). The results in [37] indicate that the highest RVE sizes for the one-scale model correspond to rigidly-reinforced media with Vv  0:49. Accordingly, the variances DZ2 .V / and DZ2 , with Z D m (mean stress or 1=3 of the trace of the stress tensor), have been computed for the two-scales Cox Boolean model when Vv1 D Vv2 D 0:7 for a contrast higher than 10,000 of the bulk and shear moduli for a scale ratio equal to 10 [38]. It is found that the integral range of the two-scales is increased by a factor close to 5:8 times, whereas the point variance

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increases by a factor 322 . Accordingly, a 10% precision(i.e.  D 0:1) is achieved when V  5503 , which corresponds to an increase of the RVE by a factor 5:73 . The drastic increase of the point variance in the case of the two-scale random medium underlies very large local stresses, that could induce more damage in the matrix if it contains defect, as compared to the one scale model for the same volume fraction.

8 Conclusion Multi scale models of random media provide a wide variety of morphologies to simulate complex microstructures for application to real materials. These models are able to capture phenomena like the observation of very low percolation thresholds explaining the enhancement of some effective properties. Combined to predictive models by analytical means, or by numerical simulations, they give access to the optimization of microstructures with respect to the required properties. Acknowledgements This work was partially supported by Institut Carnot-Mines by a grant (Nanostructures project).

References 1. Balberg, I., Anderson, C.H., Alexander, S., Wagner, N.: Excluded volume and its relation to the onset of percolation. Phys. Rev. B, 30, Nı 7, 3933 (1984) 2. Beran, M.J., Molyneux, J.: Use of classical variational principles to determine bounds for the effective bulk modulus in heterogeneous materials. Q. Appl. Math. 24, 107 (1966) 3. Bretheau, T., Jeulin, D.: Caract´eristiques morphologiques des constituants et comportement e´ lastique d’un mat´eriau biphas´e Fe/Ag. Revue Phys. Appl. 24, 861–869 (1989) 4. Cailletaud, G., Jeulin, D., Rolland, Ph.: Size effect on elastic properties of random composites. Eng. Comput. 11, N 2, 99–110 (1994) 5. Eyre, D.J., Milton, G.W.: A fast numerical scheme for computing the response of composites using grid refinement. Eur. Phys. J. Appl. Phys. 6, 41 (1999) 6. Hashin, Z.: The elastic moduli of heterogeneous materials. J. Appl. Mech., Trans. ASME 29, 143 (1962) 7. Jean, A., Jeulin, D., Forest, S., Cantournet, S., N’Guyen, F.: A multiscale microstructure model of carbon black distribution in rubber. J. Microscopy. 241(3), 243 (2011) 8. Jean, A., Willot, F., Cantournet, S., Forest, S., Jeulin, D.: Large scale computations of effective elastic properties of rubber with carbon black fillers, International Journal of Multiscale Computational Engineering 9(3), 271 (2011) 9. Jean, A., Jeulin, D., Forest, S., Cantournet, S., N’Guyen, F.: A multiscale microstructure model of carbon black distribution in rubber. J. Microsc. 241(3), 243 (2011) 10. Jeulin, D.: Modeling heterogeneous materials by random structures, Invited lecture, European Workshop on Application of Statistics and Probabilities in Wood Mechanics, Bordeaux (22–23 March 1996), N-06/96/MM, Paris School of Mines Publication (1996) 11. Jeulin, D.: Random texture models for materials structures. Stat. Comput. 10, 121 (2000) 12. Jeulin, D.: Random structure models for homogenization and fracture statistics. In: Jeulin, D., Ostoja-Starzewski, M. (eds.) Mechanics of Random and Multiscale Microstructures, p. 33. Springer, Berlin (2001)

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13. Jeulin, D.: Random structures in physics. In: Bilodeau, M., Meyer, F., Schmitt, M. (eds.) Space, Structure and Randomness, Series: Lecture Notes in Statistics, vol. 183, p. 183. Springer, Berlin (2005) 14. Jeulin, D.: Multi scale random models of complex microstructures. In: Chandra, T., Wanderka, N., Reimers, W., Ionescu, M. (eds.) Thermec 2009, Materials Science Forum, vol. 638–642, pp. 81–86 (2009) 15. Jeulin, D.: Analysis and modeling of 3D microstructures. In: Talbot, H., Najman, L. (eds.) Mathematical Morphology: From Theory to Applications, Chapter 19. ISTE/Wiley, New York (2010) 16. Jeulin, D., Moreaud, M.: Multi-scale simulation of random spheres aggregates-application to nanocomposites. In: Proceedings of the 9th European Congress on Stereology and Image Analysis, vol. I, p. 341. Zakopane, Poland (2005) 17. Jeulin, D., Moreaud, M.: Percolation of multi-scale fiber aggregates. In: Lechnerova, R., Saxl, I., Benes, V. (eds.) S4G, 6th International Conference Stereology, Spatial Statistics and Stochastic Geometry. Prague, 26–29 June 2006, Union Czech Mathematicians and Physicists, p. 269 (2006) 18. Jeulin, D., Moreaud, M.: Percolation d’agr´egats multi-´echelles de sph`eres et de fibres: Application aux nanocomposites. In: Mat´eriaux 2006. Dijon (2006) 19. Jeulin, D., Moreaud, M.: Volume e´ l´ementaire repr´esentatif pour la permittivit´e di´electrique de milieux al´eatoires. In: Mat´eriaux 2006, Dijon (2006) 20. Kanit, T., Forest, S., Galliet, I., Mounoury, V., Jeulin, D.: Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int. J. Sol. Str. 40, 3647 (2003) 21. Kanit, T., N’Guyen, F., Forest, S., Jeulin, D., Reed, M., Singleton, S.: Apparent and effective physical properties of heterogeneous materials: representativity of samples of two materials from food industry. Comput. Methods Appl. Mech. Eng. 195, 3960–3982 (2006) 22. Matheron, G.: El´ements pour une th´eorie des milieux poreux. Masson, Paris (1967) 23. Matheron, G.: The theory of regionalized variables and its applications. Paris School of Mines publications, Paris (1971) 24. Matheron, G.: Random sets and Integral Geometry. Wiley, New York (1975) 25. Matheron, G.: Estimating and Choosing. Springer, Berlin (1989) 26. McCoy, J.J.: In: On the displacement field in an elastic medium with random variations of material properties, Recent Advances in Engineering Sciences, vol. 5, p. 235. Gordon and Breach, New York (1970) 27. Mecke, K., Stoyan, D.: The Boolean model: from Matheron till today, space, structures, and randomness. In: Bilodeau, M., Meyer, F., Schmitt, M. (eds.) Space, Structure and Randomness, Series: Lecture Notes in Statistics, vol. 183, p. 151. Springer, Berlin (2005) 28. Milton, G.W.: Bounds on the elastic and transport properties of two-component composites. J. Mech. Phys. Solid. 30, 177 (1982) 29. Moulinec, H., Suquet, P.: A numerical method for computing the overall response of nonlinear composites with complex microstructure. Comp. Meth. Appl. Mech. Eng. 157, 69 (1998) 30. Paciornik, S., Gomes, O.F.M., Delarue, A., Schamm, S., Jeulin, D., Thorel, A.: Multi-scale analysis of the dielectric properties and structure of resin/carbon-black nanocomposites. Eur. Phys. J. Appl. Phys. 21, 17 (2003) 31. Parra-Denis, E., Barat, C., Jeulin, D., Ducottet, Ch.: 3D complex shape characterization by statistical analysis: application to aluminium alloys. Mater. Char. 59, 338 (2008) 32. Rintoul, M.D., Torquato, S.: Precise determination of the critical threshold and exponents in a three-dimensional continuum percolation model. J. Phys. A. Math. Gen. 30, L585–L92 (1997) 33. Savary, L., Jeulin, D., Thorel, A.: Morphological analysis of carbon-polymer composite materials from thick sections. Acta. Stereol. 18(3), 297 (1999) 34. Serra, J.: Image analysis and Mathematical Morphology, Academic Press, London (1982) 35. Smith, J.C.: The elastic constants of a particulate-filled glassy polymer: comparison of experimental values with theoretical predictions. J. Res. NBS, 80A, 45 (1976)

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36. Torquato, S., Lado, F.: Effective properties of two-phase disordered composite media: II. Evaluation of bounds on the conductivity and bulk modulus of dispersions of impenetrable spheres. Phys. Rev. B 33, 6428 (1986) 37. Willot, F., Jeulin, D.: Elastic behavior of composites containing Boolean random sets of inhomogeneities. Int. J. Eng. Sci. 47 313 (2009) 38. Willot, F., Jeulin, D.: Elastic and Electrical Behavior of Some Random Multiscale Highly Contrasted Composites, International Journal of Multiscale Computational Engineering 9(3), 305 (2010)

Part V

Modeling, Analysis and Computation of Fluid Dynamics Problems

Overview At the ECMI 2010 Conference several advances in the large field of fluid dynamics were presented. They spanned the whole range from modeling, analysis to simulation and optimization. Owing to the broad focus of ECMI, we cannot expect in this chapter a full account of all current advances in fluid dynamics. But the ECMI contributions provide a detailed description and envisage solution of dedicated applications. This will be of great importance for industry and it is the primary purpose of ECMI. The contributions of this part have been grouped under three keywords: modeling, analysis and computation—although such a partition is not always unique. The first three contributions are mainly concerned with modeling and numerical verification: The authors, T.G. Myers and S.L. Mitchell, present their work on the modeling of practical relevant Stefan problems: Mathematical modelling of phase change with a flowing thin film. They discuss models for Leidenfrost and contact melting and derive a more tractable mathematical model. This has applications in aircraft icing or ice accretion e.g. on power transmission equipments. To validate their models, they use a heat balance integral method to obtain numerical solutions and compare these to experimental data. In A. Hlod’s contribution, On Modeling of Curved Jets of Viscous Fluid Hitting a Moving Surface, appropriate mathematical descriptions are sought where the jet’s three different flow regimes can be incorporated. Based on inertia effects, longitudinal viscosity and gravity, a jet model is set up. Then boundary conditions need to be supplied especially for the jet’s orientation. Analyzing the characteristics (of an underlying hyperbolic conservation law), the author can assign boundary conditions and finds that the orientation is determined by dominant effects in the momentum transfer. Pumping milk produces bubbles (milk foam). To accurately measure the volume of milk that is delivered, M. Devreux and W. Lee introduce a novel design in their

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work Air Elimination in Milk. It consists of an extra air elimination vessel equipped with an additional pump for air. Based on St-Venant-like equations, the authors set up a mathematical model for the flow of milk in the construction. Using a model for bubble formation they are able to predict the parameters for operation. Analysis of the respective mathematical model is a core issue of the following four papers. In Quantum Navier–Stokes equations, A. J¨ungel and J.-P. Miliˇsi´c consider compressible Navier–Stokes models for quantum fluids. They can derive a viscous quantum Euler formulation from the barotropic quantum Navier–Stokes system by means of a new velocity variable. For the Euler formulation a new energy functional is derived, which enables a priori estimates of the solutions. This is employed to prove the existence of weak solutions. Numerical results complete the contribution. In the analysis paper Travelling-wave Similarity Solutions for an Unsteady Gravity-driven Dry Patch by Y.M. Yatim et al. the gravity-driven flow of a thin Newtonian flow is investigated on an inclined plane in the three dimensional space. Using the lubrication approximation of the governing equations, they find travellingwave similarity solution, where the dry patch has a parabolic shape. To measure fluid parameters as viscosity under applied forces in a laboratory, one places a small drop of liquid between a cone and plate rheometer. Rotating the cone, the parameters are computed via the transmitted rotational momentum. For the according flow simulation, an initial geometry of the drop is needed. V. Cregan et al. in Asymptotics of a Small Liquid Drop on a Cone and Plate Rheometer derive the asymptotic structure of these profils (based on the Bond number). Many industrial applications need highly regular coating processes, but finger formation endangers the quality. To investigate this phenomenon and the stability of the process, J.P.F. Charpin considers a standard thin film approximation on an inclined plane to model spin coating. For the stability of the equation governing the flow front the eigenvalues are derived in Finger formation and non-Newtonian fluids. First results underline that with increasing non-Newtonian property (Ellis model) both the wavenumber and the growth rate of fingers rise. Numerics, simulation and optimization are core issues of the last papers. The contribution Numerical Modelling and Simulation of Ship Hull Geometries by M. Hopfensitz et al. consider an optimization problem in computational fluid dynamics. They present a concept of four steps to numerically optimize a ship hull geometry with propellers for efficient navigation. Direct optimization (Hooke– Jeeves) is applied, since gradients of the target (force) are not available. The numerical method in use is fully automatic and the paper is completed by numerical results. A. Dumitrache et al. investigate coupled fluid dynamic models for combustion in Analysis of Combustion and Turbulence Models in a Cylindrical Combustion Chamber. Together with the usual conservation laws, they employ a standard k   model for turbulences and an eddy dissipation combustion model for that phenomenon. Numerical results based on a finite volume discretization and certain power law schemes are compared to other numerical results and data from experiments.

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M. Viscor and M. Stynes develop a new model for spray drying, which allows a varying temperature and moisture content inside the particle. Thereby, they track the interface that separates wet and dry phases of the particle and they face the problem of the formation of interior layers. To numerically solve this problem, they propose a new Numerical method for a Stefan-type problem with interior layers using dedicated variables and a dedicated mesh. Wuppertal, Germany

Andreas Bartel Michael G¨unther

Mathematical Modelling of Phase Change with a Flowing Thin Film Tim G. Myers and Sarah L. Mitchell

Abstract This paper concerns the practical applications and modelling of Stefan problems with a flowing thin liquid layer. The modelling will be discussed in the context of two practically important scenarios, Leidenfrost (when a liquid droplet floats above a hot surface) and contact melting. The governing equations will be derived and then reduced to a more tractable system. Along the way we will introduce an accurate variant of the Heat Balance Integral Method, which allows us to approximate the solution to the Stefan problem on a finite domain. In both cases excellent agreement between the model results and experimental data will be demonstrated.

1 Introduction Phase change in the presence of a flowing liquid layer occurs in a multitude of situations. For example, ice growth from a thin moving water layer has been studied in the context of aircraft icing and ice accretion on wind turbines and power transmission equipment. Analogous problems also arise in the modelling of lava flow, atherosclerosis (plaque build-up on artery walls) and wax deposition in oil pipelines, see [5, 10, 11] for example. In this paper we focus on two particular, related problems, namely Leidenfrost and contact melting. Leidenfrost occurs when a liquid is placed on a surface which is above the phase change temperature. The heat flux from the surface causes the fluid T.G. Myers () Centre de Recerca Matem`atica, Edifici C, 08193 Bellaterra, Barcelona e-mail: [email protected] S.L. Mitchell MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 46, © Springer-Verlag Berlin Heidelberg 2012

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to evaporate and so the fluid floats on a vapour layer. The vapour layer is constantly being renewed through continued evaporation whilst it is also being squeezed out due to the fluid’s weight. Contact melting is a similar process in that it involves a melting solid on a hot surface. This process is exploited industrially as a means of energy storage. The mathematical modelling of this class of problem involves solving the heat equation in solid, liquid or vapour layers, where the boundary position is determined through a Stefan condition and force balance. Since the Stefan problem only has an exact solution in highly idealised situations we begin our analysis by describing an approximate method for the one-dimensional Stefan problem. The accuracy of this method is then compared to the standard large Stefan number perturbation solution. It is then used in the solution of the contact melting problem. Finally, we employ a similar analysis on the Leidenfrost model.

2 Heat Balance Integral Method and Stefan Problems Consider

u.0; t/ D 1 ;

@2 u @u D 2 ; @t @x u.s; t/ D 0 ;

0 < x < s.t/ ˇ @u ˇˇ ds D ˇ ; dt @x ˇxDs

(1) s.0/ D 0 :

(2)

This is scaled in such a way that the only remaining parameter is the Stefan number ˇ D Lm =.cp u/, where Lm is the latent heat of melting, cp the specific heat capacity and u the temperature variation. The exact solution is   p x u D 1  erfc p .erf.˛//1 ; s.t/ D 2˛ t; (3) 2 t p 2 where ˛ satisfies the transcendental equation ˇ˛ erf.˛/e ˛ D 1 : The large Stefan number (ˇ  1) solution may be obtained by carrying out a perturbation analysis after re-scaling time and using a boundary fixing transformation to give s   1 23 2t sD 1 C C    : (4) ˇ 6ˇ 360ˇ 2 Of course this must coincide with the large ˇ expansion of the exact solution. Further details may be found in [4] for example. An alternative approach to find an approximate solution to this problem is via a Heat Balance Integral Method (HBIM). The basic HBIM involves approximating the temperature by a function of the form   x m x C .1  a/ 1  uDa 1 : (5) s s

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This satisfies the boundary conditions at x D 0; s. The HBIM then proceeds by integrating the heat equation over x 2 Œ0; s, with u specified by the above polynomial. This leads to an ODE for s. The value of a is also unknown and so the Stefan condition is invoked to provide a second equation. The simple one-phase analysis is described in [6]. In the original HBIM of Goodman [3] the exponent m was set to 2 or 3. However, recently Myers [14] has developed a method whereby m is chosen through minimising the error Z

s

En D

.ut  uxx /2 dx :

(6)

0

This method ensures that the HBIM approach is accurate and, for certain situations such as a constant flux boundary condition, it can reduce the error by orders of magnitude from the n D 2 solution [13, 14]. An alternative to the HBIM is the RIM [7] (or integral method by integration [4]) which simply involves integrating the heat equation twice. Mitchell & Myers [8] propose a different method to reduce the error by employing both the HBIM and RIM to provide a further equation to determine m. This is then termed the Combined Integral Method (CIM). Following the HBIM or RIM, with a minimised error, or the CIM, we may easily determine an analytical p expression for s / t, which may then be compared to the exact solution. For large Stefan number we may also expand the result and compare with the perturbation solution. For example, with the CIM we obtain s sD

2t ˇ

  1 149 1 C C : 6ˇ 2520ˇ 2

(7)

This differs from the perturbation solution at 2nd order by 7.4%. However, unlike the perturbation solution the HBIM solution is not only valid at large ˇ and so can provide a wider range for the validity of the approximate solution. In [8] it is shown that the CIM result is more accurate than the 2nd order perturbation for ˇ < 7. For water and paraffin wax typically ˇ 2 Œ1; 10, so for physically realistic problems the heat balance methods are more accurate.

3 Contact Melting Contact melting is a process frequently exploited as a method of latent heat storage in the construction industry. However, perhaps the most familiar example occurs when a block of ice is placed on a warm surface. The ice melts to form a layer of water between the ice and surface. This water is squeezed out due to the weight of the ice, however as the ice approaches the surface the melting process speeds up and so a balance forms between melting and squeezing. Now consider a two-dimensional phase change problem, where a block of initial thickness H and width 2L is placed on a warm substrate. The mathematical model must deal with (at least) two distinct stages. When the phase change material is

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placed on the warm surface there is an initial stage where the PCM heats up to the melt temperature. Subsequently melting occurs and heat propagates into the solid. The most general description is given by 

@2 u @p ; D @z2 @x

@p D0; @z

@2 T D0; @z2 ˇ dhm @ ˇˇ @T s Lm D ks ˇ  kl dt @z zDh @z

@u @w C D0; @x @z @2  @ D s 2 ; @t @z

ˇ ˇ ˇ ˇ

; zDh

  2s L H0  hm .t/ g D

(8)

(9) Z

L

p dx : (10) L

This system represents: (8) fluid flow in the melt layer, subject to the lubrication approximation; (9) heat flow in the liquid and solid (where only the Tzz term remains since the liquid layer is thin); (10a) the Stefan condition; (10b) a mass balance, where the weight of the block is balanced by the fluid pressure. An important point to note is that the observed film thickness of the liquid layer, h, is different to the height of melted liquid, hm , due to the squeeze effect, hence the Stefan condition involves hm but the derivatives are evaluated at h. Further details may be found in [12]. In the pre-melt stage the temperature profile in the solid can be found exactly, or approximated using the HBIM. In the melt stage the temperature in the liquid layer is obviously approximately linear, whilst in the solid we must solve the Stefan problem over a finite domain and so require the HBIM. The problem reduces to solving 3 equations of the form 3

dh dı c0 C D dt dt ıh

dhm c1 c2 D C dt ıh 1 C c3 h

(11)

together with the mass balance (10) to determine the unknowns h; hm ; ı. The terms ci are constants that depend on the thermal properties of the materials. The term ı is key to HBIM analyses. It is termed the heat penetration depth and represents a notional distance over which the heat penetrates into the solid. The numerical solution of the three differential equations is straightforward. In Fig. 1 we compare the results of the current theory with experiments from [9]. These experiments involved the melting of a wide (and so quasi-two-dimensional) block of n-octadecane, with initial height 5.5 cm. The sides and top were insulated, hence the experiments are ideal for comparison with our theory. Our model uses a finite heat transfer coefficient, also shown are results with an infinite heat transfer coefficient, the analytical expression from [9] and a standard quasi-steady solution (see [1]). Obviously the current theory provides excellent agreement with experiment.

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0.05 0.04 H (m) 0.03 0.02 0.01 t (s) 0

0

200

400

600

800

1000

1200

1400

1600

Fig. 1 Comparison of height H.t / with experiment (asterisks) for the current theory with (a) HTC = 7;000 W/m2 (solid line), (b) infinite HTC (dashed line), (c) theory of [9] (dash-dot line) and (d) quasi-steady solution (dotted)

4 Leidenfrost The Leidenfrost phenomenon occurs when a liquid drop is placed on a surface which is above the vaporisation temperature. This causes the droplet to float on a layer of vapour. Obviously this problem is analogous to the contact melting one, with the main differences being that the droplet shape constantly changes (large droplets are disk-like whilst small droplets are closer to hemispherical) and the temperature in the liquid is approximately constant. The governing equations are similar to those of the contact melting problem, with two important differences. Firstly, since the liquid is free to circulate we assume it is everywhere at the liquidus and so do not solve a heat equation there. The vapour forms a thin film and so the temperature is approximately linear. Secondly, we must also incorporate the Young-Laplace equation, which balances gravity and surface tension, to describe the droplet shape. In this case the numerical solution reduces to solving the Stefan condition coupled to the mass balance and the Young-Laplace equation. Comparison of numerical solutions showed that an important effect, neglected in previous studies, is evaporation from the upper surface of the droplet. Without this effect the droplet will never evaporate sufficiently rapidly, even if we take an infinite heat transfer coefficient at the base. In Fig. 2 we show a comparison of model solutions to experimental results for the height of the vapour layer thickness (taken from [2]). The curves represent the numerical solution of the governing equations, an approximate analytical solution and then solutions with different heat transfer and evaporation rates. In this case the experimental results are rather trickier to calculate due to the droplet motion and narrow gap width, however, it is clear that the current theory again provides excellent agreement.

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1.5

× 10–4

1 h(t) (m)

(a)

0.5

(c) (b)

0

(d) t (s)

0

100

200

300

400

500

600

700

800

Fig. 2 Vapour layer thickness as a function of time and comparison with Biance [2] measurements, (a) numerical solution and (b) approximate solution with HTC 900W/m2 K, and evaporation at the top surface, (c) numerical solution with HTC 900W/m2 K and no evaporation at the top, (d) numerical solution with infinite HTC and no evaporation at the top

5 Conclusion Clearly for both problems the reduction of the full set of governing equations (Navier-Stokes, heat equations and a Stefan condition) to a set of three first order ordinary differential equations significantly simplified the problem. Comparison with experimental data clearly demonstrated the accuracy of the models. Although, when the Leidenfrost model was applied to very small droplets the agreement was not so good, presumably due to the fact that the vapour layer is no longer confined to a thin region. Key to the solution of these problems is the HBIM. Previously this method could be quite inaccurate. However recent developments and variants have significantly improved the accuracy, to the extent that for physically realistic parameter values it provides more accurate solutions than the second order perturbation expansion.

References 1. Bejan A.: Convection Heat Transfer. John Wiley & Sons, New York (1984) 2. Biance, A.-L., Clane, C., Quere, D.: Leidenfrost drops. Phys. Fluid. 15(6), 1632–1637 (2003) 3. Goodman T.R.: The heat-balance integral and its application to problems involving a change of phase. Trans. ASME, 80, 335–342 (1958) 4. Hill J.M.: One-Dimensional Stefan Problems: An Introduction. Chapman & Hall (1989) 5. Lister, J.R., Dellar P.J.: Solidification of pressure-driven flow in a finite rigid channel with application to volcanic eruptions J. Fluid Mech. 323, 267–283 (1996) 6. Mitchell S.L., Myers T.G.: Heat balance integral method for one-dimensional finite ablation. J. Thermophys. Heat Transf. 22(3), 508–514 (2008); DOI: 10.2514/1.31755 7. Mitchell S.L., Myers T.G.: Application of Standard and Refined Heat Balance Integral Methods to One-Dimensional Stefan Problems SIAM Rev. 52(1), 57–86 (2010) 8. Mitchell, S.L., Myers, T.G.: Application of the combined integral method to Stefan problems. Appl. Math. Model. 35(9), 4281–4294 (2011). doi:10.1016/j.apm.2011.02.049

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9. Moallemi M.K., Webb B.W., Viskanta. R.: An experimental and analytical study of close-contact melting. J. Heat Trans. 108(4), 894–899 (1986) 10. Myers, T.G., Charpin, J.P.F., Chapman, S.J.: The flow and solidification of a thin fluid film on an arbitrary three-dimensional surface. Phys. Fluids, 14, 2788–2803 (2002) 11. Myers, T.G., Charpin, J.P.F., Thompson, C.P.: Slowly accreting ice due to super cooled water impacting on a cold surface. Phys. Fluids, 14, 240–256 (2002) 12. Myers, T.G., Mitchell, S.L., Muchatibaya, G.: Unsteady contact melting of a rectangular crosssection material on a flat plate. Phys. Fluids 20(10) (2008); DOI: 10.1063/1.2990751 13. Myers, T.G.: Optimizing the exponent in the Heat Balance and Refined Integral Methods, Int. Comm. Heat Mass Trans. 36(2), 143–147 (2009); DOI:10.1016/j.icheatmasstransfer. 2008.10.013 14. Myers, T.G.: Optimal exponent heat balance and refined integral methods applied to Stefan problems. Int. J. Heat Mass Trans. 53 (2010); DOI:10.1016/j.ijheatmasstransfer.2009.10.04

On Modeling of Curved Jets of Viscous Fluid Hitting a Moving Surface Andriy Hlod

Abstract A jet of Newtonian fluid can fall from the oriented nozzle onto the moving surface in three regimes. A flow regime depends on the process parameters and is characterized by the dominant effect in the momentum transfer through the jet cross-section. To model the three jet flow regimes we describe the jet by the effects of inertia, longitudinal viscosity, and gravity. The key issue is to prescribe the boundary conditions for the jet orientation, which follow from the conservation of momentum for the dynamic jet. If the jet is under tension, the principal part of the conservation of momentum equation is of hyperbolic type, and the boundary conditions for the jet shape follow from the directions of characteristics. From this we find that the boundary conditions for the jet orientation are determined by the dominant effect in the momentum transfer through the jet cross-section, which can be due to inertia, or due to viscosity. This choice of boundary conditions allows us to find the solution to the steady jet model for all parameters, and partition the parameter space between the three jet flow regimes.

1 Introduction Growing interest in modeling of industrial processes (such as production of glass wool [19], high-temperature thermal isolation [1], and rotor spinning process [6], [14], [12]) requires development and study of models of a fluid jet hitting a moving surface under influence of external forces. These models are used to predict the jet shape, study jet stability, and describe the influence of the process parameters. A configuration, in which the fluid jet hits a moving surface, is the jet of viscous fluid falling onto a moving belt under gravity. In this process the three flow regimes

A. Hlod () Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513 5600 MB Eindhoven, The Netherlands e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 47, © Springer-Verlag Berlin Heidelberg 2012

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408

(a) Inertial (concave) jet.

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(b) Viscous-inertial (vertical) jet.

(c) Viscous (convex) jet.

Fig. 1 The three flow regimes of the jet falling onto the moving belt

of the jet fall are distinguished and characterized by the convexity or concaveness of the jet shape i.e. concave, vertical and convex; see Figure 1. For characterization, we consider only the major part of the jet, neglecting possible bending or unstable regions near the nozzle and the belt. A flow regime is determined by the process parameters i.e. fluid kinematic viscosity , flow velocity at the nozzle vnozzle , belt velocity vbelt and falling height L. Next, we describe each flow regime, and provide the naming based on the dominant effect in the momentum transfer through the jet cross-section; see Sect. 3 for details. The inertial jet has a concave shape compared to a ballistic trajectory, and occurs, among other possibilities. for small  and large vnozzle ; see Fig. 1(a). The viscous-inertial jet has a straight vertical shape, and among other occurs for large  and L, and small vnozzle ; see Fig. 1(b). The viscous jet has a convex shape, and occurs for large  and vbelt , and small vnozzle and L; see Fig. 1(c). In this paper we address a problem of modeling the three jet flow regimes describe above. In particular we would like to model the three regimes using a model as simple as possible. To do that we use a model of [12], which includes all the essential effects (inertia, viscosity and gravity) to qualitatively describe the jet fall. Such kind of jet model is often called the string jet model analogously to the elastic strings. The system is defined by the three dimensionless quantities and the nozzle orientation. Similar problems of the curved jets under the influence of gravity, or centrifugal and Coriolis forces but no moving surface, are described in [5, 16, 18–20]. Various aspects of the problem addressed here have been studied in [2, 12, 13, 17, 21]. In this paper we start with description of the jet model, which is partly solved and transformed into a first order ODE on an unknown domain with additional scalar unknown; see Sect. 2. Next, we extensively discuss and motivate our choice of boundary conditions for the jet shape by studying the conservation of momentum equation for the dynamic jet; see Sect. 3. In Sect. 4 we present the partitioning of the parameter space between the three flow regimes, and perform some simulations. In Sect. 5 we give some conclusions.

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2 Model The jet is modeled as a curve in 2D; see Fig. 2, which is parameterized by its arc length s with the origin at the nozzle s D 0 and s D send at the belt. Here send is the unknown jet length. The local coordinate system with the basis of the tangent and the normal vectors et ; en is constructed at each point at the jet. The angle between the tangent vector and the horizontal direction is . The nozzle orientation is given by the angle ˛nozzle . The flow velocity in the jet is v. The jet at the touchdown point has the same velocity as the belt v.send / D vbelt , and the flow velocity at the nozzle is v.0/ D vnozzle . The cross-sectional area of the jet is A . The system of equations describing a thin dynamical jet in 2D can be found [8, 22, 24]. It consists of the laws of conservations of momentum and mass, and for the stationary jet is A .rs vvs C v 2 rss / D 3.vs A rs /s C A g;

(1)

.A v/s D 0;

(2)

jrs j D 1:

(3)

Here,  is the fluid density and g is gravity. Next, we perform following manipulations 1. Find A from (2) and substitute into (1); 2. Introduce a new variable vs  D v  3 ; (4) v which stands for the momentum transfer through the jet cross-section; 3. Write the components of rs in terms of the angle  in the local coordinate basis et ; en (such that (3) is automatically satisfied);

anozzle

s=0 g r

L

en

vnozzle s

Q et

s = send vbelt

Fig. 2 The fall of the viscous jet onto the moving belt

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4. Scale the system as follow: the length is scaled with respect to 3=vnozzle and the velocity with respect to vnozzle ; 5. Replace the material coordinate s by the lagrangian , according to ds D v./d :

(5)

Then, (1) together with the boundary conditions become  D A sin./; A cos./ ;  v  D v  2; v  D

v.0/ D 1;

(6) (7) (8) (9)

v.end / D Dr; Z end sin..//v./d  D Re:

(10) (11)

0 3 Here A D 3g=vnozzle , the Reynolds number Re D vnozzle L=.3/, the draw ratio Dr D vbelt =vnozzle , and end is the result of coordinate transformation (5) of send . Thus, the system is described by the three dimensionless parameters and the parameter space P is

P WD f.A; Re; Dr/ W A > 0; Re > 0; Dr > 0g:

(12)

We solve (6)–(7), using following boundary conditions for  that are derived in Sect. 3 (13) .0/ D ˛nozzle ; .end / D 0

(14)

for the inertial and viscous jets respectively. For the viscous-inertial jet   =2:

(15)

Next, we partly solve (6–11), (13–15), and find  and  explicitly. After substituting the solutions for  and  into (8)-(11) we arrive at the system for v and end 8 p 2 if convex jet, < w A .  2end /=w2 C 1 v v 2 D p w C A if vertical jet, : v A2  2 C w2 C 2Aw sin.˛nozzle / if concave jet, v.0/ D 1;

(16)

(17)

On Modeling of Curved Jets of Viscous Fluid Hitting a Moving Surface

411

v.end / D Dr;

(18)

8 Z end A.end  / ˆ ˆ p v./d  if convex jet, ˆ ˆ 2 ˆ A .  2end / C w2 0 ˆ < Z end v./d  if vertical jet, Re D ˆ ˆ Z0 end ˆ ˆ A C w sin.˛nozzle / ˆ ˆ v./d  if concave jet, p : A2  2 C w2 C 2Aw sin.˛nozzle / 0

(19)

where w D .0/ is unknown as well. The system (16–19) is solved using the shooting method; see [12] for more details.

3 Boundary Conditions for the Jet Shape In this section we extensively discuss the boundary conditions for the jet shape. As it follows from [11], demanding the alignment of the jet at the nozzle with the nozzle orientation leads to a non-solvable system for certain model parameters; see Figs. 1(a) and 1(b) for illustration. Figures 1(a) and 1(b) suggest that the tangency at the surface should be prescribed as a boundary condition in the second case, but not in the first. In such a way depending on the situation one might need to demand tangency with the belt for the viscous jet, and alignment with the nozzle orientation for the inertial jet. In this section we derive a criterion how to prescribe boundary conditions for r. The approach presented here is applicable to the string jet model in different configurations; e.g. rotary spinning. To determine the boundary conditions for r, we write the dynamic conservation of momentum equation as a semi-linear partial differential equation for r of the form rt t C 2vrst C vrss D rt t C 2vrst C vrss D Qf;

(20)

where Qf D .3.A vs /s =A  vt  vvs /rs C g. According to the classification [3, p. 422–423] the equation (20) is hyperbolic when vs > 0, parabolic when vs D 0, and elliptic when vs < 0. The sign of the variable  plays a crucial role in this equation. The quantity A v D A v 2  3A vs

(21)

represents the net momentum flux (i.e. the momentum transfer per unit of time) through a cross-section due to inertia A v 2 and viscosity 3A vs . For a positive sign of , the momentum flux due to inertia is larger than that due to viscosity, and for a negative sign it is the other way around. Let us consider only the case vs > 0 throughout the jet, such that (20) is hyperbolic. We comment on the case vs < 0 in Remark 2 at the end of this section.

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For hyperbolic equations it is well-known that the number of boundary conditions at each boundary should be equal to the number of the characteristics directed into the domain at this point [10, p. 417] and [7,15]. An easy way to understand this follows from the concept of “domain of dependence” [3, p. 438-449]. The characteristic equation [4, p. 57] for (20) is z2  2vz C v 2  3vs D 0;

(22)

where z is the velocity of a characteristic curve. Equation (22) has the solutions z1 D v C

p p 3vs ; and z2 D v  3vs :

(23)

The directions of the characteristics of (20) depend on the sign of  as follows: 1. If  < 0 then z1 > 0 and z2 < 0, i.e. one characteristic points to the left and one to the right. 2. If  D 0 then z1 > 0 and z2 D 0, i.e. one characteristic points to the right and one is stationary. 3. If  > 0 then z1 > 0 and z2 > 0, i.e. both characteristics point to the right. In this problem the characteristic z1 is identified with the information about the jet position and the characteristic z2 is identified with the information about the jet orientation. Next, we will state the monotonic properties of .s/ for the steady jet. We will use these properties to determine the characteristic directions of the dynamic jet equations for r (20) at both jet ends. From this the boundary conditions for r directly follow. Now let us consider the steady jet. By taking the inner product of (1) with rs and using (3), we obtain s D .g; rs /=v: (24) In our configuration the term .g; rs /=v is always positive (follow from the explicit solution for ), and thus the function .s/ is strictly increasing. As a consequence there are three possibilities for the sign of .s/: 1. .s/ < 0 for s 2 Œ0; send . According to (21) viscous momentum flux dominates inertial flux everywhere in the jet. Because of that we call this flow regime viscous. 2. .s/ < 0 for s 2 Œ0; s  / and .s/ > 0 for s 2 .s  ; send , where .s  / D 0 and s  2 Œ0; send . According to (21), viscous momentum flux dominates at the nozzle and inertial flux dominates at the surface. Because of that we call this flow regime viscous-inertial. 3. .s/ > 0 for s 2 Œ0; send . According to (21), inertial momentum flux dominates viscous flux everywhere in the jet. Because of that we call this flow regime inertial. Thus, the sign of  provides a classification of the three flow regimes for the jet flow.

On Modeling of Curved Jets of Viscous Fluid Hitting a Moving Surface

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Next, we select the boundary conditions for r in case of the steady jet. To do this we treat the solution of the steady jet equations as the stationary solution of the dynamic jet equations. Doing this for (20), we obtain the boundary conditions for r from the characteristic directions of (20), which are determined by the sign of . Next, we treat the three jet flow regimes separately: 1. In the case of the viscous jet, both at the nozzle and at the surface one characteristic z2 points to the left and one z1 to the right; see Fig. 3(a). Therefore, we have to prescribe one boundary condition for r at each end. At the nozzle (s D 0) we prescribe the nozzle position, and at the surface we prescribe the tangency with the surface (s D send ). The latter provides the boundary condition for the jet orientation. 2. In the case of the viscous-inertial jet, at the nozzle one characteristic z2 points to the left and one z1 to the right, and two characteristics z1 and z2 point to the right at the surface; see Fig. 3(b). Therefore, we can only prescribe one boundary condition at the nozzle (s D 0), namely the nozzle position. The missing condition will be formulated in (25) further on. 3. In the case of the inertial jet, two characteristics z1 and z2 point to the right, both at the nozzle and at the surface; see Fig. 3(c). Therefore, we prescribe two boundary conditions at the nozzle, i.e the nozzle position and orientation. The latter condition is new and provides the boundary condition for the jet orientation. Hence, for the steady jet we appoint the nozzle position as a boundary condition for all the three flow regimes, the tangency with the surface for the viscous flow, and the nozzle orientation for the inertial flow. Remark 1. The method of prescribing the boundary conditions for r according to the direction of characteristic does not cover the situation if the jet is (partly) under compression. For the jet under compression the equation for r is elliptic and the method described above is not applicable. We extend the mechanism of prescribing boundary conditions for the steady jet fully or partly under compression and prescribe the boundary conditions for r according to the sign of  in the same way as described above. Note that for the viscous-inertial jet, we prescribe only one boundary condition for the second-order differential equation (1) for r. An extra condition follows from

z2

z2

z1

z2

(a)

Viscous jet.

z1

z2

z2

z1

z2

z1

(b)

Viscous-inertial jet.

z1

z1

(c)

Fig. 3 Characteristics directions for the three flow regimes in drag spinning

Inertial jet.

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.s  / D 0, expressing that at s D s  the jet should be aligned with the direction of the external force at this point, or, as follows from (1), rs D

1 g at s D s  : vs

(25)

The analysis of characteristics, as directions of information propagation, explains why the nozzle orientation influences the jet shape only in the inertial flow, and why the surface orientation influences the jet shape only in the viscous flow. In this respect, we see that: • In viscous flow, one characteristic points into the domain at the nozzle and one at the surface. Hence, information about the direction of the surface orientation influences the jet shape; see Fig. 4(a). Therefore, the surface orientation becomes relevant in viscous flow, whereas the nozzle orientation is irrelevant for the viscous jet. • In viscous-inertial flow, only one characteristic (at the nozzle) points into the domain. Therefore, no information about the nozzle orientation or the flow orientation at the surface influences the jet shape; see Fig. 4(b). Thus, in viscousinertial flow the nozzle and the surface orientations are irrelevant for the jet. The information about the orientation travels from the point s  towards the nozzle and the surface. • In inertial flow, the information about the jet shape travels from the nozzle to the surface. Therefore, not only the nozzle position but also the nozzle orientation is relevant for the jet; see Fig. 4(c). In addition, no information on the flow orientation travels back from the surface. Remark 2. The dynamic equation for r, (20), becomes elliptic when vs < 0, and in reality a steady jet might not exist [23]. In this situation the conservation of momentum (20) becomes elliptic for r. In case vs < 0, everywhere in the jet, one has to solve a Cauchy problem for the elliptic equation. Such kind of problems are expected to be ill-posed. Analogy can be made with Hadamard’s example [9, p. 234]. This example shows that a solution to a Cauchy problem for the Laplace equation does not continuously depends on the initial data in any Sobolev norm. It is possible to show that for some arbitrarily small initial data, the solution can be arbitrary large. Because of this the dynamic string model does not describe the jet.

position & angle

position

position s*

angle

angle

(a)

Viscous jet.

(b)

Viscous-inertial jet.

(c)

Fig. 4 Directions of information propagations for the three flow regimes

Inertial jet.

On Modeling of Curved Jets of Viscous Fluid Hitting a Moving Surface

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A way to overcome the ill-posed problem is to include the bending stiffness in the jet model.

4 Results In this section we present partitioning of the parameter space, and the jet shape evolution if one physical parameter changes. The regions of inertial Pinert , viscous-inertial Pv-i , and viscous Pvisc jets, and the borders between them, are illustrated in Fig. 5. Note, that the regions of the model parameters: Pinert , Pv-i and Pvisc do not intersect and cover the admissible parameter space P. Next, we study the evolution of the jet if one of the dimensional parameters varies as to change the flow type from viscous to viscous-inertial. For a reference configuration we take the dimensional parameters L D 1 cm,  D 0:047 m2=s, vbelt D 1:4 m=s, and vnozzle D 1 m=s, for which the jet is viscous. If we increase L, decrease , decrease vbelt , or increase vnozzle , eventually the jet flow changes from viscous to viscous-inertial. Changes of the jet shape while only one of the dimensional parameters L, , vbelt , or vnozzle vary as described above are shown in Figs. 6(a), 6(b), 6(c), and 6(d),

3

2

visc

Dr 1

0

v-i

0.

inert

1. A* A

3 2

3. 1

4. 0

Re

Fig. 5 Parameter regions for three flow regimes Pinert , Pv-i and Pvisc

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y (cm)

y (cm)

3.5

1.0

3.0

0.8

2.5 2.0

0.6

1.5

0.4

1.0 0.2

0.5

x (cm)

x (cm) 0.5

1.0

1.5

2.0

0.5

2.5

(a)

1.0

1.5

2.0

2.5

(b)

Viscous jet shapes for different L: 1, 1.7, 2.2, and 3.5 cm. The shape approaches the vertical as L increases.

Viscous jet shapes for different n: 2 0.047, 0.026, 0.015, and 0.012 m / s. The shape approaches the vertical as n decreases.

y (cm)

y (cm) 1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 x (cm) 0.5

(c)

1.0

1.5

2.0

Viscous jet shapes for different nbelt: 1.4, 1.21, 1.11, and 1.08 m / s. The shape approaches the vertical as nbelt decreases.

2.5

x (cm) 0.5

1.0

1.5

2.0

2.5

(d)

Viscous jet shapes for different nnozzle : 1, 1.16, 1.24, and 1.26 m / s. The shape approaches the vertical as nnozzle increases.

Fig. 6 Shapes of the viscous jet for different values of L, , vbelt and vnozzle . The reference values are L D 1 cm,  D 0:047 m2 =s, vbelt D 1:4 m=s, and vnozzle D 1 m=s

respectively. If the point .A; Re; Dr/ approaches the boundary of Pvisc , the jet shape becomes vertical. If .A; Re; Dr/ is very close to the boundary of Pvisc the jet shape is almost vertical, except for the small region near the belt where the jet rapidly bends to the horizontal belt direction. To illustrate the change of flow from inertial to viscous-inertial, while only one of the parameters L, , vbelt , and vnozzle varies, we take the reference values L D 30 cm,  D 0:2 m2 =s, vbelt D 2 m=s, and vnozzle D 1:5 m=s. If we decrease L, increase , increase vbelt , or decrease vnozzle eventually the jet flow changes from inertial to viscous-inertial. Changes of the jet shape for ˛nozzle D =4, while only one of the dimensional parameters L, , vbelt , or vnozzle varies as described above are shown in Figs. 7(a), 7(b), 7(c), and 7(d), respectively. If the point .A; Re; Dr/ approaches the boundary of Pinert , the jet shape becomes more vertical. If .A; Re; Dr/ is very close to the boundary of Pinert the jet shape is almost vertical except for the small region near the belt where the jet rapidly bends from the nozzle direction to an almost vertical one.

On Modeling of Curved Jets of Viscous Fluid Hitting a Moving Surface y (cm) 30

30

25

417

y (cm)

25

20

20

15

15

10

10

5

5

2

4

6

x

8

(cm)

x

(cm)

x

(cm)

2 4 6 8 (b) Inertial jet shapes for different n : 0.2, 0.26 ,0.3, and 0.32 m2 / s. The shape approaches the vertical as n increases.

(a) Inertial jet shapes for different L : 30, 18, 13, and 12 cm. The shape approaches the vertical as L decreass.

y (cm)

y (cm) 30

30 25

25 20 20 15 15 10 10 5 5 2

4

6

8

x

(c) Inertial jet shapes for different

vbelt: 2, 2.57, 2.86, and 2.95 m / s. The shape approaches the vertical as vbelt increases

(cm)

2

4 6 8 (d) Inertial jet shapes for different vnozzle: 1.5, 1.4, 1.36, and 1.34 m / s. The shape approaches the vertical as vnozzle decreases.

Fig. 7 Shapes of the inertial jet for different values of L, , vbelt , vnozzle . The reference values are L D 30 cm,  D 0:2 m2 =s, vbelt D 2 m=s, and vnozzle D 1:5 m=s. The nozzle orientation is ˛nozzle D =4

5 Conclusions In this paper we present a model describing the three flow regimes of the jet of viscous fluid falling onto the moving surface. The model includes effects of inertia, longitudinal viscosity, and gravity, and describes the jet for all admissible

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parameters. The key issue is the boundary condition for the jet orientation, which follow from the conservation of momentum for the dynamic jet. This equation is of hyperbolic type if the jet is under tension. Thus, the number of characteristics pointing inside the domain at each jet end is equal to the number of boundary conditions for r at this jet end. From this follows that the choice of the boundary conditions depends on the dominant effect in the momentum transfer, which can be due to inertia, or viscosity. The way to prescribe the boundary conditions for the jet orientation based on the dominant effect in the momentum transfer follows from the mathematical properties of the conservation of momentum equation. The same way of prescribing the boundary conditions is suggested by the experimental observations of the three flow regimes, which corresponds to the model results. All these findings lead to the main conclusion of this paper, namely that the boundary conditions for r proposed in this paper are the natural one to use for the string type jet models, where the jet shape is governed by the equations of hyperbolic type. Acknowledgements The author would like to acknowledge Teijin Aramid, a part of the Teijin group of companies, for providing the experimental equipment and valuable suggestions for experiments.

References 1. Breward, C., Dyson, R., Edwards, C., Metcalfe, P., Please, C., Zyskin, M.: Modelling of melt on spinning wheels. Study group report, European Study Group with Industry 49th ESGI (Oxford 29/3/2004–4/4/2004) (2005); Thermal Ceramics UK 2. Chiu-Webster, S., Lister, J.R.: The fall of a viscous thread onto a moving surface: a ‘fluidmechanical sewing machine’. J. Fluid Mech. 569, 89–111 (2006) 3. Courant, R., Hilbert, D.: Methods of mathematical physics. Vol. II. Wiley Classics Library. Wiley, New York (1989); Partial differential equations, Reprint of the 1962 original, A WileyInterscience Publication 4. Davis, J.L.: Mathematics of Wave Propagation. Princeton University Press, Princeton, NJ (2000) 5. Decent, S.P., King, A.C., Wallwork, I.M.: Free jets spun from a prilling tower. J. Eng. Math. 42(3), 265–282 (2002) 6. den Decker, P., Knoester, H., Meerman, H., Dekker K. van Horssen, W., Vuik, C., Wesswling, P., Prokert, G., van ’t Hof, B., van Beckum, F.: The rotor spinning process for fibr production. In: Proceedings of the 48th European Study Group Mathematics with Industry (Delft, 15–19 March 2004), pp. 35–48 7. Dubois, F., LeFloch, P.: Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. Differ. Equat. 71(1), 93–122 (1988) 8. Entonov, V.M., Yarin, A.L.: Dynamical equation for a liquid jet. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza (5), 11–18 (1980) 9. Evans, L.C.: Partial differential equations, Graduate Studies in Mathematics, Vol. 19. American Mathematical Society, Providence, RI (1998) 10. Godlewski, E., Raviart, P.A.: Numerical approximation of hyperbolic systems of conservation laws, Applied Mathematical Sciences, Vol. 118. Springer, New York (1996)

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11. G¨otz, T., Klar, A., Unterreiter, A., Wegener, R.: Numerical evidence for the non-existence of stationary solutions of the equations describing rotational fiber spinning. Math. Models Meth. Appl. Sci. 18(10), 1829–1844 (2008) 12. Hlod, A.: Curved jets of viscous fluid: Interactions with a moving wall. Ph.D. thesis, Eindhoven University of Technology (2009) 13. Hlod, A., Aarts, A., v.d. Ven, A., Peletier, M.: Mathematical model of falling of a viscous jet onto a moving surface. Eur. J. Appl. Math. 18, 659–677 (2007) 14. Kolk, E.: Modelling of melt on spinning wheels. Interim report, TU Delft (2005) 15. Kreiss, H.O.: Initial boundary value problems for hyperbolic systems. Comm. Pure Appl. Math. 23, 277–298 (1970) 16. Marheineke, N., Wegener, R.: Asymptotic model for the dynamics of curved viscous fibres with surface tension. J. Fluid Mech. 622(-1), 345–369 (2009) 17. Morris, S.W., Dawes, J.H.P., Ribe, N.M., Lister, J.R.: Meandering instability of a viscous thread. Phys. Rev. E (Statistical, Nonlinear, and Soft Matter Physics) 77(6), 066218 (2008) 18. Panda, S.: The dynamics of viscous fibers. Ph.D. thesis, Technische Universit¨at Kaiserslautern (2006) 19. Panda, S., Marheineke, N., Wegener, R.: Systematic derivation of an asymptotic model for the dynamics of curved viscous fibers. Math. Meth. Appl. Sci. 31(10), 1153–1173 (2008) 20. Parau, E.I., Decent, S., Simmons, M., Wong, D.C.Y., King, A.C.: Nonlinear viscous liquid jets from a rotating orifice. J. Eng. Math. 57(2), 159–179 (2007) 21. Ribe, N.M., Lister, J.R., Chiu-Webster, S.: Stability of a dragged viscous thread: Onset of “stitching” in a fluid-mechanical “sewing machine”. Phys. Fluids 18(124), 105–1–8 (2006) 22. Roos, J.P., Schweigman, C., Timman, R.: Mathematical formulation of the laws of conservation of mass and energy and the equation of motion for a moving thread. J. Eng. Math. 7(2), 139– 146 (1973) 23. Taylor, G.: Instability of jets, threads, and sheets of viscous fluid. In: Proceedings of the 12th International Congress of Applied mechnaics (Stanford, 1968), pp. 382–388. Springer, New York (1969) 24. Yarin, A.L.: Free liquid jets and films: hydrodynamics and rheology. Interaction of Mechanics and Mathematics Series. Longman Scientific & Technical, Harlow (1993)

Air Elimination in Milk Michael Devereux and William Lee

Abstract This paper presents the work done to model a novel design for accurate volume measurement of milk. The new design proposes the addition of an air elimination vessel to the current milk pumping systems. This is an extra tank between a farmer’s milk storage tank and the pumping storage system on a lorry designed to collect the milk from farms. The purpose of the air elimination vessel is to allow a pool of milk to accumulate so that air bubbles entrained with the milk can be removed and pumped out by a separate pump for air. The paper first discusses the flow of milk inside the air elimination vessel. This is modelled by a system of partial differential equations similar to the St Venant equations but modified for a cylindrical environment. The paper then discusses bubble formation and flow and predicts the level of milk required in the air elimination vessel for bubbles of air to rise and be pumped out. This is modelled by a single ordinary differential equation derived from momentum conservation.

1 Introduction The aim of this project is to increase the measurement accuracy of the volume of milk being pumped from a farmer’s tank to a storage tank on a lorry which travels from farm to farm collecting milk. Inaccuracies are introduced to the volume measurement by air, in the form of bubbles and foam, being sucked through the volume measurement device and registering as milk. The very simplified schematic given in Fig. 1 shows that when the level of milk in the farmer’s tank is less than the width of the pipe draining it, air from the atmosphere will also be pumped. This air will create bubbles and foam which cause the inaccuracy in measurement.

M. Devereux ()  W. Lee MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland e-mail: [email protected]; [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 48, © Springer-Verlag Berlin Heidelberg 2012

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Fig. 1 Generic farmer’s tank and pump schematic

Fig. 2 Proposed design

The proposed solution to this problem is the use of an air elimination vessel (AEV) which will remove any air sucked in with the milk using a second pump. The problem consists of a number of parts: the simulation of the entire system to show that the design can empty the milk from the farmer’s tank, analysis on the flow of milk inside the AEV, and the formation of air bubbles inside the AEV and their removal.

1.1 Proposed Solution The design for the AEV is confidential so Fig. 2 is a simplified schematic for the proposed solution. The AEV functions by allowing a pool of milk to accumulate which will allow any air bubbles trapped in the milk to rise and escape. This is achieved by a number of independent pumps and valves controlling the flow of milk and air from the AEV. The vessel contains sensors for measuring the pressure of air and the level of milk. It has been verified that this design will successfully pump milk from the farmer’s tank, through the AEV and into the storage tank. This analysis will not be included in this paper. We will instead focus on the flow of milk inside the AEV and the formation of bubbles.

2 Flow of Milk Inside AEV We now consider the flow of milk inside the AEV as it enters from the pipe at the top. Figure 3 shows the pipe attached tangentially to the top of the AEV. Hence the milk circulates around the inside wall of the AEV and forms a shallow layer of milk. The flow is modelled by a continuity equation (1), a linear momentum equation (2) and an angular momentum equation (3):

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Fig. 3 Milk flow inside the AEV. R is the radius of the AEV, Q is the volumetric flow rate of milk, ui n is the inflow velocity, g is acceleration due to gravity, h is the thickness of the layer of milk on the inside wall of the AEV, v and vz are the azimuthal and axial components of the milk velocity

1 @ @h @ D .hv /  .hvz / C q.z; /; @t r @ @z

(1)

q 1 @.hv vz / @.hvz2 / 1 @.v2 h2 / @ .hvz / D    g h  C v2 C vz2 vz  ; (2) @t r @ @z 2r @z q @ 1 @.hv2 / @hv vz 1 @.v2 h2 / .hv/ D    C v2 C vz2 v  C q.z; /ui n ; @t R @ @z 2R2 @ (3) where C is the surface friction coefficient for the fluid on the inside of the AEV and q.z; / is the flux of milk into the AEV. The case of hR is used in deriving (3). These equations are based on the St Venant equations [2] but specialised to a cylindrical geometry. The main difference is the inclusion of a centrifugal force term. The system of equations is simplified by neglecting time and angular dependance to give d  .h vz / C q.z/ D 0; (4) dz d 1 d 2 2 .h vz2 /  gh C C vz2  .u h / D 0: (5) dz 2r d z i n This simplification can be made as it is expected that a steady state thin film of milk circulating around the inside walls of the AEV will occur for a continuous stream of Z Q milk from the farmer’s tank. Solving (4) for h.z/, and assuming q.z/ d z D 2 r where Q is the steady flux from the pipe and r the radius of the pipe, results in an expression for h.z/ which can be substituted into (5) to produce a single ODE 

Q g Q dvz Q2 u2in 1 dvz C D 0: C C vz2 C 2 r d z 2 r vz 4 2 r 3 vz3 d z

(6)

424

M. Devereux and W. Lee Vertical Velocity versus Milk Position inside AEV

Position (metre)

0

–0.5

–1

–1.5

–2

–1.8

–1.7

–1.6

–1.5

–1.4

–1.3

–1.2

–1.1

–1

Velocity (metre / second)

Fig. 4 Solution of (6) for velocity of milk as it flows down side of AEV

Figure 4 is a plot of the numerical solution of (6). A nonzero initial condition is used as the assumption of the shallow layer of milk swirling around the inside of the AEV breaks down due to the rapid flow of milk from the pipe from the farmer’s tank. The plot shows that the milk flowing down the sides reaches a terminal velocity. This is due to the gravity and friction terms in (6) balancing.

3 Bubble Formation In the previous section we obtained a terminal velocity for the milk flowing down the inside walls of the AEV. We will now use this to calculate how deep bubbles will be entrained into the pool of milk inside the AEV. From analysis done on the pumping system, it was found that a steady state occurs inside the AEV when the depth of milk is 60 cm. The depth these bubbles travel into the pool of milk is required to determine if they will have time to rise and be removed from the milk or sink all the way to the bottom and be pumped out with the milk. Clanet and Lasheras [1] show that when the velocity of the fluid containing bubbles decreases, due to buoyancy and drag, to 0:22 m s1 or less, the bubbles will start to rise. We need to determine the depth at which the entrained bubbles reach this velocity. We do this by considering conservation of momentum. Figure 5 shows the formation of bubbles by the entrainment of air. The positive direction of z has been reversed for convenience. Milk flows down the inside wall by gravity.

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Fig. 5 Bubble formation as milk flows down the inside wall of the AEV

Bubble Velocity (ms−1)

2

Bubble Velocity versus Depth

1.5

1

0.5

0

0.22 ms–1

0

0.2

0.4 Depth (m)

0.6

0.8

Fig. 6 Bubble velocity versus bubble depth

The bubbly liquid loses momentum to entrained fluid described by the jet angle ˛, friction with the wall and buoyancy. This is modelled by u2 d 2 .u / D dx

C

 C tan.˛/  gˇ0 h0 ; h0 C xtan.˛/

2

(7)

where h0 is the height of the thin film of milk, u is the velocity of the jet of milk and ˇ0 is the bubble volume fraction. This was derived using similar analysis to the derivation of the model used by [1] but making assumptions appropriate to a planar jet instead of a circular jet. This equation relates the velocity of the bubbly fluid in the pool of milk to its depth. The initial condition used to solve this is the terminal velocity the milk reaches as shown in Fig. 4. This was solved numerically using MATLAB. A plot of the solution is given below in Fig. 6. This shows bubble velocity decreasing with depth as expected. However, it achieves the velocity 0:22 m s1 at a much greater depth than expected. As the height of the milk reaches a steady state

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at approximately 0:6 m, as predicted by simulation of the entire system, and the bubbles here reach this velocity at approximately 0:8 m, the bubbles may not have time to escape and will instead be sucked out by the pump at the base of the AEV.

4 Conclusions First we consider the flow of milk in the AEV. A system of three PDEs is derived, simplified and solved to calculate a terminal velocity for the milk flowing down the sides of the AEV. We use this terminal velocity in the Sect. 3 when we calculate the depth at which these bubble sink to. It was shown depth is approximately 0:8 m. This is greater then the depth of milk in the AEV and would allow air to be pumped with the milk from the AEV. This suggests a redesign of the AEV is required. Due to the confidential nature of the AEV design, this redesign cannot be outlined here. Acknowledgements All contributors would like to thank Archie Hamilton (Piper Systems) for introducing the problem and assisting in answering questions and formulating the model. We also acknowledge support of the Mathematics Applications Consortium for Science and Industry (www.macsi.ul.ie) funded by the Science Foundation Ireland Mathematics Initiative Grant 06/MI/005. Michael Devereux’s postgraduate fellowship is funded by the Irish Research Council for Science, Engineering and Technology, IRCSET.

References 1. Clanet, C., Lasheras, J.: Depth of penetration of bubbles entrained by a plunging water jet. Phys. Fluid. 9(7), 1864–1866 (1997) 2. Toro, E.F.: Shock-Capturing Methods for Free-Surface Shallow Flows. Wiley, New York (2001)

Quantum Navier–Stokes Equations Ansgar Jungel ¨ and Josipa-Pina Miliˇsi´c

Abstract Compressible Navier–Stokes models for quantum fluids are reviewed. They are derived from a collisional Wigner equation by a moment method and a Chapman–Enskog expansion around the quantum equilibrium. Introducing a new velocity variable, the barotropic quantum Navier–Stokes model can be reformulated as a viscous quantum Euler system, which possesses a new Lyapunov (energy) functional. This functional provides a priori estimates which are exploited to prove the global-in-time existence of weak solutions for general initial data. Furthermore, new numerical results for the isothermal model are presented.

1 Introduction Quantum fluid modeling has become very attractive due to novel experimental discoveries in Bose–Einstein condensation. Recently, disordered superfluids and, in particular, the interplay between superfluidity and the onset of dissipative processes has been investigated [10]. Typically, the dynamics of the condensate is modeled by a nonlinear Schr¨odinger equation involving dissipative terms [3]. The normal component of the Bose–Einstein gas at low temperature may be described by using kinetic equations, such as the Wigner equation [1]. This approach has the advantage that dissipation can be included in a rather natural way via collision operators on the right-hand side of the Wigner equation. Examples, used in semiconductor modeling, are the Caldeira–Leggett scattering operator, the Fokker–Planck operator, or A. J¨ungel () Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstr. 8-10, 1040 Wien, Austria e-mail: [email protected] J.-P. Miliˇsi´c Department of Applied Mathematics, University of Zagreb, Unska 3, 10000 Zagreb, Croatia e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 49, © Springer-Verlag Berlin Heidelberg 2012

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BGK-type operators (named after Bhatnagar, Gross, and Krook); see [22] for a detailed description. The hydrodynamic dynamics of a superfluid may be modeled by the Madelung equations, which are derived from the Schr¨odinger equation via the Madelung transform [30]. The model consists of the Euler equations for a potential flow involving the quantum Bohm potential as a third-order derivative. The quantum Euler or quantum hydrodynamic equations have been also used to describe the carrier transport in open quantum systems such as semiconductor heterostructures and tunneling diodes [16, 22]. An alternative derivation is based on the Wigner equation by employing the moment method and the maximum entropy principle [13]. Quantum hydrodynamic models have the advantages that they allow for an efficient numerical discretization and that macroscopic boundary conditions in open systems may be imposed. In this review, we summarize recent progress in the derivation and mathematical analysis of certain dissipative quantum hydrodynamic equations, namely quantum Navier–Stokes models. Such systems have been already proposed in the 1960s [19]. The first derivation from a Wigner–BGK equation has been performed by Brull and M´ehats [9] for constant temperature. In [27], the full quantum Navier–Stokes system, including the energy equation, has been derived and numerically solved. The existence of weak solutions to the barotropic model has been shown in [14, 20, 23] (see Theorem 2 below). In the physical literature, quantum Navier–Stokes systems are typically motivated from the classical model by using a chemical potential obtained from the Thomas–Fermi–Dirac–Weizs¨acker density functional theory (see, e.g., [32]). Clearly, in this situation, the viscous correction is equal to the classical one and often, constant viscosity coefficients are assumed. The derivation from the Wigner– BGK equation leads to nonconstant viscosity coefficients depending on the particle density and temperature (see Theorem 1 below). Density-dependent viscosities may generate vacuum, which leads to mathematical difficulties in the analysis of the equations (see, e.g., [5, 29]). This review is organized as follows. In Sect. 2, following [27], the derivation of the quantum Navier–Stokes system from a Wigner–BGK equation using a Chapman–Enskog expansion of the Wigner function around the quantum equilibrium is sketched. Section 3 is concerned with the analysis of the compressible barotropic model (density-dependent pressure function). The existence analysis is based on the formulation of the model as a viscous quantum Euler system via a new variable, the so-called effective velocity, first used in viscous Korteweg models [6]. Finally, in Sect. 4, the isothermal equations (constant temperature) are numerically discretized by central finite differences in one space dimension, and new numerical simulations for a tunneling diode are presented.

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2 Derivation The quantum Navier–Stokes equations are derived from a Wigner–BGK model using the moment method and a Chapman–Enskog expansion. Degond et al. [12] have proposed the Wigner–BGK equation wt C p  rx w C ŒV w D

1 .M Œw  w/; ˛

.x; p/ 2 R3  R3 ; t > 0;

(1)

where w.x; p; t/ is the Wigner function in the phase-space variables .x; p/ and time t > 0, and ˛ > 0 is the scaled mean free path. The potential operator ŒV  is a pseudo-differential operator modeling the influence of the electric potential V D V .x; t/; see [22] for a definition. The kernel of the potential operator is of quantum mechanical nature and contains the scaled Planck constant ". The righthand side of (1) describes a relaxation process towards the quantum equilibrium M Œw. The equilibrium is defined as the formal maximizer of the von-Neumann entropy (or R quantum free energy) under the constraint that its moments, i.e. the integrals R3 M Œw.p/dp for some vector-valued function .p/, are the same as those for the given function w. This concept has been introduced by Degond and Ringhofer [13]. Denoting the Lagrange multipliers by .x; t/, the quantum equilibrium reads as M Œw.x; p; t/ D Exp..x; t/  .p//; where the so-called quantum exponential is given by Exp.w/ D W .exp W 1 .w//, W is the Wigner transform, W 1 its inverse, and exp is the operator exponential. When the collision operator Q.w/ D .M Œw  w/=˛ conserves mass, we prescribe the local particle density, and the quantum equilibrium becomes M Œw D Exp.A.x; t/  jpj2 =2/ for some Lagrange multiplier A.x; t/. The existence and uniqueness of this maximizer has been proved in [31] in a one-dimensional setting. When scattering conserves mass, momentum, and energy (thus, we take .p/ D .1; p; jpj2 =2/), we have   jp  v.x; t/j2 ; M Œw D Exp A.x; t/  2T .x; t/ where now A, v, and T are Lagrange multipliers. The expressions for the equilibrium distributions look similar to the corresponding classical Maxwell distributions [28]. However, M Œw is a nonlocal operator, which expresses the nonlocal nature of quantum mechanics, and the Lagrange multipliers do not correspond to the moments as in the classical model. For instance, v equals the mean velocity only up to terms of order O."2 /. We derive macroscopic equations by multiplying the Wigner equation (1) by the weight vector .p/ D .1;Rp; 12 jpj2 /. To simplify the notation, we introduce the notation hf .p/i D .2"/3 R3 f .p/dp, where f .p/ is

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a function. The collisions are assumed to conserve mass, momentum, and energy, h.M Œw  w/.p/i D 0. Then the moment equations become @t hwi C divx hpwi C hŒV wi D 0; @t hpwi C divx hp ˝ pwi C hpŒV wi D 0; @t h 12 jpj2 wi C divx h 21 pjpj2 wi C h 21 jpj2 ŒV wi D 0; where p ˝ p denotes a matrix with components pj pk (j; k D 1; 2; 3), governing the evolution of the particle density n, the momentum nu, and the energy density ne, defined by n D hwi; nu D hpwi; ne D h 21 jpj2 wi: The variable u D .nu/=n is the macroscopic velocity and e D .ne/=n the macroscopic energy. The integrals involving the potential can be expressed in terms of the moments n, nu, and ne (see [27] for details). It remains to compute the higherorder moments hp˝pwi and h 21 pjpj2 wi. For this, we employ the Chapman–Enskog expansion w D M Œw C ˛g. Introducing the quantum stress tensor P D h.p  u/ ˝ .p  u/M Œwi and the quantum heat flux q D h 21 .p  u/jp  uj2 M Œwi, a straight-forward computation leads to the following moment equations: nt C divx .nu/ D 0; .nu/t C divx .P C nu ˝ u/  nrx V D ˛divx hp ˝ pgi;   .ne/t C divx .P C neI/u C divx q  nu  rx V D ˛divx h 21 pjpj2 gi; where I is the unit matrix in R33 . In order to calculate the moments of g, we take advantage of the simple structure of the collision operator, allowing us to specify g explicitly. Indeed, inserting the Wigner equation and Chapman–Enskog expansion, we find that g D .M Œw  w/=˛ D M Œwt  p  rx M Œw  ŒV M Œw C O.˛/, where O.˛/ contains terms of order ˛. More explicit expressions are obtained by expanding the moments of M Œw in powers of the squared scaled Planck constant "2 . "2 The quantum heat flux becomes q D  24 n.x u C 2rx divx u/ C O."4 /, and the "2 quantum stress tensor expands according to P D nT I  12 nrx2 log n C O."4 /, 2 where rx log n is the Hessian of log n, Furthermore, a tedious computation shows that (see [27]) ˛divx hp ˝ pgi D ˛divx S;

5 ˛divx h 21 pjpj2 gi D ˛divx .S u/ C nT rx T; 2

where S D 2nTD.u/  23 nT divx u I C O."2 C ˛/ can be interpreted as a viscous stress tensor. Here, D.u/ D .rx uCrx u> /=2. The term 52 nT rx T is the Fourier heat term, and it adds to the quantum heat flux. This shows the following result [27].

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Theorem 1. Assume that A.u/ D .ru  ru> /=2 D O."2 / and r log T D O."2 /. Then, up to terms of order O.˛ 2 C ˛"2 C "4 /, the moment equations of the Wigner equation read as nt C div.nu/ D 0;

(2)

"2 div.nr 2 log n/  nrV D ˛div S; (3) 12   "2   .ne/t C div .ne C nT /u  div n.r 2 log n/u C div q0  nu  rV D ˛div.S u/; 12

.nu/t C div.nu ˝ u/ C r.nT / 

where q0 D q C 52 nT rT is the total heat flux and S D 2nTD.u/  23 nT divx u I is the viscous stress tensor. For later use, we remark that the third-order quantum term can be written equivp p alently as a self-interacting force term, involving the Bohm potential  n= n, div.nr 2 log n/ D 2nr

 pn  : p n

(4)

When the collisions conserve mass and momentum only, the quantum equilibrium becomes M Œw D Exp.A  jp  vj2 =2/. In this situation, a Chapman–Enskog expansion has been carried out by Brull and M´ehats [9]. They obtain (2)–(3) with T D 1 and S D 2nD.u/.

3 Analysis System (2)–(3) with T D 1 possesses a surprising property which has been exploited in [23] to prove the existence of solutions. More precisely, we consider the system nt C div.nu/ D 0;

x 2 Td ; t > 0;

(5)  p   n "  nrV D 2˛div.nD.u//; (6) .nu/t C div.nu ˝ u/ C rp.n/  nr p 6 n 2

n.; 0/ D n0 ;

.nu/.; 0/ D n0 u0

in Td ;

(7)

where Td is the d -dimensional torus (d  3). The function p.n/ D n with   1 is the pressure. Compared to (3), the quantum term is reformulated using (4). In the treatment of (5)–(7), we need to overcome several mathematical difficulties. The first problem lies in the strongly nonlinear third-order differential operator and the dispersive structure of the momentum equation. In particular, as the maximum principle is not applicable, it is not clear how to obtain the positivity

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or nonnegativity of the particle density. In the literature, some ideas have been developed to overcome this problem. For instance, some artificial diffusion has been added to the mass equation such that the maximum principle can be applied [15]. Another idea is to introduce an additional pressure with negative powers of the density, which allows one to derive an L1 bound for 1=n [7]. The second problem is the density-dependent viscosity .n/ D ˛n which degenerates at vacuum. In fact, most results for the Navier–Stokes equations in the literature are valid for constant viscosities .n/ D ˛ only, since this enables one to derive H 1 estimates for the velocity. Recently, some works have been concerned with density-dependent viscosities, see, e.g., [5, 29] and references therein. The third problem is the lack of suitable a priori estimates. Indeed, let us define the energy of (5)–(6) by the sum of the kinetic, internal, and quantum energies, Z E"2 .n; u/ D

n Td

2

juj2 C H.n/ C

"2 p 2  jr nj dx; 6

(8)

where H.n/ D n =.  1/ if  > 1 and H.n/ D n.log n  1/ if  D 1. A formal computation shows that, without electric field rV D 0, dE"2 .n; u/ C ˛ dt

Z njD.u/j2 dx D 0: Td

p This provides an H 1 estimate for n, but this seems to be insufficient to obtain p compactness for (an approximate sequence of) r n needed to define the quantum term in a weak or distributional sense. Our main idea to solve these problems is to transform the quantum Navier–Stokes system by means of the so-called effective velocity w D u C ˛r log n;

(9)

The term ˛r log n has been called in [19] the “kinematical quasivelocity”. A computation shows [23] that the system (5)–(6) can be equivalently written as nt C div.nw/ D ˛n;

 p   n "0  nrV D ˛.nw/; .nw/t C div.nw ˝ w/ C rp.n/  nr p 6 n

(10) (11)

where w0 D u0 C ˛r log n0 and "0 D "2  12˛ 2 . This formulation has two advantages. The first advantage is that it allows for an additional energy estimate if "2 > 12˛ 2 . Indeed, if rV D 0, we compute dE"0 .n; w/ C ˛ dt

Z Td

  "0 njrwj2 C H 0 .n/jrnj2 C njr 2 log nj2 dx D 0: 12

(12)

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The inequality [24] Z

p jr 2 nj2 dx  C

Td

Z njr 2 log nj2 dx

(13)

Td

p with some constant C > 0 provides an L2loc .0; 1I H 2 .Td // bound for n. This bound is the key argument of the global existence analysis. The second advantage is that we may apply the maximum principle to the parabolic equation (10) to deduce strict positivity of the density n if n0 is strictly positive and the velocity w is smooth. The viscous quantum Euler model (10)–(11) is of interest by itself. Indeed, it has been derived from a Wigner–Fokker–Planck equation by a moment method [18]. The viscous terms ˛n and ˛.nw/ arise from the moments of the Fokker– Planck collision operator. This operator also provides the momentum relaxation term nw= to the right-hand side of the momentum equation, where > 0 is the relaxation time. For existence results for the viscous quantum Euler system, we refer to [11, 17, 18, 26]. Neglecting the viscous terms (˛ D 0), the two systems (5)–(6) and (10)–(11) reduce to the so-called quantum hydrodynamic model, see, e.g., [16, 22]. First mathematical results have been concerned with the local existence of solutions or the global existence of near-equilibrium solutions. For the stationary problem, only the existence of “subsonic” solutions has been achieved so far [21]. Recently, the global existence of weak transient solutions for general initial data has been shown by Antonelli and Marcati [2]. Interestingly, the effective velocity (9) has been used also in related models. First, Bresch and Desjardins employed it to derive new entropy estimates for viscous Korteweg-type and shallow-water equations [6]. Brenner [4] suggested the modified Navier–Stokes model nt C div.nw/ D 0;

.nu/t C div.nu ˝ w/ C rp.n/ D div S:

The variables u and w are interpreted as the volume and mass velocities, respectively, and they are related by the constitutive equation u  w D ˛r log n with a phenomenological constant ˛ > 0. The variable nw D nu C ˛rn was employed in [26] to prove the existence of solutions to the one-dimensional stationary viscous quantum Euler problem with physical boundary conditions. The strategy of the existence proof for (5)–(7) is as follows. First, the viscous quantum Euler system (10)–(11) is approximated by a projection of the infinitedimensional momentum equation onto a finite system of ordinary differential equations on a Faedo-Galerkin space with dimension N , following the ideas of Feireisl in [15] and generalizing the one-dimensional approach in [17]. We need a second approximation parameter ı by adding the term ı.w  w/ to the right-hand side of (11), which allows one to derive an H 1 estimate for w. The global existence of approximate solutions .nı ; wı / follows from the energy estimate (12), which also provides some Sobolev estimates independent of .N; ı/. The limits N ! 1 and ı ! 0 then give the following existence result; for a proof we refer to [23].

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Theorem 2. Let d  3, ", ˛ > 0, p.n/ D n with  > 3 if d D 3 and   1 if d D 2, rV 2 L1 .0; 1I L1 .Td //, and .n0 ; u0 / is such that n0  0 and E" .n0 ; u0 C ˛r log n0 / is finite. Then there exists a weak solution .n; u/ to (5)–(7) with the regularity p 1 d 2 2 d n 2 L1 loc .0; 1I H .T // \ Lloc .0; 1I H .T //;

n  0 in Td ;

1  d 2 1;3 .0; 1I L2 .Td // \ L1 .Td //; n 2 Hloc loc .0; 1I L .T // \ Lloc .0; 1I W p 2 d nu 2 L1 nu 2 L2loc .0; 1I W 1;3=2 .Td //; loc .0; 1I L .T //;

njruj 2 L2loc .0; 1I L2 .Td //; satisfying (5) pointwise and, for all T0 > 0 and for all smooth test functions satisfying .; T0 / D 0, Z

Z

 Td

n20 u0

 .; 0/dx D 0

T0

Z Td

 n2 u  t  n2 div.u/u   C nu ˝ nu W r

 p  p  n C1 div   2"2  n 2 nrn   C n3=2 div   C1   C n2 rV    nD.u/ W nr C rn ˝  dxdt:

C

(14) In the above theorem, the product “A W B” means summation over both indices of the matrices A and B. In order to control the behavior of the solutions when the particle density n vanishes, we need to define test functions for the momentum equation, which are, in some sense, supported on the set fn > 0g. In fact, we have chosen in the weak formulation (14) as in [8] test functions of the form n, where  is some smooth function, in order to deal with the convection term. The restriction  > 3 is the uniform L3 bound for n (obtained pneeded to improve 1  from the H bound for n) to an L bound. This property helps us in the limit ı ! 0 to achieve a suitable weak convergence result (see [23] for details). Theorem 2 is proved inp[23] for the case "2 > 12˛ 2 . This condition is necessary to obtain H 2 bounds for n via the viscous quantum Euler model fromp the new energy estimate (12). In the case "2  12˛ 2 , we loose the H 2 control on n. The limiting case "2 D 12˛ 2 has been treated by Dong [14]. Indeed, using (an p recently p approximation of) the test function  n= n in (10) leads to d dt

Z Td

p ˛ jr nj2 dx C 4

Z njr 2 log nj2 dx  T3

1 4˛

Z njrwj2 dx:

(15)

R3

In view of the energy inequality (12), the right-hand side is uniformly bounded. By p (13), this shows the desired H 2 bound for n. Jiang and Jiang [20] have combined

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the inequalities (12) and (15) to treat the remaining case "2 < 12˛ 2 . Let "0 D "2  12˛ 2 < 0 and define Z  n 2 "0 p  jwj C H.n/  jr nj2 dx  0: F .n; w/ D 6 Td 2 Then we use (12) and (15) to conclude Z p dF dE"0 "0 d D  jr nj2 dx dt dt 3 dt Td Z   1 2 2 0 2 dx  0: .12˛  " /njrwj C H .n/jrnj  ˛ 0 2 Td 12˛ p Since 12˛ 2  "0 D "2 > 0, we obtain an L2 estimate for njrwj. Going back to (15), see that the right-hand side is bounded, which provides an L2 bound for p we p 2 2 njr log nj and hence, by (13), the desired H bound for n.

4 Numerical Simulation In this section we present the results from our numerical simulation of a simple resonant tunneling diode, computed from the one-dimensional stationary quantum Navier–Stokes model (5)–(6) with  D 1, coupled to the Poisson equation 2 Vxx D n  C.x/, x 2 .0; 1/, where  is the Debye length and C.x/ is the doping concentration. The geometry of the diode is as follows. The length of the diode is 75 nm. It consists of two highly doped 25 nm GaAs regions near the contacts and a lightly doped 25 nm middle region. The middle region contains a quantum well of 5 nm length sandwiched between two 5 nm Al0:3 Ga0:7 As barriers. This double barrier heterostructure is placed between two 5 nm GaAs spacer layers. The barrier height B is incorporated in the model by replacing V by V C B in the momentum equation. The effective electron mass in GaAs is meff D 0:067  m0 (m0 D 1031 kg) and we have chosen ˛ D ". For the unscaled model and the other physical parameters, we refer to [27]. The boundary conditions are n.0/ D C.0/;

n.1/ D C.1/;

ux .0/ D ux .1/ D 0;

nx .0/ D nx .1/ D 0;

T .0/ D T .1/ D T0 ;

V .0/ D 0;

V .1/ D U;

where T0 D 77 K is the lattice temperature and U is the applied voltage. We have discretized the stationary quantum Navier–Stokes–Poisson system using central finite differences on a uniform mesh with N D 500 points. Compared to previously approximated quantum fluid models [25, 26], we do not need any numerical stabilization. The resulting nonlinear discrete system is solved by the

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Fig. 1 Current-voltage characteristics for various values of the number N of discretization points

5

x 105 N = 250 N = 500 N = 1000 N = 1500 N = 2000

Current Density J [Am−2]

4

3

2

1

0

0

0.1

0.2

0.3

0.4

Voltage U [V]

4

x 106

15

x 104 meff = 0.126 ⋅m0

Current Density J [Am−2]

Current Density J [Am−2]

meff = 0.067⋅ m0

3

2

1

0

0

0.1

0.2

0.3

10

5

0

0.4

0

0.1

4

x 106

4

0.3

0.4

0.3

0.4

x 105 B = 0.4 eV

Current Density J [Am−2]

Current Density J [Am−2]

B = 0.3 eV

3

2

1

0

0.2 Voltage U [V]

Voltage U [V]

3

2

1

0 0

0.1

0.2 Voltage U [V]

0.3

0.4

0

0.1

0.2 Voltage U [V]

Fig. 2 Current-voltage characteristics for different effective masses meff and barrier heights B

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(undamped) Newton method, together with a continuation in the applied voltage with the voltage step 4V D 1 mV. Figure 1 shows the dependence of the current-voltage characteristics on the number of the discretization points N . As expected, there is a region of negative differential resistance (NDR) in which the current density decreases although the applied voltage increases. It seems that the characteristics converge to some “limit curve” as N ! 1, thus confirming numerical stability. The influence of the effective mass meff and the barrier height B is depicted in Fig. 2. As observed in other quantum hydrodynamic simulations [25, 26], there is no NDR region using the physical effective mass, but the NDR effect is more pronounced for larger (unphysical) values. Furthermore, larger barrier heights enhance NDR.

x 107

2

α = ε ⋅ 0.025

4000

α = ε ⋅ 0.05 α = ε ⋅ 0.1 α=ε

3500 Current Density J [Am−2]

Current Density J [Am−2]

2.5

1.5

1

0.5

α = ε ⋅ 100 α = ε ⋅ 500 α = ε ⋅ 1000

3000 2500 2000 1500 1000 500

0

0 0

0.2

0.4

0.6

0.8

0

0.1

Voltage U [V]

0.2

0.3

Voltage U [V]

Fig. 3 Current-voltage characteristics for different values of ˛

Electron density n [m−3]

1026

1024

1022

α = ε ⋅ 0.025 1020

α = ε ⋅ 0.05 α = ε ⋅ 0.1 α=ε

1018

0

20

40 Position x [nm]

Fig. 4 Electron density for different values of ˛

60

75

0.4

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Finally, we present the dependence of the model on the viscous parameter ˛; see Fig. 3. When ˛ is much smaller than ", several NDR regions occur, whereas for ˛ larger than ", we observe only one NDR region. The electron density develops “wiggles” at the right barrier which are due to the NDR effect and which have been observed in the quantum hydrodynamic model with ˛ D 0 (Fig. 4). A more complete numerical study and a numerical comparison with other quantum hydrodynamic models can be found in the work [27] in which the full quantum Navier–Stokes–Poisson system is solved. Acknowledgements The first author acknowledges partial support from the Isaac Newton Institute in Cambridge (UK), where a part of this paper has been written; the Austrian Science Fund (FWF), grants P20214 and P22108; the Doctoral Program W8 from the FWF; the Austria-France Project FR 07/2010; and the Austria-Spain Project ES 08/2010 of the Austrian ¨ Exchange Service (OAD). Both authors have been supported by the Austrian-Croatian Project ¨ of the OAD and the Ministry of Science, Education, and Sports of the Republic of Croatia (MZOS).

References 1. Allemand, T.: Derivation of a two-fluids model for a Bose gas from a quantum kinetic system. Kinet. Relat. Model. 2, 379–402 (2009) 2. Antonelli, P., Marcati, P.: On the finite energy weak solutions to a system in quantum fluid dynamics. Commun. Math. Phys. 287, 657–686 (2009) 3. Arecchi, F., Bragard, J., Castellano, L.: Dissipative dynamics of an open Bose-Einstein condensate. Optics. Commun. 179, 149–156 (2000) 4. Brenner, H.: Navier-Stokes revisited. Phys. A 349, 60–132 (2005) 5. Bresch, D., Desjardins, B.: Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model. Commun. Math. Phys. 238, 211– 223 (2003) 6. Bresch, D., Desjardins, B.: Some diffusive capillary models of Korteweg type. C. R. Math. Acad. Sci. Paris, Sec. M´ecanique 332, 881–886 (2004) 7. Bresch, D., Desjardins, B.: On the existence of global weak solutions to the Navier–Stokes equations for viscous compressible and heat conducting fluids. J. Math. Pures Appl. 87, 57–90 (2007) 8. Bresch, D., Desjardins, B., Lin, C.-K.: On some compressible fluid models: Korteweg, lubrication and shallow water systems. Commun. Part. Diff. Eqs. 28, 1009–1037 (2003) 9. Brull, S., M´ehats, F.: Derivation of viscous correction terms for the isothermal quantum Euler model. Z. Angew. Math. Mech. 90, 219–230 (2010) 10. Burger, S., Cataliotti, F., Fort, C., Minardi, F., Inguscio, M., Chiofalo, M., Tosi, M.: Superfluid and dissipative dynamics of a Bose-Einstein condensate in a periodic optimal potential. Phys. Rev. Lett. 86, 4447–4450 (2001) 11. Chen, L., Dreher, M.: The viscous model of quantum hydrodynamics in several dimensions. Math. Model. Meth. Appl. Sci. 17, 1065–1093 (2007) 12. Degond, P., M´ehats, F., Ringhofer, C.: Quantum energy-transport and drift-diffusion models. J. Stat. Phys. 118, 625–665 (2005) 13. Degond, P., Ringhofer, C.: Quantum moment hydrodynamics and the entropy principle. J. Stat. Phys. 112, 587–628 (2003) 14. Dong, J.: A note on barotropic compressible quantum Navier-Stokes equations. Nonlin. Anal. 73, 854–856 (2010)

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15. Feireisl, E.: Dynamics of Viscous Compressible Fluids. Oxford University Press, Oxford (2004) 16. Ferry, D., Zhou, J.-R.: Form of the quantum potential for use in hydrodynamic equations for semiconductor device modeling. Phys. Rev. B 48, 7944–7950 (1993) 17. Gamba, I., J¨ungel, A., Vasseur, A.: Global existence of solutions to one-dimensional viscous quantum hydrodynamic equations. J. Diff. Eqs. 247, 3117–3135 (2009) 18. Gualdani, M., J¨ungel, A.: Analysis of the viscous quantum hydrodynamic equations for semiconductors. Europ. J. Appl. Math. 15, 577–595 (2004) 19. Harvey, R.: Navier-Stokes analog of quantum mechanics. Phys. Rev. 152, 1115 (1966) 20. Jiang, F.: A remark on weak solutions to the barotropic compressible quantum Navier-Stokes equations, Nonlin. Anal. Real World Appl. 12, 1733–1735 (2011) 21. J¨ungel, A.: A steady-state quantum Euler–Poisson system for semiconductors. Commun. Math. Phys. 194, 463–479 (1998) 22. J¨ungel, A.: Transport Equations for Semiconductors. Lecture Notes in Physics, 773. Springer, Berlin (2009) 23. J¨ungel, A.: Global weak solutions to compressible Navier-Stokes equations for quantum fluids. SIAM J. Math. Anal. 42, 1025–1045 (2010) 24. J¨ungel, A., Matthes, D.: The Derrida-Lebowitz-Speer-Spohn equation: Existence, nonuniqueness, and decay rates of the solutions. SIAM J. Math. Anal. 39, 1996–2015 (2008) 25. J¨ungel, A., Matthes, D., Miliˇsi´c, J.-P.: Derivation of new quantum hydrodynamic equations using entropy minimization. SIAM J. Appl. Math. 67, 46–68 (2006) 26. J¨ungel, A., Miliˇsi´c, J.-P.: Physical and numerical viscosity for quantum hydrodynamics. Commun. Math. Sci. 5, 447–471 (2007) 27. J¨ungel, A., Miliˇsi´c, J.-P.: Full compressible Navier-Stokes equations for quantum fluids: derivation and numerical solution. Preprint, Vienna University of Technology, Austria (2010) 28. Levermore, C.D.: Moment closure hierarchies for kinetic theory. J. Stat. Phys. 83, 1021–1065 (1996) 29. Li, H.-L., Li, J., Xin, Z.: Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations. Commun. Math. Phys. 281, 401–444 (2008) 30. Madelung, E.: Quantentheorie in hydrodynamischer Form. Z. Phys. 40, 322–326 (1927) 31. M´ehats, F., Pinaud, O.: An inverse problem in quantum statistical physics. Preprint, Universit´e de Rennes, France (2010) 32. Slavchov, R., Tsekov, R.: Quantum hydrodynamics of electron gases. J. Chem. Phys. 132, 084505 (2010)

Travelling-Wave Similarity Solutions for an Unsteady Gravity-Driven Dry Patch Y. Mohd Yatim, Brian R. Duffy, and Stephen K. Wilson

Abstract Travelling-wave similarity solutions are obtained for three-dimensional unsteady gravity-driven flow of a thin film of Newtonian fluid around a moving symmetric slender dry patch on an inclined plane.

1 Introduction Dry patches in a flowing thin film occur in numerous practical situations, notably in industrial contexts such as in heat exchangers, nuclear reactors and coating processes. One of the first studies of a dry patch in a flowing film draining vertically under gravity and surface shear stress was performed by Hartley and Murgatroyd [1]. They obtained two different criteria, namely a force-balance criterion (based on the balance between surface-tension and inertia forces at the stagnation point at the apex of the dry patch) and a minimum total-energy-flow criterion (including kinetic and surface energy), which they used to predict the critical film thickness and the maximum flow rate of fluid for the dry patch to persist. Hobler [2] used a minimum total-energy criterion similar to that of Hartley and Murgatroyd [1] and included the effect of contact angle to calculate when it is energetically favourable for a film on a vertical substrate to break up into rivulets. Wilson [4] developed a more sophisticated model to incorporate the presence of the ridge of fluid near the contact line at the apex of the dry patch, which is absent in the theory of Hartley and Murgatroyd [1]. Podgorski et al. [3] observed the presence of this ridge experimentally, and derived a simple model to describe the flow which

Y.M. Yatim ()  B.R. Duffy  S.K. Wilson Department of Mathematics and Statistics, University of Strathclyde, Glasgow, UK Current address: School of Mathematical Sciences, Universiti Sains Malaysia, Penang, Malaysia e-mail: [email protected]; [email protected]; [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 50, © Springer-Verlag Berlin Heidelberg 2012

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is similar to the model proposed by Wilson [4]. More recently, Wilson et al. [5] obtained two steady similarity solutions for flow around a non-uniform slender dry patch in a thin film draining under gravity on an inclined plane, namely one for the case of weak surface tension and one for the case of strong surface tension.

2 Problem Formulation Consider unsteady flow of a thin film of Newtonian fluid with constant density  and constant viscosity  on a planar substrate inclined at an angle ˛ .0 < ˛ < / to the horizontal subject to gravitational acceleration g around a dry patch on the substrate, as sketched in Fig. 1. Cartesian coordinates Oxyz with the x axis down the line of greatest slope and the z axis normal to the substrate are adopted, with the substrate at z D 0. We denote the free surface profile of the film by z D h.x; y; t/, where t denotes time. We take the dry patch to be slender (varying much more slowly in the longitudinal (x) direction than in the transverse (y) direction), and we neglect surface-tension effects. Then with the familiar lubrication approximation, the velocity .u; v; w/ and pressure p satisfy the governing equations ux C vy C wz D 0;

(1)

uzz C g sin ˛ D 0;

(2)

 py C vzz D 0;

(3)

 pz  g cos ˛ D 0:

(4)

z

Free surface z = h(x, y, t)

y

g

x

Dry patch

a Contact lines y = ± a(x, t)

Fig. 1 Sketch of the geometry of the problem

Travelling-Wave Similarity Solutions for an Unsteady Gravity-Driven Dry Patch

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Integrating (2)–(4) subject to the boundary conditions of no slip and no penetration on the substrate z D 0: u D 0, v D w D 0, and balances of normal and tangential stresses on the free surface z D h: p D pa , uz D 0, vz D 0 (where pa denotes atmospheric pressure) yields p D pa C g cos ˛ .h  z/ ;

(5)

g sin ˛ .2h  z/ z; 2 g cos ˛ vD hy .2h  z/ z: 2 uD

(6) (7)

We introduce local fluxes uN D uN .x; y; t/ and vN D v.x; N y; t/ given by Z uN D

h 0

g sin ˛ 3 u dz D h; 3

Z vN D

h 0

v dz D 

g cos ˛ 3 h hy I 3

(8)

then the kinematic condition on z D h takes the form ht C uN x C vNy D 0, yielding the governing partial differential equation for h: ht D

g cos ˛  3  g sin ˛  3  h hy y  h x: 3 3

(9)

Once h is determined from (9) the solution for p, u and v in (5)–(7) is known. We are concerned with unsteady flow around a dry patch in a film of uniform thickness h1 at infinity (that is, in a film that would be of thickness h1 everywhere if the dry patch were absent). We shall restrict attention to dry patches that are symmetric about y D 0 (so that h is even in y) with (unknown) semi-width a D a.x; t/, so that the fluid occupies jyj  a, and h D 0 at the contact lines y D ˙a. The zero-mass-flux condition at the contact lines y D ˙a.x; t/ is vN D ˙ax uN , and from (8) we have uN D 0 at y D ˙a; therefore we have h D 0 at y D ˙a;

h3 hy ! 0 as

y ! ˙a:

(10)

3 A Similarity Solution We seek an unsteady travelling-wave similarity solution of (9) in the form h D h1 F ./; h D h1

D

y Œ`.x  ct/

1 2

9 if `.x  ct/  0; = if `.x  ct/ < 0;

;

(11)

where the velocity c (> 0) of the dry patch down the substrate and the dimensionless function F D F ./ ( 0) of the dimensionless similarity variable  are to be

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determined, and without loss of generality we write ` D 2h1 cot ˛. The dry patch lies in the region where `.xct/  0, and the fluid in the region where `.xct/ < 0 is of uniform thickness h1 ; at x D ct the thickness h and its derivative hy are continuous (so that u, v and p are continuous there), except at the apex of the dry patch at the singular point x D ct, y D 0.  and re-scale variables according p We non-dimensionalise p to x D Xx , y D     j`jXy , z D h1 z , t D .X=U /t , h D h1 h , a D j`jX a and c D Uc  , where X . h1 ) is a length scale in the x direction, which we may choose arbitrarily, and U D g sin ˛h21 =3 is an appropriate velocity scale. Then with stars dropped for clarity, the solution (11)1 takes the slightly simpler form h D F ./;

yD

q

Sg .x  ct/;

aD

q

Sg .x  ct/0 ;

 y D ; a 0

(12)

where Sg D sgn.cos ˛/. Also the (unknown) position where F D 0 is denoted by  D 0 (corresponding to the contact-line position y D a), so that the fluid lies in jj  0 . Equation (9) reduces to an ordinary differential equation for F ./, namely 

0  0 F 3 F 0 C  F 3  cF D 0;

(13)

where a dash denotes differentiation with respect to , to be integrated subject to F D0

at  D 0 ;

F 3 F 0 ! 0 as

It is found that

 ! 0 ;

F ! 1 as

1

F  Œ3c0 .  0 / 3

 ! 1: (14)

(15)

in the limit  ! C 0 , provided that c > 0, and that F has the far-field behaviour F  1 / 1 exp .c  3/2 =2 in the limit  ! 1, provided that c < 3. Conditions for the dry patch to bep thin and slender are that the length p scales in the x, y and z directions, namely X , j`jX and h1 , satisfy h1  j`jX  X , so that X  h1 j cot ˛j and X  h1 j tan ˛j. Since a closed-form solution of (13) is not available, we solved it numerically for F by shooting from a chosen value of the contact-line position  D 0 , with a chosen value of c. The solution F was monitored to see if it settled to the constant value 1 as  ! 1 to within a prescribed tolerance. In fact, the numerical computation cannot be started exactly at  D 0 (because of the singular slope there, given by (15)), so instead it was started from a position  D 0 C ı, where ı (> 0) is small; thus we solved (13) subject to the approximated boundary conditions 1

F .0 C ı/  .3c0 ı/ 3 ;

F 0 .0 C ı/ 

 c  13 0

9ı 2

;

(16)

obtained from (15). The computation was then repeated with smaller values of ı (as small as ı D 1010 ) until the solution converged to within a prescribed tolerance.

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c 1.86 1.5 1.85

0.03

0.06

1

0.5

0

1

2

3

4

5

h0

Fig. 2 Plot of c as a function of 0 , together with the asymptotic value c D c1 D 1 in the limit 0 ! 1 (shown as a dashed line). The inset shows an enlargement of the behaviour near 0 D 0

Figure 2 shows a plot of c as a function of 0 obtained in this way. As Fig. 2 shows, c is a single-valued function of 0 , but behaves non-monotonically; specifically, c decreases from its value c D c0 ' 1:8507 when  D 0 to a (local) minimum value c D cmin ' 1:8501 when 0 ' 0:0040, then increases to a (global) maximum value c D cmax ' 1:8674 when 0 ' 0:0450, and thereafter decreases monotonically towards the value c D 1 as 0 ! 1. Thus the speed of the dry patch satisfies 1 < c  cmax for any value of 0 , more restrictive than the necessary condition 0 < c < 3 found earlier. Moreover, for a given value of c there is one corresponding value of 0 if either c D cmax or 1 < c < cmin , two if either c0 < c < cmax or c D cmin , three if cmin < c  c0 , and none if either c > cmax or c  1; thus there can be zero, one, two or three different dry patches that travel at a given speed c. Figure 3 shows examples of cross-sectional profiles F ./ for various values of 0 . The cross-sectional profiles F increase monotonically with , from F D 0 at  D 0 to F D 1 as  ! 1. Figure 4 shows three-dimensional plots of the free-surface profiles h in a sessile case with 0 D 1 at times t = 1 and 3. In the limit of a wide dry patch, 0 ! 1, we write  D 0 C

Q ; 0

F D FQ ./; Q

c D c1 I

(17)

then at leading order equation (13) reduces to  3 0 0  3 0 FQ FQ C FQ  c1 FQ D 0; which is readily solved subject to the boundary conditions

(18)

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0.6

0.4

0.2

0.1 1

5

3

h

10

Fig. 3 Cross-sectional profiles F ./ for several values of 0

Fig. 4 Three-dimensional plots of the free-surface profiles h with 0 D 1 at times t = 1 and 3

FQ D 0

at Q D 0;

FQ 3 FQ 0 ! 0

as Q ! 0;

FQ ! 1

as Q ! 1

(19)

to give the implicit solution 1 C FQ 1 Q D log 2 1  FQ

!  FQ ;

c1 D 1:

The asymptotic value c D c1 D 1 is included in Fig. 2 as a dashed line.

(20)

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4 Conclusions The lubrication approximation was used to analyse unsteady gravity-driven flow of a thin film of Newtonian fluid around a moving symmetric slender dry patch. Similarity solutions were obtained in which the dry patch has a parabolic shape, its scaled semi-width a varying like .x  ct/1=2 , where 1 < c  cmax ' 1:8674. The thickness of the fluid film increases monotonically away from the dry patch. Acknowledgements The first author (YMY) wishes to thank the Ministry of Higher Education (MOHE), Malaysia and Universiti Sains Malaysia for financial support.

References 1. Hartley, D.E., Murgatroyd, W.: Criteria for the break-up of thin liquid layers flowing isothermally over solid surfaces. Int. J. Heat Mass Trans. 7, 1003–1015 (1964) 2. Hobler, T.: Minimum surface wetting. Chem. Stosow. 2B, 145–159 (1964) (in Polish) 3. Podgorski, T., Flesselles, J.M., Limat, L.: Dry arches within flowing films. Phys. Fluids 11, 845–852 (1999) 4. Wilson, S.D.R.: The stability of a dry patch on a wetted wall. Int. J. Heat Mass Trans. 17, 1607–1615 (1974) 5. Wilson, S.K., Duffy, B.R., Davis, S.H.: On a slender dry patch in a liquid film draining under gravity down an inclined plane. Eur. J. Appl. Math. 12, 233–252 (2001)

Asymptotics of a Small Liquid Drop on a Cone and Plate Rheometer Vincent Cregan, Stephen B.G. O’Brien, and Sean McKee

Abstract A cone and a plate rheometer is a laboratory apparatus used to measure the viscosity and other related parameters of a non-Newtonian liquid subject to an applied force. A small drop, of order millimetres, of the liquid is located between the horizontal plate and the shallow cone of the rheometer. Rotation of the cone ensues, the liquid begins to flow and the plate starts to rotate. Liquid parameters are inferred based on the difference in the applied rotational force and the resulting rotational force of the plate. To describe the flow of the drop, the initial drop configuration, before rotation commences, must be determined. The equilibrium drop profile is given by the solution to the well-known nonlinear Young–Laplace equation. We formulate asymptotic solutions for the drop profile based on the small Bond number. The modelling of the drop exhibits a rich asymptotic structure consisting of five distinct scalings which are resolved via the method matched asymptotics.

1 Introduction The study of surface tension and capillarity has long been an area of interest to both scientists and applied mathematicians. The importance of capillarity phenomena is highlighted by their abundance in both nature (self-cleaning behaviour of the lotus plant [1] and the water repellent properties of water striders [3]) and industry (glass fabrication [4] and in the application of coatings to surfaces such as television screens [7]).

V. Cregan ()  S.B.G. O’Brien MACSI, University of Limerick, Limerick, Ireland e-mail: [email protected]; [email protected] S. McKee University of Strathclyde, Glasgow, Scotland, UK e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 51, © Springer-Verlag Berlin Heidelberg 2012

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Early attempts at understanding surface tension include Leonardo da Vinci’s ad-hoc, intuitive explanation for capillary effects and Newton’s experiments involving the rise of a liquid up a thin tube based on the attraction of the liquid to the tube [8]. In the early nineteenth century, the independent surface tension research of Young and Laplace resulted in the Young–Laplace capillary equation  p D 

1 1 C R1 R2

 ;

(1)

which describes the equilibrium profile of a static liquid–gas interface. We observe that p is the pressure difference across the liquid interface,  is the surface tension and R1 and R2 are the principal radii of curvature. A cone and plate rheometer is a laboratory device used to study the way in which a non-Newtonian fluid flows and deforms subject to an applied force. A fluid drop is placed on the flat plate of the rheometer and the shallow cone is lowered towards and in to the drop (see Fig. 1a). Typically, the plate is rotated though in certain designs the cone may rotate. The rotational force causes the fluid to flow and thus, cone rotation ensues. On the basis of the difference of the applied force and the resulting rotational force exerted on the cone, parameters such as fluid viscosity can be established. To simulate the fluid flow associated with the drop in contact with the cone and plate rheometer the initial, static drop profile must be determined. The method of matched asymptotics is used to derive expressions for the shape of the static drop profile. The perturbation approach is based on the small Bond number—where surface tension dominates body force terms—and is similar to previous work on sessile drops and pendant drops [5].

a

b O(ε3)

Upper Neck

Cone

Upper Boundary

r *, X

O(ε3)

O(ε2)

O(ε3)

0 φ z*, Y

Plate

Outer

Lower Boundary Lower Neck

O(ε)

O(ε2)

O(ε)

O(ε3) O(ε3)

Fig. 1 (a) Schematic drop profile. (b) Drop asymptotic regions (neither drawn to scale.)

O(ε3)

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2 Mathematical Model and Nondimensionalisation Assuming that the contact angles are constant, the resulting drop is axisymmetric with profile z D z .r  / with respect to a polar coordinate system aligned such that z D 0 is located at the thinnest part of the upper neck of the drop with the z -axis pointing downwards in the direction of gravity (see Fig. 1a). The hydrostatic pressure in the drop is given by p  C gz where  is the fluid density, g is gravity and p  is the unknown pressure at z D 0 where the profile becomes vertical. Thus, at the liquid–gas interface, the hydrostatic pressure in the drop is balanced by the capillary forces and it follows from (1) that  

00

0

z z C 0 . 1 C z 2 /3=2 r  . 1 C z0 2 /1=2



D p  C gz ;

(2)

where differential geometry has been used to formulate expressions for R1 and R2 . We adopt the standard nondimensionalisation approach [5] and we nondimensionalise (2) via the fundamental dimensionless variables z D aY ; where a 

r  D aX ;

p  D gaP;

(3)

p =.g/ is the liquid capillary length, to obtain 00

0

Y Y C D P C Y: . 1 C Y 0 2 /3=2 X . 1 C Y 0 2 /1=2

(4)

From a numerical perspective a more convenient parametric formulation of (4) is dX D cos  ; ds

dY D sin  ; ds

d sin  C D P C Y; ds X

(5)

where  is the inclination (see Fig. 1a) and s is the arclength. Finally, elimination of the arclength from (5) yields dX X cos  D ; d XP C XY  sin 

dY X sin  D ; d XP C XY  sin 

(6)

which is the starting point for our asymptotic analysis. We denote L to be the maximum radius of the drop (or drop half-width) in the main body of the drop where its profile becomes vertical [9]. From previous work p [5], we assume that the width of the neck is O."3 / where "  L=a  L2 g= is the dimensionless half-width and may also viewed as a Bond number. We consider solutions for "  1 (or L  a) which represents the dominance of surface tension over body force effects in determining the drop profile. We begin the solutions from the point of minimum width in the upper neck where X D 0, Y D 0 and  D =2 and the corresponding boundary conditions are

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X. D =2/ D ˛"3 and Y . D =2/ D 0 where ˛ is an O.1/ parameter which is found via the asymptotic analysis. The dimensionless half-width condition X. D =2/ D " fixes the pressure P . In relation to the contact angles we adopt the strategy of previous authors whereby the contact angles are used to determine the points at which the drop is in contact with the cone and the plate [5]. For example if the lower contact angle is =2 we truncate the solutions at the point in the main body of the drop where the profile becomes vertical. The drop asymptotic structure consists of an upper neck, an upper boundary layer, an outer region (main body), a lower boundary layer and a lower neck (see Fig. 1b) and is based on previous work on sessile drops and pendant drops [5, 6, 8].

3 Results To reflect the balance between the surface tension curvature terms, which are opposite in sign, in the upper neck of the drop we define the rescaled neck variables X D "3 u ;

Y D "3 v ;

P D p=" ;

 D O.1/;

(7)

which upon substituting into (6) leads to the leading order equations du D u cot  ; d

dv D u; d

(8)

with boundary conditions u. D =2/ D ˛ and v. D =2/ D 0. The corresponding solutions are u D ˛ csc  ;

v D ˛ ln j tan =2j;

(9)

where we note the existence of a singularity as  ! 0 which implies that the upper rescaling is not appropriate and an alternative set of scaled variables must be defined. The upper boundary layer provides a transitional layer between the curvature dominated terms of the upper neck region and the three term balance in the main body of the drop. Moreover, Fig. 1b illustrates a change in sign in the curvature in the upper boundary layer which suggests the presence of a point of inflection. Consequently, (6) is rescaled via X D "2  ;

Y D "3  ;

P D p=" ;

 D "˚;

(10)

to obtain the leading equations d  D ; d˚ p  ˚

d ˚ D : d˚ p  ˚

(11)

Asymptotics of a Small Liquid Drop on a Cone and Plate Rheometer

We note the solution

453

p 1 ˚ ˙ ˚ 2 C 4C ; (12) 2 where p the positive root applies before the point of inflection, located at  D " 4C , and the negative root applies after inflection. The unknown integration constants are found via asymptotic matching [2]. In the main body of the drop, the basic shape is nearly spherical and the fundamental balance is between the curvature terms and the pressure term P . To highlight this balance we rescale (6) by the outer variables  D

(

z D Lx ;

)

r  D Ly;

(13)

dy x sin  D 2 ; d " xy C xp  sin 

(14)

to attain the outer equations x cos  dx D 2 ; d " xy C xp  sin 

with the condition x. D =2/ D 1. The presence of the "2 terms in (14) suggest O."2 / asymptotic expansions in x, y and p. The leading order solutions are x0 D sin  ;

y0 D 1  cos  ;

p0 D 2;

(15)

which represents a circular drop profile. Proceeding to O."2 / we have x1 D

1 .1  3˛/ cos 2  2 cos3  C 1  3˛ ; 6 sin 

(16)

which upon inspection reveals a singularity as  !  and thus an alternative rescaling for  near  D  is required. At the base of the drop we encounter a lower boundary layer analogous to the upper boundary layer and we rescale via X D "2  ;

Y D 2" C "3  ;

P D p=" ;

 D   "˚;

(17)

which upon introduction into (6) leads to a system of equations identical to (11). Noteworthy is the solution D ; (18) ˚ D    where via asymptotic matching we find D D 2=3  ˛. From (18) it is evident that if D < 0 (and thus ˛ > 2=3) then ˚ > 0 and it follows that  < . This leads to another point of inflection in the lower boundary layer and the beginning of a new drop. Hence, the magnitude of ˛ and thus the sign of D has a profound effect on the structure of the drop profile. Accordingly, we rescale (6) via a set of lower boundary layer and lower neck variables (analogous to (7) and (10), respectively) proceed to

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leading order and obtain the relevant solutions. The theoretical results pertaining to the lower boundary layer (˛ > 2=3) and the presence of another drop structure need to be validated experimentally. In reality, the static solutions outlined here may be quite difficult to achieve if the appropriate experimental configuration is not calibrated correctly. Other authors have reported on systems which exhibit a similar type of multiple drop structure as outlined here [8].

a

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4 Conclusion The method of matched asymptotic expansions has been used to derive asymptotic solutions for the profile of a liquid drop in contact with a cone and plate rheometer. A number of rescalings and boundary layers were required to fully describe the drop profile. As indicated by Fig. 2 and Fig. 3 the asymptotic solutions display excellent agreement with the corresponding numerical solutions. Acknowledgements We gratefully acknowledge the financial support of the Mathematics Applications Consortium for Science and Industry (MACSI) supported by a Science Foundation Ireland mathematics grant 06/MI/005 and an Embark Initiative postgraduate award RS/2006/41.

References 1. Dupuis, A., Yeomans, Y.: Modeling droplets on superhydrophobic surfaces: equilibrium states and transitions. Langmuir 21(6), 2264–2629 (2005) 2. Dyke, M.V.: Perturbation Methods in Fluid Mechanics. Academic Press, New York (Annotated edition from Parabolic Press, Stanford) (1975) 3. Hu, D., Bush, J.: The hydrodynamics of water-walking arthropods. J. Fluid Mech. 644, 5–33 (2010) 4. Nemchimsky, V.: Size and shape of the liquid droplet at the molten tip of an arc electrode. J. Appl. Phys. 27(7), 1433–1442 (1994) 5. O’Brien, S.: On the shape of small sessile and pendant drops by singular perturbation techniques. J. Fluid Mech. 233, 519–537 (1991) 6. O’Brien, S.: Asymptotic solutions for double pendant and extended sessile drops. Q. Appl. Math. 52(1), 43–48 (1994) 7. O’Brien, S.: The meniscus near a small sphere and its relationship to line pinning of contact lines. J. Colloid Interface Sci. 183(1), 51–56 (1996) 8. O’Brien, S.: Asymptotics of a sequence of pendant drops. SIAM J. Appl. Math. 62(5), 1569– 1580 (2002) 9. Padday, J.: The profiles of axially symmetric menisci. Phil. Trans. Roy. Soc. Lond. A Math. Phys. Sci. 269(1197), 265–293 (1971)

Finger Formation and Non-newtonian Fluids Jean P.F. Charpin

Abstract Coating is used in numerous industrial applications. A coated layer should be very regular but also as thin as possible. Fingers are likely to form at the edge of the liquid and compromise the quality of the coating. The fluids involved during the process are also likely to display non-Newtonian properties. A standard model is presented for non-Newtonian fluid films and the formation of fingers is investigated. Preliminary results show that the wavenumber and growth rate of the fingers increase when the non-Newtonian properties of the liquid become stronger.

1 Introduction The electronic industry is continuously trying to produce smaller components. Current production methods are reaching their limits. New techniques are now being developed where nano scale components are formed by self assembly [3]. In a typical process using this approach, channels in which the self assembling materials are confined, are created by lithography. A uniform layer of non-Newtonian fluid is applied uniformly over the geometry. It contains solvent and the monomers which will self assemble. The excess of solution is removed by the technique of spin coating and solvent evaporation. The present work focuses on studying the stability of this type of non-Newtonian flows. Spin coating of thin film flows has been widely studied for example see [1,4]. The stability of these flows may be investigated by only considering the moving front of the liquid. In this situation the governing equations describing spin coating are very similar to the flow down an inclined plane. In the following this simpler equation is derived using the standard thin film approximation. The non-Newtonian

J.P.F. Charpin () MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 52, © Springer-Verlag Berlin Heidelberg 2012

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fluids considered in this study are described using the Ellis model [2]. The stability of the resulting equation is investigated using an eigenvalue method. Finally results are presented for four different situations.

2 Governing Equation Figure 1 shows the typical configuration for a flow down an inclined plane. The liquid is placed at the top the inclined plane and let flow down. The reservoir is maintained at a constant level at all times. The typical length of the flow L is much larger than its typical thickness H D L where   1. The equation governing this fluid flow may be derived from the standard mass conservation and Navier Stokes equations. These equations are non-dimensionalised and simplified using lubrication theory. They may be written as:     @ @u @p  D C gx C O  2 ;  2 Re ; @z @z @x     @v @p @  D C O  2 ;  2 Re ; @z @z @y

(1) (2)

  @p D 0 C O  2 ;  2 Re ; @z

(3)

where .x; y; z/ are the standard Cartesian coordinates with x in the direction of the flow, .u; v/ are the velocities in the x and y directions, p.x; y; t/ and h.x; y; t/ are the non-dimensional pressure and liquid height,  denotes the dynamic viscosity, Re is the Reynolds number and gx represents the normalised gravity component in the flow direction. Velocities were scaled by U D gH 2 =N and g, N and  represent the gravity acceleration, typical fluid dynamic viscosity and fluid density respectively. Pressure was non-dimensionalised by the usual pressure scale P D 2 U=. N L/. The equations are solved using the following standard boundary conditions: no slip, no shear stress at the top and a pressure jump across the top surface u.0/ D v.0/ D 0;

ˇ ˇ @u ˇˇ @v ˇˇ D D 0; @z ˇzDh @z ˇzDh

 p D p0  C

L z

Fig. 1 Typical configuration

x

 @2 h @2 h ; (4) C @x 2 @y 2 H Reservoir

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and the inverse capillary number is C D  3 =.U N / where  denotes the liquid surface tension. The pressure may be written as stated in the boundary condition  p D p0  C

 @2 h @2 h C 2 : @x 2 @y

(5)

The viscosity is calculated using the Ellis model. In non-dimensional form, this may be written: D

1 0  1 C ˇ 1 jj˛1 ; N

(6)

where ˛ is the non-Newtonian coefficient, 0 corresponds to the viscosity when no shear stress is applied, ˇ D .H1=2 =U N /˛1 and 1=2 defines the shear stress corresponding to the (dimensional) viscosity  D 0 =2. The non-Newtonian component of the viscosity is expected to dominate, so the viscosity scale may be written  1=˛

N D 0

H1=2 U

.˛1/=˛ :

The shear stress, , may be calculated by integrating the mass conservation and Navier Stokes equations (1)–(3). Another two integrations of these equations with respect to the boundary conditions leads to the following mass balance @Qy @h @Qx C C D 0; @t @x @y

(7)

where  Qx D   Qy D 



2 3

2s

@p 6 h Cgx 4ˇ C 4 @x 3

@p @y





@p C gx @x

2

 C

@p @y

2

3

3˛1 5

˛C2

h 7 5 ; (8) .˛ C 2/

2

3 2s 3  2  2 ˛1 ˛C2 @p h @p 5 6 h 7  gx C 4ˇ C 4 5; 3 @x @y .˛ C 2/ 3

and @p D C @x



@3 h @3 h C 3 @x @x@2 y

 ;

@p D C @y



 @3 h d 3h C : @2 x@y dy 3

(9)

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3 One Dimensional Cross Section Before rivulets form, the flow is one dimensional and the height h .x; t/ of the cross section is therefore governed by: "  ˇ˛ ˛C2 # ˇ 3  3 ˇ h ˇ @ h @ @h @3 h h C C  ˇˇC C gx C gx ˇˇ ˇ C D 0; 3 3 @t @x @x 3 @x ˛C2

(10)

where  is the sign of @p=@x C gx . After a few time units, the fluid is moving with a constant shape at its front. The velocity of the front may be estimated as u0 D

˛C2 ˇgx h31  h3p  jgx j˛ h˛C2 1  hp C 3 h1  hp ˛ C 2 h1  hp

(11)

where h1 is the fluid height far away upstream from the front and hp is the precursor film height used to resolve the standard contradiction in the thin flow model between the non-slip boundary condition and the movement of the fluid. The equation governing the shape of the front may then be written as # "  ˇ˛ ˛C2 ˇ 3  3 ˇ h0 ˇ @ h0 @ @h0 @3 h0 h0 ˇ ˇ C C  ˇC  u0 h0 D 0; C gx C gx ˇ ˇ C @t @x @x 3 3 @x 3 ˛C2 (12) This equation may be solved purely numerically or using analytical techniques [5,7]. Figure 2 shows five different profiles at the front corresponding to five different fluids. The highest hump at the front corresponds to the Newtonian model. The

1.8 1.6

α = 1.25, β = 0.25

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α = 1.25, β = 0.05

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h

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Fig. 2 Front flow profile for ˛ D 1:25; 1:5 and ˇ D 0:05; 0:25

0

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other four models lead to lower humps. This was expected as all four cases describe shear thinning fluids. The smallest hump is obtained for ˛ D 1:5 and ˇ D 0:05. This corresponds to the case where the non-Newtonian properties of the fluid are the strongest. The stability of these profiles will now be investigated.

4 Stability Analysis When rivulets start forming, the fluid height may be described as: h.t; x; y/ D h0 .t; x/ C h1 .t; x; y/; where h0 .t; x/ corresponds to the one dimensional cross section profile and h1 .t; x; y/ represents the (first order) perturbation of the flow. The perturbation may be written as h1 .t; x; y/ D G.x/e i !y e t

(13)

where ! is the wavenumber of the perturbation in the y direction. Combining (13) with (7)–(9) leads to the equation governing the growth of the perturbation ˇ ˇ˛1 #   3  ) ˇ d 3 h0 ˇ @ G ˇh30 h˛C2 0 2 @G ˇ ˇ C C C gx ˇ ! C 3 ˛ C 2 ˇ dx 3 @x 3 @x # "  ˇ ˇ  ˇ˛ ˛C1 ˇ d 3 h0 @ d 3 h0 2 ˇ ˇ C C gx h0 G C  ˇC C gx ˇ h0 G  u0 G ˇ C @x dx 3 dx 3 ( ˇ˛1 !   3 ˇ  ) ˇ ˇ d 3 h0 @ G @G @ 2  C gx ˇˇ ! C .˛  1/ ˇˇC @x dx 3 @x 3 @x " ˇ˛1 # ˇ    ˇ ˇ d 3 h0 ˇh30 h˛C2 0 4 2 @G ˇ ˇC C C ! G ! C C g D 0 (14) x ˇ ˇ @x 3 ˛C2 dx 3

@ @G C @t @x

("

Studying the stability of the front reduces to solving the above eigenvalue problem D G, and G are the eigenvalue and eigenfunction respectively. If [5,6] where @G @t for a given value of the wavenumber !, the corresponding growth rate is positive, the perturbation will grow and fingers will form. If the growth rate is negative, no finger will be forming. Figure 3 shows the evolution of the growth rate when the wavenumber ! of the perturbation is varied for the non-Newtonian models considered in the previous section. In all the cases, considered, is positive at first, reaches a maximum, then decreases and becomes negative. Fingers are most likely to form at the wavenumber ! corresponding to the maximum of . This behaviour was

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0.6

α=1.5 α=1.25

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Fig. 3 Stability results for ˇ D 0:05 (left) and ˛ D 1:25 (right)

already observed for Newtonian flows [6]. The present results show that the finger wavenumber and their growth rate is highly dependent on the non-Newtonian parameters ˛ and ˇ: when (for different fluids) the non-Newtonian properties become stronger i.e. when ˛ increases and ˇ decreases, both the wavenumber and growth rate of the fingers increase. More numerical simulations should be carried out to confirm these results.

5 Conclusion Finger formation for a non-Newtonian fluid was investigated. The equation governing the fluid flow was derived for the non-Newtonian Ellis model. A first order perturbation of this equation leads to an eigenvalue problem governing the growth of the fingers. Initial results show that when the non-Newtonian properties of the fluid become more pronounced, both the wavenumber and the growth rate of the fingers increase. These results should be confirmed by further numerical simulations and could then be extended to the spin coating case to determine the number and growth rate of fingers likely to develop. Acknowledgement J.P.F. Charpin acknowledges the support of the Mathematics Applications Consortium for Science and Industry (www.macsi.ul.ie) funded by the Science Foundation Ireland mathematics initiative grant 06/MI/005.

References 1. Charpin, J.P.F., Lombe, M., Myers, T.G.: Spin coating of non-Newtonian fluids with a moving front. Phys. Rev. E 76, Article Number: 016,312, 1–9 (2007) 2. Cheremisinoff, N.P.: Rheology and non-Newtonian flows. Encyclopedia of Fluid Mechanics, vol. 7. Gulf Publishing, Houston (1988)

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3. Fitzgerald, T.G., Farrell, R.A., Petkov, N., Shaw, M.T., Charpin, J.P.F., Gleeson, J.P., Holmes, J.D., Morris, M.A.: A study on the combined effects of solvent evaporation and polymer flow upon block copolymer self-assembly and alignment on topographic patterns. Langmuir 25, 13,551–13,560 (2009) 4. Fraysse, N., Homsy, G.: An experimental study of rivulet instabilities in centrifugal spin coating of viscous Newtonian and non-Newtonian fluids. Phys. Fluid. 6, 1491–1504 (1994) 5. Marshall, J.S., Wang, S.: Contact-line fingering and rivulet formation in the presence of surface contamination. Comput. Fluid. 34, 664–683 (2005) 6. Troian, S., Herbolzheimer, W., Safran, S., Joanny, J.: Fingering instabilities of driven spreading films. Europhys. Lett. 10, 25–30 (1989) 7. Tuck, E., Schwartz, L.: A numerical and asymptotic study of some 3rd-order ordinary differential equations relevant to draining and coating flows. SIAM Rev. 32, 453–469 (1990)

Numerical Modelling and Simulation of Ship Hull Geometries Michael Hopfensitz, Juan C. Matutat, and Karsten Urban

Abstract We consider the numerical optimization of the geometry of ship hulls with the Voith–Schneiderr Propeller (VSP), an efficient propulsion and steering device. The presence of one or more VSP sets some requirements to the ship hull. We describe modeling, simulation and optimization of such geometries as a four step chain. First, we introduce a geometrical model of the ship hull plus attachment parts. Second, this model is transformed into a computational mesh which is used in Computational Fluid Dynamics (CFD) in the third step. The numerical optimization is the fourth part. Our method is fully automatic. We show some numerical results.

1 Introduction The Voith-Schneiderr Propeller (VSP) is an efficient propulsion and steering device [1]. It is mainly used for ships requiring high steering control such as water tractors and ferries. In the past, the efficiency of VSP has been significantly improved by using numerical optimization [4, 5, 8]. As a next step, one is interested in the optimization of the complete configuration consisting of ship hull and VSP. This allows an efficient ship design following the requirements of a particular client. The goal of the present paper is to describe a mathematical approach for this complex optimization. It consists of four main parts, namely 1. 2. 3. 4.

A geometrical model of the potential ship hull. An automatic mesh generator. A numerical scheme for simulating the flow around ship and VSP. A numerical optimization.

M. Hopfensitz  J.C. Matutat ()  K. Urban Institute of Numerical Mathematics, University of Ulm, Ulm, Germany e-mail: [email protected]; [email protected]; [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 53, © Springer-Verlag Berlin Heidelberg 2012

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Let us describe these four parts in a little more detail. On one hand, the geometrical model has to be flexible such that many different shapes of ship hulls can be represented. On the other hand, however, such a parametric model has to be as simple as possible since the number of parameters determines the dimension of the optimization problem. We introduce one such model in Sect. 2. In order to simulate the flow around such a geometry, we need a computational grid. Since this has to be done within a numerical optimization, the meshing has to be performed automatically, see Sect. 3. The simulation and optimization is briefly described in Sect. 4. As already mentioned, these four steps have to be performed automatically. Thus, the coupling of the modules in one chain is an important step which is realized by a Perl script allowing fully automatic computations. Finally, we describe some numerical results in Sect. 5. This paper is partly based upon [3, 6].

2 Geometric Ship Model In order to determine some “optimal” ship geometry, we obviously first need to setup a parametric model for a ship hull. As already mentioned, such a model needs to be both as sophisticated and as simple as possible. Sophisticated in the sense, that the model should allow to represent various kinds of geometries including in particular existing ships. Moreover, the more flexibility such a model allows, the more one can hope to determine an optimal ship. Within an optimization one hence seeks to determine specific values for the parameters in the model in order to optimize the ship w.r.t. a given target, e.g. efficiency. This means, that the number of parameters coincides with the dimension of the optimization problem. Hence, one would like to have as few parameters as possible which explains why the model should be as simple as possible. We use techniques from CAGD (Computer-Aided Geometric Design) in order to generate a smooth and watertight parametrical model. The generation of the model must be stable, robust and efficiently computable. Moreover, the model generation has to be included into an automatic optimization. In order to understand the specific requirements of a ship model, let us consider a CAD model of the VSP Water Tractor in Fig. 1. We see that the ship hull consists of different surfaces and at least one attaching part like the fin or a protection plate. Thus, we have to be able to model a hull with different attachments. Moreover, both in CAD and true building, a ship is made of different ribs. These rips are combined horizontally by so called design curves, such as the water or keel line. The surfaces are determined by the points of the design curves on the ribs. Both for construction and numerical purposes, it is necessary that the grid lines for the CFD code follow these design curves. Hence, our geometrical model must provide this. We start by describing a model for one particular rib. Due to symmetry, we can reduce ourselves to one half of it, see Fig. 2. We subdivide each rib into four parts representing different surface parts of the ship hull. We assume that the end points of each part are located on one design

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Fig. 1 Ribs in a CAD model (left) and CAD model of VSP Water Tractor composed by different surfaces and an attachment Fig. 2 Model of a single rib. The curve is divided into subcurves with different regularity. The water line is indicated by “wl”. The keel line is located at point A

curve, so that the collection of corresponding parts of different ribs in horizontal direction describe one surface. This surface is parameterized as   (1) S W D WD Œ0; 12 ! R3 ; .u; v/ 7! P .u/; s.v; Q.u// : Here, P W Œ0; 1 ! R describes the location of a rib, i.e., the scaling in x-direction. With the parameter space ˝  Rr , r 2 N, the function s W Œ0; 1  ˝ ! R2 models the surface. We choose s as a NUBS (Non uniform B-spline, [7]) of the form s.v; !/ WD

b X

n dn .!/ Nn;k .v/;

(2)

nD1

where dn are the control points depending on the parameters ! 2 ˝ (by solving a linear system of equations), Nn;k form a B-Spline basis of order k on a non-uniform grid  on Œ0; 1  R and n are the weights. Finally, Q W Œ0; 1 ! ˝ describes the design curves. Thus, we obtain a closed surface description. Inserting the NUBSrepresentation (2) into the geometric model equation (1) shows that the surface is

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parameterized in terms of the mapping Q for given mappings P and s. Each part on a rib is hence described by r parameters contained in ! 2 ˝. These parameters can describe e.g. the angle, the starting points, the area or the centroid of the particular curve. For an appropriate number of such conditions the NUBS-curve is uniquely described. The parameters used for the description of each block curve have been chosen in close corporation with the engineers from the company Voith in order to assign physical interpretation to the parameters. When putting together the model for each rib for describing the surface, we perform a reduction of parameters since there is obviously redundancy. This is done by numerical optimization w.r.t. a catalogue of existing ship models. For building up a model based on more surfaces the model can be extended in the following way. We define a vector S D .Si /i D1;:::;p 2 Rp whose components Si are surfaces as modeled in (1). In order to enforce continuity of the complete surface, we enforce Si .; 1/ D Si C1 .; 0/ for i D 1; : : : ; p  1. Hence, the whole ship hull geometry is parametrized and described by the vector S. Finally, attachment parts like a fin or a protection plate are modeled in the same rib-wise manner and these parts are attached to the ship hull by computing the intersection curves. This can be done quite efficiently by using the NUBSrepresentation of the surfaces. Before the described geometry can be used as an input for our mesh generator, we need to prepare the geometry by determining some “pre-discretized” surface description. This has been done by using the STL data format. We obtain a smooth triangulation since our geometry description allows for a piecewise smooth representation.

3 Mesh Generation Based upon the STL-representation, the next step is to generate a mesh for the numerical simulation. Such a mesh needs to fulfill some requirements, namely 1. The mesh has to be generated fast, robust and automatically without manual corrections. 2. It has to give a good representation of the geometry. 3. The mesh has to be of good quality for numerical simulations. In particular requirements 2 and 3 need to be balanced. For a flexible representation of the geometry, a mesh consisting of tetrahedra is desirable, see also Fig. 3. A good mesh quality (e.g. in terms of angles or aspect ratios) can be better realized by hexahedra. Therefore we follow a compromise by creating three layers, the inner and outer ones formed by hexahedra, the inner layer by tetrahedra. The inner layer close to the ship surface consists of possibly orthogonal hexahedra in order to optimize numerical precision. This is of particular importance since many optimization criteria (e.g. efficiency) involve physical quantities like the pressure on the surface. The outer layer is formed by hexahedra in order to reduce computational

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force in x-direction

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old model: 3 600 000 cells with wake (reference calculation) vmesh: 1 316 000 cells, adaptive surface mesh, with inner layer vmesh: 990 000 cells, non adaptive surface mesh, with inner layer vmesh: 1 300 000 cells, without inner layer

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Fig. 3 Mesh of a ship hull geometry (left). Results of the numerical simulation (right). Force in x-direction over the time step (in N scaled on model size)

complexity. Finally, the connection between these two hexahedra layers is done by tetrahedra in the inner layer. Tetrahedra allow for more flexibility here. The interface between the layers is done by introducing pyramids in order to maintain mesh quality. Details on our meshing algorithms and tools can be found in [3]. For the generation of the surface mesh and the tetrahedron mesh we use CUBIT from Sandia National Laboratory.

4 CFD Computations and Numerical Optimization The CFD computation is based on a cell centered finite volume method using the CFD code COMET from CD-Adapco [2]. The governing equation are the Reynolds averaged-Navier–Stokes equations for incompressible fluids with a k-" turbulence model. This is appropriate since we have a mach number below 0.3. The equations are well-known read ut  Re1 u C .u  r/u C rp D g, div.u/ D 0, where u denotes the unknown velocity field, p the unknown pressure, g a given exterior force and Re denotes the Reynolds number. In addition, appropriate initial and boundary conditions have to be imposed. COMET produces accurate and reliable results, that have been validated both in model experiments and real scale measurements. The simulation of the full model takes in the order of ten hours on a PC cluster with 12 processors when using a mesh with about one million cells. So far, we use a direct optimization scheme (Hooke–Jeeves) since the gradient of the target is not available. However, first results using Automatic Differentiation are quite promising and we will follow this approach in future work.

5 Numerical Results Now, we present some of our numerical results. In Fig. 3 (left), we show one example of a mesh of a ship geometry. We obtain a smooth mesh following the design curves without any hanging nodes. Of course, we have tested our tool for a variety of geometries. The results are equally satisfying.

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Fig. 4 Numerical optimization w.r.t. force in x-direction. Iteration history (left) and resulting ship geometry (right)

In Fig. 3 (right) the results of a CFD simulation are shown. The force in x-direction is plotted for each time step of the simulation. We compare different meshes. The red line corresponds to a mesh created and frequently used by Voith. This is our reference using 3.2 million cells. The other curves correspond to different meshes created by our mesh generator. The pink curve corresponds to a mesh without inner layer. Even though the mesh consists of 1.3 million cells, the result is far away from the reference curve. This indicates the importance of the inner layer. The blue and green curve correspond to meshes with different number of cells. The green one matches the reference result quite good, keeping in mind that the mesh has only about 1/3 of number of cells than the reference mesh. Increasing the number of cells even more results in a perfect matching of the curves. This clearly indicates the good performance of our mesh generator. Finally, in Fig. 4, we show the results of a numerical optimization w.r.t. the force in x-direction. The figure on the left shows the optimization steps of a direct method including the search steps. The result is shown on the left. We obtain a significant reduction of the target value. Acknowledgements The authors are grateful to Dirk J¨urgens, Michael Palm and Sebastian Singer (Voith Turbo Schneider Propulsion) for various helpful and inspiring discussions on the topic of this paper. This work was funded by the German Federal Ministry of Economics and Technology (BMWi) within the project VSPSimu.

References 1. Bartels, J.E., J¨urgens, D.: The Voith Schneider Propeller: Current applications and new developments. Technical report, Voith Turbo Marine GmbH (2006) 2. Ferziger, J., Peric, M.: Numerische Str¨omungsmechanik. Springer, Berlin (2008) 3. Hopfensitz, M.: Numerische Optimierung von Schiffen mit VSP: Automatische Gittergenerierung. Ph.D. thesis, University of Ulm (2010)

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4. J¨urgens, D.: Theoretische und experimentelle Untersuchungen instation¨arer Tragfl¨ugelumstr¨omungen und Entwicklung eines Berechnungsverfahrens f¨ur Vertikalachsmotoren. Ph.D. thesis, University of Rostock (1994) 5. J¨urgens, D., Palm, M., Singer, S., Urban, K.: Numerical optimization of the Voith-Schneider propeller. Z. Angew. Math. Mech. 87(10), 698–710 (2007) 6. Matutat, J.C.: Numerische Modellierung und Simulation von Schiffen mit VSP: GeometrieModellierung und Optimierung. Ph.D. thesis, University of Ulm (2012) 7. Piegel, L., Tiller, W.: The NURBS Book. Springer, Berlin (1997) 8. Singer, S.F.: Numerische Optimierung der hydromechanischen Parameter des Voith-SchneiderPropellers. Master’s thesis, University of Ulm (2003)

Analysis of Combustion and Turbulence Models in a Cylindrical Combustion Chamber Alexandru Dumitrache, Florin Frunzulica, and Horia Dumitrescu

Abstract The combustion process of methane downstream of an axisymmetric sudden expansion in a circular pipe with a constant wall temperature has been studied. The conservation equations of mass, momentum, energy, and species as well as additional equations due to turbulence modeling have been numerically solved. The standard k   model and eddy dissipation combustion model has been used to simulate the turbulence and combustion phenomenon, respectively. The governing equations have been discretized using a finite volume approach and power law scheme and the resulting set of algebraic equations has been solved simultaneously. The numerical results have been compared with the other’s numerical results and experimental data.

1 Introduction When the fluid path is changed abruptly in a combustion chambers, flow parameters, flow characteristics and the heat transfer rate are altered consequently. In the other hand, turbulent flows through axisymmetric sudden expansions are influenced by many parameters: inlet geometry, inlet flow Reynolds number, expansion ratio,

A. Dumitrache () Institute of Mathematical Statistics and Applied Mathematics, Calea 13 Septembrie no. 13, 050711 Bucharest, Romania e-mail: [email protected] F. Frunzulica POLITEHNICA University of Bucharest, Polizu 1-6, 011061 Bucharest, Romania e-mail: [email protected] H. Dumitrescu Institute of Mathematical Statistics and Applied Mathematics, Calea 13 Septembrie no. 13, 050711 Bucharest, Romania e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 54, © Springer-Verlag Berlin Heidelberg 2012

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step height, inlet turbulence intensity and inlet boundary condition. Combustion phenomenon and the important design parameters are governed by the interaction between turbulence and combustion. The mathematical modeling of turbulent combustion has been outlined by Magnussen et al. [1]. They have developed the eddy dissipation concept for modeling of combustion. Turbulent non-reacting flows have been briefly studied by Ramous [2]. He, later, developed a mathematical model to study turbulent, confined, swirling flows under reacting non-premixed condition [3]. It is observed that the dimensions of the recirculation zone are larger for non-premixed reacting flows than for incompressible conditions. Two fast-chemistry models, the eddy dissipation concept (EDC) and the conserved scalar (CS) approach, have been analyzed [4]. The equations are discretized using a hybrid scheme and the SIMPLE algorithm is employed to solve the resulting algebraic equations. In this work, combustion of methane in turbulent flow is studied. It is assumed that the fuel and air enter the cylindrical chamber with abrupt expansion as confined coaxial jets.

2 Governing Equations The geometry under study (Fig. 1) consists of two confined, coaxial jets; the inner jet contains pure fuel (methane) and the outer jet contains air. It is assumed here that the methane and air react by means of a one-step overall and irreversible chemical reaction rather than through a multitude of elementary reaction steps. The hydrodynamic equations governing the statistical stationary, axisymmetric, turbulent flow field in the geometry shown in Fig. 1 can be written as: – Continuity

1 @ @ .u/ C .r v/ D 0 @x r @r

Fig. 1 Schematic of flow geometry and solution domain

(1)

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– Axial momentum component     1 @ @u @u eff C reff C Su @x r @r @r

(2)

     1 @ 2 @  @u @v eff C reff  eff rV C k @x r @r @r 3 @x

(3)

1 @ @ @ .uu/ C .r vu/ D @x r @r @x where Su D 

@ @p C @x @x

– Radial momentum component @ 1 @ @ .uv/ C .rvv/ D @x r @r @x

    1 @ @v @v eff C reff C Sv @x r @r @r

(4)

where     @ 1 @ @u @v v @p eff C reff  2eff 2  C @r @x @r r @r @r r  2 1   2 @   r eff rV C k C eff rV C k  3 r @r 3r

Sv D 

(5)

The systems of equations is supplemented with a standard k " turbulence model [5] and a eddy dissipation model [1], with specific constants. The overall chemical reaction can be written as: CH4 C 2 .O2 C 3:76N2 / ! CO2 C 2H2 O C 2  3:76N2

(6)

The mean of mixture is calculated from the ideal gas equation of state . density P S Yi  D p RT N i D1 Mi , where, R, NS , and Mi are the universal gas constant, the number of species and the molecular mass of specie i, respectively. The specific heat at constant pressure, Cpm , for the mixture is computed using the constants for each specie that can be determined from  [6].  tabulated data The temperature of gas, T , is obtained from: T D H  Yf HR =Cpm , where HR is the enthalpy of combustion.

2.1 Boundary Conditions To solve the governing equations, boundary conditions must be applied to the inlet, outlet, solid walls and centerline of the solution domain as follows. Two flows were assumed to have zero radial velocities at the inlet and enter the combustion chamber with uniform but different axial velocities and temperatures.

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The inlet turbulence kinetic energy and its dissipation are taken as k D I 2 u2 and " D C k 1:5 =0:03l, in which I is the inlet turbulence intensity and l is either the radius of the fuel inlet flow, or annular height of the air inlet flow. The flow is assumed to leave the combustion chamber with zero axial gradients of flow quantities. At the solid walls, the no slip condition for normal velocity component was applied while the wall function [5, 7] was employed to calculate the tangential velocity near the wall and heat transfer rates. At the combustion chamber centerline, radial velocity component is zero, and the radial gradients of other flow quantities are zero because of symmetry.

3 Numerical Procedure The conservation equations for mass, axial and radial momentum, energy, species, kinetic energy of turbulence and its dissipation rate, are discretized using the finite volume method and the power law scheme [8]. For this purpose, the computational domain is considered as a non-uniform staggered grid, which covers the whole solution domain. The grid has a higher node concentration at the annulus-mixing layer between the fuel and the air stream, in the recirculation zone near the inlet and near the wall where steep gradients of the flow variables are expected. The values of scalar variables are calculated at the grid nodes, while the axial and radial velocity components are calculated at the grid midpoints in order to have a conservative finite-difference algorithm. The resulting algebraic equations are coupled and are simultaneously solved by Three-Diagonal Matrixes Algorithm (TDMA). The SIMPLER algorithm is used to indicate pressure gradient in the momentum equations. In order to prevent the divergence of non-linear equations, the suitable under relaxation factors are used. Because of intensive influence of temperature and pressure changes on density, the density is assumed to be constant at the first 500 iterations to prevent divergence. Then the resulting flow variables are taken as initial quantities to solve equations simultaneously. Therefore the equations are solved iteratively until a prescribed convergence criterion is achieved. The criterion for convergence of numerical solution was that the total normalized residual is less than 103 . This convergence was achieved in about 1,800 iterations.

4 Results and Discussion The geometrical and dynamical parameters of the combustion chamber and the flows have been chosen as to be comparable with those in [9]. The temperatures of combustion chamber walls, Tw , fuel inlet, Ti , and air inlet, T0 , are 500 K, 300 K and 600 K, respectively.

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To ensure the precision of computations, the numerical procedure has been thoroughly checked. First, the sensitivity analysis has been done. The proper choice of grid size has been examined in [10]; solutions have been performed with different grid densities and grid points to ensure a grid-independent solution. The axial velocity at the centerline of combustion chamber (central velocity) for two different grid points .80  60; 200  120/ has been compared. It has been observed that the difference between them is less than 5%. Then, it was found that a grid consisting of 200  120 points is sufficient in providing a grid-independent solution. Typical fluid flow and streamlines are shown in Fig. 2a, b. Two recirculation zones are observed; one, appears at the centerline immediately downstream of the fuel inlet which was produced due to high momentum flux ratio of the air and fuel flows and resulting in flame stabilization, the other, appears near the outer wall because of the sudden expansion. The temperature profiles at different sections of combustion chamber are compared with EDC model results [4] and experimental results [10] in order to verify the correctness of the numerical procedure The results of this study are in good agreement with experimental data (not shown here). Also it is observed that, increasing turbulence intensity decreases the dimensions of recirculation zone. The central velocity at different conditions are compared with each other in Fig. 3: (1) using the existing model and I D 7% (2) using the existing model and I D 17% (3) using the existing model without combustion (cold flow) (4) using the EDC model (5) using the FLUENT model EDC, and (6) using experimental data. It is observed that, the existing model is more effective in prediction of recirculation zone than the EDC model. Also the axial velocity profiles for cold flow (without combustion) and hot flow (with combustion) at different axial sections of combustion chamber are compared with experimental data. It is observed that the flow is expanded due to high temperatures caused by combustion reaction. Numerical results are in good agreement with experimental data especially at downstream of recirculation zone (the Figure not shown here).

Fig. 2 (a) Fluid flow near the inlet of combustion chamber (b) Streamlines near the inlet of combustion chamber

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Fig. 3 Central velocity for two different intensity, cold

5 Conclusion The model can predict the velocity, pressure, temperature, fuel mass fraction, mixing mass fraction and combustion products mass fraction distributions. Using these distributions, one can estimate the critical values of the above properties. As an example, using the prementioned conditions, the maximum temperature of the combustion products turns out to be approx. 2,000 K. Because of high temperatures caused by combustion reaction, fluid flow is expanded. Increasing the turbulence intensity decreases the dimensions of recirculation zone. Acknowledgement This paper was supported from the Romanian Research Contract, 81027/2008.

References 1. Magnussen, B.F., Hjertager, B.H.: On mathematical modeling of turbulent combustion with special emphasis on soot formation and combustion. In: Proceedings of 16th International Symposium on Combustion, pp. 719–728. Pittsburgh, Pennsylvania (1976) 2. Ramos, J.I.: Turbulent non-reacting swirling flows. AIAA J. 22, 846–848 (1984) 3. Ramos, J.I.: Numerical solution of non-premixed reactive flows in a swirl combustor model. Eng. Comput. 1, 173–182 (1984) 4. Nisbet, J., Davidson, L., Olsson, E.: Analysis of two fast-chemistry combustion models and turbulence modeling in variable density flow. Comput. Fluid Dynam. Conf. 1, 557–563 (1992) 5. Launder, B.E., Spalding, D.B.: The Numerical computation of turbulent flows. Comp. Meth. Appl. Mech. Eng. 3, 269–289 (1974) 6. Cengel, Y.A., Boles, M.A.: Thermodynamics an engineering approach, 2nd edn. McGraw-Hill, New York (1989) 7. Jayatilleke, C.L.V.: The Influence of Prandtl number and surface roughness on resistance of the laminar sublayer to momentum and heat transfer. Progr. Heat Mass Tran. 1, 193–329 (1969) 8. Patankar, S.V.: Numerical Heat Transfer and Fluid Flow. McGraw-Hill, Washington (1980) 9. Spaddacini, L.J., Owen, F.K., Bowman, C.T.: Influence of aerodynamic phenomena on pollutant formation in combustion (Phase I, Gaseous fuels). U.S. Environmental Protection Agency Rep. EPA-60012-76-247a, Washington (1976) 10. Mirmoosavi, M.: Numerical investigation of combustion in cylindrical tube, M.Sc. Thesis, Dep of Mechanical Eng., Shahid Bahonar University of Kerman (1998)

Numerical Method for a Stefan-Type Problem with Interior Layers Martin Viscor and Martin Stynes

1 Introduction Spray-drying is commonly used in the production of food (e.g., milk powder), detergents, mineral ores, chemicals, etc. In this process droplets of fluid that contain dissolved or suspended solids are sprayed from the top of a chamber while hot air rises in the opposite direction. As the droplets fall, the liquid evaporates and the remaining granulated dry material gathers at the bottom of the chamber. Models presented in the literature (see [1, 2] and their references) consider the process undergone by each particle (droplet) in several phases. First, the particle is heated by the air with no significant evaporation. When the particle reaches wet bulb temperature, evaporation starts and consequently the particle begins to shrink and its moisture content decreases. When the moisture content on the outer surface drops to a critical level (which is essentially zero), a dry crust starts to form. An interface that separates this dry crust from the wet core appears and propagates into the particle. Eventually this interface reaches the centre of the particle, which is now entirely dry. In the last phase, the temperature of the dry particle increases further.

2 The Phase with Moving Interface We are interested only in the moving interface phase, which is the most challenging phase to model. Towards this end we simplify the process before the interface appears by assuming that there is no change in the diameter of the particle during the

M. Viscor National University of Ireland, Cork, Dublin, Ireland e-mail: [email protected] M. Stynes () Department of Mathematics, National University of Ireland, Cork, Dublin, Ireland e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 55, © Springer-Verlag Berlin Heidelberg 2012

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entire process. Some current models (e.g., [1]) permit temperature to be nonuniform inside the particle, while others (e.g., [2]) allow nonuniform moisture content, but no model that we have seen allows both these quantities to be nonuniform inside the particle. An interface condition is needed to describe the position s D s.t/ of the moving interface at time t. Its exact form depends on the model used, but in [1, 2] and other papers it always includes the term ds=dt. In order to have a more realistic model, we allow both temperature and moisture content to vary throughout the particle. Consequently our interface condition at the point r D s.t/ at time t is  lim  HD r!s.t /

@ @ C.r; t/ D lim  T .r; t/: @r r!s.t /C @r

Here C.r; t/ is the moisture content, T .r; t/ is temperature, H is latent heat of vaporization, D is the moisture diffusivity coefficient and  is thermal conductivity. Here the term ds=dt does not appear, which is significantly different from the Stefan-type problems mentioned above; interface conditions of this form do not seem to have been considered in the research literature. We shall describe a new technique for the numerical solution of this problem.

3 Problem Formulation Let t1 , t2 , T be fixed with 0 < t1 < t2 < T . Here t1 is the time when the interface first appears and the dry crust starts to form at the surface of the particle, while t2 is the time when the interface reaches the centre of the particle—the particle is now completely dry. Thus for 0  t < t1 the whole particle is wet, for t1  t  t2 an interface exists that separates the wet and dry parts, and for t2 < t  T the particle is dry. Finally, T is the time when the heating process ends. Let s.t/ be a continuous function that satisfies s.t/ D 1 for 0  t  t1 , 0 < s.t/ < 1 for t1 < t < t2 and s.t/ D 0 for t2  t  T . This function defines the position of the interface: s.t1 / D 1 (particle surface), then s.t/ decreases for t1 < t < t2 until s.t2 / D 0 (particle centre). For convenience, we define s.t/ for all times t 2 Œ0; T . Define the sets G D .0; 1/  .0; T , G1 D f.r; t/j0 < t < t2 ; 0 < r < s.t/g and G2 D f.r; t/jt1 < t  T; s.t/ < r < 1g. Here G1 represents the wet part of the particle, while G2 represents the dry part. Let a1 > 0, a2 > 0 be fixed. We seek functions s.t/, u1 .r; t/ and u2 .r; t/ such that the following two parabolic partial differential equations are satisfied: ai @ r 2 @r

  @ @ r 2 ui .r; t/  ui .r; t/ D 0 for .r; t/ 2 Gi ; @r @t

i D 1; 2:

(1)

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Here a1 and a2 are the moisture diffusivity and heat conductivity coefficients respectively. The function u1 .r; t/ represents the moisture content in the wet part of the particle, while u2 .r; t/ represents the temperature in the dry part of the particle. The equations (1) are standard diffusion equations in spherical coordinates. For later convenience we replace u1 by u1 in (1), which does not alter this differential equation. Then we set the initial-boundary conditions to be u1 .r; 0/ D '.r/ for 0  r  1;

(2a)

u1;r .0; t/ D 0 for 0 < t < t2 ;

(2b)

u2;r .0; t/ D 0 for t2 < t  T;

(2c)

u1 .1; t/ C u1;r .1; t/ D

.t/ for 0 < t < t1 ;

(2d)

u2 .1; t/ C u2;r .1; t/ D

.t/ for t1 < t  T;

(2e)

and the interface conditions u1 .s.t/; t/ D u2 .s.t/; t/ D 0 for t1  t  t2 ;

(2f)

u1;r .s.t/; t/ D u2;r .s.t/; t/ for t1 < t < t2 :

(2g)

These initial-boundary and interface conditions are chosen to match the physical properties of the process. The point t1 is defined by the relation u1 .1; t/ < 0 for 0  t < t1 , u1 .1; t1 / D 0 and the point t2 is defined by the relation s.t/ > 0 for 0  t < t2 , s.t2 / D 0. Assume that the functions '.r/ and .r/ are sufficiently smooth and satisfy '.r/ < 0, ' 0 .r/  0 for 0  r  1 and .t/ > 0 for 0 < t  T . Finally, for .r; t/ 2 GN set  u.r; t/ D

u1 .r; t/ if .r; t/ 2 GN 1 ; u2 .r; t/ if .r; t/ 2 GN 2 :

(3)

We call u.x; t/ the solution of the problem (1)–(2). The above model was developed by Shishkin et al. [3], but only in a planar geometry, i.e., the equations (1) were replaced by ai

@ @2 ui .r; t/  ui .r; t/ D 0; 2 @r @t

i D 1; 2:

(4)

Remark 1. We will make no attempt to include real data in our model at this stage. Nevertheless, as in reality the moisture diffusivity coefficient can be several orders of magnitude smaller then the heat conductivity coefficient, our aim will be to study numerically the case a1  a2 .

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4 Interior Layer in Space When a1 becomes small, layers (i.e., regions with very steep gradients) develop in the solution u.r; t/. These layers are displayed in a numerical example with data a1 D 1=2048; a2 D 1; T D 70; '.r/ D r 2  2:1 and .t/ D t C 0:9. Its solution has two layers—one in the t-direction close to the line r D 0 (see Fig. 1 (left)) and another in the r-direction (see Fig. 2). The layer in the r-direction is located at the interface, so it moves at t changes— see Fig. 1 (right).

5 Numerical Method In [3] a numerical method was developed to solve the problem (1)–(2) on an equidistant mesh in space and time directions. Thus for a1  1 this method fails to resolve the layers that appear in the solution and needs to be modified. We outline these changes here; for more details see [4].

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Fig. 4 The mesh in .x; t / (left) and .; t / (right) variables

A non-equidistant mesh in both space and time is now used. Let  (with 0 <  < 1) be a fixed parameter which represents the width of the layer in the r-direction. Let !r be a piecewise equidistant mesh on Œ0; 1, with transition points  and 1  , and containing Nr mesh intervals (it has Nr =4 intervals on Œ0; , Nr =2 intervals on Œ; 1   and Nr =4 intervals on Œ1  ; 1—see Fig. 3). For 0  t  t1 we use the backward Euler method on this mesh with an a priori chosen time step. Standard difference approximations are used to discretize equations (1). For t1 < t  t2 , make a change of variables resembling [3]: .r; t/ ! .; t/,  D r  s.t/. This maps the interface to the straight line  D 0. Equations (1) are solved in .; t/ on the modified domain (see Fig. 4). The length of each time step is chosen in such a way that the interface moves exactly one spatial mesh interval in space towards the centre of the particle at r D 0. Finally, for t2 < t  T we proceed as for 0  t  t1 , except that the time steps used beyond t2 are the same as those before t2 but in the reverse order. This is so that one moves gradually from small time steps to large ones.

6 Numerical Results We will compare the performance of the method from [3] (with the obvious modification that (4) is replaced by (1)) with our method on a piecewise equidistant mesh with  D 0:025. Denote by Nr and Nt the number of mesh intervals in

484 Table 1 Nr t2 (1) t2 (2) Nt (2)

M. Viscor and M. Stynes Numerical approximation of t2 by method of [3] (1) and our method (2) 16 32 64 128 256 512 1; 024 2; 048 21:875 29:531 38:281 45:391 49:766 52:158 53:457 54:106 45:809 50:638 51:657 52:427 53:447 54:272 54:473 54:641 34 60 119 236 469 933 1; 862 3; 720

4; 096 54:457 54:727 7; 436

space and time respectively. Note that for our method Nt is not known a priori and depends on Nr ; in numerical experiments we have observed that typically one gets Nt ' 2Nr , so in the method of [3] we set Nt D 2Nr . For both methods there is no convergence in L1 while there is first-order convergence in L1 . Thus these error measures do not essentially distinguish between the methods. Consequently we take as a measure of accuracy how well the methods approximate the time t2 (the time when interface reaches the centre of the particle). This time is of great interest from the industrial point of view. The results are shown in Table 1. It is apparent that our modified method yields faster convergence of the computed value of t2 as Nr increases. Acknowledgement This research was supported by the Mathematics Application Consortium for Science and Industry in Ireland (MACSI) under the Science Foundation Ireland (SFI) Mathematics Initiative.

References 1. Farid, M.: A new approach to modelling of single droplet drying. Chem. Eng. Sci. 58, 2985– 2993 (2003) 2. Handscomb, C.S., Kraft, M., Bayly, A.E.: A new model for the drying of droplets containing suspended solids. Chem. Eng. Sci. 64, 628–637 (2009) 3. Shishkin, G.I., Shishkina, L.P., Cronin, K., Stynes, M., Viscor, M.: A numerical method for a stefan-type problem. Math. Model. Anal. 16(1), 119–142 (2011) 4. Viscor, M.: Numerical modelling of industrial processes exhibiting layer phenomena. Ph.D. thesis, University College Cork (2010)

Part VI

Life and Environmental Sciences

Overview As an interdisciplinary science industrial mathematics is concerned with the development of new mathematical tools needed to tackle the problems posed by industrial application. Consequently, the contributions to ECMI 2010 that relate to life and environmental sciences do not only span a huge variety of applications— from geological to medical and economic problems—, but also of the mathematical tools used in modeling, simulation and optimization. The first two papers are concerned with geological problems, above and below earth. J.M. Gambi et al. discuss a covariant method to locate radio-transmitters by means of Earth satellites in “The post-Newtonian Geolocation Problem by TDOA”. This method increases the accuracy of Ho and Chan’s Classical passive TDOA (time difference of arrival) method. In addition, they derive bounds on the smallest number of satellites needed to give unique exact locations for different scenarios. In “Analytical Method for Inverse Problems of Deep Magneto Telluric Sounding”, S.E. Guseynov analyzes an analytical-numerical scheme for the inverse problem of deep magneto telluric sounding for a fundamental three-layer model with tectonic faults. Even if the tectonic fault resistance has an influence on the total impedance of the model, the method turns out to be stable. The following four contributions deal with medical applications: tablet design, thermoacustic tomography, drug delivery and vessel growth. N. Ahmat et al. are concerned with drug design problems in “Modelling the mechanical behavior of a pharmaceutical tablet using PDEs”. Partial differential equations allow for a parameterized description of the surface of a pharmaceutical tablet. Besides the shape of a tablet, also properties like mass,

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tolerance to temperature changes and storage stability are subject to design. To take account for these parameters, the authors extend the PDE model of the surface to describe the solid tablet. In the paper “On causality of Thermoacoustic tomography of dissipative tissue”, R. Kowar proposes a causal attenuation model for dissipative media in thermoacoustic tomography. To derive attenuated pressure data from unattenuated data, an integral equation model is derived. The numerical results show a rapid decrease in resolution for increasing distance of the object of interest from the pressure detector. The optimal design of local drug delivery systems is the motivation behind the paper “A Mathematical Model For Drug Delivery by T.N.T. Vo et al. A reduced model of one scalar nonlinear diffusion equation for the total drug concentration is used to investigate the tissue residence time for strongly bound drugs by considering a problem with uniform initial drug concentration and perfect sink boundary conditions. In “Estimators of the intensity of fibre processes and applications”, P.M.V. Rancoita and A. Micheletti present statistical methods for the estimation of the intensity of fibre processes. They derive a method for the estimation of the variance of their estimators and introduce a scheme for the computation of these estimators on digital images in practice. They apply the developed scheme for the analysis of digital photos of angiogenic processes. In their example they estimate the growth of vessels in a mouse eye. The last four papers concern economic and environmental applications: optimal fishery management, optimization of river fishways, wind farm simulations and water management. An SDE model with multifractional Brownian motion is proposed by D. Filatova in the paper “Optimal control strategies for stochastic/deterministic bioeconomic models” to describe population dynamics for optimal fishery management. Using a moment approximation, the associated stochastic control problem is transformed to a deterministic one with both control and state constraints. For this problem, firstorder necessary conditions are derived. Around man-made barriers, hydraulic structures (fishways) are placed to aid the natural fish migration. In the paper “Fishway Optimization Revisited”, L.J. Alvarez– V´azquez et al. use both mathematical modeling and optimal control theory to improve the shape design of a river fishway. The results are confirmed numerically for a standard real-world situation. L. De Tommasi and M. Gibescu discuss a special reduced model to simulate wind farms in the paper “On a wind farm aggregate model based on the output rescaling of a single turbine model”. The proposed aggregate model for wind turbines turns out to be especially tailored to reproduce power fluctuations due to wind turbulences.

VI Life and Environmental Sciences

487

P. Bastian et al. present recent results of a network project aiming at the modeling and simulation of coupled surface and subsurface flows in the paper “Adaptive Modelling of Coupled Hydrological Processes with Application in Water Management”. The coupling of surface and subsurface processes is based on a heterogeneous nonlinear Dirichlet–Neumann method. The shallow water equations are solved by a discontinuous Galerkin method which includes a special treatment of wetting and drying. Wuppertal, Germany

Michael G¨unther Andreas Bartel

The Post-Newtonian Geolocation Problem by TDOA J.M. Gambi, M.C. Rodriguez-Teijeiro, and M.L. Garcia del Pino

Abstract A covariant method that increases the accuracy of Ho and Chan’s Classical passive TDOA method for the location of radio-transmitters by means of Earth satellites is derived. It is also shown that the smallest number of satellites needed to give unique exact locations must be five. In fact, we show that when the radio-transmitter is known to be on the Earth surface, not only three GEO-satellites, but three equatorial satellites are enough to improve the possible locations given in Ho and Chan’s method, whereas for a transmitter in the outer space five nonequatorial satellites are needed to give the exact unique location.

1 Introduction The time difference of arrival (TDOA) is the most outstanding technique in engineering Geolocation applications for the passive location of radio-transmitters. For a given number of satellites, the Geolocation TDOA problem consists in the determination of the space-time coordinates of the radio-transmitter by means of the time differences of arrival of the signal to the satellites [2]. In this work we show a procedure that increases the nominal Classical accuracy of the procedure given in Ho and Chan [3]. Our method is developed by means of Synge’s world function for the Earth post-Newtonian framework [4] and linear equations that are rearrangements of the equations given in [3]. In particular, for a radio-transmitter on the Earth surface we show that three equatorial satellites, not necessarily GEO-satellites, are needed to locate the radio-transmitter, whereas for a transmitter into the outer space five non-equatorial satellites are enough to give the exact unique location. J.M. Gambi ()  M.C. Rodriguez-Teijeiro  M.L. Garcia del Pino G. Mill´an Institute, Fluid Dynamics, Nanoscience and Industrial Mathematics, University Carlos III de Madrid, Spain e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 56, © Springer-Verlag Berlin Heidelberg 2012

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In terms of Synge’s world function, the basic equations from which the TDOA equations are derived are ˝.xTi ; xSi / D 0; (1) where ˝.xTi ; xSi / represents the world functions corresponding to the Newtonian and post-Newtonian approximations of the Earth surrounding space. In these equations xT˛ are the spatial coordinates of the radio transmitter, T , at the time of the signal emission, xT0 , and xS˛ are the space coordinates of any one of the satellites that receive the signal at the time of arrival, xS0 . (Latin upper indices range from 0 to 3, and Greek from 1 to 3). Inertial (ECI) coordinate system [6], the world function,  In the Earth-Centered ˝ xTi ; xSi , has the following form at the Newtonian level of approximation [1, 4]:    2 i 1h .x ˛ /2  c 2 x 0 ; ˝ xTi ; xSi D 2

(2)

where x ˛ D xS˛  xT˛ , and x 0 D xS0  xT0 . At the post-Newtonian level it takes the form    2 i 1h ˝ xTi ; xSi D .x ˛ /2  c 2 x 0 2 ! ! tg 2T ˛ C m jx j log C cos T  cos S tg 2S ! 2  m x 0 tg 2T C ; log jx ˛ j tg 2S

(3)

which, as for the most accurate versions for the Navigation problem by GPS, corresponds to the weak approximation of the Schwarzschild field for the spacetime surrounding the Earth [4]. As can be seen in (2) and (3), the second order terms in (3) are characterized by m D GM , where M is the mass of the Earth, and by T c2 and S , which are the angles which xT˛ and xS˛ make with x ˛ . From the different TDOA procedures available for passive locations, the procedure in [3] is appropriated for our purpose, since it allows give unique exact locations when it is conveniently modified. In fact, we show that the four locations given in [3] can be reduced to two, although never to less than two when, not only GEO, but equatorial satellites are used. Next, we show the modification of this procedure in order to derive the one single true location. This modification consists in increasing up to five the number of satellites, which must be nonequatorial. Finally, two numerical simulations are given to show the post-Newtonian corrections to the Newtonian locations of several radio-transmitters.

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2 TDOA Geolocation with Three Equatorial Satellites and the Radio-Transmitter on the Earth Surface The reference system most suitable to derive the TDOA equations from (1) is the ECI coordinate system [6]. This is so because Synge’s world function is half the square of the measure of the geodesic that joints any two events in any space-time. To derive a general TDOA equation at the Classical level of approximationwe start by solving (1) for x 0 under the assumption that the world functionˇ ˝ xTi ; ˇxSi is given by (2). Thus, we find that x 0 D jx ˛ j D rS , where rS D ˇxS˛  xT˛ ˇ D 1 Œ.x ˛ / .x ˛ / 2 is the Euclidean pseudo-range from the emitter to the satellite under consideration given in seconds (c D 1). And from here we have that for any two satellites, Si and Sj , the corresponding TDOA equation is xS0i  xS0j  ri;j D rSi  rSj ;

(4)

where xS0i  xS0j  ri;j is the (measured) time difference of arrival of the emitter’s signal when it is captured by these satellites. Analogously, the equation at the post-Newtonian level of approximation can be  derived when ˝ xTi ; xSi is given by (3). Thus, we find 2

xS0i  xS0j D rSi  rSj C 

0

Ti 2 m 42 log @ Si tg 2



tg

3

Sj 2 Tj tg 2

tg

1

  A C cos Ti  cos Tj

C cos Sj  cos Si 5 ;

(5)

where Ti and Si are the angles that xT˛ and xS˛i make with xS˛i  xT˛ , respectively. Now, to avoid singularities in the 3x3 coefficient matrix for the satellites Si .i D 1; 2; 3/ as derived in [3], we take the following equations for our Classical version: r1;2 D rS1  rS2 and r3;2 D rS3  rS2 . Next, to account for an emitter on the Earth surface and Geostationary satellites, we add the restrictions xT˛ xT˛ D rE2 and .xS1I /2 C .xS2I /2 D R2 , where rE and R are the Earth and the Geostationary radius, respectively. And finally, from these four equations we have that the equation to be solved for the pseudo range rS2 is ˇ !ˇˇ ˇ!  ˇR2  R3 ˇ   2 2 .rS2 /2 D K  ˇ !ˇˇ K  r1;2  2r1;2 rS2  .rS2 / ˇ!  ˇR1  R3 ˇ ˇ !ˇˇ ˇ!  ˇR1  R2 ˇ   2 2  ˇ !ˇˇ K  r3;2  2r3;2 rS2  .rS2 / ; ˇ!  ˇR1  R3 ˇ

(6)

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 ! where K D .xS1I /2 C .xS2I /2 C rE2 D R2 C rE2 and Ri .i D 1; 2; 3/ are the position vectors of the three satellites, Si , with respect to the ECI center. Here we note that the two roots of this equation are real and positive. Therefore, even though one of these roots is greater than R (or a bit smaller), so that it can be neglected, we have that the method in [3] gives always not less than two locations, except when the radio-transmitter is on the Earth Equator. In case the three satellites are not assumed at fixed positions, but yet may be considered as equatorial (which is the most realistic situation [5]), we have that the corresponding equation for rS2 is ˇ !ˇˇ ˇ!  ˇR2  R3 ˇ   2 ˇ K1  r1;2 .rS2 /2 D K2  ˇ  2r1;2 rS2  .rS2 /2 !  ! ˇ ˇ ˇR1  R3 ˇ ˇ !ˇˇ ˇ!  ˇR1  R2 ˇ   2 2  ˇ !ˇˇ K3  r3;2  2r3;2 rS2  .rS2 / ; ˇ!  ˇR1  R3 ˇ

(7)

where Ki D .Ri /2 C rE2 and Ri are the distances of the satellites to the ECI center .i D 1; 2; 3/. Therefore, we have the same conclusions with respect to the number of locations than in the previous case. Now, if the post-Newtonian corrections for the satellites, Si and Sj , are denoted by pij then the post-Newtonian TDOA equations may be made analogous to (4), since they can be written as r1;2 D .rS1  rS2 /.1  12 / and r3;2 D .rS3  rS2 /.1  pij 32 /, where ij D  ri r result in small dimensionless quantities that make the j only difference between the Classical and post-Newtonian formulations given here. This fact is due to the covariant nature of ˝.x1k ; x2k / so that, according to (5), ij account for the influence of Shapiro time delay in locating the emitter. Thus, for three equatorial satellites, not necessarily at fixed positions, we have the following post-Newtonian equation, which is analogous to the equation in (7), ˇ !ˇˇ ˇ!  ˇR2  R3 ˇ   2 ˇ  K1  r1;2 .rS2 /2 D K2  ˇ .1 C 12 /2  2r1;2 rS2 .1 C 12 /  .rS2 /2  ! ! ˇ ˇ ˇR1  R3 ˇ ˇ !ˇˇ ˇ!  ˇR1  R2 ˇ   2 2 2  ˇ !ˇˇ  K3  r3;2 .1 C 32 /  2r3;2 rS2 .1 C 32 /  .rS2 / : (8) ˇ!  ˇR1  R3 ˇ Therefore, this equation is to give two slightly different solutions from those corresponding to the equation in (7). The difference is due to the time delays of the signal in reaching the satellites. Finally, we note that the equations in (6), (7) and (8) are computationally difficult to work with when the time differences of arrival

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493

of the signal to the satellites are small, and particularly when the satellites are close to each other.

3 TDOA Geolocation with Five Non-equatorial Satellites and the Radio-Transmitter Placed Anywhere From the discussion above it becomes obvious that to discard one of the two solutions given so far more than three satellites are clearly needed. And to this end we find that it is enough to increase this number up to five, wherever the radiotransmitter may be. However, the equations to be added to the systems in [3] cannot be chosen arbitrarily, since the system to be solved must be determined. In fact, there are many possible combinations. Thus, for example, if the fifth equation to be added to the Classical approach in [3], which is characterized by the first four equations shown next, is chosen to form the following system: r3;2 r2;1 r3;1 D l1 C m1 xT1 C n1 xT2 C v1 xT3 r4;3 r3;1 r4;1 D l2 C m2 xT1 C n2 xT2 C v2 xT3 r4;2 r2;1 r4;1 D l3 C m3 xT1 C n3 xT2 C v3 xT3 r4;3 r3;2 r4;2 D l4 C m4 xT1 C n4 xT2 C v4 xT3 r5;4 r4;1 r5;1 D l5 C m5 xT1 C n5 xT2 C v5 xT3 ; where l1 D r3;2 K1 C r2;1 K3  r3;1 K2 , m1 D 2.r3;2 xS11 C r2;1 xS13  r3;1 xS12 /, n1 D 2.r3;2 xS21 C r2;1 xS23  r3;1 xS22 /, v1 D 2.r3;2 xS31 C r2;1 xS33  r3;1 xS32 /, etc., then we have that the first, second and fifth equations, all them relating the five satellites, are independent from each other. Therefore, in this case the system that gives the unique solution is 0

10 1 1 0 1 m1 n1 v1 xT r3;2 r2;1 r3;1  l1 @ m2 n2 v2 A @ x 2 A D @ r4;3 r3;1 r4;1  l2 A : T m5 n5 v5 xT3 r5;4 r4;1 r5;1  l5

(9)

Similarly, at the post-Newtonian level of approximation we have that the system to be solved is 0 10 1 1 0 1 xT m1 n1 v1 p1  l 1 @ m2 n2 v2 A @ x 2 A D @ p2  l2 A (10) T 3 m5 n5 v5 xT p5  l 5 where, for example, p1 D r3;2 r2;1 r3;1 , p2 D r4;3 r3;1 r4;1 , l1 D l1 .1C32 C21 C 31 /, m1 D m1 .1 C 32 C 21 C 31 /, n1 D n1 .1 C 32 C 21 C 31 /, etc.

494

J.M. Gambi et al. 450 400

Dif.Loc.

350 300

cm

250 200 150 100 50 0 −100

−50

0 radio transmitter

50

100

Fig. 1 Radio-transmitter on the Earth Equator

4 Numerical Simulations In order to illustrate the numerical values of the post-Newtonian corrections to the classical solutions, we use representative satellites which closely resemble actual distributions. The TDOA Geolocation is made by means of five satellites. In Fig. 1 the radio-transmitter moves along the Earth Equator, and in Fig. 2 the radio-transmitter moves radially on the equatorial plane towards the outer space. In Fig. 1 the positions of the satellites with respect to the ECEF reference frame [6] are S1 .11; 996 Km; 0:1o N; 70o W /, S2 .11; 996 Km; 0:1o S; 42o W /, S3 .41; 972 Km; 0:1o N; 0o /, S4 .29; 982 Km; 0:1o S; 42o E/, and S5 .11; 996 Km; 0:1o N; 70o E/. In Fig. 2 the respective positions are S1 .11; 996 Km; 0:1o N; 20o W /, S2 .11; 996 Km; 0:1o S; 42o W /, S3 .11; 996 Km; 0:1o N; 15o W /, S4 .11; 996 Km; 0:1o S; 42o E/, and S5 .41; 972 Km; 0:1o N; 50o E/. In this case the longitude of the transmitter is 15o E:

5 Conclusions The scope of this paper is to show the post-Newtonian corrections to the classical TDOA Geolocation of radio-transmitters by means of Synge’s world function and a modification of the Geolocation method by Ho and Chan. From the computational

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495

1200

Dif.Loc. 1000

cm

800

600

400

200

0 0.5

1

1.5

2

2.5

radio transmitter

3

3.5

4

4.5 x 104

Fig. 2 Radio-transmitter towards the outer space

point of view there is not much difference between the Classical and post-Newtonian procedures shown here. As can be seen in (5), these corrections are combinations of several Shapiro time delays that depend on a number of variables (the coordinates of the satellites, as well as of the radio transmitter) which may go from twelve to eighteen, depending on the transmitter position. Due to the big amount of variables, it is not straightforward to show characteristic patterns, but we always have bulges of different size appearing when the distances of the satellites to the ECI center are different from each other (Fig. 1). And what is more important, the corrections are larger when the radio- transmitter is flying into the outer space than when it is moving on the Earth surface, as can be seen in Figs. 1–2.

References 1. Bahder, T.B.: Navigation in Curved Space-time. Am. J. Phys. 69, 315–321 (2001) 2. Gaarder, N.T.: On estimating the location of a signal source. IEEE Trans. Inform. Theory IT15(5) (1969) 3. Ho, K.C., Chan, Y.T.: Solution and Performance Analysis of Geolocation by TDOA. IEEE Trans. Aerospace and Electronic Systems. 29(4) (1993) 4. Synge, J.L.: Relativity: The General Theory. North-Holland, New York (1960) 5. Soop, E.M.: Handbook of Geostationary Orbits. Kluwer Academic Publisher, Dordrecht (1994) 6. Tapley, B.D., Schutz, B.E., Born, G.H.: Statistical Orbit Determination. Elsevier Academic Press, Burlington, MA (2004)

Analytical Method for Inverse Problems of Deep Magneto Telluric Sounding Sharif E. Guseynov

Abstract In this work 2D inverse problem is investigated in the case of H-polarization for a fundamental three-layer model of deep magneto telluric (MT) sounding with tectonic faults taking into account that the value of impedance Z is influenced not only by the depth h until the perfectly conducting base, but also by the values of tectonic fault resistance. An analytical-numerical method for solution of the studied inverse problem of deep MT sounding is proposed and stability of the proposed method is proved.

1 Introduction As it is well known (see [9]), for exploration the depth structure of the Earth a dependence on the frequency of the impedance of the Earth’s natural field Z .!/ is used (impedance of the Earth’s natural field equals to the ratio of tangential components of the electric field to the magnetic field measured on the Earth’s surface). Theorems of unique solution of inverse problems of magneto telluric (MT) sounding are proved for 1D (see [10]) and 2D (see [7]) distribution of electrical conductivity. Obtained in the work [3] results have shown that interpretation of deep MT sounding data is severely influenced by the non-homogeneities of the nearsurface layer, which lies on the crystalline basement. Besides, the interpretation of deep MT sounding data is severely influenced by tectonic faults, which appear during the process of tectonic movements and deformations in crustal rocks under the influence of stretching, compression and shearing forces. Therefore, in the inverse problem of deep MT sounding it is necessary to find the depth H until the perfectly conducting base at the following known initial data: impedance Z .!; y/ S.E. Guseynov () Transport and Telecommunication Institute; Institute of Mathematical Sciences and Information Technologies, University of Liepaja, Latvia e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 57, © Springer-Verlag Berlin Heidelberg 2012

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is known at one point of profile y 2 .a; b/ on Earth’s surface at various frequencies !; coordinates yi .i D 1; 2/ of tectonic fault profile are known.

2 Mathematical Model of Layered Domain with Tectonic Faults What are tectonic faults? Tectonic faults are sites of localized motion, both at the Earth’s surface and within its dynamic interior (for instance, see [4]). Faulting is directly linked to a wide range of global phenomena, including long-term climate change and the evolution of hominids, the opening and closure of oceans, and the rise and fall of mountain ranges. In Tectonic Faults, scientists from a variety of disciplines explore the connections between faulting and the processes of the Earth’s atmosphere, surface, and interior. They consider faults and faulting from many different vantage points—including those of surface analysts, geochemists, material scientists, and physicists—and in all scales, from seismic fault slip to moving tectonic plates. They address basic issues, including the imaging of faults from Earth’s surface to the base of the lithosphere and deeper, the structure and rheology of fault rocks, and the role of fluids and melt on the physical properties of deforming rock. They suggest strategies for understanding the interaction of faulting with topography and climate, predicting fault behavior, and interpreting the impacts on the rock record and the human environment. Using an Earth Systems approach, Tectonic Faults provides a new understanding of feedback between faulting and Earth’s atmospheric, surface, and interior processes, and recommends new approaches for advancing knowledge of tectonic faults as an integral part of our dynamic planet (in order to receive supplementary information on tectonic faults see [4] and the list of corresponding references in this). In this work we consider 2D inverse problem in the case of H-polarization for a fundamental three-layer model of deep MT sounding with tectonic faults. Electro conductivity of such layered domain is posed by formula 8 < 1  .y; z/ D 2 D 0 : 3 D 1

if 0 < z < h.y/I if h.y/ < z < H I if z > H;

where lim h.y/ D h1  const; h .y/ 6h1 ; y 2 .a; b/ W y1 < a < b < y2 . jyj!1

The first layer describes electro conductivity of sedimentary cover. It is assumed that the electro conductivity 1 and power h.y/ of the first layer are known. Let us assume the following basic conditions: (1) out of the interval .a; b/ the integrated def

conductivity S1 .y/ 1  h.y/ of the first layer is constant; (2) the second layer lies on the perfectly conducting base, i.e. 3 D 1; (3) the second layer has zero electro conductivity, i.e. 2 D 0; (4) the first layer thickness is many fewer than the

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499

thickness of the second layer, i.e. H  h; (5) the integrated conductivity function S1 .y/ is slowly varying function; (6) impedance Z .!; y/ is known at one point of profile y 2 .a; b/ on Earth’s surface at various frequencies !; (7) coordinates yi .i D 1; 2/ of tectonic fault profile are known. The following parameters are unknown data of the considered inverse problem: the depth H until the perfectly conducting base; the values of tectonic fault resistances ri .i D 1; 2/, because the value of impedance Z is influenced not only by the depth h until the perfectly conducting base, but also by the values of tectonic fault resistances ri .i D 1; 2/. Let R2 D r1 C r2 . Since the R2 is unknown, then solving the inverse problem will be defined as the depth H and the total resistance R2 . Let us consider the first layer: 0 < z < h.y/. For this layer from Maxwell’s equations for the case H-polarization we obtain i  !    Hx D 1  Ey D

@Ez @Ey  ; @y @z

(1)

@Hx ; @z

(2)

@Hx ; @y

(3)

1  Ez D 

Hx jzD0C0 D 1I Hx jzDh0 D Hx jzDhC0 I Ez jzDh0 D Ez jzDhC0 :

(4)

Now let us consider the second layer: h.y/ < z < H I y ¤ yi .i D 1; 2/. Then for this layer from Maxwell’s equations for the case H-polarization we obtain i  !    Hx D

@Ez @Ey  ; @y @z

(5)

@Hx D 0; @z

(6)

@Hx D 0; @y

(7)

ˇ Last two continuity conditions from (4) and Ey ˇzDH 0 D 0:

(8)

From the equations (6) and (7) follows that 8 ˆ < X; Hx D X 1 ; ˆ : X 1;

if h.y/ < z < H; y1 < y < y2 I if h.y/ < z < H; y < y1 I if h.y/ < z < H; y > y2 ;

where X; X 1 are as yet unknown constants. It is not difficult to find the unknown constant X 1 from 1D problem that can be obtained for electromagnetic field under jyj ! 1 (for instance, see [3], [1]):

500

S.E. Guseynov

X1 D

1 ; k  sin .k  h1 /  H  fcos .k  h1 / C k  h1  sin .k  h1 /g

(9)

p where k D i  !    1 ; Im .k/ > 0. Since an electromagnetic field in any domain is always distributed in such a way as to minimize the amount of emit (as well as available) energy (for instance, see [5], [8]). Consequently, sought constant X can be found from the condition of minimum energy released by current, which passes through the faults and through the first layer. For this purpose we will introduce J that signifies the limit of current under jyj ! 1 which passes on the first layer. In the neighborhood of the fault it is divided to current J1 , which generates a flow on the first layer, and to current J2 , which flows in vertical conducting channel, at that J D J1 C J2 (for instance, see [5], [6]). Since according to the equation (2) we can write Zh if y … .y1 ; y2 / then J D

1 

Ey1 d z

0

Zh D 0

@Hx1 zDh0 .4/ 1 D X  1I d z D Hx1 jzD0C0 @z

Zh if y 2 .y1 ; y2 / then J1 D

Zh 1  Ey d z D

0

0

@Hx zDh0 .4/ D X  1: d z D Hx jzD0C0 @z

Therefore J2 D J  J1 D X 1  X:

(10)

On the other hand the amount of energy emitting W1 at the period in the first layer (under the condition y 2 .y1 ; y2 /) equals  W1 D  !  1

Zy2

Zh dy

y1

0

@.Hx /2 dz  i      @z

Zy2

Zh .Hx /2 d z;

dy y1

(11)

0

and the amount of energy emitting W1 at the period in tectonic faults equals W2 D

.10/    R2  jJ2 j2 D  R2  jX  X 1 j2 : ! !

(12)

From our above mentioned assumption “... the integrated conductivity function is slowly varying function ...” follows that variation of the first layer thickness is z slow. Consequently the item @E @y in the (1) is small, and we can neglect this item. As a result, we obtain the following simplified equations in the first layer under y 2 .y1 ; y2 /:

Analytical Method for Inverse Problems of Deep Magneto Telluric Sounding

i  !    Hxf0g D 

f0g

@Ey ; @z

1  Eyf0g D

501

f0g

@Hx @z

(13)

where Eyf0g and Hxf0g are approximation solutions of Maxwell’s equations. From (13) we receive the following ODE in function Hxf0g : f0g

d 2 Hx d z2

C k 2  Hxf0g D 0; 0 < z < h.y/;

(14)

p where k D i  !    1 . We must add the following boundary conditions to the ODE (14): ˇ ˇ Hxf0g ˇ

zD0C0

ˇ ˇ D 1I Hxf0g ˇ

zDh0

D X:

(15)

The analytical solution of the boundary problem (14), (15) is Hxf0g D

  e i kz  e i kz  X  e i kh.y/ C e i kz : i kh.y/ i kh.y/ e e

(16)

Substituting the expression (16) in (11), we have 2 W1 D  !

s

i !    1

Zy2

 C 2 .y/  sinh ..y// C

y1

    1  C.y/  e .y/ 1 dy 2

  ˚ D  F1  X 2 C .F2 C i  F3 /  X C F4 ; !

(17)

where C.y/ D ˛.y/  X C ˇ.y/I ˛.y/ D

1 I e i kh.y/  e i kh.y/

(18)

p e i kh.y/ I .y/ D 2  i  !    1  h.y/I (19) i kh.y/ e s s Zy2 Zy2 i ! i ! 2 ˛ .y/ sinh ..y// dyI F2 D 2  .1  e  / ˛.y/dyI F1 D 2  1 1

ˇ.y/ D 

e i kh.y/

y1

y1

(20)

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s F3 D 4  i  s F4 D 2 



i !   1

i !   1

Zy2 

Zy2

N ˛.y/  ˇ.y/  sinh ..y// dyI

y1

ˇN 2 .y/  sinh ..y// C



y1

    1 N  ˇ.y/  e .y/  1 dy: 2

Here ˇN is complex conjugate for ˇ. Finally, unknown number X can be found as the solution of the following 1D optimization problem: WD

!  .W1 C W2 / D F1  X 2 C .F2 C i  F3 /  X C F4 C R2  jX  X 1 j2 ! min:  (21)

Having solved this 1D optimization problem (21) we receive XD

def

F2 Ci F3 2R2 C RF12

X1  1

D

X 1 C G1 ; 1 C G2

(22)

def

Ci F3 ; G2   FR132 . where G1   F22R 2 According to the formulas (18)–(20) low-frequency asymptotic for F1 , which is very important in the depth MT sounding, has the following simple form: F1 D Ry2 R1 C O .!/ under ! ! 0, where R1 D S1dy.y/ is a total resistance of the first layer y1

between two tectonic faults; S1 .y/ D 1  h.y/ is an integrated conductivity of the first layer. Consequently the parameter G2 in the formula (22) is actually the ratio of resistance of the first layer to the total resistance of two tectonic faults. Thus for the case H-polarization impedance is ˇ ˇ Ey ˇˇ ZD D  Ey ˇzD0 : ˇ Hx zD0

(23)

Then from (23) and (2) we have ˇ 1 @Hx ˇˇ ZD  : 1 @z ˇzD0

(24)

Substituting the expression (16) in the obtained formula (24), and taking into account the relation (22), we can write that ZD

ˇ 1 ˇ ˇ X C G1 ˇ k  ˇˇ  cos .k  h.y//ˇˇ : 1  sin .k  h.y// 1 C G2

(25)

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From the formula (25) we receive  .y/ 

 F1 F2 C i  F3 C 1 D X 1 C ; R2 2  R2

(26)

def

 cos .k  h.y//. Now to determine unknown where .y/  1  Z  sin.kh.y// k parameters H and R2 it is enough to measure the impedance Z at two different values 1 and 2 of wavelength: then from (9) and from (26) we will receive R2 D

F2 .j /Ci F3 .j / 2 j





j

1 H j j

   F1 j

.i D 1; 2/ ;

(27)

      , j D  j , j D 1; 2I  D k  sin .k  h1 /, j ,  j D  j p where j D  D cos .k  h1 / C   h1 ; k D i  !    1 ; Im .k/ > 0. Finally, excluding the unknown quantity R2 from the system of two equations (27), we will obtain the following equation for determination of unknown parameter H : . F2 .2 / C i  F3 .2 / D F2 .1 / C i  F3 .1 / .

 .H  2  2 /  1/  .H  1  1 / : 1  .H  1  1 /  1/  .H  2  2 / 2

The last equation is quadratic equation with respect to unknown parameter H , and this equation has two roots H f1g and H :f2g What root of these is correct value of unknown parameter H ‹ To elucidate this question we must substitute each of these two roots into the equation (27) for just one value of parameter , for example, for  D 1 . Then we will receive two values R2f1g nand R2f1g of ounknown n parameter o f1g

f2g

and H f2g I R2 R2 . To choose between two obtained solutions H f1g I R2 we substitute these found solutions into the formula (25), then taking into account the relations (9) at  D 1 . As a result we obtain two theoretical values of the impedance ZnT .1 / .n D 1; 2/. The proposed approach has been applied for the first time in the work [10]. As is shown in the monograph [2] (see as well as the fundamental monograph [1]), the proposed procedure works most of the time in practice. Now we can find the needed number (i.e. true value) n 2 f1I 2g from the ˇ def ˇ condition of minimum of the discrepancy, i.e. .n/  ˇZnT .1 /  ZnE .1 /ˇ ! min . nD1; 2

Acknowledgements Author would like to express his deep gratitude to Professor of Lomonosov Moscow State University, M.N. Berdichevsky for his verbal statement of investigated problem (on January, 2009) and for multiple helpful discussions of obtained results. Unfortunately, on August 11, 2009 Professor M.N. Berdichevsky passed away. His passing is a great loss to me, Moscow

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State University and Russian Electromagnetic Induction Community, but his example of a life well lived, of commitment and service, is a legacy of which I am the beneficiary. Present work was executed within the framework of the European Social Fund (ESF) Project No. 1DP/1.1.1.2.0/09/APIA/VIAA/142, and with the pointed Project financial support.

References 1. Berdichevsky, M.N., Dmitriev, V.I.: Models and Methods of Magnetotellurics. pp. 574. Springer, Berlin (2011) 2. Dmitriev, V.I.: Electromagnetic fields in layered inhomogeneous medium. pp. 311. Moscow State University Press, Moscow (1969) 3. Guseynov, Sh.E., Dmitriev, V.I.: On allowance for severely influence tectonic faults in the deep MT sounding inverse problem. In: PHerald of the Moscow State University, Series 15: Computational mathematics and Cybernetics, issue 2, pp. 24–30 (1992) 4. Handy, M.R., Hirth, G., Hovius, N. (eds): Tectonic Faults. Agents of Change on a Dynamic Earth, Series “Dahlem Workshop Reports”. pp. 504. Cambridge, London, MIT Press (2007) 5. Landau, L.D., Lifshitz, E.M.: Theoretical Physics. Electrodynamics of continuous medium. Vol. 8, pp. 474. Pergamon Press, Oxford (1984) 6. Mahan, G.D: Many-Particle Physics. pp. 1032. Plenum Press, New York (1990) 7. Medin, A.E.: The Magnetotelluric Inverse Problem. Ph.D. thesis, University of California, San Diego (2008) 8. Sommerfeld, A.: Theoretical Physics, Vol. 3: Electrodynamics. pp. 505. “Fizmatlit”, Moscow (2007) 9. Tikhonov, A.N.: On determining electrical characteristics of the deep layers of the earth crust. In: Proc. Acad. Sci. USSR 72(2), 295–297 (1950) 10. Tikhonov, A.N.: Mathematical grounding of the electromagnetic sounding theory. In: J. Comput. Math. Math. Phys. 5(3), 545–548 (1965)

Modelling the Mechanical Behaviour of a Pharmaceutical Tablet Using PDEs Norhayati Ahmat, Hassan Ugail, and Gabriela Gonz´alez Castro

Abstract Detailed design of pharmaceutical tablets is essential nowadays in order to produce robust tablets with tailor-made properties. Compressibility and compactibility are the main compaction properties involved in the design and development of solid dosage forms. The data obtained from measured forces and displacements of the punch are normally analysed using the Heckel model to assess the mechanical behaviour of pharmaceutical powders. In this paper, we present a technique for shape modelling of pharmaceutical tablets based on the PDE method. We extended the formulation of the PDE method to a higher dimensional space in order to generate a solid tablet and a cuboid mesh is created to represent the tablet’s components. We also modelled the displacement components of a compressed PDE-based representation of a tablet by utilising the solution of the axisymmetric boundary value problem for a finite cylinder subject to a uniform axial load. The experimental data and the results obtained from the developed model are shown in Heckel plots and a good agreement is found between both.

1 Introduction Tablets are the dominant dosage form for drug delivery in the pharmaceutical industry. This type of dosage form is convenient to use by patients, has long term storage stability and good tolerance to temperature changes. The quality of the tablet is determined by several parameters such as accurate mass, height and hardness. Thus, in order to produce quality tablets, it is important to understand the mechanical properties of these tablets. Therefore, many studies have been carried out to investigate the compaction properties (compressibility and compactibility) of

N. Ahmat ()  H. Ugail  G. Gonz´alez Castro CVC, University of Bradford, BD7 1DP, UK e-mail: [email protected]; [email protected]; [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 58, © Springer-Verlag Berlin Heidelberg 2012

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various types of excipients. Compressibility refers to the ability of the powder to deform under pressure [3] whereas compactibility is the ability of a powder bed to convert from small particles into a mechanically strong tablet [8]. The Heckel model is the most popular mathematical model for measuring the compressibility of pharmaceutical powders [3, 9]. Thus, we employed this model to analyse the data of compressed powder over a PDE-based tablet. The linear Heckel equation is based on the relative density-pressure relationship,  ln

1 1  rel

 D PK C A;

(1)

where rel is the relative density, P is an axial pressure, and K and A are constants. The relative density is defined as the ratio of the density of a substance at pressure P to the true density of the material. The constant A is associated with particle rearrangements before deformation while the reciprocal of K is a measure of the particles yield pressure (Py ), which determines the hardness of powders [3]. Low values of Py indicate harder tablets [9]. Recently, there has been a rapid expansion of computer uses in medical application especially in medical image processing [4] and drugs’ design [5]. However, we have not found in the literature any work related to geometric modelling of pharmaceutical tablets based on the use of parametric surface representation. Therefore, the objective of this work is to model a solid cylindrical pharmaceutical tablet by utilising the PDE method [7] and to study the mechanical behaviour of the axially compressed PDE-based representation of a tablet.

2 Generating a Cylindrical Tablet Using the PDE Method The PDE method generates parametric surfaces and it is based on the use of elliptic PDEs. The surface is defined by the two parameters u and v, in the region comprising 0  u  1 and 0  v  2. The general form of an elliptic PDE over a two-dimensional domain is given by, 

2 @2 2 @ C ˛ @u2 @v 2

r .u; v/ D 0;

(2)

where .u; v/ is the function defining a surface in 3D space while ˛ is a smoothing parameter [7] and r defines the order of the PDE. Equation (2) is transformed to the Biharmonic Equation by taking r D 2 and ˛ D 1. A smooth surface patch can be produced by solving the fourth order PDE analytically subject to a set of four periodic boundary conditions (BCs),

Modelling the Mechanical Behaviour of a Pharmaceutical Tablet Using PDEs

.0; v/ D P 0 .v/;

.1; v/ D P 1 .v/;

u .0; v/ D d 0 .v/;

507

u .1; v/ D d 1 .v/: (3)

The overall shape of the PDE surface depends on the derivative conditions which are defined by the derivative vector along the boundary curves [7]. The analytic solution of the fourth order PDE can be written as, .u; v/ D

4 X

a0m um1 C

mD1

1 X ŒAn .u/ cos.nv/ C B n .u/ sin.nv/;

(4)

nD1

where An D .a n1 C an3 u/e ˛ nu C .an2 C an4 u/e ˛ nu ; B n D .b n1 C b n3 u/e ˛ nu C .b n2 C b n4 u/e ˛ nu :

(5)

P The first term in (4) traces the spine of the surface patch while 1 nD1 ŒAn.u/ cos.nv/C B n .u/ sin.nv/ gives the radial position of a point .u; v/ away from the spine. The BCs are expressed as Fourier series in order to identify the constants in (4). The approximate solution to (4) can be found based on the sum of the first Fourier modes (typically N D 6) and a remainder function, R.u; v/, .u; v/ D

4 X mD1

a 0m um1 C

N X

ŒAn .u/ cos.nv/ C B n .u/ sin.nv/ C R.u; v/;

(6)

nD1

where R.u; v/ D .r 1 .v/ C r 3 .v/u/e N u.˛C1/ C .r 2 .v/ C r 4 .v/u/e N u.˛C1/ ;

(7)

Figure 1 presents an example of the PDE surface defined by 4 boundary curves. Many studies have been carried out to exploit the full potential of the PDE method in visual computing since the PDE surfaces offer many advantages over other type of surfaces. Most of all, this technique is capable of blending surfaces [1] and offers modelling tools to manipulate the shape of the PDE surface [7]. Moreover, smooth surfaces with high-order continuity requirements can be defined through PDEs since the formulation is well-conditioned and technically sound.

Fig. 1 The shape of a surface generated by the PDE method; Boundary conditions in the form of curves (left). The resulting surface shape (right)

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a

c

b 10 9

8

7 6 5

1

2

3

4

Fig. 2 Cylindrical tablet generated using the PDE method; Boundary curves (a), Resulting PDE surface (b), Solid cylindrical tablet with domain 0  u; w  1 and 2  v  2 (c)

The geometric model representing a flat-faced cylindrical tablet used throughout this work has been designed using 10 generating curves to produce a surface composed of 3 patches. The adjacent surface patch is created by evaluating the BCs using the next set of curves. Each patch shares one boundary curve with either one or two different PDEs so that position continuity is guaranteed along the generated surface. As one can see in Fig. 2a, the last curve of Patch 1 is used as the first curve of Patch 2. The output shape of the generated closed cylinder with radius 5 mm and height 6 mm can be seen in Fig. 2b. Since the PDE method’s formulation used in the early part of this section only generates the tablet’s shell, then we extended (4) to a higher dimensional space by introducing a new parameter, w, .u; v; w/ D

4 X mD1

a 0m um1 C w

1 X ŒAn .u/ cos.nv/ C B n .u/ sin.nv/;

(8)

nD1

where 0  w  1. This new parameter generates the interior points of the tablet, from the spine towards the point .u; v/ on the surface. A cuboid mesh is produced to represent the tablet’s inner part as illustrated in Fig. 2c. The number of nodes and cuboids used to generate the solid object depend on parameters u, v and w. For example, the number of cuboids in Fig. 2c is 12,000 when the parameters are defined 1 1 as u D w D Œ0; 20 ; 10 ; : : : ; 1 and v D Œ 2 ; 11 ; : : : ; 2. 20

3 Compression of a Cylindrical Tablet Tablets composed of 300 mg of ˛-lactose monohydrate were prepared through the Single Ended Compression (SEC) process. The powder was poured into a cylindrical die of radius (c) 5mm with an initial height (h0 ) of the powder bed equal to 6mm. The measured true density of this powder has been reported to be 1.3 mg/mm3 . The compression pressures applied on the powder bed were ranging from 0.05 to 70 MPa in order to obtain a tablet of 3mm in height. Since the radius of the die is fixed, only the axial displacement is involved in this process. The experimental data were analysed using the Heckel model and only data from

Modelling the Mechanical Behaviour of a Pharmaceutical Tablet Using PDEs

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pressure 10 MPa up to 50 MPa have been used because this range showed the best linearity which represents particle deformation. In order to model the mechanical behaviour of an axially compressed cylindrical PDE-based tablet, a three-dimensional analytical solution for a uniformly loaded finite, homogeneous and isotropic cylinder has been adopted [2]. The solution is obtained by utilising the Biharmonic Love’s stress function .r; z/ with 8 terms, D

a6 .16z6  120z4 r 2 C 90z2 r 4  5r 6 / C b6 .8z6  16z4 r 2  21z2 r 4 C 3r 6 / 3 C b3 .z3 C zr 2 / C a3 .2z3  3zr 2 / C a4 .8z4  24z2 r 2 C 3r 4 /

(9)

C b4 .2z4 C z2 r 2  r 4 / C a2 .2z2  r 2 / C b2 .z2 C r 2 /; where ai and bi .i D 2; 3; 4; 6/ are determined from the BCs. For the axisymmetric problem, the stress and displacement components can be expressed in terms of .r; z/ [6] as, z D .2   /

r D 

@ 2 @3  r  3; @z @z

1 C  @2  ; E @z@r

rz D .1   / !z D

@ 2 @3  r  ; @r @r@z2

  1C @2  2.1   /r 2   2 ; E @z

(10)

(11)

where E is the Young modulus and  is the Poisson’s ratio. The axial displacement of the compressed PDE-based tablet is obtained by substituting (9–11) into the following set of BCs, ˇ ˇ ˇ ˇ ˇ @!z ˇˇ z ˇzD0 D 0; z ˇzDh0 D P; rz ˇzD0;h0 D 0; !z ˇzD0 D 0; r ˇrDc D 0; D 0: @r zD0 (12) Thus, the axial displacement component can be written as, !z D !z0 C

   2 c2 100P z2 .1 C  / z2  .1   / C 1  ; r > 0; (13)  Eh0 r2 2  1 r2

where z and r are any point in z and r directions respectively and !z0 is an adjustment constant which is obtained from the difference between the initial axial displacement of the compressed pharmaceutical powder and the PDE-based tablet. The change in the PDE-based tablet height due to axial pressure ranging from 10 MPa to 50 MPa are measured using (13) with E D 2:64 GPa,  D 0:21 and !z0 D 1:07 mm, and the results are plotted as a Heckel graph. The Heckel plot of the compressed lactose powder and PDE-based tablet can be seen in Fig. 3. From the graph, it is found that the Py of ˛-lactose is slightly higher than the one obtained from our model, where their values are 103.09 MPa and 93.46 MPa respectively.

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Fig. 3 Heckel plot of the simulated compression and experiment on lactose powder

This is expected because the generated PDE-based tablet does not take the particle size and the degree of porosity into account. The results shown in Fig. 3 prove that the solution of the Love’s stress function can be utilised to measure the axial displacement of the compressed PDE-based tablet. However, the validity of the developed model is only verified at the lower pressure, where it indicates the deformation of the powder. Furthermore this model can only be applied to a cylindrical tablet defined by a set of BCs that depend on the chosen tabletting process, which in this case is the SEC process. Consequently, a more general model for characterising the stress distribution must be developed.

4 Conclusions The work presented in this paper focuses on the application of the PDE method for designing a parametric representation of a cylindrical pharmaceutical tablet. Three smooth surface patches generated by a fourth order PDE have been blended together to construct a hollow cylindrical tablet. The solid PDE-based tablet is generated by extending the PDE method to a higher dimension by introducing an additional parameter, w into the analytic solution of the elliptic PDE. The axial displacement component of a compressed PDE-based tablet is measured by utilising one of the solutions of the Love’s stress function found in the literature to model compaction

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of tablets. Heckel analysis is employed to analyse the results obtained from the developed model and is compared with the experimental results. It is found that the theoretical Heckel’s parameter is quite similar to the experimental ones. However, the developed model seems to underestimate the initial volume of the particle bed. Additionally, the output of this model is sensitive to the change of the elastic properties such as the Young Modulus and the Poisson’s ratio. Acknowledgements We would like to thank Prof. Anant Paradkar and Dr. Ravindra Dhumal, Institute of Pharmaceutical Innovation, University of Bradford for valuable discussions and providing experimental data of the compressed lactose.

References 1. Gonz´alez Castro, G., Ugail, H., Willis, P., Palmer, I.J.: A survey of partial differential equations in geometric design. Vis. Comput. 24(3), 213–225 (2008) 2. Hao-Jiang, D., Xiang-Yu, L., Wei-Qiu, C.: Analytic solutions for a uniformly loaded circular plate with clamped edges. J. Zhejiang University-Science A. 6, 1163–1168 (2005) 3. Ili´c, I., K´asa, P., Dreu, R., Pintye-H´odi, K., SrLciLc, S.: The compressibility and compactibility of different types of lactose. Drug Dev. Ind. Pharm. 35, 1271–1280 (2009) 4. Peir´o, J., Formaggia, L., Gazzola, M., Radaelli, A., Rigamonti, V.: Shape reconstruction from medical images and quality mesh generation via implicit surfaces. Int. J. Numer. Meth. Fluid. 53, 1339–1360 (2007) 5. Song, C.M., Lim, S.J., Tong, J.C.: Recent advances in computer-aided drug design. Brief Bioinform. 10, 579–591 (2009) 6. Timoshenko, S.P., Goodier, J.N.: Theory of elasticity (3rd Ed.). McGraw Hill, New York (1970) 7. Ugail, H.: Method of trimming PDE surfaces. Comput. Graph. 30, 225–232 (2006) 8. Yap, S.F., Adams, M.J., Seville, J.P.K., Zhang, Z.: Single and bulk compression of pharmaceutical excipients: Evaluation of mechanical properties. Powder Tech. 185, 1–10 (2008) 9. Zhang, Y., Law, Y., Chakrabarti, S.: Physical properties and compact analysis of commonly used direct compression binders. AAPS Pharm. Sci. Tech. 4(62), (2003)

On Causality of Thermoacoustic Tomography of Dissipative Tissue Richard Kowar

Abstract Since all attenuation models for dissipative media that come into question for thermoacoustic tomography (TAT) violate causality, a causal attenuation model for TAT is proposed. A goal of this article is to discuss causality in the context of dissipative wave propagation and TAT. In the process we shortly discuss the frequency power law, a causal attenuation model (with a constant wave front speed which can be adjusted via an additional parameter) and the respective wave equation. Afterwards an integral equation model for estimating the unattenuated pressure data of TAT from the attenuated pressure data of TAT is presented and discussed. Our numerical results show a fast decrease of resolution of TAT for increasing distance of the object of interest from the pressure detector.

1 Classical Thermoacoustic Tomography The main goal of thermoacoustic tomography (TAT) is to estimate variations in the electromagnetic absorption coefficient of tissue. First, the tissue under inspection is illuminated with an electromagnetic wave in the microwave or radio-frequency range. The energy absorbed by the tissue causes heat deformations and consequently a pressure wave is generated inside the tissue. From measurements of this pressure wave on a surface  surrounding the tissue the initial pressure distribution p0 .x/ can be estimated. Usually, it is assumed that variations in p0 .x/ correspond to variations in the electromagnetic absorption coefficient. Therefore p0 .x/ is usually used as the imaging function in TAT. Classical TAT assumes that the medium:

R. Kowar () Institut of Mathematics, University of Innsbruck, Technikerstrasse 21a/2, 6020 Innsbruck, Austria e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 59, © Springer-Verlag Berlin Heidelberg 2012

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• Is non-dissipative (D no wave attenuation), • Has a constant wave front speed and • A constant density. The first two assumptions mean that the phase speed of the pressure wave does not depend on the frequency and the spatial coordinates. In case of point detectors and under the above assumptions, the inverse problem of TAT can be formulated as follows: find p0 .x/ in1 r 2 p.x; t/ 

1 @2 p .x; t/ D p0 .x/ı 0 .t/ c02 @t 2

x 2 R3 ; t > 0

(1)

from the set of pressure data fp.x; t/ j x 2 ; t 2 Œ0; T g. Here ı 0 .t/ denotes the derivative of the time delta distribution, c0 denotes the phase speed of the wave p,  denotes a surface surrounding the tissue and T is a sufficiently large time period. For more details on classical TAT, we refer to the references in [2, 3]. Meanwhile some efforts have been made to incorporate dissipation and variing wave speed into TAT. One reason for this is to reduce the resolution limit of TAT caused by the above assumptions. In this paper we present a model for TAT that takes acoustic dissipation into account for approximately homogeneous and isotropic media. In the next section we discuss the dissipative wave equation that replaces wave equation (1) in case of TAT of dissipative media. In particular, causality of attenuated waves will be mathematically defined and discussed.

2 Acoustic Dissipation and Causality 2.1 The Dissipative Wave Equation Due to conciseness we only discuss here the standard form of spherical attenuated waves (3) below. For the more general case we refer to [2, 3]. Below we present a generalization of the causal attenuation model in [2, 3] that contains an additional parameter ˛2  0 such that the attenuation is not changed, but as we prove below, the speed of the wave front is changed from cF D c0 to cF D c0 =.1 C˛2 c0 /. We note that the causal variant (and generalization) of the thermo-viscous wave equation derived in [4] is applicable for TAT if dissipation in tissue is similar to that of water. As shown in [2], an acoustic dissipative wave can be modeled by   1 @ 2 r 2 p.x; t/  D C p.x; t/ D f .x; t/ c0 @t

1

Here the use of ı 0 .t / is also an assumption that is not always accurate.

x 2 R3 ; t > 0;

(2)

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where f denotes a source term modeling the wave generation, c0 > 0 is a constant and D denotes the time-convolution operator with appropriate kernel K .t/. The respective Green function reads as follows2   1 F 1 e ˛ .!/ jxj .t/ t G0 .x; t/ G.x; t/ D p 2

(! frequency);

(3)

where G0 denotes the Green function of (1), F denotes the Fourier transform and Z 1 1 1 F .˛ .!// D ˛ .!/ e i ! t d!: (4) K .t/ D p 2 R 2 In the following ˛ is called attenuation coefficient and <.˛ / is called attenuation law. In order that G is real valued and dissipation takes place, it is necessary that: (A1)

<.˛ .!// is even and positive and =.˛ .!// is odd for all ! 2 R.

The final requirement for ˛ is causality of the pressure wave.

2.2 The Issue of Causality For clarity, we start with causality in case of a pure time dependent system. Definition 1. A distribution G.t/ is called causal if supp.G/  Œ0; 1/. The time convolution operator A with kernel G.t/ is called causal if G is a causal distribution. A linear filter A as in Definition 1 has the property (T 2 R): f .t/ D g.t/

for

t
)

A .f /.t/ D A .g/.t/

for

t
where f and g denote time signals. Indeed, this fact is the reason for the naming “causal operator” and “causal kernel”. Unfortunately, this notation is not appropriate for space and time variing systems. If G0 denotes the Green function of the standard wave equation (1), then causality of the spherical wave G0 is usually understood as G0 .x; t C jxj=c0 / D 0

if

t <0

for all x 6D 0.

(5)

In words, the speed of the wave front of G0 is finite and if one moves locally with a part of the wave front, then the wave vanishes on one side for t > 0. This is a much more restrictive condition for G0 than causality in the sense of Definition 1. Condition (5) provides a coupling between space and time which is typical for waves

2

Here G is the solution of (2) with f .x; t / WD ı.x/ı.t / and t denotes the time-convolution.

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and cannot occur in a purely time dependend system like an oscillator. So the word causal is used in mathematics and physics in an ambiguous and misleading manner. Definition 2. Let c0 > 0 be the constant in (2), G0 and G denote the Green function of (1) and (2), respectively. a) The travel time of the wave front of G is defined by T .jxj/ WD supft > 0 j G.x; / D 0 for all   tg and the speed of the wave front of G at distance r from the origin x D 0 is defined by cF .r/ WD 1=T 0 .r/. b) G is called strong causal if the wave front speed cF of G is smaller or equal to the wave front speed c0 of G0 , i.e. Z

jxj 0

T 0 .r/ dr 

jxj c0

for all x 2 R3 .

c) If G is strong causal, the space time convolution operator A (attenuation operator) with kernel G is called strong causal. The proof of the following theorem can be found in [3]. Theorem 1. The Green function (3) is strong causal if and only if the mapping     1 x K W t 7! p F 1 e ˛ .!/ jxj D 4  jxj G x; t C c0 2

(6)

is causal for every x 2 R3 . If G defined by (3) is strong causal, then T .r/ D r=cF with constant cF 2 .0; c0  and K .t/ defined as in (4) is causal.   Note cF is the smallest number 2 .0; c0  such that t 7! G x; t C cxF is causal for each x. We emphasize that in general the causality of K (which corresponds to the standard test of causality in acoustics (cf. [5])) does not imply the causality of K. Since this fact has not been realized in acoustics, there are many false and misleading statements about causality. Causality of the kernel K can be tested by Theorem 1 together with Theorem 4 in [1]. For the discussion below we formulate a simpler version of Theorem 4 in [1]. Let C WD fz 2 C j =.z/  g. Proposition 1. Let K be defined as in (6) with ˛ satisfying (A1), (A2) ˛ .z/ is holomorphic in the interior of C0 and (A3) ˛ .z/ is piecewise continuous on C0 . Let x 2 R3 be fixed. If K.x; / is a tempered distribution, then K.x; / is causal if and only if for every  > 0 there exists a polynomial P such jF .K/.x; z/j  P .jzj/

for

z 2 C :

A detailed causality analysis of common attenuation models can be found in [3]. To show the issue of causality we consider the frequency power law (cf. [5]).

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Corollary 1 (Frequency power law). Let a1 > 0, a2  0,  2 .0; 1/nN and pl

˛ .!/ WD a1

.i !/ C a2 .i !/: cos. =2/

(7)

K defined as in (6) with (7) is causal if an only if  2 .0; 1/. Proof. We perform the proof for the simplest cases namely  2 .1; 3/ [ .5; 7/ [    . For the remaining details we refer to [3]. It is easy to see that properties (A1)-(A3) pl are satisfied by ˛ .z/. For fixed x and zn WD i n with n 2 N we get   p a1 n jxj  a2 n jxj 2  jF .K/.x; zn /j D exp cos. =2/ which cannot be bounded by P .jzn j/ for any polynomial P , since  2 .1; 3/ [ .5; 7/ [    . Therefore, by Proposition 1, K.x; / cannot be causal. pl

We note that ! 2 R 7! <.˛ .!// D a1 j!j is called the frequency power law. The following Theorem is a generalization of Theorem 8 in [2]. Proposition 2 (Proposed model). Let c0 ; ˛1 > 0, ˛2  0 and  2 .1; 2 and ˛ .!/ WD

˛1 .i !/ p C ˛2 .i !/; c0 1 C .i 0 !/ 1

(8)

K defined as in (6) with (8) is causal and the wave front speed of G defined by (3) is cF D c0 =.1 C ˛2 c0 /. Proof. Let x be p fixed. In [2] it has been proven that K is causal if ˛2 D 0. This together with 2 F 1 .expf˛2 .i !/ jxjg/ D ı.t  ˛2 jxj/ implies that K with ˛2 2 Œ0; 1/ is causal. Let c1 WD c0 =.1 C ˛2 c0 /. Since cF is the smallest constant 2 .0; c0  such that t 7! G.x; t C jxj=cF / is causal, we have to show that there exists an  > 0 such that K W t 7! G.x; t C . c11 C / jxj/ is not causal. For zn WD i n with large n 2 N we get from the definition of c1 that p 2  jF .K /.zn /j D exp

(

˛1 n jxj p C  n jxj c0 1 C .0 n/ 1

)

which cannot be bounded by P .jzn j/ for any polynomial P , since  2 .1; 2. Hence cF D c1 . According to experiments soft tissue behaves for small frequencies like a power law with   1:5. Indeed, if j0 !j << 1, then (8) is approximately (7) with  1

a1 WD

˛1 0

j cos. =2/j 2 c0

and

1 a2 WD 0



 ˛1 C ˛2 : c0

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3 The Inverse Problem Taking Acoustic Dissipative into Account Let p and pat t denote the solutions of (1) and (2) with f .x; t/ D p0 .x/ı 0 .t/, i.e. p D G0 x;t f and pat t D G x;t f . From (3), it follows (cf. [2, 3]) Z pat t .x; t/ D

1 0

M .t; t 0 / p.x; t/ dt 0

1 F 1 M .t; t / W D p 2 0



with

 i ! exp f˛ .!/ c0 t 0 C i ! t 0 g : ˛ .!/ c0 C i !

(9)

The requirement that K is causal (cf. Theorem 1) implies M .t; t 0 / D 0 if t < jt 0 j (cf. [2, 3]), i.e. the upper integration limit in (9) can be replaced by t. This means that no information from the future of the unattenuated pressure wave p is required. Equation (9) allows the estimation of the unattenuated data required for TAT without dissipation. Hence the reconstruction methods of classical TAT can still be used. The numerical results presented in Table 1 show a fast decrease of resolution of TAT for increasing distance L of the object of interest from the pressure detector.

4 Conclusion We presented a generalization of the causal attenuation model in [2, 3] that contains an additional parameter ˛2  0 which does not change the attenuation. However, as we proved, the speed of the wave front is then cF D c0 =.1 C ˛2 c0 /. In particular, cF D c0 for ˛2 D 0, which sharpens up the statement cF  c0 proven in [2, 3]. Table 1 Estimation of the number of singular values of M that can be used for TAT with attenuation coefficient (8) and parameters ˛1 D 6, 1 ˛2 D 0, c0 D 0:15  cm  MHz, 0 D 104  MHz  (exponent in (8))

L D 1 cm

L D 2 cm

L D 3 cm

L D 4 cm

1:10 1:66 2

30 6 4

17 4 4

12 4 4

10 4 2

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References 1. Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 5. Springer, New York (2000) 2. Kowar, R.: Integral equation models for thermoacoustic imaging of acoustic dissipative tissue. Inverse Problems 26, 095005 (18pp), DOI: 10.1088/0266-5611/26/9/095005 (2010) 3. Kowar, R., Scherzer, O.: Attenuation Models in Photoacoustics. to appear in: Mathematical Modeling in Biomedical Imaging II Lecture Notes in Mathematics, 2012, Volume 2035/2012, 85–130. arXiv:1009.4350 http://arxiv.org/abs/1009.4350 4. Nachman, A.I., Smith, J.F.I., Waag, R.C.: An equation for acoustic propagation in inhomogeneous media with relaxation losses. J. Acoust. Soc. Am. 88(3), 1584–1595 (1990) 5. Waters, K., Mobely, J., Miller, J.G.: Causality-imposed (Kramers-Kr¨onig) relationships between attenuation and dispersion. IEEE Trans. Ultrason. Ferroelectr. Freq. Contr. 52(5), 822–833 (2005)

A Mathematical Model for Drug Delivery Vo Thi Ngoc Tuoi, Rongbing Yang, Yury Rochev, and Martin Meere

Abstract We consider a model for local drug re-distribution in a tissue that incorporates the effects of diffusion and reversible binding with immobile sites within the tissue. The model tracks the evolution of the concentration in the tissue of free drug, specifically and non-specifically bound drug, and specific binding sites. We reduce the model to a scalar nonlinear diffusion equation for the total drug. The model is used to investigate tissue residence time for strongly bound drugs by considering a problem with uniform initial drug concentration and perfect sink boundary conditions. The behaviour predicted by the model has potential implications for the design of local drug delivery systems if the drug is strongly bound.

1 Introduction and Modelling The concentration of molecules in a tissue can depend, among other things, on the rate at which they diffuse through the tissue and their propensity to bind with elements of the tissue; see, for example, Lauffenburger & Linderman [2] or Saltzman [4]. The binding can be highly specific, as is the case for receptor-ligand interactions, or non-specific. The action of many drugs is dependent on the drug molecules binding with high affinity to specific receptors in a tissue. If the binding sites are immobile then binding will clearly have an effect on the rate at which the

V.T.N. Tuoi  M. Meere () School of Mathematics, NUI Galway, Ireland e-mail: [email protected]; [email protected] R. Yang  Y. Rochev NCBES, NUI Galway, Ireland e-mail: [email protected]; [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 60, © Springer-Verlag Berlin Heidelberg 2012

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molecules can move through the tissue, and in this paper we investigate the effect of binding on drug mobility. The model incorporates drug diffusion through the tissue and reversible binding of the drug with immobile sites (both specific and non-specific) within the tissue. We denote by C; B; Bn ; A; An a free molecule, a specific binding site, a nonspecific binding site, a drug-specific binding site complex, and a drug-non-specific binding site complex, respectively. The reversible binding reactions can then be represented by: kon

C C B • A (specific binding); koff

kon,n

C C Bns • Ans (non-specific binding), koff,n

where kon ; koff ; kon,n ; koff,n are rate constants. We denote by c; b; a; an the concentrations of C; B; A; An , respectively, and each of these quantities can depend on time t and a spatial variable x. The governing equations are: @a @an @b D kon bc  koff a; D kon,n bn c  koff,n an ; D kon bc C koff a; (1) @t @t @t @2 c @c D D 2  kon bc C koff a  kon,n bn c C koff,n an ; @t @x where the constants D; bn refer to the diffusivity of the drug and the concentration of non-specific binding sites, respectively. We denote by c  a representative free drug concentration and by L a representative tissue length-scale, and define b  D maxx fF .x/g. We then define the following non-dimensional variables: tN D

t .L2 =D/

; xN D

x a b c an ; aN D  ; bN D  ; cN D  ; aNn D  ; L b b c b

to obtain the dimensionless equations (dropping overbars): @an Kbn @2 c @a Kb @ .a C an C c/ D 2 ; ı D bc a; ın D c an ; a C b D F.x/; @t @x @t  @t  (2) where F .x/ is the distribution of specific binding sites, and D

D D kon b  kon,n bn b ; ı D ; ı D ; K D ; K D ; n b bn c koff L2 koff,n L2 koff koff,n

are dimensionless parameters. The parameter Kb is sometimes referred to as the (specific) binding constant, with Kb  1 corresponding to the drug molecules having a high affinity for their target binding sites. Similar models to the one just described can be found in Lovich & Edelman [3], Sakharov et al. [5], Borghi et al. [1], and Tzafriri et al. [6].

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Model reduction. We confine our discussion to cases for which the diffusion timescale is much longer than the time-scales associated with the binding reactions, so that  ˚ L2 =D  max 1=.kon b  /; 1=koff ; 1=.kon,n b  /; 1=koff,n ; which implies that ı  min.Kb ; 1/ and ın  min.Kbn ; 1/. Hence, we neglect the terms involving ı, ın in (2)2 , (2)3 , respectively, to obtain Kb bc D a; Kbn c D an :

(3)

The total drug concentration (bound plus free), which we denote by cT , is given in dimensionless terms by cT D a C an C c:

(4)

We can write a; an ; b; c as functions of cT using the algebraic relations (2)4 , (3) and (4), so that in this model it is sufficient solve for cT . Equation (2)1 yields the following convection-reaction-diffusion equation for cT : @cT @ @cT C veff .cT ; F / D @t @x @x

  @cT Deff .cT ; F / C reff .cT ; F /; @x

(5)

where 2 Kb F 0 ..1 C Kbn C Kb F / C Kb cT / ; S.cT ; F /3   .1 C Kbn  Kb F / C Kb cT 1 1C ; (6) Deff .cT ; F / D 2.1 C Kbn / S.cT ; F /   2Kb23 cT .F 0 /2 .1 C Kbn C Kb F /  Kb cT F 00 1 C C reff .cT ; F / D ; 2.1 C Kbn / S.cT ; F / S.cT ; F /3

veff .cT ; F / D

are the effective convective velocity, diffusivity, and reaction term, respectively, and S.cT ; F / D

p ..1 C Kb F C Kbn /  Kb cT /2 C 4Kb .1 C Kbn /cT :

(7)

Equation (5) simplifies considerably if the distribution of specific binding sites is uniform, so that F .x/  1. In this case, the convection and reaction terms in (5) vanish, and it reduces to the following nonlinear diffusion equation (with Kbn D 0): @ @cT D @t @x

  @cT Deff .cT / ; @x

(8)

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where the effective diffusivity for the total drug, Deff .cT /, is now given by .1  Kb / C Kb cT 1 1C p Deff .cT / D 2 ..1 C Kb /  Kb cT /2 C 4Kb cT

! I

(9)

this form has also been recently noted in [6]. It is easily seen that Deff .cT /  1 (or Deff .cT /  D in dimensional terms) as would be expected since cT contains the immobile species a. For the case of strong binding, Kb  1, we find that:

Deff .cT / 

8 ˆ ˆ 1 < 1

for cT > ; (Fast rate)  q 1=2 1 C cT = cT2 C 42 for cT D O.1/ where cT D  CKb cT ;

2 ˆ ˆ : 1 2 Kb  =.  cT /2  1

for cT < : (Slow rate) (10)

This result is readily interpreted by noting that cT >  corresponds in dimensional terms to cT > b  , which implies that the total concentration of drug exceeds that of the available binding sites. In this case, we have that for strong binding and at leading order, the available binding sites are occupied and the free drug diffuses unhindered by binding (the fast rate). For cT < , which corresponds to the concentration of binding sites exceeding that of the total drug, the concentration of free drug is asymptotically low and at leading order the drug is bound (the slow rate). We emphasise that for very many systems of interest, Kb is large; for example, for calf vascular tissue, it has been estimated that Kb  40 for paclitaxel and Kb  140 for rapamycin (see [6]).

2 Drug Deposition: A Surface Source Problem We now indicate how the behaviour exhibited in (10) might be exploited in the design of drug delivery devices. We evaluate the speed of tissue penetration for strongly bound drugs by considering the following source problem for Kb  1:   @ @cT @cT D Deff .cT / ; cT D 1 on x D 0; cT ! 0 as x ! 1; cT D 0 at t D 0: @t @x @x (11) This problem could, for example, serve as a crude model for the release of drug from a drug eluting stent. The problem has already been discussed in [6] for the case Kb D 1 in which the diffusivity (10) becomes a step function, and so we need only briefly sketch the results of the analysis for Kb  1 here. The discussion splits into the two cases 1 >  and 1 < .

A Mathematical Model for Drug Delivery Surface source problem with Kb = 200, c* = 2b*

Surface source problem with Kb = 200, c* = b* / 2

1 Total drug concentration (cT / c*)

1 Total drug concentration (cT / c*)

525

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0 0

0.2

0.4

0.6

0.8

1 x

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1 x

1.2

1.4

1.6

1.8

2

Fig. 1 Numerical solutions of (11) for the total drug concentration at t D L2 =D in dimensional terms. Here Kb D 200 and on the left we have  D 1=2 (fast in-diffusion), and on the right  D 2 (slow in-diffusion)

1 > Á: rapid in-diffusion. This corresponds to the case c  > b  so that the surface concentration of drug exceeds the binding site concentration and in the region adjacent to the surface, there is free drug that can diffuse. In the tissue bulk, when the drug concentration becomes comparable with that of the binding sites, the drug profile falls rapidly since the effective diffusivity becomes asymptotically small— this occurs at the so-called binding site barrier, and it is clearly visible in the numerical solution corresponding to this case on the left hand side of Fig. 1. The location of this barrier is determined as part of the solution to a moving boundary problem; see [6] for details. In dimensional terms, the drug penetrates a distance x D O.L/ in time t D O.L2 =D/. 1 < Á: slow in-diffusion. We now have c  < b  which corresponds to the total drug concentration at the surface being maintained at a value below that of the available binding sites, so that the effective diffusivity is asymptotically small. In dimensional terms, the drug penetrates a distance x D O.L/ on the long time-scale t D O.Kb L2 =D/ and in-diffusion is slow; see the right hand side of Fig. 1. This suggests that for drug/tissue systems where strong binding and diffusion are the dominant mechanisms, the drug should be presented to the tissue for the purpose of loading at a sufficiently high concentration (if feasible) for the fast rate to apply.

3 Drug Removal: Uniform Initial Distribution of Drug We now consider a simple problem to investigate the removal of drug from a tissue subsequent to its deposition. We should emphasise that our model only incorporates one mechanism for drug elimination, namely, diffusion in the unbound state. However in some systems, other mechanisms such as metabolism, blood vessel uptake and convection may also be active in drug removal. The tissue is taken to occupy 1 < x < 1 and to be initially uniformly loaded with drug. We impose

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perfect sink conditions for the drug at the surfaces x D  1 and x D 1, so that: @ @cT D @t @x

  @cT Deff .cT / ; cT D 0 on x D ˙1; cT D 1 at t D 0: @x

(12)

We consider the case  < 1 so that the initial concentration of drug exceeds the concentration of binding sites, and briefly sketch the asymptotic behaviour as Kb ! 1; the details will appear in a forthcoming paper. There are three time-scales to consider, but we confine our discussion here to the two of most interest. Short time-scale, t D O.L2 =D/. This is the time-scale over which the unbound drug out-diffuses. In dimensionless terms it corresponds to t D O.1/, and in s t D O.1/, jxj < 1, we pose cT  cT0 .x; t/ as Kb ! 1 to obtain the linear problem: s s @2 cT0 @cT0 s s D ; cT0 D  on x D ˙1; cT0 D 1 at t D 0; @t @x 2

which can be solved by separating variables. There are boundary layers at x D ˙1 that we do not discuss here; see the top curve in Fig. 2 for a numerical solution corresponding to this case. Long time-scale, t D O.Kb L2 =D/. From the point of view of applications, this is perhaps the most important time-scale because it determines the maximum period over which significant amounts of the drug can be present and active in the tissue. In dimensionless terms, it occurs at t D O.Kb /, and writing t D Kb T we pose L cT  cT0 .x; T / in T D O.1/, jxj < 1 as Kb ! 1 to obtain the leading order problem Uniform initial concentration with Kb =100, c* = 2b*. 1

t = 0.2 L2 / D t = 1 L2 / D

Total drug concentration (cT / c*)

0.9

t = 40 L2 / D

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −1

−0.8

−0.6

−0.4

−0.2

0 x/L

0.2

0.4

0.6

0.8

1

Fig. 2 Numerical solutions of (12) for the total drug for various times with Kb D 100 and  D 0:5

A Mathematical Model for Drug Delivery

527 1 Fraction of drug released M(t) / M(infinity)

Fraction of drug released M(t) / M(infinity)

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 eta = 0.2

0.2

eta = 0.5

0.1

0.9 0.8 0.7 0.6 0.5 0.4 0.3 Kb = 0

0.2

Kb = 10

0.1

eta = 1

Kb = 100

eta = 10

Kb = 1000

0

0 0

10

20

30

40

50

0

10

Time t

20

30

40

50

Time t

Fig. 3 Numerical solutions of (12) for the drug fraction released as a function of time. On the left, Kb D 100 and  is varied. On the right,  D 1=2 and Kb is varied

L @cT0 @ D @T @x



L @cT0 2 L 2 @x .  cT0 /

 L L ; cT0 D 0 on x D ˙1; cT0 D  at T D 0:

L Here cT0 ! 0 as T ! 1, and so the drug is eliminated from the tissue on this time-scale; see the bottom curve of Fig. 2.

Release profiles. A quantity that is frequently measured in experiments is the fraction of the available drug that has been released from a system by a given time. Here the fraction of drug released from the tissue by time t is given by 1 M.t/ D1 M.1/ 2

Z

1 1

cT dx;

and some numerical results for this quantity are displayed in Fig. 3. On the left hand side of Fig. 3 we display release profiles for Kb D 100 and various values of . We note that the release behaviour changes dramatically as  is increased through the value 1 (corresponding to b  D c  ), and since  D b  =c  is a quantity that may be varied in experiments, this strong effect may be observable. Hence, for drug/tissue systems in which diffusion and strong reversible binding are the dominant mechanisms, significant insight may be obtained simply from a knowledge of the three parameters L2 =D, Kb  1 and b  . By overwhelming the available binding sites b  , the drug may be loaded onto the tissue via diffusion on the short time-scale t D O.L2 =D/; however, subsequent to deposition, the drug may be present and active in the tissue on the long time-scale t D O.Kb L2 =D/. Acknowledgements VTNT thanks the Mathematics Applications Consortium for Science and Industry (MACSI) and Science Foundation Ireland (SFI) for their financial support (06/MI/005). RY and YR thank SFI for its financial support (08/RFP/MTR1201). MGM thanks MACSI for its support and NUI Galway for the award of a travel grant.

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References 1. Borghi, A., Foa, E., Balossino, R., Migliavacca, F., Dubini, G.: Modelling drug elution from stents: effects of reversible binding in the vascular wall and degradable polymeric matrix. Comput. Meth. Biomech. Biomed. Eng. 11, 367–377 (2008) 2. Lauffenburger, D.A., Linderman, J.J.: Receptors: Models for Binding, Trafficking and Signaling, Oxford University Press, Oxford (2003) 3. Lovich, M.A., Edelman, E.R.: Computational simulations of local vascular herapin desposition and distribution. Am. J. Phys. 271, H2014–H2024 (1996) 4. Saltzman, W.M.: Drug Delivery: Engineering Principles for Drug Delivery, Oxford University Press, Oxford (2001) 5. Sakharov, D.V., Kalachev, L.V., Rijken, D.C.: Numerical simulation of local pharmacokinetics of a drug after intravascular delivery with an eluting stent. J. Drug Target. 10, 507–513 (2002) 6. Tzafriri, A.R., Levin, A.D., Edelman, E.R.: Diffusion-limited binding explains binary dose response for local arterial and tumor drug delivery. Cell Proli. 42, 348–363 (2009)

Estimators of the Intensity of Fibre Processes and Applications Paola M.V. Rancoita and Alessandra Micheletti

Abstract Many objects in the real world can be modeled as fibres (i.e. lines in 2D or 3D space). If the process is invariant under translations, one of its characteristics is the mean length per unit area (called intensity). Under suitable conditions, two estimators of the intensity have been shown to be asymptotically normal when the sample is “enriched” by enlarging the window of observation. We discuss the applicability of these estimators in practice, by using both simulated and real images of fibre processes.

1 Introduction Fibre processes are random geometric objects that can be used in medicine, biology, material science, to model structures like capillaries, radices and nervature of fibrous material [1, 8]. A quantity that characterize a stationary (i.e. with distribution invariant under translations) fibre process ˚ is the density of its length, called intensity and denoted by LA . Therefore, statistical methods for the estimation of the intensity of a fibre process may offer relevant tools for applications. In [3, 5], we proposed an unbiased estimator of the intensity of a stationary twodimensional fibre process, obtained by intersecting the fibre process under study ˚1 with another independent, stationary and isotropic (simulated) test fibre process ˚2 , and by considering the associated counting measure of the intersection points:

P.M.V. Rancoita () Istituto Dalle Molle di Studi sull’Intelligenza Artificiale, Manno-Lugano, Switzerland; Laboratory of Experimental Oncology, Oncology Institute of Southern Switzerland, Bellinzona, Switzerland e-mail: [email protected] A. Micheletti Dipartimento di Matematica, Universit`a degli Studi di Milano, Milano, Italy e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 61, © Springer-Verlag Berlin Heidelberg 2012

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N˚1 \˚2 .W /  b LA;1 .W / D ; 2 .W / 2 LA;2

(1)

where W is the window of observation, 2 .:/ represents the Lebesgue measure in R2 , N˚1 \˚2 .:/ is the counting measure of the point process ˚1 \˚2 and LA;1 and LA;2 are the intensity of ˚1 and ˚2 , respectively. If ˚1 \ ˚2 is ergodic, then the estimator is strongly consistent [3, 5], by enlarging the window of observation. If ˚1 \ ˚2 is also independent in Borel sets at distance greater than l (for some 0  l < 1), than the estimator is also asymptotically normal [6]. The independence at distance l can be, for example, satisfied when the fibres are generated independently and have a finite maximum length (lower than l). The proposed estimator is a generalization of a class of unbiased estimators presented in [4, 8], which had no asymptotic properties, since they were obtained via the intersection with a finite deterministic fibre system. When the fibre system consists of a grid of N circles with radius R which are all contained in the window of observation W , the estimator becomes: b Lcircles .W / D A

N˚ \ .W / ; 4NR

(2)

where ˚ is the fibre process under study and LA is its intensity. Only for this particular the estimator is proportional to the number of intersections (like for b LA;1 ), because of the isotropic shape of the circles, otherwise it generally requires information about the angle of the tangent to the fibres at the intersection points. Since the fibre system is finite, we do not obtain an enrichment of the sample by enlarging the window of observation. Thus, in the case of b Lcircles , we defined a A suitable sequence of systems of circles (corresponding to the sequence of enlarging windows) and we showed that the corresponding sequence of estimators converges in distribution to a normal random variable [6]. Here, we derive a method for the approximation of the variance of the estimators when only one window of observation is available (as in many applications). We also propose how to compute in practice these estimators on digital images and we verify that this computational method maintains the asymptotic normality of the estimators, by using simulated images of fibre processes. In the simulations, we consider Boolean fibre processes for both ˚1 and ˚2 (i.e. the center of the fibres are distributed as a Poisson point process with intensity 1 and 2 , respectively, and the fibres are i.i.d). In particular, we define ˚2 as either a Poisson segment process (i.e. the fibres are uniformly oriented segments of length l2 ) or a Poisson circle process (i.e. the fibres are circles of radius R2 ) and the corresponding estimators are called b LA;1;seg and b LA;1;circ . Finally, we show an application to real images of angiogenesis (i.e. formation of a vascular network).

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2 Variance Approximation on a Single Image Since in many applications only one or few images of the fibre process are available, we derived a method for the approximation of the variance, by using the information coming from only one image. Given a window of observation W of the process ˚1 and denoting by b L an th b estimator of the intensity (b LA;1 or b Lcircles , in our case), we call L the i estimator i A of type b L computed on W , from the i t h realization of ˚2 , in case of b LA;1 , and the b i th system of circles, in case of b L denotes the average of these estimators. Lcircles . A n b Obviously, fLi gi D1 have all the same distribution and, since they are calculated on the same window W and the same realization of the process ˚1 , they are not independent. We call  2 the variance of any estimator b Li , i D 1, : : : , n, and c the covariance of any pair of estimators (c D Cov.b Li ; b Lj /, for any i ¤ j and i; j D 1, : : : , n). It can be easily shown [6] that the sample variance of the estimators is an unbiased estimator for  2  c. Therefore, if c  c1  2 with c1 < 1, then 2 3 n  2 X 1 1 b Lj  b E4 (3) L 5; Var.b Li /  1  c1 n  1 j D1 for all i D 1, : : : , n. From Equation (3), if c1 is close to 1 (i.e. c   2 ), we cannot retrieve a good approximation of Var.b Li / from the sample variance on one image. Main results on simulations. We performed simulation of fibre processes and we computed the estimators, by varying the values of: R, 2 , R2 , l2 , 1 , the shape of the fibres in ˚1 and the side of W (which we considered squared). From the data, we obtained that c1  0:6, for both b LA;1;seg and b LA;1;circ , and c1  1, for b Lcircles . Thus, A circles b we cannot use this method to approximate Var.LA /. Remark 1. The method here proposed to estimate the variance could be compared with other methods involving the estimate of the covariance function of a stochastic geometric process. We leave this comparison to subsequent papers.

3 Computation of the Counting Measure on Digital Images Let us suppose to have a digital black and white image of ˚1 (as in many real applications), where, for example, the fibres are depicted in black and the background in white, like in Fig. 1. In this case, instead of knowing the exact coordinates of the points belonging to the fibres, we only know that such points are located inside the black pixels. Thus, we need an algorithm that is able to identify the intersection points in this situation.

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Fig. 1 From left to right: an example of simulated image of fibre process, an example of computation of an intersection point, an example of underestimation and an example of overestimation. In all images, '1 is represented in black and '2 in grey

In case of b LA;1 , after having simulated the parameters of the fibres of ˚2 (for example, their centers and orientations, if the fibres are segments), we overlap one by one the fibres of ˚2 (here each single fibre is called  ) to the image of the realization of ˚1 (here called '1 ). Then, we follow the pixels of  in a consecutive way, identifying which of them belong also to any of the fibres of '1 (Fig. 1). The number of intersections between  and '1 is given by the number of disjoint sets of consecutive pixels along  , that belong also to '1 . The computation is similar for b Lcircles . In our application to real images of angiogenic processes A (Sect. 4), the number of intersections will be derived by counting the number of extended maxima [7] to avoid the binarization of the image through a threshold, which does not allow a reliable detection of thin capillaries. Since our images of angiogenesis represent a very challenging problem in image analysis, some tools (like skeletons [7] and curve fitting [2]) are not applicable. Obviously, due to the pixel approximation of the fibre, we expect that the algorithm will not be able to identify all the intersections (underestimation) and that sometimes it will detect an intersection where it does not exist (overestimation). For example, the loss of an intersection point happens when two fibres of '1 are close at the intersection points and, in the pixel resolution, they become a unique region (Fig. 1). Viceversa, a higher number of intersections can be counted when  and the intersecting fibre of '1 are almost tangent, so that the pixels of the intersection are not contiguous on  (Fig. 1). Then the program counts as many intersections as the number of these disjoint sets of pixels. Main results on simulations. We verified that this computational method maintains the properties of the estimators, by using simulated images of fibre processes in which we varied the values of: R, 2 , R2 , l2 , 1 , the shape of the fibres in ˚1 and the side of W (which we considered squared). By applying suitable 2 -test, we found that the approximation with the normal distribution was already good when the side of W equals 200 or 300 pixels. We observed that the estimators can over/underestimate LA;1 , but the bias depends only on the shape of the fibres of ˚1 and not on the value of LA;1 . The bias is also the same for all estimators. Thus, this computational method can be used for the purpose of comparing the intensities of fibre processes having fibres with the same shape. Moreover, the convergence of the

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confidence intervals was fast even when we approximated the variance using the information of only one image (Sect. 2).

4 Example of Real Application As example of application, we analyzed digital photos of angiogenic processes obtained at IFOM (FIRC Institute of Molecular Oncology Foundation, Milan). The experiments consisted in implanting on a mouse cornea a pellet, containing an angiogenic factor (called hrFGF-2) together with an antibody: either the antibody nonimmune rat IgG (Rt-IgG), which has no effect against the angiogenic process (i.e. is a placebo), or one of the developed anti VE-Cadherin antibodies (the protein VE-Cadherin plays a fundamental role in the creation of new vessels). For each antibody, a sample of two images (of two eyes) was available. The aim was to determine which of these antibodies was more able to inhibit the formation of new vessels. In order to quantify the effect of a specific antibody in the inhibition of the angiogenic process (induced by hrFGF-2) from the images of the vessels, we estimated some parameters that characterize their geometry. We modeled the capillaries as a stationary planar fibre process and we estimated the intensity of the corresponding processes. Moreover, in order to better compare the geometry of these processes, we also estimated the mean capillary width as the ratio between the area occupied by the capillaries and their estimated length (the length is given by b LA 2 .W /). For the estimation of the intensity of the process of the vessels, we had to identify the capillaries in the images. Since the capillaries were of a vivid red color, while the background of the eye was pale red, we used the saturation values to select the capillaries. The saturation represents the degree of purity/intensity of a color and its range of values is in Œ0; 1 (1 means a pure color). Therefore, the capillaries have a higher saturation (close to 1) than the background (Fig. 2). Moreover, in order to consider the vessels as a stationary fibre process, for each eye, we selected, as window of observation W , only the part of the eye over the limbic vessel (i.e. the round vein) occupied by the capillaries (Fig. 2), where the process looked stationary.

Fig. 2 Image analysis of a mouse eye treated with Rt-IgG

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Table 1 Geometric parameters computed on the images of eyes of mice treated with antibodies: Rt-IgG (placebo), 10G4 and 6D10. The mean capillary width was computed by using the intensity LA;1;circ estimated by b

b b Mean capil. LA;1;seg LA;1;circ Antibody Eye Width [95% conf. int.] [95% conf. int.] [95% conf. int.] Rt-IgG 1 6.999 [6.994, 7.005] 0.093 [0.080, 0.106] 0.088 [0.079, 0.098] 2 10.535 [10.525, 10.545] 0.095 [0.080, 0.110] 0.089 [0.078, 0.100] 10G4 1 12.182 [12.175, 12.189] 0.078 [0.067, 0.089] 0.073 [0.065, 0.081] 2 6.468 [6.461, 6.475] 0.099 [0.086, 0.111] 0.096 [0.085, 0.106] 6D10 1 0.609 [0.608, 0.610] 0.094 [0.076, 0.113] 0.093 [0.078, 0.108] 2 1.243 [1.242, 1.244] 0.093 [0.075, 0.111] 0.088 [0.075, 0.101]

b Lcircles A 0.101 0.098 0.084 0.108 0.105 0.101

To obtain a sufficient number of intersections, we used 2 D 0:008 for both b LA;1;seg and b LA;1;circ and we set l2 =100 and R2 =50 (since the portion of eye which contains the capillaries can be always included in a rectangle of sides 250450 pixels). Concerning the estimator b Lcircles , we set R D 10, because our grid consisted A of nonintersecting and equally spaced circles. Since we had only a sample of two images per treatment, we decided to compute the confidence interval of the intensity for each single image of each eye and we used the method described in Sect. 2 to approximate the variance of the estimators. Thus, we could estimate only the intervals based on the estimators b LA;1;seg and b LA;1;circ , by simulating, and overlapping to '1 , 100 i.i.d. copies of the corresponding process ˚2 . Table 1 shows the results for the antibody Rt-IgG and two anti VE-Cadherin antibodies (10G4 and 6D10). We can observe that generally b LA;1;seg had a larger LA;1;circ was almost included variance than b LA;1;circ . Usually, the interval based on b in the one based on b LA;1;seg . Instead, b Lcircles seemed to overestimate the intensity, A maybe because its variance is high due to the small number of circles in the test system. From the results, we cannot say that the intensity of the fibre processes corresponding to the three treatments are significantly different (i.e. the mean length of the capillaries per unit area is similar in the three cases). Instead, the mean width of the capillaries is significantly lower by using 6D10, than by using the other antibodies. Thus, 6D10 is more able to inhibit the angiogenic activity of the protein VE-Cadherin as confirmed by our biologist expert.

5 Conclusions We discussed the applicability of two estimators for the intensity of a stationary fibre process, which derive from the intersection either with another independent test fibre process (b LA;1 ) or with a deterministic grid of circles (b Lcircles ). In particular, A our automatic way of counting the number of intersections (on a digital image) maintains the asymptotic normality of the estimators, but the estimators calculated in this way are biased. The bias depends only on the shape of the fibres of ˚1 , thus

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the procedure can still be used in case of comparison of intensities of fibre processes with the same shape of the fibres. We also derived a way to approximate the variance of the estimator b LA;1 when only one image of the fibre process is available and we verified that this is a good approximation even for small windows of observation. Finally, the results obtained on images of real angiogenic processes were coherent with the experimental (qualitative) evidence. Acknowledgements We acknowledge Prof. Elisabetta Dejana (IFOM, Milano) for providing us the real data and Alessandro Giusti (IDSIA, Lugano) for useful discussions regarding the computational method. A.M. has been partially supported by the Italian Project PRIN 2007 77BWEP003 Dalla modellizzazione stocastica alla statistica di processi aleatori a struttura spaziotemporale in dinamica delle popolazioni. P.M.V.R. has been supported by Swiss National Science Foundation (grant 205321-112430).

References 1. BenLes, V., Rataj, J.: Stochastic Geometry: Selected Topics. Boston Kluwer Academic Publishers, Dordrecht (2004) 2. Dierckx, P.: Curve and Surface Fitting with Splines. Oxford University Press Inc., New York (1996) 3. Micheletti, A., Rancoita, P.M.V.: Estimators of the intensity of stationary planar fibre processes. In: Capasso, V., Micheletti, A., Aletti, G. (eds.) Stereology and Image Analysis. Ecs10 Proceedings of the 10th European Congress of ISS, The MIRIAM Project Series, pp. 131–136. ESCULAPIO Pub. Co., Bologna, Italy (2009) 4. Ohser, J.: A remark on the estimation of the rose directions of fibre processes. Mathematische Operationsforschung und Statistik. Ser. Stat. 12, 581–585 (1981) 5. Rancoita, P.M.V.: Statistics of fibre processes. Applications to angiogenesis. MSc thesis, Universit`a degli Studi di Milano (2006); In Italian. 6. Rancoita, P.M.V.: Stochastic methods in cancer research. Applications to genomics and angiogenesis. Ph.D. thesis, Universit`a degli Studi di Milano (2010) 7. Soille, P.: Morphological Image Analysis. Principles and Applications. Springer, Berlin, Heidelberg, New York (1999) 8. Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic Geometry and its Applications. (2nd eds). Wiley, Chichester, New York, Brisbane, Toronto, Singapore (1995)

Optimal Control Strategies for Stochastic/Deterministic Bioeconomic Models Darya Filatova

Abstract In this work the new model, namely the stochastic differential equation with multifractional Brownian motion, is proposed to describe the dynamics of the population in the task of the optimal fishery management. To avoid the problems of the identifiability of the model and to take into account the discounted rate of the population, the stochastic control problem is transformed to the deterministic one by suitable moments approximation of the order 0 <  < 1. This transformation results the singular integral equation as the control object equation. Taking into account both control and state constraints and applying the variation calculus we derive the first-order necessary conditions in the form of the local maximum principle.

1 Introduction Bio-economic modeling helps to solve many problems of the biological system exploitation. Among these problems are the concept of harvesting and foresting strategies, their replanting, or regulation. It is clear that problems, for an example of overfishing and overexpansion of fishing capacity, are still appear. Therefore the consequent policy for resource management is required [7]. Let us formulate the optimal control problem in general form taking into account uncertainties which are common for the biological communities. Let the time interval Œt0 ; t1  be fixed, X 2 R denote the state variable (the size of the biomass), and u 2 Rp denote the vector of control variables (the harvest rate). Both variables are somehow connected with stochastic process B.t/ defined on   the filtered probability space ˝; F ; fFt gt  0 ; P , which satisfy the usual condition [6]. We also suppose that the control is taken from the set U Œt0 ; t1  , D. Filatova () Dorodnicyn Computing Centre of RAS, ul. Vavilova 40, 119991 Moscow, Russian Federation, and UJK, Kielce, Poland e-mail: daria [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 62, © Springer-Verlag Berlin Heidelberg 2012

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˚  u W Œt0 ; t1   ˝ ! U j u ./ is fFt gt  0 -adapted , where U is a given metric space, and that the cost function has the Bolza form J .u .// D E

8 t
t0

9 =

F .t; X.t/; u.t// dt C ' .X .t1 // ; ;

(1)

where F and ' are smooth (C 1 ) functions; E Πis the mathematical expectation operator. Definition 1. A control u ./ is called an admissible control, and .X./; u.// an admissible pair, if: (1) (2) (3) (4)

u.t/ 2 U Œt0 ; t1 . X.t/ is the unique solution of a dynamic equation. Some prescribed state and control constraints are satisfied. 1 1 S .t; X.t/; u.t// 2 LF .Œt0 ; t1  ; R/ and ' .X .t1 // 2 LF .˝I R/. t 1

The goal is to find uopt ./ 2 Uad Œt0 ; t1  (if it ever exists), such that   inf J .u .// . J uopt ./ D u./2Uad Œt0 ;t1 

(2)

It is clear that fishery control strategies are very sensitive to small errors in population and stock data. So, the solution of Problem (2) mostly depends on the condition (2) of Definition 1, that is to say on a mathematic model of the dynamic equation. To make the model more stable for different kinds of uncertainties we propose the stochastic differential equation (SDE) with multifractional Brownian motion (mfBm), namely: dX.t/ D f .t; X.t/; u.t// dt C

n X

qi .t; X.t// d BHi .t/;

(3)

i D1

where f W Œt0 ; t1   R  U ! R , q W Œt0 ; t1   R ! R; X .t0 / D X0 2 R, d BHi .t/ is the increment of the fractional Brownian motion BHi .t/ with the Hurst parameter Hi 2 0; 1Œ, processes BHi .t/ are assumed to be mutually independent; the solution is treated as a strong one. Actually, if one solves a real-world problem, he/she does not know results of parametric estimation of proposed model as well as of model verification. We can imagine the situations when the Hurst parameter Hi D 12 or qi .t; X.t// D 0 for all t 2 Œt0 ; t1  and any i D 1; 2; : : : ; n. That means that the same method based on stochastic calculus can not be used to solve the Problem (2) [1]. The purpose of this work is to uniform solution of the Problem (2), which does not depend on the results of the identification of the object equation. The paper is organized as follows. In Sect. 2, we formulate the problem of optimal

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539

fishery management. In Sect. 3, we derive the necessary optimality conditions, using principles of Dubovitski-Milyutin method [4], [5]. Finally, Sect. 4 provides conclusions.

2 Fishery Management Problem 2.1 Generalized Model For the simplicity, we study object (3) in the case when n D 2. The SDE (3) describes the size of a certain population of fish, with X.t/ representing the total biomass. Let the stable growth of the population be described by the stochastic logistic model dX.t/ D Œ1 X.t/ .1  2 X.t//  u.t/ dt

(4)

H2 1 C q1 .t; X.t// d BH t C q2 .t; X.t// d Bt ;

rate, where 1 is the intrinsic   growth  2 is the carrying capacity, the harvest rate 0  u.t/  umax , H1 2 0; 12 , H2 2 12 ; 1 , and X .t0 / D X0 . Set F .t; X  .t/; u.t// D e t Œp .t; u.t//  c .t; X  .t// ; u.t/, where p .; / is the inverse demand function and c .; ; / is the fishery cost function,  is the annual discount rate, and 0 <   1 is the discount rate to avoid the overexploitation of the fish resource. Then the coast function (1) means that the objective of the investor is to maximize the expected discount total utility of the used fish resource with respect to the harvest rate limitation and stability of the ecosystem.

2.2 Some Required Transformations To get uniform solution of the Problem (2) with respect to generalized fishery management model we approximate SDE (4) by the moment equation [3]  .  1/ X   1=2 2 qi t; x .t/ .dt/2Hi , 2 i D1 2

dx D f .t; x.t/; u.t// dt C

where x.t/ D E ŒX  .t/, x .t0 / D E ŒX  .t0 /, and according the the role of integration with respect to .d /2H come to the integral equation of the form

540

D. Filatova

Zt x.t/ D x .t0 / C

Zt f .; x./; u.// d  C  .1   / H1

t0

2 6 C 4H2

Zt

q

1 2

.1   / q2 .; x.// .t  /

t0

1H2

32

t0

q1 .; x.// .t  /12H1

d  (5)

7 d5 ;

or Zt

Zt f .; x./; u.// d  C  .1   / H1

x.t/D t0

t0

Zt y.t/Dy .t0 / C t0

g .x.// .t  /1  H2

q1 .; x.// .t  /1  2H1

d  C G .y.t// (6)

d

where x 2 R, y 2 R, u 2 Rp , y .t0 / 2 R, g ./ and G ./ are smooth (C 1 ) functions. So, the stochastic optimal control Problem (2) with respect to the object equation is converted into the deterministic optimal control problem. It means that the identification of the parameters of (4) does not influence strategy of the optimal solution. In the next section we are going to use Dubovitski-Milyutin method to derive the first-order necessary conditions of optimality.

3 Local Maximum Principle Let the time interval Œt0 ; t1  be fixed, x; y 2 R denote the state variables, and u 2 Rp denote the vector of control variables. The goal function (1) takes form Zt1 J .u .// D

F .t; x.t/; y.t/; u.t// dt C ' .x .t1 // ;

sup u./2Uad Œt0 ;t1 

(7)

t0

where F and ' are smooth functions, and is subjected to: • The control constraint  .u/ is a smooth vector function of the dimension p  .u.t//  0I

(8)

• The state constraint ˚ .x/ is a smooth function of the dimension m ˚ .x.t//  0I

(9)

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• The object (6). The nonlinear control problem with the integral (6) as the control object and only with the control constraint (8) was studied in [2]. Here we added the state constraint (9) and got the necessary optimality conditions in the form of the following theorem. Theorem 1. Let .x.t/; y.t/; u.t// be an optimal process on the interval Œt0 ; t1 , where x ./ 2 C .Œt0 ; t1  ; R/, y ./ 2 C .Œt0 ; t1  ; R/, u ./ 2 L 1 .Œt0 ; t1  ; Rp /. Then there exists a set of Lagrange multipliers .˛0 ; ./ ;  ./ ; / such that ˛0 is a scalar, ./ W Œt0 ; t1  ! R is a function of bounded variation continuous from the left defining the measure d ,  ./ W Œt0 ; t1  ! Rp is an integrable function, ./ W Œt0 ; t1  ! R is a function of bounded variation continuous from the left defining the measure d , and the following conditions are fulfilled: • • • •

Nonnegativity ˛0  0, .t/  0 a.e. on Œt0 ; t1 , d  0; Nontriviality ˛0 C k k C kd k > 0; Complementarity .t/ .u.t// D 0 a.e. on Œt0 ; t1 , ˚ .x.t// d .t/ D 0I Adjoint equation 2 d .t/ D

.t/fx .t/dt  k 4

Zt1 t

3 d ./ .t1 / 5  dt .  t/1 .t1  t/1

2 t Z1 0 G .y.// d ./ 4  .  t/1 t

3 .t1 / G 0 .y .t1 // 5 0 g .x.t// dt .t1  t/1

C˛0 Fx .t; x.t/; u.t// dt  ˚ 0 .x.t// d .t/I • Transversality condition .t1 / D ˛0 ' 0 .x .t1 // I • Local maximum principle .t/fu .t; x.t/; u.t// C ˛0 Fu .t; x.t/; u.t//  .t/ 0 .u.t// D 0; where k D  .1   / H1 q1 .; x .//, 1 D 1  2H1 , and 2 D 1  H2 . Proof. To prove this theorem we need to rewrite the task (6)–(9) as the auxiliary Milyutin’s task. Since the object (6) is presented by the system of the singular integral equations, the main difficulty here is the deducing of the Euler equations. Define a nonlinear operator P W .x; y; u/ 2 C  C  L 1 . The equation P .x; y; u/ D 0 is equivalent to the system (6). An arbitrary functional `, vanishing on the kernel of the operator P 0 .x; y; u/, has the form

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Zt1 ` .x; y; u/ D

Zt1 x.t/d 1 .t/ 

t0

2 4

t0

Zt1 k t0

2 4

Zt

3 .fx ./x./ C fu ./u.// d  5 d 1 .t/

t0

3 Zt1 x./ 5 d  d 1 .t/  G 0 .y.t/y.t// d 1 .t/ .t  /1

t0

t0

Zt1

Zt1 y.t/d 2 .t/ 

C

Zt

t0

t0

2 4

Zt

3 g 0 ./x./ 5 d  d 2 .t/; .t  /2

t0

where  stands for “variations”. Using classical methods of the variation calculus we get the Euler equation Zt1 ˛0

.Fx .t/x.t/ C Fu .t/u.t// dt  ˛0 ' 0 .x1 .t/x .t1 // C ` .x; y; u/ (10)

t0

˝

˛ C ;  .u .// u ./ C 0

Zt1

˚ 0 .x.t// x.t/d .t/ D 0;

t0

where  2 .L1 / ,   0, d 2 C  , d  0, ˚ .x.t// d .t/ D 0. The Euler equation (10) fulfills Tonelli’s theorem [5], then setting x D 0 and y D 0 in this equation and using the fact that u 2 L 1 is an arbitrary element, we deduce the local maximum principle, if u D 0 and x, y are arbitrary we deduce the adjoint equation and transversality conditions. Now the optimality of the solution for the task (6)–(9) can be proved by means of the Pontryagin’s theorem in the form of Dubovitski-Milyutin scheme. t u

4 Conclusions We have considered an optimal control problem of the fishery management using the new model for the fish stock description. We showed how to transform the object equation, presented as the SDE with mfBm, to overcome the problem of the identifiably and to take into account constraints of discounted fish stock, coming to the singular integral equation. New necessary optimality conditions in the form the local maximum principle were obtained by means of the Dubovitski-Milyutin method.

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References 1. Biagini, F., Hu, Y., Øksendal, B., Sulem, A.: A stochastic maximum principle for the processes driven by fractional Brownian motion. Stoch. Process. Appl. 100, 233–253 (2002) 2. Filatova D., Grzywaczewski M., Osmolovskii N.: Optimal control problem with an integral equation as the control object. Nonlinear Anal. Theor. Meth. Appl. 72(3–4), 1235–1246 (2010) 3. Jumarie G.: Lagrange mechanics of fractional order, Hamilton-Jacobi fractional PDE and Taylor’s series of nondifferentiable functions. Chaos Solut. Fractals 32, 969–987 (2007) 4. Milyutin A.A., Osmolovskii N.P.: Calculus of Variations and Optimal Control. Translations of Mathematical Monographs 180, American Mathematical Society, Providence (1998) 5. Milyutin A.A., Dmitruk A.V., Osmolovskii N.P.: Maximum Principle in Optimal Control. Moscow State University, Moscow (2004); (in Russian). 6. Shyryaev A.N.: The Basis of Stochastic Financial Mathematics: Facts and Models. FAZIS, Moscow (1998); (in Russian) 7. Sethi, G., Costello, C., Fisher, A., Hanemann, M., Karp, L.: Fishery management under multiple uncertainty. J. Environ. Econ. Manag. 50, 300–318 (2005)

Fishway Optimization Revisited Lino J. Alvarez-V´azquez, Aurea Mart´ınez, Joaquim J. Judice, ´ Carmen Rodr´ıguez, Miguel E. V´azquez-M´endez, and Miguel A. Vilar

Abstract River fishways are hydraulic structures enabling fish to overcome stream obstructions (for instance, dams in hydroelectric power plants). This paper presents a combination of mathematical modelling and optimal control theory in order to improve the optimal shape design of a fishway. The problem can be formulated within the framework of the optimal control of partial differential equations, approximated by a discrete optimization problem, and solved by using a gradienttype method (the Spectral Projected-Gradient algorithm). Numerical results are shown for a standard real-world situation.

1 Introduction Several species of fish (salmon, trout, carp, sturgeon, eel. . . ) attempt migrations between fresh and salt water on a regular basis, on time scales varying from daily to annual, and with distances ranging from a few meters to thousands of kilometers. L.J. Alvarez-V´azquez ()  A. Mart´ınez Departamento de Matem´atica Aplicada II, ETSI Telecomunicaci´on, Universidade de Vigo, 36310 Vigo, Spain e-mail: [email protected]; [email protected] J.J. J´udice Departamento de Matem´atica da Universidade de Coimbra, and Instituto de Telecomunicac¸o˜ es, 3000 Coimbra, Portugal e-mail: [email protected] C. Rodr´ıguez Departamento de Matem´atica Aplicada, Facultade de Matem´aticas, Universidade de Santiago de Compostela, 15706 Santiago, Spain e-mail: [email protected] M.E. V´azquez-M´endez  M.A. Vilar Departamento de Matem´atica Aplicada, Escola Polit´ecnica Superior, Universidade de Santiago de Compostela, 27002 Lugo, Spain e-mail: [email protected]; [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 63, © Springer-Verlag Berlin Heidelberg 2012

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g1

g0 g2

0,97 m Domain w 0,6065 m

12,13 m

Fig. 1 Ground plant (domain ! with lateral boundary 0 , inflow boundary 1 and outflow boundary 2 ) and elevation of a standard fishway

So, when we construct an artificial barrier in a river (for example, a weir or a dam in a hydroelectric power plant) European legal regulations force us to also install a fishway in order to allow fish to overcome it. Fishways are hydraulic structures placed around man-made barriers to assist the natural migration of fish. The vertical slot fishway is the more widely adopted for upstream passage of fish in stream obstructions. It consists of a rectangular channel with a sloping floor that is divided into a reduced number of pools (see Fig. 1). Water runs downstream in this channel, through a series of vertical slots from one pool to the next one below. The water flow forms a jet at the slot, and the energy is dissipated by mixing in the pool. Fish ascends, using its burst speed, to get past the slot and then it rests in the pool until the next slot is tried [4]. During recent decades much attention has been paid, both from the theoretical and the experimental viewpoint, to the hydraulic characteristics, the flow regimes, and the vorticity structures in all types of river fishways. Nevertheless, the fundamental role of a correct design in the fishway has been much less studied. As far as we know, the optimal design of a vertical slot fishway has only been previously analyzed by the authors [1, 2] in a simple case. As stated above, the objective of a fishway is enabling fish to overcome obstructions. In order to get it, water velocity in the fishway must be controlled. Specifically, this means that in the zone of the channel near the slots, the velocity must be close to a desired velocity suitable for fish leaping and swimming capabilities. In the remains of the fishway, the velocity must be close to zero for making possible the rest of the fish. Moreover, in all the channel, flow vorticity must be minimized. Water velocity can be directly controlled by determining the optimal shape of the fishway, that is, the location and length of the baffles separating the pools. In this work we use mathematical modelling and optimal control theory to address the optimal design of a fishway. In order to do this, we begin presenting a mathematical model (shallow water equations) to simulate the water velocity in a fishway and giving a mathematical expression to evaluate the quality of that velocity field in terms of the fish capabilities. Next, we study the problem of the optimal design of a fishway. We describe the problem, formulate it as a shape optimization problem, and show that it can be approximated by a discrete optimization problem. A gradienttype method (the Spectral Projected-Gradient algorithm) is proposed to solve this problem, and numerical experiences are reported.

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2 Simulation of Fishway Hydrodynamics Let !  R2 be the ground plant of a fishway consisting of a rectangular channel divided into a small number of pools with baffles and sloping floor, and two transition pools (one at the beginning and another one at the end of the channel) with no baffles and flat floor. A scheme of the standard fishway used in this paper can be seen in Fig. 1: water enters by the left side and runs downstream to the right side, while fish ascend in the opposite direction [1]. The number of pools (ten) and the dimensions of the full channel correspond to an experimental scale fishway reported by Puertas et al. [5]. Water flow inside the domain ! along the time interval .0; T / is governed by the (Saint-Venant) shallow water equations [2]: @H C r :Q D 0; @t

  @Q Q Cr: ˝ Q C gH r .H  / D f @t H

in !  .0; T /; (1)

where H.x; y; t/ is the height of water at point .x; y/ 2 ! and at time t 2 .0; T /; u.x; y; t/ D .u; v/ is the depth-averaged horizontal velocity of water, Q.x; y; t/ D uH is the flow per unit depth, .x; y/ represents the bottom geometry, and second member f collects all the effects of bottom friction, atmospheric pressure and so on. These equations must be completed with a set of initial and boundary conditions on the lateral boundary of the channel (denoted by 0 ), the inflow boundary (denoted by 1 ), and the outflow boundary (denoted by 2 ). By using these notations we can give a mathematical expression to evaluate the quality of water velocity in the fishway. We have to bear in mind two objectives: (1) In the zone of the channel near the slots (say the lower third) the velocity must be as close as possible to a typical horizontal velocity  suitable for fish leaping and swimming capabilities; in the remaining of the fishway, the velocity must be close to zero for making possible the rest of the fish. In short, the velocity must be close to the following target velocity:  v.x1 ; x2 / D

.; 0/; if x2  13 W; .0; 0/; otherwise ,

(2)

where W is the width of the channel (in our case W D 0:97 m, as shown in Fig. 1). (2) Flow vorticity must be minimized in all the channel in order to avoid fish disorientation, that is, the vorticity must be as much reduced as possible. According to this, if we fix a weight parameter ˛  0 for the role of the vorticity and define the objective function 1 J D 2

Z

T 0

Z

Q ˛ k  vk2 C 2 ! H

Z

T 0

Z !

jcurl.

Q 2 /j ; H

(3)

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the water velocity u D Q=H is better for our purposes as the value of the cost function J becomes smaller. In order to evaluate J , we need to solve the shallow water equations (1). In this work we use an implicit discretization in time, upwinding the convective term by the method of characteristics, and Raviart-Thomas finite elements for the space discretization [1]. If we consider a time step t D T =N > 0 and a Lagrange-Galerkin finite element triangulation h of the domain !; the numerical scheme provides an approximated flux Qkh and an approximated height Hhk ; which are piecewise-linear polynomials and discontinuous piecewise-constant functions, respectively, and are required to compute an approximated value of J .

3 Optimal Shape Design of a Fishway In this section we study how to improve the optimal design of a fishway. As commented before, we can control the water velocity through the location and length of the baffles in the pools. To fix ideas, we take the channel described in previous section, we assume that the structure of the ten pools with sloping floor has to be the same (the shape of the complete fishway is given by the shape of the first pool) and then, we take the three midpoints corresponding to the end of the baffles in the first pool (points a D .s1 ; s2 /, b D .s3 ; s4 / and c D .s5 ; s6 / in Fig. 2) as design variables. So, we look for points a, b and c providing the best velocity for fish (i.e. minimizing the function J given by (3)), but, previously, we must impose several design constraints on these points. First, we assume that points a, b and c are inside the dashed rectangle of Fig. 2, that is, the following twelve geometrical relations must be satisfied (in order to avoid unnecessary symmetrical duplicate solutions): (0,0.97)

3 1.213 4 c=(s5,s6) Δ4 Δ3 1.213 4

Fig. 2 Scheme of the first pool

O = (0,0)

a=(s1,s2) Δ2

b=(s3, s4)

0.97 2

Δ1 (1.213,0)

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1 3 1:213  s1 ; s3 ; s5  1:213; 4 4

0  s2 ; s4 ; s6 

1 0:97: 2

(4)

A second type of constraints is related to the fact that the vertical slot must be large enough so that fish can pass comfortably through it. This translates into the two following linear constraints: 1 D s3  s1  0:1;

2 D s2  s4  0:05:

(5)

Finally, a third type of constraints is related to structural stability, as given by the two additional linear constraints, with r D 0:0305 the half width of the baffle: 3 D s1  s5 

1 0:0305; 2

4 D s6  s2 

1 0:0305: 2

(6)

Then, the optimization problem can be formulated as follows: Problem (P): Find the optimal shape of domain !, that is, find s D .a; b; c/ D .s1 ; s2 ; s3 ; s4 ; s5 ; s6 / 2 R6 verifying constraints (4)–(6), in such a way that Q and H , given by the solution of the state system (1) on the fishway !  !.s/, minimize the objective function J  J.s/ defined by (3). (A mathematical analysis of a simpler related problem can be found in Alvarez-V´azquez et al. [1].) For its numerical resolution we propose the Spectral Projected-Gradient (SPG) algorithm, due to Birgin et al. [3], where the required derivatives are approached by finite difference approximations. This algorithm provides in each iteration an admissible point in the convex set ˝ defined by the constraints (4)–(6). To achieve this, we designed a special polynomial-time algorithm to compute the projection of a vector over this set. Moreover, global convergence for the SPG method is assured under reasonable hypotheses (see [3] for more details).

4 Numerical Results We give here some numerical results obtained for a standard situation. We considered the fishway under study, whose scheme is shown in Fig. 1. The time interval for the simulation was T D 300 s. Moreover, for the sake of simplicity, for the second member f we considered only the bottom friction stress for a Chezy coefficient of 57:36 m0:5s 1 . For the objective function we took a target velocity value  D 0:8 ms 1 , and a weighting parameter ˛ D 0:15. Finally, for the time discretization we took N D 3;000 (that is, a time step of t D 0:1 s) and, for the different space discretizations, we tried several regular triangulations of about 9,500 elements. Although we have developed many numerical experiences, we present here only one example for this realistic problem. So, applying the Spectral Projected-Gradient algorithm, we have passed, after only 6 iterations, from the initial cost J D 551:05,

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Fig. 3 Initial random horizontal velocity field in the central pool at final time T D 300 s

Fig. 4 Optimal horizontal velocity field in the central pool at final time T D 300 s

corresponding to the random points a D .0:6; 0:15/; b D .0:9; 0:07/; c D .0:5; 0:4/, to the minimum cost J  D 242:67, corresponding to the optimal design variables a D .0:71; 0:16/; b  D .0:91; 0:06/; c  D .0:41; 0:42/. The total process took about 38 hours of CPU time in a laptop with Intel Pentium 4 microprocessors. Two close-ups of the central pool are shown, respectively, in Figs. 3 and 4. In the case of the initial random shape (Fig. 3) we can identify the standard flow patterns presented, for instance, in [5]: a direct flow region where the flow circulates in a curved trajectory at high velocity from one slot to the next downstream, and two recirculation regions—the larger one located between the long baffles and the smaller one located between the short baffles—flowing in opposite directions. In the case of the optimal shape (Fig. 4) the direct flow velocity is very close to the target horizontal velocity v, the smaller recirculation region is completely removed, and the larger one is highly reduced. Acknowledgements Financial support provided by Project MTM2009-07749 of MICINN (Spain), and Project INCITE09PXIG291083PR of Xunta de Galicia is gratefully acknowledged.

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References 1. Alvarez-V´azquez, L.J., Mart´ınez, A., V´azquez-M´endez, M.E., Vilar, M.A.: An optimal shape problem related to the realistic design of river fishways, Ecological Eng. 32, 293–300 (2008) 2. Alvarez-V´azquez, L.J., Mart´ınez, A., V´azquez-M´endez, M.E., Vilar, M.A.: Vertical slot fishways: modeling and optimal management, J. Comput. Appl. Math. 218, 395–403 (2008) 3. Birgin, G., Mart´ınez, J.M., Raydan, M.: Monotone Spectral projected gradients on convex sets, SIAM J. Optim. 10, 1196–1211 (2000) 4. Clay, C.H.: Design of fishways and other fish facilities. CRC Press, Boca Raton (1995) 5. Puertas, J., Pena, L., Teijeiro, T.: Experimental approach to the hydraulics of vertical slot fishways, J. Hydraul. Eng. 130, 10–23 (2004)

On a Wind Farm Aggregate Model Based on the Output Rescaling of a Single Turbine Model Luciano De Tommasi and Madeleine Gibescu

Abstract In transmission and distribution grids with a large share of renewable and distributed resources, the potential instabilities (and possible control actions) related to large power fluctuations can be investigated by means of simulation models. In this scenario, suitable methods of aggregation have to be considered to lower the computational cost of simulations. An aggregate model of a complex system is a reduced model which allows avoiding the repetition of a component model the number of times that such a component is present in the system. This paper reviews a wind farm aggregate model especially tailored to reproduce the power fluctuations due to wind turbulence (up to about 10 min). The structure of the model is justified by means of statistical considerations.

1 Introduction An important emerging characteristic of the modern electrical power system is that the share of renewable and distributed generation (such as wind and solar energy) is increasing more and more. It is foreseen that this will lead to replacement of some conventional power plants. Renowable generation constitutes a less controllable energy source than conventional power plants. This inherent property of wind generation may have an impact on the stability of the whole electrical system which

L. De Tommasi () Energy research Center of The Netherlands, 1755ZG Petten, The Netherlands Current address: United Technologies Research Center, Lee Mills House, Prospect Row, Cork, Ireland e-mail: [email protected] M. Gibescu Department of Electrical Sustainable Energy, Delft University of Technology, 2600 CD Delft, The Netherlands e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 64, © Springer-Verlag Berlin Heidelberg 2012

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has to be studied carefully. To succeed in this task, simulation models are, of course, of paramount importance. Simulation models have to be as efficient as possible while preserving the fundamental properties of the physical objects they represent, with a level of detail that is appropriate for the type of simulation study at hand. The work reported in this paper concerns the aggregate modeling of a wind farm. To explain what an aggregate model is, and why it is useful, it is worthwhile to observe that several physical systems include a number of identical components. For example, a wind farm typically includes from some tens to hundred identical wind turbine generators (WTGs). A straightforward method to represent such a system in a computer simulation environment is to repeat the component model the number of times that such a component is present in the system. This is an accurate but very inefficient method. In the remaining of this paper we refer to this representation as full model. An alternative method is to aggregate the N different components by representing all of them with a number of instances of the component model M < N . Aggregation may be full or partial. In the latter case, only some subparts of the considered models (e.g. WTGs) are aggregated together. The basic building block of a wind farm model is the model of a single WTG [1, 2]. A WTG can be modelled as a non-linear, time-invariant, dynamical system, whose state space representation is: (

D QfŒx.t/; u.t/ Q y.t/ Q D hŒx.t/; u.t/

d dt x.t/

(1)

where x is the state vector, yQ is the output and u is the input. Following [3], we assume that the output yQ is the real power fed into the grid, whereas u is the wind speed acting on the turbine at hub height. We define: u.t/ D uo C utur .t/, where uo is the mean wind speed in a given time interval (e.g. 10 min) and utur is a stochastic function known as turbulence. We assume that u.t/ is a wide-sense stationary process. The power spectral density Stur of the turbulence utur .t/ follows the Kaimal spectrum: Stur .f / D

2 tur 4f L=uo f .1 C 6f L=uo / 53

(2)

where tur is the standard deviation of utur .t/, f is the frequency and L is the integral length scale of the wind speed at hub height. Moreover, we assume that the wind turbulence can be described by means of a Gaussian probability density function [4]: 2

 u2 1 e 2tur utur .t/  p.u/ D p 2tur

(3)

Wind simulation is included in computer programs for wind turbine design, e.g. [1].

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Assuming that the output y.t/ of a wind farm including N turbines can be obtained by adding the outputs yQi .t/ of the single turbine models [2], a state-space representation of the full model is: 8 d Q ˆ < dt xn .t/ D fŒxn .t/; un .t/; n D 1 : : : N Q n .t/; un .t/; n D 1 : : : N yQn .t/ D hŒx ˆ P : y.t/ D n yQn .t/

(4)

An aggregate model [5–7] is a model of reduced complexity which approximates (4) such that: ya .t/  y.t/

(5)

ya .t/ being the output of the approximate model. The type of approximation (5) depends on the specific application at hand. If the output of the aggregate model has to follow closely the one of the full model, it is natural to try achieving that the deviation between the aggregate model’s output and the full model’s output is less or equal than a certain quantity: jya .t/  y.t/j  ;

8t

(6)

An approximation like (6) requires that at least some of the wind speed realizations un .t/ of the full model (4) are used as inputs for the aggregate model as well. However, when it is sufficient to approximate just the qualitative behavior of the full model, then the requirement (6) can be relaxed, such that the output of the aggregate model ya approximates the output of the full model in a convenient statistical sense, e.g.: 8 ˆ < EŒya .t/  EŒy.t/ VarŒya .t/  VarŒy.t/ ˆ : rya ./  ry ./

(7)

where ry ./ D EŒy.t C /y.t/. It is remarked that EŒya .t/ and EŒy.t/, as well as VarŒya .t/ and VarŒy.t/, do not depend on time because ya .t/ and y.t/ are stationary as a consequence of the stationarity of un .t/ and the fact that the WTG is time invariant. To achieve the approximation (6), a partial aggregation method (cluster or compound representations [5]) is more suitable, whereas in the latter case a single machine representation is sufficient. A novel wind farm aggregate model based on the output rescaling of a single turbine model was introduced in [3]. It extends the work in [11] to variable speed wind turbines (doubly-fed induction generator) and enables an accurate modeling of the smoothing effect given by the wind turbulence on the wind farm’s output power. In this paper, we develop a discussion on the model [3], showing that it actually fulfills the conditions in (7).

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The reader is referred to [3] for simulation results comparing the aggregate against the full model.

2 The Proposed Aggregate Model A number of aggregate models have been proposed in literature. The model introduced in [3] differs from them by the fact that it retains the physical parameters of a single turbine model and rescales the output to account for the contribution of multiple turbines. A typical single machine aggregate model [8–10] uses “equivalent” parameters. This means that the physical parameters of the single turbine model are rescaled to take into account the presence of multiple turbines. The state space representation is: (

d dt x.t/

D fŒx.t/; u.t/

yaN .t/ D hŒx.t/; u.t/

(8)

where the effect of parameters rescaling is taken into account in the functions f and h which replace Qf and hQ of the single turbine model. The proposed model differs from the models [8–10], because it retains the physical parameters of the single turbine dynamic model and produces the aggregate model output by filtering its output (see also [11, 12]): (

d x.t/ dt

D QfŒx.t/; u.t/

ya .t/ D h0 Œx.t/; uo ; u.t/

(9)

It is remarked that the function Qf in the first of (9) is the same one used in the first of (1). The model rescales the output of a single turbine model y.t/ Q as follows: ya .t/ D

p N .y.t/ Q  O yQ / C N O yQ

(10)

Q The computation of O yQ is shown in the where O yQ is an estimation of yQ D EŒy.t/. appendix.

2.1 Statistical Properties In this section, we compare mean, variance and autocorrelation of both full and aggregate models. Wind turbulences acting on different WTGs are assumed incoherent, i.e. i ¤ j ) EŒ.ui .t/  uo /.uj .t/  uo / D 0; 8t. This implies that the WTG outputs are incoherent as well: i ¤ j ) EŒfyQi .t/  EŒyQi .t/gfyQj .t/  EŒyQj .t/g D 0; 8t. The following theorem holds.

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Theorem 1. The mean, variance and autocorrelation of the aggregate model’s output (10) tend to those ones of the full model when the estimation of the mean of the single turbine model’s output O yQ tends to the exact value yQ . In symbols: lim EŒya .t/ D EŒy.t/; lim VarŒya .t/ D VarŒy.t/; lim rya ./ D ry ./8

 O Qy !Qy

 O Qy !Qy

 O Qy !Qy

(11) Proof. The mean, variance and autocorrelation of the full model’s output are: 8 PN P ˆ Qn .t/ D NyQ.t / EŒy.t/ D EŒ N nD1 yQn .t/ D nD1 EŒy ˆ < PN PN VarŒy.t/ D VarŒ nD1 yQn .t/ D nD1 VarŒyQn .t/ D NyQ2.t / ˆ ˆ : r ./ D EŒy.t C /y.t/ D N  r ./ C .N 2  N /2 yQ

y

(12)

yQ

The mean, variance and autocorrelation of the aggregate model output are instead: 8 p p ˆ EŒya .t/ D N .EŒy.t/ Q  O yQ / C N O yQ D N .yQ  O yQ / C N O yQ ˆ < VarŒya .t/ D N  EŒ.y.t/ Q  O yQ /2  ˆ ˆ : r ./ D N  r ./ C .N 2  N /O 2 ya

yQ

(13)

yQ

It can be immediately seen that EŒya .t/ and VarŒya .t/ are both continous functions of O y and that if O y D yQ then EŒya .t/ D EŒy.t/ and VarŒya .t/ D VarŒy.t/. Furthermore, the two autocorrelation functions differ only for the second (constant) term. Also in this case, it can be seen that rya is a continous function of O yQ (for any ) and that if O yQ D yQ then rya ./ D ry ./. Moreover, since the power spectral density Sy is the Fourier transform of R C1 the autocorrelation function Sy .f / D 1 ry ./e j 2f  d  (Wiener–Kintchine Theorem), it holds that Sya .f / ! SyN .f / when O y ! yQ as well.

2.2 Fluctuation Width Yet another metric that can be used to compare the full and aggregate models is the fluctuation width [3, 13]. It is the difference between the max and the min output in a time interval [0, T]: Fy .T / D maxŒ0;T  Œy.t/  minŒ0;T  Œy.t/:

(14)

Note that in (14), y.t/ is a deterministic function (realization of a stochastic process). The following theorem holds. Theorem 2. The fluctuation width of the aggregate model is less or equal than the one of the full model, i.e.: Fya .T /  Fy .T /.

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Proof. In (14) it is convenient to express the output y.t/ in per units of the rated wind farm power, which means that y.t/ varies between 0 and 1. The maximum output of a single turbine model in p.u. of the rated wind farm power, is therefore : maxŒy.t/ Q D N1 D yQmax . It is now possible to compute the maximum output of the aggregate model and the one of the full model: maxŒy.t/ D N  maxŒy.t/ Q D1   p p 1  O yQ C N O yQ Q  O yQ g C N O yQ D N maxŒya .t/ D N fmaxŒy.t/ N

(15) (16)

It can be seen from (16) that: O yQ < N1 ) maxŒya .t/ < 1. Therefore, from (15) and the definition (14) follows the thesis of the theorem: Fya .T /  Fy .T /.

Appendix: Estimation of the Single Turbine Model Output’s Mean In order to achieve EŒya .t/  EŒy.t/, VarŒya .t/  VarŒy.t/ and rya ./  ryN ./ a good approximation of yQ is needed (O yQ  yQ ). To compute O yQ , we approximate the dynamical model (1) with its steady state part yQss W 8 0 ˆ ˆ < k  u3 yQss D ˆ yQ ˆ : max 0

if u  uci if uci < u  urat if urat  u  uco if u > uco

(17)

where k is a WTG constant. The approximation O yQ is then computed by applying the Gaussian filter method [14]: Z O yQ .uo / D

C1 1

2

.u u/  o 2 1 yQss .u/ p e 2tur d u D ŒyQss .u/  p.u/uDuo 2tur

(18)

Acknowledgements This work has been executed as part of the project: “Dynamic state Estimation and Voltage Stability of transmission and distribution grids with a large share of distributed generation capacity” (DEVS), financially sypported by the Dutch Ministry of Economic Affairs under the program EOS-LT.

References 1. ECN Control Design Tool, http://www.ecn.nl/units/wind/rd-programme/integrated-windturbine-design/design-tools/control-design-tool/ (2011). 2. Pierik, J.T.G., Morren, J., Wiggelinkhuizen, E.J., de Haan, S.W.H., van Engelen, T.G., Bozelie, J.: Electrical and Control Aspects of Offshore Wind Farms II (ERAO II) vol. 1–2, ECN-C04050/051 (2004) Technical report

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3. De Tommasi, L., Gibescu, M., Brand, A.J.: A dynamic wind farm aggregate model for the simulation of power fluctuations due to wind turbulence. J. Comput. Sci. 1(2), 75–81 (2010) 4. Freris, L.L.: Wind energy conversion systems. Prentice Hall, New York (1990) 5. Perdana, A., Carlson, O.: Aggregated Models of Large Wind Farm Consisting of Variable Speed Wind Turbines for Power System Stability Studies. Proceedings of 8th International Workshop on Large Scale Integration of Wind Power and on Transmission Networks for Offshore Wind Farms, Energynautics GmbH - M¨uhlstrasse 51 - 63225 Langen, Germany (2009) 6. P¨oller, M., Achilles S.: Aggregated wind park models for analyzing power system dynamics. In: 4th International Workshop on Large-Scale Integration of Wind Power and Transmission Networks for Offshore Wind Farms, Billund, Denmark, 2003 7. Ackermann, T. (ed.): Wind Power in Power Systems. Wiley, New York (2005) 8. Akhmatov, V., Knudsen, H.: An aggregate model of a grid-connected, large scale, offshore ¨ importance of windmill mechanical system. wind farm for power stability investigations U Electr. Power Energ. Syst. 24, 709–717 (2002) 9. Akhmatov, V.: An aggregated model of a large wind farm with variable-speed wind turbines equipped with doubly-fed induction generators. Wind Eng. 28(4), 479–486 (2004) 10. Fernandez, L.M., Jurado, F., Ramon Saenz, J.: Aggregated dynamic model for wind farms with doubly fed induction generator wind turbines. Renew. Energ. 33, 129–140 (2008) 11. Pierik, J.T.G., Zhou, Y., Bauer, P.: Wind Farm as Power Plant; Dynamic modelling studies, ECN-E-08-017 (2008) Technical Report 12. Nanahara, T., Asari, M., Sato, T., Yamaguchi, K., Shibata, M., Maejima, T.: Smoothing effects of distributed wind turbines. Part 1. Coherence and smoothing effects at a wind farm. Wind Energ. 7, 61–74 (2004) 13. Asari, M., Nanahara, T., Maejima, T., Yamaguchi, K., Sato, T.: A study on smoothing effect on output fluctuation of distributed wind power generation. Asia Pacific IEEE/PES Transmission and Distribution Conference and Exhibition (2002) 14. Noorgard, P., Holttinen, H.: A multi-turbine power curve approach, Proceedings of Nordic Wind Power Conference (2004)

Adaptive Modelling of Coupled Hydrological Processes with Application in Water Management Peter Bastian, Heiko Berninger, Andreas Dedner, Christian Engwer, Patrick Henning, Ralf Kornhuber, Dietmar Kr¨oner, Mario Ohlberger, Oliver Sander, Gerd Schiffler, Nina Shokina, and Kathrin Smetana

Abstract This paper presents recent results of a network project aiming at the modelling and simulation of coupled surface and subsurface flows. In particular, a discontinuous Galerkin method for the shallow water equations has been developed which includes a special treatment of wetting and drying. A robust solver for saturated–unsaturated groundwater flow in homogeneous soil is at hand, which, by domain decomposition techniques, can be reused as a subdomain solver for flow in heterogeneous soil. Coupling of surface and subsurface processes is implemented based on a heterogeneous nonlinear Dirichlet–Neumann method, using the dunegrid-glue module in the numerics software DUNE.

H. Berninger ()  R. Kornhuber  O. Sander Institut f¨ur Mathematik Freie Universit¨at Berlin Arnimallee 6 D-14195 Berlin e-mail: [email protected]; [email protected]; [email protected] A. Dedner  D. Kr¨oner  N. Shokina Abteilung f¨ur Angewandte Mathematik Albert-Ludwigs-Universit¨at Freiburg HermannHerder-Straße 10 D-79104 Freiburg i. Br. e-mail: [email protected]; [email protected]; [email protected] P. Bastian  C. Engwer Interdisziplin¨ares Zentrum f¨ur Wissenschaftliches Rechnen Universit¨at Heidelberg Im Neuenheimer Feld 368 D-69120 Heidelberg e-mail: [email protected]; [email protected] P. Henning  M. Ohlberger  K. Smetana Institut f¨ur Numerische und Angewandte Mathematik Universit¨at M¨unster Einsteinstraße 62 D-48149 M¨unster e-mail: [email protected]; [email protected]; [email protected]; uni-muenster.de G. Schiffler WALD C CORBE GbR Am Hecklehamm 18 D-76549 H¨ugelsheim e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 65, © Springer-Verlag Berlin Heidelberg 2012

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1 Introduction Reliable flood prediction and the design of efficient flood protection measures are tasks that engineering companies for water management have to face. Since flood control basins and water retention walls on rivers have effects on the groundwater table, sound modelling and simulation of coupled ground- and surface water processes is required. Challenges include the temporal and spatial scale differences between ground- and surface water, and the anisotropy of the groundwater domain. As the surface water model we consider the shallow water equations, given by hyperbolic conservation laws for the water height h and the discharge q @t h C div q D Sh ;

@t q C div.q2 = h C 0:5 gh2 I / D ghrb:

(1)

Here, Sh denotes a source term, g the gravity constant and I 2 R33 the identity matrix. The graph ˙ of the function b on the domain ˝s  R2 coincides with the topography. The latter is identical to the top boundary of the domain ˝p  R3 that represents the porous medium. Saturated–unsaturated groundwater flow in ˝p is modelled by the Richards equation n .p/t C div v.p/ D 0;

v.p/ D Kh kr..p//r.p  z/

(2)

in case of homogeneous equations of state for the saturation  and the relative permeability kr. It combines mass conservation on the left with a nonlinear version of Darcy’s law on the right. The water or capillary pressure head is denoted by p and z is the downward pointing component of x 2 ˝p . The porosity n 2 Œ0; 1 and the hydraulic conductivity Kh > 0 may depend on x. The functions p 7! .p/ 2 Œ0; 1 and  7! kr./ 2 Œ0; 1 are increasing with kr.1/ D 1. Conditions for coupling Equations (1) and (2) across the surface ˙ are given by mass conservation and continuity of the pressure Sh D v  n and p D h on ˙:

(3)

Here, n is the outward normal of ˝p on ˙ and the identities shall be understood upto the projection of ˙ onto ˝s induced by b. Well-posedness of the coupled problem (1)–(3) does not seem to be known. We strive for a numerical solution of (1)–(3). In the following Sects. 2 and 3, we summarize known facts concerning our approaches to solve (1) and (2), respectively. In Sects. 4 and 5, which contain new results, we present our coupling strategy and its implementation in the software framework DUNE [1] as well as first numerical examples.

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2 DG Method for the Shallow Water Equations The shallow water equations (1) are a special case of the general evolution problem @t u.t; / D L Œu.t; /./

in .Œ0; T /  ˝s /  .IR  IRd /; d 2 f1; 2; 3g;

with the spatial operator L Œw D S.w/  div F .w/, where w W ˝s !   IRr belongs to some suitable function space V ,  is the set of states for a given problem, S.w/ is a source term function and F .w/ is the analytical flux function. For a tessellation Th of ˝s we consider the discrete space Vh WD f' 2 L2 .˝s /j 'jT 2 Pp .T / 8 T 2 Th g and, for ' 2 Vh , define the discrete operator Lh W Vh ! Vh0 by Z Lh Œuh ' WD ˝s

X Z T 2Th

Z

Z

S.uh /' C T

F .uh /  r'  T

' @T

 G.uC h ; uh ; : : :/

  nT

:

 nT denotes the outward normal of T , and uC h and uh are the values of the function d uh on both sides of @T . G.u; v; x/ W V V IR ! IRd is a numerical flux function. The discontinuous Galerkin (DG) method is determined by the functions S and F , the numerical flux G (we use the Local-Lax–Friedrichs flux) and the space Vh . This d e h Œuh .t/, solved with space discretization leads to a system of ODEs dt uh .t/ D L an explicit Runge–Kutta (RK) method of order p C 1, cf. [7]. Since the RK–DG method is not stable for convection dominated nonlinear problems where strong e h of Lh is used. Our construction is shocks might appear, a stabilized version L based on the shock detection and limiting strategy studied in [6]. For a correct handling of steady states, [7] suggests a well-balancing method based on a reformulation of the topography source term in the balance law (1) for the discharge q. We rewrite this term as Sq D  12 ghrb  12 r  .ghbI / C 12 gbrh, move the second term (which is in divergence form) to the left hand side, and treat the topography b as an additional unknown. We arrive at

@t hCdiv.q/ D 0 ; @t b D 0 ; @t qCdiv.q2 = hC0:5gh .hCb/I / D 0:5g.brhhrb/ To retain the steady state q D 0 and b C h D const:, describing a “lake at rest”, we need to guarantee @t h D 0; div.h2 I / D 2hrb in the original formulation. Discretized with DG, this leads to a volume integral and boundary fluxes that have to sum up to zero, which does not hold without special treatment. The new formulation @t b D 0; @t h D 0; div.h .h C b/I / D brh  hrb, however, shows a symmetry in h and b that can be used to derive a well-balanced DG scheme for a wide range of standard numerical fluxes. In the resulting scheme, both the volume part and the boundary fluxes vanish so that no balancing of fluxes and volume integrals is required. The wetting–drying treatment is based on the thin water layer approach of [5]. Simplifying the stability condition in [5], we use the “reflection numerical flux”

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to prohibit mass transfer through an element boundary as soon as we detect an emptying of this element. After the positivity of the mean water depth in each element is guaranteed, the so-called positive depth operator [5] ensures the positivity of water depth node-wise. See [7] for details of our implementation.

3 Efficient Solver for the Richards Equation in Heterogeneous Soil Our solution technique for the Richards equation is based on Kirchhoff transformation and convex minimization for homogeneous state equations and, in addition, on domain decomposition methods for layered heterogeneous soil. For simplicity, we assume n D Kh D 1. Then, a time discretization of (2), which treats the main part implicitly and the convective part (in z-direction) explicitly, leads to the spatial problem .p/  div.kr..p//rp/ D f (4) with  W p 7! u WD R p some function f . By the Kirchhoff transformation 1 kr..q// dq and the definition M.u/ WD . .u//, we can rewrite (4) as 0 M.u/  u D f:

(5)

We can endow (5) with quite general boundary conditions including outflow conditions of Signorini-type. Since M W R ! R is increasing, the weak form of Problem (5) allows an equivalent formulation as a uniquely solvable convex minimization problem on a convex subset of H 1 .˝p /. Using linear finite elements, we construct a convergent discretization of (5), which is also meaningful in the physical variable p. It can be solved efficiently and robustly by monotone multigrid methods [3]. In layered heterogeneous soil, different i ./ and kri ./, i D 1; : : : ; m, belonging to the layers ˝i of ˝p occur. Application of different Kirchhoff transformations i in the layers and the assumption of pressure continuity as well as mass conservation across the interfaces between the layers lead to local convex minimization problems that are coupled by nonlinear transmission conditions. In fact, these may account for pressure jumps across the interfaces, too. However, pressure continuity is hydrologically reasonable in the Richards model where the nonwetting phase air has constant pressure [9]. Now, for m D 2, we obtain the transmission problem Mi .ui /  ui D f

on ˝i ;

11 u1 D 21 u2 ;

v1  n1 D v2  n1

on ; (6)

with the interface  WD ˝ 1 \ ˝ 2 and the outward normal n1 of @˝1 . The coupled problem (6) can be solved iteratively by nonlinear Dirichlet–Neumann or Robin methods. For analytical and numerical results on such methods we refer to [4].

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4 Coupling Strategy: Algorithm and Modular Implementation We obtain a spatial coupled problem by implicitly discretizing the time dependent problem (1)–(3). Typically, the time scale for (1) is considerably smaller than the time scale for (2). Therefore, our time discretization of (1)–(3) attributes a certain number N of equidistant time steps to (1) within an interval between two time steps for (2). Correspondingly, we assign 1=N of the source term Sh in (3) from one time step of (2) equally to the sub time steps of (1). In order to solve the resulting discretization of (1)–(3), we apply a Dirichlet– Neumann-type iteration technique. Given solutions p0 and u0 of the Richards equation (RE) and the shallow water equations (SWE) for a time step tk , using the water flux F˙ D v0  n as the source term, we solve the SWE at sub time steps tk < tki  tkC1 . Thus we obtain a Dirichlet value p D h1 on ˙ for the RE at time step tkC1 . The solution of the RE at tkC1 yields a water flux F1 D v1  n. Setting F˙ D F1 , the SWE are solved again in tki  tkC1 and we obtain a new Dirichlet value p D h2 for the RE at the time step tkC1 . We repeat this iteration until khj C1  hj k1 is small enough. Implementing the treatment of (1)–(3) is challenging since the surface and the subsurface solvers exist as completely separate codes on top of the DUNE libraries. Coupling is obtained via the DUNE module dune-grid-glue, which offers abstract interfaces for the geometric coupling of finite element grids. Its design follows the concept of intersections used in the DUNE grid interface [1], and it supports most domain decomposition techniques in use today including overlapping and nonoverlapping couplings between non-matching grids. In our case, by injection in R3 and projection onto ˙, the surface water domain ˝s  R2 is coupled to the top boundary ˙ of the porous media domain ˝p . The corresponding image T .˝s / of the two-dimensional (surface) grid on ˝s and the restriction T .˝p j˙ / of the porous media (subsurface) grid onto ˙ are in general non-matching. The dune-grid-glue module efficiently computes all intersections between elements of T .˝s / and T .˝p j˙ /. These so-called remote intersections [2] encapsulate the relationships between elements in the generating grids T .˝s / and T .˝p j˙ / and constitute a new partition T .˙/ of ˙. For high performance parallel computations, the concept of intersections is extended and provides parallel communication methods to exchange data associated with intersections between elements on remote processes [8]. For the Dirichlet–Neumann coupling, the fluxes F˙ are computed at quadrature points on I from the solution of (2). Communicating the fluxes, the source term Sh can be computed on the surface grid. To evaluate Dirichlet values p D hj for the Richards equation, a representation of hj on the boundary of the subsurface grid is required. On the surface grid the solution is projected onto a discontinuous function hdisc on T .˙/. The coefficients of hdisc for each intersection I are communicated and hdisc can be evaluated on the subsurface grid.

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5 Benchmark Problems and Numerical Experiments We applied realistic data (grid and soil parameters) provided by the engineering company WALD+CORBE. The geometry is a real piece of landscape (size: 2,525  2,415 m; 5  5 m resolution of the digital elevation model). Measured groundwater levels were available (see Fig. 1). The observed region consists of three separate geologic formations with different hydraulic conductivities: a hardly permeable top layer (alluvial clay), the aquifer (quaternary gravel sand) and the riverbed (river gravel). The decomposition is illustrated in Fig. 1. Numerical experiments showed that we need a resolution of about 40 nodes in gravity direction in order to capture the infiltration adequately. Since the geological layers are very thin (0:5– 15 m) in comparison to width and length of the domain, we developed a nonlinear line smoother for the multigrid method to treat the anisotropy. The numerical experiments simulate flood events in the considered domain (Figs. 2 and 3), which contains a fictional polder and a retention basin to be built for flood protection.

Fig. 1 Initial groundwater distribution for computation and decomposition of the domain in its three layers (overlying strata, aquifer and riverbed). Scaling in direction of gravity by a factor of 10

Fig. 2 Simulations of flood events, depiction of the solutions at different time steps: (a) Richards equation—visualization of capillary pressure, time steps correspond to minutes. (b) Shallow water equations—visualization of discharge with wetting and drying effect, time steps correspond to seconds

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Fig. 3 Flooding of a small subdomain, with reflexion by an artificial obstacle. The solution is depicted for three different time steps. Here, we used the coupling of the solvers for the Richards and the shallow water equation. After several flood waves, a slight infiltration can be observed (processes take place at different time scales)

Acknowledgements This work was supported by the BMBF–Programm “Mathematik f¨ur Innovationen in Industrie und Dienstleistungen”, F¨orderkennzeichen (contract numbers) 03OMPAF1, 03BAPAF2, 03DEPAF3, 03KOPAF4.

References 1. Bastian, P., Blatt, M., Dedner, A., Engwer, C., Kl¨ofkorn, R., Kornhuber, R., Ohlberger, M., Sander, O.: A generic grid interface for parallel and adaptive scientific computing. Computing 82(2–3), 121–138 (2008) 2. Bastian, P., Buse, G., Sander, O.: Infrastructure for the coupling of Dune grids. In: Proceedings of ENUMATH, vol. 9, Springer, Berlin Heidelberg (2010) 3. Berninger, H., Kornhuber, R., Sander, O.: Fast and robust numerical solution of the Richards equation in homogeneous soil. SIAM J. Numer. Anal. 49(6), 2576–2597 4. Berninger, H., Kornhuber, R., Sander, O.: Convergence behaviour of Dirichlet–Neumann and Robin methods for a nonlinear transmission problem. In: Proceedings of DD19, LNCSE 78, pp. 87–98. Springer, Berlin (2011) 5. Bunya, S., Dawson, C. et al.: A wetting and drying treatment for the Runge–Kutta DG solution to the shallow water equations. Comput. Meth. Appl. Mech. Eng. 198, 1548–1562 (2009) 6. Dedner, A., Kl¨ofkorn, R.: A generic stabilization approach for higher order Discontinuous Galerkin methods for convection dominated problems. J. Sci. Comp. 10, 1–24 (2010) 7. Dedner, A., Kr¨oner, D., Shokina, N.: Adaptive modelling of two-dimensional shallow water flows with wetting and drying. In: E. Krause, D. Kr¨oner, M. Resch, N. Shokina, Y. Shokin (eds.): Computational Science and High Performance Computing IV, vol. 115, pp. 1–15. Springer (2011) 8. Engwer, C., Sander, O.: A framework for the parallel coupling of Dune grids (in preparation) 9. Helmig, R., Weiss, A., Wohlmuth, B.: Variational inequalities for modeling flow in heterogeneous porous media with entry pressure. Comput. Geosci. 13(3), 373–389 (2009)

Part VII

Dedicated and Versatile Methods

Overview At ECMI 2010 many results and scientific findings both in analysis and numerical treatment have been presented. In “classical” fields of applications such as electromagnetics or material sciences and in techniques such as model order reduction or uncertainty quantification that are gathering more and more interest, we have seen various approaches and viewpoints. Fortunately, however, ECMI 2010 was a versatile conference with contributions from different areas of research which are beginning to gain more interest. Although the techniques and findings in the papers gathered in this chapter could most probably be applied in several “classical” fields, we give these contributions their own space and emphasize their contents. Below we list a variety of topics: coupled mechanical-heat analysis in shape optimization, modeling based on measurements, port-Hamiltonian modeling, error estimation between continuous and discrete models, delay differential equations and shocks, matrix-differential equations, coupling between models with different scales, sparse matrix techniques, boundary conditions to bound an unbounded region. R. Schneider presents “FEINS:Finite element solver for shape optimization with adjoint equations”. The scope of this finite element software package is geometry optimization of designs, e.g., pedal cranks. Usually a lot of parameters are used to describe the shape of an object. Using the discrete adjoint technique, FEINS computes the gradients of the performance functionals like weight and rigidity. This technique makes the computational cost independent of the complexity of the parameterization. M. Burger et al. construct “Invariant loading for full vehicle simulation”. When simulating a moving vehicle a road profile, i.e., an input load for the vehicle model, has to be prescribed. This input has to be invariant, hence independent of the system under consideration yielding its applicability to different variants of the vehicle. The task is to derive such an invariant road profile from a measurement of the reaction of a vehicle, hence from system dependent data, to a real road profile. The problem to be solved can be formulated in terms of an optimal control problem. The authors

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solve this optimal control problem with the method of control-constraints and present results with the help of a Porsche Cayenne model. T. Voß discusses “Structure preserving spatial discretization of a piezoelectric beam” in the port-Hamiltonian framework. In port-Hamiltonian systems coupling of subsystems is described in terms of exchange of energy. Thus, especially for multiphysical problems they offer an alternative to the modeling via classical dynamical systems as both the structure and the physical properties are preserved. Using the example of a piezoelectric beam the author describes the transfer of an infinite dimensional port-Hamiltonian model to a corresponding finite dimensional one that can be used for simulation or control design. A. Rauh et al. model in “Error estimates for finite-dimensional approximations in control of distributed parameter systems” the error between the exact dynamics, described by a partial differential equation and the corresponding finite-dimensional FEM approximation. Knowledge about this deviation is necessary when deriving optimal control strategies for, e.g., elastic multibody systems. To determine these uncertainties the authors compare a continuous time estimator and a discrete-time Extended Kalman Filter using the example of a prototypical high-speed rack feeder. F. Bernal deals in “Solving non-smooth delay differential equations with multiquadrics” with the numerical solution of a special class of delay differential equations: neutral differential equations. The differential equations of that kind depend upon the history of the solution and in addition upon the history of the derivative. As a consequence, discontinuities are not smoothened out but are propagated without damping. The travelling discontinuities cause problems within the numerical integration and reduce the accuracy of an approximation via a multiquadric interpolant (that otherwise offers exponential spatial convergence). The scheme presented determines the location of discontinuities, exploiting sensitivity, and decomposes the domain and applies the multiquadrics on subintervals. E. Defez et al. tackle in “Higher-order matrix splines for systems of secondorder differential equations” the numerical solution of nonlinear second-order matrix differential equations. Problems of this kind may arise, e.g., in moleculardynamics simulation of semiconductor devices and in mechanical vibration systems. The approach proposed here constructs matrix splines which are globally secondorder differentiable and have a higher order of differentiability on subintervals. In applying the method, the second-order differential equation does not have to be transformed to a higher dimensional first-order differential equation. K. Fackeldey presents two “Multiscale methods in time and space”. As an example of multiscale phenomena in space the problem of crack formation in solid materials is examined. In the direct vicinity of the crack, the material is resolved in atomistic scale. Outside this area, the displacement is described by continuum mechanics. Hence, in the numerical simulation molecular dynamics simulation and the finite element method are coupled. Metastability of molecule configurations serves as an example for the introduction of the conformation dynamics. Here the separation of the domain into regions of metastability might lead to a coarsening of timesteps in the solution of equations of motion in an ensemble.

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A.N. Yzelman and R.H. Bisseling propose “A cache-oblivious sparse matrixvector multiplication scheme based on the Hilbert curve”. The authors tackle the problem of efficient multiplication when the matrix is unstructured. In this case accesses to the operand, i.e., to the vector to be multiplied are unpredictable. This causes cache misses. To overcome this problem, they suggest using the Hilbert curve to store non-zeros in the matrix. By this, the multiplication becomes “local” in terms of accesses to the vector entries. Furthermore, the information about which matrix columns contain nonzeros is stored in a relative, i.e., incremental manner, rather than in an absolute way. P. Klein et al. construct “Absorbing boundary conditions for solving stationary Schr¨odinger equations” that arise, e.g., in condensed matter physics, optics and underwater accoustics. Computing the numerical solution to a PDE problem that is defined on an unbounded domain, one is faced with the need to define boundary condition on a computational, hence, bounded domain. To derive such boundary conditions for the stationary Schr¨odinger equation the authors carry forward ideas they presented for the case of the nonstationary problem. The validity of this approach is shown with the help of a traveling plane wave belonging to the class of linear scattering problems. M.M. Tung provides in “Diffusion on surfaces of revolution” analytical results for sourceless diffusion processes on surfaces of revolution, e.g., liquid mirror telescopes. A differential-geometric approach shows that for axisymmetric shapes the equations of motion reduce to a self-adjoint eigenvalue problem. By this the completeness of the fundamental solution is guaranteed. The author derives the equation of motion from a Hamiltonian system using a variational principle. Theoretical results presented in this paper may lead to the construction of effective numerical tools for related problems. E. Auer proposes in “Verified Analysis of a Biomechanics-Related System” a way to propagate uncertainties through a system using interval arithmetic. The work presented belongs to the project PROREOP within which a surgical planning system for the human skeletal system is developed. Therefore, human stance stabilization serves as an example where e.g., the weight of parts of the human body, the positions in a coordinate system and forces acting on the body are parameters with incertitudes. Numerical results are produced with the C++ software environment SmartMOBILE where, here especially, a method for computing interval evaluations of first derivatives of piecewise functions is introduced. Wuppertal, Germany

Michael Striebel E. Jan W. ter Maten

FEINS: Finite Element Solver for Shape Optimization with Adjoint Equations Ren´e Schneider

Abstract The finite element software package FEINS is introduced. One of its unique features is the design for shape optimisation problems. To enable efficient shape optimisation it provides the gradient of performance functionals with respect to the domain geometry. These gradients are computed by the discrete adjoint technique to achieve computational cost independent of the dimension of the shape parameterisation. Special attention is given to efficient solution techniques for the algebraic systems of equations which result from FE discretisation. An overview of the capabilities of FEINS is given and examples illustrate its utility.

1 Introduction In most situations where a physical phenomenon or technological process is modelled by partial differential equations (PDEs) the real aim is not only to predict or simulate the process, but to influence it in order to improve the performance of a device. In many cases the simplest and most effective way of influencing the process is by the shape of the device in which the process takes place. This gives rise to shape optimisation, which aims to find the best possible design for such a device. Examples of this are given by Fig. 1, where the shape of an object in a channel flow is optimised for drag, and in Fig. 2, where an optimal compromise between weight and rigidity of a pedal crank is illustrated. Due to the usually high (or even infinite) dimension of the design space, exploration by trial and error is prohibitive in terms of cost, and even an engineer with the best intuition for finding good designs will never know if his design is

R. Schneider () Mathematik in Industrie und Technik, Fakult¨at f¨ur Mathematik, TU Chemnitz, 09107 Chemnitz, Germany e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 66, © Springer-Verlag Berlin Heidelberg 2012

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Fig. 1 Obstacle in a channel flow (a) initial design (b) optimised shape, both with streamlines and pressure (color) around the obstacle

Fig. 2 Pedal crank (a) initial design (b) optimised shape, both with von Misses stress (color)

truly optimal. Efficient computational optimisation approaches usually require the gradient of the objective function to define search directions, but straight forward computation of these gradients results in cost comparable to k evaluations of the performance, where k is the dimension of the design space. When each of these evaluations is a time consuming simulation this becomes quickly prohibitive as well. However, the solution of adjoint problems allows to compute the gradients at a cost of approximately one evaluation of the performance, independent of k, thus enabling efficient computational optimisation even for problems where the dimension of the search space is high and the evaluation of the performance is computationally expensive. In this paper we present the finite element (FE) solver suite FEINS which was developed with the aim of enabling shape optimisation by implementing the discrete adjoint technique. The name FEINS is an acronym for Finite Elements in Incompressible Navier–Stokes, which was chosen because initially the solver was only intended for problems from fluid dynamics. However, as the code structure separates the low level general purpose finite element routines from mostly high level application specific routines, it is relatively easy to use these low level routines for other elliptic PDE problems as well. This has already been done for the Lam´e equations of linear elasticity, for the Poisson equation and for convection-diffusion problems. In all cases special attention was given to efficient solution techniques for the algebraic equation systems, which is in PDE optimisation of greater importance

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than ever, as the solution of the optimisation problems usually requires multiple evaluations of the performance criterion, each comprising of such a solve of a FE system. In the following section we summarise briefly the discrete adjoint technique. Section 3 gives an overview of FEINS and Sect. 4 discusses briefly the two application examples of Figs. 1 and 2.

2 Discrete Adjoint Technique in FEM Context The discrete adjoint technique allows efficent evaluation of gradients (sensitivities) of performance criteria with respect to problem parameters. A good introduction of the general idea can for example be found in [3]. A detailed discussion in the context of finite elements and shape sensitivities was the subject of our previous works [5, 6]. As this technique is the very heart of our approach we summarise its core here for completeness and readability. Consider a finite element discretization of an elliptic PDE (say) on a domain whose geometry is uniquely defined by a set of parameters, F , which results in the (possibly nonlinear) algebraic system R.u.F /; F / D 0;

(1)

where u is the vector of finite element coefficients and R.; / D 0 represents the FE equations on the domain defined by F . Let e I .u.F /; F / be a scalar valued function (performance function) which depends upon u.F /. Assuming (1) to have a unique solution u.F / for all F , we may regard this functional as depending only upon F , that is I.F / WD e I .u.F /; F /:

(2)

In this setting the total derivative of I with respect to F can be evaluated by @e I @R DI D  T ; DF @F @F

(3)

where  is the unique solution of the discrete adjoint equation 

@R @u

"

T D

@e I @u

#T :

(4)

The vector  is called the adjoint solution and is of the same dimension as u. The importance of this representation is that, once the original equation (1) is solved and I.F / evaluated from (2), DI =DF may be evaluated for little more

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than the cost of a single solve of the linear system (4) and a single matrix-vector product in (3), regardless of the dimension of F . This is compared to other methods of evaluating DI =DF which typically require the solution of (1) (or a linearised version) per component of F . So this method is particularly efficient if the number of parameters F is large. Evaluating the derivative by means of (4) and (3) is called the discrete adjoint technique. The specific definition of the parameter vector F in the above derivation is unimportant, and may be anything from coefficients of the PDE, to boundary data, or indeed the shape of the domain. In our approach to shape optimisation, one can think of F as being the node positions in the FE mesh, to simplify ideas. The dependence of these node positions on shape parameters, for example by the Bezier-splines used in this work, is straight forward and results only in an intermediate step in the evaluation of the sensitivities. The key difficulties are in the evaluation of the sensitivities with respect to the mesh (node positions). The expression (3) for the total derivative thus requires derivatives of the FE equation system and the performance functional with respect to the node positions in the mesh. In [6] we gave a general approach to computing these derivatives for a wide range of element types. The problem is reduced to derivatives of two common subexpressions, for which explicit formulae are given in [6, Proposition 3.4].

3 Finite Element Solver FEINS The FE solver FEINS comprises of a general purpose FE library and a collection of solvers for specific elliptic PDEs (problems). The code is written entirely in C and contains currently approximately 46,200 lines of code. It is available under the GNU General Public License (GPL) version 3 from the web site http://www.feins. org/. If required, different licenses are available on request from the author. FEINS has been developed primarily to allow shape optimisation. For this purpose it handles curved boundaries described by Bezier–Splines. Further, the triangular mesh generator triangle [7] is linked by default, such that only a description of the boundary of domains has to be provided, the interior mesh can be generated automatically. This allows great flexibility in the domains under consideration and reduces the effort to set up the optimisation problems. For the optimisation FEINS supplies performance functionals and their gradient computed by the discrete adjoint technique, as described in Sect. 2. The optimisation solver itself can be user chosen and is not part of FEINS . The library part contains low level routines for handling and arithmetic of sparse matrices, including preconditioned iterative solvers (CG, GMRES, conjugate residuals) and file output of vectors an matrices. Multigrid and multilevel techniques (BPX) are available for preconditioning, allowing optimal overall complexity of the solvers (O.n/). An optional interface to the sparse direct solver UMFPACK is also

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Table 1 Solver timings, measured on a single core of a two CPU system with Intel Xeon Dual Core CPUs at 3.0 GHz, 64 GB RAM (DDR2-667), running 64 Bit openSUSE Linux 11.1 Problem # unknowns Time solve (s) Time shape gradient (s) Lam´e Navier–Stokes

35,419,650 37,769,219

733 31,000

733 37,000

provided. Table 1 lists computational times for two different example problems to illustrate the efficency of the solvers. On the finite element side, FEINS can handle two dimensional meshes of triangles (piecewise linear (P1 ) and piecewise quadratic (P2 ) elements). Extensions to three dimensions and other element types are prepared, but time constraints have prevented their implementation so far. The handling of the meshes is responsible for a large portion of the code. Adaptive mesh refinement based on the B¨ansch green refinement [1] is implemented as well as uniform (red) refinement. The meshes together with the approximated PDE solutions can be written to files for visualisation (e.g. Paraview .vtk Format and GNUplot). The problem classes that are implemented in FEINS are the incompressible Navier–Stokes equations and Stokes equations (fluid dynamics), Lam´e equations (linear elasticity), Poisson equation and convection-diffusion equation. Performance functionals and shape gradient evaluation are so far only implemented for Navier– Stokes and Lam´e. The Poisson equation served mainly as testbed for parts of the developments, no actual application scenarios for this have been considered so far. The convection-diffusion is still in early development, it currently offers only (suboptimal) direct solvers as options. While for Poisson and Lam´e problems the generic multigrid or BPX preconditioner are suitable and deliver optimal solvers, this is more complicated for Navier–Stokes. There we use the Schur-complement preconditioning technique of [2]. This requires the solution of convection-diffusion-reaction subproblems for which we employ again multigrid preconditioned GMRES. There are further subproblems which appear in this preconditioner, but they are of a simpler nature by far. While this preconditioning works well for low Reynolds numbers and delivers optimal solvers, it deteriorates for increasing Reynolds numbers. For details on this we refer to [5, Section 2.5.2].

4 Examples and Conclusions The example of Fig. 1 comprises of an obstacle in a two dimensional channel of viscous fluid at Reynolds number 20, where the induced drag is to be minimised, subject to the area of the obstacle being greater or equal to that of the initial design. The shape parameterisation by symmetrised Bezier-splines results in ten free parameters for the optimisation. The SQP optimisation solver DONLP2 by

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P. Spellucci stops after 16 iterations. As each of these iterations may require several steps of a line search, a total of 60 function and gradient evaluations take place. The performance functional (viscous drag) is reduced from 1:4056 to 1:3714. For details on the setup we refer to [5, Section 2.7.1] where this example has been considered also. Figure 2 shows an example where elastic deformation of a crank is considered. In order to get optimal compromises between deformation energy (compliance) and weight, the functional 1 IQ.u; ˝/ WD 2

Z

Z .u/ W ".u/ d˝  ˝

g  u d C ˛j˝j

N

is minimised, where u is the deformation, .u/ the stress and ".u/ the strain. The function g is the force density acting on the circular hole at the right of the part, which is set to a constant vector pointing downward while the part is clamped at the diamond shaped hole at the left (u D 0). The geometry is parametrised by symmetrised Bezier-splines with 24 free parameters. As the von Mises stress indicates, the computed compromise is far more economical in the sense that the stress is more uniformly distributed in the domain and obviously the area of the domain is reduced. We publish FEINS in the hope that it might find use in actual industrial applications and/or related research projects. Acknowledgements We thank Peter Jimack of the university of Leeds for the guidance during the initial development of FEINS which took place while the author studied for a Ph.D. under Peters supervision [5]. Further, we thank Andreas G¨unnel who implemented the discrete adjoint and adaptivity for the Lam´e equation during his work for his Diplom thesis [4]. Finally, we thank Arnd Meyer of TU Chemnitz for many fruitful discussions and for his co-supervision of the adaptivity implementation.

References 1. B¨ansch, E.: An adaptive finite-element-strategy for the three-dimensional time-dependent Navier-Stokes-Equations. J. Comput. Appl. Math. 36(1), 3–28 (1991) 2. Elman, H., Loghin, D., Wathen, A.: Preconditioning techniques for Newton’s method for the incompressible Navier-Stokes equations. BIT 43(5), 961–974 (2003) 3. Giles, M., Pierce, N.: An introduction to the adjoint approach to design. Flow Turbulence Combust. 65(3–4), 393–415 (2000) 4. G¨unnel, A.: Adaptive Mesh Design in Shape Optimization with the Discrete Adjoint Method. Master’s thesis, TU Chemnitz, Chemnitz, Germany (2010) Permanent URL: http://nbnresolving.de/urn:nbn:de:bsz:ch1-201000390 5. Schneider, R.: Applications of the discrete adjoint method in computational fluid dynamics. Ph.D. thesis, University of Leeds (2006) http://www.comp.leeds.ac.uk/research/pubs/theses/ schneider.pdf

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6. Schneider, R., Jimack, P.: On the evaluation of finite element sensitivities to nodal coordinates. ETNA 32, Special Volume with Selected Papers from the 20th Chemnitz Finite Element Symposium, 134–144 (2008). http://etna.mcs.kent.edu/vol.32.2008, Published online 04/03/2009 7. Shewchuk, J.: Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator. In: Lin, M., Manocha, D. (eds.) Applied Computational Geometry: Towards Geometric Engineering, Lecture Notes in Computer Science, vol. 1148, pp. 203–222. Springer, Berlin (1996). Article and software available at http://www.cs.cmu.edu/quake/triangle.html

Invariant Loading for Full Vehicle Simulation Michael Burger, Klaus Dreßler, Albert Marquardt, Michael Morr, and Lothar Witte

Abstract Input loads are essential for the numerical simulation of vehicle multibody system (mbs) models. Such load data is called invariant, if it is independent of the specific system under consideration. A digital road profile, e.g., can be used to excite mbs models of different vehicle variants. However, quantities obtained by measurement such as wheel forces are typically not invariant in this sense. This leads to the general task to derive invariant loads on the basis of measurable, but system-dependent quantities. Mathematically, this can be formulated as an optimal control problem. We present a strategy to solve this problem and an application to an off-road driving simulation of a Porsche Cayenne model.

1 Introduction The numerical simulation of vehicle-(mbs-) models plays an important role in vehicle engineering. Virtual prototyping can accelerate the development process enormously and reduces costs. In order to simulate the dynamical motion of a multibody vehicle model, load signals are needed as input data. Such load data is called invariant, if it is independent of the specific system under consideration. A convenient example for invariant loads is a digital road profile used for driving simulation of a vehicle. The main benefit of invariant input signals is the possibility to transfer and use them

M. Burger ()  K. Dreßler  A. Marquardt Fraunhofer Institute for Industrial and Financial Mathematics (ITWM), Kaiserslautern, Germany e-mail: [email protected]; [email protected]; [email protected] M. Morr  L. Witte Fraunhofer Institute - Porsche AG, Kaiserslautern - Weissach, Germany e-mail: [email protected]; [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 67, © Springer-Verlag Berlin Heidelberg 2012

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as excitation for different model-variants, which may only exist as virtual model. In this way, time- and cost-intensive experiments can be avoided. Typically, output quantities such as wheel forces, accelerations or relative displacements in the vehicle are measured. However, those quantities are not invariant but highly dependent on the specific vehicle variant, that was used for the measurement. Therefore, they cannot be transferred to excite other vehicle-model variants. The general task is now to derive and calculate invariant input loads on the basis of easily measured (but system-dependent) quantities. Mathematically, this leads to an optimal control problem, see Sect. 2 for a general formulation. In this paper we present an approach for dealing with this problem. We only consider multibody system models, consequently we assume, that a mathematical description of the considered system, as differential equation, is available. We present a solution approach for the problem of deriving invariant input signals: the method of control-constraints, see also [3, 4]. The remaining part of this paper is organized as follows: In Sect. 2 we give some mathematical preliminaries and formulate the optimal control problem. In Sect. 3 we present a strategy to solve the optimal control problem. So far we consider general multibody systems, whereas we focus on a specific tire model in Sect. 4 to derive a virtual road profile to simulate a full vehicle model. We present the general principle, Sect. 4.1, and some simulation results obtained by deriving a virtual road profile to excite a mbs model of a Porsche Cayenne, Sect. 4.2.

2 Mathematical Problem Formulation 2.1 The Equations of Motion of a Multibody System A multibody system generally consists of a finite number of rigid and elastic bodies with masses and moments of inertia that are linked together by joints and force-elements (springs, dampers, driven actuators etc.), which are assumed to be massless. Further, we assume, that the system is excited by a well-defined input signal, e.g., the displacement of some bodies of the system or some forces acting on bodies of the system. The equations of motion describing such a mbs form a second order system of ordinary differential equations (ODE): M.q/qR D f .t; q; dtd q; u/;

(1)

If the mbs contains kinematically closed loops, algebraic constraint equations have to be added resulting in a differential-algebraic equation (DAE). The output quantities of the considered system are defined as a given function of the positions and/or velocities and possibly the input u: y.t/ WD gout .q.t/; dtd q.t/; u.t//:

(2)

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Now, we assume, that there are some (desired) reference output signals, yref .t/, gained by experiment and measurement, available as function of time or a sampled time-signal respectively.

2.2 The Optimal Control Problem We assume, that the mbs including the input interface description defines the input u representing invariant input quantities. Then, the problem of deriving (invariant) input signals as described in the introduction leads to the following optimal control problem (OCP): Find a state variable x.t/ WD .q.t/; v.t// 2 W 1;1 .Œ0I T / and (bounded) input signal u.t/ 2 L1 .Œ0I T /, such that the cost functional Z

T

J Œx; u WD

  gout .x.t/; u.t//  yref .t/2 C r  kuk2 dt;

(3)

0

is minimized, subject to the ODE/DAE (1), where rkuk2 is a regularization-term.

3 Solution Strategy: Method of Control-Constraints The solution algorithms to solve optimal control problems are mainly divided in direct and indirect methods. The first ones are based on evaluations of necessary conditions, whereas the latter first discretize the optimal control problem and then apply some optimization procedures. For an overview and comparison, we refer to [2, 4, 5]. In this paper, we focus on a third alternative, the method of control-constraints, [3,4]. This method seems suitable for the application, we present in the next section. One first restriction of this approach is the requirement, that the number of inputs equals the number of outputs. If this is the case, however, the main idea is simple: Require the best achievable result, namely J Œx; u D ky  yref k2L2  0. In case of existence, the corresponding u is a global minimizer. Sufficient for this requirement is the condition y.t/  yref .t/ D gout .x.t/; u.t//  yref .t/ D 0

8t 2 Œ0I T :

(4)

The solution approach consists of adding (4) to the model-equation (1) as a further algebraic constraint-equation, a so called control-constraint. The resulting equation is a differential-algebraic equation (DAE) and it can be solved by a DAE-integrator. In this context, the control variables u are algebraic variables. The (differentiation-) index of the resulting DAE, however, can be very high, depending on “where” the input goes into the system and which state variable appears in the output gout , see [4,5]. Thus, in most application cases some index reduction methods

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and stabilization techniques are necessary, which require a lot of information about the equations of motion. Therefore the whole method is only suitable and applicable for systems of moderate size and complexity, for which the equations of motions are known in an analytical form.

4 Application: Invariant Loading for Full Vehicle Simulation 4.1 General Principle In this section we describe the general principle, that we used to derive a virtual road profile, and then apply the method of control-constraints explained above to solve the arising optimal control problem. As a starting point we have measured tire forces gained by the driving of a Porsche Cayenne over an off-road test-track. In a second step we simulate an mbsmodel of the Porsche Cayenne with the vehicle-dependent tire forces as excitation. From that simulation, we obtain rim-motions as reference signals. Now comes the crucial step: We use a simple tire surrogate model including a six-dimensional road profile as input quantity, that is thought to be linked to the rim instead of the four tires. The task is to derive the input road profile in such a way, that the corresponding rim-body of the tire model moves exactly as the rim of the vehicle model. In other words: The system is the tire surrogate model, the motion of the tire model’s rim-body is the system output, the reference signals have been obtained by the simulation before as rim motions of the full vehicle. For future steps, the derived virtual road profile together with tire model shall be linked to vehicle models at their rims, and then, the full vehicle including a tire model can be simulated. Figure (1) sketches this procedure schematically. Note, that this procedure treats all four tires independently, i.e., for each tire surrogate model, a six-dimensional, virtual road profile is derived. As tire surrogate model, we take a six-dimensional two-mass-spring-dampersystem, the highest mass represents the rim, consequently, the output is the

F, M

Identify excitation as virtual road profile measured quantities

Fig. 1 Using a tire surrogate model

Æ

u

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motion of that mass. The corresponding DAE coming from the method of controlconstraints reads as follows qR 1 D f1 .q1 ; dtd q1 ; q2 ; dtd q2 ; F; M /

(5)

qR 2 D f2 .q1 ; dtd q1 ; q2 ; dtd q2 ; u/

(6)

0 D q1 .t/  yref .t/:

(7)

The variables qi 2 R6 denote the translational and rotational motion of the two masses, fi describe the dynamics of the two-mass-spring-damper-system and F; M are measured tire forces and torques, which acts on the rim, i.e., on the highest mass of the tire surrogate model. The variable u 2 R6 represents the six-dimensional virtual road profile. Corresponding to these six input variables, there are six control constraints, (7), requiring the motion of the highest mass equals the desired rim motion.

4.2 Simulation Results We close this section with some simulation results. As mentioned above, the tire forces and torques have been obtained by a Porsche Cayenne’s driving over an off-road test-track, the rim motion as reference signals by a force-excited mbs simulation of a Cayenne-mbs-model. All software-routines to implement the general principle described above have been developed at ITWM. They include the surrogate tire model and can be coupled to commercial mbs software-tools. In this special case, we use the mbs software tool ADAMS, [1]. The resulting DAE, (5)–(7), is solved by the DAE-integrator RADAU5, [6]. Figure (2) shows the displacement of the highest mass in z-direction and the corresponding reference signal. Note, that Fig. 2b represents one of the control constraint equations in (7).

x 10−9

x 10−9 2

error

1

displacement [m]

displacement [m]

2 –9

x 10

0 –1 –2 –3

error

1 x 10

–9

0 –1 –2 –3

4

6

8

10

12

14

16

times [s] (a) Simulated and reference output

18

4

6

8

10

12

14

16

18

times [s] (b) Difference between simulated and reference output

Fig. 2 Simulation results I: displacement of the highest mass

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8000 force z [N]

displacement [m]

calculated Force original section Force

6000 4000 2000 0 –2000 0

2

4

6

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0.6 0.5 0.4 0.3 0.2 0.1 0 –0.1

virtual road profile uz

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time [s]

time [s]

(a) Simulated and reference output, vertical force

(b) Derived virtual road profile in z-direction

20

Fig. 3 Simulation results II: vertical section force and derived road profile

Figure (3) shows exemplarily the vertical section force at the rim and the derived road profile in z-direction, front-left.

5 Outlook A next step in our investigation is an independence-analysis, i.e., to what extent can the derived road profile be used to excite other vehicle variants? How different can they be? To obtain better independence-properties, it could be suitable to use more advanced, physically motivated tire models instead of the two-mass-spring-damper system.

References 1. Adams, M.D.: MSC. Software GmbH. Munich, Germany (2010) 2. Betts, J.T.: Practical Methods for Optimal Control Using Nonlinear Programming. Society for Industrial and Applied Mathematics, Philadelphia (2001) 3. Blajer, W., Kolodziejczyk, K.: A geometric approach to solving problems of control constraints: Theory and a DAE framework. Multibody Syst. Dyn. 11, 343–364 (2004) 4. Burger, M., Dressler, K., Marquardt, A., Speckert, M.: Calculating invariant loads for system simulation in vehicle engineering. In: Multiobody Dynamics 2009 ECCOMAS Thematic Conference. Warsaw (2009) 5. Burger, M., Speckert, M., Dressler, K.: Optimal control methods for the calculation of invariant excitation signals for multibody systems. In: The 1st Joint International Conference on Multibody System Dynamics. Lappeenranta (2010) 6. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Springer, Berlin (1996)

Structure Preserving Spatial Discretization of a Piezoelectric Beam Thomas Voß

Abstract The modeling and control of piezoelectric materials is an important research topic due to their technological feasibility and continuously increasing applicability. In this paper we discuss the spatial discretization of piezoelectric beams modeled as infinite dimensional systems in the port-Hamiltonian (pH) framework (energy-based modeling framework). The spatial discretization is performed such that the structure and consequently the physical properties of the infinite dimensional pH model are preserved. The finite dimensional system can then be used for simulation or control design. Furthermore, the spatial discretization scheme can also be used for other infinite dimensional systems, e.g. transmission lines.

1 Introduction Port-Hamiltonian (pH) systems represent an excellent alternative to the classical dynamical systems framework due to their properties with respect to modeling and control of complex multiphysics phenomena. Initially the pH framework was developed for modeling finite dimensional systems [2], but recently this framework has been extended to modeling infinite dimensional systems [1] to be able to handle more complex systems. Typically, during the design and/or implementation of a controller for an infinite dimensional system one has to spatially discretize the system. But standard spatial discretization schemes are not suitable for the discretization of pH systems. This lays in the fact that classical schemes will certainly destroy the pH structure. Therefore, for the spatial discretization of the

T. Voß () Department of Electrical Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 68, © Springer-Verlag Berlin Heidelberg 2012

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piezoelectric beam we apply a method which will preserve the mathematical structure of the pH system and hence we preserve the physical properties of the infinite dimensional systems.

2 Short Introduction to Finite Dimensional Port-Hamiltonian Systems In this section we briefly introduce the concept of finite dimensional pH systems, see [1]. A finite dimensional pH system in local coordinates can be described as d dt x

D .J.x/  R.x//

y D B > .x/

@H .x/ C B.x/u @x

(1)

@H .x/ @x

where x D .x1 ; : : : ; xn / 2 X  Rn is the state of the system, u 2 Rm and y 2 Rm are the inputs and outputs respectively. The matrices J D J > and R D R>  0 are the interconnection and the resistance matrix respectively. The matrix B 2 Rnm is the input force matrix, whileH.x/ W X ! R with H.x/ > c; 8x 2 X is the so called Hamiltonian of the system. The Hamiltonian H.x/ represents the stored energy in a physical system. Note, that one can prove that for a pH system the energy-balancing property ( dtd H  y > u) holds—this means that the system is passive. Hence, the Hamiltonian can be used as a Lyapunov function to prove stability. Another often used formulation is the effort-flow form of a finite dimensional pH system, where one expresses (1) in terms of flows f D dtd x @ H . Another concept, which we will use, is the energy flow of the and efforts e D @x system also called the net power. The energy flow of (1) is defined as Pnet D e > f C y > u:

(2)

3 Infinite Dimensional Port-Hamiltonian Model of a Piezoelectric Beam In this section we define a system that describes the dynamics of a piezoelectric beam in distributed pH form, for more details on pH systems see [1]. Let the coordinate system in which we define the dynamics of the beam be denoted by .z1 ; z2 ; z3 /. We consider a beam which has a length of L (z1 2 Z D Œ0; L), a width of 2g (z2 2 Œg; g) and a height of h2  h1 (z3 2 Œh1 ; h2 ). Moreover, we define the boundaries of the domain Z as @Z. Then the dynamics of a piezoelectric beam in

Structure Preserving Spatial Discretization of a Piezoelectric Beam

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pH form [3] can be written as follows 2 6 6 6 6 4

d p dt d 0 u dt d dt q d dt e

3

2

3

2

0 0 0 d

0 6 6 0 7 76 6 5 d 6 4 0

  ym D B > 0 0 0 e;

d ˇ q @Z dt

0 7 7 6 J > 7D6 m 7 4 0 5 0

Jm 0 0 0

ˇ

3

2 3 B 7 7 6 7 7 607 7 C 4 5 um 7 0 5 0

ıH ıp ıH ıu0 ıH ıq ıH ıe

D V .@Z/;

(3)

ˇ

d ˇ dt e @Z

D I.@Z/

(4)

where 2

3 d 0 0 0 Jm D 4 0 d 0 0 5; 0 0  d

2

3 2g 0 B D 4 0 2g 5; 2ghb 0

p D M dtd u;

>  u D u0 ; w; ;  0 :

Furthermore, the energy stored in the piezoelectric beam can be stated as 1 H.p; u ; q; e / D 2 0

Z

L

>

1

Z



p M pC

0

Z

h15 q

A h2

"11 d z3 C

h1

 C E "211 .u0 / C 2G"213 .u0 / dA

(5)

1 2 1 q C e2 d z1 C L

where u0 and w are the displacements in the z1 and z3 direction, respectively,  is the rotation of the cross sectional area, q is the electrical charge and e is the magnetic flux of the piezoelectric beam. The variable M represents the mass matrix of our systems, C E is the Young’s modulus, G is the shear modulus, C is the capacitance, and L the inductance of the used materials. All these variables depend on the z1 coordinate and on the time t. Note that we have formulated the problem in the differential geometric framework, hence, d is the exterior derivative,  is the Hodge star operator and ıH ıx is the variational derivative of H with respect to x. Note that (4) represents the dynamics of a piezoelectric beam, which are actually the coupled dynamics of a Timoshenko beam and a transmission line—the coupling is done via an energy exchange [3]. Hence, by setting the mechanical domain to zero one obtains the equations of motion for the transmission line: "

d q dt d dt e

ˇ

#



0 d D d 0

"

ıH ıq ıH ıe

#

ˇ D V .@Z/; dtd e ˇ@Z D I.@Z/ Z 1 L 1 2 1 H.q; e / D q C e2 d z1 : 2 0 C L d ˇ dt q @Z

(6)

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Similarly one can derive the equations of motion for the Timoshenko beam by setting the electrical domain to zero.

4 Spatial Discretization The spatial discretization of the piezoelectric beam is performed under the premise that the pH structure of the infinite dimensional system is preserved during the spatial discretization procedure. This is achieved by first discretizing the interconnection structure (4) and then spatially discretizing the energy function (5) on the domain Zab D Œa; b  Z. Both results combined yield then the dynamics of the pH system on the spatial domain Zab . Moreover, assuming a power S conserving interconnection of n systems defined on n subsets Zi of Z, s.t. Z D niD1 Zi , yields then a finite dimensional approximation of the dynamics on Z in the pH form.

4.1 Spatial Discretization of the Interconnection Structure To be able to spatially discretizing the system one performs a transformation (4) as presented below. One can decompose Jm as follows 2

3 2 3 00 0 0 d 000 Jm D 4 0 0 0 0 5 C 4 0 d 0 0 5 D J0 C dJ1 : 0 0  0 0 00d We then use this results to reformulate (4) in terms modified flow defined as

which is

WD f  J0 e: The approximation of efforts and flows is done by a multiplication of a time depending scaling factor and a spatially dependent shape function: Z

e l .t; z/ D eal .t/!a .z/ C ebl .t/!b .z/; !ab D !a .a/ D !b .b/ D 1;

l

.t; z/ D

l ab .t/!ab .z/

!a .b/ D !b .a/ D 0:

(7)

Zab

The subscript l is related to the effort/flow we want to approximate. Next, the substitution of the shape function into (4) is used to derive a finite dimensional approximation for the equations of motion and to ensure that the shape functions fulfill (4) also geometrically. For the first equation of motion this can  1 p be done in the following manner. Assuming that ebu1 D 0 and c D ab1  eau1 we

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have that c!ab D d!a ) !ab D d!a (7)

Similarly, we obtain !ab D d!b : Then substituting this relation into the equation of motion an integrating over Zab yields a finite dimensional approximation for (4) on Zab , e.g. p1 ab

D ebu1  eau1 :

However, the finite dimensional equations of motion still depend on . To obtain an approximation in terms of f one performs a back transformation for f u4 and f p3 . Therefore, we substitute the approximation of the efforts and flows into the equation p for p3 , multiply by !ab3 , and integrate over Zab . This yields then p3 ab

p

 1 ˛ab eau3 C ˛ba ebu3 ˛N Z p p !ab3 ^ !ab3 ; ˛ab D

D fab3 C Z

˛N D Zab

Z p

Zab

!au3 ^ !ab3 ;

p

˛ba D Zab

!au3 ^ !ab3 :

Although one has already obtained a finite dimensional approximation of the dynamics, one still needs to find an approximation of the efforts in order to be able to define a pH system. This can be done via the net power of the system which is defined as Pnet D

Z X n

Z

m X

ei ^ fi C

Z i D1

@Z

ui ^ yi :

i

The substitution of the approximation of the efforts and flows and the integration over Zab yields then an approximation of the net power on Zab  p > p  u0 > u0 q q   net Pab D eab fab C eab fab C eab fab C eabe fabe where the total efforts are defined as l D ˛ab eal C ˛ba ebl : eab

Furthermore, since a general pH system is accusal, one needs to define the input of the system in order to be able to define a causal interconnection structure. We choose the following inputs to the system i> h  uab D eap1 ; eap2 ; eap3 ; ebu1 ; ebu2 ; ebu3 ; eaq ; eb e :

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The input vector uab represents the values of the efforts at the boundaries of the p spatial domain, e.g. ea 1 expresses the force in the z1 direction at point a. Then the finite dimensional interconnection structure can be defined as fab D Jab eab C Bab uab yab D

B> ab eab

(8)

C Dab uab

where 2

Jab

0 2

0

0 0 JeT

3

0

0

0 0 0

3 2 0 I3 0 0 7 6 7 0 7 6 I3 0 0 0 7 7; 7 ; Dab D 6 4 0 0 0 1 5 Be 5 0

3

0 Be 0

0 2

3

1 1 1 1 6 7 m m 60 0 0 5 ; Je D Be D ; B1 D  I3 ; B2 D ˛ ˛ab ˛ab 4 0 1 1 0 ˛N ˛ba 0   p1 p2 p3 > q D eb ; eb ; eb ; eau1 ; eau2 ; eau3 ; eb ; eae : 1 ˛ba

6 Jm D 4 0 0 yab

0

2 m B1 0 6 0 Bm 7 0 7 6 2 7 ; Bab D 6 4 0 0 Je 5

0

0 Jm 6 J > 0 6 m D6 4 0 0

0 0 0

1 ˛ba

0 1 0 3 00 1 07 7; 0 05 01

By setting the mechanical domain to zero one can derive the finite dimensional approximation of (6), in the same way as it has been done for the infinite dimensional system: " Jab D

0

1 ˛

 ˛1 0

"

# ; Bab D

0

1 ˛

 ˛1 0

#

 ; Dab D

 0 1 ; 1 0

i> h >  q  uab D eaq ; eb e ; yab D eb ; eae : The last step is now to spatially discretize the energy function.

4.2 Spatial Discretization of the Energy Function The discretization of the energy function is compared to the discretization of the input-state-output structure rather straightforward. Because the flow f is related to the state x via the time derivative (f D dtd x), it is clear that the spatial approximation of the state has to be done in the same way as for the flows. Hence, we l 1 have the following approximation for the states, x l  xab !ab . By substituting this approximation in the definition of our energy function H.x/ and then integrating over Zab we obtain a finite dimensional approximation of the energy function which l we denote by Hab . Hence, we define the efforts as eab D @x@l Hab .xab /: Finally, by ab

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combing these results with (8) we obtain a finite dimensional approximation for the dynamics of (4) on Zac in pH form.

5 Conclusion The structure preserving spatial discretization of port-Hamiltonian (pH) systems offers, from the control perspective, a very suitable alternative to the classical spatial discretization scheme, where one cannot ensure a structure preservation and hence cannot guarantee that the physical properties of the pH system are preserved. Therefore, in this paper we have presented a spatial discretization scheme in the pH framework. This scheme has been developed for piezoelectric beams which are modeled as infinite dimensional systems. Moreover, the spatial discretization has been performed such that the structure and consequently the physical properties of the infinite dimensional pH model are preserved. Then the finite dimensional system can be used for simulation or control design. Furthermore, the spatial discretization scheme can also be used for other infinite dimensional systems such as transmission lines. Acknowledgements The here presented research was funded by MicroNed programme.

References 1. Duindam, V., Macchelli, A., Stramigioli, S., Bruyninckx, H. (eds.): Modeling and Control of Complex Physical Systems. Springer, Berlin (2009) 2. Van der Schaft, A., Maschke, B.: The Hamiltonian formulation of energy conserving physical ¨ systems with external ports. AEU. Archiv f¨ur Elektronik und Ubertragungstechnik 49(5–6), 362–371 (1995) 3. Voß, T., Scherpen, J.: Modeling for control of a nonlinear Timoshenko beam with piezo actuation. (submitted)

Error Estimates for Finite-Dimensional Approximations in Control of Distributed Parameter Systems Andreas Rauh, J¨oran Ritzke, and Harald Aschemann

Abstract Distributed parameter systems such as elastic multibody systems or heat and mass transfer processes are described by partial differential equations. However, for the control design of such systems, finite-dimensional approximations are usually applied. To derive accurate control strategies both for stabilization of fixed operating points and tracking control, it is essential to quantify deviations between the exact dynamics and its finite-dimensional approximation. Using experimental data, modeling uncertainties are estimated in real time for a finite-dimensional mathematical representation of the motion of an elastic beam employing a continuous-time Luenberger-type observer and a discrete-time stochastic filtering approach.

1 Modeling and Control of a Light-Weight Rack Feeder In previous work, it has been shown that elastic multibody models can be employed advantageously to design control laws for flexible mechanical systems [1, 2]. In this paper, a prototypical test rig of a light-weight high-speed rack feeder is considered. For the system depicted in Fig. 1, both tracking controllers including active oscillation damping by the system’s main drive and continuous-time as well as discrete-time state and disturbance estimators are developed. The estimators are applied to determine deviations between the system’s finite-dimensional representation and its real-life behavior. To design the decentralized control structure and the state estimators, the rack feeder is represented by a multibody model with three rigid bodies—the carriage with mass mS , the cage (mass mK , mass moment of inertia K ,

A. Rauh ()  J. Ritzke  H. Aschemann University of Rostock, D-18059 Rostock, Germany e-mail: [email protected]; [email protected]; [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 69, © Springer-Verlag Berlin Heidelberg 2012

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Fig. 1 High-bay rack feeder: Elastic multibody model (left) and decentralized control (right)

position xK .t/) movable vertically on the beam structure, and a point mass mE at the tip of the beam—and an elastic Bernoulli beam (density , cross section A, Youngs modulus E, second moment of area IzB , and length l). In the following, the vertical cage position is denoted by the dimensionless parameter .t/ WD xKl.t / . The elastic degrees of freedom of the beam with respect to its bending deflection are represented by the Ritz ansatz v .x.t/; t/ D

    3 x 2 1  x 3 v1 .t/ ;  2 l 2 l

(1)

in which only the first bending mode is considered. The equations of motion for the rack feeder with the generalized coordinates   T T q WD yS .t/ v1 .t/ and the input vector h D 1 0 can then be stated as M q.t/ R C D q.t/ P C Kq.t/ D h  .FM .t/  FF yPS .t//

(2)

with the motor and friction forces FM and FF as well as the mass matrix " M D

mS C Al C mK C mE 3 8 Al

C

mK  2 2

3 8 Al

C

.3  / C mE

mK  2 2

.3  / C mE

# ;

(3)

m22

  4 2 2 33 where m22 D 140 Al C 6I5lzB C mK4 .3  /2 C 9lK2 1   C 4 C mE holds. The stiffness and damping matrices K and D, respectively, are given by " KD

0 0

3EIzB l3

0  3mK g 3 3 1C  8 Ag  l

# 3 2 20



3 4





6mE g 5l

" and D D

#

0

0

0

3kd EIzB l3

:

(4)

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The decentralized control for the rack feeder consists of a proportional controller for the vertical cage position .t/, see [2], and an underlying velocity control for the electric drive in horizontal direction—implemented on its current converter— with the resulting dynamics T1y yRS .t/ C yPS .t/.t/ D vS .t/. Taking into account this first-order lag system, the equations of motion are given by " xP y D

0

#

I

My1 Ky My1 Dy

"  xy C

xy D

0 My1 hy

# vS DW Ay xy C by uy ; (5)

" # q qP

with " My D

T1y 3 8 Al

C

mK  2 2

0

.3  / C mE m22

"

# and Dy D

1

0

0

3kd EIzB l3

# :

(6)

Using the state-space representation (1), a gain scheduled linear feedback controller uy;FB D kyT ./xy is determined by minimization of the time-weighted quadratic cost function [1] 1 J D 2

Z1 h

i xyT Qy xy C ry u2y e 2˛y t dt;

(7)

0

in which the state variables and the control effort are weighted by Qy and ry . In (7), the parameter ˛y guarantees an absolute stability margin of the closedloop eigenvalues, i.e., the maximum real part of the eigenvalues is bounded by ˛y for fixed values of . Introducing Ay˛ ./ WD Ay ./ C ˛y I , the positive definite solution of the parameter-dependent algebraic Riccati equation ATy˛ ./Py ./ C Py ./Ay˛ ./  ry1 Py ./by ./byT ./P ./ C Qy D 0

(8)

leads to a gain scheduled optimal feedback control law with kyT ./ D ry1 byT ./Py ./. Two representative controller gains determined using this procedure are depicted in Fig. 2. They are computed offline for different values of  2 Œ0 I 1. In the online application, the results are interpolated linearly using the measured position . achieve a sufficiently accurate tracking for the end effector position yK .t/ D  To  1 12  2 .3  / 0 0 xy .t/ D cyT ./xy .t/ in horizontal direction, the controller is extended by the state-dependent feedforward part in Fig. 1. It is determined in such a way that the control transfer function is inverted approximately. An exact inversion is not possible since the position yK .t/ corresponds to a non-flat system output [1].

A. Rauh et al. −4.4

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ky,2(κ)

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Fig. 2 Gain scheduled coefficients of the feedback control law: Coefficients ky;2 ./ and ky;3 ./

2 State and Disturbance Estimation To determine deviations between the dynamics ofthe test rigand its mathematical PO model, both a continuous-time estimator x.t/ Q Df xQO .t/ ; u .t/ CL  .y .t/  yO .t// with yO .t/ D C xQO .t/ and the observer gain matrix L, where xQO represents the estimates for the extended state vector xQ (containing xy as well as uncertain parameters and model errors) and a discrete-time Extended Kalman Filter (EKF) are applied [4]. The disturbances and model errors are estimated by assuming piecewise constant (i.e., slowly varying) parameters described by integrator disturbance models. The continuous-time estimator is parameterized by minimization of the cost function 1 J D 2

Z1

 x.t/ Q T QJ x.t/ Q C y.t/T RJ y.t/ dt

(9)

0

with QJ D QJT  0 and RJ D RJT > 0 for constant parameters . Like for the controller design, the observer gain matrix L./ D PJ ./ cy ./ RJ1 is determined offline for various parameters  2 Œ0 I 1 by solving the algebraic Riccati equation PJ ./cy ./RJ1 cyT ./PJ ./  Ay ./PJ ./  PJ ./ATy ./  QJ D 0:

(10)

Alternatively, stochastic estimates are determined by an EKF with the discretized system model xQ kC1 D ak .xQ k / C wk , yOk D hk .xQ k / C vk and the Gaussian system and measurement noise wk , vk as well as piecewise constant inputs. The EKF makes use of a two-stage procedure. First, state estimates are updated in a filter step by p

e fx;k Q

.xQ k jyOk / D R Rn x

fx;k .xQ k /  fv;k .yOk  hk .xQ k // Q p

fx;k .xQ k /  fv;k .yOk  hk .xQ k // d xQ k Q

(11)

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as soon as new measured data yOk are available at the point of time tk . Second, this probability density is propagated until the next sampling point by the prediction step p

Z

.xQ kC1 / D fx;kC1 Q

e fx;k Q k /  fw;k .xQ kC1  ak .xQ k // d xQ k : Q .x

(12)

Rn x

To approximate both probability densities by closed-form normal distributions i h exp  12 kxQ  xQ k2C 1 xQ N .xQ ; CxQ / D p ; xQ 2 Rnx ; nx .2/ jCxQ j

(13)

the discrete-time system model is linearized for the current state estimate according to xOQ kC1 D Ak  xOQ k C wQ k and yOk D Hk  xOQ k C vQ k .  1 p p Introducing Kk D Cx;k HkT Hk Cx;k HkT C Cv;k , the posterior density after Q Q the filter step is given by   p p Ok  Hk x;k ex;k Q D x;k Q C Kk  y Q  v;k

p

p

e and Cx;k Q D Cx;k Q  Kk Hk Cx;k Q ;

(14) while the prediction step is approximated by p

D Ak ex;k x;kC1 Q C w;k Q

p

e T and Cx;kC1 D Ak Cx;k Q Ak C Cw;k : Q

(15)

p

e and Cx;kC1 in addition Since the EKF provides the covariance matrices Cx;k Q Q p e to the mean values x;k and  , it quantifies the level of confidence in the Q x;kC1 Q computed estimates which is not possible using the continuous-time estimator derived above.

3 Results of Online State and Disturbance Estimation Based on the measured carriage position, the state vector xy is reconstructed using different observer configurations. First, the continuous-time observer is used (Fig. 3). Its disadvantage is the lack of information concerning the reliability of the estimates. For that reason, the EKF is used (Figs. 4 and 5) to reconstruct the system states as well as either the additive error ad .t/ in the differential equation for vP 1 .t/ or the damping coefficient kd , improving the results as compared to the continuoustime estimator with kd D 0. To initialize the EKF, kd is also set to zero. Now, the three- ranges (dotted lines) provide a means to quantify the reliability of the estimates. To guarantee full state observability during estimation of ad and kd , the

600

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0.05 Δ x2 in m

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Fig. 3 Optimization-based state reconstruction without disturbance estimation: Carriage position x1 , estimation error x1 of carriage position, bending deflection x2 , estimation error x2 of bending deflection 0.05

0.01 0

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0

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10 15 20 25 t in s

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10 15 20 25 t in s

0

30

5

10 15 20 25 t in s

0 −0.02 −0.04 −0.06 −0.08 −0.10

30

0

5

10 15 20 25 t in s

30

Fig. 4 Stochastic state reconstruction using EKF with estimation of acceleration errors: Estimation error x1 of carriage position, bending deflection x2 , estimation error x2 of bending deflection, acceleration error ad

0

0

0.05 damping kd

0.01

Dx2 in m

x2 in m

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20 25 t in s

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0.03 0.02 0.01 0 –0.01

−0.05

−0.05

−0.02

0.04

0

5

10 15

20 25 t in s

30

0

5

10 15

20 25 t in s

30

Fig. 5 Stochastic state reconstruction using EKF with estimation of the damping coefficient kd : Estimation error x1 of carriage position, bending deflection x2 , estimation error x2 of bending deflection, damping coefficient kd

complete closed-loop system model of the horizontal motion is used in the EKF, where the reference signal w.t/ replaces the input u.t/ in the previous section.

4 Conclusions and Outlook on Future Work In this paper, the control design for an elastic multibody model describing the motion of a prototypical high-speed rack feeder has been summarized. This modelbased approach improves the tracking accuracy as compared to classical gain scheduled PI controllers. This is achieved by systematic adaptation of the controller gains in such a way that minimization of a quadratic cost function becomes possible. However, to guarantee high accuracy, it is also necessary to quantify model errors resulting from finite-dimensional system representations. Therefore, different

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observer approaches have been compared which yield estimates for the deviations between the model and the real system behavior and quantify the estimation quality. In future work, we aim at the development of nonlinear stochastic estimators to reduce linearization errors introduced by the EKF for multiplicative couplings between non-measured state variables and parameters such as the unknown damping coefficient. A possible extension of the EKF is based on Carleman linearization embedding the nonlinear dynamics in a higher dimensional linear model [3].

References 1. Aschemann, H., Ritzke, J.: Gain-scheduled tracking control for high-speed rack feeders. In: Proceedings of 1st Joint International Conference on Multibody System Dynamics (IMSD). Lappeenranta, Finland (2010) 2. Aschemann, H., Schindele, D.: Model Predictive Trajectory Control for High-Speed Rack Feeders, Model Predictive Control, Tao Zheng (Ed.), ISBN: 978-953-307-102-2, InTech, Available from: http://www.intechopen.com/articles/show/title/model-predictive-trajectory-controlfor-high-speed-rack-feeders 2010, last date of access: February 06, 2012 3. Rauh, A., Minisini, J., Aschemann, H.: Carleman linearization for control and for state and disturbance estimation of nonlinear dynamical processes. In: CD-Proceedings of IEEE International Conference on Methods and Models in Automation and Robotics MMAR 2009. Miedzyzdroje, Poland (2009) 4. Stengel, R.: Optimal Control and Estimation. Dover Publications, New York (1994)

Solving Non-smooth Delay Differential Equations with Multiquadrics Francisco Bernal

Abstract We put forward a discretization scheme for the numerical solution of neutral differential equations (NDEs). The solution to the NDE in an interval I D Œa; b is approximated by a multiquadric (MQ) interpolant, whose coefficients are found by collocation on a set of N nodes in I . This approach, also known as Kansa’s method, enjoys an exponential rate of convergence and great flexibility regarding the location of the nodes, as long as the solution to the differential equation is smooth. However, the critical difficulty posed by NDEs is precisely that they propagate, forward in time and without damping, low-order discontinuities of the history function. Here, we exploit the sensitivity of the MQ interpolant to discontinuities in order to detect them in computing time. This allows for a partition of I into smooth subintervals, which are then sequentially solved by Kansa’s method.

1 Introduction Differential equations with delayed arguments are increasingly being used for modeling phenomena appearing in biology, economics, ecology, and engineering, to name but a few relevant fields of application [2]. The sophistication of many such models calls for accurate and reliable numerical solvers [9]. In turn, the growing availability of general-purpose numerical software encourage researchers to incorporate delay effects into their mathematical models. Delayed differential equations (DDEs), however, are difficult to solve numerically. They inherit all of the potential challenges posed by ordinary differential

F. Bernal () Instituto Superior Tecnico, UTL Lisboa, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal e-mail: [email protected]

M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 70, © Springer-Verlag Berlin Heidelberg 2012

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equations (which can, in fact, be seen as a subset of DDEs), such as stiffness, instability, etc., but also have some difficulties of their own. Two of them are particularly relevant. In DDEs, there is often the need to interpolate the delayed argument between discretization nodes, which makes necessary a good interpolant. Moreover, DDEs often propagate discontinuities forward in time. We stress the fact that the arising discontinuities are part of the solution, not a numerical artifact. In order to understand why, consider the following simple DDE: (

y 0 .t/ D y.t  /;

t 0

y.t/ D h.t/;

t < 0:

(1)

where h.t/ is the history function and  is the delay. If h.0/ ¤ y.0/, a discontinuity occurs at y.0/. The solution progresses smoothly until t D , when y 0 .  / D h.0/ but y 0 . C / D y.0/. This brings about a discontinuity in y 0 —which, in turn, will propagate as discontinuities in y 00 .2/; y 000 .3/ : : :, and so on. Neutral differential equations are a type of delayed differential equation of the form: y 0 .t/ D F Œt; y.t/; y.t  1 /; y 0 .t  2 /

(2)

Because of the presence of the delayed argument in the derivative (which is the hallmark of NDEs), singularities may propagate without “smoothing out”, as was the case in the previous example. The correct tracking of such discontinuities make NDEs especially challenging from a numerical point of view, since the performance of most numerical methods degrade when they step over discontinuities. If the NDE is linear and has a constant delay, the location of the derivative jumps can be predicted, and measures which preserve the accuracy of the numerical scheme can be prepared in advance—typically, splitting the interval by the jump. For nonlinear NDEs (or non-constant delays), however, this may not be possible, and discontinuities must be detected in computing time. Therefore, a discontinuity detector is a critical ingredient in a NDE solver. MQ interpolants have advantageous interpolation properties such as exponential spatial convergence [7]. In exchange, they behave poorly in the presence of singularities. In this work, we benefit from the sensitivity of MQ interpolants to non-smooth features of the solution in a twofold way: to detect them, first, and to approximate the smooth solution of the NDE in each of the singularity-free subintervals which lie in between. This paper is organized as follows. In Sect. 2, the features of MQ interpolation which are of interest for this work are reviewed. In Sect. 3, Kansa’s method is adapted to linear NDEs with singularities. Section 4 concludes the paper with a numerical example.

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2 MQ Interpolation of 1D Functions Consider f W Œa; b ! R and N scattered nodes  D fx1 < x2 < : : : < xN 1 < xN g. Hardy’s multiquadric (MQ) centered at xj is defined as: q jc .x  xj / D .x  xj /2 C cj2 (3) whose derivative is jc .x  xj /0 D q

x  xj .x  xj /2 C cj2

(4)

The shape of the MQ depends on the free parameter cj (hence the name of shape parameter for it). Notice that it can be thought of as a characteristic distance. An MQ interpolant of f can be written as f .x/ 

N X

˛j jc .x  xj /

(5)

j D1

where the coefficients are found by collocation, i.e. by solving the linear system 2

32 3 3 2 : : : Nc .x1  xN / ˛1 f .x1 / 6 7 6 :: 7 6 :: 7 :: :: 4 54 : 5 D 4 : 5 : : c c N .0/ ˛N 1 .xN  x1 / : : : f .xN / 1c .0/ :: :

(6)

The fact that the MQ has global support leads to fully populated matrices. It is a hallmark of this method that the best accuracy can only be obtained at the expense of extreme ill-conditioning [8]. Notice also that, since the interpolation space is made up of infinitely smooth basis functions jc .x/; j D 1 : : : N , it cannot be expected to capture well non-smooth features in f . Let us assume that the N nodes are equispaced in Œa; b D Œx1 ; xN , and that c1 D : : : D cN D c. We are interested in the case when f is piecewise smooth with a single jump discontinuity ı, located at a < x0 < b (i.e., jf .x0C /  f .x0 /j D ı). Then, the MQ interpolant will exhibit oscillations around x0 —the well known Gibbs phenomenon. The problem just described was studied in detail in [4]. Let us define ˛max D maxj D1:::N j˛j j, and xmax is the center of the MQ with coefficient ˛max . Here, we will focus on the following observations from that paper: • For fixed N , the magnitude of the spurious oscillations grow with increasing c and ı. • If ı D 0, ˛max takes place in the neighbourhood of the boundaries. • If ı > 0, ˛max takes place in the neighbourhood of x0 .

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We will exploit these observations to detect jump discontinuities in piecewise smooth functions, based on xmax  x0 . Since the fj˛j jg tend to form a “peak” around the discontinuity, the full-width at half maximum (FWHM) of it can be used to estimate the error: jx0  xmax j . FWHM

(7)

We can devise a scheme to improve xmax iteratively: MQ Discontinuity Detection Algorithm .0/

.0/

• Define I0 D Œa0 ; b0  WD Œa; b, N equispaced MQs 0 D fa0 D x1 ; : : : ; xN D b0 g, and c D c0 > 0; and compute and store Œ c0 1 0 (the matrix in (6)). • At iteration k: 1. 2. 3. 4. 5. 6. 7. 8.

.k/

.k/

.k/

T Find ˛max D maxj D1;:::;N jŒ ck 1 k Œf .x1 /; : : : ; f .xN / j. Estimate FWHMk . .k/ .k/ If xmax  ak < FWHMk or bk  xmax < FWHMk , then let x0  xmax . END. .k/ .k/ Let akC1 D max.ak ; xmax  FWHMk / and bkC1 D min.bk ; xmax C FWHMk /. Let ˇkC1 D .bkC1  akC1 /=.bk  ak /; 0 < ˇkC1 < 1, and ckC1 D ˇkC1 ck . ck 1 Then, Œ ckC1 1 kC1 D Œ k =ˇkC1 . .kC1/ .kC1/ ; : : : ; xN D bkC1 g. Let IkC1 D ŒakC1 ; bkC1  and kC1 D fakC1 D x1 Iterate until the desired precision is reached or FWHMk  .bk  ak /=2. (The condition number is the same throughout the iterations).

3 Solving Linear NDEs by Kansa’s Method Kansa modified the MQ interpolation scheme to solve PDEs [5, 6]. Kansa’s method was adapted to smooth DDEs in [1]. Consider the following linear NDE: y 0 .x/  p.x/y.x/  q.x/yŒx  .x/  r.x/y 0 Œx  .x/ D s.x/ if x 2 Œa; b (8) y.x/ D h.x/ if x  a

(9)

It will be convenient to split (9) into a NDE and an ODE y 0 .x/  p.x/y.x/  q.x/yŒx  .x/  r.x/y 0 Œx  .x/ D s.x/

if x  .x/ > a (10)

y 0 .x/  p.x/y.x/ D q.x/hŒx  .x/ C r.x/h0 Œx  .x/ C s.x/

if x  .x/ < a (11)

y.a/ D h.a/

(12)

Solving Non-smooth Delay Differential Equations with Multiquadrics

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We seek an approximate solution to (10)–(12) in the form: y.x/ D

N X

˛j jc .x  xj /

(13)

j D1

In order to solve for the coefficients, (10)–(12) are enforced over (13). For i D 1; : : : ; N : N X

fjc 0 .xi  xj /  p.xi /jc .xi  xj /  q.xi /jc .xi  .xi /  xj /

j D1

(14)

r.xi /jc 0 .xi  .xi /  xj /g˛j D s.xi / if xi  .xi / > a N X fjc 0 .xi  xj /  p.xi /jc .xi  xj /g˛j

(15)

j D1

D q.xi /hŒxi  .xi / C r.xi /h0 Œxi  .xi / C s.xi / if xi  .xi /  a N X

˛j jc .xi  xj / D h.a/ if xi D a

(16)

j D1

This scheme can be extended, in a straightforward manner, to the case of instances of y and y 0 being present with different delays. For nonlinear NDEs, the system of collocation equations above is nonlinear and a nonlinear solver must be used to find a root. Finally, an important yet open issue in Kansa’s method is the optimal number and location of RBF centers/collocation nodes. A recommended approach is the residual subsampling algorithm (RSA) from [3] (see also [1]). For a general NDE containing several discontinuities in a finite interval, we first try to find an estimate of the two leftmost discontinuities by looking at the MQ coefficients. Next, we define a subinterval which contains just the first one (x0I ), and apply the Algorithm in Sect. 2 to accurately locate it. Then Kansa’s method is used to approximate the discontinuity-free solution up to x0I . We then repeat the procedure to bracket and find the second discontinuity x0II —whereby we use the previous MQ approximation as “history function”—and so on.

4 A Numerical Example The following NDE is also used as a demo in MATLAB’s NDE solver ddeNsd [10]. It is a crafted example with a non-trivial exact solution and known discontinuities which allows us to check our method (Table 1).  0 y .t/ D y 0 .t  2/; t 0 (17) 5 t < 0: y.t/ D h.t/ D .t C 1/ ;

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Table 1 Approximate discontinuities, error estimations and accuracy for the example. RMS./ stands for the root mean square of the error to the exact solution Discontinuity Exact location MQ Approximation FWHM Subinterval RMS () 6 I 1.0 0.99999703 2:0  10 Œa; x0I / 6:50  107 II 2.0 1.99999931 1:2  106 .x0I ; x0II / 7:46  106 III 3.0 3.00000021 1:0  106 .x0II ; x0III / 1:73  105 IV III 7 IV 4.0 4.00000070 9:9  10 .x0 ; x0 / 2:95  105 IV .x0 ; b 4:36  107 4.5 4 3.5 3 2.5

history y(t)

2 1.5 1 0.5 0 −1

0

1

2 t

3

4

5

Fig. 1 The exact solution of the NDE (17). The MQ approximation cannot be distinguished with the naked eye

The exact solution is yEX .t/ D Œt C .t  Œt/5 ; t  0. It has a jump discontinuity at t D 0 and first-derivative discontinuities at t D n for n  0. For simplicity, (17) was solved in each discontinuity-free subinterval using N D 80 equispaced nodes. Equispaced nodes are a suboptimal distribution, and results can be expected to improve by orders of magnitude using the RSA adaptive algorithm, as shown in [1] (Fig. 1).

References 1. Bernal, F., Gutierrez, G.: Solving delay differential equations through RBF collocation. Adv. Appl. Math. Mech. 1, 257–272 (2009) 2. Erneux, T.: Applied Delay Differential Equations. Springer, Berlin (2009)

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3. Driscoll, T.A., Heryudono, A.: Adaptive residual subsampling methods for radial basis function interpolation and collocation problems. Comput. Math. Appl. 53, 927–939 (2007) 4. Jung, J.H.: A note on the Gibbs phenomenon with multiquadric radial basis functions. Appl. Numer. Math. 57(2), 213–229 (2007) 5. Kansa, E.J.: Multiquadrics - a scattered data approximation scheme with applications to computational fluid-dynamics. I. Surface approximations and partial derivative estimates. Comput. Math. Appls. 19, 127–145 (1990) 6. Kansa, E.J.: Multiquadrics - a scattered data approximation scheme with applications to computational fluid-dynamics. II. Solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput. Math. Appls. 19, 147–161 (1990) 7. Madych, W.R.: Miscellaneous error bounds for multiquadric and related interpolators, Comput. Math. Appl. 24, 121–38 (1992) 8. Schaback, R.: Error estimates and condition numbers for radial basis function interpolation. Adv. Comput. Math. 3, 251–264 (1995) 9. Shampine, L.F.: Solving ODEs and DDEs with residual control. Appl. Numer. Math. 52, 113– 127 (2005) 10. Shampine, L.F.: Dissipative approximations to neutral DDEs. Appl. Math. Comput. 203, 641– 648 (2008)

Higher-Order Matrix Splines for Systems of Second-Order Differential Equations ˜ and Luna Soler Emilio Defez, Michael M. Tung, Jacinto Javier Ib´anez,

Abstract We present a method based on matrix splines of higher order to approximate the solutions of Y 00 .x/ D f .x; Y .x//. Our approach is a generalization of previously developed methods employing matrix-cubic splines for similar matrix equations. An estimation of the approximation error and an illustrative example are presented.

1 Introduction and Preliminaries Initial value problems of the matrix type Y 00 .x/ D f .x; Y .x//; Y .a/ D Y0 ; Y 0 .a/ D Y1 ; a  x  b;

(1)

are frequently encountered in various areas of engineering and industrial modeling, such as molecular-dynamics simulation of semiconductor devices and mechanical vibration systems [3,5]. Matrix splines are a convenient tool to tackle such problems. The simplest, that is, scalar case of (1) produces already considerable difficulties with divergences for splines of order m > 3, see [4]. These difficulties emerge in a similar fashion for the general matrix case as outlined in [2] and references therein. Table 1 clearly demonstrates such disastrous behaviour for the standard test problem ([2, p.952] and references therein) with divergences for matrix splines of order

E. Defez ()  M.M. Tung  L. Soler Universidad Polit´ecnica de Valencia, Camino de Vera, s/n, 46022, Valencia, Spain e-mail: [email protected]; [email protected] J.J. Ib´an˜ ez Instituto de Instrumentaci´on para Imagen Molecular, Universidad Polit´ecnica de Valencia, Camino de Vera, s/n, 46022, Valencia, Spain e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 71, © Springer-Verlag Berlin Heidelberg 2012

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Table 1 Matrix spline error for method proposed in ([2] and references therein) with m D 6, h D 0:1 Interval

Œ0; 0:1

Œ0:1; 0:2

Œ0:2; 0:3 Œ0:3; 0:4 Œ0:4; 0:5 Œ0:5; 0:6 Œ0:6; 0:7 Œ0:7; 0:8 Œ0:8; 0:9 Œ0:9; 1

Max. error 1:71  107 2:74  106 2  105 2  104 0:0025

0:024

0:245

2:432

24:090

238:655

m D 6 with h D 0:1. Surprisingly, this makes clear that higher-order splines (m D 6) do not necessarily improve the quality of approximation (for the problem considered m D 3 yields a maximum error of only 6:2838  104 ). We will show that these problems can be avoided using splines S.x/ of order m  3 with differentiability class C 2 , and not C m1 . There are several important advantages of our proposed method. Firstly, the algorithm is straightforward to implement on numerical and symbolical computer systems. Secondly, in the case of higher-order differential matrix systems, our method does not require to disentangle the system at hand and reduce it to a higher dimensional system of lower order, common practice in problems of this kind. Finally, all approximate solutions are easy to evaluate. The work is organized as follows. In Sect. 2 we first develop a new method to tackle the aforementioned problem, and in Sect. 2.1 we then conclude with a test example. Throughout the text, we will follow the notation established in [1, 2].

2 Construction of the Method We now focus on problem (1), where f W Œa; b  Rrq ! Rrq , f 2 C s .T /, with T D f.x; Y /I a  x  b; Y 2 Rrq g and Y0 ; Y1 ; Y .t/ 2 Rrq . Furthermore, f will have to satisfy the Lipschitz condition kf .x; Y1 /  f .x; Y2 /k  L kY1  Y2 k ; a  x  b; Y1 ; Y2 2 Rrq :

(2)

A partition of the interval Œa; b is given by Œa;b D fa D x0 < : : : < xn D bg, where xk D a C kh with k D 0; 1; : : : ; n, and the step size is h D .b  a/=n for n any fixed positive integer. Then, to approximate the solution for (1), we construct in each subinterval Œa C kh; a C .k C 1/h a matrix spline of order m  s, where s denotes the order of differentiability of f . In the first interval Œa; a C h, the spline shall take the form: m1

SjŒa;aCh .x/ D A0

.x  a/m X .l/ .x  a/l C ; Y .a/ mŠ lŠ

(3)

lD0

where A0 2 Rrq is a matrix parameter still to be determined. To calculate the third-order derivative Y .3/ .x/, we follow the notation and procedure of [1] to obtain

Higher-Order Matrix Splines for Systems of Second-Order Differential Equations

Y .3/ .x/ D

613

i @f .x; Y .x// @f .x; Y .x// h C Œvec f .x; Y .x//T ˝ Ir D g1 .x; Y .x//: @x @ vec Y .x/ (4)

Here g1 2 C s1 .T / is a function of the given data, and hence Y .3/ .a/ D g1 .a; Y .a//. Next, we may assume that f 2 C s .T / for s  2 and obtain the higher-order derivatives: Y .4/ .x/ D g2 .x; Y .x// 2 C s2 .T /; : : : ; Y .m1/ .x/ D gm3 .x; Y .x// 2 C s.m2/ .T /: (5) All of these derivatives may readily be calculated employing suitable computer algebra systems. Substituting x D a in (5), yields Y .4/ .a/, : : :, Y .m1/ .a/. This way all matrix parameters of the spline (3) are known, except for A0 . It is now straightforward to prove that .a/ D Y 0 .a/; S 00 .a/ D Y 00 .a/ D f .a; SjŒa;aCh .a//: SjŒa;aCh .a/ D Y .a/; S 0 jŒa;aCh jŒa;aCh Thus, the spline defined in (3) satisfies the equation of problem (1) at point x D a. Suppose that (3) is also the solution of problem (1) for  x D aCh. As a consequence,  00 one has the matrix equation: S .a C h/ D f a C h; SjŒa;aCh .a C h/ ,which jŒa;aCh in its expanded form produces the following implicit matrix equation " ! # m1 X Y .l/ .a/hl hm .m2/Š Y .m1/ .a/hm3 00 CA0 A0 D m2 f aCh; Y .a/   : h lŠ mŠ .m  3/Š lD0

(6) If for the moment we accept that the given equation has a unique solution, the spline (3) is completely determined in Œa; a C h. In the subsequent interval Œa C h; a C 2h we define: SjŒaCh;aC2h .x/ D

1 X i D0

C

S

.i /

jŒa;aCh

m1 X j D2

.a C h/.x  .a C h//i

1 Y .j / .a C h/.x  .a C h//j C A1 .x  .a C h//m ; (7) jŠ mŠ

  where, using (4)–(5) one defines Y 00 .a C h/ D f a C h; SjŒa;aCh .aCh/ ;   Y .3/ .a C h/ D g1 a C h; SjŒa;aCh .aCh/ ; : : : ; Y .m1/ .a C h/ D   gm3 a C h; SjŒa;aCh .a C h/ .

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Contrary to the approach of [2] now we do not obtain these coefficients via the derivatives of the spline in the preceding interval, but make use of (5). This way S.x/ 2 C 2 .Œa; a C h [ Œa C h; a C 2h/ and all the coefficients of (7) are known, except A1 2 Rrq . By construction, (7) satisfies the differential equation (1) at x D a C h. We obtain A1 enforcing that thedifferential equation is also fulfilled .a C 2h/ D f a C 2h; SjŒaCh;aC2h .a C 2h/ . After at x D a C 2h: S 00 jŒaCh;aC2h expanding, we again obtain an implicit matrix equation: 1 2 0 1 m1 m .j / .a C h/hj X X Y h .m  2/Š 4 @ .i / A1 D C A1 A f a C 2h; S .a C h/hi C jŒa;aCh hm2 jŠ mŠ i D0 j D2 3 m3 h 5:  Y 00 .a C h/      Y .m1/ .a C h/ .m  3/Š

(8)

As before, we assume that this equation has a unique solution A1 , which implies that the spline is well defined in Œa C h; a C 2h. Repeating this process, we presuppose that the spline in the interval Œa C .k  1/h; a C kh exists. For the subsequent interval Œa C kh; a C .k C 1/h, one defines the corresponding spline accordingly: Sˇˇˇ

ŒaCkh;aC.kC1/h

.x/ D

1 X i D0

C

.i /

Sˇˇ

ˇŒaC.k1/h;aCkh

m1 X j D2

again

(9)

Y .j / .a C kh/.x  .a C kh//j .x  .a C kh//m C Ak ; jŠ mŠ

Y 00 .a

C kh/ D  Y .m1/ .a C kh/ D gm3 a C kh; Sˇˇˇ where

.a C kh/.x  .a C kh//i

 f a C kh; Sˇˇˇ ŒaC.k1/h;aCkh

 ŒaC.k1/h;aCkh



.aCkh/ ; : : : ;

.aCkh/ . This way we find

S.x/ 2 C 2 .Œa; a C .k C 1/h/ which satisfies (1) at x D a C kh. Let us assume that it is also true for x D a C .k C 1/h: Sˇˇ00 .a C .k C 1/h/ D ˇŒaCkh;aC.kC1/h   f aC.k C1/h; Sˇˇˇ .aC.k C1/h/ , from which we derive the following ŒaCkh;aC.kC1/h

matrix equation for Ak :

2 0 1 m1 X X Y .j / .a C kh/hj .m  2/Š .i / i 4 @ ˇ S .a C kh/h C Ak D f a C kh; ˇ ˇŒaC.k1/h;aCkh hm2 jŠ i D0 j D2 1 3 m3 h hm 5: C Ak A  Y 00 .a C kh/      Y .m1/ .a C kh/ mŠ .m  3/Š

(10)

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615

Note that this equation is similar to equations (6) and (8) with k D 0 and k D 1, respectively. For a fixed value h > 0, we consider the following matrix function of a matrix variable gk W Rrq ! Rrq defined by 20 1 m1 X X Y .j / .a C kh/hj .m  2/Š i 4f @a C kh; ˇ.i / S .a C kh/h C gk .T / D ˇ ˇ hm2 jŠ i D0 ŒaC.k1/h;aCkh j D2 CT

m

1

m3

3

h h A  Y 00 .a C kh/      Y .m1/ .a C kh/ 5: mŠ .m  3/Š

(11)

Obviously, expression (11) is fulfilled if and only if Ak D gk .Ak /, i.e. when Ak is a fixed point of the function gk . Using the Lipschitz condition (2), it is Lh2 straightforward to show that kgk .T1 /  gk .T2 /k  m.m1/ kT1  T2 k. Taking h < p m.m  1/=L, the matrix function gk .T / is therefore contractive and thus the equation (10) has a unique solution for k D 0; 1; : : : ; n  1. Hence, the matrix spline is now completely determined in the interval Œa; b under consideration. Taking into account [4, Theorem 5], the following result was proved: p m.m  1/=L, then the matrix spline S.x/ of order m Theorem 1. If h < exists by successive construction in each subinterval Œa C kh; a C .k C 1/h, with k D 0; 1; : : : ; n  1, as has been outlined before. Moreover, if f 2 C m .T /, then it is kY .x/  S.x/k D O.hm1 /8x 2 Œa; b, where Y .x/ is the solution of problem (1).

10–8 10–10 10–14

error

10–14 10–16 10–18 10–20 10–22 10–24 0.001

h = 0.1 h = 0.01 h = 0.001 0.1

0.2

0.3

0.4

0.5

0.6

approximation interval

0.7

0.8

0.9

1

Fig. 1 Error margins for the test case of Sect. 2.1, with m D 6, h D 0:1, h D 0:01 and h D 0:001

616

Table 2 Results for the proposed method for the test case of Sect. 2.1 with m D 7 and h D 0:1 Interval Approximation   1:  0:5x 2 C 0:0417x 4  0:0014x 6 1:x 2 C 0:1667x 4  0:0083x 6 Œ0; 0:1 2 4 6 0 1:  0:5x C 0:0417x  0:0014x   1:  0:5x 2 C 0:0417x 4  0:0014x 6 1:x 2 C 0:1667x 4  0:0084x 6 Œ0:1; 0:2 2 4 6 0 1:  0:5x C 0:0417x  0:0014x   1:  0:5x 2 C 0:0417x 4  0:0014x 6 1:x 2 C 0:1666x 4  0:0086x 6 Œ0:2; 0:3 0 1:  0:5x 2 C 0:0417x 4  0:0014x 6   2 4 6 1:x 2 C 0:1665x 4  0:0089x 6 1:  0:5x C 0:0417x  0:0015x Œ0:3; 0:4 0 1:  0:5x 2 C 0:0417x 4  0:0015x 6   2 4 6 1:x 2 C 0:1663x 4  0:0092x 6 1:  0:5x C 0:0416x  0:0015x Œ0:4; 0:5 0 1:  0:5x 2 C 0:0416x 4  0:0015x 6   1:  0:5x 2 C 0:0416x 4  0:0016x 6 1:0001x 2 C 0:1658x 4 C 0:0014x 5  0:0097x 6 Œ0:5; 0:6 2 4 6 0 1:  0:5x C 0:0416x  0:0016x   1:  0:5x 2 C 0:0414x 4  0:0016x 6 1:0003x 2 C 0:165x 4 C 0:0022x 5  0:0102x 6 Œ0:6; 0:7 2 4 6 0 1:  0:5x C 0:0414x  0:0016x   0:1  0:5001x 2 C 0:0413x 4  0:0017x 6 1:0006x 2 C 0:0016x 3 C 0:1638x 4 C 0:0033x 5  0:0107x 6 C 0:001x 7 Œ0:7; 0:8 2 4 6 0 0:1  0:5001x C 0:0413x  0:0017x   0:1  0:5002x 2 C 0:0411x 4  0:0018x 6 1:0012x 2 C 0:0029x 3 C 0:1621x 4 C 0:0046x 5  0:0113x 6 C 0:0011x 7 Œ0:8; 0:9 0 0:1  0:5002x 2 C 0:0411x 4  0:0018x 6   0:1  0:5003x 2 C 0:0408x 4  0:0019x 6 1:0021x 2 C 0:0046x 3 C 0:1601x 4 C 0:0060x 5  0:0118x 6 C 0:0012x 7 Œ0:9; 1 0 0:1  0:5003x 2 C 0:0408x 4  0:0019x 6

1:06297  1010

8:9077  1011

7:2331  1011

5:651  1011

4:2023  1011

2:9239  1011

1:8478  1011

1:0004  1011

4:0217  1012

6:7111  1013

Max. error

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617

2.1 An Illustrative Test Example 00 For numerical illustration we  choose  the problem Y .t/ C AY .t/ D 0 for t 2 12 Œ0; 1, with the matrix A D , and initial conditions Y0 D I22 ; Y1 D 022 , 01 where Ipp ; 0pp are the identity and null matrices of order p, respectively. For test purposes, we will compareour approximate results with the exact solution of  cos .t/ t sin .t/ the problem given by Y .t/ D . In this case L  2:41, which 0 cos .t/ implies by Theorem 1, if we take m D 6 that the step size has to be h < 3:52. The results are summarized in Fig. 1, where we also compare the error margins of this test case for variable step sizes h D 0:1; 0:001; 0:0001. The quality of the approximation becomes evident as for h D 0:001 its accuracy already approaches the limits of machine precision. For m D 7 we require that h < 4:17, so we conveniently choose h D 0:1 again. The results are listed in Table 2, where the approximation have been rounded to the fourth relevant digit. In each subinterval, we evaluated the difference between the approximation and the exact solution. The maximum of these errors are indicated in the third column.

Acknowledgements This work has been partially supported by the Universidad Polit´ecnica de Valencia under the grants PAID-06-11-2020, and PAID-05-09-4338, and by the Spanish Ministerio de Educaci´on under the grant MTM2009-08587.

References 1. Defez, E., Soler, L., Herv´as, A., Tung, M.M.: Numerical solutions of matrix differential models using cubic matrix splines II. Math. Comput. Model. 46, 657–669 (2007) 2. Defez, E., Soler, L., Tung, M.M., Herv´as, A.: Cubic-matrix splines and second order matrix models. In: Bonilla, L.L., Moscoso, M., Platero, G., Vega, J.M. (eds.) Progress in Industrial Mathematics at ECMI 2006, Mathematics in Industry, vol. 12, pp. 949–953. Springer, Berlin (2008) 3. Kim, E., Lee, Y.H., Lee, J.Y.: Fragmentation of C60 and C70 clusters. Phys. Rev. B 48(24), 18,230–18,234 (1993) 4. Loscalzo, F.R., Talbot, T.D.: Spline function approximations for solutions of ordinary differential equations. SIAM J. Numer. Anal. 4(3), 433–445 (1967) 5. Zhang, J.F.: Optimal control for mechanical vibration systems based on second-order matrix equations. Mech. Syst. Signal Process. 16(1), 61–67 (2002)

Multiscale Methods in Time and Space Konstantin Fackeldey

Abstract The simulation of events on the atomistic scale is even with the increasing computer power still beyond the means. Thus in the last decades multiscale methods have been developed in order to cope with these problems. Here we present two different multiscale methods. The first method bridges multiscale phenomena in solids (e.g. cracks) from atomistic to continuum by assigning a partition of unity to the atoms. The second method copes with time steps in biomolecular simulations (drug design) by using a conformation dynamics approach.

1 Multiscale Methods for Atomistic Simulations In the last few decades, multiscale methods have been developed with the aim to couple physical phenomena on different length and time scales. Here, we focus on multiscale methods for atomistic simulations, i.e. methods with a fine scale, which provides detailed informations of the atoms and a coarse scale which is less accurate than the fine scale. The necessity of atomistic multiscale methods is given by the fact that atomistic simulations offer a high resolution of the phenomenon but are often computational not feasible. In Fig. 1 biological phenomena with their different time scale are shown. It can be clearly seen, that for a classical molecular dynamics simulation with a time step in femtoseconds a protein folding simulations (seconds or minutes) is hardly practicable. Difference in the length scale (cp. Fig. 2) can have a deep impact e.g. in structural mechanics. The fact that one millimeter of metal contains approximately 1019 atoms confines full atomistic simulations to very small and for engineering purposes improper length scales.

K. Fackeldey () Zuse Institut Bertlin, Takustrasse 7, 14195 Berlin, Germany e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 72, © Springer-Verlag Berlin Heidelberg 2012

619

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Fig. 1 Different biological phenomena and their time scale

Fig. 2 The characteristic length scales of different phenomena

We come to the conclusion that atomistic simulations suffer from the very small time scale and the vast number of atoms and are thus a computational burden. As a matter of fact, the coarse scale contains less information than the fine scale does. However, with a focus to a certain application, often not all atomistic details are needed or—at least—not needed on the whole domain but only on a part of the domain of interest. Here, we give a brief overview of two strategies, to make atomistic simulations more efficient. The goal of this article is to show that this topic is not only confined to a few certain problems, but a rich area of applications spreading from crack problems in solid mechanics to the efficient simualtion of biomolecules for drug design. For the first category of application (in solid mechanics) we review the weak coupling method [5, 7] which offers a strategy to couple finite element methods, representing continuum mechanics with molecular dynamics. The conformation dynamics method [2, 3, 9, 10] is a technique to overcome the time gap in molecular modelling and is devoted to the latter category (biomolecules).

Multiscale Methods in Time and Space

621

2 The Weak Coupling Method A problem which is in great demand in the engineering community is the simulation of crack growth in solids. However, for an accurate description of the crack a simulation down to atomistic size in a zone close to the crack tip is needed, whereas the rest, outside of this zone can be approximated well by a continuum model. This serves as a motivation, to separate the displacement field of the solid into a coarse and a fine scale, with the aim only to use the fine scale near the crack tip zone and the coarse scale for the rest. This total displacement field can be decomposed [8, 12] by q D qN C q 0 :

(1)

Thus the total displacement field q, is decomposed into a coarse part qN which can be represented by finite elements and q 0 as the fine scale part which can not be represented by the finite elements. However any scale decomposition in the fashion of (1) has to deal with the difficulty, that the atomistic displacements are given as point values in R3N , whereas the coarse scale displacements qN are assumed to be elements of a function space. Consequently, for a proper definition of the decomposition (1) a suitable space has to be chosen. In contrast to all existing methods, the weak coupling method applies a function space setting for the decomposition. This function space approach provides several advantages with respect to the development of the scale transfer operator ranging between the MD scale and the CM scale. As afore mentioned the molecular dynamics are given in a Euclidean space as scattered data, i.e. .D/ D f.x˛ ; q˛ /j˛ D 1; : : : ; N; x˛ 2 Rd g, where D  Rd is a domain. In order to perform the decomposition in a function space we employ a mapping { W .D/ ! L2 .D/, which transfers the molecular displacement into a function space. It is given by { W .Rd  Rd /N ! V  L2 .D/; with {.x; q/ D

N X

q˛ '˛ .x/;

˛ D1

the .'˛ .x//˛ D 1;:::;N form a partition of unity [11]. These meshfree basis functions are defined by Shepard’s approach: '˛ .x/ D PNW˛ .x/ , where W˛ W Rd ! RC are ˇ D 1 Wˇ .x/

non negative weight functions attached to each atoms x˛ . For a detailed explanation we refer to [5, 7]. In order to identify the coarse scale displacement, we employ an L2 -Projection  W L2 .D/ ! S h , ranging from the total displacement into the finite element space S h , being the union of all elements having an non-empty cut with supp '˛ . More precisely the coarse scale displacement is given by .w/ 2 S h W ..w/; /L2 .D/ D .w; /L2 .D/

8 2 M h ;

(2)

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here, M h is the multiplier space which is spanned by finite element basis functions span.p / over the domain D. In [7] we pointed out, that there are different choices for the multiplier space, which has consequences for the extension of the displacement outside of the coupling region. In order to compute the algebraic representation of  in (2), we need to assemble two (scaled) mass matrices [6]. Summing up, on the basis of the two operators { and  the scale decomposition in function space is given by {.x; q/ D .{.x; q/  .{.x; q/// C .{.x; q//: This allows, to switch between two scales and only to use expensive atomistic simulation where needed. For more details we refer to [5]

3 Conformation Dynamics In this section, we introduce another multiscale method ,which has celebrated great success in the area of biomlecular simulations [1]. In contrast to crystalline structures, where the bond interaction mainly determines the structure, in biomolecules the torsion angles can be considered as the essential degrees of freedom. More precisely it is the configuration of the torsion angles in a biomolecule that determines its functionality. The arrangement of these torsion angles in a molecule show a metastable behavior. Metastability can be considered as a “weak form” of stability, which is invariant against small perturbations but instable when the perturbation excess a certain amount of perturbation. From the mathematical point of view, a metastability can be described as a “region” i in the phase space, for which Pii ./  Pij ./; where Pii is the probability to stay in “region” i for a time span  and Pij denotes the probability of a molecule changing from “region” i to “region” j . In the forthcoming we give this very raw description a precise meaning, by defining a suitable transfer operator T for calculating P and a reasonable definition of the “regions” in the phase space. Let us consider a canonical ensemble (constant number of particles, constant volume and constant temperature), where, the positions q and the momenta p of all atoms are given according to the Boltzmann distribution: .q; p/ / exp.ˇH.q; p//: Here ˇ D 1=kB T is the inverse temperature T multiplied with the Boltzmann constant kB and H denotes the Hamiltonian function which is given by

Multiscale Methods in Time and Space

623

H.q; p/ D V .q/ C K.p/, where V .q/ is the potential and K.p/ is the kinetic energy. The canonical density can be split into a distribution of momenta .q/ and positions .p/ where .q/ / exp.ˇV .q// and .p/ / exp.ˇK.p//. In order to employ Hamiltonian dynamics we consider the deterministic Hamiltonian, which is given by d q dt

D p;

d p dt

D rV .q/:

For some pk chosen randomly, a Markov process can be described by qkC1 D ˘q ˚  .qk ; pk / where ˚  denotes the flow for some time span  and ˘q is the projection of a state .q; p/ onto its position coordinate q. Thus a transition function between the two densities f and g can be given by Z P ./.f; g/ D

T  .f .q//g.q/dq; ˝

where T  is given by Z T  f .q/ D Rd

f .˘q ˚  .q; p//.p/dp:

Thus, the operator T allows us to describe a dynamical process. In next step we are concerned with a suitable discretization of the phase space. To do so, we consider the projection of the spatial density .q/ onto the essential degrees of freedom. This projection decomposes the space into nearly non-overlapping metastabilities. More precisely, the metastabilities can be defined as almost characteristic functions  W ˝ ! Œ0; 1 on the position space, which are almost invariant under the transfer operator T , i.e. T  i .q/  i .q/;

i D 1; : : : ; nc ;

(3)

With this definition, the position space can be decomposed into nc metastable conformations 1 ; : : : ; nc , which form a partition of Unity, i.e. nc X

i .q/ D 1

8q 2 ˝; i D 1; : : : ; nc :

i D1

In order to identify these metastabilities, the phase space ˝ has to be discretized. However due to the high dimensional structure of the phase space, standard mesh based discretization methods fail. For suitable chosen .q` /N ` D 1 in the phase space,

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is partitioned by a setP of functions '` W ˝ ! R  Œ0; 1; 8` D 1; : : : ; N' ; nc  N'  N such that ` '` D 1. There are several choices for the construction of the '` ’s: as characteristic functions [9, 10] or as meshfree basis functions [13]. The metastabilites can then be written as a linear combination of the basis functions by j .q/ D

N X

dj .'` /'` .q/;

j D 1; : : : ; nc :

`D1

Here, dj .'` / is a membership vector with entries ranging between Œ0; 1 and reflects the degree of membership of '` to conformation j [4]. A Galerkin discretization leads to the following eigenvalue problem P  j  Sj ;

j D 1; : : : ; nc

where P  2 Rnc nc is a stochastic transition probability matrix, given by Z Pij

D

Z T 'j .q/i .q/dq and Sij D 

˝

'j .q/i .q/dq; ˝

where 'i .q/.q/ : ˝ 'i .q/.q/dq

.q/ D R

The metastabilities can be identified by applying a robust Perron Cluster Analysis [4] on the stochastic transition matrix P  . Summing up, the conformation based approach coarsens the deterministic time step of a molecular simulation, by generating efficiently transition probabilities between metastabilities.

4 Applications The application for this method can be found e.g. in crack propagation problems, by employing the following strategy: Since close to the crack tip linear elastic models fail it is favorable to simulate a region lose to the tip by molecular dynamics, whereas the rest of the domain (outside of the crack zone) can be simulated by finite elements. The weak coupling method thus provides a matching method for the MD crack zone simulation and the outer linear elastic approximation. The conformation dynamics with the meshfree methods approach are used to simulate the dynamics of biomolecules like ˇ-secretase which plays an important role in Alzheimer’s disease. Thus conformation dynamics helps to understand the behavior of biomolecules and thus opens the door for new strategies in drug design.

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References 1. Deuflhard, P.: From molecular dynamics to conformation dynamics in drug design. In: Kirkilionis, M., Kr¨omker, S., Rannacher, R., Toni, F. (eds.) Trends in Nonlinear Analysis, pp. 269–288. Springer, New York (2003) 2. Deuflhard, P., Dellnitz, M., Junge, O., Sch¨utte, C.: Computation of essential molecular dynamics by subdivision techniques. In: Deuflhard, P., Hermans, J., Leimkuhler, B., Mark, A., Reich, S., Skeel, R. (eds.) Computational Molecular Dynamics: Challenges, Methods, Ideas., LNCSE, vol. 4, pp. 98–115. Springer, New York (1998) 3. Deuflhard, P., Sch¨utte, C.: Molecular conformation dynamics and computational drug design. In: Hill, J., Moore, R. (eds.) Applied Mathematics Entering the 21st Century, Invited Talks from the ICIAM 2003 Congress, Sydney, Australia, pp. 91–119 (2004) 4. Deuflhard, P., Weber, M.: Robust Perron Cluster Analysis in Conformation Dynamics . Lin. Alg. App. 398c, 161–184 (2005) 5. Fackeldey, K.: The weak coupling method for coupling continuum mechanics with molecular dynamics. Ph.D. thesis, Universit¨at Bonn (2009) 6. Fackeldey, K., Krause, D., Krause, R.: Concepts and implementation of the weak coupling method. In: Proceedings of 4th International Conference on Multiscale Materials Modeling 2008, pp. 62–65 (2008) 7. Fackeldey, K., Krause, R.: Multiscale coupling in function space - weak coupling between moelcular dynamics and continuum mechanics. Int. J. Num. Meth. Engrg, 79(12), 1517–1535 (2008) 8. Hughes, T., Feijoo, G., Mazzei, J., Quincy, J.: The variational mutiscale method - a paradigm for computational mechanics. Comput. Meth. Appl. Mech. 166, 3–24 (1998) 9. Sch¨utte, C.: Conformational dynamics: Modelling, theory, algorithm, and application to biomolecules. habilitation thesis (1999) 10. Sch¨utte, C., Fischer, A., Huisinga, W., Deuflhard, P.: A direct approach to conformational dynamics based on hybrid Monte Carlo. J. Comput. Phys. 151, 146–169 (1999) 11. Schweitzer, M.: A Parallel Multilevel Partition of Unity Method for Elliptic Partial Differential Equations. No. 29 in LNCSE. Springer, New York (2003) 12. Wagner, G., Liu, W.: Coupling of atomistic and continuum simulations using a bridging scale decomposition. J. Comp. Phy. 190, 1261–1289 (2003) 13. Weber, M.: Meshless methods in conformation dynamics. Ph.D. thesis, Freie Universit¨at Berlin (2006)

A Cache-Oblivious Sparse Matrix–Vector Multiplication Scheme Based on the Hilbert Curve Albert-Jan N. Yzelman and Rob H. Bisseling

Abstract The sparse matrix–vector (SpMV) multiplication is an important kernel in many applications. When the sparse matrix used is unstructured, however, standard SpMV multiplication implementations typically are inefficient in terms of cache usage, sometimes working at only a fraction of peak performance. Cache-aware algorithms take information on specifics of the cache architecture as a parameter to derive an efficient SpMV multiply. In contrast, cache-oblivious algorithms strive to obtain efficiency regardless of cache specifics. In earlier work in this latter area, Haase et al. (2007) use the Hilbert curve to order nonzeroes in the sparse matrix. They obtain speedup mainly when multiplying against multiple (up to eight) right-hand sides simultaneously. We improve on this by introducing a new datastructure, called Bi-directional Incremental Compressed Row Storage (BICRS). Using this datastructure to store the nonzeroes in Hilbert order, speedups of up to a factor two are attained for the SpMV multiplication y D Ax on sufficiently large, unstructured matrices.

1 Introduction Given an mn sparse matrix A and a dense vector x, we consider the sparse matrix– vector (SpMV) multiply y D Ax, with y a dense result vector. A standard way of storing a sparse matrix A is the Compressed Row Storage (CRS) format [1], which stores data in a row-by-row fashion using three arrays: j , v, and r. The first two arrays are of size nz.A/, with nz.A/ the number of nonzeroes in A, whereas r is of length mC1. The array j stores the column index of each nonzero in A, and v stores the corresponding numerical values. The ranges Œri ; ri C1 / in those arrays correspond

A.-J.N. Yzelman ()  R.H. Bisseling Utrecht University, 3508 TA Utrecht, The Netherlands e-mail: [email protected]; [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 73, © Springer-Verlag Berlin Heidelberg 2012

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Algorithm 3 SpMV multiplication algorithm calculating y D Ax using CRS for i D 0 to m  1 do for k D rŒi  to rŒi C 1  1 do yŒi  D yŒi  C vŒk  xŒj Œk end for end for

to the nonzeroes in the i th row of A. A standard SpMV multiply algorithm using CRS is given in Algorithm 3. It writes to y sequentially, and thus performs optimally (regarding y) in terms of cache efficiency. Accesses to x, however, are unpredictable in case of unstructured A, causing cache misses to occur on its elements. This is the main reason for inefficiencies during the SpMV multiply [3, 5, 6, 12]. A way to increase performance is to force the SpMV multiply to work only on smaller and uninterrupted subranges of x, such that the vector components involved fit into cache. This can be done by permuting rows and columns from the input matrix so that the resulting structure forces this behaviour when using standard CRS. Results on this method have been reported in [16], using a onedimensional (1D) method, and in [17], where the method has been extended to two dimensions (2D). It must be noted that the 2D method theoretically requires a different datastructure than CRS, but results show that CRS can still outperform more complex datastructures when an appropriate permutation strategy is used. Gains can be as large as 50% for the 1D method and 63% for the 2D method. What we consider in this paper is a change of datastructure instead of a change in the input matrix structure. This means finding a datastructure which accesses nonzeroes in A in a more “local” manner; that is, an order such that jumps in the input and output vector remain small and thus yield fewer cache misses. Earlier work in this direction includes the Blocked CRS format [11], the auto-tuning sparse BLAS library OSKI [13], exploiting variable sized blocking [10, 14], and several other approaches [2, 12]. In the dense case, relevant are the work by Goto et al. [4], who hand-tuned dense kernels to various different architectures, and the ATLAS project [15], which strives to do the same using auto-tuning. Of specific interest is the use of space-filling curves to improve cache locality in the dense case, in particular the use of the Morton (Z-curve) ordering [9], more recently combined with regular row-major formats to form hybrid-Morton formats [8]. In the sparse case, the work by Haase et al. in [5], which already contains the foundation of the main idea presented here, is of specific interest. They propose to store the matrix in an order defined by the Hilbert curve, making use of the good locality-preserving attributes of this space-filling curve. Figure 1 shows an example of a Hilbert curve within 2  2 and 4  4 matrices. This locality means that, from the cache perspective, accesses to the input and output vector remain close to each other when following the Hilbert curve. The curve is defined recursively as can be seen in the figure: any one of the four “super”-squares in the two-by-two matrix can readily be subdivided into four subsquares, onto which a rotated version of the original curve is projected such that the starting point is on a subsquare adjacent to

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Fig. 1 The Hilbert curve drawn within two-by-two and four-by-four matrices

where the original curve entered the super-square, and similarly for the end point. A Hilbert curve thus can be projected on any 2dlog2 me  2dlog2 ne matrix, which in turn can embed the sparse matrix A, imposing a 1D ordering on its nonzeroes. Haase et al. [5] stored these nonzeroes in triplet format: three arrays i; j; v of length nz.A/ are defined, such that the kth nonzero of A with value vŒk is stored at the location .i Œk; j Œk/, as determined by the Hilbert ordering. The main drawback is the difference in storage space required; this is 3nz.A/, an increase of nz.A/  m compared to the standard CRS datastructure. The number of cache misses prevented thus must overtake this amount of extra data movement before any gain in efficiency becomes noticeable. A new datastructure is proposed in Sect. 2 to alleviate this problem, and results of experiments using the Hilbert curve and this new data format are presented in Sect. 3. These are followed by the conclusions in Sect. 4.

2 Bi-directional Incremental CRS If using the Hilbert curve to store the nonzeroes of a sparse matrix can be said to be the first of two main ideas around this cache-oblivious method, the second enabling idea is the Bi-directional Incremental CRS datastructure (BICRS). It is capable of efficiently storing the nonzeroes of A in the Hilbert order. We will introduce BICRS by deriving it from the Incremental CRS datastructure (ICRS), which can be viewed as an alternative implementation of the standard CRS datastructure, as presented by Koster [7]. Instead of storing the j array, an incremental version j is stored instead; that is, j Œ0 D j Œ0 and j Œq D j Œq  j Œq  1 for 1  q < n. This means that an SpMV multiplication kernel, upon processing the kth nonzero, simply increases its current column index with j Œk to find the column index of the next nonzero to be processed. A row change can be signalled by overflowing the column index such that subtracting n from the overflowed index yields the starting column index on the next row. The row array r can then be exchanged for an incremental row

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array r as well, so that rŒk yields the distance between the kth nonempty row and the next nonempty row. rŒ0 specifies which row contains the first nonzero. Note that when there are no empty rows, r contains only 1-values except at rŒ0, which equals 0. This means the array does not have to be stored, bringing the total storage requirement down to 2nz.A/. When the row increment array is stored, the storage requirement is equal to that of CRS with 2nz.A/ C m, worst case; in the case where A has empty rows, the required storage is less. The main gain is that the SpMV multiply can be efficiently written using pointer arithmetic, which yields a decrease in machine code instructions [7]. As described, ICRS is not capable of storing nonzeroes in any ordering other than the CRS ordering. A simple extension, however, is to allow negative increments, thus facilitating jumping through nonzeroes of the sparse matrix in any bi-directional, possibly non-CRS, order. Overflows in the column-direction still trigger row changes, as with ICRS. We refer to this generalised datastructure as Bi-directional ICRS. An immediate disadvantage is that the row increments array now can become larger than the number of nonempty rows if nonzeroes are not traversed in a row-by-row manner. This hampers efficiency since the number of memory accesses required to traverse A increases to 2nz.A/Cmjumps, where mjumps is the number of row jumps stored in r, with m  mjumps  nz.A/. It is, however, a definite improvement over the triplet structure used in [5]. In case of a dense matrix, the number of row jumps made when nonzeroes are ordered according to the Hilbert curve is about nz.A/=2, but this gives no guarantee for the number of jumps in the sparse case; this is entirely dependent on the nonzero structure. Note that while this datastructure is bi-directional, the datastructure orientation still matters.

3 Experiments Experiments have been performed on two quad-core architectures, using only one of the four cores available for the sequential SpMV multiplications. The first is an Intel Core 2 Q6600 with a 32 KB L1 data cache, and a 4 MB semi-shared L2 cache. No L3 cache is available. The second architecture is an AMD Phenom II 945e on which each core has a private 64 KB L1 and 512 kB L2 data cache, while all four cores together share a 6 MB L3 cache. The SpMV kernels,1 based on CRS, ICRS and BICRS using Hilbert ordering, are each executed 100 times on given matrices, and report an average running time of a single SpMV multiplication. Experiments have been performed on 9 sparse matrices, all taken to be large in the sense that the input and output vector do not fit into the L2 cache; see Table 1(top). All matrices are available through the University of Florida sparse matrix collection. Tests on smaller matrices were performed as well, but, in contrast to when using the reordering methods, any decrease in L1-cache misses did not

1

The source code is freely available at http://albert-jan.yzelman.net/software

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Table 1 Matrices used in our experiments (top) and SpMV timings (bottom). An S (U) indicates that a matrix is considered structured (unstructured). Experiments were done on the Intel Q6600 (bottom-left) and the AMD 945e (bottom-right). Timings are in milliseconds Name Rows Columns Nonzeroes Symmetry, origin Stanford cont1 l Stanford-berkeley Freescale1 Wikipedia-20051105 cage14 GL7d18 Wikipedia-20060925 Wikipedia-20070206

CRS 30:22 44:02 35:29 122:27 366:45 136:19 774:55 812:42 1012:73

ICRS 40:24 46:41 34:56 131:52 374:82 141:07 856:16 831:17 994:35

281903 1918399 683446 3428755 1634989 1505785 1955309 2983494 3566907

Hilbert BICRS 25:74 62:85 45:82 210:10 253:45 165:21 372:25 576:67 776:48

Extra build 1456 5085 5578 14458 12632 20453 22126 23839 27345

281903 1921596 683446 3428755 1634989 1505785 1548650 2983494 3566907

2312497 7031999 7583376 17052626 19753078 27130349 35590540 37269096 45030389

Matrix Stanford cont1 l Stanford-berkeley Freescale1 Wikipedia-20051105 cage14 GL7d18 Wikipedia-20060925 Wikipedia-20070206

U S U S U S U U U

U S U S U S U U U

Link matrix Linear programming Link matrix Circuit design Link matrix Struct. symm., DNA Combinatorial problem Link matrix Link matrix

CRS ICRS 22:15 27:52 31:07 26:99 26:05 24:52 98:55 95:00 368:36 387:39 116:44 110:69 716:32 824:32 823:53 879:53 1033:95 1124:02

Hilbert BICRS 18:48 48:05 34:29 148:04 250:30 140:20 452:89 550:00 591:08

Extra build 832 3084 3415 8913 5850 12095 10064 11814 14753

result in a faster SpMV execution. Results on larger matrices in terms of wall-clock time are reported in Table 1 for the Q6600 system (bottom-left), as well as for the AMD 945e (bottom-right). Also reported is the extra build time, that is, the time required to build the Hilbert BICRS structure minus the time required to build a CRS datastructure.

4 Conclusions The cache-oblivious SpMV multiplication scheme works very well on large unstructured matrices. In the best case (the GL7d18 matrix on the Q6600), it gains 51% in execution speed. On both architectures, 5 out of the 6 unstructured matrices show significant gains, typically around 30%–40%. The only exception is the stanford-berkeley matrix, taking a performance hit of 32%, on both architectures. Interestingly, the 1D and 2D reordering methods also do not perform well on this matrix [16, 17]. The method also shows excellent performance regarding preprocessing times, taking a maximum of 28 s for wikipedia-2007 on the Q6600 system. This is in contrast to 1D and 2D reordering methods, where pre-processing times can take hours for larger matrices, e.g., 21 h for wikipedia-2006 [16, 17].

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Gains in efficiency when reordering, however, are more pronounced than for the Hilbert-curve scheme presented here. Note that the methods do not exclude each other: 1D or 2D reordering techniques can be applied before loading the matrix into BICRS using the Hilbert ordering to gain additional efficiency. The results also show that, as expected, the method cannot outperform standard CRS ordering when the matrix already is favourably structured, resulting in slowdowns. For future improvement of the Hilbert-curve method, we suggest applying the Hilbert ordering to small (e.g., 8 by 8) sparse submatrices of A instead of its nonzeroes, and imposing a regular CRS ordering on the nonzeroes contained within each such submatrix. Such a hybrid scheme has also been suggested for dense matrices [8], although the motivation differs; in our case, since BICRS can still be used, the number of row jumps required is reduced in case the rows of the submatrices contain several nonzeroes, thus increasing performance further.

References 1. Bai, Z., Demmel, J., Dongarra, J., Ruhe, A., van der Vorst, H. (eds.): Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. SIAM, Philadelphia, PA (2000) 2. Bender, M.A., Brodal, G.S., Fagerberg, R., Jacob, R., Vicari, E.: Optimal sparse matrix dense vector multiplication in the I/O-model. In: Proceedings 19th Annual ACM Symposium on Parallel Algorithms and Architectures, pp. 61–70. ACM Press, New York (2007) 3. Dennis, J.M., Jessup, E.R.: Applying automated memory analysis to improve iterative algorithms. SIAM J. Sci. Comput. 29(5), 2210–2223 (2007) 4. Goto, K., van de Geijn, R.: On reducing TLB misses in matrix multiplication. Technical Report TR-2002-55, University of Texas at Austin, Department of Computer Sciences (2002) 5. Haase, G., Liebmann, M., Plank, G.: A Hilbert-order multiplication scheme for unstructured sparse matrices. Int. J. Parallel, Emergent Distr. Syst. 22(4), 213–220 (2007) 6. Im, E.J., Yelick, K.A.: Optimizing sparse matrix computations for register reuse in SPARSITY. In: Proceedings International Conference on Computational Science, Part I, Lecture Notes in Computer Science, vol. 2073, pp. 127–136, Springer, Berlin (2001) 7. Koster, J.: Parallel templates for numerical linear algebra, a high-performance computation library. Master’s thesis, Utrecht University, Department of Mathematics (2002) 8. Lorton, K.P., Wise, D.S.: Analyzing block locality in Morton-order and Morton-hybrid matrices. SIGARCH Comput. Archit. News 35(4), 6–12 (2007) 9. Morton, G.M.: A computer oriented geodetic data base and a new technique in file sequencing. Technical report, IBM, Ottawa, Canada (1966) 10. Nishtala, R., Vuduc, R.W., Demmel, J.W., Yelick, K.A.: When cache blocking of sparse matrix vector multiply works and why. Appl. Algebra Engrg. Comm. Comput. 18(3), 297–311 (2007) 11. Pinar, A., Heath, M.T.: Improving performance of sparse matrix-vector multiplication. In: Proceedings Supercomputing 1999, p. 30. ACM Press, New York (1999) 12. Toledo, S.: Improving the memory-system performance of sparse-matrix vector multiplication. IBM J. Res. Dev. 41(6), 711–725 (1997) 13. Vuduc, R., Demmel, J.W., Yelick, K.A.: OSKI: A library of automatically tuned sparse matrix kernels. J. Phys. Conf. Series 16, 521–530 (2005) 14. Vuduc, R.W., Moon, H.J.: Fast sparse matrix-vector multiplication by exploiting variable block structure. In: High Performance Computing and Communications 2005, Lecture Notes in Computer Science, vol. 3726, pp. 807–816, Springer, Berlin (2005)

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15. Whaley, R.C., Petitet, A., Dongarra, J.J.: Automated empirical optimizations of software and the ATLAS project. Parallel Comput. 27(1–2), 3–35 (2001) 16. Yzelman, A.N., Bisseling, R.H.: Cache-oblivious sparse matrix–vector multiplication by using sparse matrix partitioning methods. SIAM J. Sci. Comput. 31(4), 3128–3154 (2009) 17. Yzelman, A.N., Bisseling, R.H.: Two-dimensional cache-oblivious sparse matrix–vector multiplication, Parallel Comput. 37(12), 806–819 (2011)

Absorbing Boundary Conditions for Solving Stationary Schr¨odinger Equations Pauline Klein, Xavier Antoine, Christophe Besse, and Matthias Ehrhardt

Abstract Using pseudodifferential calculus and factorization theorems we construct a hierarchy of novel absorbing boundary conditions (ABCs) for the stationary Schr¨odinger equation with general (linear and nonlinear) exterior potential V .x/. Doing so, we generalize the well-known quantum transmitting boundary condition of Lent and Kirkner to the case of space-dependent potential. Here, we present a brief introduction into our new approach based on finite elements suitable for computing scattering solutions and bound states.

1 Introduction The solution of the Schr¨odinger equation occurs in many applications in physics, chemistry and engineering (e.g. quantum transport, condensed matter physics, quantum chemistry, optics, underwater acoustics, . . . ). The considered problem can appear in different forms: time-dependent or stationary equation, linear or nonlinear equation, inclusion of a variable potential among others.

P. Klein ()  X. Antoine Institut Elie Cartan Nancy, Nancy-Universit´e, CNRS UMR 7502, INRIA CORIDA Team, Boulevard des Aiguillettes B.P. 239, 54506 Vandoeuvre-l`es-Nancy, France e-mail: [email protected]; [email protected] C. Besse Equipe Projet Simpaf – Inria CR Lille Nord Europe, Laboratoire Paul Painlev´e, Unit´e Mixte de Recherche CNRS (UMR 8524), UFR de Math´ematiques Pures et Appliqu´ees, Universit´e des Sciences et Technologies de Lille, Cit´e Scientifique, 59655 Villeneuve d’Ascq Cedex, France e-mail: [email protected] M. Ehrhardt Lehrstuhl f¨ur Angewandte Mathematik und Numerische Analysis, Fachbereich C Mathematik und Naturwissenchaften, Bergische Universit¨at Wuppertal, Gaußstr. 20, 42119 Wuppertal, Germany e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 74, © Springer-Verlag Berlin Heidelberg 2012

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One of the main difficulties when solving the Schr¨odinger equation, and most particularly from a numerical point of view, is to impose suitable and physically admissible boundary conditions to solve numerically a bounded domain equation modelling an equation originally posed on an unbounded domain. Concerning the time-domain problem, many efforts have been achieved these last years. We refer the interested reader e.g. to the recent review paper [2] and the references therein. Here we focus on the solution to the stationary Schr¨odinger equation. For a given potential V , eventually nonlinear (V WD V .x; '/), we want to solve the equation   d2 ˛ 2 C V ' D E'; x 2 R; (1) dx with a parameter ˛ that allows for flexibility. More precisely, we study the extension of the recently derived time-domain boundary conditions [3] to the two situations: • Linear and nonlinear scattering: E is a given value and the potential V being linear (independent of ') or nonlinear, we want to compute ' as solution of (1). • Stationary states: we determine here the pair .'; E/, for a given linear or nonlinear potential V . The energy of the system is then the eigenvalue E and the associated stationary state is the eigenfunction '. In particular, we seek the fundamental stationary state which is linked to the smallest eigenvalue. For the stationary Schr¨odinger equation (1), boundary conditions for solving linear scattering problems with a constant potential outside a finite domain have been proposed e.g. by Ben Abdallah, Degond and Markowich [6], by Arnold [5] for a fully discrete Schr¨odinger equation and in a two-dimensional quantum waveguide by Lent and Kirkner [8]. The case of bound states can be found for the onedimensional linear Schr¨odinger equation with constant potential in [9]. Finally, let us point out that these absorbing boundary conditions can be extended to higher dimensional problems [7] and other situations like variable mass problems.

2 Absorbing Boundary Conditions: From the Time-Domain to the Stationary Case In order to derive some absorbing boundary conditions (ABCs) for the stationary Schr¨odinger equation (1), let us first start with the time-domain situation. In case of e the time-dependent Schr¨odinger equation with a linear or nonlinear potential V ( eu D 0; 8.x; t/ 2 R  RC ; i@t u C @2x u C V (2) u.x; 0/ D u0 .x/; x 2 R; the following second- and fourth-order ABCs p  e u D 0; ABC22 @n u  iOp  C V

(3)

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ABC42

p  e @n V e u C 1 Op @n u  iOp  C V e 4  C V

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! u D 0;

(4)

on ˙  RC were derived recently in [3]. Here, Op denotes a pseudodifferential operator,  denotes the dual time variable and the fictitious boundary ˙ is located at the two interval endpoints x` and xr . The outwardly directed unit normal vector to the bounded computational domain ˝ Dx` I xr Œ is denoted by n. To obtain some ABCs for (1), we consider it supplied with a new potential: e WD V =˛. Moreover, we are seeking some time-harmonic solutions u.x; t/ WD V E E '.x/ e i ˛ t and since i@t u D E=˛'.x/ e i ˛ t , the variable  can be identified with E=˛ since formally  corresponds to i@t . This yields some stationary ABCs on ˙ that we designate by SABCM (“S” stands for stationary and M denotes the order): SABC2 SABC4

1 p @n ' D i p E  V '; on ˙; ˛ 1 p 1 @n V ': @n ' D i p EV 'C 4E V ˛

(5) (6)

Let us remark that we constructed for the time-dependent case two families M of ABCs, denoted by ABCM 1 and ABC2 [3]. These ABCs all coincide if the potential is time-independent. In the stationary case, all the potentials fall into this category and thus the ABCs are equivalent. Hence, we get the unique class of stationary ABCs, SABCM (without subscript index). For convenience, the form of the boundary conditions (5)–(6) is based on ABCM 2 (we refer to [3] for more technical details).

3 Application to Linear Scattering Problems Let us consider an incident right-traveling plane wave ' inc .x/ D e ikx ;

k > 9;

x 2  1I x` ;

(7)

coming from 1. The parameter k is the real valued positive wave number and the variable potential V models an inhomogeneous medium. We consider a bounded computational domain ˝ Dx` I xr Œ and assume that the wave '  ' inc is perfectly reflected back at the left endpoint x` . Furthermore, we assume that the wave is totally transmitted in Œxr I 1Œ, propagating then towards C1. As a consequence, we have to solve the following boundary value problem   d2 ˛ 2 C V ' D E'; for x 2 ˝; dx (8) @n ' D gM;` ' C fM;`

at x D x` ;

@n ' D gM;r '

at x D xr ;

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with fM;` D @n ' inc .x` /gM;` ' inc .x` /. Here, the order M is equal to 2 or 4 according to the choice of SABCM (5) or (6) and thus we have 1 p E  V`;r ; g2;.`;r/ WD i p ˛ g4;.`;r/ WD g2;.`;r/ C

(9)

1 @n VjxDx`;r : 4 E  VjxDx`;r

(10)

In the sequel of this paper, we will also use the following other concise writing @n ' D gM ' C fM ;

on ˙;

(11)

for each function being adapted with respect to the endpoint. Finally, for a plane wave, we have the dispersion relation: E D ˛k 2 C V` , where V` D V .x` /. We use a finite element method (FEM) to solve numerically this problem. One benefit of using FEM in this application is that the ABCs can be incorporated directly into the variational formulation. The interval Œx` I xr  is decomposed into nh elementary uniform segments of size h. Classically, the ABCs are considered as (impedance) Fourier-Robin boundary conditions. Let ' 2 Cnh C1 denote the vector of nodal values of the P1 interpolation of ' and R let S 2 Mnh C1 .R/ the P1 stiffness matrix associated with the bilinear form ˝ @x ' @x ' dx. Next we introduce MV E 2 Mnh C1 R .R/ as the generalized mass matrix arising from the linear approximation of ˝ .V  E/' dx, for any test-function  2 H 1 .˝/. Let BM 2 Mnh C1 .C/ be the matrix of the boundary terms related to the ABC SABCM .  > The right-hand side bM 2 Cnh C1 is given by b D ˛fM;` ; 0; : : : ; 0 and the linear system reads .˛S C MV E C BM /' D bM :

(12)

Example 1. We study the stationary Schr¨odinger equation (1) with ˛ D 1=2: 

1 d2 ' C V ' D E'; 2 dx 2

x 2 R;

(13)

and consider an incident right-traveling plane wave with wave number k D 10. We analyze the results for a Gaussian potential V .x/ D A expf.x xc /2 =w2 g, centered at xc D 20 with the amplitude A D  5 and the parameter w D 3. The numerical reference solution is computed on the large domain 0I 58Πusing the fourth-order ABC. At the fictitious boundary points x` and xr of the computational domain, the values of the potential are V .58/  1069 and V .0/  1019 , i.e. from a numerical point of view, the potential can be considered as compactly supported in this reference domain. Then, the ABCs are highly accurate [2] yielding a suitable reference solution 'ref with spatial step size h D 5  103 .

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We next compute the solution obtained by applying the ABCs on a smaller computational domain by shifting the right endpoint to xr D 18, now the potential being far from vanishing at this endpoint. In the negative half-space x < x` D 0, the potential is almost equal to zero and hence the second-order ABC is very accurate. Figure 1 shows the computed solutions (denoted by 'num ), superposed on the potential and reference solution, with the second-order (green) and fourth-order (cyan) ABCs placed at the right endpoint xr . The ABCs give quite good results as it can be clearly observed in Fig. 1. Next we plot in Fig. 2 the error curves on the real part x 7! jRe.'.x//j with the error ' D 'num  'ref . We can see that the approximation error by using the SABC2 is roughly 5  104 while the error associated with ABC4 is almost 106 , which is also the linear finite element approximation error h2  106 . Hence, not only the results are precise but they are also of increasing accuracy as the order of the SABC increases.

Conclusion We have proposed some accurate and physically admissible absorbing boundary conditions for modeling linear (and nonlinear) stationary Schr¨odinger equations with variable potentials. Based on numerical schemes, these boundary conditions have been validated for linear scattering computations.

Fig. 1 Real parts of the numerical solutions (zoomed around the boundary xr D 18)

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Fig. 2 Real part of errors j'num  'ref j for the potential V .x/ D 5e .x20/

2 =9

A more detailed discussion and examples including the consideration of linear and nonlinear eigenstate computation with applications to many possible given variable potentials and nonlinearities can be found in [4, 7]. Acknowledgements This work was supported partially by the French ANR fundings under the project MicroWave NT09 460489 (http://microwave.math.cnrs.fr/).

References 1. Antoine, X., Besse, C., Mouysset, V.: Numerical schemes for the simulation of the twodimensional Schr¨odinger equation using non-reflecting boundary conditions. Math. Comp. 73, 1779–1799 (2004) 2. Antoine, X., Arnold, A., Besse, C., Ehrhardt, M., A. Sch¨adle, A.: A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schr¨odinger equations. Commun. Comput. Phys. 4, 729–796 (2008) 3. Antoine, X., Besse, C., Klein, P.: Absorbing boundary conditions for the one-dimensional Schr¨odinger equation with an exterior repulsive potential. J. Comp. Phys. 228, 312–335 (2009) 4. Antoine, X., Besse, C., Ehrhardt, M., Klein, P.: Modeling boundary conditions for solving stationary Schr¨odinger equations. Preprint 10/04, University of Wuppertal (2010) 5. Arnold, A.: Mathematical concepts of open quantum boundary conditions. Trans. Theory Stat. Phys. 30, 561–584 (2001) 6. Ben Abdallah, N., Degond, P., Markowich, P.A.: On a one-dimensional Schr¨odinger-Poisson scattering model. Z. Angew. Math. Phys. 48, 135–155 (1997)

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7. Klein, P., Antoine, X., Besse, C., Ehrhardt, M.: Absorbing boundary conditions for solving N -dimensional stationary Schr¨odinger equations with unbounded potentials and nonlinearities. Commun. Comput. Phys. 10, 1280–1304 (2011) 8. Lent, C., Kirkner, D.: The quantum transmitting boundary method. J. Appl. Phys. 67, 6353–6359 (1990) 9. Moyer, C.: Numerical solution of the stationary state Schr¨odinger equation using transparent boundary conditions. Comput. Sci. Engrg. 8, 32–40 (2006)

Diffusion on Surfaces of Revolution Michael M. Tung

Abstract Diffusion processes play a fundamental role in mathematical models and simulation for particle and fluid dynamics, or heat transport. Many industrial devices have simple design, in particular the shape of surfaces of revolution. For sourceless diffusion on surfaces of revolution, we introduce a Hamiltonian system on a manifold and derive the governing diffusion equation from a variational principle. For axisymmetric shapes these equations of motion reduce to self-adjoint eigenvalue problems. This property is important as it guarantees completeness of the corresponding fundamental solutions. Also, we prove that the dependence on the height is given by a Sturm–Liouville equation. As an example, we tackle diffusion on a circular paraboloid as found in mirror telescopes or antennas among other applications. These analytical results may provide a valuable theoretical tool for related diffusion models.

1 Introduction and Overview Diffusion processes govern many fundamental processes in nature and as such play an important role in the mathematical modelling and simulation of most processes involving particle and fluid dynamics, or heat transfer. In industry, many devices have rather simple design shapes, especially possessing axisymmetric features. This work discusses sourceless diffusion phenomena on surfaces of revolution, taking a general differential-geometric approach in combination with a variational principle argument. We will first set up a Hamiltonian system on a manifold with all necessary ingredients to describe diffusion [5]. This leads to a Lagrangian formulation of M.M. Tung () Instituto Universitario de Matem´atica Multidisciplinar, Universidad Polit´ecnica de Valencia, Camino de Vera, s/n, E-46022 Valencia, Spain e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 75, © Springer-Verlag Berlin Heidelberg 2012

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diffusion phenomena, which then allows for a direct derivation of the governing equation of motion via a variational principle. This has the advantage that the resulting general diffusion equation, a non-homogeneous parabolic differential equation, is expressed in manifestly covariant form. In the next step, axisymmetry is imposed on the metric of the underlying space, and it is shown that this constraint will imply certain well-behaved properties for the corresponding system. Most importantly, the Laplace–Beltrami operator, which emerges naturally in the covariant diffusion equation, is shown to be self-adjoint. As a consequence, the differential equation for the dependence along the symmetry axis of the diffusion problem takes the form of a classical Sturm–Liouville equation. In the past, analytical solutions for diffusion on a variety of axisymmetric shapes have been found, e.g. involving Bessel functions Jn .z/ and Yn .z/ for the cylinder [1], Legendre functions Pnm .z/ for the sphere [7], and Bessel functions Jˇ .˛z/ and Jˇ .˛z/ for the cone [6], where as usual z 2 R denotes the position on the axis of symmetry and the remaining parameters are integer or real constants. Parabolic surface shapes are found in a wide range of industrial devices, such as liquid mirror telescopes or antennas, among many other applications. Surprisingly, analytical solutions, to the best of our knowledge, have not been studied before. Therefore, as an application of our approach, we will investigate the fundamental solutions for diffusion on surfaces with an underlying axisymmetry corresponding to a circular paraboloid. In the course of the derivation, we will prove that for diffusion on the paraboloid the differential equation describing the dependence along the axis has as solution the confluent Heun’s functions.

2 Hamiltonian System and Lagrangian Framework In order to describe diffusion processes, we introduce concentration C.x; t/ on a smooth two-dimensional manifold M with metric g, where t  0 denotes the time and x i 2 R, i D 1; 2, are the two parameters to specify the location on the surface S  M . The driving force of diffusion is a non-vanishing gradient of the concentration, expressed as covariant derivatives CIi . For a schematic representation see Fig. 1. Any particular configuration of the system is then identified by a mapping B ! N , where B D M  RC is the base space and N D B  P is the configuration space [3]. Here, the ambient space P is defined by the two concentrations C; C  W B ! RC [6]. We seek a general principle to describe the evolution of C.x; t/ for diffusion on a surface. Thus, the goal is to find a Lagrangian function L which produces the equations of motions as stationary solutions of an action integral Z L

ı S

p g d D 0:

(1)

Diffusion on Surfaces of Revolution Fig. 1 Schematic view of the basic elements to describe diffusion: surface S being a subset of the underlying Riemannian manifold .M; g/, and the concentration given by C.x; t /, where x 2 S and t 2 RC . The driving force behind diffusion is the non-vanishing gradient or covariant derivative of the concentration CIi

645

diffusion (M, g)

x∈ S⊆M C(x,t)

substance

∂S

gradient C;i medium

p p The invariant area element is denoted by g d D g dx 1 dx 2 , where g is the determinant of the metric. The Lagrangian for diffusion is then a function on the jet bundle J 1 over the previously defined configuration space N , namely, L W J 1 N ! R, and takes for source-free diffusion the following explicit form [5]: L D D ij C Ii C  Ij 

1 2

  CC  CC ;

(2)

where the dot denotes the time derivative and the diffusion tensor at p 2 S is defined by Dp W .Tp S /2 ! R. It is natural, however, to assume isotropy for the underlying space, such that Dp  gp for all p 2 S:

(3)

Substituting (2) and (3) in (1) and then carrying out the variation for the generalized coordinate C  readily yields Z h ı S

 g ij C Ii C Ij 

1 2



CC  CC D

ip g d

Z h  i p g ij C Ii Ij  C ıC  g d D 0; (4) S

where we have used integration by parts and the usual assumption that all variations vanish on the boundary @S . Since the metric g is covariantly constant and ıC  ¤ 0 inside the domain S  M , one obtains   gij C Ii Ij  C D 0 ) g ij C Iij  C D 0: (5) The fundamental differential equation for sourceless isotropic diffusion then is  1 p ij g g C ;i ;j D C; S C D g ij C Iij D p g

(6)

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Fig. 2 An axisymmetric surface is conveniently represented as a Monge patch with coordinates ' and z: 0 1 .z/ cos ' r.'; z/ D @ .z/ sin ' A, z where the function .z/ determines the radial distance perpendicular to the z axis

where S is the Laplace–Beltrami operator on the surface S with metric g and the comma is the usual notation for a partial derivative w. r. t. one of the coordinates. For surfaces of revolution a convenient representation is a Monge patch with coordinates x 1 D O ' and x 2 D O z, see Fig. 2. It is straightforward to derive the metric ! 2 0 2 ;  (7) gij D 0 1 C 0 .z/   which implies g D 2 1 C 02 for any given function .z/.

3 Formal Self-adjointness and Fundamental Solutions We are now in the position to investigate further properties of (6) considering the axisymmetric metric (7). With an inner product consistent with (1), one can show hS f1 ; f2 i.S;g/ D hf1 ; S f2 i.S;g/ for all f1 ; f2 2 ff W S ! RI f 2 C 2 .S /g; (8) which means that the operator S is formally self-adjoint.1 As a consequence the time-independent eigenvalue problem .S C /˚.; x/ D 0 for all x 2 S , has a continuous non-negative spectrum with a complete and orthogonal set of eigenfunctions ˚.; x/, with  2 RC and x 2 S . Continuity follows from the unbounded integration domain of the differential equation [2]. Thus, we may expand the solutions of the diffusion equation in terms of these eigenfunctions and use the method of separation of variables (see e.g. [7]), which readily yields the ansatz: Z1 C.'; z; t/ D

a./ e t f .; z/ h.; '/ d;

a./ 2 R:

(9)

0

1 Quite interestingly C. Lanczos already conjectured in 1949 that “Any physical law which can be expressed as a variational principle describes an expression which is self-adjoint.”

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The functions f and h are still to be determined; all we require is sufficient differentiability, namely of class C 2 , on their corresponding parameter domain. Substitution of (9) in (6) with the metric (7) produces after some simplification: Z1 a./ e

t



1 d p g dz



 2 0 1 00 p f .z/ h.'/ C 2 f .z/ h .'/ C f .z/ h.'/ d D 0: g 

0

(10) For arbitrary coefficients a./, it is the bracket term which has to vanish, and h00 = h D m2 must be a constant. This directly implies for the polar part of the fundamental solution h.'/ D

1 h X

i Am ./ cos.m'/ C Bm ./ sin.m'/ ;

(11)

mD0

where as usual the real coefficients Am and Bm may be a function of , depending on the explicit initial or boundary conditions. The behaviour of the fundamental solution along the z-axis is also obtained from (10) and can be recast in the form of a Sturm–Liouville equation: 

 p m2 d  2 p g f: p f0 C g 2 f D „ƒ‚… dz g  „ ƒ‚ … „ƒ‚… w.z/ q.z/ p.z/

(12)

Here, p.z/; q.z/ and the  weight  function w.z/ can be shown to be all positive. Using  D .z/ and g D 2 1 C 02 ¤ 0 gives the more practical expression: 2 f 00 C 

2 0 g  12 g 0 0 m2 f C g   2 f D 0: g 

(13)

Finally, the solutions fm .; z/ of (13), in combination with (11), completely determine the general structure of the fundamental solution (9) for diffusion on a surface of revolution S .

4 Diffusion on a Paraboloid p Let S be an open paraboloid with its peak at the origin given by .z/ D 2 z. For obtaining a description of diffusion on this surface, our proposed differentialgeometric framework reduces all further efforts to only finding the solutions to the

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corresponding Sturm-Liouville problem. Equation (13) then becomes zf 00 C



m2 2Cz 1Cz f0C z  f D 0: 2.1 C z/ z 4

(14)

The substitution fm .; z/ D e i

p

z

m z 2 fQm .; z/

(15)

is useful to render (14) as z2 ˛ C .2 C ˇ C   ˛/ z  1  ˇ Q0 fQ00 C f z.z  1/ 1 ..2 C ˇ C  / ˛ C 2 ı/ z  .ˇ C 1/ ˛ C . C 1/ ˇ C  C 2  Q f D 0; C 2 z .z  1/

(16)

where fQ.0/ D 1 and fQ0 .0/. Equation (16) is the confluent Heun’s differential equation for fQ in standard form[4]. Explicit calculation enables us to identify p ˛ D 2i ; ˇ D ˙m;  D 3=2; ı D  C m2 =4;  D  C 3=4  m2 =4; and the two independent solutions (15) can now be expressed in terms of the confluent Heun’s functions:  p  fQ˙m .; z/ D Hc 2i ; ˙m; 3=2;  C m2 =4;  C 3=4  m2 =4I z D Hc ˙m .I z/:

(17)

Substituting results (17) and (11) into (9), finally provides the full formal solution Z1 C.'; z; t/ D 0

e t i

p

z

1 X

h i m z˙ 2 Am ./ cos.m'/ C Bm ./ sin.m'/

mD0

i h  Cm ./Hc m .I z/ C Dm ./Hc m .I z/ d:

(18)

5 Conclusion and Outlook In summary, we have presented a differential-geometric approach to deal with diffusion on axisymmetric surfaces. Starting point was a Lagrangian on a Riemannian manifold using a variational principle. The dependence of the solution along the axis of symmetry is non-trivial. It was shown to be given by a Sturm–Liouville equation.

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Apart from the standard trigonometric functions, confluent Heun’s functions emerge in the analytical treatment of sourceless diffusion on a parabolic surface. These results and their approximations may provide a valuable tool for related diffusion models, and one might develop new semi-numerical models based on them. Acknowledgements This work has been supported by the Universidad Polit´ecnica de Valencia under grants PAID-06-08-3307 and PAID-05-09-4338, and by the Ministerio de Educaci´on of Spain under grant MTM2009-08587.

References 1. Carslaw, H.S., Jaeger, J.C.: Conduction of Heat in Solids. Clarendon Press, London (1959) 2. Courant, R., Hilbert, D.: Methoden der Mathematischen Physik. Springer, Berlin (1993) 3. Lew, A., Marsden, J.E., Ortiz, M., West, M.: Asynchronous variational integrators. Arch. Ration. Mech. Anal. 167, 85–146 (2003) 4. Ronveaux, A. (ed.): Heun’s Differential Equations. Oxford University Press, Oxford (1995) 5. Tung, M.M.: Basics of a differential-geometric approach to diffusion: Uniting Lagrangian and Eulerian models on a manifold. In: Bonilla, L.L., Moscoso, M.A., Platero, G., Vega, J.M. (eds.) Progress in Industrial Mathematics at ECMI 2006, Mathematics in Industry, vol. 12, pp. 897– 901. Springer, Berlin (2007) 6. Tung, M.M., Herv´as, A.: A differential-geometric approach to model isotropic diffusion on circular conic surfaces in uniform rotation. In: Fitt, A.D., Norbury, J., Ockendon, H., Wilson, E. (eds.) Progress in Industrial Mathematics at ECMI 2008, Mathematics in Industry, vol. 15, pp. 1053–1060. Springer, Berlin (2010) 7. Zwillinger, D.: Handbook of Differential Equations. Academic Press, San Diego (1998)

Verified Analysis of a Biomechanics-Related System Ekaterina Auer

Abstract Biomechanics is an interesting application field for verified computations. For example, it is difficult to measure parameters of living organisms adequately, and simultaneously it is very important to do that as precisely as possible. Interval methods offer a solution for such problems. We can enhance models themselves, account for floating-point errors and guarantee the correctness of a numerical solution, propagate input uncertainty through the system, or compute parameter sensitivities. In this paper, we consider how the problem of stance stabilization that came into existence in the course of the recent project PROREOP can be analyzed with the help of interval methods.

1 Introduction Interval arithmetic is a well developed area of numerics with applications in engineering, medical science or mechanics. It belongs to the group of the verified methods, that is, methods that guarantee the correctness of the outcome of a computer simulation using mathematically exact proofs based, for example, on fixed point theorems. examples of such methods. The beginnings can be dated back as far as at least the nineteenth century, but the actual break-through came with the work [9] by R. Moore. Beside proving the correctness of the computed result, verified methods can take care of rounding errors and propagate bounded uncertainties through systems. In this paper, we show how to make use of the latter property of interval arithmetic in biomechanics. In this field of science, principles from mechanics are applied to living organisms in order to solve medical problems. One of the

E. Auer () University of Duisburg-Essen, 45141 Essen, Germany e-mail: [email protected] M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9 76, © Springer-Verlag Berlin Heidelberg 2012

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difficulties in this area is the uncertainty arising, for example, from the impossibility to measure parameters of humans such as lengths of bones adequately. That is the reason why methods to propagate uncertainties through the system gain more and more importance. As mentioned before, we can deal with bounded uncertainty in input parameters and in the outcome by specifying upper and lower bounds on all possible results using interval or other verified methods. One example of biomechanical software is the tool MOBILEBODY [7], a diagnose program for human musculoskeletal system, that was developed to help surgeons during their operation planing in the course of the project PROREOP (Development of a new prognosis system to optimize patient-specific preoperative surgical planning for the human skeletal system) [10] on the basis of the multibody modeling and simulation software MOBILE [4]. It combines information gathered in the gait lab using a marker-based technology with MRT and Xray recordings into a patient-specific mechanical model. There are a lot of subtasks in this process, of which we consider the problem of human stance stabilization from the verified point of view. The paper is structured as follows. In Sect. 2 we overview verified methods, the tool MOBILE and its verified version SMARTMOBILE [1] in brief. In the next section, we describe the problem of human stance stabilization and the corresponding biomechanical model, then report on our first analysis of uncertainties in its parameters from the verified perspective, and conclude by an outlook on what still has to be done in this direction. Finally, we recapitulate the main results and point out our future work.

2 Background To be able to consider uncertainties in the model of human stance stabilization, we use SMARTMOBILE, a verified modeling and simulation tool based on MOBILE. In this section, we describe briefly the theory and libraries which make verified computations in SMARTMOBILE possible and give a short overview of the involved multibody software itself.

2.1 Interval Methods and Libraries To model and simulate stabilization of human stance, we rely on interval arithmetic [8] in our first verified analysis. An interval Œx; x, where x is the lower, x the upper bound, is defined as X D Œx D Œx; x D fx 2 Rjx  x  xg: Elementary operations and functions can be defined on intervals in such a way as to result in intervals. To be able to work with this definition on a computer

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using a finite precision arithmetic, the concept of machine intervals is necessary. They are represented by floating point numbers for the lower and upper bounds. To obtain the corresponding machine interval for the real interval Œx; x, the lower bound is rounded down to the largest representable machine number equal or less than x, and the upper bound is rounded up to the smallest machine number equal or greater than x. These notions can be extended to define interval vectors and matrices. There exist verified analogs to higher-level numerical algorithms such as those for solving linear, nonlinear or differential systems of equations. Almost all of them need at least one derivative of the right side of system model equations to be able to work. That is, it is necessary to obtain derivatives of code automatically [3]. There are several libraries implementing this concept which employ either overloading or code transformation. Another concept important for this paper is the sensitivity. We understand it as a linear measure of uncertainty influence. If we have a bounded uncertain parameter p 2 Œp, which our characteristic of interest Œx depend on, then the sensitivity is defined as Œs D @Œx=@Œp, an interval evaluation of the partial derivative of Œx wrt. p over Œp. If this definition does not produce a meaningfulPresult in interval arithmetic, we might use a reference from engineering: Œr D i j@x=@pi j  Œpi  (with interval operations). Here, x and pi are reference values, for example, midpoints of the uncertain quantities. Note that Œs is proved to contain all possible values of derivatives over the parameter interval Œp under consideration of all other parameter uncertainties, whereas local or sampling-based sensitivity methods from engineering do not have this guarantee and provide derivatives for certain parameter values in Œp. If 0 … Œs and jsj, jsj are large numbers, we can state that x is highly sensitive to p. If 0 2 Œs but maxfjsj; jsjg is large, there are regions with different sensitivities within the parameter interval Œp. However, as we work in interval arithmetic, this might also be due to overestimation. In either case, the parameter bounds for Œp should be reduced. This situation is not detected by local methods and might be accidentally missed by sampling methods. In SMARTMOBILE, we use PROFIL/BIAS [5] for basic interval operations and FADBAD++[11] for algorithmic differentiation in this paper.

2.2 Piecewise Functions Many functions which are to have physical meaning are in fact only piecewise smooth. For example, the force should be negative, which is usually expressed as an “if-then-else”condition in code. Such characteristics are not easily handled by algorithmic differentiation. There are several libraries, for example [2], which use their own versions of “if-then-else” conditions to produce correct derivatives of the corresponding functions. However, they only work pointwise, that is, they cannot be applied if their arguments are proper intervals. To deal with this problem, we

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implemented a class pwFunc for computation of enclosures and first derivatives of piecewise functions for interval arguments. Let the piecewise function be defined in the following way: 8 f0 .x/; if c1 D 1 < x  c0 ; ˆ ˆ ˆ < f .x/; if c < x  c ; 1 0 1 ; where ci are constants. f .x/ D ˆ : : : : : : ˆ ˆ : fn .x/; if cn1 < x < cn D C1 For such functions, we define the first derivative as

0

f .X/ D

8 0 f .X/; ˆ ˆ < i ˆ ˆ : hul l

jS 1 kDi C1

!

if X  .ci 1 ; ci ;

fk0 .Œck1 ; ck / [ fi0 .Œx; ci / [ fj0 .Œcj 1 ; x/ ; if X  .ci 1 ; cj 

:

(1)

Note that the function f .x/ should be continuous in x D ci . Besides, f .x/ is not differentiable in general, and f 0 .X / for X 3 ci encloses both left and right derivatives if the definition from (1) is used. hul l is the convex hull of all relevant intervals, which the class pwFunc implements pairwise.

2.3 MOBILE and SmartMOBILE SMARTMOBILE [1] is a C++ object-oriented software for verification of various classes of mechanical systems based on MOBILE [4] which employs usual numerics. Models in both tools are executable C++ programs built of the supplied classes for transmission elements such as rigid links for modeling of rigid bodies, scalar or spatial objects (e.g. coordinate frames) and solvers (e.g. for differential equations). SMARTMOBILE is one of the first integrated environments providing result verification for kinematic and dynamic simulations of mechanical systems. The advantage of this software is its flexibility due to the template structure: the user can choose the kind of (non)verified arithmetics according to his task. Advanced users are not limited to the already defined classes for these arithmetics and are free to plug in their own implementations. For most kinematical problems, it is sufficient to use a certain basic data type as the parameter of all the template classes implementing a particular model. The main idea for dynamical and such kinematical tasks as finding of system equilibria is to use pairs basic data type/specialized solver. Besides verified modeling and simulation, SMARTMOBILE offers techniques for sensitivity analysis and uncertainty management.

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3 Characterizing Uncertainty in Stance Stabilization Problem In this section, we describe how the uncertainties in several measured parameters influence the model for the human stance stabilization. The problem of modeling the stance can be divided into three stages [6]. First, human skeleton has to modeled. The proposed model consists of nine segments: the pelvis representing the whole upper body, then right and left femur, right and left tibia as well as right and left foot composed of a forefoot and hindfoot each. These segments are connected by appropriate joints. The second stage is the modeling of the foot contact. It is achieved by choosing two cylinders as contact surfaces for the foot and using a Hunt-Crossley based contact scheme. Finally, a PID controller is applied to stabilize the stance. At each of these stages, there are parameters known with some large or small incertitude (cf. Table 1). For example, the pelvis mass of [35, 65]kg or its position on the x axis ([0.05, 0.1]m) constitute the group of mass parameters and belong roughly to the first stage. Such parameters as the radii of forefoot or hindfoot or static and dynamic friction coefficients influence mainly the second stage. Forces along the x and y axes can be counted to the last stage. This problem has 26 degrees of freedom. We implemented the above model in SMARTMOBILE. We used the basic data type F for the interval based sensitivity analysis and the class pwFunc to represent non smooth functions such as jxj. The goal was to obtain the equations of motion for the problem of stance stabilization at the first simulation time interval to study the influence of the uncertainty in parameters on them. Note that the exact expressions for the equations of motion are unknown a priori and constructed from the involved transmission elements, which methodically means interval evaluation. Partial derivatives are simultaneously computed exactly by algorithmic differentiation (also for piecewise smooth functions) and evaluated for the corresponding intervals.

Table 1 Some uncertain parameters in stance stabilization Force Contact

Mass

! [0.5, 6.28] frequency

mp [35, 65]kg pelvis mass

Fx [0, 200]N force along x Fy [0, 50]N force along y

rff [0.04, 0.2]m radius forefoot rhf [0.02, 0.15]m radius hindfoot eN [0.01, 0.2] normal restitution eT [0.01, 0.2]tangential restitution st [0.5, 2.0] static friction coefficients d [0.08, 2.3] dynamic friction coefficients

px [0.05, 0.1]m x-position of pelvis py [0.1, 0.5]m y-position pz [0.05, 0.05]m z-position

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E. Auer Table 2 The abridged force vector Œf1 f2 f4 f6  for different sets of uncertain parameters (directed rounding to the second digit after the decimal point) All from Table 1 mp , px and Fx Nominal f1 [0, 200] [0, 200] [99.99, 100.00] f2 [940.00, 595.69] [915.00, 620.69] [767.85, 767.84] f4 [31.89, 31.89] [0, 0] [0, 0] f6 [50.17, 45.49] [50.17, 45.49] [1.33, 1.34]

Table 3 Interval sensitivity (Columns 2–5) and reference uncertainty (Column 6), directed rounding to the second digit after the decimal point @./=@mp @./=@px @./=@mrf @./=@Fx [r] f1 0.0 (0) 0.0 (0) 0.0 (0) [0.99, 1] (1) [0.00, 200.00] f2 [9.81, 9.80] 0.0 (0) [9.81, 9.80] 0.0 (0) [444.43, 738.44] (9.81) (9.81) 0.0 (0) [0.78, 0.79](0.78) 0.0 (0) 8.07 f4 0.0 (0) f6 [9.82, 0.50] [637.66, 343.34] [0.49, 0.5] (0.5) 0.0 (0) [38.44, 70.47] (0.25) (490.5)

The parameters of interest are the pelvis mass mp , the position of the pelvis’ center of mass on the x axis px , the applied force along the x axis Fx and the mass of the right femur mrf D 10:34 kg. We consider the first, second, fourth and sixth coordinates of the force vector from the equations of motion. In Table 2, we show interval evaluations for these characteristics under influence of two sets of uncertain parameters and for nominal parameters (midpoints of corresponding intervals). The sensitivity of f1 , f2 , f4 and f6 under uncertainty in mp , px and Fx is shown in Table 3. The first value in each cell is the interval sensitivity Œs, the value in parentheses denotes the sensitivity for nominal parameters. The Tables show that the force-induced part of equations of motion depends most substantially on the position and the mass of the pelvis. This holds especially for f6 , which is most sensitive to px . This is also confirmed by the interval evaluation. If we consider only the uncertainty in mp (the diameter of the input uncertainty is therefore 30), then the enclosure for f6 is Œ2:35; 5:02 N (of the diameter 7:37). However, the input uncertainty in px (diam.px / D 0:05) leads to the output Œ35:46; 38:15 N of the diameter 73:61 (this corresponds to the values of sensitivities in Table 3). In this case, even the sign of the characteristic cannot be derived any longer. The major challenge while simulating dynamics of the stance stabilization in a verified way is the foot contact stage. The main reason is that the equations of motion for it change their right side in dependance on the zeros of a certain choice function. That is, we have to deal with a hybrid system. Verified treatment is infrequent, but has some advantages, for example, for contact area modeling. The contact area between a cylinder and a plane is not a point for small angles between the corresponding normals, and is presently projected into a point. Verified methods

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offer a possibility to work with the original contact area without the projection by representing it as an interval. We presented a first verified sensitivity analysis of the stance stabilization model from PROREOP. We used SMARTMOBILE for this purpose, a tool providing verified kinematics, dynamics and sensitivity analysis options for several classes of (bio)mechanical systems. Besides, we introduced an implementation pwFunc of a class for computing interval evaluations and first derivatives of piecewise functions. We showed that the equations of motion for the stance stabilization are particularly sensitive to the position of pelvis and the pelvis mass. Our future work will include the development of a verified solver for hybrid systems of the type we considered in this paper and modeling of the contact area between a cylinder and a plane with the help of intervals.

References 1. Auer, E., Luther, W.: SmartMOBILE and its Applications to Guaranteed Modeling and Simulation of Mechanical Systems. In: Lecture Notes in Electrical Engineering, vol. 24 (2009), pp. 139–151, Springer Berlin Heidelberg 2. Bell, B.M.: www.seanet.com/Bradbell/cppad, Web page (2012) 3. Griewank, A.: Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. SIAM (2000) 4. Kecskem´ethy, A.: Objektorientierte Modellierung der Dynamik von Mehrk¨orpersystemen mit Hilfe von u¨ bertragungselementen. Ph.D. thesis, Gerhard Mercator Universit¨at Duisburg (1993) 5. Kn¨uppel, O.: PROFIL/BIAS — A fast interval library. Computing 53, 277–287 (1994) 6. Liu, X., Kecskem´ethy, A., T¨andl, M.: A self-stablilized foot-ground contact model using two segments and cylinder-plane pairs. (2008). I-FAB Poster 7. MobileBody: Patientenindividuelle Ganganalyse. http://www.uni-due.de/mechanikb/ forschung/projekte.php (2010), last access date 2012 8. Moore, E., Kearfott, B., Cloud, M.: Introduction to Interval Analysis, vol. 1. Society for Industrial Mathematics, Philadelphia (2009) 9. Moore, R.E.: Interval analysis. Prentice-Hall, Englewood Cliffs (1966) 10. fsPROREOP: http://www.uni-due.de/proreop/. Online document (2007–2008), last access date 2012 11. Stauning, O., Bendtsen, C.: Fadbad++ web page. http://www.fadbad.com/, last access date 2012

Subject Index

0-missing, 179

Absorbing boundary condition, 635 Accelerator physics, 93 Acoustic dissipation, 513 Adaptive approximation, 5 Adiabatic Berry’s phase approximation, 373 Adjoint, 573 Aerodynamic, 239 Aggregate model, 553 Air elimination, 421 Air-knife stripping, 311 Analog Insydes, 187 Angiogenic process, 529 Anisotropic thermal conductivity, 303 Annealing, 303 Arnold tongue, 21 Artificial satellites, 489 Asymptotic analysis, 349, 521 Asymptotic solutions, 441 Asymptotics, 327, 449 Automotive design, 295 Axisymmetric deformation, 505

B-splines, 357 Balanced truncation, 169 Batch annealing furnace, 303 Beam, 587 Behavioral parameter, 195 Bilinear systems, 153 Black-Scholes equation, 231 nonlinear, 207, 223 Bloch-Boltzmann-Peierls equations, 69 Blow moulding, 319

BMBF network project, 561 Boltzmann transport equation, 54, 61 Bordered matrices, 29, 195 Boundary conditions, 341 Boundary value problems, 319 Branch-oriented model, 13 Bubble formation, 421

Cache efficiency, 627 oblivious, 627 Cauchy problem, 407 Causal wave attenuation, 513 CFD, 465 Channel stress, 85 Characteristic equation, 407 Cholesky factor ADI for preconditioning, 131 Circuit simulation, 5, 13, 145, 169, 187 integrated circuits, 161 nonlinear, 145, 153 Classical and post-Newtonian locations, 489 Closed form solution, 489 Coating deformations, 311 Collision detection, 335 Combination technique, 231 Compact finite differences, 215 Compressed row storage (CRS) format, 627 Compression, 505 Computational finance, 231 Cone and plate rheometer, 449 Conformation dynamics, 619 Conservation laws, 247 Consistency, 107 Contact modeling, 335 Control constraints, 537

M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9, © Springer-Verlag Berlin Heidelberg 2012

659

660 Control design, 263, 595 Control theory, 255 Control-constraints, 581 Convection-reaction-diffusion equation, 521 Cosserat rods, 349 Cosserat strings, 349 Counting measure in digital images, 529 Coupled PDE-ODE systems, 247 Coupling surface and subsurface flows, 561 Cubic matrix splines, 611 Curvature, 449 Cylindrical combustion chamber, 473

Data, 295 Deep magnetotelluric sounding, 497 Delay differential equations, 283 neutral, 603 Design, 573 Design for yield, 187 Differential algebraic equation, 13, 169, 263 Differential flatness, 255 Differential geometry, 643 Diffusion model, 365 phenomena, 643 Diffusive limit, 69 Direct simulation Monte Carlo, 61 Discontinuity detection, 603 Discontinuous Galerkin method, 561 Discrete adjoint, 573 Dissipative wave equation, 513 Domain decomposition methods, 561 Drag, 239 Drift-diffusion equations, 45, 161 Driven flow, 441 Drug delivery, 521 Drug release profile, 521 Dry patches in a flowing film, 441 Dubovitski-Milytin method, 537 DUNE, 561 Dunegrid-glue module, 561

Early exercise boundary, 207 Eddy dissipation combustion model, 473 Effective diffusivity, 521 Eigenmode analysis, 113 Eigenvalue problem, 457, 635 generalized, 29, 195 Elastic fibers, 335 Elastic multibody systems, 595 Elastic thread, 239 Elasticity, 573

Subject Index Electromagnetic absorption coefficient, 513 Electromagnetic field calculation, 93 Electromagnetic transient analysis, 107 Electromagnetics, 37 Electron phonon transport, 61 Electronic design automation, 187 Energy-transport models, 69 Entrainment, 21 Error detection, 357 estimation, 137, 595 ESVDMOR, 137 Etching, 327 Evolutionary algorithm, 77 Existence analysis, 427 Extended Kalman filter, 595

FFT, 29 Fiber dynamics, 335 Fiber spinning, 349 Fiber-flow interactions, 239 Finger formation, 457 Finite difference method, 207 Finite element, 573 Finite volume, 85, 473 Finite-dimensional approximation, 595 Fishery management, 537 Fishway, 545 Floquet theory, 21 Fluids, 421, 465 Fractional Brownian motion, 537 Free boundary, 207, 341 Frequency analysis, 113 Full-vehicle simulation, 581

Galvanisation, 311 Geolocation, 489 Geometry parameterization, 465 Gravity, 441

H-polarization, 497 Hamiltonian system, 643 Heat balance integral method, 399 Heckel model, 505 Hermoacoustic tomography, 513 Heterogeneous nonlinear Dirichlet-Neumann method, 561 Heun’s differential equation, 643 Heun’s functions, 643 High order, 215 Higher-order matrix splines, 611

Subject Index Hilbert curve, 627 Holographic grating, 365 Homogenization, 381 Hopf bifurcation, 13 Hotspot detection, 77 Hybrid discretization, 93 Hydrodynamical models, 69 Hyperbolic, 13 Ill-posed problem, 407 Image analysis, 381 Incomplete factorizations multilevel ILU, 131 Inertial jet, 407 Inertial flow regime, 407 Infinite dimensional systems, 587 Injection locking, 21 Input-loads, 581 Intensity estimator, 529 Interpolation, 145, 153 bulky data, 295 Interval arithmetic, 263 methods, 255, 651 Inverse problems, 319, 497 Kronecker product, 54 Krylov subspace methods, 153 Lagrangian function, 643 Lambert W function, 283 Layout analysis, 77 LC-block, 13 Linear descriptor systems, 137 Linear irreversible thermodynamics, 69 Liquid jets, 421 Locking range, 21 Logistic equation with delay, 283 Loscalzo-Talbot method, 611 Low rank Krylov subspace methods, 131 Low rank tensor product approximation, 131 Lubrication theory, 399, 441 Lyapunov equations generalized, projected, 131 Macromodeling, 37, 145 Magnetic levitation train, 113 Magnetoquasistatic models, 93 Markovian switching, 247

661 Matched asymptotic expansions, 449 Mathematical morphology, 381 Matrix approximation, 611 Maximum entropy principle, 69 Maxwell equations, 93, 497 Meshfree, 107, 619 Meshing, 465 Metal alloy coating, 311 Method, 473 Microwave structures, 37 Milk, 421 Minimal polynomial (vector) extrapolation, 29 Mixed finite element methods, 161 Model order reduction, 137, 145, 153, 161, 169, 187 circuit simulation, 131 many terminals, 137 Modeling, 319, 465, 545 3D, 85 industrial processes, 327 multiphysics, 113 physical, 587 symbolic, 187 Modified nodal analysis, 45, 169 Momentum flux, 407 Monte-Carlo simulations, 239 Moving surface, 341 Multi-band Wigner-Moyal formalism, 373 Multi-component etching, 327 Multi-dimensional device-circuit coupled model, 45 Multibody modeling and simulation, 651 Multibody systems, 113, 581 Multiquadrics, 603 Multiscale, 619 Multiscale random structure, 381

Nano PMOSFETs, 85 Nanoelectronics, 187 Navier-Stokes equation, 427, 573 NBI, 195 Newtonian fluid, 441 Non-Newtonian fluids, 457 Nonlinear dynamical systems, 255 Nonlinear metamodeling, 295 Nonlinear systems, 153 NUBS, 465 Numerical simulations, 381, 427 Numerical solution, 215

Operator splitting method, 207 Optical interference patterns, 365

662 Optical surfaces, 357 Optimal control, 537, 545, 581 Optimization algorithm, 77 layout, 77 multi-objective, 195 numerical, 465, 545 shape, 573 Option pricing, 215 multi-dimensional, 231 Ordinary differential equations, 421

P-structure, 13 PABTEC, 169 Parameter variation, 187 Parametric surfaces, 505 Parametrized Dynamical Systems, 161 Partial differential algebraic equations, 45 Partial differential equations, 545 method, 505 nonlinear, 223 Particle-hole dynamics in graphene, 373 Passive radio-transmiters, 489 Passive systems, 37 Pendant drop, 449 Perturbation, 457 Phase change, 399 Phase constraints, 537 Photonic crystals, 365 Photopolymerisation, 365 Piecewise deterministic processes (PDPs), 247 Piecewise smooth functions, 651 Piezoelectricity, 587 Poincar´e method, 29 Polynomial chaos, 271, 283 Pontryagin’s Theorem, 537 Port-Hamiltonian, 587 Power law, 513 Power MOSFET, 77 Preconditioning, 93 Production networks, 247 Pseudo differential operator, 635 Purely imaginary eigenvalue (PIE), 13

Quadratic-bilinear systems, 153 Quantum corrections to the semi-classical equation of motion, 373 Quantum equilibrium, 427 Quantum kinetic approach, 373 Quantum mechanics, 635 Quasiinterpolation, 357

Subject Index Radial basis functions, 295, 603 Random breakdowns, 247 Random differential equations, 271 Random sets, 381 Rational approximation, 37 Reduced basis methods, 161 Representative volume element (RVE), 381 RF simulation, 5 Richards equation, 561 Riemannian manifolds, 643 River, 545 Rotary spinning, 341

Saturated unsaturated groundwater flow, 561 Schr¨odinger equation, 635 Schur complement, 93 Second-order differential systems, 611 Semiconductor device simulation, 54 Semiconductors, 61, 69 Sensitivities, 573 Sensitivity analysis, 187, 651 Sessile drop, 449 Shallow water equations, 561 Shape design, 545 Shapiro time-delay, 489 Sheet steel production, 303, 311 Similarity solution, 441 Simulation, 85, 465 Singular integral equation, 537 Singular value decomposition, 295 Slenderbody theory, 349 Smoothed particle electromagnetics (SPEM), 107 hydrodynamics (SPH), 107 Smoothing effect, 553 Software, 573 Solver, 573 Sparse grid, 231 Sparse matrix, 54 Sparse matrix-vector multiplication, 627 Spatial discretization, 587 SPEA2, 195 Special mesh, 479 Specific binding, 521 Spectral Method, 54 Spherical harmonics expansion, 54 Spin coating, 457 Splines, 5, 465 B-splines, 357 matrix splines, 611 Spray drying, 479 Stability, 215, 283

Subject Index Stance stabilization, 651 Standard k-", 473 State and disturbance estimation, 595 State estimation, 263 Stationary fibre process, 529 Stationary solutions, 223 Statistical fluctuations, 381 Steel coils, 303 Steepest descents, 327 Stefan problem, 399, 479 Stochastic differential equation, 247, 537 Stochastic partial differential equations, 239 Stochastic volatility, 215, 223 Stokes flow, 319 String jet model, 341 Structural analysis, 263 Structure preservation, 137, 587 Sturm-Liouville problem, 643 Superconductive magnet, 93 Surface tension, 319, 449 Surrogate model, 195 Synchronisation, 21 Synge’s world function, 489

Taylor models, 255 TDOA method, 489 Technical textiles, 335 Tectonic faults, 497 Thermal effects, 61, 69 Thin film flows, 441, 457 Thin jet model, 341 Three-dimensional unsteady flow on an inclined plane, 441 Timoshenko, 587 Tractability index 2, 45 Transaction costs, 223 Travelling wave, 441 Tunneling diode, 427

663 Turbulence, 239, 473, 553 intensity, 473 model, 473 wind, 553 Turbulent combustion process, 473

Uncertainty quantification, 271, 283 Undamped oscillations, 13 Upper and lower solution method, 223

Variance of intensity estimator, 529 Variational principle, 643 Verified computations, 263 integration, 255 Viscous jet, 407 Viscous flow regime, 407 Viscous jet, 349 curved, 341 Viscous quantum Euler equations, 427 Viscous-inertial flow regime, 407 Viscous-inertial jet, 407

Wavelets, 5, 357 Weak coupling, 619 Wet chemical etching, 327 Wetting and drying, 561 Wiener calculus, 271 Wigner-BGK equation, 427 Wind farm, 553 Wind power fluctuations, 553

Zernike polynomials, 357 Zero crossing, 29

Authors Index

Ahmat, N., 505 Ala, G., 107 Al`ı, G., 45, 69 Alvarez-V´azquez, L.J., 545 Antoine, X., 635 Antritter, F., 255 Arne, W., 349 Aschemann, H., 263, 595 Auer, E., 651 Augustin, F., 271

Babeva, T., 365 Barry, S.I., 303 Bartel, A., 45 Bastian, P., 561 Beelen, T.G.J., 29, 195 Benk, J., 231 Benner, P., 137, 153 Bernal, F., 603 Berninger, H., 561 Besse, C., 635 Bisseling, R.H., 627 Bittner, K., 5 Bollh¨ofer, M., 131 Brachtendorf, H.-G., 21 Breiten, T., 153 Breward, C., 311 Bungartz, H.-J., 231 Burger, M., 581

Celis, O.S., 37 Charpin, J.P.F., 457 Clees, T., 295

Cregan, V., 449 Cuyt, A., 37

Dautbegovic, E., 5 De Gersem, H., 93 De Tommasi, L., 195, 553 Dedner, A., 561 Defez, E., 611 Dellnitz, M., 113 Deschrijver, D., 37 Devereux, M., 421 Dhaene, T., 37 Di Stefano, V., 61 Dignath, F., 113 Dreßler, K., 581 Duffy, B.R., 441 Dumitrache, A., 473 Dumitrescu, H., 473 D¨uring, B., 215

Ehrhardt, M., 635 El Boukili, A., 85 Engwer, C., 561 Eppler, A.K., 131

Fabi˜ao, F., 223 Fackeldey, K., 619 Filatova, D., 537 Fitt, A.D., 311 Flaßkamp, K., 113 Fourni´e, M., 215 Francomano, E., 107 Frunzulica, F., 473

M. G¨unther et al. (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, DOI 10.1007/978-3-642-25100-9, © Springer-Verlag Berlin Heidelberg 2012

665

666 Gambi, J.M., 489 Garcia del Pino, M.L., 489 Giannopapa, C.G., 319 Gibescu, M., 553 Gonz´alez Castro, G., 505 G¨ottlich, S., 247 Grasser, T., 54 Greco, G., 77 Groot, J.A.W.M., 319 Grossinho, M.R., 223 Guinness, M.M., 303 Guseynov, S.E., 497

Hauser, M., 187 Henning, P., 561 Hering-Bertram, M., 335 Hessel-von Molo, M., 113 Hinze, M., 161 Hlod, A., 341, 407 Hocking, G.C., 311 Honkala, M., 29 Hopfensitz, M., 465 Hulkkonen, M., 29

Authors Index Mattheij, R.M.M., 319 Matthes, U., 161 Matutat, J.C., 465 McKee, S., 449 Meere, M., 521 Menke, C., 357 Micheletti, A., 529 Miliˇsi´c, J.-P., 427 Mitchell, S.L., 399 Morais, E., 223 Morandi, O., 373 Morr, M., 581 Muscato, O., 61 Myers, T.G., 399

Nagy, A.-E., 231 Naydenova, I., 365 Nikitin, I., 295 Nikitina, L., 295

O’Brien, S.B.G., 449 Ohlberger, M., 561 Olawsky, F., 335

Ib´an˜ ez, J.J., 611 Peletier, M.A., 341 Jester, P., 357 Jeulin, D., 381 J´udice, J.J., 545 J¨ungel, A., 54, 427

Klein, P., 635 Kletting, M., 255 Koch, S., 93 Kornhuber, R., 561 Kowar, R., 513 Kr¨oner, D., 561 Kr¨uger, M., 113 Kunkel, M., 161

Laur, R., 21 Lee, W., 421

Mackey, D., 365 Marheineke, N., 239, 335, 349 Marquardt, A., 581 Martin, S., 247 Mart´ınez, A., 545

Rancoita, P.M.V., 529 Rauh, A., 263, 595 Rentrop, P., 271 Riaza, R., 13 Rinaudo, S., 77 Ritzke, J., 595 Rochev, Y., 521 Rodr´ıguez, C., 545 Rodriguez-Teijeiro, M.C., 489 Romano, V., 69 Rommes, J., 145, 195 Rongbing, Y., 521 Rotundo, N., 45, 69 Rupp, K., 54

Salzig, C., 187 Sander, O., 561 Schiffler, G., 561 Schmeißer, A., 335 Schmidt, O., 179 Schneider, A., 137 Schneider, R., 573 Schraufstetter, S., 231

Authors Index Sch¨urrer, F., 373 Sevat, M.F., 195 ˇ coviˇc, D., 207 Sevˇ Shokina, N., 561 Sickenberger, T., 247 Sim˜oes, O.A., 223 Smetana, K., 561 Soler, L., 611 Spagnuolo, A., 107 Steinbrecher, A., 169 Striebel, M., 145 Stykel, T., 169 Stynes, M., 479 Sweatman, W.L., 303, 311

ter Maten, E.J.W., 29, 195 Timmermann, R., 113 Tischendorf, C., 13 Toal, V., 365 Tung, M.M., 611, 643

Ugail, H., 505 Urban, K., 357, 465

667 van de Ven, A.A.F., 341 V´azquez-M´endez, M.E., 545 Venturi, A., 187 Vilar, M.A., 545 Villegas Caballero, M., 283 Virtanen, J., 29 Viscor, M., 479 Vo, T.T.N., 521 Voß, T., 587

Ward, J., 327 Wegener, R., 239, 349 Weiland, T., 93 Wever, U., 271 Wilson, S.K., 441 Witte, L., 581

Yatim, Y.M., 441 Yzelman, A.N., 627

Zheng, Q., 113