Computer Methods in Mechanics: Lectures of the CMM 2009 (Advanced Structured Materials)

Advanced Structured Materials Series Editors: Andreas Öchsner (Editor-in-chief), Lucas Filipe Martins da Silva, and Hol...

Author: Mieczyslaw Kuczma | Krzysztof Wilmanski

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Advanced Structured Materials Series Editors: Andreas Öchsner (Editor-in-chief), Lucas Filipe Martins da Silva, and Holm Altenbach

Mieczyslaw Kuczma · Krzysztof Wilmanski

Computer Methods in Mechanics Lectures of the CMM 2009

With 292 Figures

ABC

Editors Prof. Mieczyslaw Kuczma University of Zielona Góra Inst. Building Engineering ul. Z. Szafrana 1 65-516 Zielona Góra Poland E-mail: [email protected]

Prof. Krzysztof Wilmanski Rue Diderot 4 13469 Berlin Germany E-mail: [email protected]

ISBN 978-3-642-05240-8

e-ISBN 978-3-642-05241-5

DOI 10.1007/978-3-642-05241-5 Advanced Structured Materials

ISSN 978-3-642-05241-5

Library of Congress Control Number: 2009940899 c 2010 Springer-Verlag Berlin Heidelberg 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. Typeset design: Scientiﬁc Publishing Services Pvt. Ltd., Chennai, India. Cover design: WMX Design, Heidelberg Printed in acid-free paper 987654321 springer.com

Dedicated to the memory of Professor Olgierd Cecil Zienkiewicz: outstanding scholar, our dear teacher, colleague, and friend

Foreword

This book contains collected plenary and keynote papers presented during the 18th International Conference on Computer Methods in Mechanics - CMM 2009, which took place in Zielona Gra on May 18-21, 2009. The CMM 2009 was organized by the Polish Academy of Sciences (PAS), the Polish Association for Computational Mechanics and the University of Zielona Gra. The chairmen of the conference were Prof. M. Kuczma and Prof. K. Wilma´ nski. The Honorary Chairman of the CMM 2009 was Professor O.C. Zienkiewicz who passed away during the ﬁnal preparation stage of the Conference. The conference was organized under the auspices of the European Community on Computational Methods in Applied Sciences (ECCOMAS) and the Central European Association for Computational Mechanics (CEACM). The CMM 2009 continued an over 35-year tradition of the Polish Conferences on Computer Methods in Mechanics. The 1st conference of this series was held in Pozna´ n in 1973, and ever since every two years in diﬀerent academic centres of Poland the conferences on computational mechanics have been organized under the auspices of PAS and a selected local university of technology. In 2001, instead of the regular CMM conference, the European Conference on Computational Mechanics (ECCM 2001) took place in Cracow. The tradition of these conferences on computer methods in mechanics is one of the longest in Europe. Polish scientists and scientists of Polish descent have had great inﬂuence on the development of the ﬁeld of computer methods in mechanics. One of the most distinguished scientist in this ﬁeld was Prof. O.C. Zienkiewicz. He was a guest of numerous CMM conferences. Since 2003 he was Honorary Chairman of the CMM conferences. This book is dedicated to His memory. The CMM 2009 brought together 300 researchers, practitioners, professors, and students from European countries as well as scholars from overseas. Its aim was to promote the development of computational methods in civil, mechanical, material science and bioengineering, and their applications in engineering practice. In particular, it reﬂects the state-of-the-art of computational mechanics in science. The main focus was on applied mathematics

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Foreword

and computational methods, meshless and boundary element methods, solid mechanics, structural mechanics, coupled problems, mechanics of materials, multiscale modelling and nanomechanics, optimization and identiﬁcation, geomechanics, heat transfer, biomechanics, contact problems, computational intelligence, experimental mechanics and industrial applications. The warm gratitude is given to the members of the Organizing Committee for their great eﬀort and their enthusiasm, which contributed to a big success of this scientiﬁc meeting. The reviewing of all contributed papers by the members of the International Scientiﬁc Committee and the National Scientiﬁc Committee is also gratefully acknowledged. August 2009

Tadeusz Burczy´ nski

Professor Olgierd Cecil Zienkiewicz (1921 - 2009)

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Foreword

Professor Olgierd Cecil Zienkiewicz, Professor of applied mechanics, a civil engineer, one of the pioneers of the ﬁnite element method, one of the greatest mechanicians of the 2nd half of the 20th century, the recipient of many honours and medals died on January 2nd, 2009 in Swansea. Prof. O.C. Zienkiewicz was the highly respected Nestor of the world-wide computational mechanics and far beyond this ﬁeld and the very inﬂuential great scientist. Moreover he had a humanistic view of the world, and he was a reliable personal friend. We loose an outstanding scientist and true friend whose memory will surely remain vivid in all our minds. Prof. O.C Zienkiewicz was born in 1921 in Caterham, Great Britain, as a son of Kazimierz Zienkiewicz and Edith Violet Penny. Since 1922 he lived in Katowice, Poland, where his father was a district judge. He ﬁnished secondary school in Katowice in 1939. Due to the outbreak of the World War Two he did not start studies at Warsaw University of Technology. He studied in Great Britain. In 1943 he graduated with ﬁrst class honours from Imperial College and obtained his B.Sc. there. In 1945 he obtained his Ph.D. and DIC (Diploma Imperial College) as a member of Sir Richard Southwell’s research team. In 1946 he obtained his engineer’s degree at Polish University of Technology in London and in 1965 his D.Sc. (Eng) at the University of London. In the late 1940s Prof. Zienkiewicz worked as a Consulting Engineer, from 1949 till 1957 he was a lecturer at the University of Edinburgh. From 1957 till 1961 he was a professor of structural mechanics in Northwestern University, Evanstone, Illinois, USA. Since 1961 he was associated with the Department of Civil Engineering, the University of Wales, Swansea, UK. Prof. Zienkiewicz established the Institute for Numerical Methods in Engineering and was its Director till 1988, when he retired. He created a scientiﬁc school in Swansea, which soon became the world leading centre in the ﬁeld of numerical methods. Here he developed the ﬁnite element method (FEM) – the main work of his life. After retirement from Swansea in 1987, Olek spent two months each year at the International Center for Numerical Methods in Engineering (CIMNE) at Universitat Politecnica de Catalunya (UPC) in Barcelona, Spain. In 1989 Olek was appointed to the UNESCO Chair of Numerical Methods in Engineering at UPS. This was the very ﬁrst UNESCO Chair in the world. FEM has become an irreplaceable method in the analysis of complex problems of solid and ﬂuid mechanics, in all ﬁelds of engineering, e.g. acoustics, heat conduction, biomedical engineering, electromagnetics, coupled and multiphysics problems. Professor’s activities aiming at the inner development of FEM are also worth mentioning. Among many problems developed in Swansea one can ﬁnd the relations between FEM and other approximation methods (like ﬁnite diﬀerence method, the boundary element method, meshless methods), especially in the ﬁeld of coupling these methods and taking into account an adaptive FE choice.

Foreword

XI

Prof. Zienkiewicz’s genius caused that his knowledge and intuition, his abilities to model and formulate algorithms opened not only new scientiﬁc horizons but also encompassed application possibilities of solving diﬀerent engineering problems. Additionally, his capability to build research teams attracted outstanding students and young academics. It is worth emphasizing that Professor Zienkiewicz school soon achieved world standards and was distinguished by openness and wide international co-operation. Software created in Swansea was from the beginning in the public domain. It was continuously modernized and widened to ﬁt the growing applications and abilities of better and better computers and computer networks. Professor Zienkiewicz published nearly 600 papers and wrote or edited 25 books. His ﬁrst paper published in 1947 dealt with numerical stress analysis of dams. He wrote the ﬁrst book on FEM (The Finite Element Method in Structural Mechanics, McGrow Hill, 1967). His next book was published in 1971 and then many times revised till its three-volume 6th edition in 2005. It was edited with Prof. R.L. Taylor from Berkeley University, California – a former student of Prof. Zienkiewicz – is regarded as a basic reference and the best textbook on FEM. Another ﬁeld in which Professor Zienkiewicz was very active is the organization of science. In 1968 in Swansea he founded with Prof. R.H. Gallagher from USA International Journal for Numerical Methods in Engineering, which has became the major journal concerning FEM and other computational methods. Professor was the editor-in-chief of the journal till 1988, he was also a member of editorial boards of other leading journals dealing with numerical methods. Professor Zienkiewicz was the founder and the 1st President of the International Association of Computational Mechanics (IACM) 1986-1990. Nowadays, national associations of computational mechanics from all states that count on the world scientiﬁc map and regional associations are aﬃliated to IACM. He was also one of the founders of the European Community of Computational Mechanics and Applied Science (ECCOMAS). He was a member, the honorary chairman and the author of a huge number of general lectures on many regional and national computational mechanics conferences, as well as on conferences dealing with the applications of numerical methods FEM in science and technology. Professor Zienkiewicz great prestige is the result of not only his achievements and scientiﬁc activity but also of his personal features, among which it is worth to mention the communicativeness of his papers and lectures, the criticism concerning his own results, kindness to young academics, openness and will to share his achievements with others. He supervised over 70 Ph.D. students, not only in Swansea, but also in many other universities, where he was Visiting Professor or during short-term visits. Professor Zienkiewicz was also a scientiﬁc consultant of many companies and concerns (e.g. Rolls Royce, English Electric, Sir William Halerow & Partners).

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Foreword

He was a member of many academies of science, among others: Fellowship of the Royal Society, U.K., 1979; Fellowship of the Royal Acad. of Engng., U.K., 1979; US National Acad. of Engng., 1981; the Polish Academy of Science, 1985; Accademia Nazionale dei Lincei, Roma, 1999; Accademia di Science Lettere, Milano, 1999. He received Honorary degrees from many universities: Laboratorio Nacional de Engen. Civil, Lisboa, Portugal, 1972; Univ. of Ireland, 1975; Vrije Univ. Brussel, Belgium, 1982; Northwest Univ., USA, 1984; Techn. Univ. Trondheim, Norway, 1985; Chalmers Univ. of Technol., Goteborg, Sweden, 1987; Univ. of Dundee, Scotland, 1987; Dalion Inst. of Technology, Chine, 1987; Warsaw University of Technology, Poland, 1989; Cracow University of Technology, Poland, 1990; Techn. Univ. Budapest, Hungary, 1992; Univ. of Hong Kong, 1992; Compiegne Univ. of Technol., France, 1992; Univ of Wales, UK,1993; Brunel Univ., London, UK, 1993; Aristotle Univ. of Thessaloniki, Greece 1993; Imperial College of Sci., Technol and Medicine, London, UK, 1993; Ecole Normale Sup. de Cachan, Paris, France, 1997; Uniwersitat Politechnica de Madrid, Spain, 1998; Univ. Buenos Aires, Argentina 1998; Chinese Acad. Sci., 1998; Techn. Univ. Lisboa, Portugal, 2001; Silesian University of Technology, Gliwice, Poland, 2001; Politecnico di Milano, Italy, 2001; Czestochowa University of Technology, Poland, 2005. He received also many awards and distinctions. Professor Zienkiewicz was in touch with his family in Poland and his schoolmates in Katowice. In Swansea he received trainees and Ph.D. students from Poland. Since the mid 1960s he was in close contact with Prof. J. Szmelter from Lodz University of Technology, later Military University of Technology, Warsaw . the creator of the Polish school of FEM and Prof. I. Kisiel from Wroclaw University of Technology, who translated into Polish Professor’s book The Finite Element Method (Arkady, 1972). Prof. Zienkiewicz attended Polish Conferences on Computational Mechanics since 1981, was its honorary chairman since 2003, the invited honorary member of the Scientiﬁc Committee of the 2nd European Conference on Computational Mechanics ECCM2001, Cracow, 2001, promoted continuous cooperation with the Institute of Fundamental Technological Research of the Polish Academy of Science, and Warsaw, Cracow and Silesian Universities of Technology. Researchers and engineers on mechanics and computational methods regard Professor Zienkiewicz as one of the most outstanding scientists of the 20th century. He spoke beautiful and rich Polish and often emphasized his Polish roots. Professor’s undisturbed activity after he had retired from University of Wales, Swansea in 1989 is worth mentioning. For instance, he chaired the UNESCO Chair of Numerical Methods in Engineering at University of Technology of Catalunya, Barcelona, Spain. He was an honorary member of the Polish Association for Computational Mechanics (PACM). In recognition of his achievements PACM founded in 2007 Prof. O.C. Zienkiewicz Medal for outstanding Polish and foreign

Foreword

XIII

scientists who contributed signiﬁcantly to the development of computational methods. Gliwice, May 2009 Barcelona, May 2009

Tadeusz Burczy´ nski, PACM President Eugenio O˜ nate, IACM President

Preface

This book contains the written versions of some general and keynote lectures given at the 18th International Conference on C omputer Methods in Mechanics, CMM 2009, which took place from 18 to 21 May 2009 at the University of Zielona G´ ora, Poland. This prestigious event was overshadowed by the death of one of the pioneers of computer sciences, the honorary chairman of the Conference, prof. dr. Olgierd Cecil Zienkiewicz, Professor Emeritus of the Swansea University (United Kingdom) who passed away on 2nd January 2009. His vivid memories were perceptible during the whole meeting in Zielona G´ ora. From thirty nine outstanding scientists presenting the general and keynote lectures at the conference, twenty seven decided to contribute their presentations in the form of articles to this volume. Among them are students or coworkers of Professor Zienkiewicz, a highly respected researcher and author. We all have greatly beneﬁted from his seminal works and activities. Therefore, we and the authors would like to pay tribute to the late Professor Zienkiewicz by dedicating the present book to his memory. The Conference was attended by 300 participants from 24 countries. It served both as a forum for the review and dissemination of recent scientiﬁc developments and technical information regarding all aspects of computer methods (250 presentations in 6 parallel sessions), and as a means to encourage cooperation and stimulate future research. The general lecturers and the keynote speakers are leading world-renowned scientists of this ﬁeld of science, carefully selected by the International and National Scientiﬁc Committees. We have divided the contributions, rather arbitrarily, into 6 groups which appear as Parts of the book. Since most of the contributed articles are interdisciplinary, we do not claim this classiﬁcation to be deﬁnite. However, we hope our grouping of papers will be helpful for readers who look for a particular issue in the ﬁeld of computer science. Part I, Mathematical Methods, encompasses works where emphasis is laid upon mathematical analysis of the considered models supported with numerical experiments for ﬁnite dimensional approximations. These works are concerned with wave propagation and coupled multiphysics problems,

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meshless methods, non-smooth non-convex frictional contact problems, shape and topology sensitivity analysis, boundary integral equation on a sphere, hp-adaptivity and adaptive hierarchical modelling. The tools used include ﬁnite, spectral, and boundary elements, variational equations and inequalities, Clarke subdiﬀerentials, hemivariational inequalities and evolution inclusions, as well as topological derivatives. A variety of types of materials were considered: elastic, poroelastic, viscoelastic, and piezoelectric, subjected to static or dynamic deformation processes. Part II, Soft Computing and Optimization, contains papers concerned with sensor network design and identiﬁcation problems, and application of immune computing in computational bioengineering and in structural optimization. The techniques used include bio-inspired approaches like Artiﬁcial Neural Networks, Evolutionary Methods, Particle Swarm Optimization as well as classical Finite Element Method and optimization algorithms, which allow for diﬀerent types of numbers (interval, fuzzy, random) and box constraints. In Part III, Multiscale Methods, we have collected the contributions which are strongly related to multilevel description. They combine molecular dynamics and an extended ﬁnite element method to simulate dynamic fracture, atomistic/continuum models to investigate crystal defects in semiconductor structures, or make use of the theory of Cosserat point to develop a 3-D brick element for ﬁnite elasticity, the particle ﬁnite element method to analyse coupled ﬂuid/structure interaction problems, and ﬁnally of a meso-macro homogenization procedure to study electro-mechanically coupled material behaviour. Part IV, Geomechanics, is primarily related to the numerical analysis of multicomponent and porous media. The physical coupling of solids and ﬂuids in those media requires special computational methods. These are a combination of percolation and forest ﬁre algorithms for modelling cementitious materials at early age, mesh-free and strain enhancement procedures for material point and low-order element methods to solve problems involving incompressibility, and enhancements of a micro-polar hypoplastic constitutive model for granular materials in the framework of ﬁnite element study of shear localization. Computational problems presented in Part V, Biomechanics, are even more special. These deal with tissue – implant interactions accounting for bone adaptation to mechanical stresses, sequential stages of the tooth-implant life cycle design process with focus on mechanical behavior and optimization of dental implants using ﬁnite elements and a genetic optimization algorithm, and mechanobiology to model the response of tissues to changes in their mechanical environment by combining ﬁnite elements and a lattice with algorithms for cell activities. Part VI, Structural Mechanics, features the most common application ﬁeld of computer mechanics in engineering sciences. This part is devoted to complex practical problems: unilateral frictional contact of beams, parameter indentiﬁcation of material/structural responses for real-life structures and

Preface

XVII

infrastructures based on a combination of computational and non-destructive measurements methods exempliﬁed by analyses of steel pipelines, large concrete dams, and of marine propulsion system’s alignment of aged ships, as well as assessment of impact-induced damage in laminate composites, and pre- and postbuckling of composite shells. The list of contributors, which is also included in the book, follows the alphabetic order organized according to the names of the speakers who presented a particular paper. Next to them the names of their co-authors are included as well. We would like to thank all authors for the cooperation on this book. At the same time, we would also like to express our gratitude to the members of the International and National Scientiﬁc Committees of CMM 2009 for their kind help in the reviewing procedures, as well as to the members of the Organizing Committee for their commitment to the organization of the Conference. Finally, we would like to thank Springer Verlag and particularly Dr. Dieter Merkle and Dr. Christoph Baumann, who kindly agreed to publish this book and further helped us to bring it to its present ﬁne printed form. We greatly appreciate the publishing of the book in the new series Advanced Structured Materials. Zielona G´ ora, Poland August 2009

Mieczyslaw Kuczma Krzysztof Wilma´ nski

Contents

Part I Mathematical Methods Explicit Discrete Dispersion Relations for the AcousticWave Equation in d-Dimensions Using Finite Element, Spectral Element and Optimally Blended Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mark Ainsworth, Haﬁz Abdul Wajid

3

hp-Adaptive Finite Elements for Coupled Multiphysics Wave Propagation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Leszek Demkowicz, Jason Kurtz, Frederick Qiu

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Nonconvex Inequality Models for Contact Problems of Nonsmooth Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stanislaw Mig´orski, Anna Ochal

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Quadrature for Meshless Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . John E. Osborn

59

Shape and Topology Sensitivity Analysis for Elastic Bodies with Rigid Inclusions and Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˙ Jan Sokołowski, Antoni Zochowski

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A Boundary Integral Equation on the Sphere for High-Precision Geodesy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ernst P. Stephan, Thanh Tran, and Adrian Costea

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Unresolved Problems of Adaptive Hierarchical Modelling and hp-Adaptive Analysis within Computational Solid Mechanics . . . . . . . . . . 111 Grzegorz Zboi´nski

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Contents

Part II Soft Computing and Optimization Granular Computing in Evolutionary Identiﬁcation . . . . . . . . . . . . . . . . . 149 Witold Beluch, Tadeusz Burczy´nski, Adam Długosz, Piotr Orantek Immune Computing: Intelligent Methodology and Its Applications in Bioengineering and Computational Mechanics . . . . . . . . . . . . . . . . . . . . . . 165 Tadeusz Burczy´nski, Michał Bereta, Arkadiusz Poteralski, Mirosław Szczepanik Bioinspired Algorithms in Multiscale Optimization . . . . . . . . . . . . . . . . . . 183 Wacław Ku´s, Tadeusz Burczy´nski Sensor Network Design for Spatio–Temporal Prediction of Distributed Parameter Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Dariusz Uci´nski Part III Multiscale Methods A Multiscale Molecular Dynamics / Extended Finite Element Method for Dynamic Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Pascal Aubertin, Julien R´ethor´e, Ren´e de Borst Nonlinear Finite Element and Atomistic Modelling of Dislocations in Heterostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Paweł Dłu˙zewski, Toby D. Young, George P. Dimitrakopulos, Joseph Kioseoglou, Philomela Komninou Accuracy and Robustness of a 3-D Brick Cosserat Point Element (CPE) for Finite Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 M. Jabareen, M.B. Rubin Possibilities of the Particle Finite Element Method in Computational Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Eugenio O˜nate, Sergio R. Idelsohn, Miguel Angel Celigueta, Riccardo Rossi, Salvador Latorre A Framework for the Two-Scale Homogenization of ElectroMechanically Coupled Boundary Value Problems . . . . . . . . . . . . . . . . . . . 311 J¨org Schr¨oder, Marc-Andr´e Keip Part IV Geomechanics Modeling Concrete at Early Age Using Percolation . . . . . . . . . . . . . . . . . . 333 Lavinia Stefan, Farid Benboudjema, Jean Michel Torrenti, Benoˆıt Bissonette

Contents

XXI

Simulation of Incompressible Problems in Geomechanics . . . . . . . . . . . . . 347 Dieter Stolle, Issam Jassim, Pieter Vermeer Effect of Boundary, Shear Rate and Grain Crushing on Shear Localization in Granular Materials within Micro-polar Hypoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 Jacek Tejchman Part V Biomechanics Biomechanical Basis of Tissue–Implant Interactions . . . . . . . . . . . . . . . . . 379 Romuald Bedzinski, Krzysztof Scigala Tooth-Implant Life Cycle Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 Tomasz Łodygowski, Marcin Wierszycki, Krzysztof Szajek, Wiesław He¸dzelek, Rafał Zagalak Predictive Modelling in Mechanobiology: Combining Algorithms for Cell Activities in Response to Physical Stimuli Using a Lattice-Modelling Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 Sara Checa, Damien P. Byrne, Patrick J. Prendergast Part VI Structural Mechanics The Beam-to-Beam Contact Smoothing with Beziers Curves and Hermites Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 Przemysław Litewka Synergic Combinations of Computational Methods and Experiments for Structural Diagnoses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 Giulio Maier, Gabriella Bolzon, Vladimir Buljak, Tomasz Garbowski, Bartosz Miller Optimization of Marine Propulsion System’s Alignment for Aged Ships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 Lech Murawski, Wieslaw Ostachowicz Experimental-Numerical Assessment of Impact-Induced Damage in Cross-Ply Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 Gigliola Salerno, Stefano Mariani, Alberto Corigliano, Francesco Caimmi, Luca Andena, Roberto Frassine Finite Element Modeling of Stringer-Stiffened Fiber Reinforced Polymer Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 Werner Wagner, Claudio Balzani Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

List of Contributors

Mark Ainsworth University of Strathclyde, 26 Richmond Street, Glasgow, G1 1XH, Scotland [email protected] Haﬁz Abdul Wajid University of Strathclyde, 26 Richmond Street, Glasgow, G1 1XH, Scotland [email protected] ´ Krzysztof Sciga la Division of Biomedical Engineering and Experimental Mechanics, Wroclaw University of Technology, ul. Lukasiewicza 7/9, 50-371 Wroclaw, Poland Romuald Bedzi´ nski Division of Biomedical Engineering and Experimental Mechanics, Wroclaw University of Technology, ul. Lukasiewicza 7/9, 50-371 Wroclaw,

Poland [email protected] Witold Beluch Department of Strength of Materials and Computational Mechanics, Silesian University of Technology, Gliwice, Poland [email protected] Tadeusz Burczy´ nski Department of Strength of Materials and Computational Mechanics, Silesian University of Technology, Gliwice, Poland, and Institute of Computer Modelling, Cracow University of Technology, Cracow, Poland [email protected] Adam Dlugosz Department of Strength of Materials and

XXIV

Computational Mechanics, Silesian University of Technology, Gliwice, Poland [email protected] Piotr Orantek Department of Strength of Materials and Computational Mechanics, Silesian University of Technology, Gliwice, Poland Sara Checa Trinity Centre for Bioengineering, School of Engineering, Trinity College, Dublin, Ireland Damien P. Byrne Trinity Centre for Bioengineering, School of Engineering, Trinity College, Dublin, Ireland Patrick J. Prendergast Trinity Centre for Bioengineering, School of Engineering, Trinity College, Dublin, Ireland [email protected] Alberto Corigliano Politecnico di Milano, Dipartimento di Ingegneria Strutturale, Piazza L. Da Vinci 32, 20133 Milano (Italy) [email protected] Luca Andena Politecnico di Milano,

List of Contributors

Dipartimento di Chimica, Materiali e Ingegneria Chimica ”Giulio Natta”, Piazza L. Da Vinci 32, 20133 Milano (Italy) [email protected] Francesco Caimmi Politecnico di Milano, Dipartimento di Chimica, Materiali e Ingegneria Chimica ”Giulio Natta”, Piazza L. Da Vinci 32, 20133 Milano (Italy) [email protected] Roberto Frassine Politecnico di Milano, Dipartimento di Chimica, Materiali e Ingegneria Chimica ”Giulio Natta”, Piazza L. Da Vinci 32, 20133 Milano (Italy) [email protected] Stefano Mariani Politecnico di Milano, Dipartimento di Ingegneria Strutturale, Piazza L. Da Vinci 32, 20133 Milano (Italy) [email protected] Gigliola Salerno Politecnico di Milano, Dipartimento di Ingegneria Strutturale, Piazza L. Da Vinci 32, 20133 Milano (Italy) [email protected] Ren´ e de Borst Eindhoven University of Technology, Department of Mechanical

List of Contributors

Engineering, P.O. Box 513, 5600 MB Eindhoven, Netherlands [email protected] Pascal Aubertin Universit´e de Lyon, CNRS INSA-Lyon, LaMCoS UMR 5259, France [email protected] Julien R´ ethor´ e Universit´e de Lyon, CNRS INSA-Lyon, LaMCoS UMR 5259, France [email protected] Leszek Demkowicz Institute for Computational Engineering and Sciences, University of Texas at Austin, ACES 6.326, 201 E. 24th Street, Austin, TX 78712, USA [email protected] Jason Kurtz Institute for Computational Engineering and Sciences, University of Texas at Austin, ACES 6.326, 201 E. 24th Street, Austin, TX 78712, USA [email protected] Frederick Qiu Institute for Computational Engineering and Sciences, University of Texas at Austin, ACES 6.326, 201 E. 24th Street, Austin, TX 78712, USA [email protected]

XXV

Pawel Dlu˙zewski Instytut Podstawowych Problemow Techniki PAN, ul. Swietokrzyska 21, 00-049 Warszawa, Poland [email protected] George P.Dimitrakopulos Department of Physics, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece Joseph Kioseoglou Department of Physics, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece Philomela Komninou Department of Physics, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece [email protected] Toby D. Young Instytut Podstawowych Problemow Techniki PAN, ul. Swietokrzyska 21, 00-049 Warszawa, Poland Waclaw Ku´ s Department for Strength of Materials and Computational Mechanics, Silesian University of Technology, ul. Konarskiego 18a, 44-100 Gliwice, Poland Waclaw:[email protected] Tadeusz S. Burczy´ nski Department for Strength of Materials and Computational Mechanics,

XXVI

Silesian, University of Technology, ul. Konarskiego 18a, 44-100 Gliwice, Poland [email protected] and Institute of Computer Modelling, Cracow University of Technology, ul. Warszawska 24, 31-155 Krak´ ow, Poland Przemyslaw Litewka Institute of Structural Engineering, Poznan University of Technology, ul. Piotrowo 5, 60-965 Poznan, Poland przemyslaw.litewka@ put.poznan.pl Tomasz L odygowski Department of Structural Mechanics, Poznan, University of Technology, ul. Piotrowo 5, 65 246 Poznan, Poland [email protected] Wieslaw H¸ edzelek University of Medical Sciences, ul. Fredry 10, 61-701 Poznan, Poland [email protected] Krzysztof Szajek Department of Structural Mechanics, Poznan, University of Technology, ul. Piotrowo 5, 65 246 Poznan, Poland [email protected]

List of Contributors

ul. Piotrowo 5, 65 246 Poznan, Poland [email protected] Rafal Zagalak Foundation of University of Medical Sciences, ul. Teczowa 3, 60-275 Poznan [email protected] Giulio Maier Department of Structural Engineering, Politecnico (Technical University) di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy [email protected] Gabriella Bolzon Department of Structural Engineering, Politecnico (Technical University) di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy [email protected] Vladimir Buljak Department of Structural Engineering, Politecnico (Technical University) di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy [email protected]

Tomasz Garbowski Department of Structural Engineering, Politecnico Marcin Wierszycki Department of Structural Mechanics, (Technical University) di Milano, piazza Leonardo da Poznan, University of Technology,

List of Contributors

XXVII

Vinci 32, 20133 Milano, Italy [email protected]

Gdansk, Poland [email protected]

Bartosz Miller Department of Structural Mechanics, Rzesz´ow University of Technology, ul. Pozna´ nska 2, 35-959 Rzesz´ow, Poland [email protected]

Eugenio O˜ nate International Center for Numerical Methods in Engineering (CIMNE) Universidad Politecnica de Cataluna, Campus Norte UPC, 08034 Barcelona, Spain [email protected]

Stanislaw Mig´ orski Jagiellonian University, Institute of Computer Science, Faculty of Mathematics and Computer Science, ul. Stanislawa L ojasiewicza 6, 30-348 Krakow, Poland {migorski,ochal}@ii.uj.edu.pl Anna Ochal Jagiellonian University, Institute of Computer Science, Faculty of Mathematics and Computer Science, ul. Stanislawa L ojasiewicza 6, 30-348 Krakow, Poland Lech Murawski Institute of Fluid Flow Machinery, Polish Academy of Sciences, ul. Fiszera 14, 80-952 Gdansk, Poland [email protected] Wieslaw Ostachowicz Institute of Fluid Flow Machinery, Polish Academy of Sciences, ul. Fiszera 14, 80-952

M.A. Celigueta International Center for Numerical Methods in Engineering (CIMNE) Universidad Politecnica de Cataluna, Campus Norte UPC, 08034 Barcelona, Spain S.R. Idelsohn ICREA Research Professor at CIMNE S. Latorre International Center for Numerical Methods in Engineering (CIMNE) Universidad Politecnica de Cataluna, Campus Norte UPC, 08034 Barcelona, Spain R. Rossi International Center for Numerical Methods in Engineering (CIMNE) Universidad Politecnica de Cataluna, Campus Norte UPC, 08034 Barcelona, Spain

XXVIII

John E. Osborn Department of Mathematics, University of Maryland, College Park, MD 20742, USA [email protected] Miles B. Rubin Faculty of Mechanical Engineering, Technion - Israel Institute of Technology, 32000 Haifa, Israel, [email protected] Mahmood Jabareen Faculty of Civil and Environmental Engineering, Technion - Israel Institute of Technology, 32000 Haifa, Israel, [email protected] J¨ org Schr¨ oder Institute for Mechanics, Faculty of Engineering Sciences, Department of Civil Engineering, University of Duisburg-Essen, Universit¨ atsstraße 15, 45141 Essen, Germany [email protected] Marc-Andr´ e Keip Institute for Mechanics, Faculty of Engineering Sciences, Department of Civil Engineering, University of Duisburg-Essen, Universit¨ atsstraße 15, 45141 Essen, Germany [email protected] Ernst P. Stephan Institute for Applied Mathematics, Leibniz Universit¨ at Hannover, Welfengarten 1, Hannover,

List of Contributors

Germany [email protected] Adrian Costea Institute for Applied Mathematics, Leibniz Universit¨at Hannover, Welfengarten 1, Hannover, Germany [email protected] Thanh Tran School of Mathematics and Statistics, Sydney 2052 , Australia [email protected] Dieter Stolle Department of Civil Engineering, McMaster University, Hamilton, Ontario, Canada, L8S4L7 [email protected] Issam Jassim Institute of Geotechnical Engineering, Stuttgart University, Pfaﬀenwaldring 35D, 70569 Stuttgart, Germany issam.jassim@ igs.uni-stuttgart.de Pieter Vermeer Institute of Geotechnical Engineering, Stuttgart University, Pfaﬀenwaldring 35D, 70569 Stuttgart, Germany pieter.vermeer@ igs.uni-stuttgart.de Jacek Tejchman Faculty for Civil and Environmental Engineering, Gdansk University

List of Contributors

of Technology, 80-952 Gda´ nsk-Wrzeszcz, Narutowicza 11/12, Poland [email protected] Jean Michel Torrenti LCPC, 58 boulevard Lefebvre, 75732 Paris cedex 15, France [email protected] Farid Benboudjema LMT, ENS Cachan, 61 Avenue du pr´esident Wilson, 94230 CACHAN, France Benoˆıt Bissonette D´epartement de G´enie Civil Pavillon Adrien-Pouliot local 2928B Universit´e Laval Qu´ebec, Canada, G1V 0A6 Lavinia Stefan LMT, ENS Cachan, 61 Avenue du pr´esident Wilson, 94230 CACHAN, France, and D´epartement de G´enie Civil, Pavillon Adrien-Pouliot, local 2928B, Universit´e Laval, Qu´ebec, Canada, G1V 0A6 [email protected] Dariusz Uci´ nski Institute of Control and Computation Engineering, University of Zielona G´ ora, ul. Licealna 9, 65–417 Zielona G´ora,

XXIX

Poland [email protected] Werner Wagner KIT – Karlsruhe Institute of Technology, Institute for Structural Analysis, Kaiserstr. 12, D-76131 Karlsruhe, Germany werner.wagner@ bs.uni-karlsruhe.de Claudio Balzani KIT – Karlsruhe Institute of Technology, Institute for Structural Analysis, Kaiserstr. 12, D-76131 Karlsruhe, Germany claudio.balzani@ bs.uni-karlsruhe.de Grzegorz Zboi´ nski Polish Academy of Sciences, Institute of Fluid Flow Machinery, ul. Fiszera 14, 80-952 Gda´ nsk, Poland, and University of Warmia and Mazury, Faculty of Technical Sciences, ul. Oczapowskiego 11, 10-736 Olsztyn, Poland [email protected] ˙ Antoni Zochowski Systems Research Institute of the Polish Academy of Sciences, ul. Newelska 6, 01-447 Warszawa, Poland [email protected] Jan Sokolowski Institut Elie Cartan,

XXX

UMR 7502 Nancy-Universit´e-CNRS-INRIA, Laboratoire de Math´ematiques, Universit´e Henri

List of Contributors

Poincar´e Nancy 1, B.P. 239, 54506 Vandoeuvre l´es Nancy Cedex, France [email protected]

Part I

Mathematical Methods

Chapter 1

Explicit Discrete Dispersion Relations for the Acoustic Wave Equation in d-Dimensions Using Finite Element, Spectral Element and Optimally Blended Schemes Mark Ainsworth and Hafiz Abdul Wajid

Abstract. We study the dispersive properties of the acoustic wave equation for finite element, spectral element and optimally blended schemes using tensor product elements defined on rectangular grid in d-dimensions. We prove and give analytical expressions for the discrete dispersion relations for the above mentioned schemes. We find that for a rectangular grid (a) the analytical expressions for the discrete dispersion error in higher dimensions can be obtained using one dimensional discrete dispersion error expressions; (b) the optimum value of the blending parameter is p/(p + 1) for all p ∈ N and for any number of spatial dimensions; (c) the optimal scheme guarantees two additional orders of accuracy compared with both finite and spectral element schemes; and (d) the absolute accuracy of the optimally blended scheme is O(p−3 ) and O(p−2 ) times better than that of the pure finite and spectral element schemes respectively.

1.1 Introduction Partial differential equations model many physical processes and real world problems posed over complex domains encountered in different fields of life. Most of the time it is not possible to obtain closed form solutions to these problems and Mark Ainsworth University of Strathclyde, 26 Richmond Street, Glasgow, G1 1XH, Scotland e-mail: [email protected] Hafiz Abdul Wajid University of Strathclyde, 26 Richmond Street, Glasgow, G1 1XH, Scotland e-mail: [email protected] Support of MA by the Engineering and Physical Sciences Research Council under grant EP/E040993/1 and of HAW by COMSATS Institute of Information Technology, Pakistan through a research studentship is gratefully acknowledged.

M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 3–17. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

4

M. Ainsworth and H. Abdul Wajid

numerical methods are widely used. Amongst the most common numerical methods, finite element [5] and spectral element [6] methods have been extensively used. The latter one is highly attractive as it can be used to solve problems with complex geometrical domains and attains the desired accuracy with less computational cost, thanks to the fact that the mass matrix is diagonal [16, 17]. Both finite and spectral element methods have been widely used to study wave propagation [8, 13, 14, 15] but fail to propagate waves with the correct physical speed [1, 3] and numerical dispersion is introduced. The finite element scheme results in phase lead whereas with the spectral element scheme phase lag is observed [3]. Mulder [19] studied the dispersive properties of the acoustic wave equation in one dimension using both finite and spectral element methods and concluded that spectral element methods with Gauss-Lobatto quadrature rules perform better than both the spectral element method with Chebyshev quadrature points and standard finite element methods. In [8] explicit expressions for the dispersion error were obtained using eigenvalues and Taylor series for low order elements for the transient wave equation. Basabe and Sen [9] studied dispersion in 2D with both finite and spectral elements for both acoustics and elastic wave equations. They provided analytical expressions for the dispersion error and stability conditions for first order elements and showed dispersion curves numerically for higher-order elements. The first detailed study of the dispersive properties of the standard finite element and spectral element methods valid for elements of arbitrary order appeared in [1, 3] where an extension of these schemes to any number of dimensions using tensor product meshes is also given. Numerical approximations obtained with linear finite and spectral elements lead and lag respectively with an equal magnitude of the dispersion error [3, 20]. Superior phase accuracy is obtained with the spectral element method compared with the finite element method for higher order elements. Marfurt [18] suggested that the most cost-effective scheme for computational wave propagation would be obtained by forming an appropriate weighted averaging of the spectral and finite element schemes. This idea was taken up in [4] where the optimal blending of the methods was obtained and shown to give an additional two orders of accuracy in the dispersion error. The case of first and second order elements had been previously considered by Fried [10]. Moreover, in [4] an equivalence between the blended scheme and non-standard quadrature rules was established. In particular this meant that the optimally blended scheme could be efficiently implemented merely by replacing the standard Gaussian quadrature rules by non-standard quadrature rules. Following an entirely different line of reasoning, Challa [7] arrived at the same non-standard quadrature rules in his thesis in the particular cases of linear and quadratic finite elements. Subsequently, Guddati and Yue [11, 12] studied such schemes for linear finite elements and commented on the relation with blended spectral-finite element schemes in the case of first order elements. In the present work we adopt the same approach used in [2] where we showed that the discrete dispersion relation in higher dimensions may be expressed in terms of the approximation of the scalar Helmholtz equation in one dimension. We use tensor product meshes to study the dispersive properties of the d-dimensional acoustic

1 Explicit Discrete Dispersion Relations

5

wave equation and obtain explicit expressions for the dispersion error for finite, spectral and the optimal blending of both finite and spectral element schemes in terms of the results in the special case of d = 1 dimensional approximation. The remainder of the chapter is organised as follows. In Section 1.2, we develop discrete dispersion relations using tensor product meshes valid for d-dimensions. In Section 1.3, discrete dispersion relations are derived for finite, spectral and optimally blended schemes. In the final section the numerical results obtained with these schemes are shown.

1.2 Acoustic Wave Equation Consider the acoustic wave equation in d-dimensions

∂ 2u − u = 0 ∂ t2

in Rd .

(1.1)

We seek time-harmonic solutions of the form u(x,t) = eiω t U(x) to the above equation, so that U satisfies the Helmholtz equation

ω 2U + U = 0

in Rd

(1.2)

where ω ∈ R is the given angular frequency.

1.2.1 Continuous Dispersion Relation Observe that in one dimension, the function u(ω ; x) = eiω x satisfies (u , v ) = ω 2 (u, v) where

1 ∀v ∈ Hloc (R)

(1.3)

1 Hloc (Rd ) = {v : Rd → R, v ∈ H 1 (Ω ) for all Ω ⊂⊂ Rd }

and (u, v) =

R

uvdx

is the L2 -inner product on R. We note that (1.3) is the variational formulation of (1.2) in one dimension. Furthermore, it is trivial to verify that the function u(ω ; x) satisfies u(ω ; x + nh) = eiω hn u(ω ; x) ∀n ∈ Z, x, h ∈ R which is the so called Bloch wave property . To obtain the dispersion relation in d-dimensions for the acoustic wave equation (1.2), we start with the variational formulation of equation (1.2) which is given by

6

M. Ainsworth and H. Abdul Wajid

ω2

Rd

Uvdx −

Rd

gradU · gradvdx = 0

1 ∀v ∈ Hloc (Rd ).

(1.4)

Now choose U and v as the product of uni-variate functions given by d

d

=1

=1

U(x1 , x2 , · · · , xd ) = ∏ u(ω ; x ) and v(x1 , x2 , · · · , xd ) = ∏ v (x ) 1 (R) and u(ω ; ·) where {ω }d=1 ∈ R are constants to be determined, {v }d=1 ∈ Hloc is defined above. Substituting U and v into (1.4), we get d

d

=1

r=1

ω 2 ∏ (u(ω ; ·), v ) − ∑ (u (ωr ; ·), vr ) ∏(u(ω ; ·), v ) = 0.

(1.5)

=r

Now, exploiting the identity (1.3), and performing straightforward manipulations, the above equation simplifies to d

ω 2 − ∑ ωr2 r=1

d

∏(u(ω; ·), v ) = 0,

(1.6)

=1

from which we see that non-trivial solutions of (1.6) exist only when the parameters {ω }d=1 satisfy ω 2 = ω12 + ω22 + · · · + ωd2 . (1.7) Equation (1.7) is the well known continuous dispersion relation of the acoustic wave equation (1.1) which is usually derived by inserting u directly into the differential equation (1.1).

1.2.2 Framework for Discrete Dispersion Relation To obtain the dispersion relation for the discrete case, we partition the real line R into infinitely many subintervals of uniform length h > 0 with nodes located at 1 (R) is the corresponding space of continuous piecewise hZ. The space Vhp ⊂ Hloc polynomials of degree p relative to the grid. We seek an approximation uhp ∈ Vhp such that (uhp , v ) − ω 2 uhp , v = 0 ∀v ∈ Vhp where ·, · is an appropriate discrete L2 -inner product on Vhp. Examples of suitable choices for ·, · will be given later, but we shall require that for v, w ∈ Vhp , v , w = (v , w ). To obtain the corresponding bilinear form in d-dimensions we consider the tensor product grid where each side of the grid has length h > 0, = 1, . . . , d. Let d ⊂ H 1 (Rd ) denote the space of continuous piecewise polynomials of degree p Vhp loc in each variable relative to the grid in d-dimensions, then we seek an approximate d such that udhp ∈ Vhp

1 Explicit Discrete Dispersion Relations

7

∇udhp , ∇v d − ω 2 udhp , v d = 0

d ∀v ∈ Vhp

(1.8)

where ·, · d is the tensor product bilinear form obtained from ·, · . Motivated by the arguments leading to the dispersion relation in the continuous case, we have the following theorem for the discrete dispersion relation. Theorem 1. Suppose there exists a non-trivial function uhp(ω ; ·) ∈ Vhp such that uhp (ω ; ·) has 1. the discrete Bloch wave property uhp (ω ; x + nh) = einhξ uhp (ω ; x),

∀n ∈ Z, x, h ∈ R

(1.9)

with discrete frequency ξ = ξ (ω ) and satisfies 2.

(uhp , v ) = ω 2 uhp , v ,

Let

∀v ∈ Vhp.

(1.10)

Ehp (ω ) = ξ 2 (ω ) − ω 2,

(1.11)

then the discrete dispersion relation for the acoustic wave equation in d-dimensions is given by d

ωh2 = ω 2 + ∑ Ehp (ω ).

(1.12)

=1

Proof. By analogy with the derivation of the dispersion relation in the continuous (d) case, we seek a non-trivial solution Uhp ∈ Vhp of the form: d

Uhp (x1 , x2 , · · · , xd ) = ∏ uhp (ω ; x )

(1.13)

=1

where {ω }d=1 ∈ R are again constants to be determined. The corresponding discrete variational formulation of (1.2) is given by (1.8). Now, substituting v of the form d

v = ∏ v (x )

(1.14)

=1

and Uhp from (1.13) into (1.8), we obtain

ω

2

d

d

=1

r=1

∏ uhp(ω ; ·), v − ∑

(uhp (ωr ; ·), vr )

∏ uhp(ω ; ·), v

=r

Now, exploiting the property (1.10), we get d 2 2 ω − ω1 + ω22 + · · · + ωd2 ∏ uhp (ω ; ·), v = 0 =1

which has a non-trivial solution only when

= 0.

(1.15)

8

M. Ainsworth and H. Abdul Wajid

ω 2 = ω12 + ω22 + · · · + ωd2 .

(1.16)

Now, consider d

Uhp (x1 + n1h1 , x2 + n2 h2 , · · · , xd + nd hd ) = ∏ uhp(ω ; x + nh ) =1

which, on using property (1.9), gives Uhp (x1 + n1 h1 , x2 + n2h2 , · · · , xd + nd hd ) = ei[h1 n1 ξ (ω1 )+h2 n2 ξ (ω2 )+···+hd nd ξ (ωd )]Uhp (x1 , x2 , · · · , xd ).

(1.17)

This is the discrete Bloch wave property for Uhp , and hence, the frequency ωh of the discrete solution satisfies

ωh2 = ξ (ω1 )2 + ξ (ω2 )2 + · · · + ξ (ωd )2 . Finally, upon using (1.11) together with (1.16) and after applying simple manipulations, the above equation gives (1.12) which is what the claimed result. Theorem 1 means that we can obtain the discrete dispersion relation for a scheme on a tensor product mesh in Rd using results for the discrete dispersion relation for the scheme in R1 . We use this result in the following section to analyse the finite element, spectral element and a novel, so-called optimally blended scheme that was introduced in [4].

1.3 Higher Order Discrete Dispersion Relations for Finite Element, Spectral Element and Optimally Blended Schemes in d-Dimensions In this section we will derive the explicit expressions of the discrete dispersion relations valid in d-dimensions for finite, spectral and optimally blended schemes of arbitrary order.

1.3.1 Standard Finite Element Scheme For finite elements we evaluate the stiffness and mass matrices using the Gaussian quadrature rule 1 −1

(p)

f (x)dx ≈ QG ( f ) =

p

∑ w f (ζ )

=0

(1.18)

1 Explicit Discrete Dispersion Relations

9

p where {ζ }=0 are the zeros of L p+1 and L p+1 is the (p + 1)-th order Legendre p polynomial. Moreover, weights {w }=0 are given by

2

w =

∀ ∈ {0, 1, . . ., p}.

(1 − ζ2)[Lp+1 (ζ )]2

(1.19)

The Gaussian quadrature rule (1.18) is exact for all polynomials of degree at most 2p + 1, and as a consequence 1 −1

1

u v dx = QG (u v ) and (p)

−1

(p)

uvdx = QG (uv) (p)

for all u, v ∈ P p . Now a composite quadrature rule IG on R given by (p)

R

f (x)dx ≈ IG ( f ) =

h 2

p

∑ ∑ w f (ζj,h )

(1.20)

j∈Z =0

h 1 j,h is constructed using (1.18) where ζ = ( j + )h + ζ , ∀ j ∈ Z and = 0, 1, . . . , p. 2 2 The discrete L2 -inner product is taken to be (p)

u, v G = IG (uv) and will be exact for u, v ∈ Vhp. Theorem 2. Let ω ∈ R be given. There exists a non-trivial uhp ∈ Vhp which satisfies (uhp , v ) = ω 2 (uhp , v),

∀v ∈ Vhp

(1.21)

and the Bloch wave property (1.9) with frequency ξ (ω ), where

p! ξ (ω ) = ω − (2p)! 2

2

2

h2p ω 2p+2 + O(h)2p+2. 2p + 1

(1.22)

Consequently, the discrete dispersion relation for finite elements in Rd is given by

ωh2 = ω 2 −

p! (2p)!

2

d 1 h2p ω 2p+2 + O(h)2p+2 ∑ 2p + 1 =1

(1.23)

where ω12 + ω22 + · · · + ωd2 = ω 2 . Proof. The existence of uhp is proved in Theorem 3.1 of [1] where it is also shown that p! 2 h2p ω 2p+2 2 2 + O(h)2p+2. ξ (ω ) = ω − (2p)! 2p + 1 Hence, applying Thorem 1, we obtain (1.23) at once.

10

M. Ainsworth and H. Abdul Wajid

1.3.2 Spectral Element Scheme The only difference for spectral elements compared with finite elements is the replacement of the Gaussian quadrature rule (1.18) with the Gauss-Lobatto quadrature (p) rule QGL defined by 1 −1

p

(p)

f (x)dx ≈ QGL ( f ) =

∑ w f (ζ˜ )

(1.24)

=0

p are taken to be the zeros of Lp (x)(1 − x2 ) with weights w given by where {ζ˜ }=0

w =

2

∀ ∈ {0, 1, . . . , p}.

p(p + 1)[L p(ζ˜ )]2

(1.25)

The Gauss-Lobatto quadrature rule (1.24) is exact for all polynomials of degree at most 2p − 1. Hence the stiffness matrix is integrated exactly 1 −1

u v dx = QGL (u v ) (p)

whereas the mass matrix is underintegrated 1 −1

(p)

uvdx ≈ QGL (uv).

We use this quadrature rule to develop a composite quadrature rule on R, which (p) we denote by IGL (·), following the same construction used in the case of finite elements. The discrete L2 -inner product is taken to be (p)

u, v L = IGL (uv). The only difference now is that the mass matrix will be under-integrated. Theorem 3. Let ω ∈ R be given. There exists a non-trivial uhp ∈ Vhp which satisfies (uhp , v ) = ω 2 uhp , v L ,

∀v ∈ Vhp

(1.26)

and the Bloch wave property (1.9) with frequency ξ (ω ), where

ξ (ω )2 = ω 2 +

p! 2 h2p ω 2p+2 1 + O(h)2p+2. p (2p)! 2p + 1

(1.27)

Consequently, the discrete dispersion relation for spectral elements in Rd is given by

ωh2 = ω 2 +

d p! 2 1 1 2p+2 + O(h)2p+2 ∑ h2p ω p (2p)! 2p + 1 =1

where ω12 + ω22 + · · · + ωd2 = ω 2 .

(1.28)

1 Explicit Discrete Dispersion Relations

11

Proof. The existence of uhp is established in Theorem 4.1 of [3] where it is also shown that p! 2 h2p ω 2p+2 1 + O(h)2p+2. ξ (ω )2 = ω 2 + p (2p)! 2p + 1 Equation (1.28) then follows at once from Theorem 1. Interestingly, from (1.28) it is clear that for higher orders the spectral element scheme provides p-times better phase accuracy as compared to the phase accuracy obtained with finite element scheme (1.23).

1.3.3 Optimally Blended Scheme We now apply Theorem 1 to a novel scheme introduced in [4] for the wave equation, whereby the finite and spectral element schemes are blended in such a way that the order of accuracy of the resulting discrete dispersion relation is optimised. If the blending parameter is denoted by τ ∈ [0, 1], then we base the blended scheme on the blended quadrature rule 1 −1 (p)

(p)

(p)

(p)

f (x)dx ≈ Qτ ( f ) = (1 − τ )QG ( f ) + τ QGL ( f ) (p)

where QG and QGL are the (p + 1)-point Gauss-Legendre and Gauss-LegendreLobatto quadrature rules defined in the previous sections and give us the standard finite and spectral element schemes for τ = 0 and τ = 1 respectively. Furthermore, (p) Qτ is the (p+1)-point non-standard quadrature rule given in [4] valid for elements p of arbitrary order with nodes {ζτ }=0 chosen as the zeros of L p+1 − τ L p−1 , where L p+1 and L p−1 are the Legendre polynomials of degrees p + 1 and p − 1 respectively, with weights given by w =

2[p(1 + τ ) + τ ] , ∀ = 0, 1, . . . , p. p(p + 1)L p(ζτ )[Lp+1 (ζτ ) − τ Lp−1 (ζτ )] (p)

Furthermore, Qτ

(1.29)

satisfies the following identity [4]

(p)

(p)

(p)

Qτ ( f ) = (1 − τ )QG ( f ) + τ QGL ( f )

∀ f ∈ P2p+1

(1.30)

and is exact for all polynomials of degrees at the most 2p − 1. We use this quadrature (p) rule to develop a composite quadrature rule on R, which we denote by Iτ (·), and follow the same construction used in the previous sections for finite and spectral element schemes. The discrete L2 -inner product is taken to be (p)

u, v τ = Iτ (uv). Once again the mass matrix is under-integrated.

12

M. Ainsworth and H. Abdul Wajid

Theorem 4. Let ω ∈ R be given. There exists a non-trivial uhp ∈ Vhp which satisfies (uhp , v ) = ω 2 uhp , v τ ,

∀v ∈ Vhp

(1.31)

and the Bloch wave property (1.9) with frequency ξ (ω ), where

p! 2 h2p ω 2p+2 1 −1 + O(h2p+2). ξ (ω ) = ω + τ 1 + p (2p)! 2p + 1 2

2

(1.32)

Consequently the discrete dispersion relation for optimally blended scheme in Rd is given by

d 1 p! 2 1 2p 2p+2 −1 ωh2 = ω 2 + τ 1 + h ω ∑ p (2p)! 2p + 1 =1

(1.33)

where ω12 + ω22 + · · · + ωd2 = ω 2 . Proof. The existence of uhp is proved in Theorem 3.1 of [4] where it is also shown that

p! 2 h2p ω 2p+2 1 −1 + O(h2p+2). ξ (ω )2 = ω 2 + τ 1 + (1.34) p (2p)! 2p + 1 Now applying Theorem 1, we obtain (1.33) at once. It is not difficult to check that the above expressions leads to expression (1.23) for τ = 0 and (1.28) for τ = 1 which are the discrete dispersion relations corresponding to finite and spectral element schemes respectively. More importantly, the first term in expression (1.33) vanishes if we choose blending parameter τ = p/(p + 1) which shows that the optimal blending parameter is independent of the number of spatial dimensions. Theorem 4 gives rise to the following corollary. Corollary 1. Let p ≥ 2. Then for the optimal choice of the blending parameter τ = p/(p + 1), the error in the discrete dispersion relation (1.33) is given by

ωh2 = ω 2 +

d 8 (p + 1)! 2 1 h2p+2ω2p+4 + O(h2p+4). ∑ (2p − 1) (2p + 2)! 2p + 3 =1

Proof. Substituting τ = p/(p + 1) in (1.33) and applying trivial manipulations gives us the required result. Whilst the cost of all of the schemes is virtually identical, remarkably the leading error term for the optimal scheme is two orders more accurate compared with the standard spectral and finite element schemes given in the previous sections. Moreover, the coefficient of the leading term in the error obtained with the blended scheme for the optimum value of τ is −2/(4p2 − 1)(2p + 3) and 2p/(4p2 − 1)(2p + 3) times

1 Explicit Discrete Dispersion Relations

13

better compared with the leading terms in the error obtained with finite and spectral element schemes respectively. 2

1.5

Exact Solution Finite Element Solution Spectral Element Solution Optimal Solution

1.5

Exact Solution Finite Element Solution Spectral Element Solution Optimal Solution

1

1 0.5

0

u

u

0.5

−0.5

0

−0.5

−1 −1 −1.5 −1.5

−2 0

0.5

1

1.5 x

2

2.5

3

1.3

1.35

1.4

1.45 x

(a)

1.5

1.55

1.6

(b)

1.5

3 Exact Solution Finite Element Solution Spectral Element Solution Optimal Solution

Exact Solution Finite Element Solution Spectral Element Solution Optimal Solution

2.5

1

2 1.5

u

1

u

0.5

0

0.5 0 −0.5

−0.5

−1 −1.5

−1 0

0.1

0.2

0.3

0.4

0.5 x

0.6

0.7

0.8

0.9

1

−2 2

2.1

2.2

(c)

2.3

2.4

2.5 x

2.6

2.7

2.8

2.9

3

(d)

Fig. 1.1 Numerical approximations of the solution to equation (1.35) obtained for p = 1 with ω = 30 and ρ = 9. Furthermore 35 and 300 elements are used outside and inside the slab respectively.

1.4 Numerical Examples In order to study the behaviour of finite, spectral and optimally blended schemes in practical computations, we consider a simple one dimensional scattering problem on the interval Ω = (0, 3) with fixed ω ∈ R, and ωs ∈ R given by −u − ω 2 (x)u = f (x) where

ω (x) =

ω , for x ∈ / (1, 2), and f (x) = ωs , for x ∈ (1, 2)

(1.35)

0, for x ∈ / (1, 2), (ω 2 − ωs2 )eiω x , for x ∈ (1, 2)

14

M. Ainsworth and H. Abdul Wajid

with the following non-reflecting boundary conditions applied at both ends of the domain u (0) + iω u(0) = 0 and u (3) − iω u(3) = 0. Evidently, the model problem corresponds to scattering of an incoming plane wave by a slab of relative density ωs2 /ω 2 located on (1, 2). In Figure 1.1 (a), we approximate scattered wave using 35 and 300 linear elements outside and inside the slab respectively for spectral, finite and optimal schemes with given frequency ω = 30 and relative density ρ = 9. Scattered waves on the left and right side of the slab are shown in Figure 1.1 (c) − (d) to analyse better the numerical approximations obtained with all the schemes. The phase lead and lag of equal magnitudes are clearly visible and correspond to finite and spectral element schemes which is consistent with error expressions given in (1.23) and (1.28). The same observation was made in [3, 4, 20] in the case of linear elements. Furthermore the numerical approximation corresponding to the optimal scheme is noticeably better than that of finite and spectral element schemes which was also observed in [4]. Figure 1.1 (b), represents the scattered wave inside the slab and once again optimal scheme performs better than that of finite and spectral element schemes nonetheless phase lead and lag of equal magnitudes with linear elements are still prominent even inside the slab. In 2

2 Exact Solution Finite Element Solution Spectral Element Solution Optimal Solution

1.5

1

1

0.5

0.5

0

0

u

u

1.5

−0.5

−0.5

−1

−1

−1.5

−1.5

−2 0

0.5

1

1.5 x

(a)

2

2.5

3

−2 1

Exact Solution Finite Element Solution Spectral Element Solution Optimal Solution

1.1

1.2

1.3

1.4

1.5 x

1.6

1.7

1.8

1.9

2

(b)

Fig. 1.2 Numerical approximations of the solution to equation (1.35) obtained for p = 2 with ω = 10 and ρ = 49. Furthermore 10 and 30 elements are used outside and inside the slab respectively.

Figures 1.2 and 1.3, we show numerical approximations obtained for all the schemes using quadratic and cubic elements. It is clear from Figures 1.2(a) and 1.3(a) that with piecewise quadratic and cubic elements both spectral and optimal schemes are performing much better than that of finite element scheme. This conjecture is consistent with analytical results (1.28) and (1.33) of dispersion error obtained for spectral and optimal schemes. The magnitude of the leading order error term for spectral and optimal schemes are O(p−1 ) and O(p−3 ) times better than that of the pure finite element scheme. Moreover, the numerical approximation obtained with finite element scheme is unresolved both for quadratic and cubic elements in each

1 Explicit Discrete Dispersion Relations

15

2

2 Exact Solution Finite Element Solution Spectral Element Solution Optimal Solution

1.5

1

1

0.5

0.5

0

u

u

1.5

−0.5

0

−0.5

−1

−1

−1.5

−1.5

−2 0

Exact Solution Finite Element Solution Spectral Element Solution Optimal Solution

0.5

1

1.5 x

2

2.5

−2 1

3

1.1

1.2

1.3

1.4

(a)

1.5 x

1.6

1.7

1.8

1.9

2

(b)

Fig. 1.3 Numerical approximations of the solution to equation (1.35) obtained for p = 3 with ω = 10 and ρ = 115. Furthermore 10 and 30 elements are used outsides and inside the slab respectively.

region. The same conjecture is observed even inside the slab which is presented in Figures 1.2(b) and 1.3(b) for quadratic and cubic elements respectively. In Figure 2

2 Exact Solution Finite Element Solution Spectral Element Solution Optimal Solution

1.5

1

1

0.5

0.5

0

u

u

1.5

−0.5

0

−0.5

−1

−1

−1.5

−1.5

−2 0

Exact Solution Finite Element Solution Spectral Element Solution Optimal Solution

0.5

1

1.5 x

(a)

2

2.5

3

−2 0

0.5

1

1.5 x

2

2.5

3

(b)

Fig. 1.4 Numerical approximations of the solution to equation (1.35) obtained with ω = 30 and ρ = 9 for (a) 35 linear and 50 cubic elements (b) 5 quartic and 15 fifth order elements used outside and inside the slab respectively.

1.4, we show the effect of using polynomials of different orders in different regions. In Figure 1.4(a), we show numerical results approximated outside the slab with first order (p = 1) elements whereas cubic elements are used inside the slab. We use the same number of elements i.e. n1 = n3 = 35 outside the slab as we used in Figure 1.1 but inside the slab using n2 = 50 cubic elements instead of 300 linear elements gives us much better results but phase leads and lags of equal magnitude are visible outside the slab as we are using linear elements there. Now using n1 = n3 = 5 quartic elements outside the slab and n2 = 15 elements of fifth order provides very accurate

16

M. Ainsworth and H. Abdul Wajid 1.5

1

0.5

0.5

0

0

u

u

1

1.5 Exact Solution Finite Element Solution Spectral Element Solution Optimal Solution

−0.5

−0.5

−1

−1

−1.5 0

0.5

1

1.5 x

2

2.5

−1.5 0

3

Exact Solution Finite Element Solution Spectral Element Solution Optimal Solution

0.5

1

(a)

1.5 x

2

2.5

3

(b)

Fig. 1.5 Numerical approximations of the solution to equation (1.35) obtained with ω = 10 and ρ = 9 using 10 linear elements outside the slab and using (a) ten cubic (b) ten fifth order elements inside the slab.

2.5 2

2 Exact Solution Finite Element Solution Spectral Element Solution Optimal Solution

1.5

Exact Solution Finite Element Solution Spectral Element Solution Optimal Solution

1.5 1 1 0.5

0

u

u

0.5

−0.5

0

−0.5

−1 −1 −1.5 −1.5

−2 −2.5 0

0.5

1

1.5 x

(a)

2

2.5

3

−2 0

0.5

1

1.5 x

2

2.5

3

(b)

Fig. 1.6 Numerical approximations of the solution to equation (1.35) obtained with ω = 10 and ρ = 10 for (a) 10 and 50 (b) 200 and 50 linear elements used outside and inside the slab respectively.

results as shown in Figure 1.4 (b). In Figure 1.5, we show that when the waves are fully resolved inside the slab than increasing the order or increasing the number of elements inside the slab does not help the waves outside the slab to converge. Hence when the waves are almost resolved inside the slab then waves outside the slab can be resolved either increasing the number of elements or using the higher order elements. Now consider the case where the wave is not resolved inside the slab then increasing the number of elements or using the higher order elements do not give us a completely resolved wave where this behaviour is shown in Figure 1.6.

1 Explicit Discrete Dispersion Relations

17

References [1] Ainsworth, M.: Discrete dispersion relation for hp-version finite element approximation at high wave number. SIAM J. Numer. Anal. 42(2), 553–575 (2004) (electronic) [2] Ainsworth, M.: Dispersive properties of high-order Nédélec/edge element approximation of the time-harmonic Maxwell equations. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 362(1816), 471–491 (2004) [3] Ainsworth, M., Wajid, H.A.: Dispersive and dissipative behaviour of the spectral element method. SIAM J. Numer. Anal. (accepted, 2009) [4] Ainsworth, M., Wajid, H.A.: Optimally blended spectral-finite element scheme for wave propagation, and non-standard reduced integration. SIAM J. Numer. Anal. (submitted, 2009) [5] Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. In: Texts in Applied Mathematics, 3rd edn., vol. 15. Springer, New York (2008) [6] Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral methods. In: Evolution to complex geometries and applications to fluid dynamics. Springer, Berlin (2007) [7] Challa, S.: High-order accurate spectral elements for wave propagation. Masters thesis in mechanical engineering, Clemson University (1998) [8] Cohen, G.C.: Higher-order numerical methods for transient wave equations. In: Scientific Computation. Springer, Heidelberg (2002) With a foreword by R. Glowinski [9] De Basabe, J., Sen, M.: Grid dispersion and stability criteria of some common finiteelement methods for acoustic and elastic wave equation. Geophysics 72(6), T81–T95 (2007) [10] Fried, I., Chavez, M.: Superaccurate finite element eigenvalue computation. Journal of sound and vibration 275, 415–422 (2004) [11] Guddati, M., Yue, B.: Modified integration rules for reducing dispersion error in finite element methods. Comput. Methods Appl. Mech. Engrg. 193, 275–287 (2004) [12] Guddati, M., Yue, B.: Dispersion-reducing finite elements for transient acoustics. J. Acoust. Soc. Am. 118(4), 2132–2141 (2005) [13] Ihlenburg, F.: Finite element analysis of acoustic scattering. In: Applied Mathematical Sciences, vol. 132. Springer, New York (1998) [14] Ihlenburg, F., Babuška, I.: Finite element solution of the Helmholtz equation with high wave number. I. The h-version of the FEM. Comput. Math. Appl. 30(9), 9–37 (1995) [15] Ihlenburg, F., Babuška, I.: Finite element solution of the Helmholtz equation with high wave number. II. The h-p version of the FEM. SIAM J. Numer. Anal. 34(1), 315–358 (1997) [16] Komatitsch, D., Tromp, J.: Spectral-element simulations of global seismic wave propagation-I. Validation. Geophys. J. Int. 149(2), 390–412 (2002) [17] Komatitsch, D., Tromp, J.: Spectral-element simulations of global seismic wave propagation-II. 3-D models, oceans, rotation, and self-gravitation. Geophys. J. Int. 150(1), 303–318 (2002), doi:10.1046/j.1365-246X.2002.01716.x [18] Marfurt, K.J.: Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations. Geophysics 49(5), 533–549 (1984) [19] Mulder, W.A.: Spurious modes in finite-element discretizations of the wave equation not be all that bad. Appl. Numer. Math. 30(4), 425–445 (1999) [20] Thompson, L.L., Pinsky, P.M.: Complex wavenumber Fourier analysis of the p-version finite element method. Comput. Mech. 13(4), 255–275 (1994)

Chapter 2

hp-Adaptive Finite Elements for Coupled Multiphysics Wave Propagation Problems Leszek Demkowicz, Jason Kurtz, and Frederick Qiu

Abstract. The paper describes a generalization of hp-adaptive finite elements technology to coupled multiphysics problems. Three representative examples are used: dual-mixed formulation with weakly imposed symmetry for linear elasticity, coupled acoustics/viscoelasticity and coupled acoustics/poroelasticity problem, to illustrate variational formulations and concept of weak couplings. We discuss then necessary changes in data structures, constrained approximation and the hp mesh optimization algorithm. Sample numerical results for the three problems illustrate the new methodology.

2.1 Introduction The hp-adaptive Finite Elements have been successfully applied to a large number of wave propagation problems including time-harmonic acoustics and electromagnetics, see [3, 5] for numerous examples. The energy-, or goal-oriented automatic hp-adaptivity enables solution of challenging problems with large dynamic range and strong material contrasts resulting in point and edge (3D) singularities and boundary layers. In this paper, we report on an extension of the hp-technology to coupled, multiphysics wave propagation problems. Discretization of multiphysics problems necessitates frequently the simultaneous use of H 1-, H(curl)-, H(div)- and L2-conforming1 elements that enforce different continuity across interelement boundaries. In the classical H 1 -conforming discretizations FE solution is globally continuous. In contrast, the L2 -conformity implies no condition on continuity across interelement boundaries whatsoever. Contrary to H 1 - and L2 -conforming discretizations that are based on use Leszek Demkowicz, Jason Kurtz, and Frederick Qiu Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712, USA e-mail: [email protected], [email protected], [email protected] 1

Frequently referred to as continuous, edge, face and discontinuous elements.

M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 19–42. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

20

L. Demkowicz, J. Kurtz, and F. Qiu

of scalar-valued shape functions, the H(curl)- and H(div)-conforming discretizations use vector-valued shape functions, and enforce a partial continuity across the interelement boundaries: tangential components of H(curl)- and normal components of H(div)-conforming vector fields are globally continuous. These continuity requirements are not merely a mathematical construction but they reflect the worst possible scenarios in the case of nonhomogeneous data, e.g. a jump in dielectric constant results in a discontinuity of normal component of electric field across the material interface. The properly constructed H 1 -, H(curl)-, H(div)- and L2 -conforming FE element spaces form the so-called exact sequence, both at a single element and the whole finite element mesh levels (including curvilinear elements). The exact sequence structure is absolutely crucial for constructing stable discretizations of multiphysics problems including examples discussed in this paper. Our new 2D hp code [4] uses a single hp mesh that supports the use of H 1 -, H(curl)-, H(div)- and L2 -conforming elements forming the exact sequence. The polynomial order for H 1 - conforming elements implies the corresponding order for the other involved elements. Additionally, the nature of coupled problems calls for the possibility of supporting different fields over different parts of the domain. In the discussed implementation we have restricted ourselves only to the so-called weak couplings where coupling of different physical models results in the presence of integrals over the interface boundaries. (Hybrid of mortar methods are examples of more general techniques.) From the point of view of data structures, this means that different fields may overlap at interfaces. The structure of the paper is as follows. We begin in Section 2.2 with a few examples of coupled multiphysics problems illustrating the points made in this introduction. Section 2.3 is devoted to a short discussion of our implementation including data structures, constrained approximation, and a generalization of hp algorithm to the coupled multiphysics problems. In Section 2.4, we present a few illustrative numerical results for sample problems from Section 2.2. A short discussion in Section 2.5 concludes the presentation.

2.2 Examples of Coupled Multiphysics Problems We start with the classicial example of linear elasticity.

2.2.1 Dual Mixed Formulation for Elasticity with Weakly Imposed Symmetry We begin with the time-harmonic linear elasticity problem. Let Ω ⊂ R I n , n = 2, 3, denote a bounded domain occupied by the elastic body with the boundary Γ = ∂ Ω split into two disjoint subsets Γ1 and Γ2 .

2 hp-Adaptive for Multiphysics Wave Propagation

21

We seek: • displacement vector ui (x), x ∈ Ω , • linearized strain tensor εi j (x), x ∈ Ω , • stress tensor σi j (x), x ∈ Ω that satisfy the following system of equations and boundary conditions. • Cauchy’s geometrical relation between the displacement and strain,

εi j = 12 (ui, j + u j,i ),

x∈Ω

• Equations of motion resulting from the principle of linear momentum, −σi j, j − ρ (x)ω 2 ui = fi (x),

x∈Ω

• Symmetry of the stress tensor being a consequence of the principle of angular momentum, σi j = σ ji , x ∈ Ω , • The constitutive equations of linear elasticity,

σi j = Ei jkl (x)εkl ,

x∈Ω

• kinematic boundary conditions, ui = u0i ,

x ∈ Γ1

• Traction boundary conditions,

σ ji n j = gi (x),

x ∈ Γ2

Here: • • • • • •

ρ is the density of the body, fi are volume forces prescribed within the body, gi are tractions prescribed on Γ2 part of the boundary, u0i are displacements prescribed on Γ1 part of the boundary, n j is the unit outward normal vector for boundary Γ Ei jkl = μ (δik δ jl + δil δ jk ) + λ δi j δkl is the tensor of elasticities for an isotropic solid, where μ , λ are the Lamé constants.

The constitutive equation can be inverted to represent strains in terms of stresses,

εkl = Ckli j σi j where Ckli j = Ei−1 jkl is the compliance tensor. Of particular interest is the case of nearly incompressible material corresponding to λ → ∞. Notice that the norm of the elasticities blows up then to infinity but the norm of the compliance tensor remains uniformly bounded. This suggests that

22

L. Demkowicz, J. Kurtz, and F. Qiu

formulations based on the compliance relation have a chance to remain uniformly stable for nearly incompressible materials.

Classical, displacement-based formulation The discussed equations can be treated in a strong, pointwise sense, or can be interpreted in the sense of distributions. The equations understood in the strong sense imply then the possibility of eliminating some of the unknowns. Different choices lead to different ultimate variational formulations, and different energy spaces. In the classical formulation corresponding to the Lamé equations, the momentum equations are treated in a distributional sense. In other words, we multiply the momentum equations with a test function vi (sum up with respect to i), integrate over domain Ω , and integrate the first term by parts, to obtain

Ω

σi j vi, j −

Γ

σi j n j vi − ω 2

Ω

ρ u i vi =

Ω

f i vi

The symmetry of the stress tensor implies that the first term can be rewritten in the form, σi j vi, j = σi j 12 (vi, j + v j,i ) + σi j 12 (vi, j − v j,i ) = σi j εi j (v) where by εi j (v) we understand the strain tensor (symmetric part of the gradient) corresponding to test function vi . All the remaining equations are treated in the strong sense. We use the strain-displacement relations to eliminate the strain tensor, and the constitutive equation to eliminate the stress tensor in the domain integral. We obtain,

Ω

Ei jkl εi j (u)εkl (v) −

Γ

σi j n j vi − ω 2

Ω

ρ u i vi =

Ω

f i vi

This leads to the H 1 energy setting for displacement ui . The displacement boundary conditions can then be understood in the sense of the Trace Theorem. The same energy setting is used for the test functions. We assume that the test functions vanish (in the sense of traces) on Γ1 , and reduce the boundary term that has resulted from integration by parts, to Γ2 part only. Finally, we “build in” the traction boundary condition into the formulation assuming that

Γ2

σi j n j vi =

Γ2

g i vi ,

for all eligible test functions. The traction boundary condition is thus satisfied also in a weak sense only. The ultimate formulation looks as follows.

2 hp-Adaptive for Multiphysics Wave Propagation

23

⎧ ⎪ ui ∈ H 1 (Ω ), ui = u0i on Γ1 ⎪ ⎪ ⎪ ⎪ ⎨ Ei jkl εi j (u)εkl (v) − ω 2 ρ u i vi = f i vi + g i vi ⎪ Ω Ω Ω Γ2 ⎪ ⎪ ⎪ ⎪ ∀vi ∈ H 1 (Ω ) : vi = 0 on Γ1 ⎩ In the static case ω = 0, the variational problem reduces to the classical Principle of Virtual Work, and it is equivalent to the problem of minimizing the total potential energy (Lagrange’s Theorem). The H 1 energy setting leads to the use of classical continuous (H 1 -conforming) finite element discretization. The classical dual formulation for the static problem in terms of stresses leading to the Castigliano’s Principle (maximization of complementary energy) is expressed in terms of stresses σi j that satisfy a-priori the equilibrium equations in the strong sense. The principle does not lead to practical discretization schemes as the assumption cannot be satisfied easily for piece-wise polynomial functions.

Dual formulation We start by eliminating the strain tensor from the constitutive equation in the compliance form, Ci jkl σkl = 12 (ui, j + u j,i ) In this alternative, dual approach, we solve the momentum equations in the strong sense, but treat the combined constitutive/geometry relations above in a distributional way. We multiply the equation with a symmetric tensor-valued test function τi j , integrate over Ω , and integrate by parts to obtain,

Ω

Ci jkl σkl τi j = −

Ω

ui τi j, j +

Γ

ui τi j n j

We treat the momentum equations in a strong form and solve them for the displacements, 1 ui = [−σi j, j − fi ] ρω 2 Substituting the expression into the weak statement, we get,

Ω

Ci jkl σkl τi j = −

Ω

1 [−σi j, j − fi ] τi j, j + ρω 2

Γ

ui τi j n j

We restrict ourselves now to stress fields that satisfy the traction boundary conditions in a strong sense and assume the corresponding homogeneous boundary conditions for test functions, τi j n j = 0 on Γ2 Finally, substituting the kinematic boundary condition into the boundary term, we arrive at the alternative variational formulation,

24

L. Demkowicz, J. Kurtz, and F. Qiu

⎧ σi j n j = gi on Γ2 ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

Ω

Ci jkl σkl τi j −

Ω

1 σi j, j τi j, j = ρω 2

Ω

1 fi τi j, j + u0i τi j n j ρω 2 Γ1 ∀τi j : τi j n j = 0 on Γ2

The formulation leads to the energy setting:

σ , τ ∈ H(div, Ω , S) where H(div, Ω , S) denotes the space of square integrable symmetric tensor-valued functions, whose divergence is also in L2 . The dual formulation degenerates as ω → 0 and it does not cover the static case.

Dual-mixed formulation The idea is based on satisfying the momentum equations in the strong sense, but without eliminating the displacements. We multiply the momentum equations with a test function vi and integrate over Ω to obtain2, −

Ω

σi j, j vi − ω 2

Ω

ρ u i vi =

Ω

f i vi

The final formulation reads as follows. ⎧ σ ∈ H(div, Ω , S) : σi j n j = gi on Γ2 , u ∈ L2 (Ω , V) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Ci jkl σkl τi j + ui τi j, j = u0i τi j n j ∀τ ∈ H(div, Ω , S) : τi j n j = 0 on Γ2 Ω Ω Γ1 ⎪ ⎪ ⎪ ⎪ 2 ⎪ ρ u i vi = f i vi ∀v ∈ L2 (Ω , V) ⎩ − σi j, j vi −ω Ω

Ω

Ω

Here L2 (Ω , V) denotes the square integrable vector-valued functions. The traction conditions in both dual and dual-mixed formulations are satisfied in the sense of traces for functions from H(div, Ω ) (they live in H −1/2(Γ )). The corresponding static case can be derived formally by considering the socalled Hellinger-Reissner functional, and the variational formulation is frequently identified as the Hellinger-Reissner variational principle.

Dual-mixed formulation with weakly imposed symmetry The symmetry condition is difficult to enforce on the discrete level. This has led to the idea of relaxing the symmetric function and satisfying it in a weaker, integral 2

Notice that we do not integrate by parts.

2 hp-Adaptive for Multiphysics Wave Propagation

25

form. This is obtained by introducing tensor-valued test functions q with values in the space of antisymmetric tensors K := {qi j : qi j = −q ji }, and replacing the symmetry condition with an integral condition,

Ω

σi j qi j = 0,

∀q ∈ L2 (Ω , K)

On the continuous level, the integral condition implies the pointwise condition (understood in the L2 sense), but on the discrete level, with an appropriate choice of spaces, the integral condition does not necessary imply the symmetry condition pointwise. The extra condition leads to an extra unknown. The derivation of the weak form of the constitutive equation has to be revisited. We start by introducing the tensor of infinitesimal rotations, pi j = 12 (ui, j − u j,i ) Upon eliminating the strain tensor, the constitutive equation in the compliance form is now rewritten as, Ci jkl σkl = 12 (ui, j + u j,i) = ui, j − pi j Multiplication with a test function τ (now, not necessarily symmetric), integration over Ω , and integration by parts, leads to a new relaxed version of the equation,

Ω

Ci jkl σkl τi j = −

Ω

ui τi j, j +

Γ

ui τi j n j −

Ω

pi j τi j

After similar considerations as before, we obtain a new variational formulation in the form: ⎧ σ ∈ H(div, Ω , M) : σi j n j = gi on Γ2 , u ∈ L2 (Ω , V), p ∈ L2 (Ω , K) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ci jkl σkl τi j + ui τi j, j + pi j τi j = u0i τi j n j ⎪ ⎪ ⎪ Ω Ω Ω Γ ⎪ 1 ⎨ ∀τ ∈ H(div, Ω , M) : τi j n j = 0 on Γ2 ⎪ ⎪ ⎪ ⎪ ⎪ − σi j, j vi −ω 2 ρ u i vi = f i vi ∀v ∈ L2 (Ω , V) ⎪ ⎪ ⎪ Ω Ω Ω ⎪ ⎪ ⎪ ⎪ ⎩ σi j qi j =0 ∀q ∈ L2 (Ω , K) Ω

In the above, H(div, Ω , M) denotes the space of (general) tensor-valued square integrable fields with square integrable divergence. The formulation can be derived formally by looking for a stationary point of the so-called Generalized HellingerReissner functional, and it is frequently identified as the Generalized HellingerReissner Variational Principle. Summarizing, in two space dimensions, the dual-mixed variational formulation with the weakly imposed symmetry involves the use of two H(div)-conforming

26

L. Demkowicz, J. Kurtz, and F. Qiu

fields representing rows of the stress tensor σi, j , j = 1, 2, i = 1, 2, and three L2 conforming fields representing two components of the displacement vector ui , and scalar variable p representing the antisymmetric tensor of infinitesimal rotations

0p pi j = −p 0 All fields are supported over the whole domain Ω . Notice that the problem is symmetric.

2.2.2 Acoustics Coupled with (visco-)Elasticity In this example, we present a coupled acoustics/visco-elasticity problem encountered in modeling of streamers. Streamers are several kilometer long hollow plastic cylinders strengthened with a couple of ropes (strength elements) and filled with a (very) soft viscoelastic “filler”. Placed in the filler is a periodic array of spacers (made of hard plastic) housing hydrophones. Periodically spaced buoyancy system (the “birds”) maintains the streamer floating below the surface of a sea. An array of six to ten streamers, deployed on the sea, and pulled by a tugboat is used then to record acoustical waves initiated near the surface and reflected from layers of the ocean floor. The recorded signals are ultimately used to solve an inverse problem to reconstruct the geological formation and produce survey maps necessary for oil exploration. A simple, axisymmetric mechanical model of the streamer is presented in Fig. 2.1. The strength elements have been lumped into a single rope placed in the middle of the streamer. The z axis coincides with the axis of symmetry and the picture has been magnified by a factor of 10 in the radial direction to enable visualization.

Fig. 2.1 Axisymmetric model of a streamer

2 hp-Adaptive for Multiphysics Wave Propagation

27

Sample material properties of the streamer and geometrical dimensions are recorded in Table 2.1. Under a large drag force resulting from pulling the streamers through water, the strength elements become very stiff. The spacers and streamer’s skin are only moderately stiff but the filler is very soft. Additionally, the viscoelastic properties of the filler depend upon the temperature. Table 2.1 Material properties of streamer components component E(GPa)

ρ (kg.m−3 ) ν

length(m)

height(m)

Spacer

1.8

1200

0.30

0.075

0.021843

Rope Skin

41.0 0.02

1400 1200

0.30 0.45

75 75

0.005657 0.0032

Filler

Egel (ω , T ) 1040

0.45

0.165

0.021843

E f iller (ω , T ) = Er (ω , T ) + iEr (ω , T ) . At T = 10◦C, we have Er (ω , T ) = 2.9 × 100.4125 log10 (ω )+3.0871 [Pa] and Ei (ω , T ) = 2.9 × 100.3977 log10 (ω )+2.9707 [Pa]. In the real, three-dimensional scenario, the filler extends into cavities in the spacers housing the microphones. The critical quantity of interest in simulating vibrations of such structures is thus the distribution of the pressure in the soft part of the structure - the gel. The streamer project is focusing on studying unwanted vibrations of the streamers propagating from the tugboat and polluting the recorded signals. The problem is formulated as a coupled acoustics-(visco)elasticity problem. The domain occupied by the streamer is denoted by Ωe and the rest of the domain, occupied by the acoustical fluid, is denoted by Ωa . In the acoustic domain, we wish to determine pressure p and velocity wi satisfying: • conservation of mass,

c−2 iω p + ρ f w j, j = 0

and • conservation of linear momentum,

ρ f iω w j + p, j = 0 where ρ f is the density of the fluid, c is the sound speed, ω is the angular frequency and i denotes the imaginary unit. In the (visco)elastic domain, we wish to determine stresses σi j and displacements ui satisfying the equations discussed in our first example: • conservation of linear momentum equations, −ρs ω 2 ui − σi j, j = 0

28

L. Demkowicz, J. Kurtz, and F. Qiu

and • constitutive equations combined with geometrical relations,

σi j = μ (ui, j + u j,i) + λ uk,k δi j The system of equations is accompanied by the interface boundary conditions representing the continuity of normal velocity and tractions, iω ui ni = wi ni

and σi j n j = −pni

on interface ΓI

As the acoustic domain representing the ocean is infinite, the system must be accompanied by a radiation condition eliminating waves coming from infinity. Finally, we will assume that the system is driven by a kinematic boundary condition imposed on part of the streamer boundary representing the connector with the tugboat. ui = ui0

on Γ1e

Variational formulation and the weak coupling We proceed in the following steps. Step 1. Formulate conservation of mass (acoustics) and balance of momentum (elasticity) in a weak form: −

ω 2 Ωa

c Ωe

pq + iωρ f vi q,i −

ΓI

−ω 2 ρ ui vi + σi j vi, j −

iωρ f vi ni q = 0 ∀q ΓI

σi j n j vi = 0 ∀vi : vi = 0 on Γe1

Step 2. Use the remaining equations in the strong form to eliminate fluid velocity and elastic stresses: ω 2 − pq + p,iq,i − iωρ f vi ni q = 0 ∀q Ωa c ΓI Ωe

−ω 2 ρ ui vi + μ (ui, j + u j,i)vi, j + λ uk,k vk,k −

ΓI

σi j n j vi = 0 ∀vi : vi = 0 on Γe1

Step 3. Use the interface conditions to couple the two variational formulations: − Ωe

ω 2 Ωa

c

pq + p,iq,i + ω 2

ΓI

−ω 2 ρ ui vi + μ (ui, j + u j,i )vi, j + λ uk,k vk,k +

ρ f ui ni q = 0 ∀q ΓI

pvi ni = 0 ∀vi : vi = 0 on Γe1

We end up with the following variational formulation.

2 hp-Adaptive for Multiphysics Wave Propagation

29

⎧ u ∈ u˜ 0 + V e , p ∈ Va ⎪ ⎪ ⎪ ⎪ ⎨ bee (u, v) + bae(p, v) = le (v) ∀v ∈ V e ⎪ ⎪ ⎪ ⎪ ⎩ bea (u, q) + baa(p, q) = la (q) ∀q ∈ Va where

V e = {v ∈ H 1 (Ωe ) : v = 0 on Γe1 } Va = H 1 (Ωa ) bee (u, v) =

Ωe

bae (p, v) =

ΓI

bea (u, q) = − baa (p, q) =

Ei jkl uk,l vi, j − ρs ω 2 ui vi dx

pvn dS un q dS

ΓI

1

ω 2ρ

f

Ωa

∇p∇q − k2 pq dx

le (v) = 0 la (q) = 0 Above, u˜ 0 represents a lift of the kinematic boundary data. The problem is driven by the kinematic boundary condition only. The radiation condition is handled by a Perfectly Matched Layer resulting in complex material data but preserving the symmetry of the formulation, see [5] for details. The problem is thus complex-symmetric (not Hermitian !). Discretization of a two-dimensional axisymmetric version of the problem requires the use of three H 1 -conforming components: two components of the elastic displacement ui , and one component for the pressure p. The elastic displacement components are supported in the elastic domain Ωe , and the pressure lives over Ωa . All three components must be supported over the elastic/acoustic interface ΓI .

2.2.3 Coupled Acoustics and Poroelasticity In this section we present an example of acoustics coupled with poroelasticity, a problem encountered in modeling of sonic tools used in a borehole environment. The hp technology was used simulate the experiments performed in [7]. A porous sandstone cylinder with a borehole is placed in a bath of saturating liquid. A transmitter excites waves within the borehole, and measurements of the velocity and attenuation of Stoneley waves along the borehole wall are made. The axisymmetric geometry is shown in Figure 2.2.

30

L. Demkowicz, J. Kurtz, and F. Qiu

z

r

Fig. 2.2 Axisymmetric geometry for sandstone cylinder in liquid bath.

Assuming the time dependence eiω t , with angular frequency ω , the liquid subdomain Ωa is modeled by the linear acoustics equations, iω p = −c2 ρ ∇ · v

(2.1)

iωρ v = −∇p

(2.2)

involving the pressure p, velocity v, density ρ and sound speed c. The variational formulation is obtained by multiplying (2.1) by a test function δ p, integrating over Ωa , integrating by parts, and using equation (2.1) in the strong form to eliminate the velocity:

Ωa

1 ∇p · ∇δ p − k2 pδ p dx + iω ρ

Γi

vn δ p dS = −iω

Γs

vsn δ p dS

(2.3)

In fact, (2.3) is modified to incorporate a perfectly matched layer (PML) using the procedure described in [5]. The surface integrals in (2.3) are over the source Γs at the bottom of the borehole (where the normal velocity vsn is prescribed), and the interface Γi with the poroelastic cylinder. The poroelastic subdomain Ω p is modeled by the two-velocity approach [2]. The system involves the solid velocity u, liquid velocity v, pressure p and partial stress h (the total stress is σ = −h − pI)

2 hp-Adaptive for Multiphysics Wave Propagation

31

ρs ∇p − ∇ · h − ρl2 χ (u − v) ρ ρl iωρl v = − ∇p + ρl2 χ (u − v) ρ iω p = ρ a3 ∇ · u − ρ a4∇ · v

ρs ρl K − λ ∇ · u + K ∇ · v I − 2 με (u) iω h = ρ ρ

iωρs u = −

(2.4) (2.5) (2.6) (2.7)

where ε (u) = (∇u + ∇uT )/2 is the linear strain operator. The total density ρ is the sum of the partial densities of liquid and solid, ρl and ρs , respectively. The parameter controlling dissipation, η χ= , kρρl is given in terms of the viscosity η and permeability k. The Lamé parameters, λ and μ , and a third parameter α3 , are determined by the three wave speeds for the medium. Finally, K = λ + 2 μ /3 ρl a3 = K 2 − α3 ρρs ρ ρl a4 = K 2 + α3 ρρl ρ The bottom of the cylinder Γb is fixed so that u = 0, vn = 0 and the variational formulation is Ωp

−

ρl a4 (∇ · v)(∇ · δ v) − ω 2 ρl v · δ v dx

Ωp

ρl a3 (∇ · u)(∇ · δ v) dx + iω

Γi

ρl p (δ v · n) dS = 0 ρ

(2.8)

K 2με (u) : ε (δ u) + λ − ρs + a3 (∇ · u)(∇ · δ u) − ω 2 ρs (u · δ u) dx ρ Ωp

ρs h + pI δ u · n dS = 0 (2.9) − ρl a3 (∇ · v)(∇ · δ u) dx + iω ρ Ωp Γi for all test functions δ u = 0, δ vn = 0 on Γb . The two formulations are weakly coupled by introducing a surface porosity d and the interface conditions van = dvn + (1 − d)un

ρl p = d pa ρ ρs hn + pn = (1 − d)pan ρ

(2.10) (2.11) (2.12)

32

L. Demkowicz, J. Kurtz, and F. Qiu

where we have introduced the superscript “a” to denote quantities from the acoustic subdomain. The material parameters were set to model the Berea sandstone experiments in [7].

2.2.4 Exact Sequence and the de Rham Diagram The energy spaces form the classical grad-curl-div exact sequence: ∇ ∇× ∇◦ H 1 → H(curl) → H(div) → L2 that can be reproduced on the discrete level using hp finite element spaces for elements of all shapes: tetrahedra, hexahedra, prisms and pyramids, including curvilinear elements. The continuous and discrete spaces are connected through special Projection-Based Interpolation (PB) operators that make the following diagram commute. ∇ ∇× ∇◦ H 1 → H(curl) → H(div) → L2 ⏐ ⏐ ⏐ ⏐ ⏐ grad ⏐ curl ⏐ div ⏐ Π Π Π P ∇ Whp →

Qhp

∇× ∇◦ → V hp → Yhp

In the two-dimensional case, the 3D sequence reduces to two 2D sequences: ∇ ∇× R I −→ H 1 −→ H(curl) −→ L2 −→ 0 ∇× ∇◦ R I −→ H 1 −→ H(div) −→ L2 −→ 0 The discussed code supports hybrid meshes based on Nédélec triangles of the second type and Nédélec quads of the first type, see [5] for detailed explanations.

2.3 hp Technology In this section we discuss shortly the main implementation changes required to update the hp technology presented in [3, 5] to coupled, multiphysics problems.

2 hp-Adaptive for Multiphysics Wave Propagation

33

2.3.1 Handling Multiphysics The code supports an arbitrary number of H 1 -, H(curl)-, H(div)- and L2 -conforming components. User specifies involved fields along with the dicretization type and number of components. For instance, some characteristics of the dual-mixed formulation for elasticity with weakly imposed symmetry are gathered in Tab. 2.2. Table 2.2 Characteristics of the dual-mixed formulation #

name

discretization type

number of components

1

stress

H(div)

2

displacement

L2

2

rotation

L2

1

2 3

Recall that the main idea of the hp data structure presented in [3, 5] is based on the concept of a node object, an abstraction for element vertices, edges, faces (3D) and interiors. Mesh refinements involve breaking of the nodes and growing nodal trees, in contrast to h-adaptive methods that “grow” only trees for elements. The nodal trees provide a sufficient information for a number of efficient algorithms supporting looping through elements in the current mesh (natural order of elements), determination of element neighbors for edges and faces (3D) and, most importantly, element to nodes connectivities. The nodal connectivities include information on constrained (hanging) nodes. Only 1-irregular meshes are supported and the minimum rule is enforced through the whole adaptivity process. Operations on nodes are identical for all types of discretization. The difference between different discretizations has to be accounted for only at the level of nodal degrees-of-freedom (d.o.f.). This two-level logic is crucial for the simultaneous support of the exact sequences and reduction of complexity of the code. The definition of a node has been modified to include a partial list of variables to be supported and pointers to arrays storing the corresponding d.o.f. (dynamically allocated arrays). For instance, for the coupled acoustics/elasticity problem, a node in the interior of acoustics subdomain supports only the pressure, a node inside of the elastic domain supports only the elastic displacement, but a node on elastic/acoustic interface supports both variables. The dimension of the arrays depends upon the (polynomial) order of the node and the actual allocation takes place during the initial mesh generation or mesh modifications. A C preprocessing is used to produce a real-valued or complex-valued version of the code. Our Geometrical Modeling Package (GMP) provides the Mesh Based Geometry (MBG) description [3] without any changes. The logic of GMP blocks is adopted to specify on input different fields to be supported in each block. The initial mesh generator automatically identifies then interfaces and the fields to be supported at interface vertex and edge nodes.

34

L. Demkowicz, J. Kurtz, and F. Qiu

Postprocessing: Expanded Mode In order to simplify mathematical and graphical postprocessing, all solution d.o.f. are output elementwise in the so-called expanded mode that includes all fields specified by the user. Thus, e.g., for an element in the acoustical subdomain, we return also the d.o.f for the elastic displacement. For an element in the interior of the acoustical domain, all returned elastic d.o.f. are simply zero. However, for an element adjacent to the acoustic/elastic interface, the returned d.o.f. will represent a FE lift of the values on the interface.

2.3.2 Constrained Approximation The constrained approximation package described in [3, 5] is a result of many years of improvements and we have been very hesitant to modify it. The package utilizes the element to nodes connectivity information (including constraints), and generates the corresponding generalized d.o.f. connectivity information enabling two tasks: • Assembly of modified element matrices • Computation of local element geometry and solution d.o.f. Two versions of those routines were available: one for the H 1 -, and another one for H(curl)-conforming discretization. As in 2D, the H(div)-conforming elements are obtained from H(curl)-conforming elements by a 90 degrees rotation, the corresponding constrained approximation coefficients are identical. Finally, the L2 conforming elements involve only interior d.o.f. and do not involve any constraints. This has led us to the idea of computing the multiphysics element stiffness matrices and load vectors in the block form. Thus, for example, solution of the elasticity problem with the dual-mixed formulation with weakly imposed symmetry, involves computation of nine different stiffness matrices involving H(div) and L2 shape functions. The original, single physics, constrained approximation arrays are then used to transform every block into the corresponding block for the modified element. For example, for the coupled acoustics/elasticity problem, for an element in acoustic domain but adjacent to acoustic/elastic interface, we generate all four multiphysics element matrices even though the two field interact through the boundary. In order to eliminate the zero entries and, at the same time, account for Dirichlet boundary conditions, we perform one more logical step on element matrices, changing the ordering of the modified element d.o.f. from the multiphysics ordering to the geometrical ordering: • interior node H 1 , H(curl),H(div) and L2 d.o.f.; • edge nodes H 1 ,H(curl) and H(div) d.o.f.; • vertex nodes H 1 d.o.f.

2 hp-Adaptive for Multiphysics Wave Propagation

35

The process of reordering of d.o.f. is accompanied by elimination of “empty nodes” (e.g. elastic interior node for an element in acoustic domain or nodes whose d.o.f. are determined by Dirichlet boundary conditions.) With the use of frontal solvers, the new ordering is equivalent to the static condensation of interior d.o.f., a well known trick to improve the conditioning.

2.3.3 Automatic hp Adaptivity The 3D package supporting automatic hp-adaptivity described in [5] has been modified and expanded to adopt the multiphysics setting. The hp-algorithm is a black box based on a two-grid paradigm and it is problem independent. Given a coarse mesh, we perform a global hp-refinement (each element is broken into four elementsons, and the order is raised uniformly by one) and solve the problem on the fine mesh. The logical information on the coarse grid and the corresponding fine grid solution is then entered into the black box which returns the optimal hp-refinement. The execution of the optimal refinement is accompanied with a “closing operation” enforcing 1-irregularity and minimum order mesh regularity rules. The principle of the energy-based hp algorithm is as follows. Given the fine grid solution uh/2,p+1, corresponding to the current coarse grid hp, we determine the next coarse grid hpopt by solving the following maximization problem: uh/2,p+1 − Πhpuh/2,p+12 − uh/2,p+1 − Πhpopt uh/2,p+12 → max Nhpopt − Nhp Here Πhp , Πhpopt denote the Projection Based Interpolation (PBI) operators mentioned in the Section 2.2, corresponding to the coarse grid, and the new, optimal coarse grid to be determined, and Nhp , Nhpopt denote the total number of d.o.f for the two meshes. The principle is widely accepted - we maximize the rate of our investment. The actual algorithm follows the logic of PB Interpolation. Step 0: We subtract from the H 1 components of the fine grid solution, the coarse grid vertex interpolants. Similarly, we subtract from the H(curl) and H(div) components of the fine grid solution, the coarse grid edge averages of tangential or normal components, respectively. Step 1: We determine optimal refinements for coarse grid element edges. This involves making three decisions on: • how to refine an edge, h or p ? • which edges to refine ? • and, in the case of an h-refined edge, what order for edge sons should be used, i.e. how many d.o.f. to invest ?

36

L. Demkowicz, J. Kurtz, and F. Qiu

The only difference between the single- and multi-physics versions is that we have to interpolate now multiple fields, with a number and type of those fields varying from edge to edge. We conclude the edge optimization step by subtracting from the fine grid solution the corresponding edge interpolants corresponding to the optimal edge refinements. Step 2: We determine optimal refinements for the coarse grid element interiors. This involves now not only the choice between h and p-refinements but also a choice between different h-refinements (anisotropic and isotropic refinements). The established already optimal refinements for edges set the stage for the second step, and significantly limit the number of cases to be considered. For instance, if an edge of an element is to be h-refined, this eliminates the possibility of a p-refinement for the element, see [5] for details. In conclusion, we have been able to keep the logic of our hp algorithm practically unchanged.

2.4 Numerical Examples In this section, we present a few representative numerical results for sample coupled multiphysics problems discussed in Section 2.2.

2.4.1 Mixed-Dual Elasticity The hp discretization of this problem extends the theory developed by Arnold et al. [1] for uniform p, to the case of variable order of approximation. A stability analysis presented in [6] extends the results from [1] to the variable p but, at present, a convergence analysis for p- and hp-methods is not available. This does not stop us from numerical experimentation. The classical Airy stress function approach combined with separation of variables has been used to construct an analytical solution with finite energy and strongest singularity for an infinite L-shape sector of a plane (see [6]) for details). The solution is then used as a manufactured solution on the standard (bounded) L-shape domain with the corresponding traction boundary conditions on truncating boundary driving the problem. Although manufactured, the solution displays the same singular behavior at the origin as actual elasticity boundary-value problems set up in the same domain. Fig. 2.3 presents an optimal coarse hp mesh corresponding to relative energy error below 0.6 percent. Fig. 2.5 presents the corresponding convergence history for the hp algorithm and, for illustration, Fig. 2.4 displays the distribution of σ11 component of the stress tensor on the optimal mesh. Notice the use of algebraic scale on the convergence plot with which the straight line corresponds to exponential convergence.

2 hp-Adaptive for Multiphysics Wave Propagation

31 22

33

42

34

90

44 45

97

100

99

106 109

37

108

55

117

23

56

20

153 146 156155162 165 164 173 118 209 211 218 220 212 221 229 147202 174 144 265 267 274 276 258 268 277 285 203 230 200 228 172 259 266 275 286 256 257 283 284 292 294 295 301 238 182 304 239 293 302 303 183 248 237 246 247 192 127 181 190 191 136

91 88

54 116 126 66 67 125

134

135 78 65 76

y z x

77

Fig. 2.3 Mixed-dual elasticity: L-shape domain problem. Optimal hp mesh corresponding to 0.6 percent error.

31 22

33

42

34

90

44 45

97

100

99

106 109

108

55

117

23

56

20 91 88

153 146 156155162 165 164 173 118 209 211 218 220 212 221 229 147202 174 144 265 267 274 276 258 268 277 285 203 230 200 228 172 259 266 275 286 256 257 283 284 292 294 295 301 238 182 304 239 293 302 303 183 248 237 246 247 192 127 181 190 191 136

54 116 126 66 67 125

134

135 78 65 76

y z x

77

Fig. 2.4 Mixed-dual elasticity: L-shape domain problem. Distribution of stress σ11 on the optimal mesh.

15.51

error SCALES: nrdof^0.33, log(error)

8.88 5.09

adaptive hp

2.91 1.67 0.96 0.55 0.31 0.18 0.10 0.06 243

594

1184

2075

3332

5018

7197

9933

nrdof 13292

Fig. 2.5 Mixed-dual elasticity: L-shape domain problem. Convergence history.

38

L. Demkowicz, J. Kurtz, and F. Qiu

2.4.2 The Streamer Problem Fig. 2.6 presents an optimal hp coarse mesh corresponding to (estimated) relative H 1 error of 3.6 percent. Each junction of three or more different materials results in a singularity. Additional singularities occurred at reentrant corners at elastic/acoustic interface. The use of PML results in development of strong boundary layers whose resolution with hp-adaptivity is also critical for an high accuracy solution. Fig. 2.7 presents the pressure in both the structure and water in terms of decibels.

Fig. 2.6 Streamer problem: Optimal hp mesh corresponding to 3.6 percent error.

Fig. 2.7 Streamer problem: Pressure distribution on the optimal mesh. Range: -20 - 133.6 dB.

2 hp-Adaptive for Multiphysics Wave Propagation

39

The next two figures illustrate the need for resolving the singularities for problems with large material contrasts. Fig. 2.8 presents the pressure distribution along a horizontal section through the soft filler, just next to the stiff strength element (rope). One should emphasize that, in order to avoid locking, we have started with horizontal order p = 5 and vertical order q = 2. In other words, singularities but not locking presents the main difficulty with getting an accurate solution. Fig. 2.9 presenting the same distribution but on the optimal mesh, is dramatically different, although the small oscillations indicate that the accuracy is still insufficient (the problem was solved on a laptop with 1GB memory only...).

Fig. 2.8 Streamer problem: Pressure profile for the initial mesh.

Fig. 2.9 Streamer problem: Pressure profile for the optimal mesh.

40

L. Demkowicz, J. Kurtz, and F. Qiu

2.4.3 Modeling of Sonic Tools The following figures present illustrative examples for the Berea’s experiment. The history of hp-refinements is shown in Fig. 2.10. Final hp fine grid with the corresponding distribution of acoustic pressure is shown in Fig. 2.11. Note that only the acoustic part of the domain is meaningful (recall the discussion on the expanded mode). Figures 2.12 and 2.13 present the corresponding distribution of elastic displacements and fluid velocity (in pores) in the poroelastic region. Clearly, the entire action happens along the interface.

2

Percent relative error in energy norm

10

1

10

0

10

6162

7269

7775 8098 8226 8295 8700 91449420

10139

10843 11440 11439 1147111911

12715

Number of dof in algebraic scale N1/3

Fig. 2.10 Simulating Berea’s experiment: Sequence of hp coarse grid errors for the sandstone cylinder.

The results display a clear exponential convergence for which again, at present, we do not have any supporting stability analysis.

2 hp-Adaptive for Multiphysics Wave Propagation

41

Fig. 2.11 Simulating Berea’s experiment: Final hp fine grid and acoustic pressure p for the sandstone cylinder.

Fig. 2.12 Simulating Berea’s experiment: r and z components of solid velocity u for the sandstone cylinder.

2.5 Conclusions The paper reports on our current effort to extend the technology of hp-adaptive finite elements to a class of coupled, multiphysics problems. Three representative examples presented in the paper reflect well the complexity and technical problems encountered in the course of the project. In order to account for the multiphysics and coupled problems setting, we had to develop a new code including a new data structure, initial mesh generator, constrained approximation, mesh modification package, postprocessing etc. However, we have been able to preserve the overall logical structure

42

L. Demkowicz, J. Kurtz, and F. Qiu

Fig. 2.13 Simulating Berea’s experiment: r and z components of fluid velocity v for the sandstone cylinder (rescaled to obtain a nice picture: observed max |vr | ≈ 0.065, max |vz | ≈ 0.0074).

of or hp implementations and, through compromises discussed in the text, recycle the most difficult part of the constrained approximation package. The hp-algorithm based on the two-grid paradigm and the family of Projection-Based Interpolation operators seems to extend well to the class of discussed problems. Presented numerical examples document exponential convergence and illustrate the necessity of using adaptive methods for problems with high material contrasts. A detailed manual for our new 2D code is under development [4] and we hope to report on new numerical and theoretical results in forthcoming papers.

References [1] Arnold, D.N., Falk, R.S., Winther, R.: Mixed finite element methods for linear elasticity with weakly imposed symmetry. Mathematics of Computation 76, 1699–1723 (2007) [2] Blokhin, A., Dorovsky, V.: Mathematical Modelling in the Theory of Multivelocity Continuum. Nova Science Publishers, Inc., Bombay (1995) [3] Demkowicz, L.: Computing with hp Finite Elements. I. One- and Two-Dimensional Elliptic and Maxwell Problems. Chapman & Hall/CRC Press, Taylor and Francis (2006) [4] Demkowicz, L., Kurtz, J.: An hp framework for coupled multiphysics problems. Technical report, ICES (2009) (in preparation) [5] Demkowicz, L., Kurtz, J., Pardo, D., Paszy´nski, M., Rachowicz, W., Zdunek, A.: Computing with hp Finite Elements. II. Frontiers: Three-Dimensional Elliptic and Maxwell Problems with Applications, October 2007. Chapman & Hall/CRC (2007) [6] Qiu, W., Demkowicz, L.: Mixed h p-finite element method for linear elasticity with weakly imposed symmetry. Comput. Methods Appl. Mech. Engrg (2009); in print, see also ICES Report 2008/30 [7] Winkler, K.W., Liu, H.L., Johnson, D.L.: Permeability and borehole Stoneley waves: Comparison between experiment and theory. Geophysics 54(1), 66–75 (1989)

Chapter 3

Nonconvex Inequality Models for Contact Problems of Nonsmooth Mechanics Stanislaw Migórski and Anna Ochal

Abstract. This review paper deals with selected nonsmooth and nonconvex inequality problems for dynamic frictional contact between a viscoelastic body and a foundation. The process is modeled by general nonmonotone possibly multivalued multidimensional Clarke subdifferential contact boundary conditions. The problems of frictional contact with both short and long memory, thermoviscoelastic frictional contact, bilateral frictional contact and bilateral contact for piezoelectric materials with adhesion are considered. The formulations and results on existence, and uniqueness of solutions are presented.

3.1 Introduction In this paper we present an up-to-date review of a part of our current results on dynamic contact problems described by nonmonotone subdifferential boundary conditions. The related results on static and quasistatic contact problems with and without additional effects are today quite well covered by still growing literature [13, 39, 40, 41], however, their nonconvex counterparts as well as dynamic models need further investigation. There is a fundamental need for existence theory of contact problems. Even when the constitutive laws are linear, the problems are nonlinear because of contact conditions. The aim of our study is to contribute towards this goal and present in a unified way, results on analysis that deal with models for contact processes involving multivalued contact laws. It is clear that such contact problems do not have classical solutions. We mention that the existence of a solution to the dynamic contact problem between an elastic body and a rigid obstacle, described by a unilateral contact Stanislaw Migórski · Anna Ochal Jagiellonian University, Institute of Computer Science, Faculty of Mathematics and Computer Science, ul. Stanisława Łojasiewicza 6, 30-348 Krakow, Poland e-mail: {migorski,ochal}@ii.uj.edu.pl

M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 43–58. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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condition of the Signorini type, is still an open problem. Today the literature on various mathematical and numerical aspects of dynamic contact with or without friction is growing rapidly. We restrict ourselves to the description of viscoelastic contact problems with the Kelvin-Voigt constitutive law (Section 3.2), frictional contact with both short and long memory (Section 3.3), thermoviscoelastic frictional contact (Section 3.4), bilateral frictional contact (Section 3.5) and bilateral contact for viscoelastic piezoelectric materials with adhesion (Section 3.6). The frictional contact problem in Section 3.2 is presented with more details and for other models we give their classical and weak formulations and refer to the references for a precise statement of existence and uniqueness results and further explanations. We treat the problems within the infinitesimal strain theory. For all of these problems with nonconvex superpotentials, effective numerical methods are needed and reliable general three-dimensional codes should be delivered. Such studies allow for better understanding and more accurate, and realistic prediction of the long time behavior of the considered system. We present nonsmooth and nonconvex inequality problems for dynamic frictional contact between a viscoelastic body and a foundation. We establish the existence of weak solutions to different models by employing a unified approach based on evolution hemivariational inequalities. The latter can, in turn, be formulated as the Cauchy problem for the following evolution inclusion of second order t

u (t) + A(t, u(t)) + B(t, u(t)) +

C(t − s) u(s) ds + F(t, u(t), u (t)) f (t).

0

Here A, B and C correspond to viscosity, elasticity and relaxation operators in the mechanical problems and the multifunction F contains different subdifferential boundary contact conditions. The notion of hemivariational inequality was introduced and studied by P.D. Panagiotopoulos ([37, 38]) in the early 1980s as variational formulation for certain classes of mechanical problems with the nonconvex, nondifferentiable and nonsmooth energy functionals. Such inequality results from the d’Alembert principle for a dynamic mechanical system. During years a hemivariational inequality proved to be an effective tool to give positive answers to unsolved or partially unsolved problems, cf. [38, 35]. In this respect our approach can be considered as a continuation of works of Panagiotopoulos. In this paper the main interest lies in general nonmonotone and multivalued Clarke subdifferential boundary conditions. More precisely, it is supposed that on the boundary of the body under consideration, the subdifferential relations hold, for instance, between the normal component of the velocity and the normal component of the stress, between the tangential components of these quantities and between temperature and the heat flux vector. These subdifferential boundary conditions are the natural generalizations of the normal damped response condition, the accociated friction law and the well known Fourier law of heat conduction, respectively. They allow to incorporate in our models several types of boundary conditions considered earlier e.g. in [13, 38, 39, 40, 41].

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3.2 Physical Setting of Dynamic Viscoelastic Problem In this section we present a dynamic viscoelastic frictional contact problem and describe a method which can be used to obtain results on its solvability and unique solvability. This method uses various concepts of nonlinear analysis including evolution inclusions and hemivariational inequalities. The main novelty of the approach arises in the fact that we deal with a large class of multivalued contact and friction boundary conditions for nonconvex superpotentials. In fact, it is the nonconvexity which makes the problem more difficult than those already studied in the literature. After the description of the mechanical problem, we will show that it leads to a hemivariational inequality for the displacement field. We begin with the description of the mechanical problem. In its reference configuration, a viscoelastic body occupies a subset Ω in R d , d = 2, 3 in applications. We suppose that Ω has a Lipschitz boundary Γ , thus the unit outward normal vector ν exists a.e. on Γ and the boundary is divided into three mutually disjoint parts ΓD , ΓN and ΓC such that m(ΓD ) > 0. The body is acted upon by volume forces and surface tractions and, as a result, its state is evolving. We are interested in dynamic evolution process of the mechanical state of the body on the time interval [0, T ] with T > 0. We assume that the body is clamped on ΓD , so the displacement field vanishes there. Volume forces of density f1 act in Ω and surface tractions of density f2 are applied on ΓN . The body may come in contact with an obstacle over the potential contact surface ΓC . We denote by Sd the linear space of second order symmetric tensors on Rd and we use the notation Q = Ω × (0, T ). For simplicity we skip the dependence of various functions on the spatial variable x ∈ Ω ∪ Γ . Then, the frictional contact problem under consideration can be stated as follows: find the displacement field u : Q → Rd and the stress tensor σ : Q → Sd such that u (t) − div σ (t) = f1 (t) in Q

(3.1)

σ (t) = A (t, ε (u (t))) + B ε (u(t)) in Q

(3.2)

u(t) = 0 on ΓD × (0, T )

(3.3)

σ (t)ν = f2 (t) on ΓN × (0, T )

(3.4)

−σν (t) ∈ ∂ jν (t, uν (t)),

(3.5)

−στ (t) ∈ ∂ jτ (t, uτ (t)) on ΓC × (0, T )

u(0) = u0 , u (0) = v0 in Ω .

(3.6)

Equation (3.1) is the equation of motion and div denotes the divergence operator for tensor valued functions. Relation (3.2) represents the Kelvin–Voigt viscoelastic constitutive law, where A is a nonlinear operator describing the purely viscous properties of the material and B is the linear elasticity operators. Note that the operator A may depend explicitly on the time variable and this is the case when the viscosity properties of the material depend on the temperature field. Conditions (3.3) and

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(3.4) are the displacement and the traction boundary conditions, respectively. Conditions (3.5) represent the frictional contact condition in which jν and jτ are given contact and frictional potentials and the subscripts ν and τ for σ and u indicate normal and tangential components of tensors and vectors. The symbol ∂ j denotes the Clarke subdifferential of j with respect to the last variable. Note also that the explicit dependence of the functions jν and jτ with respect to the time variable allows to model situations when the frictional contact conditions depend on the temperature, which plays the role of a parameter, and which evolution in time is prescribed. Finally, conditions (3.6) represent the initial conditions where u0 and v0 denote the initial displacement and the initial velocity, respectively. We recall two notions introduced by Clarke (cf. [3, 9]) which are the basic tools in our studies. Let ϕ : X → R be a locally Lipschitz function defined on a Banach space X . Then the generalized directional derivative of ϕ at x ∈ X in the direction v ∈ X, denoted by ϕ 0 (x; v), is defined by

ϕ 0 (x; v) = lim sup

y→x, λ ↓0

ϕ (y + λ v) − ϕ (y) λ

and the generalized gradient of ϕ at x, denoted by ∂ ϕ (x), is a subset of a dual space X ∗ given by ∂ ϕ (x) = ζ ∈ X ∗ | ϕ 0 (x; v) ≥ ζ , v X ∗ ×X for all v ∈ X . A locally Lipschitz function ϕ is called regular (in the sense of Clarke) at x ∈ X if for all v ∈ X the one-sided directional derivative ϕ (x; v) exists and satisfies ϕ 0 (x; v) = ϕ (x; v) for all v ∈ X. Weak formulation of the problem. Throughout the paper, indices i, j, k, l, etc. run from 1 to d. In order to provide the weak form of the problem under consideration, we consider the following spaces H = L2 (Ω ; R d ),

H = L2 (Ω ; Sd )

and let V be the closed subspace of H 1 (Ω ; Rd ) given by V = v ∈ H 1 (Ω ; Rd ) | v = 0 on ΓD . The deformation and the divergence operators are given by 1 ε (u) = {εi j (u)}, εi j (u) = (∂ j ui + ∂i u j ), div σ = {∂ j σi j }. 2 We denote by V ∗ the dual space of V and by ·, · the duality pairing of V and V ∗ . Let V = L2 (0, T ;V ) and W = {w ∈ V | w ∈ V ∗ }, where the time derivative is understood in the sense of vector valued distributions. We obtain W ⊂ V ⊂ L2 (0, T ; H) ⊂ V ∗ , where all the embeddings are continuous (cf. [10]).

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Moreover, let Z = H 1/2 (Ω ; Rd ) and let γ denote the norm of the trace operator γ : Z → L2 (Γ ; Rd ) which has adjoint γ ∗ : L2 (Γ ; R d ) → Z ∗ . In the study of problem (3.1)–(3.6) we need the following assumptions on the data. H(A ) :

A : Q × Sd → Sd is such that

(i) A (·, ·, ε ) is measurable on Q, for all ε ∈ Sd ; (ii) A (x,t, ·) is continuous on Sd for a.e. (x,t) ∈ Q; (iii) A (x,t, ε ) ≤ a(x,t) + c1 ε for all ε ∈ Sd , a.e. (x,t) ∈ Q with a ∈ L2 (Q), a ≥ 0 and c1 > 0; (iv) (A (x,t, ε1 ) − A (x,t, ε2 )) : (ε1 − ε2 ) ≥ m1 ε1 − ε2 2 for all ε1 , ε2 ∈ Sd , a.e. (x,t) ∈ Q with m1 > 0; (v) A (x,t, ε ) : ε ≥ c2 ε 2 for all ε ∈ Sd , a.e. (x,t) ∈ Q with c2 > 0. H(B) :

B : Ω × Sd → Sd is such that B(x, ε ) = b(x)ε and

(i) b(x) = {bi jkl (x)} with bi jkl = b jikl = blki j ∈ L∞ (Ω ); (ii) bi jkl (x)εi j εkl ≥ 0 for all ε = {εi j } ∈ Sd , a.e. x ∈ Ω . H( jν ) :

jν : ΓC × (0, T ) × R → R satisfies

(i) jν (·, ·, r) is measurable for all r ∈ R and jν (·, ·, 0) ∈ L1 (ΓC × (0, T )); (ii) jν (x,t, ·) is locally Lipschitz for a.e. (x,t) ∈ ΓC × (0, T ); (iii) |∂ jν (x,t, r)| ≤ cν (1 + |r|) for a.e. (x,t) ∈ ΓC × (0, T ), all r ∈ R with cν > 0; (iv) jν0 (x,t, r; −r) ≤ dν (1 + |r|) for all r ∈ R, a.e. (x,t) ∈ ΓC × (0, T ) with dν ≥ 0; (v) (η1 − η2 )(r1 − r2 ) ≥ −mν |r1 − r2 |2 for all ηi ∈ ∂ jν (x,t, ri ), ri ∈ R, i = 1, 2, a.e. (x,t) ∈ ΓC × (0, T ) with mν ≥ 0. H( jτ ) : jτ : ΓC × (0, T ) × R d → R satisfies (i) jτ (·, ·, ξ ) is measurable for all ξ ∈ R d and jτ (·, ·, 0) ∈ L1 (ΓC × (0, T )); (ii) jτ (x,t, ·) is locally Lipschitz for a.e. (x,t) ∈ ΓC × (0, T ); (iii) ∂ jτ (x,t, ξ ) ≤ cτ (1 + ξ ) a.e. (x,t) ∈ ΓC × (0, T ), all ξ ∈ Rd with cτ > 0; (iv) jτ0 (x,t, ξ ; −ξ ) ≤ dτ (1 + ξ ) a.e. (x,t) ∈ ΓC × (0, T ), all ξ ∈ Rd with dτ ≥ 0; (v) (η1 − η2 ) · (ξ1 − ξ2 ) ≥ −mτ ξ1 − ξ2 2 a.e. (x,t) ∈ ΓC × (0, T ), all ηi ∈ ∂ jτ (x,t, ξi ), ξi ∈ Rd , i = 1, 2 with mτ ≥ 0. H(0) :

f1 ∈ L2 (0, T ; H), f2 ∈ L2 (0, T ; L2 (ΓN ; R d )), u0 ∈ V , v0 ∈ H.

Now, we introduce the operators A : (0, T ) ×V → V ∗ and B : V → V ∗ defined by A(t, u), v = A (t, ε (u)), ε (v) H Bu, v = B ε (u), ε (v) H

for u, v ∈ V and t ∈ (0, T ),

for u, v ∈ V.

We also consider the function f : (0, T ) → V ∗ given by

(3.7) (3.8)

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f (t), v = f1 (t), v H + ( f2 (t), v)L2 (ΓN ;Rd ) for v ∈ V, a.e. t ∈ (0, T ). The weak formulation of the contact problem (3.1)–(3.6) is obtain by the standard method. Assume that (u, σ ) is a couple of regular functions which solve (3.1)–(3.6), v ∈ V and t ∈ (0, T ). Using the equation of motion (3.1) and the Green formula, one finds u (t), v + σ (t), ε (v) H = f1 (t), v H + σ (t) ν · v dΓ . Γ

We take into account the boundary conditions (3.3) and (3.4) to get

Γ

σ (t) ν · v dΓ =

ΓN

f2 (t) · v d Γ +

ΓC

(σν (t)vν + στ (t) · vτ ) d Γ

and from the definition of the Clarke subdifferential and (3.5), we have −σν (t)vν ≤ jν0 (x,t, uν (x,t); vν ), −στ (t) · vτ ≤ jτ0 (x,t, uτ (x,t); vτ ) on ΓC × (0, T ), which imply that − ≤

ΓC

ΓC

(σν (t)vν + στ (t) · vτ ) d Γ ≤

jν0 (x,t, uν (x,t); vν (x)) + jτ0 (x,t, uτ (x,t); vτ (x)) dΓ .

Combining the above, we obtain u (t), v + σ (t), ε (v) H + 0 jν (x,t, uν (x,t); vν (x)) + jτ0 (x,t, uτ (x,t); vτ (x)) dΓ ≥ + ΓC

≥ f (t), v for all v ∈ V and a.e. t ∈ (0, T ). (3.9) We use (3.9), the constitutive law (3.2), (3.7)–(3.8) and the initial conditions (3.6) to obtain the following weak formulation of the mechanical problem (3.1)–(3.6): ⎧ ⎪ find u : (0, T ) → V such that u ∈ V , u ∈ W and ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u (t) + A(t, u(t)) + Bu(t), v + ⎪ ⎪ ⎨ 0 (3.10) jν (x,t, uν (x,t); vν (x)) + jτ0 (x,t, uτ (x,t); vτ (x)) dΓ ≥ + ⎪ ⎪ Γ ⎪ C ⎪ ⎪ ⎪ ≥ f (t), v for all v ∈ V and a.e. t ∈ (0, T ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩u(0) = u , u (0) = v . 0 0 Unique weak solvability of the problem. The following result concerns the existence and uniqueness of solutions to problem (3.10). Theorem 1. Assume that H(A ), H(B), H( jν ), H( jτ ), H(0) hold and

3 Nonconvex Models in Contact Mechanics

49

m1 > (mν + mτ )γ 2 .

(3.11)

Then problem (3.10) admits at least one solution. If, in addition, ⎧ ⎨either jν (x,t, ·) or − jν (x,t, ·) is regular and ⎩either j (x,t, ·) or − j (x,t, ·) τ τ

is regular,

then problem (3.10) has a unique solution. We comment on the proof of Theorem 1. The method of the proof is based on the existence and uniqueness results obtained recently in [20, 26] for evolution inclusions of the form ⎧ ⎪ find u ∈ V with u ∈ W such that ⎪ ⎪ ⎨ (3.12) u (t) + A(t, u(t)) + Bu(t) + γ ∗ ∂ J(t, γ u (t)) f (t) a.e. t ∈ (0, T ), ⎪ ⎪ ⎪ ⎩u(0) = u , u (0) = v . 0 0 Theorem 10 of [20] provides hypotheses on the data under which the inclusion (3.12) is solvable while Proposition 15 ibid. gives conditions which guarantee the uniqueness of solution. Let us mention that the application of these results to the hemivariational inequality (3.10) requires the introduction of the integral functional J : (0, T ) × L2 (ΓC ; R d ) → R defined by J(t, v) =

ΓC

j(x,t, v(x)) d Γ a.e. t ∈ (0, T ) and v ∈ L2 (ΓC ; R d ),

where j : ΓC × (0, T ) × Rd → R is given by j(x,t, ξ ) = jν (x,t, ξν ) + jτ (x,t, ξτ )

a.e. (x,t) ∈ ΓC × (0, T ), all ξ ∈ Rd .

Furthermore, we can associate with the hemivariational inequality (3.10) the abstract inequality of the form ⎧ ⎪ find u ∈ V with u ∈ W such that ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ j0 (x,t, u (t); v) d Γ ≥ f (t), v

u (t) + A(t, u(t)) + Bu(t), v + Γ C ⎪ ⎪ ⎪ for all v ∈ V and a.e. t ∈ (0, T ), ⎪ ⎪ ⎪ ⎪ ⎩ u(0) = u0 , u (0) = v0 . We refer to [20, 26] to see that existence of solutions to (3.12) implies the existence of solution to the hemivariational inequality (3.10) and that both these problems are equivalent under the condition that either J(t, ·) or −J(t, ·) is regular. On the other hand, the solvability of the inclusion (3.12) is based on the known surjectivity result

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for a class of pseudomonotone operators in a reflexive Banach space (cf. Theorem 6.3.73 of [10]). The existence result of Theorem 1 solves an open problem formulated by Panagiotopoulos in Chapter 7.2 of [38]. We also mention that the strong monotonicity of the operator A(t, ·), cf. H(A )(iv), in the existence part of Theorem 1 can be relaxed, cf. [20]. Finally, we underline that if in addition jν (x,t, ·) and jτ (x,t, ·) are convex potentials, then the hemivariational inequality (3.10) reduces to hyperbolic variational inequality and the conditions H( jν )(v) and H( jτ )(v) hold with mν = mτ = 0. This is due to a fact that the subgradient of a convex function is a maximal monotone operator (cf. Theorem 1.3.19 of [10]). In this case the condition (3.11) is trivially satisfied.

3.3 Integrodifferential Hemivariational Inequalities and Viscoelastic Frictional Contact The physical setting of the problem considered in this section is analogous to the one of (3.1)–(3.6) with one exception, namely the constitutive relation (3.2) is now replaced by the following law t

σ (t) = A (t, ε (u (t))) + B ε (u(t)) +

C (t − s) ε (u(s)) ds in Q.

0

The assumptions also remain the same as in Section 3.2 and, in addition, we assume that the relaxation operator C is linear and continuous (cf. [32] for details). One of the features of this viscoelastic model consists in gathering short and long memory effects in the same constitutive law. This leads to a new class of hemivariational inequalities which involve a Volterra integral term t

u (t) + A(t, u(t)) + Bu(t) +

C(t − s) u(s) ds + γ ∗ ∂ J(t, γ u (t)) f (t).

0

For this kind of hemivariational inequalities, the existence and uniqueness results have been delivered in ([32]), while the dependence of the solution on the memory term and a convergence result can be found in [15]. It can be shown that a sequence of solutions corresponding to a long memory material converges to a solution of the problem with short memory as the relaxation coefficient tends to zero.

3.4 Thermoviscoelastic Frictional Contact Problem In this section we give the mathematical formulation of a frictional contact problem of linear thermoviscoelasticity. The physical setting and notation remain the same as in Section 3.2. The volume Ω is occupied by a viscoelastic body and we assume

3 Nonconvex Models in Contact Mechanics

51

that the temperature changes accompanying the deformations are small and they do not produce any changes in the material parameters which are regarded temperature independent. We assume with no loss of generality that the material density and the specific heat at constant deformation are constants, both set equal to one. The system of equations of motion assuming small displacements and the law of conservation of energy takes the form ui − ∂ j σi j = f1i

in Q,

θ + ∂i qi = −ci j ∂ j ui + g in Q. Further the heat flux vector q ∈ Rd satisfies the Fourier law of heat conduction qi = −ki j ∂ j θ in Q. The behavior of the body is governed by the thermoviscoelastic constitutive equation of the Kelvin-Voigt type

σi j = ai jkl ∂l uk + bi jkl ∂l uk − ci j θ in Q, where {ai jkl } and {bi jkl } are the viscosity and elasticity tensors, respectively, and {ci j } are the coefficients of thermal expansion. We study the evolution of the system state in a time interval [0, T ]: find a displacement field u : Q → R d and a temperature θ : Q → R such that ui − ∂ j ai jkl ∂l uk − ∂ j bi jkl ∂l uk + ∂ j (ci j θ ) = f1i in Q

θ − ∂i (ki j ∂ j θ ) + ci j ∂ j ui = g in Q u(t) = 0 on ΓD × (0, T )

σ (t)ν = f2 (t) on ΓN × (0, T ) −σν (t) ∈ ∂ jν (t, uν (t)),

−στ (t) ∈ ∂ jτ (t, uτ (t)) on ΓC × (0, T )

−ki j ∂ j θ νi ∈ ∂ j(t, θ (t)) on ΓC × (0, T )

θ (t) = 0 on (ΓD ∪ ΓN ) × (0, T ) u(0) = u0 , u (0) = v0 , θ (0) = θ0 in Ω . The weak form of the problem consists of a system of coupled evolution inclusions: ⎧ ∗ ⎪ ⎪u (t) + A u (t) + B u(t) + C1 θ (t) + γ ∂ J1 (t, γ u (t)) f (t) a.e. t ⎪ ⎨ (3.13) θ (t) + C2 θ (t) + C3 u (t) + γs∗ ∂ J2 (t, γs θ (t)) g(t) a.e. t ⎪ ⎪ ⎪ ⎩u(0) = u , u (0) = v , θ (0) = θ . 0 0 0 We refer to [8] for a detail explanation of the notation used for this system. Similarly to our description in Section 3.2, the system of inclusions (3.13) is related to the system of the hemivariational inequality of hyperbolic type for the displacement

52

S. Migórski and A. Ochal

and the parabolic hemivariational inequality for the temperature. The existence and uniqueness of solutions to (3.13) is proved in [5].

3.5 Bilateral Frictional Contact Problem in Viscoelasticity In the following, under the notation of Section 3.2, we consider a dynamic viscoelastic model with the equation of motion (3.1), the Kelvin-Voigt constitutive law (3.2), the displacement and traction boundary conditions (3.3) and (3.4), the initial conditions (3.6) and the following boundary conditions on the contact surface: ⎧ ⎨uν (t) = 0 (3.14) ⎩−σ (t) ∈ μ (t) p(|Rσ (t)|) ∂ j(t, u (t)) on Γ × (0, T ). τ ν C τ Here μ ∈ L∞ (ΓC × (0, T )), μ ≥ 0 a.e. on ΓC × (0, T ) is the coefficient of friction and R ∈ L (H −1/2 (Γ ); L2 (Γ )) is a regularization operator. The symbol ∂ j stands for the subdifferential of the contact superpotential j(x,t, ·) which is now assumed to be convex. We do not state here various possible hypotheses on the friction function p (we refer to [7]). Instead we comment on examples. In (3.14) the potential j : ΓC × (0, T ) × R d → R can be given by j(x,t, ξ ) = ξ for a.e. (x,t) ∈ ΓC × (0, T ) and all ξ ∈ R d . Since the subdifferential of the function ϕ (ξ ) = ξ is the unit vector in the direction of ξ when ξ = 0, and is the whole unit ball when ξ = 0, in this case we obtain the Coulomb law of friction, and the contact boundary condition −στ (x,t) ∈ μ (x,t) p(x, |Rσν (x,t)|) ∂ uτ (x,t) on ΓC × (0, T ) is equivalent to ⎧ ⎪ στ ≤ μ p(|Rσν |) with ⎪ ⎪ ⎨ στ < μ p(|Rσν |) =⇒ uτ = 0, ⎪ ⎪ ⎪ ⎩ σ = μ p(|Rσ |) =⇒ ∃ λ ≥ 0 : σ = −λ u on Γ × (0, T ). τ ν τ C τ

(3.15)

If p is a known function which is independent of |Rσν |, i.e. p(x, r) = h(x) with h ∈ L∞ (ΓC ), h ≥ 0, then the conditions (3.15) become the Tresca friction law (cf. Section 2.6 of [39] for a detailed discussion). If p(x, r) = |r|, then (3.15) reduces to the usual regularized Coulomb friction boundary condition which was extensively used in the literature (cf. e.g. [11, 13, 14, 22, 26, 37, 39, 40, 41, 43]). If p(x, r) = |r|(1 − δ |r|)+ with (·)+ = max{·, 0}, where δ is a small positive coefficient related to the wear and hardness of the surface, then we obtain a modification of the Coulomb law of friction. Such a modification, called the SJK model, consists of the factor (1 − δ | · |)+ and was derived in [42] from the thermodynamical considerations. It leads to the condition

3 Nonconvex Models in Contact Mechanics

53

⎧ ⎪ ⎪στ ≤ μ |Rσν |(1 − δ |Rσν |)+ with ⎪ ⎨ στ < μ |Rσν |(1 − δ |Rσν |)+ =⇒ uτ = 0, ⎪ ⎪ ⎪ ⎩ σ = μ |Rσ |(1 − δ |Rσ |) =⇒ ∃ λ ≥ 0 : σ = −λ u on Γ × (0, T ). τ ν ν + τ C τ For the discussion of the SJK generalization of the Coulomb law, we refer to [42, 39, 40]. Finally, if the potential is given by j(x,t, ξ ) = α1 |ξ1 | + . . . + αd |ξd |, where αi ≥ 0, we are lead to the orthotropic friction. In this case the relations −στi ∈ μ p(|Rσν |) αi ∂ |uτi | can be interpreted as follows ⎧ ⎪ |στi | ≤ μi p(|Rσν |) with ⎪ ⎪ ⎨ στi < μi p(|Rσν |) =⇒ uτi = 0, ⎪ ⎪ ⎪ ⎩ στi = μi p(|Rσν |) =⇒ στi = −λi uτi with λi ≥ 0 on ΓC × (0, T ) for μi = μ αi . Since j(x,t, ·) is convex, the viscoelastic contact model (3.1), (3.2), (3.3), (3.4), (3.6) and (3.14) is described by a hyperbolic variational inequality. The results on the existence and uniqueness of weak solutions to this problem when the friction coefficient is sufficiently small can be found in Theorem 12 of [7].

3.6 Bilateral Contact Problem for Viscoelastic Piezoelectric Materials with Adhesion In this section we describe the viscoelastic problem of piezoelectricity with adhesion. We present its classical and variational formulations. We complete the notation of Section 3.2. We assume that the set Ω is occupied by a viscoelastic piezoelectric body that remains in adhesive contact with an insulator obstacle, the so-called foundation. The contact is supposed to be bilateral, so during the process there is no gap between the body and the foundation. We consider now two partitions of Γ = ∂ Ω . A first partition is given (as in Section 3.2) by three disjoint measurable parts ΓD , ΓN and ΓC such that m(ΓD ) > 0, and a second one consists of two disjoint measurable parts Γ1 and Γ2 such that m(Γ1 ) > 0 and ΓC ⊂ Γ2 . In addition to the notation of Section 3.2, for simplicity, we assume free electric charges. Due to the adhesive contact, we suppose a nonmonotone possibly multivalued law between the shear and the tangential displacement. This law depends also on a bonding field, denoted by β , which describes the pointwise fractional density of active bonds on ΓC . Following [12], the evolution of the bonding field is governed by an ordinary differential equation depending on the displacement and considered on contact surface. When β = 1 at a point of the contact part, the adhesion is complete and all

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the bonds are active, when β = 0 all bonds are inactive and there is no adhesion, when 0 < β < 1 the adhesion is partial and a fracture β of the bonds is active. The dynamical model for the process under consideration is as follows: find a displacement field u : Q → Rd , an electric potential ϕ : Q → R and a bonding field β : ΓC × (0, T ) → [0, 1] such that u (t) − divσ (t) = f1 (t) in Q

(3.16)

divD(t) = 0 in Q σ (t) = A ε (u (t)) + B ε (u(t)) − P E(ϕ (t)) in Q

(3.17) (3.18)

D(t) = DE(ϕ (t)) + P ε (u(t)) in Q

(3.19)

u(t) = 0 on ΓD × (0, T ) σ (t)ν = f2 (t) on ΓN × (0, T )

(3.20) (3.21)

ϕ (t) = 0 on Γ1 × (0, T ) D(t) · ν = 0 on Γ2 × (0, T )

(3.22) (3.23)

uν (t) = 0 on ΓC × (0, T ) −στ (t) ∈ ∂ j(β (t), uτ (t)) on ΓC × (0, T )

(3.24) (3.25)

β (t) = F(t, u(t), β (t)) on ΓC × (0, T ) β (0) = β0 on ΓC u(0) = u0 , u (0) = v0 in Ω .

(3.26) (3.27) (3.28)

We shortly comment on the above model. Equations (3.16) and (3.17) are the equation of motion for the stress field and the Gauss equilibrium equation for the electric displacement field, respectively. Recall that div stands for the divergence operator for vector valued functions, i.e. divD = ∂i Di . The relations (3.18) and (3.19) represent the electroviscoelastic constitutive law of the material in which A , B, D and P are respectively the (fourth order) viscosity tensor, the (fourth order) elasticity tensor, the (second order) electric permittivity tensor and the (third order) piezoelectric tensor. The equation (3.18) describes the converse effect and (3.19) models the direct effect of piezoelectricity. Furthermore, D : Q → Rd , D = {Di } denotes the electric displacement field and P is the tensor transposed to P. The decoupled state (purely viscoelastic and purely electric deformations) can be obtained by setting the piezoelectric tensor P = 0. The electric field–potential relation is given by E(ϕ ) = {−∂i ϕ } in Q. The equations (3.20) and (3.21) are the displacement and traction boundary conditions, respectively, while (3.22) and (3.23) represent the electric boundary conditions. Condition (3.24) shows that the contact is bilateral and (3.25) is the subdifferential boundary condition with a nonconvex nonsmooth superpotential j. This relation says that the tangential traction στ depends on the intensity of adhesion β and the tangential displacement uτ . The evolution of the adhesion (bonding) field β is governed by the ordinary differential equation (3.26) on the contact surface and (3.27) represents a given initial bonding field (cf. [12]). The adhesive evolution rate function F depends on both the bonding field and the displacement and may change sign. This allows for rebonding after debonding took

3 Nonconvex Models in Contact Mechanics

55

place, and it allows for possible cycles of debonding and rebonding (cf. [39, 29] for examples of F). Finally, the initial values for the displacement and the velocity are prescribed in (3.28). in Section 3.2 and introducing the space Φ = Using1 the notation introduced ϕ ∈ H (Ω ) | ϕ = 0 on Γ1 , the problem (3.16)–(3.28) in its weak form is formulated as follows: find u ∈ V with u ∈ W , ϕ ∈ C(0, T ; Φ ) and β ∈ W 1,∞ (0, T ; L2 (ΓC )) such that ⎧ u (t), v + a(u(t), v) + b(u(t), v) + e(ϕ (t), v)+ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ j0 (x, β (t), uτ (t); vτ ) dΓ ≥ f (t), v for a.e. t, all v ∈ V + ⎪ ⎪ ⎨ ΓC (3.29) d(ϕ (t), ψ ) = e(u(t), ψ ) for a.e. t ∈ (0, T ), all ψ ∈ Φ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ β (t) = F(t, u(t), β (t)) on ΓC , for a.e. t ∈ (0, T ) ⎪ ⎪ ⎪ ⎪ ⎩ β (0) = β0 on ΓC , u(0) = u0 , u (0) = v0 in Ω . This problem is a system coupled with the evolution hemivariational inequality for the displacement, the time dependent stationary equation for the electric potential and the ordinary differential equation for the bonding field. The existence of a weak solution to (3.29) can be found in [29]. The other frictional contact problems in piezoelectricity have been studied in [16, 17, 23, 34].

3.7 Comments on Other Nonconvex Inequality Models In this section we give further comments on results for contact problems with nonsmooth and nonconvex superpotentials. 1. A general approach to study viscoelastic contact problems with subdifferential boundary conditions has been introduced in [20, 26]. This approach allows to unify several methods for models considered in contact mechanics and obtain new existence results which are not available in the literature. The existence results for the dynamic hemivariational inequalities of hyperbolic type can be found in [21, 22, 27, 36]. 2. Optimal control problems for dynamic hemivariational inequalities have been investigated in several papers, cf. e.g. [8, 4, 6]. The results on inverse and identification problems for such inequalities can be found in [31]. 3. For the contact problems with slip dependent friction, we refer to [24] and the references therein. The hemivariational inequalities modeling contact with wear and adhesion have been studied in [1, 2, 29]. 4. Homogenization results for stationary elasticity problems described by hemivariational inequalities are presented in [19, 18].

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5. A class of inequality problems for the stationary Navier-Stokes type operators related to the model of motion of a viscous incompressible fluid through a tube or channel have been studied in [25]. 6. Vanishing viscosity method and asymptotic behavior of solutions to hemivariational inequalities have been considered in [28, 18]. 7. The quasistatic hemivariational inequalities which describe the contact problems have been treated by a vanishing acceleration method in [30] while the antiplane frictional contact problems have been considered in [33]. Acknowledgements. The authors would like to express their gratitude to Prof. M. Kuczma and Prof. K. Wilmanski for their invitation to deliver a Keynote Lecture and to Prof. P. Litewka and Prof. A. Zmitrowicz for an invitation to participate in the Minisymposium on Computational Contact Mechanics at the 18th International Conference on Computer Methods in Mechanics held in Zielona Gora, Poland, May 18–21, 2009. Research supported in part by the Ministry of Science and Higher Education of Poland under grant no. N201 027 32/1449.

References [1] Bartosz, K.: Hemivariational inequality approach to the dynamic viscoelastic sliding contact problem with wear. Nonlinear Anal. 65, 546–566 (2006) [2] Bartosz, K.: Hemivariational inequalities modeling dynamic contact problems with adhesion. Nonlinear Anal. 71, 1747–1762 (2009) [3] Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, Interscience, New York (1983) [4] Denkowski, Z., Migórski, S.: Sensitivity of optimal solutions to control problems for systems described by hemivariational inequalities. Control Cybernet. 33, 211–236 (2004) [5] Denkowski, Z., Migórski, S.: A system of evolution hemivariational inequalities modeling thermoviscoelastic frictional contact. Nonlinear Anal. 60, 1415–1441 (2005) [6] Denkowski, Z., Migórski, S.: On sensitivity of optimal solutions to control problems for hyperbolic hemivariational inequalities. Lect. Notes Pure Appl. Math. 240, 145–156 (2005) [7] Denkowski, Z., Migórski, S., Ochal, A.: Existence and uniqueness to a dynamic bilateral frictional contact problem in viscoelasticity. Acta Appl. Math. 94, 251–276 (2006) [8] Denkowski, Z., Migórski, S., Ochal, A.: Optimal control for a class of mechanical thermoviscoelastic frictional contact problems. Control Cybernet. 36, 611–632 (2007) [9] Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Theory. Kluwer Academic/Plenum Publishers, Boston (2003) [10] Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Applications. Kluwer Academic/Plenum Publishers, Boston (2003) [11] Duvaut, G., Lions, J.-L.: Les Inéquations en Mécanique et en Physique. Dunod, Paris (1972) [12] Frémond, M.: Adhérence des solides. J. Mécanique Théorique et Appliquée 6, 383–407 (1987)

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[13] Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. American Mathematical Society, International Press, Providence (2002) [14] Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1987) [15] Kulig, A.: Evolution Inclusions and Hemivariational Inequalities for Unilateral Contact in Viscoelasticity. PhD Thesis, Jagiellonian University, Krakow (2009) [16] Li, Y., Liu, Z.: Dynamic contact problem for viscoelastic piezoelectric materials with slip dependent friction. Nonlinear Analysis, Theory, Methods and Applications 71, 1414–1424 (2009) [17] Liu, Z., Migórski, S.: Noncoercive damping in dynamic hemivariational inequality with application to problem of piezoelectricity. Discrete Contin. Dyn. Syst. Ser. B 9, 129–143 (2008) [18] Liu, Z., Migórski, S., Ochal, A.: Homogenization of boundary hemivariational inequalities in linear elasticity. J. Math. Anal. Appl. 340, 1347–1361 (2008) [19] Migórski, S.: Homogenization technique in inverse problems for boundary hemivariational inequalities. Inverse Problems in Eng. 11, 229–242 (2003) [20] Migórski, S.: Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction. Appl. Anal. 84, 669–699 (2005) [21] Migórski, S.: Boundary hemivariational inequalities of hyperbolic type and applications. J. Global Optim. 31, 505–533 (2005) [22] Migórski, S.: Evolution hemivariational inequality for a class of dynamic viscoelastic nonmonotone frictional contact problems. Comput. Math. Appl. 52, 677–698 (2006) [23] Migórski, S.: Hemivariational inequality for a frictional contact problem in elastopiezoelectricity. Discrete Contin. Dyn. Syst. 6, 1339–1356 (2006) [24] Migórski, S., Ochal, A.: Hemivariational inequality for viscoelastic contact problem with slip dependent friction. Nonlinear Anal. 61, 135–161 (2005) [25] Migórski, S., Ochal, A.: Hemivariational inequalities for stationary Navier-Stokes equations. J. Math. Anal. Appl. 306, 197–217 (2005) [26] Migórski, S., Ochal, A.: A unified approach to dynamic contact problems in viscoelasticity. J. Elasticity 83, 247–275 (2006) [27] Migórski, S., Ochal, A.: Existence of solutions for second order evolution inclusions with application to mechanical contact problems. Optimization 55, 101–120 (2006) [28] Migórski, S., Ochal, A.: Vanishing viscosity for hemivariational inequality modeling dynamic problems in elasticity. Nonlinear Anal. 66, 1840–1852 (2007) [29] Migórski, S., Ochal, A.: Dynamic bilateral contact problem for viscoelastic piezoelectric materials with adhesion. Nonlinear Anal. 69, 495–509 (2008) [30] Migórski, S., Ochal, A.: Quasistatic hemivariational inequality via vanishing acceleration approach. SIAM J. Math. Anal. 41, 1415–1435 (2009) [31] Migórski, S., Ochal, A.: An inverse coefficient problem for a parabolic hemivariational inequality. Appl. Anal. (in press, 2009) [32] Migórski, S., Ochal, A., Sofonea, M.: Integrodifferential hemivariational inequalities with applications to viscoelastic frictional contact. Math. Models Methods Appl. Sci. 18, 271–290 (2008) [33] Migórski, S., Ochal, A., Sofonea, M.: Solvability of dynamic antiplane frictional contact problems for viscoelastic cylinders. Nonlinear Anal. 70, 3738–3748 (2009) [34] Migórski, S., Ochal, A., Sofonea, M.: Weak solvability of a piezoelectric contact problem. Eur. J. Appl. Math. 20, 145–167 (2009) [35] Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, Inc., New York (1995)

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[36] Ochal, A.: Existence results for evolution hemivariational inequalities of second order. Nonlinear Anal. 60, 1369–1391 (2005) [37] Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions. Birkhäuser, Basel (1985) [38] Panagiotopoulos, P.D.: Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer, Berlin (1993) [39] Shillor, M., Sofonea, M., Telega, J.J.: Models and Analysis of Quasistatic Contact. Springer, Berlin (2004) [40] Sofonea, M., Han, W., Shillor, M.: Analysis and Approximation of Contact Problems with Adhesion or Damage. Chapman-Hall/CRC Press, New York (2006) [41] Sofonea, M., Matei, A.: Variational Inequalities with Applications. A Study of Antiplane Frictional Contact Problems. Springer, New York (2009) [42] Strömberg, N., Johansson, L., Klarbring, A.: Derivation and analysis of a generalized standard model for contact, friction and wear. Internat. J. Solids Structures 33, 1817– 1836 (1996) [43] Wriggers, P.: Computational Contact Mechanics. Wiley, Chichester (2002)

Chapter 4

Quadrature for Meshless Methods John E. Osborn

The results in this paper are joint work with I. Babuška, U. Banerjee, Q. Li, and Q. Zhang. Abstract. In this paper we discuss quadrature for meshless methods. We present results of two analyses. The first of these was developed in [1] and the second was developed in [2]. Using a simple example we show that a natural quadrature scheme that satisfies no special properties, leads to severe inaccuracies in the approximate solution. We then show in the first analysis that a simple correction to the stiffness matrix controls the effect of the quadrature. In the second analysis we show that if the quadrature scheme satisfies Green’s formula, then the effect of the quadrature is controlled.

4.1 Introduction Meshless Methods (MM) have been the focus of considerable interest in recent yeas, especially in the engineering community. This interest was mainly stimulated by difficulties with mesh generation with the usual Finite Element Method (FEM), which is delicate in many situations; for example, when the domain has complicated geometry or when the mesh changes with time, as in crack propagation, and remeshing is required at each time step. It was recognized early (see, e.g., [5], [6]) that the important problem of creating effective quadrature schemes for MM is more challenging that that for FEM. The FEM was completely analyzed 30 year ago (see, e.g., [7]). The major feature of the FEM is that the shape functions are piecewise polynomials of degree p, and hence their k-th order derivatives vanish on each element for k ≥ p + 1. This permits the John E. Osborn Department of Mathematics, University of Maryland, College Park, MD 20742 e-mail: [email protected] M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 59–73. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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J.E. Osborn

exact calculation of the stiffness matrix for PDEs with constant coefficients. With the MM, the shape functions are generally not polynomials, and their k-th orderderivatives grow with k, and essentially no quadrature scheme will be accurate. In this paper we discuss two developments, referred to below as the First Analysis and the Second Analysis, in the creation of effective quadrature schemes for MM. These analyses are fully developed in [1] and [2], respectively. Several engineering papers ([3], [4], [5], [6]) discuss quadrature for MM, concentrating mainly on implementational issues, but to the best of our knowledge, a careful mathematical analysis of the effect of quadrature in MM was first reported in [1]. It is shown in [1] that MM with standard quadrature—assuming nothing special about the quadrature scheme—is erratic and practically no reasonably accuracy is obtained, and the row sum condition is identified and shown to be the key hypothesis in creating quadrature schemes that perform well. Specifically, it is shown that if the stiffness matrix satisfies the zero row sum condition, the MM reproduces polynomials of degree k = 1, and certain other conditions are satisfied, then the energy norm of the error in the approximate solution is O(h + η ), where h is the standard discretization parameter related to the diameters of the supports of the shape functions and η is a parameter indicating the accuracy of the underlying numerical quadrature. Furthermore, a simple “recipe” is identified for correcting the elements of the stiffness matrix so that zero row sum condition is satisfied. In [2], our second analysis is presented. The major hypotheses in this analysis are that the MM reproduce polynomials of degree k ≥ 1 and that the quadrature satisfies a Green’s formula. Under these and certain other conditions it is shown that the energy norm of the error in the approximate solution is O(hk−1 (h + η )), where η is a parameter related to the accuracy of the numerical quadrature and h is the standard discretization parameter. We also indicate how to obtain numerical quadrature schemes satisfying these conditions. Thus MM does not yield optimal order of convergence for η = O(h). Certainly if η = O(h), we have the optimal order of convergence. It is important to note that the parameter η (see (4.16)) associated with the quadrature scheme usually used in the FEM, namely the Gauss rule, is O(h). We mention that the numerical integration used in this paper yields a nonsymmetric stiffness matrix. But this does not pose a serious problem since nonsymmetric linear systems can be solved efficiently by iterative methods. Both analyses are for MM approximation of a second order Neumann boundary value problem. In both analyses, we see that MM does not yield optimal convergence if η , respectively, η does not equal O(h). The outline of this paper is as follows: In Section 4.2, we state the main results of the First Analysis, and in Section 4.3 we present the main results of the Second Analysis. In Section 4.4 a comparison of the results of the two analyses is presented. Computational results (plots) are included in Sections 4.2 and 4.3. The author wishes to thank Wiley InterScience for permission to use Figures 4.1 and 4.2 in this paper; they were published in International Journal for Numerical Methods in Engineering. And to thank Elsevier Science for permission to use Figures 4.3–4.5 in this paper; they were published in Computer Methods in Applied Mechanics and Engineering.

4 Quadrature for Meshless Methods

61

4.2 First Analysis The results stated in this section are fully developed and proved in [1].

4.2.1 Preliminaries Throughout the paper Ω will be a bounded domain in Rd with boundary Γ = ∂ Ω , H m (Ω ) will be the usual Sobolev space with norm and seminorm, um,Ω and |u|m,Ω , respectively. And uL2(Ω ) , uL∞ (Ω ) , uL2 (Γ ) , and uL∞ (Ω ) will denote the usual norms on L2 (Ω ), L∞ (Ω ), L2 (Γ ), and L∞ (Γ ), respectively, whereas uE = 1/2 2 denotes the energy norm on H 1 (Ω ). Ω |∇u| dx We consider the model problem, −Δ u = f in Ω , (4.1) ∂u ∂ n = g on Γ = ∂ Ω with variational formulation, u ∈ H 1 (Ω ) , B(u, v) = L(v), ∀v ∈ H 1 (Ω )

(4.2)

where B(u, v) =

Ω

∇u · ∇v dx ,

L(v) =

Ω

f v dx +

Γ

gv ds.

If the compatibility condition, L(1) = 0, is satisfied, the solution exists and is unique up to a constant. We assume, in addition to the compatibility condition, that Γ , f , and g are such that u is in H 2 (Ω ). Consider first an MM based on uniformly distributed particles and associated shape functions:

x1 − j1 h x2 − j2 h h h x j = ( j1 h, j2 h), φ j (x1 , x2 ) = φ , , with φ given, h h where j = ( j1 , j2 ) ∈ Z2 , with Z the integer lattice, and 0 < h. Let Vh = span{φ hj : j ∈ Nh }, where Nh is an index set and {φ hj : j ∈ Nh } is a basis for Vh . Then we consider the Galerkin method corresponding to the approximation space Vh . Our MM approximation uh to u is characterized by uh ∈ Vh . (4.3) B(uh , v) = L(v), ∀v ∈ Vh If we let

γi,h j =

Ω

∇φih · ∇φ hj dx =

ωih ∩ω hj

∇φih · ∇φ hj dx

62

J.E. Osborn

and lih =

Ω

f φih dx +

Γ

gφih ds =

f φih dx +

ωih

Γ ∩ω hi

gφih ds

be the stiffness matrix and the right-hand side vector, and we write uh (x) = ∑ j∈Nh uh, j φ hj (x), then uh satisfies (15.3) if and only if

∑ γi,h j uh, j = lih , for all i ∈ Nh .

j∈Nh

We also consider the quadrature version of these integrals:

γi,h∗j = −

ωih ∩ω hj

∇φi · ∇φ j dx and lih∗ = − f φih dx + − ωih

Γ ∩ω hi

gφih ds,

where − is a quadrature version of . Our MM quadrature approximation u∗h to u is characterized by ∗ uh ∈ Vh (4.4) B∗ (u∗h , v) = L∗ (v), ∀v ∈ Vh , where

B∗ (uh , vh ) =

∑

γi,h∗j uh,i vh, j and L∗ (vh ) =

i, j∈Nh

∑ lih∗ vh,i .

i∈Nh

If we write u∗h = ∑ j∈Nh u∗h, j φhj (x), then u∗h satisfies (4.4) if and only if ∑ j∈Nh γi,h∗j u∗h, j = lih∗ , for all i ∈ Nh . We will see that u − u∗h E behaves erratically and is not small, whereas u − uh E ≤ O(h). Later we will consider a corrected stiffness matrix, γi,h∗∗ j , and corrected right-hand side vector, lih∗∗ , and the corresponding quadrature approximate solution, u∗∗ h , and show that it provides an accurate approximation to u.

4.2.2 An Example / Numerical Results

Consider

−u = cos x, x ∈ Ω ≡ (0, π ) , u (0) = u (π ) = 0

with variational formulation, u ∈ H 1 (0, π ) B(u, v) = F(v), ∀v ∈ H 1 (0, π ),

π

π

where B(u, v) = 0 u v dx and F(v) = 0 cos x v dx. This is a one-dimensional version of the model problem (4.1). To construct a meshless method, we let xhj = jh, for j = . . . , −1, 0, 1, . . . , where h = π /n, for n = 1, 2, . . . , be a family of uniformly distributed particles. And we

4 Quadrature for Meshless Methods

63

use the Reproducing Kernel Particle (RKP) construction, with respect to a window function, to construct associated shape functions that are reproducing of order 1, i.e., that satisfy (4.5) ∑ (xhj )i φ hj (x) = xi , ∀x and i = 0, 1. j∈Nh

Plot of ||u−u* || /||u|| for

Plot of ||u−u* || /||u|| for

Plot of ||u−u* || /||u|| for

Trapezoid Rule, m=10

Trapezoid Rule, m=100

Trapezoid Rule, m=200

h E

E

h E

0.2 0

E

2 1 0

0.5

1 1.5 2 h Plot of ||u−u* || /||u|| for h E

3

h E

0.4

||u−u* || /||u||

||u−u*h||E/||u||E

E h E

||u−u* || /||u||

0.6

0

E

0.5

1

E

||u−u* || /||u||

h E

0

0.2 1.5

0.4 0.2 0

E

2

h E

E

15

h E

10 5 0

0

0.5

1 h

1 2 h Plot of ||u−u* || /||u|| for E

quad, tol=10−4 ||u−u* || /||u||

||u−u*h||E/||u||E

E h E

||u−u* || /||u||

0.4

E

0.6

quad, tol=10−2

0.6

1 1.5 h Plot of ||u−u* || /||u|| for

0.8

0

1 2 h Plot of ||u−u* || /||u|| for h E

0.5

h E

0.2

E

0.8

1 h

0

Gauss Rule, p=10

0.4

quad, tol=10−1

0.5

0.05

E

0.6

h Plot of ||u−u* || /||u|| for

0

0.1

0

1.5

0.8

0

1.5

h E

1 h E

||u−u*h||E/||u||E

||u−u*h||E/||u||E

2

0

0.5

0.2

Gauss Rule, p=4

4

0

E

0.15

h Plot of ||u−u* || /||u|| for

Gauss Rule, p=3

0

h E

4

0.8

0

E

1.5

0.8 0.6 0.4 0.2 0

0

1 h

2

u−u∗

Fig. 4.1 The plot of uh E with respect to h for various quadrature schemes. We observe E that the behavior of the relative error is erratic and that practically no reasonable accuracy was obtained. This figure is from [1], and has been used with permission of the publisher.

Note that (15.5) implies that ∑ j∈Nh φ hj (x) = 1, ∀x, i.e., {φ hj } is a partition of unity. Specifically, we use the window function

1 w(x) = exp 2 , −r < x < r, with r = 1.1. x − r2

64

J.E. Osborn u−u∗

In Fig. 4.1, we plot uhE E with respect to h for various quadrature schemes: the m-panel Trapezoid Rule, the p-point Gauss Rule, and MATLAB’s quad (adaptive Simpson quadrature) with various tolerances, tol. We observe that the behavior of the relative error is erratic, and that practically no reasonable accuracy is obtained. On the other hand, when we use exact integration, we get good accuracy. What feature distinguishes the stiffness matrix for exact integration from that for quadrature? Here is one possibility: With exact integration the Row Sums of the stiffness matrix are 0; while with quadrature, they differ from 0. For exact integration, this is seen from the calculation,

∑ γi,h j = ∑

j∈Nh

j∈Nh Ω

∇φih · ∇φ hj dx = =

Ω

Ω

∇φih · ∇(

∑ φ hj ) dx

j∈Nh

∇φih · ∇1 dx dy = 0;

(4.6)

and for quadrature, it is seen by examining the calculated stiffness matrix. This observation suggests a possibility when using quadrature: We consider a corrected stiffness matrix, γi,h∗∗ j , formed by modifying just the diagonal elements; h∗∗ h∗ we set γi,i = − ∑ j=i γi, j , so that ∑ j γi,h∗∗ j = 0, ∀i.

4.2.3 Theoretical Results We suppose Vh ⊂ H 1 (Ω ) is a one parameter family of finite dimensional approximation subspaces, and suppose {φ hj (x) : j ∈ Nh } is a basis for Vh . The φ hj (x) are our shape functions. We impose certain assumptions (Axioms) on the spaces Vh and on the quadrature schemes. We mention a few of the axioms: • Axiom 4.1 (Reproducing Polynomial Property; cf. (15.5))

∑ p(xi )φih (x) = p(x) for x ∈ Ω , ∀ polynomials of deg ≤ 1.

i∈Nh

• Axiom 4.2 There is a constant C, independent of h, such that

∑

i, j∈Nh

|γi,h j |(vi − v j )2 ≤ C

∑

(−γi,h j )(vi − v j )2 , ∀v =

i, j∈Nh

∑ vi φih ∈ Vh .

i∈Nh

This axiom is easily seen to be true for certain finite element spaces, but is hard to verify in general. • Axiom 4.3 (Row Sum Condition)

∑ γi,h∗j = 0.

j∈Nh

4 Quadrature for Meshless Methods

65

• Axiom 4.4 h h γi,h∗j = γi,h j + ηi,h j , fih∗ = fih + εih , and gh∗ i = gi + τi , with

|ηi,h j | ≤ η max(|γi,h j |, ν hd ), |εih | ≤ ε max(| fih |, ν hd f L∞ (Ω ) ), and |τih | ≤ τ max(|ghi |, ν hd−1 gL∞ (Γ ) ), where φih L∞(Ω ) ≤ ν and d = dimension. These are the relative and absolute quadrature error assumptions, with error parameters η , ε , and τ . Theorem 4.1 Suppose our shape functions and our quadrature scheme satisfy Axioms 4.1–4.4 and certain other axioms. Then for small η , there is a constant C > 0, independent of u, η , ε , τ , and h such that uh − u∗hE ≤ C η uE + ε f L∞ (Ω ) + τ gL∞(Γ ) , ∀h. (4.7) Theorem 4.2 Suppose our shape functions and our quadrature scheme satisfy Axioms 4.1–4.4 and certain other axioms. Then for small η , there is a constant C > 0, independent of u, η , ε , τ , and h such that u − u∗hE ≤ C hu2,Ω + η uE + ε f L∞ (Ω ) + τ gL∞(Γ ) , ∀h. (4.8)

Correction Process Axiom 4.3 (the Row Sum Condition) is unlikely to be satisfied. To see this note that the Row Sum Condition is satisfied for γi,h j (cf. (4.6)), and that ∑i, j γi,h∗j , which differs from ∑i, j γi,h j due to the quadrature error, is unlikely to be 0, and so we do not, in fact, have an estimate for u − u∗hE . We thus consider a corrected stiffness matrix γi,h∗∗ j , redefined by modifying the diagonal elements: h∗∗ γi,i =−

∑

j∈Nh , j=i

γi,h∗j .

The Row Sum Condition is now satisfied. Axiom 4.4 also holds, but with modified error parameters cη , c(ε + τ ), and c(ε + τ ). So we have the following error estimates for the corrected approximation. Theorem 4.3 Suppose our shape functions and our quadrature scheme satisfy Axioms 4.1-4.4 and certain other axioms. Then for small η , there is a constant C, independent of u, η , τ , η , and h, such that uh − u∗∗ h E ≤ C[η uE + (ε + τ ) f L∞ (Ω ) + (ε + τ )gL∞ (Γ ) ], ∀h.

(4.9)

Theorem 4.4 Suppose our shape functions and our quadrature scheme satisfy Axioms 4.1-4.4 and certain other axioms. Then for small η , there is a constant C, independent of u, η , τ , η , and h, such that u − u∗∗ h E ≤ C[hu2,Ω + η uE + (ε + τ ) f L∞ (Ω ) + (ε + τ )gL∞ (Γ ) ], ∀h. (4.10)

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J.E. Osborn

4.2.4 Numerical Results u −u∗∗

u−u∗∗

In Figure 2.2 we present log-log plots of the relative errors huhE E and uhE E with respect to h for the one dimensional problem discussed earlier in the section. The stiffness matrix γh∗ is computed with the same quadrature methods, but the matrix is then corrected to satisfy the Row Sum Condition (Axiom 4.3), and denoted by γh∗∗ . The right-hand side vector fih + ghi has been computed exactly (ε = τ = 0).

Loglog Plot of ||uh−u**h ||E/||u||E for

Loglog Plot of ||uh−u**h ||E/||u||E for

Loglog Plot of ||uh−u**h ||E/||u||E for

Trapezoid Rule with correction

Gauss Rule with correction

quad with correction

−6 −8 −3

−2

−1 h

0

E

−4

E

tol=10−4

h

−6

−2

−1 h

−6 −8

p=100

−10

0

−12 −3

1

−5

tol=10

tol=10−8 −2

−1 h

0

1

Loglog Plot of ||u−u**|| /||u|| for

Trapezoid Rule with correction

Gauss Rule with correction

quad with correction

h E

0

−3

−2

E

p=10

−3

−1

−1 h

Fig. 4.2 The log-log plot of

0

1 uh −u∗∗ h E uE

−4 −3

−2

−1 h

0

1

−1

tol=10

−2

tol=10 tol=10−4

−2 −3

tol=10−5 −8 tol=10

−4 −3

−2

p=100

m=100 −4 −3

E

−2

E

0

p=2 p=3 p=4

−1

h E

h

m=4

E

||u−u**|| /||u||

m=2 m=3 m=10

−1 −2

E

||u−u** || /||u||E h E

E E

tol=10−2

Loglog Plot of ||u−u**|| /||u|| for

0

h

−2

−4

−8 −3

1

−1

tol=10

Loglog Plot of ||u−u**|| /||u|| for h E

||u−u**|| /||u||

−2

h

m=100

h

h

−4

0

p=2 p=3 p=4 p=10

||u −u**|| /||u||

−2

0 ||uh−u** || /||u||E h E

m=2 m=3 m=10 m=4

E

||u −u**|| /||u||

E

0

−1 h

0

1

u−u∗∗ h E with respect to h with correction for uE u−u∗∗ h E error u first decreases with decreasing h, E

and

various quadrature schemes. The relative and then approaches a constant as h → 0. This figure should be compared with Fig. 4.1. This figure is from [1], and has been used with permission of the publisher.

4 Quadrature for Meshless Methods

67

We observe that the relative error uh − u∗∗ h E /uE becomes nearly constant as h → 0; this constant reflects the accuracy of the quadrature (η ). On the other hand, the relative error u − u∗∗ h E /uE first decreases with decreasing h and then levels off, becoming nearly constant as h → 0. These figures and estimate (15.10) indicate that the error has two components: one due to the MM approximation (see the estimate (15.8) for u − uhE when η = ε = τ = 0) and the other due to quadrature (see estimate (15.9)). From (15.10) and the second row in Fig. 4.2 we see that for given quadrature accuracy η , ε , and τ , the error for small h is completely governed by the quadrature accuracy. We have to set η , ε , and τ equal to o(1) if we want the relative error to converge, and we have to set η , ε , and τ equal to O(h) if we want the relative error to be O(h).

4.3 Second Analysis The results stated in this section are fully developed and proved in [2].

4.3.1 Preliminaries We again consider the problem introduced in Section 4.2: u ∈ H 1 (Ω ) B(u, v) = L(v), ∀v ∈ H 1 (Ω ). As said there, if the compatibility condition is satisfied, this problem has a unique solution up to a constant. A standard way of specifying a unique solution is to consider a linear functional Φ : L2 (Ω ) → R with Φ (1) > 0, and seek the unique solution u satisfying Φ (u) = 0. We will use an alternate variation formulation for our problem. Given Φ and letting HΦ = {(v, μ ) ∈ H 1 (Ω ) × R : (v, μ )2HΦ = |v|2H 1 (Ω ) + |Φ (v)|2 + μ 2 < ∞}, we consider the variation formulation, (u, λ ) ∈ HΦ BΦ (u, λ ; v, μ ) = L(v), ∀(v, μ ) ∈ HΦ , where

(4.11)

BΦ (u, λ ; v, μ ) = B(u, v) + λ Φ (v) + μΦ (u),

which can equivalently be written B(u, v) + λ Φ (v) = L(v), ∀v ∈ H 1 (Ω ) μΦ (u) = 0, ∀μ ∈ R.

(4.12)

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J.E. Osborn

Remark The second equation specifies the constraint Φ (u) = 0, and the first equation is the Euler-Lagrange equation for the constrained extremal problem min J(v),

u∈H 1 (Ω ) Φ (v)=0

where J(v) = 12 B(v, v) − L(v), for which λ is the Lagrange multiplier. Problem (4.12) has a unique solution, whether or not the compatibility condition is satisfied. If it is satisfied, then λ = 0 and u ∈ H 1 (Ω ) satisfies B(u, v) = L(v), ∀v ∈ H 1 (Ω ) Φ (u) = 0. We next define the discrete problem. We define a linear functional on Vh (Vh as defined in Section 4.2) by

Ψ (vh ) =

Φ (1) ∑ vi , ∀vh = ∑ vi φih ∈ Vh, |Nh | i∈N i∈Nh h

where |Nh | = cardinality of Nh , and define the space VΨh = {(vh , μ ) ∈ Vh × R : (vh , μ )2HΦ = |vh |V2 h + |Ψ (vh )|2 + μ 2 < ∞}. Ψ

Then the meshless method approximation (uh , λh ) to (u, λ ) is defined by (uh , λh ) ∈ VΨh BΨ (uh , λh ; v, μ ) = L(v), ∀(v, μ ) ∈ VΨh , where

(4.13)

BΨ (uh , λh ; v, μ ) = B(uh , v) + λhΨ (v) + μΨ (uh ).

Problem (4.13) has a unique solution. It is easily seen that λh = 0 and that uh ∈ Vh satisfies B(uh , v) = L(v), ∀v ∈ Vh Ψ (uh ) = 0. If we write uh = ∑ j∈Nh c j φ hj , (4.13) can be written as

∑ γi, j c j + λhΨ (φih ) + μ

j∈Nh

Ψ (1) |Nh |

∑ c j = lih , ∀i ∈ Nh , μ ∈ R,

j∈Nh

where

γi,h j = and

Ω

∇φ hj · ∇φih dx =

ω hj ∩ωih

∇φ hj · ∇φih dx =

ωih

∇φ hj · ∇φih dx

(4.14)

4 Quadrature for Meshless Methods

lih =

Ω

f φih dx +

Γ

69

gφih ds =

ωih

f φih dx +

Γ ∩ω hi

gφih ds = fih + ghi.

Problem (4.14) has a unique solution: λh = 0, and

∑ γi, j c j = lih , ∀i ∈ Nh , Ψ ( ∑ c j φ hj ) = 0.

j∈Nh

j∈Nh

We impose several axioms on the discretization and on the quadrature. First an axiom on the discretization: • Axiom 4.5 (Reproducing Polynomial Property)

∑ p(xi )φih (x) = p(x) for x ∈ Ω , ∀ polynomials of deg ≤ k.

i∈Nh

This should be compared to Axiom 4.1. We next consider the quadrature version of (4.14):

∑ γi,h∗j c∗j + λh∗Ψ (φih ) + μ

j∈Nh

Ψ (1) |Nh |

∑ c∗j = lih∗ , ∀i ∈ Nh , μ ∈ R,

(4.15)

j∈Nh

We note, however, that we have two formulas for γi,h j , and one can base the quadraof them. In Section 4.2, we based the quadrature version on ture versionh on either h dx; here we will base our quadrature version on h · ∇φ h dx. ∇ φ · ∇ φ ∇ φ j i j i ω h ∩ω h ωh j

i

i

So, we define

γi,h∗j = − ∇φ hj · ∇φih dx, lih∗ = − f φih dx + − ωih

ωih

Γ ∩ω hi

gφih ds.

With these values for γi,h∗j and lih∗ we solve (4.15) for u∗h = ∑ j∈Nh c∗j φ hj and λh∗ , and view u∗h as the approximation to u : u ≈ u∗h . Remark As a consequence of this choice for γi,h∗j , we have ∑ j∈Nh γi,h∗j = 0, ∀i ∈ Nh , i.e., the Row Sum Condition is satisfied. We also note that

γi,h∗j = − ∇φ hj · ∇φih dx = − ∇φ hj · ∇φih dx = γ h∗ ji , ωih

ω hj

i.e., the matrix {γi,h∗j } is not symmetric. This does not pose a serious problem since non-symmetric linear systems can be solved efficiently by iterative methods. Furthermore we observe that ∑ fih∗ + ∑ gh∗ i = 0, i∈Nh

i∈Nh

i.e., the compatibility condition with quadrature is not satisfied. This is due to the use of quadrature to evaluate fih and ghi . It is here that we see the advantage of the alternate variational formulation based on the Lagrange multiplier.

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J.E. Osborn

Next, two axioms on quadrature: • Axiom 4.6 There are small constants η and τ , independent of i and h, such that |

ωih

ρ dx − − ρ dx| ≤ η |ωih |ρ L∞(ωi )

(4.16)

ωih

and |

ω hi ∩Γ

θ ds − −

ω hi ∩Γ

θ dx| ≤ τ |ω hi ∩ Γ |θ L∞(ω h ∩Γ ) i

for appropriate classes of functions ρ and θ . • Axiom 4.7 For each i ∈ Nh , let G∗i : C2 (ω i ) → R be a linear functional given by G∗i (v) = − ∇v · ∇φih dx + − Δ vφih dx − − ωih

ωih

ω hi ∩Γ

∇v · n φih ds.

Then G∗i (p) = 0, ∀polynomials p of degree ≤ k and ∀i ∈ Nh ,

(4.17)

i.e., it is assumed that this Quadrature version of Green’s formula holds for polynomials p of degree ≤ k.

4.3.2 Theoretical Results Theorem 4.5 Suppose the approximating subspace Vh and the quadrature schemes satisfy Axioms 4.1-4.3 and certain other axioms. Then for small η , there is a constant C, independent of u, η , τ , and h, such that |u − u∗h|H 1 (Ω ) ≤ C hk + (η + τ )hk + η hk−1 uW k+1,∞ (Ω ) , ∀h.

4.3.3 Numerical Results We have developed quadrature schemes satisfying Axiom 4.3 and the other axioms. Remark on Axiom 4.7 If k = 1, for (4.17) to hold, the quadrature scheme ωih must satisfy ∂ φih − dx − − n1 φih ds = 0 ωih ∂ x1 Γ ∩ω hi and −

ωih

∂ φih dx − − n2 φih ds = 0, ∂ x2 Γ ∩ω hi

4 Quadrature for Meshless Methods

71

where n = (n1 , n2 ). In particular, when Γ ∩ ωi h = 0, / then the quadrature must satisfy − ∇φih dx = 0.

(4.18)

ωih

Consider d = 1, and suppose we have a p-point integration rule on ωi = (αi , βi ): βi αi

p

f dx =

∑ w j f (x j ).

j=1

We then define a corrected rule βi αi

p

f dx =

∑ w j f (x j ),

j=1

where

− ∑ j=1 [φih ] (x j ) p

w j = w j + θi w j [φih ] (x j ),

with θi =

∑ j=1 w j [φih ]2 p

.

Then − [φih ] dx = 0, ωih

when Γ ∩ ω hi = 0. /

Suppose the shape function φih is symmetric on ωih , and we start with the standard Gauss rule on ωih . Then, since the Gauss weights and points are symmetric on ωih , we find that θi = 0, and we do not have to correct the rule. Gauss rule (uncorrected): k=1

−1

10

8−point

−2

*

|u−uh|H1(Ω)

10

−3

10

16−point −4

10

32−point −5

10

−4

10

−3

−2

10

10

−1

10

h

Fig. 4.3 The log-log plot of |u − u∗h |E with respect to h. u∗h is the approximate solution using the p-point standard Gaussian quadrature (symmetric) with p = 8, 16, and 32. This figure is from [2], and has been used with permission of the publisher.

72

J.E. Osborn

In Fig. 4.3 we present the log-log plot of the error |u − u∗h|E with respect to h for the one-dimensional problem (4.1) with exact solution u(x) = ex − (e − 1) using the Gauss rule, which doesn’t have to be corrected. In Fig. 4.4 we present the log-log plot of |u − u∗h|E , when using a non-symmetric Gauss rule; this quadrature rule does not satisfy (4.18), and we see the the results are not satisfactory. Finally, in Fig. 4.5 we present a plot for the corrected non-symmetric Gauss rule. Non−symmetric Gauss rule (uncorrected): k=1

0

10

8−point

16−point

−1

10

32−point

*

|u−uh|H1(Ω)

−2

10

−3

10

64−point

−4

10

−5

10

−4

−3

10

−2

10

10

−1

10

h

Fig. 4.4 The log-log plot of |u − u∗h |E with respect to h. u∗h is the approximate solution obtained using the p-point non-symmetric Gaussian quadrature (uncorrected) with p = 8, 16, 32, and 64. This figure is from [2], and has been used with permission of the publisher. Corrected non−symmetric Gauss rule: k=1

−2

10

8−point −3

10

*

|u−uh|H1(Ω)

16−point

−4

10

32−point 64−point

−5

10

−4

10

−3

−2

10

10

−1

10

h

Fig. 4.5 The log-log plot of |u−u∗h |E with respect to h. u∗h is the approximate solution obtained using corrected non-symmetric Gaussian quadrature with p = 8, 16, 32, and 64 points. This figure is from [2], and has been used with permission of the publisher.

4 Quadrature for Meshless Methods

73

4.4 Comparison of the Results of the Two Analyzes • The main hypotheses in the First Analysis: the basis function reproduce polynomials of degree 1; the Row Sum Condition is satisfied. Results when ε = τ = 0 : ∗∗ u − u∗∗ h E ≤ C(h + η ), u − uh E ≤ Ch if η ≤ Ch. • The main hypotheses in the Second Analysis: the basis functions reproduce polynomials of degree k ≥ 1; the Quadrature Green’s formula is satisfied for all polynomials of degree ≤ k. The Row Sum Condition is automatically satisfied. The stiffness matrix is non-symmetric. Results when ε = τ = 0 : u − u∗h E ≤ C(hk + η hk−1 ), u − u∗hE ≤ Chk if η ≤ Ch. • The numerical results support the theorems in both analyzes. The results indicate that increased quadrature accuracy is required as h → 0 to obtain optimal order of convergence.

References [1] Babuška, I., Banerjee, U., Osborn, J., Li, Q.: Quadrature for meshless methods. Int. J. Numer. Math. Engng. 76, 1434–1470 (2008) [2] Babuška, I., Banerjee, U., Osborn, J., Zhang, Q.: Effect of numerical integration on meshlesss methods. Comput. Methods Appl. Mech. Engrg. 198, 2886–2897 (2009) [3] Beissel, S., Belytschko, T.: Nodal integration of the element-free Galerkin method. Comput. Methods Appl. Mech. Engrg. 139, 49–74 (1996) [4] Carpinteri, A., Ferro, G., Ventura, G.: The partition of unity quadrature in meshless methods. Computers and Structures 54, 987–1006 (2002) [5] Chen, J.-S., Wu, C.-S., Yoon, S., You, Y.: A stabilized conformal nodal integration for a Galerkin mesh-free method. Int. J. Numer. Meth. Engng. 50, 435–466 (2001) [6] Chen, J.-S., Wu, S., You, Y.: Non-linear version of stabilized conforming nodal integration Galerkin mesh-free methods. Int. J. Numer. Meth. Engng. 53, 2587–6515 (2002) [7] Ciarlet, P.G.: The Finite Element Method. North Holland, Amsterdam (1978)

Chapter 5

Shape and Topology Sensitivity Analysis for Elastic Bodies with Rigid Inclusions and Cracks ˙ Jan Sokołowski and Antoni Zochowski

Abstract. In the keynote lecture we describe how the asymptotic analysis in singularly perturbed domains can be employed to determine effectively the influence of nucleation of small voids on some shape functionals. To this end the classical shape sensitivity analysis is combined with asymptotic expansions in order to determine the singular limits of shape derivatives which are called topological derivatives of shape functionals. The topological derivatives are determined for elastic bodies weakened by cracks on boundaries of rigid inclusions. On the crack faces the nonpenetration conditions are prescribed, such conditions are non linear and assure that the displacements of the crack lips or surfaces cannot penetrate each other. Small voids are located on the finite distance from the crack, so there is no interaction between the crack and the voids. In such a way nucleations of small voids can be implemented in the numerical procedures of optimum design or solution of inverse problems. A nonlinear model in the framework of damage theory is presented in details for modeling and sensitivity analysis. The example of an elastic body with a rigid inclusion and a crack located at the boundary of the inclusion is considered. The asymptotic analysis which leads to topological derivatives is performed in two and three spatial dimensions. The derived formulas can be used in numerical methods of shape and topology optimization.

Jan Sokołowski Institut Elie Cartan, UMR 7502 Nancy-Université-CNRS-INRIA, Laboratoire de Mathématiques, Université Henri Poincaré Nancy 1, B.P. 239, 54506 Vandoeuvre lés Nancy Cedex, France e-mail: [email protected] ˙ Antoni Zochowski Systems Research Institute of the Polish Academy of Sciences, ul. Newelska 6, 01-447 Warszawa, Poland e-mail: [email protected] M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 75–98. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

76

˙ J. Sokołowski and A. Zochowski

5.1 Introduction Optimum shape design in structural mechanics is a classical domain of research for many years. One of the goals in the field is to devise numerical methods for shape and topology optimization. In the framework of boundary variations technique, since the shape transformations are smooth and regular in the rigorous mathematical fashion, the topology changes during the process of shape optimization are quite difficult to be included. Therefore, the singular limits of shape or material derivatives of shape functionals turn out to provide a remedy for this difficulty to propose reasonable topology changes for the structure under design procedure. From one point of view, by the singular limit procedure we mean the derivation of expressions, within the framework of matched and compound asymptotic expansions in asymptotic analysis in singularly perturbed domains, for the first nontrivial terms in expansions of associated shape functionals. In another words, in the case of small void which is included in the elastic body, the asymptotic analysis is performed with respect to the size of small void which becomes the small parameter. Such an analysis of the specific shape functional in the optimum design problem under considerations, e.g., the potential energy of the body, results in an expansion of the functional with respect to the parameter, and the first nontrivial term in the expansion is called the topological derivative of the specific shape functional. The topological derivative is a function defined inside the elastic body, depending on the location of the material point which becomes the center of the small void. We point out that such a derivation can also be performed by means of the classical shape sensitivity analysis and an appropriate singular limit passage of the shape derivatives with the size of the void tending to zero. Singular perturbations of elastic bodies are analyzed, and the influence of small defects in the body on the displacement and stress fields as well as on the energy functionals is determined for variational inequalities by a domain decomposition method, combined with asymptotic analysis of the Steklov-Poincaré operator in singularly perturbed domains. In the paper we propose an example illustrating the recent results obtained in the framework of asymptotic analysis in singularly perturbed geometrical domains for the purposes of shape and topology sensitivity analysis [20].

5.1.1 Cracks and Rigid Inclusions The problem associated to cracks in elastic bodies on boundaries of rigid inclusions appears in a vast number of applications in civil, mechanical, aerospace, biomedical and nuclear industries. In particular, some classes of materials are composed by a bulk phase with inclusions inside. When the inclusions are much stiffer than the bulk material, we can treat them as rigid inclusions. In addition, it is quite common to have cracks between both phases. Thus, in this paper we review the mechanical

5 Shape and Topology Sensitivity Analysis

77

modeling as well as the topology sensitivity analysis associated to the limit case of rigid inclusions in elastic bodies with a crack at the interface. The mechanical modeling is based on the assumption of non-penetration conditions at the crack faces between the elastic material and the rigid inclusion, which do not allow the opposite crack faces to penetrate each other, leading to a new class of variational inequalities. For the topological sensitivity analysis, we obtain the topological derivatives of the energy shape functional associated to the nucleation of a smooth imperfection in the bulk elastic material. The paper is organized as follows. The problem formulation associated to cracks in elastic bodies on boundaries of rigid inclusions is presented in Section 5.2. The topological derivatives associated to the energy shape functional are calculated in Section 5.3. We provide some closed formulas for the case of nucleation of spherical holes in 3D and circular elastic inclusions in 2D. In this last case, we present the limit cases in which the elastic inclusion becomes a hole (void) and also a rigid inclusion. The authors are indebted A.M. Khludnev and A. Novotny for scientific collaboration on modeling and sensitivity analysis of problems with rigid inclusions and cracks.

5.2 Problem Formulation Let Ω ⊂ R3 be a bounded domain with smooth boundary Γ , and ω ⊂ Ω be a subdomain with smooth boundary Ξ such that ω ∩ Γ = 0. / We assume that Ξ consists of two parts γ and Ξ \ γ , meas(Ξ \ γ ) > 0, where γ is a smooth 2D surface described as xi = xi (y1 , y2 ), (y1 , y2 ) ∈ D, i = 1, 2, 3, with bounded domain D ⊂ R2 having a smooth boundary ∂ D, and the rank of the matrix ∂∂ xy is equal to 2. Denote by ν = (ν1 , ν2 , ν3 ) a unit outward normal vector to Ξ , see Fig. 5.1. The subdomain ω is assumed to correspond to a rigid inclusion, and the surface γ describes a crack located on Ξ . Domain Ω \ ω corresponds to the elastic part of the body. For the further use we introduce the space of infinitesimal rigid displacements R(ω ) = {ρ = (ρ1 , ρ2 , ρ3 ) | ρ (x) = Bx + C, x ∈ ω }, where

⎛

⎞ 0 b12 b13 B = ⎝ −b12 0 b23 ⎠ , −b13 −b23 0

C = (c1 , c2 , c3 );

bi j , ci = const, i, j = 1, 2, 3.

Denote Ωγ = Ω \ γ . Problem formulation describing an equilibrium of the elastic body with the rigid inclusion ω and the crack γ is as follows. In the domain Ωγ , we

˙ J. Sokołowski and A. Zochowski

78

Fig. 5.1 Domain Ω with rigid inclusion ω

have to find functions u = (u1 , u2 , u3 ), u = ρ0 on ω ; ρ0 ∈ R(ω ); and in the domain Ω \ ω we have to find functions σ = {σi j }, i, j = 1, 2, 3, such that −divσ = F σ − Aε (u) = 0

in in

Ω \ ω, Ω \ ω, u = 0 on Γ ,

(u − ρ0) · ν ≥ 0

on γ ,

(5.4)

στ = 0, σν ≤ 0 on γ ,

(5.5)

σν (u − ρ0) · ν = 0 on γ ,

(5.6)

+ + +

σν · ρ =

−

F ·ρ

∀ρ ∈ R(ω ).

(5.1) (5.2) (5.3)

(5.7)

ω

Ξ

Here F = (F1 , F2 , F3 ) ∈ L2 (Ω ) is a given function,

σν = σi j ν j νi , στ =

στ 1 2 3 (στ , στ , στ ),

= σ ν − σν ν ,

σ ν = {σi j ν j }i=3 i=1 ,

1 εi j (u) = (ui, j + u j,i), i, j = 1, 2, 3. 2 All functions with two lower indices are assumed to be symmetric in those indices. Summation convention over repeated indices is accepted throughout the paper. Elasticity tensor A = {ai jkl }, i, j, k, l = 1, 2, 3, is given, and it satisfies usual symmetry and positive definiteness properties, ai jkl = akli j = a jikl ,

ai jkl ∈ L∞ (Ω ), i, j, k, l = 1, 2, 3,

ai jkl ξkl ξi j ≥ c0 |ξ |2 ,

∀ ξi j = ξ ji ,

c0 = const.

5 Shape and Topology Sensitivity Analysis

79

In addition, we consider the isotropic case, namely A = 2mI + l (I ⊗ I) ,

(5.8)

where I and I respectively are the second and fourth order identity tensors and, m and l are the Lamé coefficients, which can be defined in terms of the Young modulus E and the Poisson ratio υ as m=

E 2(1 + υ )

and

l=

υE . (1 + υ )(1 − 2υ )

(5.9)

Relations (5.1) are equilibrium equations, and (5.2) corresponds to the Hooke’s law. Inequality (5.4) describes a mutual non-penetration between crack faces γ ± . The first relation in (5.5) means a zero friction between the crack faces. For simplicity we assume clamping condition (5.3) on Γ . Note that external forces F are applied to Ω \ ω as well as to ω , but there are no equilibrium equations in ω . Influence of these forces is taken into account through (5.7). If we have no crack γ on Ξ , relations (5.4)-(5.6) should be omitted. This specific problem formulation for the particular case F = 0 in ω can be found in [27]. The problem formulation with the crack and non-penetration conditions seems to be new. First of all we provide a variational formulation of problem (5.1)-(5.7). To this end, let us consider the Sobolev space HΓ1,ω (Ωγ ) = {v ∈ H 1 (Ωγ )3 | ε (v) = 0 on ω ; v = 0 on Γ }

(5.10)

and define the set of admissible displacements Kω = {v ∈ HΓ1,ω (Ωγ ) | ε (v) = 0 on ω ; (v+ − v−) · ν ≥ 0 on γ }.

(5.11)

Let (·, ·)Ω \ω be the inner product in L2 (Ω \ ω ). Consider the energy functional 1 Π (v) = (σ (v), ε (v))Ω \ω − (F, v)Ωγ , 2

(5.12)

where the stress field σ (v) = σ is defined in (5.2) for u = v, and introduce the following minimization problem inf Π (v).

v∈Kω

(5.13)

The convex cone Kω is weakly closed in the space HΓ1,ω (Ωγ ), and the functional Π is coercive and weakly lower semi-continuous on the same space. Hence, by the standard result in the calculus of variations problem (5.13) admits a solution satisfying the variational inequality

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u ∈ Kω , (σ (u), ε (u − u))Ω \ω ≥ (F, u − u)Ωγ

(5.14) ∀ u ∈ Kω .

(5.15)

Since bilinear form is coercive, the solution u of problem (5.14)-(5.15) is unique and Lipschitz continuous with respect to data.

5.2.1 Dual Problem Formulation We introduce the dual formulation of problem (5.14)-(5.15) in stresses. By this approach a solution of dual problem σ = {σi j } in the domain Ω \ ω is defined, and moreover, we show that a solution of dual problem given by stresses σi j coincides with the solution σi j = σi j (u) given by (5.14)-(5.15). Below we provide rigorous explanations of the procedure. First, we need the deformations in terms of stresses, thus we write Hooke’s law (5.2) in the inverted form A−1 σ = ε (u) in Ω \ ω .

(5.16)

Note that the tensor A−1 enjoys the properties similar to those of A, i.e., it is symmetric and positive definite. Consider the space of stresses H = {σ = {σi j } | σi j ∈ L2 (Ω \ ω ), i, j = 1, 2, 3} and the quadratic functional G defined on H, 1 G(σ ) = (A−1 σ , σ )Ω \ω . 2 The set of admissible stresses is a cone in the space H with the elements which satisfy the sign condition for normal stresses on the crack as well as the global equilibrium condition over the inclusion, thus it is defined as follows M = {σ ∈ H | equations (5.1) and conditions (5.5), (5.7) hold}. The above cone is well defined since equations (5.1) in the definition of M are satisfied in the sense of distributions, and conditions (5.5), (5.7) hold in a weak sense. Let us consider now the dual problem in the form of the minimization problem for quadratic functional over a convex and weakly closed cone, inf G(σ ).

σ ∈M

(5.17)

Under our assumptions there exists a unique solution σ 0 of this problem which satisfies the following variational inequality

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σ 0 ∈ M, −1

(5.18)

(A σ , σ − σ )Ω \ω ≥ 0 ∀ σ ∈ M. 0

0

(5.19)

5.2.2 Passage from Elastic Inclusion to Rigid Inclusion In fact, problem (5.1)-(5.7) can be considered as a limit problem for a family of elasticity problems with the crack γ formulated in the domain Ωγ . This means that we can construct a family of problems depending on a positive parameter λ such that for any fixed λ > 0 the problem describes the equilibrium state of an elastic body occupying the domain Ωγ with the crack γ . We expect that a rigid inclusion ω is obtained for λ → 0 , i.e., for such a limit any point x ∈ ω has a displacement ρ0 (x), ρ0 ∈ R(ω ). In what follows we provide a rigorous proof of the above statement. Introduce the tensor Aλ = {aλi jkl }, i, j, k, l = 1, 2, 3, aλi jkl =

ai jkl in Ω \ ω λ −1 ai jkl in ω ,

and consider the following problem. In the domain Ωγ , we have to find functions uλ = (uλ1 , uλ2 , uλ3 ), σ λ = {σiλj }, i, j = 1, 2, 3, such that −divσ λ = F

in Ωγ ,

(5.20)

σ − A ε (u ) = 0 in Ωγ ,

(5.21)

λ

λ

λ

λ

u =0

on Γ ,

(5.22)

[uλ ] · ν ≥ 0, [σνλ ] = 0, σνλ [u] · ν = 0

on γ ,

(5.23)

σνλ

≤ 0,

στλ

=0

±

on γ .

(5.24)

Here we use notations of the previous section, and [v] = v+ − v− is a jump of v on γ , where ± fit positive and negative crack faces γ ± with respect to the normal vector ν. For any fixed λ > 0 problem (5.20)-(5.24) is well known and admits a variational formulation ([22], [23], [19]). Indeed, introduce the set of admissible displacements K = {v ∈ HΓ1 (Ωγ )3 | [v] · ν ≥ 0 on γ }, where

HΓ1 (Ωγ ) = {v ∈ H 1 (Ωγ ) | v = 0 on Γ }.

There exists a unique solution uλ of the minimization problem 1 inf { (σ λ (v), ε (v))Ωγ − (F, v)Ωγ } 2

v∈K

(5.25)

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with the stress field σ λ (v) determined by (5.21) for uλ = v. Solution uλ of the minimization problem satisfies the variational inequality uλ ∈ K,

(5.26)

(σ λ (uλ ), ε (u − uλ ))Ωγ ≥ (F, u − uλ )Ωγ ∀ u ∈ K.

(5.27)

By the convexity of the quadratic functional in (5.25) with respect to v, it follows that problems (5.25) and (5.26)-(5.27) are equivalent. Moreover, all relations (5.20)(5.24) can be obtained from (5.26)-(5.27), and conversely, relations (5.20)-(5.24) imply (5.26)-(5.27). Below we justify the limit passage with λ → 0 in (5.26)-(5.27). Substitute u = 0, u = 2uλ as test functions in (5.27), and sum up the obtained relations. It implies the equality (σ λ (uλ ), ε (uλ ))Ωγ = (F, uλ )Ωγ . (5.28) Assuming that λ ∈ (0, λ0 ), from (5.28) we obtain uλ H 1 (Ωγ )3 ≤ c1 , Γ

1 λ

ai jkl εkl (uλ )εi j (uλ ) ≤ c2

(5.29) (5.30)

ω

with constants c1 , c2 being uniform with respect to λ ∈ (0, λ0 ). Choosing a subsequence, if necessary, it can be assumed as λ → 0 uλ → u weakly in HΓ1 (Ωγ )3 . Then by (5.30)

εi j (u) = 0 in ω , i, j = 1, 2, 3.

This means that a function ρ0 exists such that u = ρ0 in ω ; ρ0 ∈ R(ω ). Since uλ converge weakly in HΓ1 (Ωγ )3 , the limit function u satisfies the inequality (u+ − ρ0 ) · ν ≥ 0 on γ . In particular, u ∈ Kω . Let us take any fixed element u ∈ Kω . Then, there exists ρ ∈ R(ω ) such that u = ρ in ω , and u can be taken as a test function in (5.27). In such a case, inequality (5.27) implies (σ λ (uλ ), ε (u − uλ ))Ωγ ≥ (F, u − uλ )Ωγ . (5.31) By using the equality Aλ = A in Ω \ ω , we can pass to the limit in (5.31) as λ → 0 which implies

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u ∈ Kω , (σ (u), ε (u − u))Ωγ \ω ≥ (F, u − u)Ωγ ∀ u ∈ Kω , what is precisely (5.14)-(5.15). Hence a passage from the elastic inclusion to the rigid inclusion is shown. We formulate the obtained result as follows Theorem 1. The solution uλ of problem (5.26)-(5.27) weakly converge in HΓ1 (Ωγ )3 to the solution u of problem (5.14)-(5.15). Observe that there is no limit in ω for the stress tensor σ λ with λ → 0.

5.3 Topological Asymptotic Analysis The topological derivative introduced in [37] quantifies the sensitivity of a given shape functional with respect to the introduction of a non-smooth perturbation (hole, inclusion, source term, for instance) in a ball Bδ (x0 ) ⊂ Ω of radius δ > 0 and center at x0 ∈ Ω , that is Bδ (x0 ) = {x ∈ R3 : x − x0 < δ }, Bδ (x0 ) is the closure of Bδ (x0 ). Therefore, this derivative can be seen as a first order correction on the shape functional J (Ω ) to estimate J (Ωδ ), where Ωδ is the perturbed domain. Thus, we have the following topological asymptotic expansion for functional J , J (Ωδ ) = J (Ω ) + f (δ )DT (x0 ) + o( f (δ )) ,

(5.32)

where f (δ ) is a positive function that decreases monotonically such that f (δ ) → 0 when δ → 0+ and the term DT (x0 ) is defined as the topological derivative of J . Then, from (5.32) we have that the classical definition of the topological derivative is given by [31, 38] DT (x0 ) = lim

δ →0

J (Ωδ ) − J (Ω ) 1 d = lim J (Ωδ ) . f (δ ) δ →0 f (δ ) d δ

(5.33)

We point out, that even if formula (5.32) looks very different from the classical shape derivatives currently used in shape optimization, however its nature is the same, since it is defined by [37] in the form of a singular limit of shape derivatives evaluated on boundaries of small voids with respect to the radius of the voids δ → 0. In this way the topological derivative is a generalization of the classical shape derivative in smooth case to the singular boundary perturbations. We refer the reader to [28] for the asymptotic analysis in singularly perturbed domains by means of the matched and compound asymptotic expansions which leads to the topological derivatives of shape functionals in elasticity with complete proofs in general case. On the other hand, from recent developments of shape optimization in fluid dynamics for the drag functional presented in [34], it turns out that the shape derivative of the drag functional can be derived in the form of a singular limit for the appropriate volume integrals, so in the framework of classical shape sensitivity analysis the

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same principle of singular limit passage can be applied in order to identify the shape gradients. We recall here, with few references, that the topological derivative has been successfully applied in the context of • topology optimization: [2], [1], [6], [10], [11], [32], [33], • inverse problems: [3], [8], [26], • image processing: [4], [5], [16], [24]. Concerning the theoretical development of the topological asymptotic analysis, the reader may refer to [28], for instance. The review of available techniques used for asymptotic analysis with respect to the size of small cavities for the elastic energy functional is presented in [25]. In our particular case, we consider a regular perturbation of the domain given by the nucleation of a small elastic inclusion with Young modulus Eη = η E, where E is the Young modulus of the bulk material and η ∈ [0, ∞) represents the contrast. We assume that there is a small elastic inclusion Bδ (x0 ) in the elastic region Ωω = Ω \ ω . If the elastic inclusion becomes a cavity, it is denoted by ωδ = Bδ (x0 ). The cavity can be obtained from the elastic inclusion by the limit passage η → 0, in the limit case we have a singular perturbation of the domain. In the case of elastic inclusion the elastic region Ωω is decomposed into two disjoint parts Ωω \ Bδ (x0 ) and Bδ (x0 ) with different material properties, namely E and η E, respectively. The other limit passage with the contrast η → ∞ results in the small rigid inclusion ωδ = Bδ (x0 ). See Fig. (5.2). We are also interested in the topological asymptotic expansion of the energy shape functional of the form 1 1 Πδ (v) = (σ (v), ε (v))Ωω \B (x0 ) + (σ (v), ε (v))Bδ (x0 ) − (F, v)Ωγ , δ 2 2

(5.34)

where we have to find function v = uδ such that −divσ = F

in

σ − Aε (uδ ) = 0 σ − Aη ε (uδ ) = 0 [uδ ] = 0 [σ ]ν = 0 uδ = 0

in in

Ω \ ω,

Ωω \ Bδ (x0 ), Bδ (x0 ), on ∂ Bδ (x0 ) on ∂ Bδ (x0 ) on Γ ,

(5.38) (5.39) (5.40) (5.41)

στ = 0, σν ≤ 0 on γ ,

(5.42)

σν (uδ − ρ0 ) · ν = 0 on γ ,

(5.43)

+ +

σν · ρ = Ξ

(5.36) (5.37)

(uδ − ρ0 ) · ν ≥ 0 on γ , +

−

(5.35)

F ·ρ ω

∀ρ ∈ R(ω ).

(5.44)

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d

85

x0

R

x0

Fig. 5.2 Domain Ω with rigid inclusion ω and an elastic inclusion Bδ (x0 ).

with A such as before and Aη = η A (since Eη is the Young modulus of the inclusion).

5.3.1 Domain Decomposition Since the problem is non-linear, let us introduce a domain decomposition given by ΩR = Ωω \ BR (x0 ), where BR (x0 ) is a ball of radius R > δ and center at x0 ∈ Ω , that is BR (x0 ) = {x ∈ R3 : x − x0 < R}, BR (x0 ) is the closure of BR (x0 ), as shown in Fig. (5.2). For the sake of simplicity, we assume that F = 0 in BR (x0 ). Thus, we have the following linear elasticity system defined in BR (x0 ) with an inclusion Bδ (x0 ) inside −divσ = 0

in

BR (x0 ),

(5.45)

σ − Aε (wδ ) = 0 σ − Aη ε (wδ ) = 0 wδ = v [wδ ] = 0 [σ ]ν = 0

in

BR (x0 ) \ Bδ (x0 ),

(5.46)

in Bδ (x0 ), on ∂ BR (x0 ),

(5.47) (5.48)

on ∂ Bδ (x0 ), on ∂ Bδ (x0 ).

(5.49) (5.50)

We are interested in the Steklov-Poincaré operator on ∂ BR , that is Aδ : v ∈ H 1/2 (∂ BR ) → σ (wδ )ν ∈ H −1/2 (∂ BR ) .

(5.51)

Then we have σ (uR )ν = Aδ (uR ) on ∂ BR , where uR is solution of the variational inequality in ΩR , that is uR ∈ Kω : aΩR (uR , ϕ − uR) ≥ (F, ϕ − uR)Ωγ \BR (x0 )

∀ϕ ∈ Kω

(5.52)

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and the bilinear form aΩR is such that aΩR (u, ϕ ) =

σ (u) · ε (ϕ ) +

ΩR

∂ BR

Aδ (u) · ϕ .

(5.53)

Finally, in the disk BR (x0 ) we have

BR \Bδ

σ (w) · ε (w) +

σ (w) · ε (w) = Bδ

∂ BR

Aδ (w) · w ,

(5.54)

where w = wδ is the solution of the elasticity system in the disk (5.45)-(5.50) or equivalently solution of the following variational problem wδ ∈ W :

BR \Bδ

σ (wδ ) · ε (ϕ ) +

Bδ

σ (wδ ) · ε (ϕ ) = 0 ∀ϕ ∈ W0 ,

(5.55)

with W and W0 such that W = {w ∈ H 1 (BR )3 | [w] = 0 on ∂ Bδ , w = v on ∂ BR } ,

(5.56)

W0 = {ϕ ∈ H (BR ) | [ϕ ] = 0 on ∂ Bδ , ϕ = 0 on ∂ BR } .

(5.57)

1

3

5.3.2 Shape Sensitivity Analysis of the Energy Functional Let us introduced the energy-based shape functional defined in the disk BR (x0 ), that is 1 1 Eδ (wδ ) := σ (wδ ) · ε (wδ ) + σ (wδ ) · ε (wδ ) . (5.58) 2 BR \Bδ 2 Bδ We need to calculate d E (w ) = dδ δ δ

BR \Bδ

+ BR \Bδ

σ (wδ ) · ε (w˙ δ ) + Σ (wδ ) · ∇V +

Bδ

Bδ

σ (wδ ) · ε (w˙ δ )

(5.59)

Σ (wδ ) · ∇V ,

which was obtained using the Reynold’s transport theorem and the concept of material derivatives of spatial fields ([14, 41]). Some of the terms in (5.59) require explanation. Vector V represents the shape change velocity field defined on the disk BR (x0 ) and such that V = 0 on ∂ BR and V = ν on ∂ Bδ . Thus, w˙ δ ∈ W0 is the material (total) derivative with respect to δ . Finally, the Eshelby energy-momentum tensor Σ takes the form ([7, 15]) 1 Σ (wδ ) := σ (wδ ) · ε (wδ )I − (∇wδ )T σ (wδ ) . 2

(5.60)

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Since w˙ δ ∈ W0 and considering that Σ (wδ ) is a free-divergence tensor field (divΣ (wδ ) = 0), the shape derivative of the energy functional becomes d E (w ) = − dδ δ δ

∂ Bδ

[Σ (wδ )]ν · ν .

(5.61)

5.3.3 Topological Derivatives Calculation The topological derivative can be evaluated by a singular limit of shape derivatives. Singular limit in such a case means that the support of of the speed velocity field tends to a point. In other words the shape derivative of a given functional is concentrated on the boundary ∂ Bδ of small void and the passage to the limit with δ → 0 is performed. Since in asymptotics for displacement and stress fields there are specific coefficients in function of the small parameter δ > 0, the precise analysis of asymptotic expansions for the fields provides the factor in (5.62) such that the limit is finite and non trivial. Therefore, the pure shape sensitivity analysis in the framework of the speed method is not sufficient in order to determine the limit which we call the topological derivative. The necessary ingredient for such derivation is the method of matched and compound asymptotic expansions in singularly perturbed domains [28], however the resulting formulas are of the same nature as the shape derivatives in smooth case, which is an important mathematical result. We refer the reader to [34, 35, 36] in the case of compressible fluids and singular limits which govern the shape derivatives of the drag functional. We turn back to the evaluation of topological derivatives for the elasticity boundary value problems with respect to nucleation of small voids or cavities. By introducing (5.61) in (5.33), we have DT (x0 ) = − lim

1

δ →0 f (δ )

∂ Bδ

[Σ (wδ )]ν · ν .

(5.62)

We shall consider here several particular cases: energy functional and spherical cavity, general stress functional and cavity of arbitrary shape, and finally energy functional and inclusion of arbitrary shape.

5.3.3.1 Topological Derivative of the Energy Functional in Three Spatial Dimensions for a Small Cavity In this case we assume that the system corresponds to three dimensional isotropic elasticity and we are interested in the energy change due to the nucleation of a spherical cavity. Thus, for the convenience of the reader we recall here the results derived in [12], [17], [33] for the three dimensional elasticity case.

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Theorem 2. Let us consider the contrast η → 0. Thus, the elastic inclusion degenerates to a spherical cavity with homogeneous Neumann boundary condition. In this case, the energy shape functional admits for δ → 0 the following topological asymptotic expansion

Πδ (uδ ) = Π (u) + πδ 3DT (x0 ) + o(δ 3 ) ,

(5.63)

with the topological derivative DT (x0 ) given by DT (x0 ) = Hσ (u(x0 )) · ε (u(x0 )) ∀x0 ∈ Ω \ ω ,

(5.64)

where u is solution of the variational inequality (5.14)-(5.15) and H is a forth-order tensor defined as

1−υ 1 − 5υ H= 10I − I⊗I . (5.65) 7 − 5υ 1 − 2υ 5.3.3.2 Topological Derivative of the General Stress Functional for Anisotropic Elasticity and a Small Cavity Let us consider the elasticity problem written in the matrix/column form L u = D(−∇x ) AD(∇x )u = 0 N

Ω ω

u = D(n) AD(∇x )u = g

N u = D(n) AD(∇x )u = 0

Ω

in Ωδ ,

(5.66)

on Ω ,

(5.67)

on ωδ ,

(5.68)

where A is a symmetric positive definite matrix√of size 6 × 6, consisting of the elastic material moduli (the Hooke’s matrix) α = 1/ 2 and D(∇x ) is 6 × 3-matrix of the first order differential operators (ξi = ∂ /∂ xi ), ⎡ ⎤ ξ1 0 0 0 αξ3 αξ2 D(ξ ) = ⎣ 0 ξ2 0 αξ3 0 αξ1 ⎦ (5.69) 0 0 ξ3 αξ2 αξ1 0 u is displacement column, n = (n1 , n2 , n3 ) is the unit outward normal vector on ∂ Ωδ . We assume here that the origin O of the coordinate system lies in ω , O ∈ ω . Then ωδ denotes ω scaled by δ , that is ωδ = δ · ω , ω1 = 1 · ω . In this notation the strain and stress columns are given respectively by ε (u) = D(∇x )u and σ (u) = AD(∇x )u, which gives √ √ √ ε (u) = ε11 , ε22 , ε33 , 2ε23 , 2ε31 , 2ε12 √ √ √ σ (u) = σ11 , σ22 , σ33 , 2σ23 , 2σ31 , 2σ12

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89

The load gΩ is supposed to be self equilibrated in order to assure the existence of a solution to the elasticity problem,

∂Ω

where

d(x) gΩ (x)dsx = 0 ∈ IR6

(5.70)

⎡

⎤ 1 0 0 0 − α x3 α x2 0 − α x1 ⎦ d(x) = ⎣ 0 1 0 α x3 0 0 1 − α x2 α x1 0

(5.71)

represents rigid body motion. The theory presented here can be applied to a broad class of shape functionals. Let us consider the functional J1δ (u) =

Ωδ

σ (u; x) B(x)σ (u; x)dx ,

(5.72)

Functional (5.72) looks like the elastic energy functional but can contain a certain symmetric 6 × 6–matrix function B. In the case of constant, diagonal matrix B functional (5.72) is related to square of the L2 (Ω )–norm of the stress tensor or of its components. On the other hand, if A(x)−1 B(x)A(x)−1 becomes a constant diagonal matrix with our choice of B, then in (5.72) the similar strain norms are obtained. From condition (5.70) follows that both problems, problem (5.66)-(5.68) in the body Ωδ with the cavity ωδ , and the first limit problem in the entire body Ω , D(−∇x ) AD(∇x )v = 0 in Ω ,

(5.73)

Ω

D(n) AD(∇x )v = g in ∂ Ω , admit the solutions u(δ , x) ∈ C2,α (Ωδ )3 and v ∈ C2,α (Ω )3 , respectively, under the loading gΩ ∈ C1,α (∂ Ω )3 . Freedom in selection of such solutions up to the rigid motions has no influence on functional (5.72) and therefore can be neglected (using additional conditions we can pass to uniquely solvable problems). The adjoint state w ∈ C2,α (Ω )3 has the form D(−∇x ) AD(∇x )w = − 2D(∇x ) BAD(∇x )v in Ω , D(n)AD(∇x )w =

(5.74)

2D(n)AD(∇x ) BAD(∇x )v on ∂ Ω . Furthermore, we define the special functions z j solving the exterior elasticity problem D(−∇ξ ) AD(∇ξ )z j = 0 in G = IR3 \ ω1 ,

(5.75)

D(n(ξ )) AD(∇ξ )z j = g j on ∂ ω1

(5.76)

with the special right hand sides

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90

g j (ξ ) = −D(n(ξ )) Ae j ,

(5.77)

where j = 1, ..., 6 and e j = δ j,1 , ..., δ j,6 is an element of the canonical basis in IR6 . Theorem 3. The following formula holds true, [28] J1ρ (u) = J10 (v) + δ 3 {ε 0 (v) ABAε 0 (v)|ω1 |+ + (ABAD(∇ξ )zε 0 (v), D(∇ξ )zε 0 (v))G + + (ε 0 (w) − 2BAε 0 (v)) mω ε 0 (v)}+

(5.78)

+ O(δ 3+ε ) , where ε 0 (v) = D(∇ξ )v(O) and ε 0 (w) = D(∇ξ )w(O) are strain columns evaluated at the point x = O for the solutions of problems (5.73) and (5.74); mω is the polarization matrix of size 6 × 6 for the cavity ω in the elastic space with the Hooke’s matrix A, and z = (z1 , ..., z6 ) is the row of energy components of the special solutions to homogeneous exterior elasticity problem (5.75)-(5.76). The term standing at δ 3 corresponds to topological derivative. The 6×6 polarization matrix may be computed explicitly using the result given below. Theorem 4. The following integral representation holds true mωjk = AD(∇ξ )z j , D(∇ξ )zk + A jk |ω1 | . G

(5.79)

The results remain true for operators with variable coefficients. For further developments on the shape sensitivity analysis for the polarization matrices which are integral attributes of small cavities required in derivation of topological derivatives, we refer the reader to [30, 29].

5.3.3.3 Topological Derivative of the Energy Functional in Two Spatial Dimensions for a Small Inclusion In two spatial directions we derive also an exterior expansion for the solutions of the variational inequality. Therefore, the result obtained is more precise compared to the general case of the cavity in three spatial dimensions. First, we repeat the model description, and then we develop the asymptotic analysis in linear elasticity to derive the equivalent form of perturbation of the bilinear form. Since in this Section we are dealing with a two dimensional elasticity problem, then the domain Ω ⊂ R2 . Thus, all indices introduced in the Section 5.2 take values from 1 to 2, instead of 1 to 3. In the particular case of plane stress, the Lamé coefficient l = l ∗ , where υE . l∗ = 1 − υ2

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91

2

r d

q

1

x0

R

x0

Fig. 5.3 Domain Ω with rigid inclusion ω and an elastic inclusion Bδ (x0 ).

In addition, the crack γ is represented now by a smooth 1D curve described as xi = xi (y),

y ∈ D, i = 1, 2,

with bounded domain D ⊂ R. The space R(ω ) of infinitesimal rigid displacements is redefined simply by setting

0 b and C = (c1 , c2 ); b, ci = const, i = 1, 2. B= −b 0 The displacement field u = (u1 , u2 ); u = ρ0 in ω ; ρ0 ∈ R(ω ); and in the domain Ω \ ω we have to find the stress tensor components σ = {σi j }, solution of (5.1)(5.7) in Ω ⊂ R2 for i, j = 1, 2. Hence, all definitions and results presented in the previous Sections hold. We use the existence of the asymptotic expansions for wδ , solution of the elasticity system (5.45)-(5.50) now defined in the disk BR (x0 ) ⊂ R2 , in the neighborhood of Bδ (x0 ), namely wδ (x) = w0 (x) + w∞ (x) + o(δ ) . (5.80) In addition, w∞ is proportional to δ , w∞ R2 = O(δ ), on the surface ∂ Bρ of the ball. The expansion of σ (wδ ) corresponding to (5.45)-(5.50) has the form

σ (wδ (x)) = σ ∞ (w0 (x0 ), x) + O(δ ) .

(5.81)

where σ ∞ is the stress distribution around a circular inclusion in an infinite medium and w0 is solution of the elasticity system (5.45)-(5.50) defined in the disk BR (x0 ) ⊂ R2 for δ = 0. Thus, σ ∞ can be calculated explicitly and it is given in a polar coordinate system (r, θ ) by:

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92

• for r ≥ δ

1−η δ 2 1−η δ 2 1−η + b 1 − 4 1+ σrr∞ (r, θ ) = a 1 − 1+ 2 ηα r ηβ r2 + 3 1+ηβ 1−η δ 2 1−η δ 4 σθ∞θ (r, θ ) = a 1 + 1+ ηα r2 − b 1 + 3 1+ηβ r4 cos 2θ 1−η δ 2 1−η δ 4 sin 2θ σr∞θ (r, θ ) = −b 1 + 2 1+ − 3 2 4 ηβ r 1+ηβ r

δ4 r4

cos 2θ (5.82) (5.83) (5.84)

• for 0 < r < δ

σrr∞ (r, θ ) = 2 1+ηαηα 1−a υ + 4 1+ηβηβ

b 3−υ

cos 2θ

(5.85)

σθ∞θ (r, θ ) = 2 1+ηαηα 1−a υ − 4 1+ηβηβ

b 3−υ

cos 2θ

(5.86)

σr∞θ (r, θ ) = −4 1+ηβηβ

b 3−υ

sin 2θ

(5.87)

In the above formulas, coefficients a and b are given respectively by 1 1 a = (σ1 + σ2 ) and b = (σ1 − σ2 ) , 2 2

(5.88)

where σ1,2 are the eigenvalues of tensor σ (w0 (x0 )). In addition, constants α and β are respectively given by

α=

1+υ 1−υ

and β =

3−υ . 1+υ

(5.89)

The jump condition of the stress field σ (wδ ) can be written as [σ (wδ )]ν = 0 ⇒ [σrr (wδ )] = 0

and [σrθ (wδ )] = 0 on ∂ Bδ .

(5.90)

In the same way, the continuity condition of the displacement field wδ implies [wδ ] = 0 ⇒ [εθ θ (wδ )] = 0 on ∂ Bδ .

(5.91)

The Eshelby tensor flux through the boundary of the inclusion is given by 1 σ (wδ ) · ε (wδ ) − σ (wδ )ν · (∇wδ ) ν 2 1 = (σθ θ (wδ )εθ θ (wδ ) − σrr (wδ )εrr (wδ ) 2

Σ (wδ )ν · ν =

+ σrθ (wδ ) ∂θ wrδ − ∂r wδθ

(5.92)

.

From the jump and continuity conditions on the boundary ∂ Bδ given by (5.90, 5.91) and considering the constitutive relation (5.36)-(5.37) for l = l ∗ , the jump of the Eshelby tensor flux in the normal direction results in (see, for instance, [13])

5 Shape and Topology Sensitivity Analysis

1 ([σθ θ (wδ )]εθ θ (wδ ) − σrr (wδ )[εrr (wδ )] 2 + 2(1 − δ )σrθ (wδ )εrθ (wδ )) .

[Σ (wδ )]ν · ν =

93

(5.93)

Finally, considering (5.94) in (5.62) and also formulas (5.82)-(5.86) we can calculate the integral on ∂ Bδ explicitly, which allows to identify function f (δ ) = πδ 2 . Then, after calculate the limit δ → 0, we obtain the following result: Theorem 5. The energy shape functional admits for δ → 0 the following topological asymptotic expansion

Πδ (uδ ) = Π (u) + πδ 2DT (x0 ) + o(δ 2 ) ,

(5.94)

with the topological derivative DT (x0 ) given by DT (x0 ) = Hη σ (u(x0 )) · ε (u(x0 )) ∀x0 ∈ Ω \ ω ,

(5.95)

where u is solution of the variational inequality (5.14)-(5.15) in Ω ⊂ R2 and Hη is a fourth-order tensor defined as Hη =

1 (1 − η )2 4 1+βη

1+β α −β 2 I+ I⊗I . 1−η 1 + αη

(5.96)

Corollary 1. Let us consider the contrast η → 0. Thus, the elastic inclusion degenerates to a circular cavity with homogeneous Neumann boundary condition and the tensor H0 becomes H0 =

1 (2(1 + β )I + (α − β )I ⊗ I) . 4

(5.97)

Corollary 2. Let us consider the contrast η → ∞. Thus, the elastic inclusion degenerates to rigid one and the tensor H∞ takes the form

α −β 1+β 1 2 H∞ = − I− I⊗I . (5.98) 4 β αβ Remark 1. From equality (5.54) we observe that the result given by theorem 5 represents the topological derivative of the Steklov-Poincaré operator (5.51). In addition, since solution u ∈ Kω of the variational inequality (5.14)-(5.15) in Ω ⊂ R2 is a H 1 (Ωγ )2 function, then it is convenient to compute the topological derivative from quantities evaluated on the boundary ∂ BR . In particular, we have the following exact analytic representation for the strain tensor ε (u(x0 )) ([40])

˙ J. Sokołowski and A. Zochowski

94 0.6

0.15

after correction

0.5 0.1

0.4

exact value

uncorrected value 0.05

0.3

after correction

uncorrected value 0.2

0

0.1 −0.05

0

exact value −0.1

0

10

20

30

40

50

60

−0.1

0

10

20

30

40

50

60

Fig. 5.4 Comparison of strains along some section of elastic body: ε11 (left), ε12 (right) – exact values, for k = 0 and with correction corresponding to k = 0.

ε11 + ε22 =

1 π R3

ε11 − ε22 =

1 π R3

2ε12 =

1 π R3

(u1 x1 + u2 x2 ) , (5.99)

12k (1 − 9k)(u1x1 − u2x2 ) + 2 (u1 x31 − u2 x32 ) (5.100) R ∂ BR

12k (1 + 9k)(u1x2 + u2x1 ) − 2 (u1 x32 + u2 x31 ) (5.101) R ∂ BR ∂ BR

where

l∗ + m . l ∗ + 3m In Fig. 5.4 we see the graphs of strains along some, irrelevant here, crossection of the elastic body. They give numerical illustration of the performance of the above formulas. The naive approach (k = 0, uncorrected values in figures) is shown to lead to substantial errors. The terms with k = 0 are needed, because u1 , u2 satisfying elasticity equations are not harmonic. Once the above integrals are evaluated e.g. numerically, then we can use the constitutive relation (5.36) to compute the stress tensor σ (u(x0 )). Finally, these results can by used to compute the topological derivative through formula (5.95). k=

5.3.4 Approximation of Solutions for Variational Inequalities We define a variational inequality for the crack problem with a perturbed bilinear form. The bilinear form is defined in the whole domain of integration, it is bounded and coercive on the energy space for the crack problem without any inclusion, and provides the first order topological sensitivity for the solutions of nonlinear elasticity boundary value problem with the nonlinear crack.

5 Shape and Topology Sensitivity Analysis

95

Approximation of crack problem in Ωδ . We determine the modified bilinear form as a sum of two terms, as it is for the energy functional, the first term defines the elastic energy in the domain Ω , the second term is a correction term, determined in Section 5.3.3. The correction term is quite complicated to evaluate, and we provide its explicit form, such a form is actually defined by the formulas in Section 5.3.3. The values of the symmetric bilinear form a(δ ; ·, ·) are given by the expression a(δ ; v, v) = a(u, u) + δ 2b(v, v) .

(5.102)

The derivative b(v, v) of the bilinear form a(δ ; v, v) with respect to δ 2 at δ = 0+ is given by the expression b(v, v) = −2π ev (0) −

2π m l ∗ + 3m

σII δ1 − σ12 δ2 ,

(5.103)

where all the quantities are evaluated for the displacement field v according to formulas in Section 5.3.3, where we provide the line integrals which defines all terms in (5.99), (5.100) and (5.101). Hence, we can determine the bilinear form a(δ ; v, w) for all v, w, from the equality 2a(δ ; v, w) = a(δ ; v + w, v + w) − a(δ ; w, w) − a(δ ; v, v) . In the same way the bilinear form b(v, w) is determined from the formula for b(v, v). The convex set is defined in this case by Kδ = {v ∈ HΓ1 (Ωδ )2 | [v]ν ≥ 0 on γ } .

(5.104)

Let us consider the following variational inequality which provides a sufficiently precise for our purposes approximation uδ of the solution u(Ωδ ) to crack problem defined in singularly perturbed domain Ωδ , uδ ∈ Kδ :

a(δ ; u, v − u) ≥ (F, v − u)Ωδ

∀v ∈ Kδ .

(5.105)

The result obtained is the following, for simplicity we assume that the linear form L(δ ; ·) is independent of δ . Theorem 6. For δ sufficiently small we have the following expansion of the solution uδ with respect to the parameter δ at 0+, uδ = u(Ω ) + δ 2 q + o(δ 2) in H 1 (Ω )2 ,

(5.106)

where the topological derivative q of the solution u(Ω ) to the crack problem is given by the unique solution of the following variational inequality q ∈ SK (u) = {v ∈ (HΓ1 (Ωγ ))2 | [v] · ν ≥ 0 on Ξ (u) , a(0; u, v) = 0} a(q, v − q) + b(u, v − q) ≥ 0 ∀v ∈ SK (u) .(5.107)

˙ J. Sokołowski and A. Zochowski

96

The coincidence set Ξ (u) = {x ∈ γ | [u(x)] · ν (x) = 0} is well defined ([9]) for any function u ∈ H 1 (Ω )2 , and u ∈ K is the solution of variational inequality (5.104) for δ = 0. For the proof of theorem we refer the reader to [39]. For the convenience of the reader we provide the explicit formulas for the terms in b(v, v) defined by (5.103), we refer to section 5.3.3 and to [39], [40] for details. We have

2π ev (0) =

π (l ∗ + m) π 2 R6

ΓR

2 (v1 x1 + v2 x2 ) ds

(5.108)

2 12k m 3 3 (1 − 9k)(v x − v x ) + (v x − v x ) ds 1 1 2 2 1 1 2 2 π 2 R6 R2 ΓR

2 m 12k 3 3 (1 + 9k)(v1x2 + v2 x1 ) − 2 (v1 x2 + v2 x1 ) ds + 2 6 , π R R ΓR +

with m σII = π R3

σ12 =

m π R3

12k 3 3 (1 − 9k)(v1 x1 − v2 x2 ) + 2 (v1 x1 − v2 x2 ) ds, R ΓR 12k 3 3 (1 + 9k)(v1x2 + v2 x1 ) − 2 (v1 x2 + v2 x1 ) ds, R ΓR

and 9k δ1 = π R3

δ2 =

9k π R3

4 3 3 (v1 x1 − v2 x2 ) − 2 (v1 x1 − v2 x2 ) ds, 3R ΓR 4 (v1 x2 + v2 x1 ) − 2 (v1 x32 + v2 x31 ) ds. 3R ΓR

Acknowledgements This research is partially supported by the Brazilian-French research program CAPES/COFECUB under grant 604/08 between LNCC in Petropolis and IECN in Nancy and by the Brazilian agencies CNPq under grant 472182/2007-2 and FAPERJ under grant E-26/171.099/2006 (Rio de Janeiro). The work is also supported by the grant N51402132/3135 Ministerstwo Nauki i Szkolnictwa Wyzszego: Optymalizacja z wykorzystaniem pochodnej topologicznej dla przeplywow w osrodkach scisliwych in Poland. These supports are gratefully acknowledged.

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97

References [1] Allaire, G., Gournay, F., Jouve, F., Toader, A.: Structural optimization using topological and shape sensitivity via a level set method. Control and Cybernetics 34(1), 59–80 (2005) [2] Amstutz, S., Andrä, H.: A new algorithm for topology optimization using a level-set method. Journal of Computational Physics 216(2), 573–588 (2006) [3] Amstutz, S., Horchani, I., Masmoudi, M.: Crack detection by the topological gradient method. Control and Cybernetics 34(1), 81–101 (2005) [4] Auroux, D., Masmoudi, M., Belaid, L.: Image restoration and classification by topological asymptotic expansion. In: Variational formulations in mechanics: theory and applications, Barcelona, Spain (2007) [5] Belaid, L., Jaoua, M., Masmoudi, M., Siala, L.: Application of the topological gradient to image restoration and edge detection. Engineering Analysis with Boundary Element 32(11), 891–899 (2008) [6] Burger, M., Hackl, B., Ring, W.: Incorporating topological derivatives into level set methods. Journal of Computational Physics 194(1), 344–362 (2004) [7] Eshelby, J.: The elastic energy-momentum tensor. Journal of Elasticity 5(3-4), 321–335 (1975) [8] Feijóo, G.: A new method in inverse scattering based on the topological derivative. Inverse Problems 20(6), 1819–1840 (2004) [9] Frémiot, G., Horn, W., Laurain, A., Rao, M., Sokolowski, J.: On the analysis of boundary value problems in nonsmooth domains. Dissertationes Mathematicae. Institute of Mathematics of the Polish Academy of Sciences 462, 149 pages (2009) [10] Fulmanski, P., Laurain, A., Scheid, J.-F., Sokolowski, J.: A level set method in shape and topology optimization for variational inequalities. Int. J. Appl. Math. Comput. Sci. 17(3), 413–430 (2007) [11] Fulmanski, P., Laurain, A., Scheid, J.-F., Sokolowski, J.: Level set method with topological derivatives in shape optimization 85(10), 1491–1514 (2008) [12] Garreau, S., Guillaume, P., Masmoudi, M.: The topological asymptotic for pde systems: the elasticity case. SIAM Journal on Control and Optimization 39(6), 1756–1778 (2001) [13] Giusti, S., Novotny, A., Padra, C.: Topological sensitivity analysis of inclusion in twodimensional linear elasticity. Engineering Analysis with Boundary Elements 32(11), 926–935 (2008) [14] Gurtin, M.: An introduction to continuum mechanics. Mathematics in Science and Engineering, vol. 158. Academic Press, New York (1981) [15] Gurtin, M.: Configurational forces as basic concept of continuum physics. Applied Mathematical Sciences, vol. 137. Springer, New York (2000) [16] Hintermüller, M.: Fast level set based algorithms using shape and topological sensitivity. Control and Cybernetics 34(1), 305–324 (2005) ˙ [17] Hlaváˇcek, I., Novotny, A., Sokołowski, J., Zochowski, A.: On topological derivatives for elastic solids with uncertain input data. Journal Optimization Theory and Applications 141(3), 569–595 (2009) [18] Hoffmann, K.-H., Khludnev, A.: Fictitious domain method for the Signorini problem in linear elasticity. Advanced Mathematical Science and Applications 14(2), 465–481 (2004) [19] Khludnev, A.: Crack theory with possible contact between the crack faces. Russian Surveys in Mechanics 3(4), 41–82 (2005) ˙ [20] Khludnev, A.M., Novotny, A.A., Sokołowski, J., Zochowski, A.: Shape and topology sensitivity analysis for cracks in elastic bodies on boundaries of rigid inclusions. Journal of the Mechanics and Physics of Solids 57(10), 1718–1732 (2009)

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[21] Khludnev, A., Sokolowski, J.: Griffith formulae for elasticity systems with unilateral conditions in domains with cracks. European Journal on Mechanics A/Solids 19, 105– 119 (2000) [22] Khludnev, A., Sokolowski, J.: On differentation of energy functionals in the crack theory with possible contact between crack faces. Journal of Applied Mathematics and Mechanics 64(3), 464–475 (2000b) [23] Khludnev, A., Sokolowski, J.: Smooth domain method for crack problem. Quarterly Applied Mathematics 62(3), 401–422 (2000c) [24] Larrabide, I., Feijóo, R., Novotny, A., Taroco, E.: Topological derivative: a tool for image processing. Computers & Structures 86(13-14), 1386–1403 (2008) [25] Lewinski, T., Sokołowski, L.: Energy change due to the appearance of cavities in elastic solids. Int. J. Solids Struct. 40(7), 1765–1803 (2003) [26] Masmoudi, M., Pommier, J., Samet, B.: The topological asymptotic expansion for the Maxwell equations and some applications. Inverse Problems 21(2), 547–564 (2005) [27] Morassi, A., Rosset, E.: Detecting rigid inclusions, or cavities, in an elastic body. Journal of Elasticity 72, 101–126 (2003) [28] Nazarov, S., Sokołowski, J.: Asymptotic analysis of shape functionals. Journal de Mathématiques Pures et Appliquées 82(2), 125–196 (2003) [29] Nazarov, S.: Elasticity polarization tensor, surface enthalpy, and Eshelby Theorem. Journal of Mathematical Sciences 159(2), 133–167 (2009) [30] Nazarov, S., Sokołowski, J., Specovius-Neugebauer, M.: Asymptotic analysis and polarization matrices (to appear) [31] Novotny, A., Feijóo, R., Padra, C., Taroco, E.: Topological sensitivity analysis. Computer Methods in Applied Mechanics and Engineering 192(7-8), 803–829 (2003) [32] Novotny, A., Feijóo, R., Padra, C., Taroco, E.: Topological derivative for linear elastic plate bending problems. Control and Cybernetics 34(1), 339–361 (2005) [33] Novotny, A., Feijóo, R., Taroco, E., Padra, C.: Topological sensitivity analysis for threedimensional linear elasticity problem. Computer Methods in Applied Mechanics and Engineering 196(41-44), 4354–4364 (2007) [34] Plotnikov, P.I., Sokołowski, J.: Inhomogeneous boundary value problems for compressible Navier-Stokes and transport equations. Journal des Mathématiques Pure et Appliquées 92(2), 113–162 (2009) [35] Plotnikov, P.I., Sokołowski, J.: Inhomogeneous boundary value problems for compressible Navier-Stokes equations, well-posedness and sensitivity analysis. SIAM Journal on Mathematical Analysis 40(3), 1152–1200 (2008) [36] Plotnikov, P.I., Sokołowski, J.: Shape derivative of drag functional (submitted, 2009) ˙ [37] Sokołowski, J., Zochowski, A.: On the topological derivatives in shape optmization. SIAM Journal on Control and Optimization 37(4), 1251–1272 (1999) ˙ [38] Sokołowski, J., Zochowski, A.: Topological derivatives of shape functionals for elasticity systems. Mechanics of Structures and Machines 29(3), 333–351 (2001) ˙ [39] Sokołowski, J., Zochowski, A.: Modeling of topological derivatives for contact problems. Numerische Mathematik 102(1), 145–179 (2005) ˙ [40] Sokołowski, J., Zochowski, A.: Topological derivatives for optimization of plane elasticity contact problems. Engineering Analysis with Boundary Elements 32(11), 900– 908 (2008) [41] Sokołowski, J., Zolésio, J.: Introduction to shape optimization. Shape sensitivity analysis. Springer, New York (1992)

Chapter 6

A Boundary Integral Equation on the Sphere for High-Precision Geodesy Ernst P. Stephan, Thanh Tran, and Adrian Costea

Abstract. Spherical radial basis functions are used to approximate the solution of a boundary integral equation on the unit sphere which is a reformulation of a geodetic boundary value problem. The approximate solution is computed with a corresponding meshless Galerkin scheme using scattered data from satellites. Numerical experiments show that this meshless method is superior to standard boundary element computations with piecewise constants. If we increase the element order, BEM might be competitive but then we also have to approximate appropriately the surface otherwise the convergence rate will be spoiled.

6.1 Boundary Integral Equation We consider the linearized Molodensky problem in geodesy for the disturbing gravity potential u. For a gravity g given at scattered points find u satisfying:

Δu = 0 in Ω = R3 \Ω˜ ∂u 2 = g on the boundary ∂ Ω˜ − u− r ∂r Ernst P. Stephan Institute for Applied Mathematics, Leibniz University Hannover, Welfengarten 1, Hannover, Germany e-mail: [email protected] Thanh Tran School of Mathematics and Statistics, Sydney 2052 , Australia e-mail: [email protected] Adrian Costea Institute for Applied Mathematics, Leibniz University Hannover, Welfengarten 1, Hannover, Germany e-mail: [email protected] M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 99–110. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

(6.1)

100

E.P. Stephan, T. Tran, and A. Costea

where ∂ Ω˜ denotes the telluroid and r denotes the radius of x ∈ R3 . In the following we consider the model where the telluroid coincides with the surface of the earth which is given by S, the boundary of the unit ball Ω˜ = B1 (0). As described in the paper by Heck [2] a single layer potential ansatz leads to a pseudodifferential equation (a second kind Fredholm integral equation) on S which will be solved numerically in the following by use of radial basis functions. By inserting the approximation of the density back into the single layer potential we thus obtain an approximation for the gravity potential u, the solution of the above geodetic boundary value problem GBVP (6.1). In detail we proceed as follows: We write u as single layer potential S with unknown density μ for X ∈ /S u(X) = S μ (X) =

1 4π

μ (y) ds(y) |X − y| S

(6.2)

and compute 1 1 (grad S μ (x))+ = − μ (x)nx − p.v. 2 4π

x−y μ (y)ds(y) 3 S |x − y|

where + denotes the limit on the surface S from the exterior domain Ω and n is the normal on S pointing into Ω . Furthermore

∂ (S μ ) 1 x 1 (x) = gradS μ (x)· = − μ (x)cos (nx , x)− p.v. ∂r |x| 2 4π and

2 2 1 S μ (x) = r |x| 4π

(x − y) · x μ (y)ds(y) |x||x − y|3 S

μ (y) ds(y). |x − y| S

Inserting these expressions into the boundary condition of (6.1) yields (see [2])

∂ (S μ ) 1 2 (x) = μ (x)cos (nx , x)+ − S μ (x) − (6.3) r ∂r 2 (x − y) · x 2 1 − μ (y)ds(y) + p.v. 3 4π |x||x − y| S |x||x − y| 1 = μ (x) cos(nx , x)+ 2 |x|2 − |y|2 − 3|x − y|2 1 μ (y)ds(y) + p.v. 4π 2|x||x − y|3 S = g(x)

6 A Boundary Integral Equation on the Sphere for High-Precision Geodesy

101

This becomes

μ (y) 1 1 1 dsy + μ (x) + Lμ (x) : = 2 − 4π S ||x − y|| 2 4π −3μ (y) 1 1 dsy = μ (x) + p.v. 2 4π S 2||x − y|| = g(x)

(x − y) · x μ (y)dsy 3 S ||x − y|| (6.4)

Now we observe that the action of the above pseudodifferential operator L can be rewritten via its discrete symbol Lˆ and the Fourier coefficients μˆ l,m of μ with respect to spherical harmonics Yl,m of degree l as Lμ =

∞

l

∑ ∑

ˆ μˆ l,mYl,m L(l)

l=0 m=−l

ˆ = l−1 , and μˆ l,m = μ ,Yl,m L (S) ( compare (6.15) below). As a with symbol L(l) 2l+1 2 pseudodifferential operator of order zero, L maps H s into itself for any s ∈ R where the Sobolev space H s is defined by $ % Hs =

∞

v : S → R| ∑

l

∑

(l + 1)2s |vˆl,m |2 < ∞ .

l=0 m=−l

Let us abbreviate the pseudodifferential equation (6.4) as Lμ = g We observe that

on S.

(6.5)

ˆ = l − 1 = 0 for l = 1. L(l) 2l + 1

So ker (L) = span {Y1,m , m = −1, 0, 1}. Therefore to ensure unique solvability of (6.5) we must impose side conditions :

γ jμ = aj

( j = 1, 2, 3)

(6.6)

with given a j ∈ R and a unisolvent set of linear functionals {γ j }, i.e., for any v ∈ ker(L), if γ j v = 0, j = 1, 2, 3, then v = 0. Application of classical Riesz-Schauder theory gives the following theorem. Theorem 1. Equations (6.5) and (6.6) have a unique solution if g, Φ = 0 ∀Φ ∈ kerL. Next we comment on the above side conditions. First let us rewrite the single layer potential as μ (y) 1 S μ (X) = ds(y) (6.7) 4π S 2 sin ψ2

102

E.P. Stephan, T. Tran, and A. Costea

where ψ is the angle between X and y (c.f. [2]). We expand the disturbing potential outside the boundary sphere S into solid spherical harmonics as

n+1 1 ∑ r un(x), n=0 ∞

u(X) = where

X = r · x, r > 1,

(6.8)

n

un (x) =

∑

uˆn,mYn,m (x).

m=−n

Correspondingly we expand the functions g(x) and μ (x) in surface spherical harmonics as ∞

g(x) =

∑ gn(x),

n=0

∞

∑ μn (x).

(6.9)

n+1 1 un (x). r

(6.10)

μ (x) =

n=0

Note that due to (6.2) 1 4π

μn (y) ψ ds(y) = 2 S sin 2

Furthermore with (6.7) the integral equation (6.4) becomes 1 1 μ (x) + 2 4π

−3μ (y) ψ ds(y) = g(x). S 4 sin 2

(6.11)

Then inserting (6.9), (6.10) and (6.8) and equating the coefficients we obtain

μn (x) − 3r−n−1un (x) = 2gn (x)

(6.12)

Setting r = 1 we have

μ0 (x) − 3u0(x) = 2g0 (x) for n = 0, μ1 (x) − 3u1(x) = 2g1 (x) for n = 1. Therefore with u1 (x) = ∑1m=−1 uˆ1,mY1,m (x) and μ1 , g1 correspondingly we obtain

μˆ 1,m − 3uˆ1,m = 2gˆ1,m ,

m = −1, 0, 1.

(6.13)

ˆ Next we comment on the discrete symbol L(l) of the pseudodifferential operator L. This can be computed by simply inserting the expansion for u and g into the boundary condition of the GBVP (6.1). From

2 ∂u − u− =g r ∂r S

6 A Boundary Integral Equation on the Sphere for High-Precision Geodesy

103

we have

∞ ∞ 2 ∞ 1 n+1 1 − ∑ un (x) + ∑ (n + 1) n+2 un (x) = ∑ gn (x) r n=0 r r n=0 n=0 or

∞

∞ (n − 1) un (x) = ∑ gn (x). n+2 n=0 r n=0

∑

Equating the coefficients and then inserting in (6.12) gives with r = 1 3 gn (x) = 2gn (x) n−1

(6.14)

3 2n + 1 gn (x) = gn (x) n−1 n−1

(6.15)

μn (x) − hence

μn (x) = 2 +

from which we deduce the result on the discrete symbol Lˆ of the pseudodifferential operator L ( compare Heck [2]).

6.2 Meshless Galerkin Method with Boundary Integral Equations The solutions of (6.5), (6.6) are approximated by spherical radial basis functions. Let $ (1 − r)m+2 , 0 < r ≤ 1, , m = 0, 1, 2 ρm (r) = 0, r > 1. √ We define φ : [−1, 1] → R by φ (t) = ρm ( 2 − 2t). This is called the Wendland function. For a set of data points {x1 , x2 , . . . , xN } on the sphere we define a set of sperical radial basis functions as:

Φi (x) :=

∞

n

∑ ∑

φˆ (n)Yn,m (xi )Yn,m (x) = φ (x · xi )

n=0 m=−n

with Fourier-Legendre coefficients

φˆ (n) = 2π

1 −1

φ (t)Pn (t)dt,

and Legendre polynomials Pn of degree n. It is shown in [5, Proposition 4.6] that for m = 0, 1, 2 c1 (1 + n2)−m−3/2 ≤ φˆ (n) ≤ c2 (1 + n2)−m−3/2 for all n = 0, 1, 2, . . ., where c1 and c2 are positive constants.

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Now the numerical scheme under consideration is the Galerkin method (G): Find μ˜ N = μ˜ 1 + μ˜ 0 with μ˜ 1 = ∑Ni=1 ci Φi∗ Lμ˜ 1 , Φk∗ L2 (S) = g, Φk∗ L2 (S) ,

1≤k≤N

where Φi∗ is obtained from Φi by deleting the Fourier terms for n = 0 and n = 1. The part μ˜ 0 of the Galerkin solution μ˜ N must satisfy the side conditions

γ 0 μ˜ 0 =

S

μ˜ 0Y0,0 = a0 ,

γ 1 μ˜ 0 =

S

μ˜ 0Y1,−1 = b−1 , (6.16)

γ 2 μ˜ 0 =

S

μ˜ 0Y1,0 = b0 ,

γ 3 μ˜ 0 =

S

μ˜ 0Y1,1 = b1 .

Note that the term for n = 0 in the expansion for Φi is dropped to make all Fourier modes in the entries of the Galerkin stiffness matrix positive. The side conditions γ 1 , γ 2 , γ 3 are just the side conditions (6.6), whereas the side condition γ 0 is needed since we dropped the expansion term for n = 0 in Φi . Theorem 2. For N sufficiently large the Galerkin scheme (G) is uniquely solvable and the Galerkin solution μ˜ N = μ˜ 1 + μ˜ 0 converges to the exact solution μ ∈ H s , 0 ≤ s ≤ m + 3, ||μ − μ˜ N ||H 0 = O(hs ), where H 0 = L2 (S). Proof. The convergence of the Galerkin scheme follows due to the fact that the pseudodifferential operator L can be written as identity plus a compact operator. Therefore it is a strongly elliptic operator in the sense of Wendland [9]. Due to the general convergence results in Stephan and Wendland [6] strong ellipticity together with the side conditions guarantee convergence of general Galerkin schemes including the case of radial basis functions considered here.The convergence estimate follows by applying a coresponding approximation result of functions in the Sobolev space H s by radial basis functions from [4] together with the quasi optimality of the Galerkin error. The latter quasi optimality follows from the analysis in [6]. Next we comment on the computation of the Galerkin stiffness matrix and the right hand side. Note that LΦi∗ , Φ ∗j L2 (S) = =

∞

n−1 & |φ (n)|2Yn,m (xi )Yn,m (x j ) 2n + 1 n=2 m=−n n

∑ ∑

n−1 & |φ (n)|2 Pn (xi · x j ), 4π n=1,n=2

∑

where we have used '∗ )n,m = φˆ (n)Yn,m (xi ) (Φ i

(6.17)

6 A Boundary Integral Equation on the Sphere for High-Precision Geodesy

105

and the addition theorem Pn (x · y) =

n 4π Yn,m (x)Yn,m (y). ∑ 2n + 1 m=−n

For the right hand side we have g, Φk∗ L2 (S) =

∞

n

∑ ∑

' φˆ (n)Yn,m (xk ). (g) n,m

(6.18)

n=0 m=−n

6.3 Numerical Example In the following we compute approximations of the Galerkin solution μ˜ N of (G) by truncating the series expansion of Φi∗ at n = 500. We consider u(X) =

1 , ||X − p||

p = (0, 0, 0.5)

and compute the right hand side via g = −2u −

∂u ∂n

on S.

In the side conditions (6.16) we choose √ √ a0 = 2 π , b1 = b−1 = 0, b0 = 3π . This gives

μ˜ 0 = a0Y0,0 + b−1Y1,−1 + b0Y1,0 + b1Y1,1 ( ( √ 1 √ 3 3 =2 π + 3π cos θ = 1 + cos θ . 4π 4π 2 In Table 6.1 we have listed |(uN (q) − u(q))| for q = (1.10227, 1.10227, 0.9) with uN (q) :=

1 4π

μ˜ N (y) ds(y) S ||q − y||

where μ˜ N is the Galerkin solution of our Galerkin system (G) computed with the Wendland radial basis functions [8] mentioned in Section 6.2, namely √ φ (t) = (1 − 2 − 2t)m+2 .

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We denote this Galerkin approximation “meshless” in Table 6.1 to distinguish from the standard boundary element solution computed with piecewise constants on the uniform mesh of Fig. 6.2. In the tables below we list the respective experimental orders of convergence for the pointwise error of the gravity potential at the point q outside of the unit sphere S. Table 6.1 Meshless Galerkin approximation uN of gravity potential u; single layer density computed by spherical radial basis functions centered at N Saff points

m=0

m=1

m=2

N =number of Value points

|(uN (q) − u(q))|

EOC

200 500 1000 2000 4000 8000

0.69442 0.67349 0.65323 0.62413 0.62213 0.62198

0.07305 0.05212 0.03186 0.00276 0.00076 0.00061

0.36844 0.71009 3.52900 1.86060 0.31691

200 500 1000 2000 4000 8000

0.79285 0.62781 0.62731 0.62326 0.62228 0.62197

0.17148 0.00644 0.00594 0.00189 0.00091 0.00059

3.58177 0.11660 1.65208 1.05445 0.60814

200 500 1000 2000 4000 8000

0.71406 0.62638 0.62603 0.62283 0.62192 0.62156

0.09269 0.00501 0.00466 0.00146 0.00055 0.00019

3.18439 0.10448 1.67436 1.40846 1.53343

The numerical experiments are performed on a uniform grid of Saff points c.f. Fig.6.1. The errors are plotted in Fig.6.3. For comparison we present here also the numerical experiments when solving approximately the integral equation (6.4) with standard BEM when using piecewise constant basis functions on triangles which approximate the surface c.f. Fig.6.2. Table 6.2 shows the corresponding results ( again q = (1.10227, 1.10227, 0.9)). In Fig.6.3 we see that radial basis functions give better convergence than standard BEM. Since data are not available at the poles we must use for the standard BEM with piecewise constants the grid shown in Fig. 6.4 with holes at the poles. Table 6.3 and Table 6.4 and Fig. 6.4 show clearly that scaled radial basis functions give much better results than standard boundary elements.

6 A Boundary Integral Equation on the Sphere for High-Precision Geodesy

107

Fig. 6.1 Uniformly distributed Saff points on S (N = 1000 Saff points), c.f. [1]

1

0.8

0.6

0.4

0.2

0

−0.2

−0.4

−0.6

−0.8 1 −1 1

0.5 0.8

0.6

0 0.4

0.2

0

−0.5 −0.2

−0.4

−0.6

−0.8

−1 −1

Fig. 6.2 Boundary element mesh consisting of triangles with vertices at N = 1000 Saff points

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Table 6.2 Standard BEM Galerkin approximation uN of gravity potential u; single layer density computed by pw. constants on triangles with N Saff points N =number of points

Value

|(uN (q) − u(q))| EOC

500 1000 2000 4000 8000

0.61808 0.61932 0.62010 0.62028 0.62051

0.00329 0.00205 0.00127 0.00109 0.00086

0.68246 0.69080 0.22050 0.34192

0.1 m=0 m=1 m=2 Saff pw.const.

|(u_n-u)(q)|

0.01

0.001

0.0001 100

1000 Number of points

10000

Fig. 6.3 Pointwise error |(uN (qq) − u(qq))| μ˜ N computed with radial basis functions (m = 0, 1, 2) and piecewise constants in the Saff points in Fig 6.1 and Fig 6.2. Table 6.3 Meshless Galerkin approximation with spherical radial basis functions at scattered points on S Number of points

Value

|(uN (q) − u(q))| EOC

m=0 scale=20.5

2133 3458 4108 7663 10443

0.62932 0.62507 0.62288 0.62254 0.62167

0.00795 0.00369 0.00151 0.00117 0.00030

1.75153 6.74972 0.40891 1.40226

m=0 scale=20

2133 3458 4108 7663 10443

0.62915 0.62471 0.62241 0.62203 0.62112

0.00778 0.00334 0.00104 0.00066 0.00024

1.58290 5.18667 0.74527 3.20844

6 A Boundary Integral Equation on the Sphere for High-Precision Geodesy

109

Table 6.4 BEM Galerkin approximation with pw. constants on triangles with vertices at scattered points Number of points

Value

|(uN (qq) − u(qq))| EOC

2133 7699 10643

0.67059 0.64751 0.63797

0.04922 0.02614 0.01660

0.49303 1.40226

0.1 m=0,scale=20 m=0,scale=20.5 pw

|(u_n-u)(q)|

0.01

1 0.8 0.6

0.001

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1

0.0001 1000

10000 Number of points

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

100000

Fig. 6.4 Pointwise error |(uN (qq) − u(qq))| for μ˜ N computed on N=3458 scattered points with scaled radial basis functions or piecewise constants

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E.P. Stephan, T. Tran, and A. Costea

6.4 Conclusion We have shown that for geodetic boundary value problems the reduction to boundary integral equations leads to a fast numerical method when the Galerkin scheme is performed with spherical radial basis functions. It is a meshless method and therefore it can be applied efficiently to problems where the satellite data are given at scattered points. Here standard boundary elements are not appropriate. As we have shown in [3] the approach can be extended to spheroids and is not restricted to the unit sphere as might be expected since in our method the density of the integral equation and also the Galerkin solution are expanded in spherical harmonics. As numerical experiments in [7] have shown the overlapping additive schwarz method can be applied as an efficient preconditioner to reduce the computing time for the numerical method.

Acknowledgements The authors Ernst P. Stephan and Adrian Costea would like to acknowledge the financial support provided by Excellence Cluster QUEST (Centre for Quantum Engineering and Space-Time Research), Leibniz University Hannover.

References [1] Hardin, D.P., Saff, E.B.: Discretizing manifolds via minimum energy points. Notices of the AMS 51, 1186–1194 (2004) [2] Heck, B.: Integral Equation Methods in Physical Geodesy. In: Grafarend, E.W., Krumm, F.W., Schwarze, V.S. (eds.) Geodesy - The Challenge of the 3rd Millenium, pp. 197–206. Springer, Heidelberg (2002) [3] Le Gia, Q.T., Tran, T., Stephan, E.P.: Solution to the Neumann problem exterior to a prolate spheroid by radial basis functions. Advances in Computational Mathematics (submitted) [4] Le Gia, Q.T., Tran, T., Sloan, I.H., Stephan, E.P.: Boundary Integral Equations on the Sphere with Radial Basis Functions: Error Analysis. Applied Numerical Mathematics (in print), http://dx.doi.org/10.1016/j.apnum.2008.12.033 [5] Narcowich, F.J., Ward, J.D.: Scattered data interpolation on spheres: error estimates and locally supported basis functions. SIAM J. Math Anal. 33, 1393–1410 (2002) [6] Stephan, E.P., Wendland, W.L.: Remarks to Galerkin and Least Squares Methods with Finite Elements for General Elliptic Problems. Manuscripta geodaetica 1, 93–123 (1976) [7] Tran, T., Le Gia, Q.T., Sloan, I.H., Stephan, E.P.: Preconditioners for Pseudodifferential Equations on the Sphere with Radial Basis Functions. Numer. Math. (to appear) [8] Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial basis functions of minimal degree. Advances in Computational Mathematics 4, 389–396 (1995) [9] Wendland, W.L.: Asymptotic accuracy and convergence. In: Brebbia, C.A. (ed.) Progress in Boundary Element Methods, vol. 1, pp. 289–313. Pentech Press, London (1981)

Chapter 7

Unresolved Problems of Adaptive Hierarchical Modelling and hp-Adaptive Analysis within Computational Solid Mechanics Grzegorz Zboi´nski

Abstract. In this chapter of the book we present some chosen problems of adaptive hierarchical modelling and adaptive hp-analysis of problems of computational solid mechanics. We consider simple and complex structures, i.e. structures described by one mechanical model or more than one mechanical model, respectively. We are interested in three fundamental problems of solid mechanics: the equilibrium (static) problem, eigenvalue (free vibration) problem, and stationary forced vibration problem as well. In the context of static analysis, we consider problems faced while applying: the hierarchical models, hp-approximations, residual error estimation methods, and three- or four-step non- or iterative adaptive strategies, oriented on the target admissible value of the modelling and approximation errors. We also address possibilities and difficulties of generalization of the mentioned techniques onto free and forced vibration analyses. In this contribution we are interested mainly in still unresolved or open problems. We will show some ways, either potentially available or checked by us numerically, to cope with all the mentioned issues.

7.1 Introduction In this book chapter we would like to discuss some theoretical and implementation difficulties of the adaptive hierarchical modelling and hp-adaptive analysis within computational solid mechanics. We present these difficulties in the context of our uniform methodology that can be applied to the equilibrium, free vibration and stationary forced vibration problems. This methodology [19] can be applied to both, simple or complex structures, i.e. structures of simple or complex mechanical Grzegorz Zboi´nski Polish Academy of Sciences, Institute of Fluid Flow Machinery, ul. Fiszera 14, 80-952 Gda´nsk, Poland e-mail: [email protected] University of Warmia and Mazury, Faculty of Technical Sciences, ul. Oczapowskiego 11, 10-736 Olsztyn, Poland

M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 111–145. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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description, based on one or more mechanical models, respectively. These two descriptions can be applied within both, simple and complex geometries. Our simple geometries, as well as different parts of complex geometries, can be of either solid, shell, or transition character. Note that our approach is much more general than the approaches of the predecessors, who usually consider one problem of solid mechanics, single mechanical model, and one type of the applied geometry. In other words, we search for one generalizing approach, suitable for a wide range of problems, rather than for specialized approaches, assigned for the specific mechanical problems.

7.1.1 Considered Problems of Solid Mechanics As mentioned above, we consider three fundamental problems of solid mechanics: the static equilibrium problem, and the dynamic problems of free and stationary forced vibrations.

7.1.1.1 Static Equilibrium Problem Let us start with the equilibrium problem. In this case we search for the static solution in displacements u = u(x). The local formulation within the elastic body V under consideration (Fig. 7.1) is

σi j, j + f0i (x) = 0, εi j = 1/2(ui, j + u j,i ), σi j = Di jkl εkl , x ∈ V,

(7.1)

where Di jkl , i, j, k, l = 1, 2, 3 is the elastic constants tensor, σi j and εkl are stress and strain tensors, f0i is the vector of static body forces, while ui is the ith component of the vector u. The above set has to be completed with the traction and kinematic boundary conditions on parts SP and SD respectively, of the body surface S

σi j n j = p0i , x ∈ SP , ui = d0i , x ∈ SD ,

(7.2)

with n j denoting the vector normal to this surface, p0i standing for the static surface tractions, and d0i being the components of the prescribed surface displacements. In order to obtain the finite element equations of the problem we have to take advantage of the variational formulation corresponding to the above local formulation. The former formulation reads Di jkl vi, j uk,l dV = V

vi f0i dV + V

vi p0i dS, SP

(7.3)

7 Unresolved problems of hierarchical modelling...

113 f

0

SD SP

V

M M

Fig. 7.1 The elastic body under consideration (the static case)

p

0

with vi standing for the admissible values of the trial displacement functions, conforming to the kinematic boundary conditions. The vector form of the finite element formulation, obtained from (7.3) after the discretization and the introduction of the hpq-interpolation shape functions, is Kqhpq = F0 ,

(7.4)

where K is the global nodal stiffness matrix, F0 is the global nodal static forces vector, while qhpq stands for the global nodal vector of the displacement dofs correM sponding to the hpq finite element formulation (see the next subsections).

7.1.1.2 Free Vibration Problem In the case of the free vibration problem (eigenproblem) of the elastic body, the solution, we search for, takes the form u = a(x)e jω0t , where a(x) represents free vibration amplitudes, ω0 is the natural frequency of the body, t is the time variable, while j is the imaginary unit. The local formulation reads now ..

σi j, j = ρ ui , εi j = 1/2(ui, j + u j,i ), σi j = Di jkl εkl , x ∈ V,

(7.5)

..

jω t with ρ standing for the material density and ui = −ω02 ai 0 being the second time derivatives of the displacements (the acceleration components). The usual boundary conditions for the free vibration problem are

σi j n j = 0, x ∈ SP , ui = 0, x ∈ SD ,

(7.6)

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G. Zboi´nski

The corresponding variational formulation, necessary for derivation of the finite element equations, and expressed through the amplitudes, is

V

Di jkl vi, j ak,l dV − ω02

ρ vi ai dV = 0

(7.7)

V

In the above relation, the values vi represent the kinematically admissible values of the amplitudes, conforming to the boundary conditions (7.6) expressed through the amplitudes. Note, that now we search for the all eigenpairs (eigenvalues and eigenvectors), ω02 and a(x), fulfilling (7.7). In order to obtain the set of the finite element method relations, resulting from the above variational formulation, one has to perform division of the body into finite elements and introduce the hpq-interpolation. This leads to the discretized eigenproblem of N degrees of freedom. The characteristic equation, necessary for deter2 , n = 1, 2, ..., N, is mination of N eigenvalues ω0n det(K − ω02 M) = 0

(7.8)

where M stands for the global mass (inertia) matrix. The dynamic equilibrium equation, necessary for determination of the nth eigenvector of amplitudes is 2 (K − ω0n M)qn

hpq

hpq T hpq qn Mqn

= 0,

=1

(7.9)

So as to obtain the unique values of the eigenvectors, the first equation (7.9) has been completed with the normalization condition (the second equation (7.9)), here corresponding to the M-orthonormal normalization. In the above relations the amplitude hpq dofs vector qn has been employed. Note that the displacements corresponding to nth mode of vibration can be calculated form the amplitudes and the natural frejω t quencies as qhpq n e 0n .

7.1.1.3 Forced Vibrations We consider here the stationary forced vibration problem for which the superposition theorem is valid. In the case of a single force of vibration the local formulation reads .

..

σi j, j + α ρ ui + fi (x,t) = ρ ui , εi j = 1/2(ui, j + u j,i ), . σi j = Di jkl εkl + β ε i j , x ∈ V,

(7.10)

where the solution vector for displacements is u = a(x)e j(ω t+φ +ϕ ) , while the given body force is of the form f(x,t) = f0 (x)e j(ω t+φ ) , with f0 being the force amplitude vector, ω and φ standing for the given force frequency and force phase angle, and

7 Unresolved problems of hierarchical modelling...

115

ϕ denoting unknown phase angles of the vibration amplitudes. The second and first time derivatives of the displacement vector (the acceleration and velocity vectors) .. . are .ui = −ω ai e j(ω t+φ +ϕ ) , ui = jω ai e j(ω t+φ +ϕ ) , while the strain velocity is equal to ε i j = jω εi j . The coefficients α and β characterize the material dumping due to viscous internal friction and can be treated as equivalent to Rayleigh dumping coefficients. In the case of forced vibrations, the form of the surface tractions is p(x,t) = p0 (x)e j(ω t+φ ) , with p0 standing for the traction amplitude vector. The related traction boundary conditions, as well as the kinematic boundary conditions, are σi j n j = pi (x,t), x ∈ SP , ui = 0, x ∈ SD ,

(7.11)

The variational formulation for the stationary forced vibration problem, corresponding to (7.10) and (7.11), is Di jkl vi, j ak,l dV + jω V

(β Di jkl vi, j ak,l + α ρ vi ai ) dV − ω 2 V

vi f0i e− jϕ dV +

= V

ρ vi ai dV V

vi p0i e− jϕ dS,

(7.12)

SP

with vi representing the kinematically admissible field of the displacement amplitudes. Note that the solution to (7.12) is searched in the complex functions domain. The physical unknowns of the problem are the amplitudes and their phase angles. After performance of the discretization and hpq-interpolation of the filed of unknowns, the matrix form of the finite element equations, corresponding to the above variational formulation, and assigned for any of m (m = 1, 2, ..., M) independent forces of vibration, is (K + jωm C − ωm2 M)qm = F0m e− jϕm , hpq

(7.13)

where F0m is the global vector of the amplitudes of the mth force, while the global dumping coefficient matrix C is composed of two components, i.e. C = β K + α M. The total solution in displacements can be obtained with the superposition theorem, hpq taking advantage of m consecutive solutions for nodal amplitudes qm and phase hpq j(ωm t+φm +ϕm ) . angles ϕm . The mth displacement contribution is qm e

7.1.2 Considered Types of Complexity within the Elastic Bodies In order to retain the general character of the analysis, we allow geometrical and mechanical complexity of the elastic bodies, analyzed in three solid mechanics problems described in the previous subsections.

116

G. Zboi´nski

solid geometry

transition geometry mid-surface shell geometry top surface mid-surface

bottom surface Fig. 7.2 Different types of the geometry

7.1.2.1 Geometrical Complexity Let us start with exemplary simple geometries. Three of such geometries, the solid, shell, and transition ones, are shown in Fig. 7.2. The solid geometry is bounded with surfaces. The shell geometry is based on the mid-surface and thickness concepts. The transition geometry is partly defined with the equations of the bounding surfaces and partly described by means of the mid-surface equation and the thickness vector normal to this surface. The numerical representation of the simple shell geometry (the plate example) is shown in Fig. 7.4. Let us pass to complex geometry of the body. We deal with such geometry when more than one type of geometry is necessary for the description of the shape of the body. Such a situation is presented in Fig. 7.3, where the solid, shell, and transition geometries are employed within one body. The numerical representation of the complex geometry can also be seen in Fig. 7.5. A thorough mathematical description and the specific definitions of the above mentioned simple and complex geometries can be found in our work [13].

7.1.2.2 Mechanical Complexity The structure of simple mechanical description is characterized with the application of one mechanical model. The complex mechanical description is a result of the necessity of application of more than one mechanical model within the structure. The application of one or more than one model refers to both, simple and complex geometries, of course. The typical situation for solid mechanics problems, where the solid, shell, and transition mechanical models are employed, is presented in

7 Unresolved problems of hierarchical modelling...

117

solid geometry transition geometry

shell geometry 3D-model

Fig. 7.3 Complex mechanical description

transition model shell model

Fig. 7.3. It can be seen from the figure that the division of the structure into zones (domains) of the different mechanical description is independent of the division into geometrical parts (members). It is obvious, however, that some models do not appear in certain parts, e.g. shell models are not suitable for solid parts of the body. The admissible neighbourhood of the different models and the appearance of the models in the different geometrical parts are explained in detail in [13]. The relations between the simple and complex geometries and the simple and complex mechanical descriptions are also illustrated in Fig. 7.4 and Fig. 7.5, where the complex mechanical models are applied within the simple and complex geometries, respectively. The first of these two figures shows the numerical representation of the symmetric quarter of the clamped plate, while in the second figure the numerical idealization of the symmetric quarter of the square floor supported by four columns is displayed. The models applied in both examples are: the theory of three-dimensional elasticity (3D) or the hierarchical shell models (MI) of the transverse order I (I = 1, 2, 3, ...), the first-order Reissner-Mindlin shell model (RM), and the solid-to-shell transition models, (3D/RM) or (MI/RM). The description of these models can be found in our work [13].

7.2 Assessed Methodology In this section we would like to present the assessed methodology for the adaptive modelling and adaptive analysis of the simple and complex structures within solid mechanics. Our methodology covers the 3D-based hierarchical modelling and the hierarchical hp-, hpq-, and hpq/hp-approximations, which all allow generating the hierarchy of numerical finite element models of the local character. The methodology includes also the error estimation with the equilibrated residual method (ERM), and the error-controlled adaptivity, based on the three- or four-step adaptive strategies, with possible iterations within the h- and p-steps.

118

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model

3D MI

260 250 180

280

170 100 90 20

270 200 190

120

10

300

110 40

290 220 210

140

30

320 310 240

130 60

RM/3D RM/MI

230 160

50

150 80

RM

70

z x

y

Fig. 7.4 Numerical representation of the complex mechanical description within simple geometry

7.2.1 3D-Based Hierarchy of Numerical Models In this subsection we discuss three major components necessary for generation of our 3D-based hierarchy of numerical models assigned for the adaptive hierarchical modelling and hpq-adaptive analysis of the simple and complex structures within computational solid mechanics. The first component is the 3D-based approach which lies in the application of the same 3D degrees of freedom for any mechanical model. The second component is the hierarchical modelling, including such different mechanical models as: the 3D-elasticity, the higher-order hierarchical shell models, the first-order shell model, and the solid-to-shell or shell-to-shell transition models. The third component is the set of hpq hierarchical approximations applied to all models of the hierarchy of mechanical models. The combination of theses three components leads to the mentioned hierarchy of numerical models. Note that the hierarchical numerical models can be applied locally, i.e. on the finite element level. The idea is illustrated in Fig. 7.6, where in each element e of the vole

e

ume V , a different model M from the set of mechanical models M, the different size e e e h, and the different longitudinal and transverse approximation orders, p and q, can be chosen.

7 Unresolved problems of hierarchical modelling...

119

model

3D MI

160

150

140

180

170

130 100 90

RM/3D RM/MI

120 20 110 10

RM

z x

y

Fig. 7.5 Numerical representation of the complex mechanical description within complex geometry

SD SP

e

e

e e

M∈M, h, p, q e

V

Fig. 7.6 The idea of hierarchical numerical modelling

7.2.1.1 3D-Based Approach The idea of the 3D-based approach lies in application of only one type of degrees of freedom within all mechanical models included into the hierarchy of models. In our case only the three-dimensional degrees of freedom (three displacements at a point of the body) are employed, regardless of the model type. The conventional mid-surface displacements and rotations of the first-order shell model, as well as the generalized mid-surface dofs of the higher-order shell models are replaced with the equivalent through-thickness dofs. One more difference between both approaches is the direction of the dofs. In the conventional case we usually apply the local

120

G. Zboi´nski

thickness

thickness

top

ξ' 3 1

d' 2j d' 0j

x'3

d' 3j

top

d'3n middle

x'2

d'2n

middle

u' j

d' 1j

ξ' 30 d' 0j ξ'31d' 1j

0

ξ'32 d' 2j

d'1n

bottom

u' j (ξ' 3 )

ξ'33 d' 3j x'1

-1

bottom − d' 1j

− d' 3j

d' 2j

d' 0j

Fig. 7.7 Mid-surface dofs (left) and the displacement field (right) of the conventional approach

directions, normal and tangent to the mid-surface, while in the case of the throughthickness dofs the global directions are employed. The equivalence of the local displacement fields uj , j = 1, 2, 3 in the cases of the mid-surface and through-thickness dofs is shown in Fig. 7.7 and Fig. 7.8. This equivalence can also be expressed through the relation uj =

I

∑ ξ3

n=0

n n d j

I

=

∑ fn (ξ3 ) u j

n

(7.14)

n=0

In the figures and the equation (7.14) the local directions are denoted as xj , j = 1, 2, 3, while the global ones as xi , i = 1, 2, 3. In the case of the third local direction we also introduce the auxiliary dimensionless coordinate ξ3 = 2x3 /t, where t is the thickness of the shell. The corresponding nth mid-surface dof in the jth local direction is d nj , while the nth through-thickness dof in the local direction j and the global direction i are u nj and uni , respectively. Note that the numbering of dofs in both cases is n = 0, 1, 2, ..., I, where I is the order of the transverse displacement field, equivalent to the order of the applied shell theory. More details on the 3Dbased approach can be found in our works [13, 21]. 7.2.1.2 Hierarchical Modelling Our methodology utilizes the 3D-based adaptive hierarchical models M for complex structures, including the first-order Reissner-Mindlin (RM) shell theory, the hierarchical higher-order shell theories MI of order I, the three-dimensional theory of elasticity (3D), the solid-to-shell transition model (3D/RM), and the shell-to-shell transition models (MI/RM) as well. The complete set of the models is defined as:

7 Unresolved problems of hierarchical modelling...

x1

1

top

x'3

u' 3j f1u' 1j

u' 2j

x3

top

u3n

u'3n middle

u1n

ξ' 3

thickness

thickness

121

u' 2n

middle

f 0u' 0j + f 3u' 3j

0

f 0u' 0j u' j (ξ' 3 )

x'2

f 2u' 2j

u2n

bottom u'1n

f 3u' 3j

x2

u' 1j

-1

bottom

u' j

u' 0j

x'1

Fig. 7.8 Through-thickness dofs (left) and the displacement field (right) of the 3D-based approach

M ∈ M,

M = {3D, MI, RM, 3D/RM, MI/RM}

(7.15)

The subset MI of the hierarchical shell models as well as the corresponding subset MI/RM of the transition models are MI = { M2, M3, M4, ...}, MI/RM = {M2/RM, M3/RM, M4/RM, ...}

(7.16)

The elements of our hierarchy can be ordered with respect to the order I (or J), I = 1, 2, 3, ..., ∞ of the shell theory , i.e. M = RM ⇒ I = 1, J = I M = M2/RM, M3/RM, M4/RM ⇒ I = 2, 3, 4, ..., J = 1 M = M2, M3, M4, ... ⇒ I = 2, 3, 4, ..., J = I M = 3D/RM ⇒ I → ∞, J = 1 M = 3D ⇒ I → ∞, J = I is

(7.17)

The fundamental property of our hierarchy of the 3D-based mechanical models lim lim uI/J(M) U,V = u3D U,V , (7.18) J=1,I

I→∞

where the global norm of the solutions uI/J(M) , equal to the strain energy U within the body volume V , is uI/J(M) U,V =

1 2

σ T (uI/J(M) ) ε (uI/J(M) ) dV, V

(7.19)

122

G. Zboi´nski

with σ and ε being the vectors of stresses and strains, respectively. Note that the subsequent solutions of the models of the hierarchy tend in the limit to the solution u3D of the model of 3D-elasticity, i.e. the highest model of the hierarchy. More information on the definitions of the models and the properties of our hierarchy can be found in our works [21, 13].

7.2.1.3 Hierarchical hpq-Approximations For each of the mentioned models we introduce the two-dimensional, three-dimensional, or mixed (corresponding to the shell, solid and transition models, respectively) hpq-adaptive approximations, where h is the characteristic dimension of an element, while p and q represent longitudinal and transverse orders of approximation within the element, respectively. In other words, the subsequent solutions uI/J(M) to our subsequent models M of the hierarchy M are now approximated with the solutions uq(M),hp , in which the transverse order of approximation q is equivalent to the order I of the hierarchical model M, i.e. q ≡ I. Taking into account the specific character of the approximations for the cases of the first-order shell, hierarchical shell, and 3D-elasticity models, we can denote the specific solutions, corresponding to these models, in the following way M = RM, I ≡ J = 1 ⇒ uq(M),hp = uhp M ∈ MI/RM, I ≥ 2, J = 1 ⇒ uq(M),hp = uhpq/hp M ∈ MI, I ≡ J ≥ 2 ⇒ uq(M),hp = uhpq M = 3D/RM, I → ∞, J = 1 ⇒ uq(M),hp = uhpp,hp M = 3D, I ≡ J → ∞ ⇒ uq(M),hp = uhpp

(7.20)

In the above notation we simply skip q = 1 for the first-order shell model, and assume q = p for the model of 3D-elasticity. In the case of the transition models we perform accordingly. The evolution of the concept and the corresponding definitions of the hierarchical hpq-approximations for the structures of complex mechanical description can be investigated in our subsequent works [19, 16, 21, 13]. The main feature of the introduced approximations is that, with the increasing p and the increasing inverse of h, the numerical solutions tend in the limit to the exact solution uI/J(M) of the corresponding model M, i.e. lim uq(M),hp

1/h,p→∞

)e

U, V

= uI/J(M) U,V

(7.21)

Combination of the proposed hierarchy of mechanical models and the hierarchical approximations for these models leads to the hierarchy of adaptive numerical models for complex structure analysis (see [13]). The useful property of the solutions within this hierarchy is

7 Unresolved problems of hierarchical modelling...

lim uq(M),hp lim lim

J=1,I I→∞

1/h,p→∞

)e

U, V

123

= lim

J=1,I

lim uI/J(M) U,V = u3D U,V ,

I→∞

(7.22) which means that with the increasing discretization parameters p and 1/h and the increasing order I of the hierarchy of mechanical models we reach the solution corresponding to the exact solution of the highest model of the hierarchy.

7.2.1.4 Hierarchy of the hp-, hpq- and hpq/hp-Adaptive Finite Elements The hierarchy of the numerical models, introduced in the previous subsection, is encoded into a hierarchy of the adaptive prismatic finite elements. The hierarchy includes: the hpp solid element, acting also as the hpq hierarchical shell elements, a family of the hpp/hp solid-to-shell transition elements, acting also as the hpq/hp shell-to-shell transition elements, and the hp first-order shell element. The local (element-level) model-, h-, as well as p- and q-adaptive capabilities of the hierarchy of elements results from the application of: the hierarchy of 3D-based mechanical models, the constrained approximation (hanging nodes) idea, as well as the hierarchical shape functions associated with the corresponding incremental degrees of freedom, respectively. These three ideas are presented in detail in our work [19]. It should be mentioned here that our works on the constrained approximation and hierarchical shape functions were inspired and took advantage of the works by the predecessors [7, 8]. Application of these concepts on the element level is elucidated in our work [20] for the solid and hierarchical shell elements, [23] for the transition elements, and [21] for the first-order shell element. The hierarchical character of the all elements is explained in [19, 22]. As the all above aspects of the assessed methodology are documented very thoroughly in the all mentioned works, we skip the corresponding details in this presentation.

7.2.2 Error Estimation The applied error-controlled model- and hpq-adaptivity is based on the estimated values of the modelling and approximation errors, obtained from the residual equilibration method (ERM) [2, 3]. The method is applied twice, firstly for estimation of the approximation error and secondly for the modelling or total error. The method provides the upper bounds of the global approximation error for all the applied models as well as the upper bound of the global modelling error for the hierarchical shell models (thus also the upper bound of the global total error for the latter models can be proved). In the case of the first-order shell elements, and the corresponding solidto-shell transition elements, one cannot prove the upper bound of the modelling and total errors. As an undesired consequence, only the global modelling error indicator can be obtained from the proposed approach. In the case of 3D elasticity model, the

124

G. Zboi´nski

total and approximation errors are equivalent, and the upper bound of the errors can be shown. The starting point for the equilibrated residual method is the variational formulation in which the global error functional in the energy norm is employed. The error energy norm is defined as the strain energy of the difference of the exact and numerical solutions. Then, the functional is decomposed into the local (element-level) functionals for each finite element. In this step the upper bound property can be proved, based on the observation that the sum of the minimized potential energies from the decomposed local (element-level) variational problems is always grater than the global minimized potential energy of the body as a whole. As a consequence, the global strain energy of the error, for the whole body, is always smaller than the sum of the strain energies of the errors from the decoupled local problems. Finally, we employ this global strain energy as an exact value of the error, and the element sum of the errors as the estimator to this exact error. This way the element residual approach to error estimation is formed. Details of our implementation of the method can be found in [19] for the whole hierarchy of our adaptive finite elements, and in [17, 18, 14] for the first-order shell element. Note that, from the implementation point of view, the obtainment of the solutions to the local (element-level) variational problems is very important, as these solutions contribute to the global estimator of the error. The local problems to be solved can be presented in the following form e

Q(M),HP Q(M),HP

B(u

,v

e

)− L(vQ(M),HP ) −

T e

e

e

vQ(M),HP r(uhp ) dS = 0,

S\(SP ∪SD )

(7.23) where the bilinear B and linear L forms represent the virtual strain energy and the virtual work of the external forces of the body, both restricted to the element e. The solution and trial functions from the proper space of the element kinematically admissible displacements are denoted as uQ(M),HP and vQ(M),HP , with H, P, and Q standing for the element size, and longitudinal and transverse orders of approximation in the discretized local problem. The last left-hand side term of (7.23) represents the virtual work of the interelement stress reactions. The equilibrated version e of these reactions is denoted as r . The equilibration means that the external and internal forces of the element are in equilibrium, and this is done with the vectors e

ef

e

α of the splitting functions, to be determined on the common faces of the element e and any adjacent element f . Our definitions of these functions can be found in [2, 3, 19]. The formal definitions of the terms of the above relation are

7 Unresolved problems of hierarchical modelling... e

Q(M),HP

B(uQ(M),HP, v

)=

e

125 e

e

T e e

e

e

T e

e

T e

ε T (vQ(M),HP )Dε (uQ(M),HP ) dV = v Q(M),HP k q Q(M),HP,

V e

L(vQ(M),HP ) =

e

T

e

v Q(M),HP f dV +

V

T

e

e

v Q(M),HP p dS = v Q(M),HP (fM + fP ),

SP e

S\(SP ∪SD )

v

e

Q(M),HP T e

r(uq(M),hp ) dS = v Q(M),HP fR

(7.24)

These terms can be also expressed in the language of the finite elements, by means e

of the element stiffness matrix k (defined with the elasticity matrix D and the strain e

e

e

vector ε ), and the element forces vectors fM , fP , and fR due to the body, surface, e and element reaction loads f, p, and r , respectively. Denoting the searched nodal e Q(M),HP displacements as q , the relation (7.23) can be equivalently written as e e

e

e

e

k q Q(M),HP = fM + fP + fR

(7.25)

We will take advantage of the above form of the local problems later on in this chapter of the book.

7.2.3 Adaptive Strategy Our adaptive strategy is based on Texas three-step strategy [9]. The original strategy lies in solution of the global problem thrice, on the initial, intermediate, and final meshes. The intermediate and final meshes are obtained through the local hrefinements (h-step) and local p-enrichments (p-step), respectively. In principle, the method leads to the solution of the assumed accuracy, as the error level is related to the discretization parameters through convergence theories. The original approach is enriched by us with the possible two or three iterations within the h- and p-steps. Additionally, we introduce one additional step proceeding the h- and p-steps, called modification one (see Fig. 7.9), in order get rid of such unpleasant phenomena as: • the improper solution limit of 3D elasticity model for q = 1 and thin structures, • numerical locking (for low values of p and 1/h within thin structures), • and boundary layers. In our additional step some global modifications (the global change of the model, the global p-enrichment, and the introduction of exponential mesh subdivision in the direction normal to the boundary) within the initial mesh are performed. This additional step needs special tools for detection of the mentioned three phenomena. The proposed tools take advantage of the algorithms of the applied error estimation method. They allow qualitative assessment of two local solutions obtained for the models for which these phenomena exist or not in accordance with the theory.

126

G. Zboi´nski Start Reading data i=1 Initial mesh generation

Formation and solution of the problem equations Detecion of three phenomena (i=1) or error estimation (i=2,3,4) Final mesh generation Initial mesh modification Intermediate mesh generation Yes Yes

One of three phenomena detected?

i=2 i=1

i=2

i=3

No

i=2

No

i=2

No Yes

No

i=3

i=3 No

No

Enough intermediate error iterations? Yes

No

Enough final error iterations?

Yes

i=4

Yes

Target error achieved? Yes

Printing results Stop

Fig. 7.9 Adaptivity control with four-step iterative strategy

7.2.4 Possibility of Other Approaches One should be aware that the proposed approach is not the only option in adaptive analysis within computational solid mechanics. Within the modelling and approximation methods we can mention the methodologies based on one mechanical model, either the first-order shell [4], higher-order shells [1, 5], or 3D elasticity [11, 12]. Possible alternatives concerning the error estimation and adaptive strategies are: the goal-oriented adaptivity [10] and the so called automatic hp-adaptive strategy based on the two-mesh paradigm [6, 12], respectively. Even though, these methodologies were developed as an answer to the problem of poor effectiveness of other techniques, they do not answer to all questions concerning the adaptive modelling and analysis within solid mechanics. For example, the application of the automatic hpadaptive strategy to the problems of complex mechanical description and problems with boundary layers is still an open question. Also, the goal-oriented approach, useful for practical applications, still does not give all answers to the problem of upper and lower boundedness of the errors.

7 Unresolved problems of hierarchical modelling...

127

7.3 Chosen Problems to Be Resolved and Some Remedies In this section we would like to present some chosen unresolved problems within the assessed methodology. Before the presentation of the problems and the related remedies we would like to state what follows. • Our choice of the problems to be presented is rather subjective. We consider difficulties faced by us during implementation of our specific methodology. • The order in which we present these problems corresponds to the subsequent steps of the algorithm, not to the significance of the problems. • We search for the solution of these problems within the presented methodology, taking advantage of the potential hidden within the applied techniques. We do not consider the escape to other techniques (we do not want to introduce other quantities of interest, apply the goal-oriented adaptivity, or take advantage of the two-mesh paradigm). The short list and the characterization of the main problems, we would like to address in this chapter, is as follows. • Geometry modelling within 3D-based hierarchy of models is important because of its influence on the orthogonality of the modelling and approximation errors. • Excessive growth of the number of dofs in 3D can appear (especially in the case of corner or edge high solution gradients – they may need application of the 2-irregular constrained approximations). • Our approach does not give formal upper bounds for the free and forced vibration problems, however the effectiveness is practically identical as in the case of the equilibrium problem. • Poor or worse effectivity of the ERM estimator can be observed in the case of elongated elements and the meshes of locally varying order and size (the uniform meshes provide higher error but better effectivity than the adapted ones). • Changing convergence rates disturb the obtainment of the target error.

7.3.1 Hierarchical Modelling Issues In the context of the hierarchical modelling, we chose the specific questions of the robust formulations of the hierarchical shell models and transition models as well. One of the key issues within our 3D-based formulation is geometry modelling based on the middle surface and thickness concepts. Such geometry modelling is necessary for orthogonal decomposition of the total error into the modelling and approximation error components. The problem of the shell geometry modeling is illustrated in Fig. 7.10, where two systems of local coordinates are introduced. The first one corresponds to any arbitrary point of the shell, and is based on the assumption that the two local directions, determined by the vectors w1 and w2 , are tangent to the mid-shell surface,

128

G. Zboi´nski any point of the shell

x3

point j

ξ3 w

ξ3

j 3

top surface

x'3 w3 x'2

w 1j w 2j

w2 x2

x1

ξ2

w1 ξ1

x'1

bottom surface shell mid-surface

ξ2 ξ1

Fig. 7.10 The proper shell geometry definition

or to any other shell surface determined by the shell natural coordinates ξ1 and ξ2 , and additionally that the third direction, determined by w3 , is perpendicular to the mentioned surface. The second system of coordinates, corresponding to any point j of the lateral boundary of the shell, is defined in a different way. Here, the vector j w3 is defined first, so as to coincide with the direction x3 , i.e. to come through the corresponding points of the top and bottom surfaces of the shell. The surface determined with the two remaining vectors, w1j and w2j , is now defined as perpendicular to the first vector. Note that when the first system of coordinates is generated in the point j, both the systems are not identical, unless the mid-shell surface is perpendicular to the direction x3 . There is no problem with fulfillment of this requirement in the case of analytical description, provided that the shell geometry is defined as the one of the symmetric thickness in the direction perpendicular to the mid-surface. Note, however, that in the finite element approximation, the interpolated (polynomial) representation of the shell mid-surface may not be perpendicular to the lateral boundary of the shell. Fortunately, the higher the interpolation order, the better coincidence between the directions x3 and ξ3 at any point j can be observed. Summing up, if one wants to take advantage of all the features of the shell theories, one has to construct geometry of the 3D shell body so as the lateral boundary of the shell is perpendicular to its mid-surface. In the finite element approximation of such shells, careful interpolation of their geometry is required.

7 Unresolved problems of hierarchical modelling...

129

7.3.2 Problems within hp-Approximation What makes the implementation of the hp-adaptive approach difficult is the constrained approximation necessary for the local (element) h-refinements. In the case of mechanical systems of the complex mechanical description, the transition models and approximations make the issue even more complicated. Because of that below we discuss the chances of getting rid of these two techniques. In this subsection we also address the problem of the excessive growth of the dof number in the case of 3D meshes. 7.3.2.1 The Constrained Approximation In this case we deal with the broken elements equipped with the so called hanging nodes. The general answer to the question if one can get rid of the hanging nodes is no. In the general h- or hp-approach, based on element refinement (as opposed to the remeshing technique), it is not possible. Note that the specific approach called Rivara’s refinement, where the division by four or two is led through the existing vertices of the rectangular faces of the elements, is applicable only to such faces. The approach requires the implementation of two types of elements within one mesh, with the rectangular and triangular faces in the undivided and divided elements, respectively. Note also that the original Rivara’s approach can be applied once. This is because, when one wants to repeat the division, he deals with the triangles, and the avoidance of hanging nodes requires the division by two of the element we want to divide and the adjacent element as well. In other words, in the case of triangular faces such an approach does not work unless we divide also the adjacent element. This makes things much more complicated and leads to over-divided meshes. 7.3.2.2 Necessity of the Transition Approximations In the case of the transition approximations, resulting from the application of the transition models joining the basic models together, the transition elements corresponding to such transition models are necessary. As an example of such a situation within solid mechanics we can mention the solid-to-shell transition elements acting between the 3D-elasticity model (or the higher-order shell models) and the firstorder Reissner-Mindlin shell model. The general answer to the question if one can get rid of the transition elements is no, again. There exist, however, some prosthesis approaches. For example, in the case of the bending-dominated shells, the ReissnerMindlin model can be replaced by the 3D-elasticity model of the transverse order q = 1, with changed elastic constants. In the case of the membrane-dominated shell structures, the Reissner-Mindlin model can be changed to the 3D-elasticity model of q = 1, without any modifications of the elastic constants. In both cases, the transition model and the corresponding transition elements are not necessary. The problem with such approaches is that, in the majority of technical applications, the character of the strain dominance (either bending or membrane one) is not known a priori.

130

G. Zboi´nski

There are also some structures where a balance between the bending and membrane strains exists. Then, the only correct model is the Reissner-Mindlin one, which unfortunately requires the transition model, when combined with the 3D-elasticity one. As we search for a general approach to hierarchical modelling, we will not consider here the imperfect approaches mentioned above, suitable for the specific states of strains only.

7.3.2.3 Growth of the Dofs Number in 3D Problems There are at least three reasons for the excessive growth in the dofs number for 3D problems. • Application of only 1-irregular hanging nodes due to the constrained approximation. • Poor effectivity of the estimation (overestimation). • Wrong h- or p-convergence exponents in the adaptivity control procedures. Here we would like to address the first problem, while the remaining two will be discussed in the section concerning adaptivity control. So as to make the first problem less severe, we propose to extend the idea of the constrained approximation onto 2irregular meshes. Note that the problem is especially important for large 3D structures with high edge or corner solution gradients. Application of 2-irregular meshes allow the avoidance of the excessive growth of dofs occurring when only 1-irregular subdivisions are possible. Such a limitation in mesh generation causes that 1-irregular mesh penetrates deeply into the regions that could have been undivided, if a sequence of 1irregular subdivisions had not been necessary due to solution continuity reasons. The example of the excessive growth of the number of dofs in the h-adapted, 1-irregular mesh, both in the interior and in the vicinity of the boundary, is shown in Fig. 7.11. The difference between the hanging nodes of 1- and 2-irregular meshes is illustrated in Fig. 7.12, where the node of the first type is marked with 1, while the node of the second type is denoted with the number 2. In order to retain solution continuity, the displacements of the node 1 of the element f have to be equal to the interpolated displacements of the undivided, adjacent element e, at the location e

ξ 1 = 1/2 of the node 1 in the element e f

e

e

q i1 = u i (ξ 1 ),

e

ξ 1=

1 2

(7.26)

In the case of the element g, obtained through a double division, the displacements dofs at the node 2 can be expressed with either the interpolated displacements of the element e1 , that could have been obtained through a single division of the element e, or better directly with the interpolated displacements of the undivided element e, e

at the location ξ 2 = 3/4 of the node 2 in the element e g

e

e1

e

e

q i2 = u1 i (ξ 2 ) = u i (ξ 2 ),

e1 1 ξ 2= , 2

e

ξ 2=

3 4

(7.27)

7 Unresolved problems of hierarchical modelling... p,q

131

28645883 2884 2874 2343 28545863 2183 5903 5843 2323 2223 2383 2203 2363 2023 2663 2163 862 6023 5683 2063 5823 2643 2043 2703 2503 2003 852 3574 3264 2683 5703 5663 3184 6063 6043 2483 3604 2543 3254 3584 3204 3284 3194 2523 25445523 692 652 3594 3504 3104 5643 6003 3274 2564 36633174 3494 2554 6603 702 3524 5503 3124 4064 3683 3114 2743 662 5183 2704 2534 332 2783 3514 2624 863 370336233094 5543 5483 2763 5223 6623 2694 2724 2723 382 2644 4084 3603 5203 3643 2634 342 463 4054 2464 883 5163 732 5463 2714 4074 5043 372 3823 32232614 1264 782 3843 2484 4143 3243 2474 50235003 5443742 3863 1424 1284 5063 326331832454 1254 4144 963 3463 4183 3163 823 3203 1434 1444772 3803 4983 4123 141412741184 4163 2863 33834803 4164 1204 903 1194 4963 39833403 4154 1174 4134 4783 3423 6743 4003 4023 6463 4823 4643 4623 3003 6123 3363 4663 2823 3963 6383 703 3704 4763 3023 2983 6443 6143 1474 2963 4603 2843 3724 6363 763 5303 6543 3714 3694 6283 1803 1783 1723 1703 2803 5283 6523 6163 1823 1863 1743 6843 4463 4303 1763 1903 1683 1883 1843 6303 6183 1643 1623 6823 4503 4343 1663 4483 4443 4323 4283 1603

8

7

6

5

4

3

2

1

z x

y

Fig. 7.11 Edge and interior excessive growth of dofs number (the first mode of free vibration)

element f

1

element g

2

element e 2 1 2

1

1

2 element g

element f element e Fig. 7.12 1- and 2-irregular constrained nodes for vertical (left) and horizontal (right) subdivisions

132

G. Zboi´nski

As it can be seen, the procedure of the obtainment of the continuity conditions for the 1- and 2-irregular nodes is generally the same. As the interpolated displacements of the undivided element e can be expressed with the active displacement dofs of this element, then also the constrained dofs of the 1- and 2-irregular nodes can be expressed by them, and the corresponding constraint coefficient matrices can be defined. The general and detailed information on how to construct such matrices can be found in [8, 19, 14].

7.3.3 A Posteriori Error Estimation In the error estimation problems, we would like to pay our attention to two issues. The first group of problems is theoretical and concerns the obtainment of the upper bound property of the residual estimators in the free and forced vibration problems. The second, implementation issue concerns poor or worse effectivity of the estimation.

7.3.3.1 Error Estimation for the Free and Forced Vibration Problems In the free vibration problem the difficulty results from the fact that the exact solution in frequency (and in the corresponding strain energy) gives the lower bound of the numerical solution, opposite to the static case, where the exact solution in strain energy constitutes the upper bound. In the stationary forced vibration problem the main difficulty arises from the appearance of the phase angles in the acting forces. The solution to such a problem has to be obtained in the domain of complex functions. The phase angles have to be introduced into the formulation of the ERM estimators. Note that when these angles are equal to zero, the solution becomes real and the method suitable for the equilibrium problem can be applied directly. The similarities, differences, and the above mentioned theoretical difficulties within three analyzed problems of solid mechanics, in the frame of the presented methodology, can be summarized as follows. • The equilibrium problem: – the upper bound exists for the approximation, modelling and total errors (3Delasticty and 3D-based hierarchical shell models), and the approximation error (3D-based Reissner-Mindlin model). • The eigenproblem (free vibration): – no formal upper bound of the error can be proved, however effectivity of the estimation of the errors in the energy norm is practically the same as for the equilibrium problem (compare Fig. 7.13 and Fig. 7.14), – the main difficulty results from the lower boundedness of the numerical solution by the exact solution, opposite to the equilibrium (static) case,

7 Unresolved problems of hierarchical modelling...

133 (X) 1.372 1.372

it

20 10

180 170 40 30

340 330 200 190 60 50

500 490 360 350 220 210 80 70

660 650 520 510 380 370 240 230 100 90

820 810 680 670 540 530 400 390 260 250 120 110

980 X 970 840 830 700 690 560 550 420 410 280 270 140 130

1140 1130 1000 990 860 850 720 710 580 570 x 440 430 300 290 160 150

max

1.208 1.167 1.063 1160 1150 1020 1010 880 870 740 730 600 590 460 450 320 310

avr

0.936 1180 1170 1040 1030 900 890 760 750 620 610 480 470

0.824 1200 1190 1060 1050 920 910 780 770 640 630

0.726 1220 1210 1080 1070 940 930 800 790

0.639 1240 1230 1100 1090 960 950

0.563 1260 1250 0.496 1280 1270 1120 0.436 1110 0.384 0.338 0.298 0.262 0.231 0.203 (x)

0.179 0.179

min

z x

y

Fig. 7.13 Total error effectivities for the equilibrium problem

– the true error is the result of the error in the natural frequency and the error of the mode of vibration (the eigenvalue and eigenmode errors), – our methodology accounts only for the eigenmode error, – our methodology can be converted into the goal-oriented approach, with the normalized strain energy being the quantity of interest, and solutions of the ERM local problems forming the dual solution. • The forced-vibration problem: – the problem is solved for the amplitudes and phase angles – this requires the complex domain, – there is still no formal answer to the question of the upper boundedness of the errors (the work is in progress), – one can expect the effectivity of the estimation to be similar as for the equilibrium and free vibration problems (to be demonstrated numerically).

7.3.3.2 Effectivity of the Estimation The second issue has an implementation character and deals with the poor effectivity of the residual approach in the case of elements elongated in order to resolve the boundary layers. The corresponding exemplary effectivities for the regular meshes

134

G. Zboi´nski (X) 1.774 1.774

it

max

1.551

20 10

180 170 40 30

340 330 200 190 60 50

500 490 360 350 220 210 80 70

660 650 520 510 380 370 240 230 100 90

820 810 680 670 540 530 400 390 260 250 120 110

980 970 840 830 700 690 560 550 420 410 280 270 140 130

1140 1130 X 1000 990 860 850 720 710 580 570 440 430 300 290 160 150

1160 1150 1020 1010 880 870 740 730 600 590 460 450 320 310

1.356 1.188 1.185 1180 1170 1040 1030 900 890 760 750 620 610 480 470

avr

1.036 1200 1190 1060 1050 920 910 780 770 640 630

0.906 1220 1210 1080 1070 940 930 800 x 790

0.792 1240 1230 1100 1090 960 950

0.692 1260 1250 0.605 1280 1270 1120 0.529 1110 0.462 0.404 0.353 0.309 0.270 0.236 (x)

0.206 0.206

min

z x

y

Fig. 7.14 Total error effectivities for the first mode of free vibration

of the same degrees of freedom, with square and elongated elements, are presented in Fig. 7.15 and Fig. 7.16, where the global effectivities are 1.2 and 11.5, respectively. Worse effectivity appears also for the locally hp-adapted elements, in comparison with the uniform elements. The meshes to be compared, of the similar number of dofs and the same uniform order of approximation, are presented in Fig. 7.15 and Fig. 7.17. The global effectivities are about 1.2 and 1.7, respectively. One of the available remedies, available but not implemented yet in 3D, is the application of the equilibration of the higher order. Such a method might include the equilibration not only at the vertices of the elements. For the first case, when only the vertex equilibration is performed, the directional components i = 1, 2, 3 of the vectors of splitting functions can be expressed through the six (in the case of the applied prismatic elements) vertex splitting coefficients. For the higher-order nodes no equilibration is performed, and the splitting coefficients, equal to 1/2, reflect the averaging of the nodal reactions between the elements. This type of equilibration can be characterized with the relation ef

αi =

6

∑

j=1

ef e 1e α ij χ j + ∑ ∑ χ j,m , j>6 m 2

(7.28)

7 Unresolved problems of hierarchical modelling...

135 (X) 6.287 6.287

ia

max

5.267 4.412 260

3.697 3.097

250 180

280

170

270

100

2.594

200

2.174

300

1.821 90

190

20

120

10

290 220

110

320 1.526

240

1.278 310 X 1.205 1.071

230

0.897

210

40

140

30

130 60

160

avr

0.752 50 x

150 0.630

80

0.527

70

0.442 (x)

0.370 0.370

min

z x

y

Fig. 7.15 Effectivities for the uniform mesh

e

where χ j represents the shape function of the vertex node j. Note that the non-zero contributions to the splitting functions correspond to four out of six vertex nodes, located on the common face of the elements e and f (see Fig. 7.18). In the higher order equilibration also the higher order mid-edge and/or mid-side nodes are included in the equilibration procedure (see [2, 19]). The respective definition of the splitting function components are ef

αi =

6

∑

j=1 e

ef

e

ef

e

α ij χ j + ∑ ∑ α ij,m χ j,m ,

(7.29)

j>6 m

with χ j,m standing for the shape function corresponding to the higher-order node j, and dof m, defined in this node. Now, the non-zero contributions to the face splitting function correspond to four vertices, four mid-edge nodes and the mid-side node of the common face. The presented idea looks very simple but our numerical tests show that the effective higher-order equilibration is not a trivial task. The results can be even worse than in the case of linear equilibration. The second suggested remedy for poor or worse effectivity of the estimation can be the constraining of the element displacement field while solving local problems of the ERM. This approach is not very common and it needs further theoretical and numerical studies.

136

G. Zboi´nski (X) 33.069 33.069

ia

max

23.115 260 250 180 170 270 200 x280 190 100 300 120 220 90 290 110 140 210

16.158 11.545 11.295

130

7.895

20 40

5.519

320 240

60

avr

160

3.858

1030

310 2.697 230 50 X

80

150

1.885 1.318 0.921 0.644

70

0.450 0.315 0.220 0.154 (x)

0.107 0.107

min

z x

y

Fig. 7.16 Effectivities for the non-uniform mesh

7.3.4 A Posteriori Detection of the Undesired Phenomena The question on how to detect a posteriori the improper solution limit, the numerical locking, and the boundary layers is still another problem. The first phenomenon leads to the incorrect solution, while the latter two cause varying convergence rates and disturb the desired exponential convergence. Also the problem of coping with these phenomena via automatic, error-controlled, adaptive approach is still open. Below we present our numerical tools [15] for the detection of the mentioned phenomena. These tools take advantage of the algorithms of the residual equilibration method applied to error estimation.

7.3.4.1 The Improper Solution Limit and Numerical Locking The improper solution limit and the locking is illustrated in Fig. 7.19. The first phenomenon appears in thin structures modelled with the 3D model, when the transverse approximation order equals unity (q = 1). Then, the solution is the fraction of the correct solution, represented by the value of 1 in the figure, regardless the structure length to thickness ratio t/l. Note that in the figure the relative value of the solution is calculated as the strain energy U, related to the reference, analytical value Ur of this energy.

7 Unresolved problems of hierarchical modelling...

137 (X) 8.628 8.628

ia

max

7.326 6.221 114

5.283 4.486

124 94 174

204 184

154 164 134

104 194 334 254

144

354 364 284 264

294 344 274

324 304

3.810 314 3.235

234

2.747

244 214

224 x 2.333 1.981

22

34

12

42 X

32

1.682 1.658 1.428

44 14

24

avr

1.213 1.030

30

0.875 0.743 (x)

0.631 0.631

min

z x

y

Fig. 7.17 Effectivities for the hp-adapted mesh

element f v4

v4 e3

element e v3

v3

e4 s

e2

v1

e1

element f v2

v1

v2

element e Fig. 7.18 Linear (left) and higher-order (right) equilibrations

The numerical locking occurs in thin structures when the longitudinal approximation order is low (p = 1 in the figure) and/or the mesh density is not high ( number of longitudinal edge subdivisions, m, equals 4 in the figure), regardless the value of

138

G. Zboi´nski 5.0E+0

MI, m=4 q=2, p=1 4.0E+0

q=2, p=4,5,6,7,8 q=1, p=4,5,6,7,8

U/Ur

3.0E+0

2.0E+0

1.0E+0

Fig. 7.19 Illustration of the improper solution limit and numerical locking

0.0E+0 1.00

10.00

l/t

100.00

1000.00

the transverse approximation order q. In such situations, the locked solutions tend to zero in the thin limit. Our numerical tool for the detection of the improper solution limit, lies in the comparison of two solutions obtained from the local problems for the element e chosen from the interior of the thin structure (see Fig. 7.20). The first solution is characterized with the model and discretization data for which the phenomenon may appear, and the second one with the parameters for which the phenomenon does not appear, i.e. e e

e

e

e e Q(RM),HP

e fM

e

k q Q(3D),HP = fM + fP + fR , kq

=

e + fP

Q = 1,

P = 8,

Q = 1,

P=8

e

+ fR ,

(7.30)

The analogous detection strategy is applied to the locking phenomena, either the shear or membrane ones (for the explanation see [19]). The local solutions to be compared are now obtained from the following problems e e

e

e e Q(M),HP

e fM

e

e

k q Q(M),HP = fM + fP + fR , kq

=

e + fP

P = p,

e

+ fR ,

P=8

(7.31)

Note that the same approach can be also utilized for the assessment of the intensity of locking. For this purpose we perform a kind of sensitivity analysis with the changing value of the longitudinal order of approximation P. The information on how to interpret the results from the local problems, concerning the detection of the numerical locking and the improper solution limit, and the

7 Unresolved problems of hierarchical modelling...

139

element e

Fig. 7.20 Single-element local problem for the detection of the improper solution limit and numerical locking

assessment of the intensity of the locking, can be found in [19, 15]. The main difficulty in this interpretation consists in the dependence of the obtained local solutions on the element dimensions resulting from the density of the global mesh (the quality of the detection is better for dense meshes).

7.3.4.2 The Boundary Layer Phenomenon The phenomenon appears in thin and thick structures, when the analytical solution is the sum of the smooth part, corresponding to the interior of the analyzed domain, and the boundary part of high gradients. In the numerical solution of such problems, the interpolation functions suitable for the smooth part of the analytical solution, may not be appropriate for modelling solution gradients in the vicinity of the boundary. The situation is illustrated in Fig. 7.21, where the convergence curves for three cases are presented. The curves relate the numerical error obtained as a difference of the numerical and reference (exact) values of the strain energy, U and Ur , with the number N of degrees of freedom within the numerical model. The first case corresponds to the plate problem described with the Reissner-Mindlin model (q = 1). This model is not very much prone to the boundary layer phenomenon and the corresponding convergence is very high. Note that the uniform mesh of the division number m = 8 is applied in this case. The second case corresponds to the same plate and the model changed to 3D-elastic one. We apply the same uniform mesh of m = 8. It appears that the convergence is very poor in comparison to the previous case. The reason is the application of the 3D model of the transverse approximation order q = 2, very much prone to the edge effect. In order to resolve the problem of the poor convergence, we introduce the mesh of constant density in the interior of the plate, and varying density in the part adjacent to the boundaries, where the exponential subdivisions towards the boundary are applied. This mesh is denoted

140

G. Zboi´nski 3.0 p=1

p=1 p=2

2.0 p=2

p=2

1.0 p=3 p=4 p=3

log (U-Ur)

0.0

p=3

-1.0

-2.0 p=4

t/l=0.33%

-3.0

RM, q=1, m=8

p=4

3D, q=2, m=8 -4.0

3D, q=2, m=4+4

-5.0

Fig. 7.21 Convergence affected by the boundary layer

2.0

2.5

3.0

log N

3.5

4.0

4.5

with the division number m = 4 + 4. The application of this mesh restores the high convergence of the solution, as it can be seen from the third presented curve. Our numerical tool, for the detection of the boundary layers, is based on the comparison of two solutions obtained from the local problems for the chosen pair of elements adjacent to the structure boundary (see Fig. 7.22). The first solution is obtained from the problem characterized with the uniform subdivision of the pair of elements into four smaller elements. For such a subdivision the edge effect may appear. In the second problem we apply the exponential subdivision (see Fig. 7.22, again) and we can expect the solution to be free of the edge effect. Our local problems to be solved are 4 fi f i

4

fi

fi

fi

i=1 4 fi

fi

fi

∑ k q Q(M),HP = ∑ ( f M + f P + f R ), Hn,i = h/2, i = 1, 2, 3, 4,

i=1 4 fi f i Q(M),HP

∑ kq

i=1

=

∑ ( f M + f P + f R ), Hn,1 = Hn,2 = h/10, Hn,3 = Hn,4 = 9h/10,

i=1

(7.32) where Hn,i , i = 1, 2, 3, 4 represent the mesh dimensions of the four smaller elements in the direction normal to the boundary. The element dimensions in the direction tangent to the boundary are kept unchanged and equal Ht = h, with h standing for the dimension of the chosen pair of elements.

7 Unresolved problems of hierarchical modelling...

141

element f4 element f3

element f2

element f1

Fig. 7.22 Four-element local problem for the detection of the boundary layer phenomenon

We would like to add that the similar approach can be utilized for the assessment of the intensity and range of the phenomenon. For this purpose, the performance of a kind of sensitivity analysis is necessary, with the changing value of the elements dimension Hn,i (in the normal direction) in the exponential subdivision of the chosen pair of elements. The information on how to interpret the results from the local problems for the detection and the assessment of the intensity and range of the phenomenon are presented in [19, 15]. The main difficulty in this interpretation is the same as for the detection of the improper solution limit and the numerical locking (the dependence on the global mesh density). The additional difficulty is that in the case of the exponential subdivisions we deal with the elongated elements, and the quality of the boundary layer detection worsens.

7.3.5 Adaptive Procedures Finally, we will address the adaptive strategy issues. In particular, we will discuss the problem of changing convergence rates, which makes achieving the target admissible value of the error difficult within the original three- or our four-step strategies. Having a closer look at the p- or h-convergence curves (Fig. 7.23 and Fig. 7.24, respectively) of the numerical solutions to the problems subject to the numerical locking and boundary layers, one can distinguish three different regions. The first region corresponds to the numerical locking (the horizontal parts of the curves in both figures), the second one to the exponential or algebraic convergence (the middle parts of the curves), and the third region to the lost of regularity due to boundary layers (the third parts of the curves). Though, for each of these three regions some convergence theories exist, the transition from one state to another is unclear and difficult to be determined analytically as a function of the structure thickness t and the discretization parameters h, p, and q. The problem is very similar to that mentioned in Sect. 7.3.2.3, where we dealt with the wrong or unknown values of the exponents of the convergence curves. Note that one of the remedies for the

142

G. Zboi´nski

problems of this type can be the application of the corrective iterations within the hand p-steps of the adaptive procedure. This idea has already been implemented in our algorithms and programs. 2.00 MI, q=1, t/l=1% m=1

p=1 p=1

p=2 p=3

m=4

p=2

0.00

log (U-Ur)

p=1

p=4

m=7

p=2

p=5 p=3

-2.00 p=3 p=6 p=4

-4.00 p=5

p=4 p=6

Fig. 7.23 Changing pconvergence rates for thin structure problems

p=5 p=6

-6.00 1.00

2.00

3.00

log N

4.00

The significance of the issue can be better understood when the standard relations (see [9, 19]) controlling the h- and p-adaptivity are taken into account. The convergence exponents μ0 and ν0 from the initial mesh, and the estimated values of the element error, η0 and ηI , from the initial and intermediate meshes, influence very much the number nI of the intermediate mesh elements replacing the initial mesh element η 2 EI 2 μ /d+1 nI 0 = 20 2 (7.33) γI u0 U and the value of the approximation order pT in the target (final) mesh pT2ν0 =

p02ν0 ηI2 EI , γT2 u0 U2

(7.34)

where d = 3 is the dimensionality of the 3D problem, γI and γT are the assumed, admissible, relative values of the intermediate and target (final) errors, and EI is the total number of elements in the intermediate and final meshes. Note also that the global strain energy norm of the global solution u0 from the initial mesh is utilized in the above equations.

7 Unresolved problems of hierarchical modelling...

143

2.00 MI, q=1, t/l=1% m=1

m=2

m=4

m=6 m=8

p=1

m=1

p=3

0.00

p=5

m=2

log (U-Ur)

m=1

m=3 m=4

-2.00 m=6 m=8 m=2

-4.00

m=3 m=4 m=5 m=6 m=8

Fig. 7.24 Changing hconvergence rates for thin structure problems

-6.00 1.00

2.00

3.00

log N

4.00

7.4 Conclusions The main problems of the adaptive hierarchical modelling and hp-adaptive analysis within computational solid mechanics have been presented. This has been done in the context of the proposed methodology, consisted of: the 3D hierarchical modelling, the hierarchical approximations, the a posteriori error estimation with the equilibrated residual method, and the four-step adaptive strategy. We have addressed such problems as: the geometry modelling within 3D-based thin structures, the over-divided meshes, the lack of the upper bound property for the free and forced vibration problems, the poor or worse effectivity of the estimation for the meshes with the elongated or non-uniform elements, the unsatisfactory quality of the detection of three undesired numerical phenomena, and the changing convergence rates for thin structures. The remedies for overcoming these problems have been indicated, basing on the described theoretical premises and our numerical experiments.

References [1] Actis, R.L., Szabó, B.A., Schwab, C.: Hierarchic models for laminated plates and shells. Comp. Methods Appl. Mech. Engng. 172, 79–107 (1999) [2] Ainsworth, M., Oden, J.T.: A posteriori error estimation in finite element analysis. Comput. Methods Appl. Mech. Engng. 142, 1–88 (1995) [3] Ainsworth, M., Oden, J.T.: A unified approach to a posteriori error estimation using element residual methods. Numer. Math. 65, 23–50 (1993)

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[4] Chinosi, C., Della Croce, L., Scapolla, T.: Hierarchic finite elements for thin plates and shells. Computer Assisted Mechanics and Engineering Sciences 5, 151–160 (1998) [5] Cho, J.R., Oden, J.T.: Adaptive hpq-finite element methods of hierarchical models for plate- and shell-like structures. Comput. Methods Appl. Mech. Engng. 136, 317–345 (1996) [6] Demkowicz, L.: Computing with hp-adaptive finite elements. CRC Press, New York (2007) [7] Demkowicz, L., Bana´s, K.: 3D hp Adaptive Package. Report No. 2/1993. Cracow University of Technology, Section of Applied Mathematics, Cracow (1993) [8] Demkowicz, L., Oden, J.T., Rachowicz, W., Hardy, O.: Towards a universal hp adaptive finite element strategy. Part 1. A constrained approximation and data structure. Comput. Methods Appl. Mech. Engng. 77, 79–112 (1989) [9] Oden, J.T.: Error estimation and control in computational fluid dynamics. The O. C. Zienkiewicz Lecture. In: Proc. Math. of Finite Elements – MAFELAP VIII, pp. 1–36. Brunnel Univ., Uxbridge (1993) [10] Oden, J.T., Prudhome, S.: Goal-oriented error estimation and adaptivity for finite element method. Comp. Math. Appl. 41, 735–756 (2001) [11] Szabó, B.A., Sahrmann, G.J.: Hierarchic plate and shell models based on p-extension. Int. J. Numer. Methods Engng. 26, 1855–1881 (1988) [12] Tews, R., Rachowicz, W.: Application of an automatic hp-adaptive finite element method for thin-walled structures. Comput. Methods Appl. Mech. Engng. (in press) [13] Zboi´nski, G.: Adaptive hpq finite element methods for the analysis of 3D-based models of complex structures. Part 1. Hierarchical modeling and approximations. Comput. Methods Appl. Mech. Engng. (to be published) [14] Zboi´nski, G.: 3D-based hp-adaptive first order shell finite element for modelling and analysis of complex structures – Part 2. Application to structural analysis. Int. J. Numer. Methods Engng. 70, 1546–1580 (2007) [15] Zboi´nski, G.: Numerical tools for a posteriori detection and assessment of the improper solution limit, locking and boundary layers in analysis of thin walled structures. In: Wiberg, N.-E., Diez, P. (eds.) Adaptive Modeling and Simulation 2005. Proceeding of the Second International Conference on Adaptive Modeling and Simulation, Barcelona (Spain), pp. 321–330 (2005) [16] Zboi´nski, G.: Adaptive modelling and analysis of complex structures with use of 3Dbased hierarchical models and hp-approximations. In: Wiberg, N.-E., Diez, P. (eds.) Adaptive Modeling and Simulation. Proceeding of the First International Conference on Adaptive Modeling and Simulation, p. 50, and: CD-ROM, Göteborg (Sweden), pp. 1–24 (2003) [17] Zboi´nski, G.: A posteriori error estimation for hp-approximation of the 3D-based first order shell model. Part I. Theoretical aspects. Applied Mathematics, Informatics and Mechanics 8(1), 104–125 (2003) [18] Zboi´nski, G.: A posteriori error estimation for hp-approximation of the 3D-based first order shell model. Part II. Implementation aspects. Applied Mathematics, Informatics and Mechanics 8(2), 59–83 (2003) [19] Zboi´nski, G.: Hierarchical modelling and finite element method for adaptive analysis of complex structures (in Polish). Zesz. Nauk. IMP PAN w Gda´nsku. Studia i Materiały, 520/1479/2001. IFFM, Gda´nsk (2001) [20] Zboi´nski, G.: Application of the three-dimensional triangular-prism hpq adaptive finite element to plate and shell analysis. Computers & Structures 65, 497–514 (1997)

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[21] Zboi´nski, G., Jasi´nski, M.: 3D-based hp-adaptive first order shell finite element for modelling and analysis of complex structures – Part 1. The model and the approximation. Int. J. Numer. Methods Engng. 70, 1513–1545 (2007) [22] Zboi´nski, G., Ostachowicz, W.: A family of 3D-based, compatible, shell, transition and solid elements for adaptive hierarchical modelling and FE analysis of complex structures. In: Abstracts of European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona, Spain, p. 1011 (2000), and: CD-ROM Proceedings of European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona (Spain), pp. 1–20 (2000) [23] Zboi´nski, G., Ostachowicz, W.: An algorithm of a family of 3D-based, solid-to-shell, hpq/hp-adaptive finite elements. Journal of Theoretical and Applied Mechanics 38, 791–806 (2000)

Part II

Soft Computing and Optimization

Chapter 8

Granular Computing in Evolutionary Identification Witold Beluch, Tadeusz Burczy´nski, Adam Długosz, and Piotr Orantek

Abstract. The paper deals with the application of the Two–Stage Granular Strategy (TSGS) to the identification problems. Identification of selected parameters of the structures is performed. The identification problem is formulated as the minimization of some objective functionals which depend on measured and computed fields. It is assumed that identified constants and measurements have non–deterministic character. Three forms of the information granularity are considered: interval numbers, fuzzy numbers and random variables. The strategy combines the following techniques: Evolutionary Algorithms (EAs), Artificial Neural Networks (ANNs), local optimization methods (LOMs) and Finite Element Method (FEM). All techniques are appropriately modified to deal with non–deterministic data. The EA is used in the first stage to perform the global optimization. The LOM supported by ANN is used in the second stage. The FEM computations are performed to solve the boundary–value problem. Numerical examples presenting the efficiency of the TSGS in different applications are attached. Witold Beluch Department of Strength of Materials and Computational Mechanics, Silesian University of Technology, Gliwice, Poland e-mail: [email protected] Tadeusz Burczy´nski Department of Strength of Materials and Computational Mechanics, Silesian University of Technology, Gliwice, Poland Institute of Computer Modelling, Cracow University of Technology, Cracow, Poland e-mail: [email protected] Adam Długosz Department of Strength of Materials and Computational Mechanics, Silesian University of Technology, Gliwice, Poland e-mail: [email protected] Piotr Orantek Department of Strength of Materials and Computational Mechanics, Silesian University of Technology, Gliwice, Poland

M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 149–163. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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8.1 Introduction In many engineering problems it is necessary to identify some unknown parameters, like material parameters, shape parameters, boundary conditions etc. Identification problems belong to the inverse problems, which are ill–possed ones [4]. To solve such problems it is necessary to collect measurement data (state fields’ values) from the considered structure and compare them with the values of the state fields calculated from the model of a structure. As a result, the identification can be treated as a minimization of the functional J with respect to a design variables vector x: min J(x) x

(8.1)

In order to calculate the state fields value the direct boundary–value problem must be solved. In the present paper the finite element method (FEM) in the granular form is used to solve the direct boundary–value problem. It can be assumed that identified parameters as well as measurements have non– precise character. If it is not possible to determine precisely the parameters of the system, the uncertain parameters which describe granular character of data may be introduced. There exist different models of the information granularity [3]. In the present paper the granularity of information is represented in the forms of the interval numbers, fuzzy numbers or random variables. Selection of the proper model of granularity typically depends on the measurement data [6]. If only a few measurements exist and the measurements have unknown probability density function then the interval approach is suitable. Stochastic approach is convenient if the statistical data exist. If some linguistic description is used to evaluate the parameters of the system, the fuzzy is more appropriate. Different identification tasks are considered: the identification of shape and position of voids in isotropic structures subjected to dynamical load, the identification of boundary conditions in thermo–mechanical problems and the identification of laminate elastic constants.

8.2 Formulation of the Granular Identification Problem The aim of the identification problem is to find a vector x, describing identified parameters. Depending on the kind of granularity each design variable xij consists of: • for interval numbers: 2 values, representing edges of the interval: j j j xi = a(xi ), b(xi )

(8.2)

• for fuzzy numbers: 5 values, representing central value cv and edges of two α – cuts of the trapezoidal fuzzy number (Fig. 8.1):

8 Granular Computing in Evolutionary Identification

151

j j j j j j xi = aL (xi ), aU (xi ), cv(xi ), bL (xi ), bU (xi )

(8.3)

• for random variables: 2 values, representing mean value m and standard deviation σ of the stochastic variable (assuming the that the random genes are independent random variables with Gaussian probability density function): xij = m(xij ), σ (xij ) (8.4)

Fig. 8.1 Trapezoidal fuzzy number described by 5 parameters

8.2.1 Identification of Voids in Dynamical Systems Let us consider an elastic isotropic structure containing a void of a circular shape of unknown size and position described by a vector x (Fig. 8.2). The structure is subjected to the dynamic loading.

W

G p( z, t )

Fig. 8.2 Elastic body containing void

sensor points

The vector of displacements u(z,t) is described by equation:

μ ∇2 u + (λ + μ )grad divu + Z = ρ u¨ (z,t),

z ∈ Ω , t ∈ T ∈ [0,t f ]

(8.5)

where: μ , λ – Lame constants, Z – vector of body forces with boundary conditions:

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W. Beluch et al.

u(z,t) = u(z,t), z ∈ Γu ≡ ∂ Ωu p(z,t) = p(z,t), z ∈ Γp ≡ ∂ Ω p Γu ∪ Γp = Γ ≡ ∂ Ω , Γu ∩ Γp = 0/

(8.6)

and initial conditions: u(z,t)|t=0 = uo (z),

o u(z,t)| ˙ t=0 = v (z),

z∈Ω

(8.7)

It is assumed that boundary conditions and material parameters have the granular character. The identification of the geometrical parameters of the void is treated as the minimization of the objective functional J, depending on measured uˆ and computed displacements u at n sensor points zi , with respect to x: (u(z,t) − uˆ (z,t))2 δ z − zi d Γ dt

n

min J, J = ∑ x

i=1

(8.8)

T Γ

where: δ – the Dirac function. It is assumed that parameters describing void as well as measurements have granular character.

8.2.2 Identification of Boundary Conditions in Thermo–Mechanical Systems Let us consider an elastic body occupying a domain Ω and having a boundary Γ . The steady–state thermoelasticity problem is considered. It is assumed, that the strain field depends on the temperature field but the temperature field does not depend on the strain field. The governing equations of the linear elasticity and steady–state heat conduction problem is expressed by following equations [9]: G ui, j j +

G 2G(1 − v) u j, ji + α T,i = 0 1 − 2v 1 − 2v

λ T,ii + Q = 0

(8.9) (8.10)

where: α – heat conduction coefficient, λ – thermal conductivity, T – temperature, Q – internal heat source. The boundary conditions are:

ΓT : Ti = T i ,

Γp : pi = pi , Γu : ui = ui Γq : qi = qi , Γc : qi = α (Ti − T ∞ )

(8.11)

where ui , pi , T i , qi , α , T ∞ – known displacements, tractions, temperatures, heat fluxes, heat conduction coefficient and ambient temperature, respectively.

8 Granular Computing in Evolutionary Identification

153

Separate parts of the boundary satisfy relations:

Γ = Γp ∪ Γu = ΓT ∪ Γq ∪ Γc Γp ∩ Γu = 0/ ΓT ∩ Γq ∩ Γc = 0/

(8.12)

The boundary conditions and measurements are assumed to have the granular character. Identification of the boundary conditions in thermo–mechanical systems can be formulated as a minimization of the funcional J: (q(z) − qˆ (z))2 δ z − zi d Γ

n

min J, J = ∑ x

i=1

(8.13)

Γ

where: qˆ – the measured values of state fields (e.g. displacements, temperatures); q – the values of the same state fields calculated from the solution of the direct problem, z = zi – i-th sensor point, n – a number of sensor points, δ – the Dirac function.

8.2.3 Identification of Elastic Constants in Laminates The identification problem in hybrid multilayered laminates is considered. Hybrid laminates have plies composed of different materials [1]. In present paper interply hybrids are considered. In interply hybrid laminates external plies are made of more expensive material having better properties while core plies are made of ’worse’, but cheaper material. The main reason of using hybrid laminates is looking for a balance between cost and other properties of the laminate. Fibre–reinforced multilayered laminates can be typically treated as 2–dimensional orthotropic structures with 4 independent elastic constants: axial Young modulus E1 , transverse Young modulus E2 , axial–transverse shear modulus G12 and axial– transverse Poisson ratio ν12 . The constitutive equation for a single layer of the laminate in the in–axis orientation has the following form [7]: ⎡ ⎧ ⎫ ⎨ σ11 ⎬ ⎢ ⎢ σ22 = ⎢ ⎩ ⎭ ⎢ ⎣ σ12

E1 ν21 E1 1−ν12 ν21 1−ν12 ν21

ν12 E2 E2 1−ν12 ν21 1−ν12 ν21

0

0

0

⎤

⎧ ⎫ ⎥ ⎨ ε11 ⎬ ⎥ 0 ⎥ ⎥ ⎩ ε22 ⎭ ⎦ ε12 G12

(8.14)

where: σi j – stress vector; εi j – strain vector; ν21 – transverse–axial Poisson ratio.

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The Poisson ratio ν21 can be expressed as:

ν21 = ν12

E2 E1

(8.15)

Apart from the identification of the elastic constants it is also necessary to identify densities ρ of particular materials in the hybrid laminate. As a result the design variables vector for a hybrid laminate can be written as: 1 2 x = {E11 , E21 , G112 , ν12 , ρ 1 , E12 , E22 , G212 , ν12 , ρ 2}

(8.16)

where superscripts denote the material number. Identification of the laminates’ elastic constants can be formulated as a minimization of the funcional J(x): N

min J, J = ∑ [qi − qˆi ]2 x

(8.17)

i=1

where: qˆi – the measured values of state fields; qi – the values of the same state fields calculated from the solution of the direct problem; The identification procedure is usually performed for the data obtained from the structure response (e.g. strain field) to the static load. This attitude is not efficient enough for the laminate structures due to problems with obtaining of the uniform stress and strain fields, boundary effects and scale effect. To avoid mentioned problems and reduce the number of sensor points the modal analysis methods are employed [13]. Two types of dynamic data are considered: i) eigenfrequencies; ii) accelerations in one sensor point in form of the frequency response. It is assumed that identified material parameters and measurements have granular character.

8.3 Two–Stage Granular Strategy In order to solve the identification problem, the Two–Stage Granular Strategy (TSGS) is proposed. Gradient optimization methods are fast and give precise results but they can be used for continuous problems. If the multimodal problems are considered gradient methods usually lead to the local optima. In such cases the evolutionary algorithms, which are the global optimization methods, can be used [10]. As the evolutionary algorithms are time–consuming and they have problems with obtaining the optimum value precisely, it can be convenient to merge both methods in a hybrid algorithm. The TSGS couples the advantages of global and local optimization methods. A Granular Evolutionary Algorithm (GEA) with granular genes is applied in the first stage. The local optimization method supported by an Artificial Neural Network

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(ANN) is used in the second stage to find the precise value of the optimum.The block diagram of the TSGS is presented in Fig. 8.3.

START GRANULAR EA (global optimization)

TRANSITION?

NO

YES

LOCAL OPTIMIZATION with ANN

Fig. 8.3 Block diagram of the Two–Stage Granular Strategy

STOP

8.3.1 The Global Optimization Stage The mean idea of GEA is similar to the classical evolutionary algorithm [2], but some modifications are required due to the granular character of the data. In the GEA the data representation, the evolutionary operators as well as the selection procedure are granular. Each chromosome, being a vector of identified variables (genes), represents one granular solution. The block diagram of the GEA is presented in Fig. 8.4. To solve the boundary–value problem the Granular (interval, fuzzy or stochastic) FEM is employed [8],[11]. Two special evolutionary operators are introduced: i) granular Gaussian mutation operator; ii) granular arithmetic crossover operator [12]. After evaluation of the solution a granular (interval, fuzzy or stochastic) fitness function value of the chromosome is obtained. The granular selection is based on the tournament selection method. The granular fitness function values are compared to select the best individual in the tournament. The better chromosome wins with a probability depending on the introduced parameter β . The cluster of granular solutions is achieved as a result of the first stage of the strategy.

8.3.2 The Local Optimization Stage Assuming that the cluster of points obtained in the previous stage is situated close to the global optimum, the local optimization is performed by means of the steepest descent method. The cluster of points is used to generate the training vector for

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Fig. 8.4 Block diagram of the Granular Evolutionary Algorithm

Stop condition YES STOP

ANN. ANN is used to evaluate the granular fitness function value [5]. The sensitivity of the fitness function is approximated by means of the ANN to avoid problems with the calculation of the objective function gradient. The special multi–level ANN is introduced. The number of levels is dependent on the data representation and is equal to the number of parameters in each gene in the first stage. The number of neurons in the input layer (for each level) equals the number of design variables. The number of neurons in hidden layers depends on the complexity of the identification problem. The single output (at each level of the output layer) represents one parameter of the granular fitness function value. The exemplary ANN for fuzzy representation of the data is presented in Fig. 8.5. The central level of the multilevel ANN corresponds with the central value cv of the fuzzy number (black colour), while the other levels correspond with other parameters of the fuzzy number: the grey levels correspond with the parameters ai , the white levels correspond with the parameters bi .

Fig. 8.5 The scheme of the multilevel ANN (fuzzy case)

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After training the multilevel ANN the local optimization process is performed until the termination condition is satisfied. If not, the point is calculated by using the Granular FEM and added to the training vector.

8.4 Numerical Examples The Two–Stage Granular Strategy is applied to solve some identification problems assuming different types of information granularity: interval numbers, fuzzy numbers or random variables. TSGS is applied to identify: • the position and the radius of the circular void in isotropic structure subjected to the dynamical loading; • the boundary conditions in thermomechanical systems; • the elastic constants in laminate structures.

8.4.1 Identification of Void A square plate 0.2x0.2 m made of an isotropic material (Fig. 8.6) is considered. The aim of the identification is to find the parameters of the circular defect. The chromosome x = [x, y, r] consists of 3 genes representing position x, y and radius r of the void. The plate is loaded by the continuous dynamical loading p(z,t) = p0 (z) H(t) , where p0 (z)=const=10 kN and H(t) is the Heaviside’s function. p ( z, t )

r x,y

sensor point

Fig. 8.6 The structure with identified circular defect

200 time–steps with δ = 1μ s are considered. It is assumed that the Young modulus of the plate material E and loading p0 have granular character. The displacements are measured in 21 sensor points on the boundary. The parameters of the GEA are: the number of chromosomes: psize=20; the number of generations: gen=40; the arithmetic crossover probability pac =0.2; the Gaussian mutation probability: pgm =0.4. The number of iterations of the local method

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is assumed to be 200. The variable ranges, actual values and results after the first and second stages for interval and fuzzy cases (for two α –cuts) are collected in Tables 8.1– 8.3 . Table 8.1 Void identification results: interval numbers

Min Max Actual Stage 1 Stage 2

x (m) a

b

y (m) a

b

r (m) a

b

0.005 0.2 0.029 0.0292 0.029

0.005 0.2 0.031 0.0311 0.031

0.005 0.2 0.029 0.0297 0.029

0.005 0.2 0.031 0.0307 0.031

0.005 0.1 0.019 0.0187 0.019

0.005 0.1 0.021 0.0209 0.021

Table 8.2 Void identification results: fuzzy numbers, lower α –cut

Min Max Actual Stage 1 Stage 2

x (m) a

b

y (m) a

b

r (m) a

b

0.005 0.2 0.025 0.023 0.025

0.005 0.2 0.035 0.034 0.035

0.005 0.2 0.025 0.028 0.025

0.005 0.2 0.035 0.032 0.035

0.005 0.1 0.015 0.018 0.015

0.005 0.1 0.025 0.023 0.025

Table 8.3 Void identification results: fuzzy numbers, upper α –cut

Min Max Actual Stage 1 Stage 2

x (m) a

b

y (m) a

b

r (m) a

b

0.005 0.2 0.025 0.022 0.025

0.005 0.2 0.035 0.037 0.035

0.005 0.2 0.025 0.026 0.025

0.005 0.2 0.035 0.031 0.035

0.005 0.1 0.015 0.015 0.015

0.005 0.1 0.025 0.021 0.025

8.4.2 Identification of Boundary Conditions A box–shaped steel structure presented in Fig.8.7 is considered. The structure is subjected to the thermomechanical loading. One surface of the box is supported

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while the point load is applied at every node on the opposite one. The total load is equal to 224 kN. The temperature T = 10◦C is applied on the supported surface of the structure. The third type thermal boundary condition (convection) is specified on the internal surfaces. The aim of the identification is to find the ambient temperature T ∞ and heat convection coefficient α on the internal surfaces. The material properties are: Young modulus E=2e11 MPa, Poisson ratio ν =0.3, thermal expansion coefficient αT = 12.5 e−61/◦C and thermal conductivity λ = 25W /m◦C.

T u

ts poin ents sor sen placem is of d

0.32

8

sen of t sor po em per ints atu re

a

T

s

1.0

P

0.5

Fig. 8.7 Geometry, boundary conditions and location of the sensor points

To gather measurement data 4 displacement sensors and 4 temperature sensors located on external surfaces of the structures are used. The parameters of the GEA are: the number of chromosomes: psize=30; the number of generations: gen=200; the arithmetic crossover probability pac =0.2; the Gaussian mutation probability: pgm =0.1. The number of iterations of the local method is assumed to be 500. The variable ranges, actual values and results after the first and second stages for interval and fuzzy cases (for 2 α –cuts) are collected in Table 8.4– 8.6. Table 8.4 Boundary conditions identification results: interval numbers

Min Max Actual Stage 1 Stage 2

α (W /m2◦C) a

b

T ∞ (◦C) a

b

1.00 25.00 3.00 2.61 3.00

1.00 25.00 7.00 6.76 7.00

0.00 105.00 45.00 49.99 45.00

0.00 105.00 52.00 53.33 52.00

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Table 8.5 Boundary conditions identification results: fuzzy numbers, lower α –cut

Min Max Actual Stage 1 Stage 2

α (W /m2◦C) a 1.00 25.00 3.00 2.06 3.00

b

T ∞ (◦C) a

b

1.00 25.00 7.00 8.45 7.00

0.00 105.00 45.00 43.54 45.00

0.00 105.00 52.00 54.44 52.00

Table 8.6 Boundary conditions identification results: fuzzy numbers, upper α –cut

Min Max Actual Stage 1 Stage 2

α (W /m2◦C) a

b

T ∞ (◦C) a

b

1.00 25.00 4.00 4.98 4.00

1.00 25.00 6.00 6.45 6.00

0.00 105.00 48.00 49.95 48.00

0.00 105.00 51.00 51.93 51.00

8.4.3 Identification of Laminates’ Elastic Constants The aim of the identification is to find the elastic constants of multi–layered, fibre– reinforced laminate. It is assumed, that identified constants are not precise and can represented by different types of information granularity. Dynamical data are collected to perform the identification procedure. A rectangular, symmetrical hybrid laminate plate of shape and dimensions presented in Fig. 8.8a) is considered. Each ply of the laminate has thickness h=0.002 m. The stacking sequence of the laminate is (0/15/-15/45/-45)s. The external plies are made of material M1 , the internal plies are made of the material M2 (Fig. 8.8 b). y

0.2

excitation point

x

M1 { M2

{

symmetry

sensor point 0.5 a)

b)

Fig. 8.8 The hybrid laminate plate: a) shape and dimensions; b) materials location

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Each chromosome in population consists of 10 genes representing identified constants and material densities (8.16). The FEM model consists of 200 4–node plane finite elements.

8.4.3.1 Interval Case It is assumed that each variable xi has granular character represented by interval number. The plate is excited in one point (Fig. 8.8a) by the the sinusoidal signal. The frequency of the excitation varies from 100 Hz to 2000 Hz. 200 samples of the acceleration amplitudes at one sensor point are measured. The parameters of the GEA are: the number of chromosomes: psize=100; the number of generations: gen=400; the arithmetic crossover probability pac =0.2; the Gaussian mutation probability: pgm =0.4. The number of iterations of the local method is assumed to be 2500. The variable ranges, actual values and results after both stages for materials M1 and M2 are collected in Tables 8.7– 8.8. Table 8.7 Hybrid laminate identification results: interval numbers, material M1 E1 (Pa) a b Min Max Actual Stage 1 Stage 2

E2 (Pa) a b

G12 (Pa) a b

1.50E10 1.50E10 4.20E9 4.20E9 0.90E9 1.50E11 1.50E11 26.00E9 26.00E9 9.00E9 1.80E11 1.82E11 10.00E9 10.04E9 7.10E9 1.72E11 1.83E11 19.87E9 10.36E9 7.01E9 1.80E11 1.82E11 10.00E9 10.04E9 7.10E9

0.90E9 9.00E9 7.18E9 7.43E9 7.18E9

ν12 a

b

ρ (kg/m3 ) a b

0.190 0.410 0.277 0.273 0.277

0.190 0.410 0.283 0.274 0.283

1.00E3 3.00E3 1.60E3 1.58E3 1.60E3

1.00E3 3.00E3 1.65E3 1.69E3 1.65E3

Table 8.8 Hybrid laminate identification results: interval numbers, material M2 E1 (Pa) a b Min Max Actual Stage 1 Stage 2

E2 (Pa) a b

G12 (Pa) a b

1.50E10 1.50E10 4.20E9 4.20E9 0.90E9 1.50E11 1.50E11 16.00E9 16.00E9 9.00E9 3.82E10 3.90E10 8.23E9 8.31E9 4.10E9 3.74E10 4.12E10 8.21E9 8.22E9 4.01E9 3.82E10 3.90E10 8.23E9 8.31E9 4.10E9

0.90E9 9.00E9 4.18E9 4.54E9 4.18E9

ν12 a

ρ (kg/m3 ) a b

b

0.190 0.410 0.257 0.273 0.257

0.190 1.00E3 0.410 3.00E3 0.263 1.80E3 0.278 1.58E3 0.263 1.80E3

1.00E3 3.00E3 1.81E3 1.69E3 1.81E3

8.4.3.2 Stochastic Case It is assumed that each variable xi and measurements have granular character represented by random variable with normal distribution described by mean value m and

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standard deviation σ . The first 10 eigenfrequencies of the plate are the measurement data. The measurements were repeated 200 times to collect data. The parameters of the GEA are: the number of chromosomes: psize=400; the number of generations: gen=1200; the arithmetic crossover probability pac =0.2; the Gaussian mutation probability: pgm =0.4. The number of iterations of the local method is assumed to be 5000. The variable ranges, actual values and results after both stages for materials M1 and M2 are collected in Tables 8.9– 8.10. Table 8.9 Hybrid laminate identification results: random variables, material M1

Min Max Actual Stage 1 Stage 2

E1 (Pa) a b

E2 (Pa) a b

G12 (Pa) a b

ν12 a b

ρ (kg/m3 ) a b

1.20E11 2.50E11 1.80E11 2.01E11 1.80E11

0.80E10 2.00E10 1.00E10 0.89E10 1.00E10

2.00E9 9.00E9 7.14E9 6.89E9 7.14E9

0.00 0.50 0.28 0.31 0.28

1.00E3 3.00E3 1.60E3 1.63E3 1.60E3

0.00E10 0.30E10 0.12E10 0.18E10 0.12E10

0.00E10 0.30E10 0.20E10 0.18E10 0.20E10

0.10E8 0.70E8 0.50E8 0.52E8 0.50E8

0.00 0.10 0.02 0.01 0.02

0.00E2 0.50E2 0.20E2 0.19E2 0.20E2

Table 8.10 Hybrid laminate identification results: random variables, material M2 E1 (Pa) σ m

E2 (Pa) m σ

Min

2.00E10 0.00E9 4.00E9 0.00E9 0 Max 6.00E10 0.30E9 9.00E9 0.30E9 Actual 3.86E10 0.12E9 8.28E9 0.20E9 Stage 1 3.75E10 0.04E9 8.12E9 0.28E9 Stage 2 3.86E10 0.12E9 8.28E9 0.20E9

G12 (Pa) m σ

ν12 m σ

ρ (kg/m3 ) m σ

2.00E9 0.10E8 0.00 0.00 1.00E3 0.00E2 6.00E9 4.14E9 4.21E9 4.14E9

0.70E8 0.50E8 0.61E8 0.50E8

0.50 0.26 0.25 0.26

0.10 0.02 0.04 0.02

3.00E3 1.80E3 1.81E3 1.80E3

0.50E2 0.20E2 0.17E2 0.20E2

8.5 Final Conclusions The Two–Stage Granular Strategy coupling the Granular Evolutionary Algorithm, the multilevel Artificial Neural Networks and local optimization methods has been presented. The application of global optimization method (GEA) in the first stage of the TSGS decreases the possibility of finding the local minimum of the objective function. The gradient method applied in the second stage is supported by ANN. ANN is used to calculate the fitness function gradient and to evaluate the granular fitness function value. The granular version of the FEM is used to solve the boundary–value problems. The identified values and the fitness functions can be represented by different forms of granularity: interval numbers, fuzzy numbers or random variables.

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The strategy gives positive results for the identification of various parameters in isotropic and orthotropic structures, for static and dynamical cases as well as for thermo–mechanical problems with different variants of uncertainties. Acknowledgements. The research is financed from the Polish science budget resources in the years 2008–2011 as the research project.

References [1] Adali, S., et al.: Optimal design of symmetric hybrid laminates with discrete ply angles for maximum buckling load and minimum cost. Compos. Struct. 32, 409–415 (1995) [2] Arabas, J.: Lectures on Evolutionary Algorithms (in Polish). WNT, Warsaw (2001) [3] Bargiela, A., Pedrycz, W.: Granular Computing: An introduction. Kluwer, Boston (2002) [4] Bui, H.D.: Inverse Problems in the Mechanics: An Introduction. CRC PRess, Bocca Raton (1994) [5] Burczy´nski, T., Orantek, P.: The evolutionary algorithm in stochastic optimization and identification problems. In: Arabas, J. (ed.) Evolutionary Computation and Global Optimization 2007, Warsaw, pp. 309–320 (2006) [6] Burczy´nski, T., Orantek, P.: Uncertain Identification Problems in the Context of Granular Computing. In: Bargiela, A., Pedrycz, W. (eds.) Human-Centric Inforamtion Processing. SCI, vol. 182, pp. 329–350. Springer, Heidelberg (2009) [7] German, J.: The basics of the fibre-reinforced composites’ mechanics (in Polish). Publ. of the Cracow University of Technology, Cracow (2001) [8] Kleiber, M., Hien, T.D.: The Stochastic Finite Element Method. John Wiley & Sons, New York (1992) [9] Maugin, G.A.: The Thermomechanics of Nonlinear Irreversible Behaviors: An Introduction. World Scientific, Singapore (1999) [10] Michalewicz, Z.: Genetic Algorithms + Data Structures = Evolutionary Programs. Springer, Berlin (1996) [11] Moens, D., Vandepitte, D.: Fuzzy Finite Element Method for Frequency Response Function Analysis of Uncertain Structures. AIAA Journal 40(1), 126–136 (2002) [12] Orantek, P.: The optimization and identification problems of structures with fuzzy parameters. In: 3rd European conference on computational mechanics ECCM 2006. CDEdition, Lisbon (2006) [13] Uhl, T.: Computer-Aided Identification of Constructional Models (in Polish). WNT, Warsaw (1997)

Chapter 9

Immune Computing: Intelligent Methodology and Its Applications in Bioengineering and Computational Mechanics Tadeusz Burczy´nski, Michał Bereta, Arkadiusz Poteralski, and Mirosław Szczepanik

Abstract. The aim of this paper is to provide a set of carefully selected problems connected with the current research directions of Immune Computing. This approach belongs to biology inspired methods. Due to the complexity of functioning of the natural immune system, extracting higher level paradigms which could serve as the basis of constructing computational models and algorithmic solutions is made. Applications of this intelligent methodology to bioengineering and computational mechanics problems are presented.

9.1 Introduction Nature inspired computing has proved to be useful in various application areas. Evolutionary methods, neural networks, swarm intelligence and many other approaches have been applied to many technical and engineering problems, such as optimization, learning, data analysis, knowledge engineering and many others. Some of these methods perform better in the given application areas and some work better in others. However, it can hardly be assumed that there exists a problem domain in which nature inspired techniques are not employed, at least as a part of the proposed solution. After decades of development, biologically inspired methods are well Tadeusz Burczy´nski Cracow University of Technology, Institute of Computer Modeling, Artificial Intelligence Division, Warszawska 24, 31-155 Cracow, Poland Silesian University of Technology, Department of Strength of Materials and Computational Mechanics, Konarskiego 18a, 44-100 Gliwice, Poland e-mail: [email protected] Michał Bereta Cracow University of Technology, Institute of Computer Modeling, Artificial Intelligence Division, Warszawska 24, 31-155 Cracow, Poland Arkadiusz Poteralski, Mirosław Szczepanik Silesian University of Technology, Department of Strength of Materials and Computational Mechanics, Konarskiego 18a, 44-100 Gliwice, Poland

M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 165–181. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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established and appreciated tools. On the other hand, many novel approaches arise to solve problems in innovative ways, hopefully more effectively. One of such novel areas is the field of Artificial Immune Systems (AIS) [2][4][6][7]. The question arises as to what AIS can offer as problem solving techniques and whether the paradigms proposed by AIS researchers are in fact novel. The aim of this paper is to provide a set of carefully selected problems connected with the current research directions of Immune Computing.

9.2 Immunology as a Metaphor for Computational Information Processing An immune system, especially this of vertebrates, is a very complicated system of interacting cells, organs and mechanisms, whose purpose is to protect the host body against any danger, either exterior or internal. To achieve that goal the immune system has to decide not only about what is not part of the host but also what can cause a damage. This is a very difficult task, as not all that comes from outside is dangerous. On the other hand, autoimmune diseases are examples of internal threats. To protect the host against all such dangers is not an easy task, especially in a changing environment. It is obvious that the immune system has to develop some sense of self, i.e., the sense of what is part of the host. How the immune system achieves this is hard to explain, as the host body changes its functioning and structure over time. Nonetheless, the immune system is able to perform its task effectively. To deal with such a difficult task the immune system needs the ability to learn new threats, to remember previous experiences and to develop specialized responses to different pathogens. Taking a closer look at all these features one can state that the immune system can be considered as a cognitive system. For that reason the immune system gained an interest of computational sciences. The artificial immune systems [4][6][7] are developed on the basis of a mechanism discovered in biological immune systems. The immune system is a complex system which contains distributed groups of specialized cells and organs. The main purpose of the immune system is to recognize and destroy pathogens - funguses, viruses, bacteria and improper functioning cells. The lymphocytes cells play a very important role in the immune system. The lymphocytes are divided into several groups of cells. There are two main groups B and T cells, both contains some subgroups (like B-T dependent or B-T independent). The B cells contain antibodies, which could neutralize pathogens and are also used to recognize pathogens. There is a big diversity between antibodies of the B cells, allowing recognition and neutralization of many different pathogens. The B cells are produced in the bone marrow in long bones. A B cell undergoes a mutation process to achieve big diversity of antibodies. The T cells mature in thymus, only T cells recognizing non self cells are released to the lymphatic and the blood systems. There are also other cells like macrophages with presenting properties, the pathogens are processed by a cell and

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presented by using MHC (Major Histocompatibility Complex) proteins. The recognition of a pathogen is performed in a few steps (Fig. 9.1). Pathogenreceptor binding Pathogen presentation as MHC signal MHC protein T cell receptor

Pathogen B cell receptor B cell

B cell activation signal

a)

T cell

b) Activated B cell Proliferation

Memory cell Proliferation Memory cell

Memory cell Secreted antibodies

c)

d)

Secreted antibodies

Macrophage

e)

Pathogen with bound antibodies

Fig. 9.1 An immune system, a) B cell and pathogen, b) the recognition of pathogen using Band T-cells, c) the proliferation of activated B-cells, d) the proliferation of a memory cell secondary response, e) pathogen absorption by a macrophage

First, the B-cells or macrophages present the pathogen to a T-cell using MHC (Fig. 9.1b), the T-cell decides if the presented antigen is a pathogen. The T-cell gives a chemical signal to B-cells to release antibodies. A part of stimulated B-cells goes to a lymph node and proliferate (clone) (Fig. 9.1c). A part of the B-cells changes into memory cells, the rest of them secrete antibodies into blood. The secondary response of the immunology system in the presence of known pathogens is faster because of memory cells. The memory cells created during primary response, proliferate and the antibodies are secreted to blood (Fig. 9.1d). The antibodies bind to pathogens and neutralize them. Other cells like macrophages destroy pathogens (Fig. 9.1e). The number of lymphocytes in the organism changes, while the presence of pathogens increases, but after attacks a part of the lymphocytes is removed from the organism. Given all the complexity of functioning of the immune system, it is necessary to extract higher level paradigms which could serve as the basis of constructing computational models and algorithmic solutions. The most important paradigms in the filed of Artificial Immune Systems are a Clonal Selection (CS), a Immune Network Theory (IMT), a Negative Selection (NS) and recently emerged a Danger Theory (DT).

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The question is whether AIS can offer anything really new and/or useful. The Clonal Selection can seem to be another exemplification of evolutionary approach to problem solving. The fact of existing of immune networks in biological immune systems is questioned by biologists. The Negative Selection appears to be truly novel approach, but one could ask whether it is enough to invest time and resources to develop AIS. In the first examinations of the immune ideas, the researchers developed several algorithmic solutions based on immune paradigms, often separately. The Clonal Selection and the Immune Network Theory have been applied to optimization, data analysis or clustering. The Negative Selection has been applied to computer security, anomaly or fault detection. Many of the proposed algorithms have successfully dealt with the tasks appointed to them. However, their usefulness according to their robustness and scalability has been under dispute when compared to other well established computational methods. The broad application areas and the new ideas proposed, show the vitality of the still young research field of Immune Computing in intelligent problem solving techniques. Applications of AIS to bioengineering problems as feature selection and classification of ECG signals are presented. Possibility of a coupling immune computing with the finite element method in solving several 2-D and 3-D optimization problems is also considered.

9.3 Applications of AIS in Computational Bioengineering 9.3.1 Negative Selection In natural immune systems T-lymphocytes are produced in thymus and cannot leave it if they do not pass the test of recognizing none of self-molecules presented to them. If a T-cell recognizes any of self-cells, it is eliminated. T-cells which leave the thymus are capable of recognizing non-self antigens while being neutral to selfcells. This paradigm is called the Negative Selection (NS) and is used in the AIS which deal with the problem of anomaly detection. NS algorithms have two stages: (i) a training stage and (ii) a detection stage (Fig. 9.2). In the training stage, detectors are created only by means of normal samples, no negative samples are presented to candidate T-cells during training. This is a great advantage of negative selection as in many tasks it is hard to prepare a representative collection of negative samples for training purpose. After the training stage, the created T-cells are used for monitoring new samples, which can be categorized as normal (self) or as antigens non-self, anomalies). A sample is detected by a T-cell if it matches to the given T-cell well enough. The matching process depends on the representation of both T-cells, self-samples and antigens. In the case of binary representation there are two manners for matching: (i) checking the number of common bits and (ii) checking the longest common substring between two binary strings.

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Fig. 9.2 Two stages of negative selection: training stage and detection stage

The sample is bound by the detector if the matching is above a given threshold. The matching process is presented in Fig. 9.3.

Fig. 9.3 Matching process in the case of binary representation of T-cells, self-data, and antigens

Assume, that in an anomaly detection task, the normal state of the process over sometime can be represented as one-dimensional signal. Having the records of the normal state of the process, one can create a set of self-cells as a set of binary strings using the method of a moving window (Fig. 9.4).

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Fig. 9.4 Moving window method for binary strings creation from real-valued signals

For each window’s position, one binary string is created. Its length depends on the window’s size and on the number of divisions of the [min, max] range. Each real value in the window is first coded as the number of range in the [min, max], in which it falls. Then, the numbers of range of each real value in the window are concatenated creating one binary string. Created binary strings are the self-samples used during the training stage of negative selection. After learning, while monitoring the process, created T-cells should be capable of recognizing unusual and abnormal states of the process. The negative selection can be also modelled by real-valued representation [1].

9.3.2 Clonal Selection The clonal selection is a process characteristic for B-lymphocytes. B-cells move freely in the body and create clones while they are stimulated. The stimulation of B-cells by antigens can be simulated by means of binary representation in the same fashion as simulating bindings of T-cells, as described earlier. The stronger the binding is, the more clones the given B-cell can be produce. Clones are mutated at very high rates (somatic hyper-mutation) which allows the system to find better lymphocytes. The important mechanisms of the immune systems is the suppression mechanism (crowding mechanism) (Fig. 9.5). B-cells react not only to the presence of the antigens but to the presence other B-cells, too, trying the eliminate them. In this way, the weakest cells are eliminated from the population. The suppression is used in AIS as a mechanisms to eliminate worse solutions represented by less stimulated artificial cells. The proper suppression mechanism plays the important role in the whole process as it is able to keep the tractable population size and necessary population diversity.

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Fig. 9.5 The clonal selection algorithm

9.3.3 Two Level Immune Classification of ECG Signals The concept of two level immune algorithm is used in the proposed system for detectors creation and feature selection (Fig. 9.6). An initial set of subpopulations P is generated randomly. Each subpopulation Pi is characterized by a binary string. This algorithm consists of negative and clonal selections.

Fig. 9.6 The concept of two level immune algorithm

The detailed description of the algorithm is given in [1]. This algorithm is summarized in Fig. 9.7.

9.3.4 Results of Feature Selection and Classification of ECG Signals The immune algorithm was tested in a task of recognizing heart diseases in ECG signals. The signals were taken from the MIT-BIH database [5]. Different types of

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Fig. 9.7 The main steps of two level algorithm

samples representing normal and pathological ECG signals were extracted from this database. Each sample’s length was 128 with middle of QRS complex as the central point. QRS complex is a part of ECG signal (Fig. 9.8) that is supposed to have the most useful medical information for diagnosis.

Fig. 9.8 A fragment of ECG record with QRS complex selected by a rectangular

ECG samples of each type used for learning and testing as well as the explanations of samples’ symbols are contained in Tab. 9.1. Fig. 9.9 presents ECG samples of different types used in the research. The samples were divided into a normal signals set used for creating self-strings and a set of pathological signals. All signals were filtered using window mean method with window size 4 and step size 4 and normalized to have zero mean and

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Fig. 9.9 Examples of ECG signals of normal N type and different pathological types

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Table 9.1 ECG samples used for training and testing Type

Train

Test

Total

Disease

N A E L R V

1226 270 93 236 249 270

23654 4796 119 15898 14261 13914

24880 5066 212 16134 14510 14186

Normal Atrial premature beat Ventricular escape beat Left bundle branch block beat Right bundle branch block beat Premature ventricular contraction

unit variance, resulting in a new set of samples of length 32. Pi were trained to discriminate between normal state and heart diseases. Then Haar wavelets coefficients were counted resulting in 32-dimensional search space. Table 9.2 Results for Haar wavelets coefficients of window -mean filtered ECG signals Type

Train

Recognised Test (% )

Recognised (% )

N (self) A E L R V Non-self

1226 270 93 236 249 270 1120

0 92.2 100 100 99.6 99.3 97.9

41.1 86.4 100 99.4 99.6 98 97.8

23654 4796 119 15898 14261 13914 48988

The results presented in Table 9.2 show that the proposed immune system is able to select the suitable subset of features.

9.4 Applications in Computational Mechanics 9.4.1 Immune Optimization The Clonal Selection mechanism is very useful in global optimization problems, especially in computational mechanics [3]. The unknown global optimum is the most dangerous searched pathogen. The memory cells contain design variables and proliferate during the optimization process. The B-cells created from memory cells undergo a mutation. The crowding (suppression) mechanism forces the diverse between memory cells. The objective functions for B-cells are evaluated by means of the finite element method. The selection process exchanges some memory cells for better B-cells (Fig. 9.10).

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START Creation of memory cells

Memory cells proliferation with hipermutation Evaluation of objective function for B cells Selection

Crowding mechanism STOP

Stop condition CONTINUE

STOP

Fig. 9.10 An artificial immune system for optimization problems

The presented approach is based on the application of the artificial immune system and the finite element method to topology optimization of 2-D and 3-D structures. The fitness function is calculated for each B-cell in each iteration by solving a boundary-value problem by means of the finite element method (FEM). In order to solve the optimization problem the fitness function, design variables and constraints should be formulated.

9.4.2 Formulation of Immune Topology Optimization of Structures Consider a structure which, at the beginning of an immune process, occupies a domain Ω0 (in E d ,d = 2 or 3), bounded by a boundary Γ0 . The domain Ω0 is filled by elastic homogeneous and isotropic material of a Young’s modulus E0 , a mass density ρ0 and a Poisson ratio ν . The structures are considered in the framework of the linear theory of elasticity. During the immune optimization process the domain Ωi , its boundary Γi and the field of mass densities ρ (x) = ρi , x ∈ Ωi can change for each iteration t (for t = 0, ρ0 = const). The immune process proceeds in the environment in which the structure fitness is described by the minimization of the mass of the structure J= ρ dΩ (9.1) Ω

with constraints imposed on displacements of the structure |u(x)| ≤ uad , x ∈ Ω or with constraints imposed on equivalent stresses of the structure

(9.2)

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σeq (x) ≤ σ ad , x ∈ Ω

(9.3)

In order to solve the formulated problems the finite element models of 2-D and 3-D structures are considered. A 2-D structure domain is divided into shell finite elements and a 3-D structure domain is divided into solid finite elements. In order to solve direct problems for 2-D and 3-D elastic structures the professional program MSC.NASTRAN is used. The distribution of mass density ρ (x), x ∈ Ω , (Fig. 9.11) in the structure is described by a surface Wp (x), x ∈ H 2 (for 2-D) and a hyper surface Wp (x), x ∈ H 3 (for 3-D). The surface (hyper surface) Wp (x) is stretched under H d ⊂ E d , (d = 2, 3) and the domain Ωi is included in H d , i.e. (Ωt ⊆ H d ). The shape of the surface (hyper surface) Wp (x) is controlled by B-cell receptors d j , j = 1, 2, ..., G which create a B-cell B − cell =< d1 , d2 , ..., dG > (9.4) ≤ d j ≤ d max d min j j

(9.5)

where d min , d max - are minimum and maximum values of the B-cell receptor, rej j spectively. B-cell receptors are the values of the function Wp (x) in the control points x j of the surface (hyper surface), i.e. d j = Wp (x j ), j = 0, 1, 2, ..., G.

Fig. 9.11 The illustration of the idea of immune topology optimization for a 2-D structure

The finite element method is applied in analysis of the structure. The domain Ω of the structure is discretized using the finite elements, Ω =

E )

Ωe . The assignation

e=1

of the mass density to each finite element Ωe is performed by the mappings:

ρe = Wp (xe ), xe ∈ Ωe , e = 1, 2, ..., E

(9.6)

It means that each finite element can have different mass density. When the value of the mass density for the e-th finite element is included in:

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• the interval 0 ≤ ρe < ρmin , the finite element is eliminated and a void is created, • the interval ρmin ≤ ρe < ρmax , the finite element remains. In the next step the Young’s modulus for the e − th finite element is evaluated using the following equation Ee = Emax (

ρe r ) ρmax

(9.7)

where: Emax and ρmax - Young’s modulus and mass density for the same material, respectively, r - parameter which can change from 1 to 9 [3]. By means of the proposed method, the material properties of finite elements are changed and some of them are eliminated. As a result the optimal shape, the topology and the material of the structures are obtained. The multinomial interpolation of the hyper surface is expressed as follows ⎤ ⎡ d1 −1 Wρ (x) = Φ X ⊗ Y−1 ⊗ Z−1 ⎣ div ⎦ (9.8) d27 where

Φ = [1,x,x2 ] ⊗ [1,y,y2 ] ⊗ [1,z,z2 ] = [1,z,z2 ,y,yz,yz2 ,y2 , y2 z,y2 z2 ,x,xz,xz2 ,xy,xyz,xyz2 ,xy2 ,xy2 z,xy2 z2 , x2 ,x2 z,x2 z2 ,x2 y,x2 yz,x2 yz2 ,x2 y2 ,x2 y2 z,x2 y2 z2 ]

and X, Y and Z are matrices described as follows ⎤ ⎡ 100 X = Y = Z = ⎣1 1 1⎦ 124

(9.9)

(9.10)

Fig. 9.12 Spacing of control points

The structure is under the immune optimization process inserted into a cube whose edges have the length A=2, B=2, C=2 (Fig. 9.12). The model of the structure is scaled by means of following expressions: x = (A* coordinate x)/length of the model y = (B* coordinate y)/width of the model z = (C* coordinate z)/height of the model

(9.11)

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The final structure obtained after the optimization process has a rough boundary. In order to get a smooth shape of the boundary, a procedure which smoothes it has to be used. The artificial immune system works on the group of B-cells. The operations described above are performed for a single B-cell from the population and lead to the evaluation of the fitness function value (Fig. 9.13).

Fig. 9.13 Operation scheme performed for a single Bcell

9.4.3 Numerical Examples of Immune Topology Optimization Two numerical examples of topology optimization are presented: • a 2-D structure (plane stress) (Example 1), • a 3-D solid body (Example 2), for minimization of the structures mass (9.1) with displacement (9.2) and stress (9.3) constraints. Structures are considered in the framework of the theory of elasticity. Results of topology optimization are obtained by using the artificial immune system with following parameters (Tab. 9.3) Table 9.3 Parameters of the artificial immune system Parameter

Value

number of memory cells number of clones crowding (suppression) factor Gaussian mutation

8 4 25% 20%

9.4.3.1 Example 1 A rectangular 2-D structure (plane stress), of dimensions 100 200 [mm], loaded with the concentrated force P in the center of the lower boundary and fixed on the bottom corners is considered (Fig. 9.14a). Due to the symmetry a half of the structure is considered. The input data to the optimization program and parameters of the

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artificial immune system are included in Tab. 9.3 and 9.4, respectively. The geometry, the distribution of the control points of the interpolation surface is shown in the Fig. 9.14b. The results of the optimization process are presented in the Fig. 9.15. Table 9.4 The input data to the optimization task of for a 2-D structure (Example 1)

σ ad

thickness [mm] P [kN]

range of ρe [g/cm3 ]

80.0

4.0

7.3≤ ρe < 7.5 (FE elimination) 7.5≤ ρe < 7.86 (FE leaving)

2

a)

b)

Fig. 9.14 A 2-D structure (Example 1): a) the geometry; b) the distribution of the control points of the interpolation surface

a)

c)

b)

d)

Fig. 9.15 Results of immune topology optimization of 2-D structure: a) the solution of the optimization task; b) the map of mass densities; c) the map of stresses; d) the map of the displacement

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9.4.3.2 Example 2 A 3-D structure with dimensions and loading is presented in the Fig. 9.16a and 9.16b. The input data to the optimization program are included in Tab. 9.5. The geometry, the distribution of the control points of the interpolation hyper surface is shown in the Fig. 9.16c. The results of the optimization process are presented in Fig. 9.17a and Fig. 9.17b.

Fig. 9.16 Two cases of loading with the hyper surface a) first case (compression), b) second case (tension), c) the distribution of the control points of the interpolation hyper surface

Table 9.5 The input data to the optimization task of for a 3-D structure (Example 2) a

Dimensions [mm] b c

100

100

100

Loading/Constraints Compression (a) Tension (b) Q = −36.3[kN] uad = 0.05[mm] σ ad = 35[MPa]

Q = −36.3[kN] uad = 0.03[mm] σ ad = 55[MPa]

It is worth noticing that different constraints can lead to the total different final solutions of topology optimization of the structure.

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Fig. 9.17 Immune optimization of 3-D structure: a) first case (compression), b) second case (tension)

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a)

b)

9.5 Conclusions The Artificial Immune Systems (AIS) as novel areas of computing can be considered as computational systems based on natural immune system metaphors. They are composed by intelligent methodologies and can be regarded as data manipulation, classification, representation and reasoning strategies that follow a biological paradigm - the human immune system. The growing interest for AIS has been observed recently. Biological inspiration provides utility and extension and also improves comprehension of natural phenomena. Examples presented in the paper show that immune computing can be very useful in computational engineering, especially for bioengineering and computational optimization of structures. AIS have also several interesting features, lake: high degree of parallelism, robust-ness with low redundancy and possibility of using heuristics to improve convergence and scope of applications. Immune computing is strongly related to other intelligent approaches as artificial neural networks and evolutionary computing.

References [1] Bereta, M., Burczy´nski, T.: Comparing binary and real-valued coding in hybrid immune algorithm for feature selection and classification of ECG signals. Engineering Applications of Artificial Intelligence 20, 571–585 (2007) [2] Burczy´nski, T. (Guest Editor): Special Issue on Artificial Immune Systems. Information Sciences 179(10) (2009) [3] Burczy´nski, T.: Evolutionary and immune computations in optimal design and inverse problems. In: Waszczyszyn, Z. (ed.) Advances of Soft Computing in Engineering. Springer, Heidelberg (2009) (in press) [4] Castro, L.N., Timmis, J.: Artificial immune systems as a novel soft computing paradigm. Soft Computing 7(8), 526–544 (2003) [5] Goldberger, a.I., Amaral, L.A.N., Glass, L., Hausdorff, J.M., Ivanov, P.C., Mark, R.G., Mietus, J.E., Moody, G.B., Peng, C.K., Stanley, H.E.: PhysioBank, PhysioToolkit, and PhysioNet: components of a new research resource for complex physiologic signals. Circulation 101(23), e215–e220 (2000); circulation Electronic Pages: http://circ.ahajournals.org/cgi/content/full/101/23/e215 [6] Perelson, A.S., Weisbuch, G.: Immunology for physicists. Reviews of Modern Physics 69(4), 1219–1267 (1997) [7] Wierzcho´n, S.T.: Artificial Immune Systems. Theory and Applications. Exit, Warsaw (2001) (in Polish)

Chapter 10

Bioinspired Algorithms in Multiscale Optimization Wacław Ku´s and Tadeusz Burczy´nski

Abstract. The paper is devoted to bioinspired optimization in multiscale problems. The composite modeled as a macrostructure with a local periodic microstructure is considered. The multiscale analysis is performed with the use of the homogenization method. The evolutionary algorithm, the artificial immune system and the particle swarm optimization are used in computations. The objective function evaluation with the use of the parallel homogenization algorithm is considered. The paper contains a description of the evolutionary algorithm, artificial immune system, particle swarm optimization, the homogenization method, the optimization formulation.

10.1 Introduction The paper is an extension of previous authors paper [1]. The multiscale model of a structure is considered. The goal of optimization presented in this paper is minimization of a functional depending on state field in one of a structure scale with respect to design variables described on another scale. The optimization problem is solved by using bioinspired algorithms. Evolutionary, immune system and particle swarm optimizations are used in computations. The objective function is calculated for each set of design parameters on the basis of results of a direct problem analysis. The direct problem is modeled with the use of a computational homogenization. The home made programs based on the Finite Element Method (FEM) [8] are used. Wacław Ku´s Silesian University of Technology, Department of Strength of Materials and Computational Mechanics, Konarskiego 18a, 44-100 Gliwice, Poland e-mail: [email protected] Tadeusz Burczy´nski Cracow University of Technology, Institute of Computer Modeling, Artificial Intelligence Division, Warszawska 24, 31-155 Cracow, Poland Silesian University of Technology, Department of Strength of Materials and Computational Mechanics, Konarskiego 18a, 44-100 Gliwice, Poland e-mail: [email protected] M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 183–192. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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10.2 The Multiscale Model The multiscale model of the structure is considered. One of the numerical techniques which enables multiscale analysis of structures is the computational homogenization. The detailed description of the computational homogenization can be found in [4]. The local periodicity is assumed. It means that there are areas of the structure with the same microstructure. The example of macrostructure with periodic microstructures is shown in Fig. 10.1.

W Fig. 10.1 Two scale model of macrostructure with periodic microstructure

macroscale

microscale

The microstructure can also be built from the lower scale of a locally periodic microstructure. The goal of the computational homogenization is analysis of the structure taking into account the local periodicity of the microstructure. The main advantage of the computational homogenization is an possibility of performing analysis in a few scales. It allows to use models with at least a few orders of degrees of freedom lower than model created in one scale. The material parameters for each integration point in finite elements depend on the solution of a representative volume element (RVE) in the lower scale. The RVE is a model of the microstructure, voids, inclusions and other properties of the microstructure can be included in the model. The RVE is in most cases modeled as a cube or a square. The numerical method like FEM is used to solve the boundary value problem for RVE. The periodic displacements boundary conditions are taken into account. The strains from the higher level are prescribed as additional boundary conditions. The RVE for each integration point of the higher level model must be created and stored for the next iteration steps if the nonlinear problem with plasticity is considered. The transfer of information both form lower to higher and higher to lower scales is needed in most cases. The one way transfer of results (from lower to higher scales) is possible if the linear problem is considered. The material parameters for the higher scale are obtained on the basis of solving a few direct problems for RVE in the lower scale. The homogenized material parameters depend on average stress values in RVE obtained after applying average strains to RVE. The stress-strain relation obtained using RVE is

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used in the higher level model. Strains in the integration point from the higher level are considered as average strains.

10.3 The Optimization Problem Formulation The design parameters describe the properties of the microscale model, and an objective function value depends on the macroscale model. The objective function is defined as: F(ch) = max{u} (10.1) where ch = [ch1 , ch2 , ...chn ]T is a vector containing design parameters values, u is a vector of reduced displacements in the macroscale. The optimization goal considered in the paper is minimization of maximal displacements in the macroscale level formulated as: min F(ch) ch

(10.2)

The design parameters describe the shape of an inclusion in the microscale. The constraints on design parameters values are imposed in the form chimin ≤ chi ≤ chimax

(10.3)

where chimin is minimal and chimax maximal value of design parameter i. The constraints on a inclusion area RA are used RA ≤ RAmax

(10.4)

where RAmax is maximal area of the inclusion. The objective function value is computed by using multiscale FEM (Fig. 10.2). The microstructure finite element mesh is created on the base of NURBS curves.

10.4 The Bioinspired Optimization Algorithms The evolutionary algorithms (EA) [6] are based on mechanisms taken from biological evolution of species. The selection based on the individual fitness, mutations in chromosomes and individuals crossover are adopted. The evolutionary algorithms operate on a population of individuals. The individuals contain one chromosome in most cases. The evolutionary operators change the chromosomes during an iterative process. The selection is performed in every iteration. The changed by mutation or crossover individuals need fitness function evaluation. The flowchart of the evolutionary algorithm is shown in Fig. 10.3. The detailed description of used EA can be found in [5].

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design parameters vector Prepare NURBS curve for microscale model

Mesh the microscale model

Compute multiscale problem by using FEM Compute objective function value based on analysis results Objective function value

Fig. 10.2 The objective function evaluation algorithm START Starting population creation Fitness function evaluation for each chromosome

Selection Evolutionary operators Fitness function evaluation for each chromosome STOP

Stop condition CONTINUE

Fig. 10.3 The evolutionary algorithm

STOP

The artificial immune systems (AIS) are developed on the basis of mechanism discovered in biological immune systems. An immune system is a complex system which contains distributed groups of specialized cells and organs. The main purpose of the immune system is to recognize and destroy pathogens - funguses, viruses, bacteria and improper functioning cells. The artificial immune systems [2] take only a few elements from the biological immune systems. The mutation of the B cells, proliferation, memory cells, and recognition by using the B and T cells are applied very

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frequently. The artificial immune systems have been used to optimization problems, classification and also computer viruses recognition. The cloning algorithm Clonalg presented by von Zuben and de Castro uses some mechanisms similar to biological immune systems to global optimization problems. The unknown global optimum is the searched pathogen. The memory cells contain project variables and proliferate during the optimization process. The B cells created from memory cells undergo mutation. The B cells are evaluated and better ones exchange memory cells. The flowchart of AIS is shown in Fig. 10.4.

START Creation of memory cells

Memory cells proliferation with hipermutation Evaluation of objective function for B cells Selection

Crowding mechanism STOP

Stop condition CONTINUE

Fig. 10.4 The artificial immune system

STOP

The particle swarm optimization (PSO) algorithm is also based on observation done in biology [3]. The algorithm has similar behavior as a birds flocking or fish schooling. The individual bird (in PSO particle) changes velocity and position taking into account neighbor birds. This social behavior shown by some animals can be very efficient in nature. PSO incorporated some biological elements in the numerical algorithm, such as velocity change of the particle on the base of neighbors. The PSO is the iterative algorithm. The positions of the particles in the search space and starting velocities are defined randomly on the beginning of the algorithm. The change of particle velocities are due to the velocities of best ever found, best in neighborhood and previous particle values. The algorithm updates position of all particles in every iteration on the basis of computed velocities. The flowchart of PSO is shown in Fig. 10.5.

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W. Ku´s and T. Burczy´nski START Randomly choose particles positions Randomly choose starting velocities for particles

Compute objective function value for particles Update particles velocities Update particles positions STOP

Stop condition CONTINUE

Fig. 10.5 The particle swarm optimization

STOP

10.5 Numerical Example The numerical example of optimization of a 2D composite structure is considered (Fig. 10.6). The optimization criterion is to minimize the maximal reduced displacement of the structure.

a)

b)

Fig. 10.6 The macromodel a) geometry, b) finite element mesh

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The multiscale analysis was performed by using home made software based on FEM package MSC.Nastran [7]. The shape of the macro level structure is shown in Fig. 10.6. The inclusion in the microstructure is described using a NURBS curve (Fig. 10.7. The 8 design variables (g1 − g8 ) represent coordinates of the NURBS curve polygon control points. The size of RVE is 1 by 1. The constraints on design parameters values are shown in Tab. 10.1. The maximum area of inclusion and matrix RAmax was equal to 0.3. Table 10.1 The constraints on design parameters values The design parameter

Minimum

Maximum

g1 g2 g3 g4 g5 g6 g7 g8

0.10 0.10 0.50 0.10 0.50 0.50 0.10 0.50

0.47 0.47 0.90 0.47 0.90 0.90 0.47 0.90

(g5,g6)

NURBS curve NURBS control polygon (g3,g4) NURBS control point

(g7,g8)

(g1,g2)

Fig. 10.7 The Representative Volume Element (micromodel)

The optimization problem was solved by using EA, AIS and PSO. The parameters of the algorithms are shown in Tab. 10.2. The numerical tests were performed 5 times for each algorithm. The best obtained and reference solution are shown in Fig. 10.8. The displacement map for the macroscale is shown in Fig. 10.9. The comparison of the best objective functions obtained by bioinspired algorithms and reference solutions are shown in Tab. 10.3.

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Table 10.2 The bioinspired algorithms parameters Parameter

Value The Evolutionary Algorithm

Number of chromosomes Prob. of Gaussian mutation Prob. of uniform mutation Prob. of simple crossover Selection mechanism

50 90% 10% 90% rank selection

The Artificial Immune System Number of memory cells Number of cloned B cells Prob. of Gaussian mutation

5 50 100%

The Particle Swarm Optimization Number particles in swarm Influence of the best ever found particle Influence of the previous state of particle Influence of the best particle in neighborhood

a)

50 0.33 0.33 0.33

b)

Fig. 10.8 The microstructure: a) starting, b) the best found. The dashed lines represent NURBS curves control polygons Table 10.3 The results of optimization Algorithm

The objective function value for the best found solution

The RA value

Reference solution (Fig. 10.8a) The Evolutionary Algorithm The Artificial Immune System The Particle Swarm Optimization

0.2605 0.2482 0.2433 0.2435

0.30 0.30 0.30 0.30

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Fig. 10.9 The macrostructure reduced displacements map for the best design vector

10.6 Conclusions The paper presents the application of bioinspired optimization techniques in multiscale modeling. The three presented algorithms can be used in optimization procedure. The use of all presented algorithms allows to choose the best solution. The minimal value of the objective function was found by AIS in considered optimization problem. The performance of the algorithms depends on objective function shape and there is not the best general algorithm. The application of different algorithms for optimization problem can increase the probability of finding global optimum. Acknowledgements. The research is financed from the Polish science budget resources in the years 2008-2010 as the research project.

References [1] Burczy´nski, T., Ku´s, W.: Microstructure optimisation and identification in multi-scale modeling. In: ECCOMAS Multidisciplinary Jubilee Symposium on New Computational Challenges in Material, Structures and Fluids, Computational Methods in Applied Sciences, vol. 14, pp. 169–181. Springer, Heidelberg (2009) [2] de Castro, L.N., Timmis, J.: Artificial Immune Systems as a Novel Soft Computing Paradigm. Soft Computing 7(8), 526–544 (2003)

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[3] Kennedy, J., Eberhart, R.C., Shi, Y.: Swarm Intelligence. Morgan Kaufmann Publishers, San Francisco (2001) [4] Kouznetsova, V.G.: Computational homogenization for the multi-scale analysis of multiphase materials. Ph.D. Thesis, TU Eindhoven (2002) [5] Ku´s, W.: Grid-enabled evolutionary algorithm application in the mechanical optimization problems. Engineering Applications of Artificial Intelligence 20, 629–636 (2007) [6] Michalewicz, Z.: Genetic algorithms + data structures = evolutionary algorithms. Springer, Berlin (1996) [7] MSC. Nastran, Users guide. (2005) [8] Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method for Solid and Structural Mechanics. Elsevier, Oxford (2005)

Chapter 11

Sensor Network Design for Spatio–Temporal Prediction of Distributed Parameter Systems Dariusz Uci´nski

Abstract. An activation strategy of pointwise sensors used for estimating unknown parameters in models described by partial differential equations is addressed. In contrast to the common approach based on parameter-space criteria, attention is paid here to a criterion in output space, which is of interest if the purpose of parameter estimation is to accurately predict system outputs. The problem is formulated as the determination of the density of gaged sites so as to minimize the adopted design criterion, subject to inequality constraints incorporating a maximum allowable sensor density in a given spatial domain. The search for the optimal solution is performed using a simplicial decomposition algorithm. The use of the proposed approach is illustrated by a numerical example involving sensor selection for a two-dimensional diffusion process.

11.1 Introduction Building models of dynamic systems is a key activity in process engineering. Modern process control frequently demands using very accurate models in which spatial dynamics has to be included in addition to the temporal one. The processes in question are often termed distributed parameter systems (DPSs) and they are described by partial differential equations. One of the basic and most important questions in DPSs is parameter estimation, which refers to the determination from observed data of unknown parameters in the system model such that the predicted response of the model is close, in some well-defined sense, to the process observations made by some suitable collection of sensors termed the measurement or observation system. A major difficulty here is related to the impossibility to measure process variables over the entire spatial domain. Moreover, the measurements are inexact by virtue Dariusz Uci´nski Institute of Control and Computation Engineering, University of Zielona Góra, ul. Licealna 9, 65–417 Zielona Góra, Poland e-mail: [email protected]

M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 193–207. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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of inherent errors of measurement associated with transducing elements and also because of the measurement environment. The inability to take distributed measurements of process states leads to the question of where to locate sensors so that the information content of the resulting signals with respect to the distributed state and PDE model be as high as possible. This is an appealing problem since in most applications these locations are not prespecified and therefore provide design parameters. The location of sensors is not necessarily dictated by physical considerations or by intuition and, therefore, some systematic approaches should still be developed in order to reduce the cost of instrumentation and to increase the efficiency of identifiers. An example which is particularly stimulating in the light of the results reported in this note constitutes optimization of air quality monitoring networks. One of the tasks of environmental protection systems is to provide expected levels of pollutant concentrations. But to produce such a forecast, a smog prediction model is necessary, which is usually chosen in the form of an advection-diffusion partial differential equation. Its calibration requires parameter estimation, e.g., the unknown spatially-varying turbulent diffusivity tensor should be identified based on the measurements from monitoring stations whose number can be quite large. Then designers must address the question of how to optimize sensor locations in order to obtain the most precise model. This issue acquires especially vital importance in the context of recent advances in distributed sensor networks [2]. Over the past years, applications have stimulated laborious research on the development of strategies for efficient sensor placement (for reviews, see papers [8, 16] and comprehensive monographs [14, 13, 12]). Nevertheless, although the need for systematic methods was widely recognized, most techniques communicated by various authors usually rely on exhaustive search over a predefined set of candidates and the combinatorial nature of the design problem is taken into account very occasionally [16]. An approach to alleviate problems with the combinatorial nature of sensor selection consists in operating on the spatial density of sensors (i.e., the number of sensors per unit area), rather than on the sensor locations. It is proved reasonable for a sufficiently large number of sensors and potential solutions would be satisfactory for many processes. The underlying idea has its origins in the concept of replication-free designs in spatial statistics, cf. [4], and over the past few years, successful attempts have been made at adapting it for use in problems ranging from maximization of observability [14] to optimization of measurement strategies for scanning observations [14, Ch. 4.1.1]. Consequently, convenient and efficient mathematical tools of convex programming theory made it possible to derive interesting characterizations of optimal solutions. All the same, some generalizations are still expected in this aspect of the sensor location problem, particularly regarding computational procedures which were left without due consideration. Although some simple exchange-type algorithms have been proposed to determine optimal designs [14], they are only locally convergent and still need a thorough refinement. In most existing approaches sensor locations are determined in experiments performed for the most accurate determination of parameter values which may have

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some physical significance. By contrast, the reliability of model predictions is the main focus of interest here. This is because in many applications, especially when a control scheme is to be built, the accuracy of model predictions is more important than the accuracy of model parameters, because the ultimate objective in modeling is the prediction or forecast of the system states. The aim of the research reported here was thus to develop a computational algorithm to determine optimum sensor densities which, while being independent of a particular model of the dynamic DPS in question, would be versatile enough to cope with large-scale monitoring networks. For that purpose, the original problem is reduced to minimization of the mean variance of the prediction over the set of all convex combinations of a finite number of nonnegative definite matrices subject to additional box constraints on the weights of those combinations. Then simplicial decomposition is applied which is a simple and direct method for dealing with large-scale convex optimization problems [7, 9]. The decomposition iterates by alternately solving a linear programming subproblem within the set of all feasible points and a nonlinear master problem within the convex hull of a subset previously generated points. As a result, an uncomplicated computational scheme is obtained which can be easily implemented without resorting to sophisticated numerical software. This extends the results of [15]where D-optimal designs were considered. Notation. Throughout the paper, R+ and R++ stand for the sets of nonnegative and positive real numbers, respectively. We use Sm to denote the set of symmetric m × m matrices, The curled inequality symbol (resp. ) is used to denote generalized inequalities. More precisely, between vectors, it represents a componentwise inequality, and between symmetric matrices, it represents the Löwner ordering: given A, B ∈ Sm , A B means that A − B is nonnegative definite (resp. positive definite). The symbols 1 and 0 denote vectors whose all components are one and zero, respectively. We call a point of the form α1 a1 + · · · + α a , where α1 + · · · + α = 1 and αi ≥ 0, i = 1, . . . , , a convex combination of the points a1 , . . . , a . Given a set of points A, co(A) stands for its convex hull, i.e., the set of all convex combinations of elements ofA. The probability (or canonical) simplex in Rn is defined as Sn = co e1 , . . . , en where e j is the usual unit vector along the j-th coordinate of Rn .

11.2 Optimal Sensor Location Problem 11.2.1 Quantification of Prediction Accuracy Let y = y(x,t; a) denote the scalar state of a given DPS at a spatial point x ∈ Ω ⊂ Rd and time instant t ∈ T = [0,t f ], t f < ∞. Here a represents an unknown constant parameter vector which must be estimated using observations of the system. In what follows, we consider the observations provided by N stationary pointwise sensors, namely

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zmj (t) = y(x j ,t; a) + ε (x j ,t),

t ∈ T,

(11.1)

zmj (t)

where is the scalar output and x j ∈ X stands for the location of the j-th sensor ( j = 1, . . . , N), X signifies the part of the spatial domain Ω where the measurements can be made and ε (x j ,t) denotes the measurement noise. A customary assumption is that the measurement noise is zero-mean, Gaussian, spatially uncorrelated and white [8]. We assume that the parameter estimate aˆ , defined as the solution to the usual output least-squares formulation of the parameter estimation problem, is to provide a basis for prediction of certain variables depending on spatial location and/or time. Let the solution to the prediction problem in context be a scalar quantity q = q(x,t; a). We are interested in selecting the sensor configurations in such a way as to maximize the accuracy of q in a given compact spatio-temporal domain Q = X × T . Clearly, in order to compare different configurations, a quantitative measure of the ‘goodness’ of particular configurations is required. A logical approach is to choose a measure related to the expected accuracy of prediction. For a given (x,t) ∈ Q, the variance of q obtained by a first-order expansion around a preliminary estimate a0 of a has the form var(q(x,t; aˆ )) = E (q(x,t; a) − q(x,t; aˆ ))2 T (11.2) ≈ ∇a q(x,t; a0 ) cov(ˆa)∇a q(x,t; a0 ) 0 T −1 0 ∼ ∇a q(x,t; a ) M ∇a q(x,t; a ) where we write ∇a q for the gradient of q with respect to a. It is customary to choose a0 as a nominal value of a or a result of a preliminary experiment. As for cov(ˆa), we used the fact that it can be approximated by the inverse of the Fisher Information Matrix (FIM) whose normalized version can be written down as [10] M=

1 Nt f

N

∑

tf

g(x j ,t)gT (x j ,t) dt,

(11.3)

j=1 0

/ where g(x,t) = ∇a y(x,t; a)/a=a0 stands for the so-called sensitivity vector. A criterion may now be set up such that the ‘optimal’ sensor positions x j minimize var(q(x,t; aˆ )) over Q. Based on the suggestions of (Fedorov and Hackl, 1997, p.25), in the sequel the following V-optimality criterion is considered:

Ψ [M] = where C=

Q

Q

var(q(x,t; aˆ )) dx dt = trace CM−1

T ∇a q(x,t; a0 ) ∇a q(x,t; a0 ) dx dt

(11.4)

(11.5)

The introduction of an optimality criterion renders it possible to formulate the sensor location problem as an optimization problem: Minimize Ψ M(x1 , . . . , xN ) with respect to x j , j = 1, . . . , N belonging to the admissible set X .

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11.2.2 Problem of Finding Optimal Sensor Densities When the number of sensors N is large, which becomes a common situation in applications involving sensor networks, the optimal sensor location problem becomes extremely difficult from a computational point of view. In order to overcome this predicament, we can operate on the spatial density of sensors (i.e. the number of sensors per unit area), rather than on the sensor locations. The density of sensors over X can be approximately described by a probability measure ξ (dx) on the space (X, B), where B is the σ -algebra of all Borel subsets of X . Feasible solutions of this form make it possible to apply convenient and efficient mathematical tools of convex programming theory. As regards the practical interpretation of the so produced results (provided that we are in a position to calculate at least their approximations), one possibility is to partition X into non-overlapping subdomains Xi of relatively small areas and then to allocate to each of them the number 0 1 Ni = N ξ (dx) (11.6) Xi

of sensors (here ρ ! is the smallest integer greater than or equal to ρ ). Accordingly, we define the class of admissible designs as all probability measures ξ over X which are absolutely continuous with respect to the Lebesgue measure and satisfy by definition the condition

ξ (dx) = 1.

(11.7)

X

Consequently, we replace (11.3) by M(ξ ) =

G(x) ξ (dx),

(11.8)

X

where G(x) =

tf

1 tf

g(x,t)gT (x,t) dt.

0

The integration in (11.7) and (11.8) is to be understood in the Lebesgue-Stieltjes sense. This leads to the so-called continuous designs which constitute the basis of the modern theory of optimal experiments [5, 17]. We impose the crucial restriction that the density of sensor allocation must not exceed some prescribed level. For a design measure ξ (dx) this amounts to the condition ξ (dx) ≤ ω (dx), (11.9) where ω (dx) signifies the maximal possible ‘number’ of sensors per dx [5, 14, 13] such that ω (dx) ≥ 1. (11.10) X

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Consequently, we are faced with the following optimization problem: Problem 1. Find

ξ = arg min Ψ (M(ξ )) ξ ∈Ξ (X)

(11.11)

subject to

ξ (dx) ≤ ω (dx),

(11.12)

where Ξ (X ) denotes the set of all probability measures on X . The design ξ above is then said to be a (Ψ , ω )-optimal design [5]. Let us make the following assumptions: (A1) X is compact, (A2) g ∈ C(X × T ; Rm ), (A3) There exists a finite real α such that ξ : Ψ [M(ξ )] < α = Ξ2 (X ) = 0, / (A4) ω (dx) is atomless, i.e., for any Δ X ⊂ X there exists a Δ X ⊂ Δ X such that Δ X

ω (dx) <

ΔX

ω (dx).

(11.13)

In what follows, we write Ξ (X) for the collection of all the design measures which satisfy the requirement $ ω (Δ X) for Δ X ⊂ supp ξ , (11.14) ξ (Δ X ) = 0 for Δ X ⊂ X \ supp ξ . (Recall that the support of a measure ξ is defined as the closed set supp ξ = X \ ) {G : ξ (G) = 0, G – open}, cf. [11, p.80].) Given a design ξ , we will say that the function ψ ( · , ξ ) defined by

ψ (x, ξ ) =

1 tf

tf

gT (x,t)M−1 (ξ )CM−1 (ξ )g(x,t) dt

(11.15)

0

separates sets X1 and X2 with respect to ω (dx) if for any two sets Δ X1 ⊂ X1 and Δ X2 ⊂ X2 with equal non-zero measures we have Δ X1

ψ (x, ξ ) ω (dx) ≥

Δ X2

ψ (x, ξ ) ω (dx).

(11.16)

We can now formulate the folowing characterization of (Ψ , ω )-optimal designs, see [13, 14].

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Theorem 1 ([14, 13]). Let Assumptions (A1)–(A4) hold. Then: (i) There exists an optimal design ξ ∈ Ξ (X), and (ii) A necessary and sufficient condition for ξ ∈ Ξ (X ) to be (Ψ , ω )-optimal is that ψ ( · , ξ ) separates X = supp ξ and its complement X \ X with respect to the measure ω (dx). From a practical point of view, the above result means that at all the support points of an optimal design ξ the mapping ψ ( · , ξ ) should be greater than anywhere else, i.e., preferably supp ξ should coincide with maximum points of ψ ( · , ξ ). In practice, this amounts to allocating observations to the points at which we know least of all about the system response.

11.3 Reduction to a Weight Optimization Problem We are now confronted with the question of how to discretize Problem 1 to make it tractable by a computer. In what follows, the basic idea is to make use of the partition of X into a union of small disjoint subdomains Xi , i = 1, . . . , n, i.e., X = )n i=1 Xi , as discussed in Section 11.2.2. Observe that a measure ξ ∈ Ξ (X ) assigns each subdomain Xi a weight pi = ξ (X). Owing to (11.6), the knowledge of the pi ’s suffices to determine an optimal distribution of sensor nodes. Assuming that the variation of G( · ) over each Xi is negligible (this can be achieved by constructing a sufficiently fine partition of X), we have n

M(ξ ) ≈ M(p) := ∑ pi Mi ,

(11.17)

i=1

where p = [p1 , . . . , pn ]T , Mi = G(xi ), xi being an arbitrary point in Xi (e.g., the centre of gravity). Thus, setting b = [ω (X1 ), . . . , ω (Xn )]T , we arrive at the following approximation to Problem 1: Problem 2. Find a vector of weights p to minimize Φ [M(p)] = trace CM−1 (p)

(11.18)

subject to 0 " p " b, T

1 p = 1.

(11.19) (11.20)

In what follows, we let P be the bounded polyhedral set of feasible weights defined by (11.19) and (11.20). Moreover, without restriction of generality, we shall further assume that b 0. index Φ is convex over the canonical simplex Sn = Notenthat Tthe performance p ∈ R+ | 1 p = 1 . What is more, it is differentiable at points in Sn yielding nonsingular FIMs, with φ (p) := ∇Φ (p) given by

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T φ (p) = − trace M(p)−1 CM(p)−1 M1 , . . . , − trace M(p)−1 CM(p)−1 Mn . (11.21) Accordingly, numerous computational methods can potentially be employed for solving Problem 2, e.g., the conditional gradient method or a gradient projection method. Unfortunately, for large n, these algorithms may lead to unsatisfactory computational times. In what follows, it will be shown how simplicial decomposition can employed to build a very simple and efficient computational scheme for solving Problem 2.

11.4 Simplicial Decomposition for Problem 2 11.4.1 Algorithm Model Simplicial decomposition (SD) proved extremely useful for large-scale pseudoconvex programming problems encountered, e.g., in traffic assignment or other network flow problems [9]. In its basic form, it proceeds by alternately solving linear and nonlinear programming subproblems, called the column generation problem (CGP) and the restricted master problem (RMP), respectively. In the RMP, the original problem is relaxed by replacing the original constraint set P with its inner approximation being the convex hull of a finite set of feasible solutions. In the CGP, this inner approximation is improved by incorporating a point in the original constraint set that lies furthest along the antigradient direction computed at the solution of the RMP. This basic strategy has been discussed in numerous references [7, 6, 9], where possible extensions have also been proposed. A marked characteristic of the SD method is that the sequence of solutions to the RMP tends to a solution to the original problem in such a way that the objective function strictly monotonically approaches its optimal value. The SD algorithm may be viewed as a form of modular nonlinear programming, provided that one has an effective computer code for solving the RMP, as well as access to a code which can take advantage of the linearity of the CGP. A principal aim of this paper is to show that this is the case within the framework of Problem 2. What is more, since we deal with minimization of the convex function Φ over a bounded polyhedral set P, this will automatically imply the convergence of the resulting SD scheme in a finite number of RMP steps [7]. Tailoring the SD scheme to our needs, we obtain Algorithm 1. In the sequel, its consecutive steps will be discussed in turn.

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Algorithm 1 Algorithm model for solving Problem 2 via simplicial decomposition. Step 0: (Initialization) Guess an initial solution p(0) ∈ P such that M(p(0) ) is nonsingular. Set I = 1, . . ., n , Q(0) = (0) and k = 0. p Step 1: (Termination check) Set (k) (k) Iub = i ∈ I | pi = bi ,

(k) (k) Iim = i ∈ I | 0 < pi < bi ,

If

⎧ ⎪ ⎨≤ λ φi (p(k) ) = λ ⎪ ⎩ ≥λ

(k) (k) Ilb = i ∈ I | pi = 0 . (11.22)

(k)

if i ∈ Iub , (k)

if i ∈ Iim ,

for some λ ∈ R+ , then STOP and p(k) is optimal. Step 2: (Solution of the column generation subproblem) Compute q(k+1) = arg min φ (p(k) )T p p∈P

and set Step 3: Find

(11.23)

(k)

if i ∈ Ilb

Q(k+1) = Q(k) ∪ q(k+1) .

(11.24)

(11.25)

(Solution of the restricted master subproblem) p(k+1) = arg

min

p∈co(Q(k+1) )

trace CM−1 (p)

(11.26)

and purge Q(k+1) of all extreme points with zero weights in the resulting expression of p(k+1) as a convex combination of elements in Q(k+1) . Increment k by one and go back to Step 1.

11.4.2 Termination Criterion for Algorithm 1 In the original SD setting, the criterion for terminating the iterations is checked only after solving the column generation problem. The computation is then stopped if the current point p(k) satisfies the “basic” optimality condition of nondecrease, to first order, in performance measure value in the whole constraint set, i.e., min φ (p(k) )T (p − p(k) ) ≥ 0. p∈P

(11.27)

The condition (11.23) is less costly in terms of the number of floating-point operations. It results from the following characterization of p which has the property that Φ (p ) = minp∈P Φ (p). Proposition 1. Suppose that the matrix M(p ) is nonsingular for some p ∈ P. The vector p constitutes a global minimum of Φ over P if, and only if, there exists a number λ such that

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⎧ ⎪ ⎨≤ λ φi (p ) = λ ⎪ ⎩ ≥ λ

if pi = bi , if 0 < pi < bi , if pi = 0

(11.28)

for i = 1, . . . , n. This result follows from direct application of Lemma 1 (see the Appendix) after setting f (p) = Φ (M(p)).

11.4.3 Solution of the Column Generation Subproblem Setting c = φ (p(k) ), in Step 2 we deal with the linear programming problem minimize cT p subject to p ∈ P.

(11.29)

The following assertion is a direct consequence of Lemma 1 in the Appendix. Proposition 2. A vector q ∈ P constitutes a global solution to the problem (11.29) if, and only if, there exists a scalar ρ such that ⎧ ⎪ ⎨≤ ρ if qi = bi , ci = ρ if 0 < qi < bi , (11.30) ⎪ ⎩ ≥ ρ if qi = 0 for i = 1, . . . , n. We thus see that, in order to solve (11.29), it is sufficient to pick the consecutive smallest components ci of c and set the corresponding weights qi as their maximal allowable values bi . The process is repeated until the sum of the assigned weights exceeds one. Then the value of the last weight which was set in this manner should be corrected so as to satisfy the constraint (11.20) and the remaining (i.e., unassigned) weights are set as zeros. This straightforward scheme is implemented as Algorithm 2. Note that its correctness requires satisfaction of the condition b " 1, which is by no means restrictive.

11.4.4 Solution of the Restricted Master Subproblem Suppose that in the (k + 1)-th iteration of Algorithm 1, we have Q(k+1) = q1 , . . . , qr ,

(11.34)

possibly with r < k + 1 owing to the built-in deletion mechanism of points in Q( j) , 1 ≤ j ≤ k, which did not contribute to the convex combinations yielding the corresponding iterates p( j) . Step 3 involves minimization of (11.18) over

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Algorithm 2 Algorithm model for solving the column generation subproblem. Step 0: (Initialization) Set j = 0 and v(0) = 0. Step 1: (Sorting) Sort the elements of c in nondecreasing order, i.e., find a permutation π on the index set I = 1, . . ., n such that (11.31) cπ (i) ≤ cπ (i+1) , i = 1, . . ., n − 1 Step 2:

(Identification of nonzero weights)

Step 2.1:

If v( j) + bπ ( j+1) < 1 then set v( j+1) = v( j) + bπ ( j+1) .

(11.32)

Otherwise, go to Step 3. Step 2.2: Increment j by one and go to Step 2.1. Step 3: Set

(Form the ultimate solution) ⎧ ⎪ ⎨bπ (i) qπ (i) = 1 − v( j) ⎪ ⎩ 0

$ co(Q

(k+1)

)=

for i = 1, . . ., j, for i = j + 1, for i = j + 2, . . ., n.

% / / T ∑ w j q j / w 0, 1 w = 1 .

(11.33)

r

(11.35)

j=1

From the representation of any p ∈ co(Q(k+1) ) as r

p=

∑ w jq j,

(11.36)

j=1

or, in component-wise form, r

pi =

∑ w j q j,i ,

i = 1, . . . , n,

(11.37)

j=1

q j,i being the i-th component of q j , it follows that n

M(p) = ∑ pi Mi = i=1

r

n

∑ w j ∑ q j,iMi

j=1

i=1

r

=

∑ w j M(q j ).

(11.38)

j=1

From this, we see that the RMP can equivalently be formulated as the following problem: Problem 3. Find the sequence of weights w ∈ Rr to minimize Ψ (w) = trace CH−1 (w)

(11.39)

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D. Uci´nski

subject to the constraints 1T w = 1, w 0, where

(11.40) (11.41)

r

H(w) =

∑ w jH j,

H j = M(q j ),

j = 1, . . . , r.

(11.42)

j=1

This formulation has captured close attention in convex optimization. Specifically, the efficient algorithm proposed in [1] has been employed here.

11.5 Computer Example Consider the two-dimensional diffusion equation

∂y = ∇ · (μ ∇y) in Ω × T ∂t

(11.43)

where Ω ⊂ R2 is the spatial domain with boundary Γ which is shown in Fig. 11.1 and T = (0, 1). The assumed form of the diffusion coefficient is

μ (x) = a1 + a2 ρ (x),

ρ (x) = x1 + x2 ,

(11.44)

where a1 and a2 are unknown parameters which have to be estimated based on the measurements from a large-scale sensor network. Throughout the design, a01 = 0.1 and a02 = 0.3 are to be used as nominal values of a1 and a2 , respectively. The PDE (11.43) is supplemented with the initial and boundary conditions y(x, 0) = 5

in Ω ,

y(x,t) = 5(1 − t) on Γ × T .

(11.45) (11.46)

The upper bound in (11.9) is given by ω (A) = c|A|/|X | for any Borel subset of X , where c ≥ 1 is fixed, i.e., ω corresponds to a uniform distribution on X , and |A| stands for the area (i.e., the Lebesgue measure) of A. The M ATLAB PDE toolbox [3] was used to generate the triangular mesh T of 1811 nodes and n = 3440 triangles shown in Fig. 11.1. The values of the sensitivities g = col[g1 , g2 ] are defined as solutions to the following system of PDEs [14, 13]: ⎧ ∂y ⎪ ⎪ = ∇ · (μ ∇y), ⎪ ⎪ ⎪ ∂t ⎪ ⎨ ∂ g1 (11.47) = ∇ · ∇y + ∇ · (μ ∇g1), ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ g2 = ∇ · (ρ ∇y) + ∇ · (μ ∇g2), ∂t

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in which the first equation constitutes the original state equation and the second and third equations result from its differentiation with respect to a1 and a2 , respectively. The initial and Dirichlet boundary conditions for the sensitivity equations are homogeneous. We numerically solved (11.47) using some routines of the M ATLAB PDE toolbox and stored matrices G(xi ) computed based on g1 and g2 interpolated at the gravity centres of individual triangles, cf. Appendix I in [14] for details. 1

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(c) Fig. 11.1 Support of the V-optimal design measure: (a) c = 2, 50% coverage of X, (b) c = 4, 25% coverage of X, (c) c = 10, 10% coverage of X.

The triangular mesh T was also used when solving Problem 2 for Q = X × T = Ω × T . All computations were performed using a low-cost laptop (Intel Centrino Duo T9300, 3 GB RAM) running Windows Vista Home Premium and Matlab 7 (R2008a). Simplicial decomposition implemented in accordance with Algorithm 1 lead to solutions in which the weights pi associated with the respective triangles approximately satisfy the ‘bang-bang’ principle formulated in Theorem 1: they are either

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D. Uci´nski

zero, or equal to the corresponding upper bound bi = c|Xi |/|X |. Thus, the support of the optimal measure ξ covers approximately c−1 · 100% of the area of |X |, cf. Fig. 11.1.

11.6 Concluding Remarks Note that some refinements may still improve the algorithm performance. First of all, the basic simplicial decomposition scheme outlined here can be replaced by the so-called restricted simplicial decomposition [6] which is based on the observation that a particular feasible solution, such as the optimal one, can be represented as the convex combination of an often much smaller number of extreme points than that implied by Carathéodory’s Theorem. Apart from that, some improvements aimed at removing nonoptimal support points can be incorporated in the restricted master problem to speed up its solution. But even the basic scheme proposed here performs well in practice and the aforementioned refinements may be necessary only when the time of computations is a truly critical factor. Acknowledgements. The research was supported by the Polish Ministry of Science and Higher Education under Grant N N519 2971 33.

Appendix Given a vector b ∈ Rn++ such that 1T b ≥ 1, consider the problem minimize

f (p)

subject to p ∈ P,

(11.48)

where we assume throughout that:

n n (a) P = S n ∩ B for Sn being the probability simplex in R and B = p ∈ R | 0 " p"b . (b) The function f : Sn → R is convex and continuously differentiable over P. The Karush-Kuhn-Tucker optimality conditions yield the following:

Lemma 1. A vector p constitutes a global minimum of the constrained problem (11.48) if, and only if, there exists a number λ such that ⎧ ⎪≤ λ if pi = bi , ∂ f (p ) ⎨ (11.49) = λ if 0 < pi < bi , ∂ pi ⎪ ⎩ ≥ λ if pi = 0 for i = 1, . . . , n.

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References [1] Botkin, N.D., Stoer, J.: Minimization of convex functions on the convex hull of a point set. Mathematical Methods of Operations Research 62(2), 167–180 (2005) [2] Cassandras, C.G., Li, W.: Sensor networks and cooperative control. European Journal of Control 11(4–5), 436–463 (2005) [3] COMSOL AB: Partial Differential Equation Toolbox for Use with Matlab. User’s Guide. The MathWorks, Inc., Natick, MA (1995) [4] Fedorov, V.V.: Optimal design with bounded density: Optimization algorithms of the exchange type. Journal of Statistical Planning and Inference 22, 1–13 (1989) [5] Fedorov, V.V., Hackl, P.: Model-Oriented Design of Experiments. Lecture Notes in Statistics. Springer, New York (1997) [6] Hearn, D.W., Lawphongpanich, S., Ventura, J.A.: Finiteness in restricted simplicial decomposition. Operations Research Letters 4(3), 125–130 (1985) [7] von Hohenbalken, B.: Simplicial decomposition in nonlinear programming algorithms. Mathematical Programming 13, 49–68 (1977) [8] Kubrusly, C.S., Malebranche, H.: Sensors and controllers location in distributed systems — A survey. Automatica 21(2), 117–128 (1985) [9] Patriksson, M.: Simplicial decomposition algorithms. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, vol. 5, pp. 205–212. Kluwer Academic Publishers, Dordrecht (2001) [10] Rafajłowicz, E.: Optimum choice of moving sensor trajectories for distributed parameter system identification. International Journal of Control 43(5), 1441–1451 (1986) [11] Rao, M.M.: Measure Theory and Integration. John Wiley & Sons, New York (1987) [12] Sun, N.Z.: Inverse Problems in Groundwater Modeling. Theory and Applications of Transport in Porous Media. Kluwer Academic Publishers, Dordrecht (1994) [13] Uci´nski, D.: Measurement Optimization for Parameter Estimation in Distributed Systems. Technical University Press, Zielona Góra (1999) [14] Uci´nski, D.: Optimal Measurement Methods for Distributed-Parameter System Identification. CRC Press, Boca Raton (2005) [15] Uci´nski, D.: An algorithm to configure a large-scale monitoring network for parameter estimation of distributed systems. In: Proceedings of the European Control Conference 2007, Kos, Greece, July 2–5 (2007); Published on CD-ROM [16] van de Wal, M., de Jager, B.: A review of methods for input/output selection. Automatica 37, 487–510 (2001) [17] Walter, É., Pronzato, L.: Identification of Parametric Models from Experimental Data. Communications and Control Engineering. Springer, Berlin (1997)

Part III

Multiscale Methods

Chapter 12

A Multiscale Molecular Dynamics / Extended Finite Element Method for Dynamic Fracture Pascal Aubertin, Julien Réthoré, and René de Borst

Abstract. A multiscale method is presented which couples a molecular dynamics approach for describing fracture at the crack tip with an extended finite element method for discretizing the remainder of the domain. After recalling the basic equations of molecular dynamics and continuum mechanics the discretization is discussed for the continuum subdomain where the partition-of-unity property of finite element shape functions is used, since in this fashion the crack in the wake of its tip is naturally modelled as a traction-free discontinuity. Next, the zonal coupling method between the atomistic and continuum models is described, including an assessment of the energy transfer between both domains for a one-dimensional problem. It is discussed how the stress has been computed in the atomistic subdomain, and a two-dimensional computation is presented of dynamic fracture using the coupled model. The result shows multiple branching, which is reminiscent of recent results from simulations on dynamic fracture using cohesive-zone models.

12.1 Introduction Modern research into fracture commences with the seminal work of Griffith [1]. Later, Irwin [2] and Rice [3] established the relation between the stress intensity factors and the energy release rate, and gave linear elastic fracture mechanics a firm Pascal Aubertin Université de Lyon, CNRS INSA-Lyon, LaMCoS UMR 5259, France e-mail: [email protected] Julien Réthoré Université de Lyon, CNRS INSA-Lyon, LaMCoS UMR 5259, France e-mail: [email protected] René de Borst Eindhoven University of Technology, Department of Mechanical Engineering, P.O. Box 513, 5600 MB Eindhoven, Netherlands e-mail: [email protected]

M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 211–237. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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basis. However, linear elastic fracture mechanics only applies to crack-like flaws in an otherwise linear elastic solid and when the singularity associated with that flaw is characterized by a non-vanishing energy release rate. The fracture and any dissipative processes must also be confined to a small region in the vicinity of the crack tip. Linear elastic fracture mechanics provides a challenge to standard finite element approaches, since the polynomials that are conventionally applied in finite element methods cannot easily capture the stress singularity at the crack tip which is predicted in linear elastic fracture mechanics. However, methods have been developed to overcome this difficulty, e.g. the so-called quarter-point elements [4, 5], and more recently, the advent of meshless methods [6, 7] and partition-of-unity based finite element methods [8, 9, 10] have provided elegant solutions to incorporate stress singularities in domain-based discretization methods. On the other hand, boundary integral methods can naturally incorporate such singularities [11]. When the region in which the separation and dissipative process take place is not small compared to a structural dimension, but any nonlinearity is confined to a surface emanating from a classical crack tip, i.e. one with a non-vanishing energy release rate, cohesive zone models as introduced by Barenblatt [12] and Dugdale [13] apply. The cohesive zone approach was extended by Hillerborg et al. [14] and Needleman [15] to circumstances where: (i) an initial crack-like flaw need not be present or, if one is present, it need not be associated with a non-vanishing energy release rate; and (ii) non-linear deformation behaviour may occur over an extended volume. Initially, cohesive-zone models were incorporated in finite element methods via special-purpose interface elements [16, 17], but more recently, partition-ofunity finite element methods have shown to be very amenable to the incorporation of cohesive-zone models, e.g. [18, 19, 20]. In particular, they naturally enable crack propagation, also in dynamics [21, 22, 23, 24, 25] and in multi-phase continua [26]. In spite of the power of the cohesive-zone approach, and its wide applicability on a range of scales, it remains a phenomenological approach. Probably, quantum mechanics is physically the most appropriate theory to describe fracture, but the difficulties to relate quantum mechanics to continuum mechanics, e.g. via Density Functional Theory [30, 31] presently seem insurmountable. One scale of observation higher is to use Molecular Dynamics to describe fracture processes from a more fundamental physics point of view. Indeed, researchers have recently used this approach to describe fracture, e.g. [27, 28, 29]. A disadvantage of the approach is that it is computationally demanding. For this reason multi-scale approaches have been introduced, in fracture [32], as well as in plasticity [33], in which only a part of the body is analysed using molecular dynamics, while the remaining part of the body is modelled using continuum mechanics and discretized using a finite element method. This manuscript furthers along this line and combines molecular dynamics for modelling the fracture process at the crack tip with an extended finite element method (XFEM), where the partition-of-unity property of the polynomial shape functions is exploited to model the crack in the wake of the tip as a traction-free discontinuity. It is noted that recently another approach has been published that couples atomistics

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and extended finite elements [34], but the current paper makes a further advancement in that it includes dynamic loadings. A major issue in multi-scale approaches as discussed above is the accurate coupling of both domains, especially when different descriptions are assumed on either domain. While the coupling can, in principle, either be achieved at a discrete interface, or on a zone of a finite size (overlap or zonal coupling), it is believed that zonal approaches, which include the Arlequin method [35, 36], the bridging domain method [37, 38, 39, 40], discrete-to-continuum bridging [42], the discontinuous enrichment method [41], and bridging scale decomposition [43, 44] enable a more gradual transition from one domain to the other. The ability of a gradual transition is especially important for highly dissimilar domains and when wave propagation phenomena are considered, where preservation of the energy and avoiding spurious reflections when a wave exits one domain and enters the other can become an issue. Inspired by earlier work by Ben Dhia and Rateau [35] and Xiao and Belytschko [37] we have chosen a weak coupling between the models in the two adjacent domains. This paper is organized as follows. First, we briefly list the equations of molecular dynamics and ways to ensure equilibrium of the atomistic domain before starting the computation that involves dynamic propagation of an existing, starter crack. This is followed by a succinct recapitulation of the governing equations of continuum mechanics, both in the strong and the weak forms. The discretization of the continuum subdomain is carried out using the extended finite element method, where the partition-of-unity property of finite element shape functions is used to model the traction-free discontinuity in the wake of the crack tip. Subsequently, it is discussed how both domains can be coupled, see also [45] and an analysis is presented of the energy conservation properties of the coupling scheme. The paper concludes with a full two-dimensional coupled analysis of dynamic crack propagation which shows multiple branching and suggests the formation of dislocations, which shows similarities with recent simulations on dynamic fracture using cohesive-zone models [17, 25].

12.2 Molecular Dynamics 12.2.1 Governing Equations in the Atomistic Domain For the discrete domain, i.e. Ωm , we build a grid of Na atoms, and, accordingly, the initial value problem in this domain can be written as: For 1 ≤ i ≤ Na (t) and t ∈ [0; T ] , given the initial conditions d (0) , d˙ (0) , find (d, f) ∈ D ad × F ad such that: mi d¨ i = fi (12.1) with mi the mass of atom i and:

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D ad = d = di (t) 1≤i≤Na , ∀t ∈ [0, T ] F ad

∇i U d (t) = f = fi (t) = −∇

1≤i≤Na

, ∀t ∈ [0, T ]

(12.2)

from where it transpires that the interatomic forces are derived from a potential energy U . d and f assemble the discrete displacements di and forces fi of the individual atoms, respectively. The internal energy of the discrete domain can be viewed as the sum of each atomic contribution U j : U = ∑ U j (d)

(12.3)

j

and the force fi acting on atom i can be written as the sum of all elementary forces: fi = −

∂U = ∑ fi j ∂ di j=i

(12.4)

In order to limit the cost of computing such a force, we reduce the summation by only including so-called “nearest” neighbours, within a cut-off radius rc : fi #

∑

ri j
fi j

(12.5)

where ri j is the interatomic distance. In the subdomain Ωm the weak formulation becomes: ∀w∗ ∈ D˙ ad,0 , given the initial conditions d (0) , d˙ (0) , find d ∈ D ad such that: am (d, w∗ ) = 0 with

w∗

(12.6)

the test function, and Na

Na

i=1

i=1

am (d, w∗ ) = ∑ mi d¨ i · w∗i + ∑ ∇ i U (d) · w∗i

12.2.2 Equilibrium of the Atomistic Domain The atomistic domain has to be embedded in a continuum and we cannot consider periodic conditions. Thus, the zone of atoms has an internal boundary with the continuum zone, which has no physical character. Atoms that are on the boundary between the coupling domain and the continuum domain do not have neighbouring atoms in the continuum. This can produce non-physical forces that break the equilibrium. In order to obtain a global stable system that is in equilibrium before loading

12 A Multiscale Method for Dynamic Fracture

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the specimen, we therefore have to introduce so-called “ghost” atoms in the continuum, which act as neighbours for the atoms on the boudaries. Their positions are directly derived from the continuum domain, and their velocities and accelerations do not have to be computed. Their creation merely aims to satisfy the equilibrium of the atomistic domain. For crack propagation we introduce a pre-crack or starter crack in the atomistic domain. For this purpose we remove the atoms along a line that defines the position of the starter crack. This initial crack crosses the continuum as well as the atomistic zone. The removal of atoms causes that the initial lattice is no longer at equilibrium and there is an additional boundary to consider. However, this boundary has a physical character because it stands for a crack surface. As alluded to above, equilibrium in the atomistic zone has to be found prior to applying any load to the specimen. A widely used technique is the use of dynamic relaxation that stabilizes a given configuration with adaptive damping. As shown in [46] the use of dynamic relaxation in molecular simulations can be effective in obtaining an equilibrium state. The solution of the problem is viewed as the steadystate solution of a damped wave equation. The classical equations of motion are augmented by a damping term to give: M¨x + C˙x − ∇x U (x) = 0

(12.7)

When the steady-state solution is found, the acceleration and the velocity are zero. Therefore, the matrices M and C have no physical meaning and can be chosen such that the computation is accelerated in an optimal sense. At each time step of the resolution, the local atomistic stiffness is computed, using the potential energy function, and a critical time step is calculated. Then, either the mass or the damping can be adapted to obtain an optimal convergence. This method has been used for finding equilibrium in the atomistic zone and gives converged results within an acceptable computational time.

12.3 Continuum Model Assuming small deformation gradients for simplicity, the governing equations in the continuum subdomain ΩM can be written in a standard manner as: For x ∈ ΩM (t) and t ∈ [0; T ] , given the initial conditions u (x, 0) , u˙ (x, 0) , find (u, σ ) ∈ U ad × S ad such that:

ρ u¨ = div σ + gd (12.8) with ρ the mass density and u the continuum displacement vector, σ the stress tensor, and gd the body force vector applied in ΩM , subject to the boundary conditions

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3 U ad = u = u (x,t) ∈ H 1 (ΩM ) ; u = ud on ∂1 Ω , ∀t ∈ [0, T ] 6 S ad = σ = K : ∇u (x,t) ∈ L 2 (ΩM ) ; σ · n = Fd on ∂2 Ω , ∀t ∈ [0, T ] (12.9) where K is the fourth-order stiffness tensor, n the outward normal vector to ∂ Ω2 , and ud and Fd the prescribed displacements and tractions at ∂ Ω1 and ∂ Ω2 , respectively. To allow for a discretization of the continuum subdomain we next specify the weak formulation: ∀v∗ ∈ U˙ ad,0 , given the initial conditions u (x, 0) , u˙ (x, 0) , find u ∈ U ad such that: aM (u, v∗ ) = lM (v∗ )

(12.10)

∗

with v the test function, and aM (u, v∗ ) =

ΩM

lM (v∗ ) =

Fig. 12.1 Discretized domain with the coupling region. The dark, “pearshaped” area is the domain where an MD calculation is carried out (Ωm ). The coupling region Ωc consists of elements that are surrounded by a bold line

ρ u¨ · v∗ dΩ +

∂2 Ω

Fd · v∗ dS +

ΩM

∇u : K : ∇v∗ dΩ

ΩM

gd · v∗ dΩ

(12.11)

(12.12)

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12.4 Coupling Scheme 12.4.1 Coupling Functions In order to enable an efficient coupling between the two domains, a coupling zone is defined and a coupling function is computed which is used to obtain the global energy. First a coupling length Lc is chosen that will be the characteristic length of the coupling region. Subsequently, the patch of atoms, Ωm , is included in the continuum at a given position. Atoms at a distance ≤ Lc from the Molecular Dynamics Box (MD-Box) boundary are considered to be in the coupling zone Ωc . The finite elements in this zone are named “coupling elements”. Inside the MD-Box, where only the atomistic model applies, elements are removed. The resulting, discretized domain is shown in Figure 12.1. In principle, each atom must be coupled to the finite element discretization of the underlying continuum. It suffices to couple only a limited number of atoms to the underlying continuum without loss of accuracy. Since this directly affects the size of the coupling matrices that will be derived next, such a limited coupling markedly decreases the computational effort, and therefore, a bigger scale transition becomes possible at reasonable computational costs. In the coupling zone Ωc , we enforce a velocity coupling in a weak sense, and we distribute the energy between both models via a partition of unity [35]. For this purpose we define the following functions, see Figure 12.2:

such that:

α : ΩM → [0, 1]

(12.13)

β : Ωm → [0, 1]

(12.14)

⎧ α (x) = 1 for x ∈ ΩM \Ωc ⎪ ⎪ ⎪ ⎪ ⎨ β (x) = 1 for x ∈ Ωm \Ωc ⎪ ⎪ ⎪ ⎪ ⎩ α (x) + β (x) = 1 for x ∈ Ωc

(12.15)

Fig. 12.2 Partition of unity for the energy distribution

The displacement and velocity fields in the domains ΩM and Ωm have a different nature. In ΩM we have a continuum field, while in Ωm we have a discrete

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field, which is only defined at the geometrical points corresponding to the atoms. Therefore we construct a so-called “mediator space”, denoted by M , on which we project the fields u˙ and d˙ in order to be able to compare them. The nature of M is constrained by the discrete character of the atomistic field. Indeed, its displacements cannot be extrapolated outside the atomic positions if we want to maintain a physical interpretation at the fine scale. Accordingly, M has to be a subspace of the physical atomistic space. More precisely, we project the velocities using an operator Π on a discrete subset Ω c of the atomic positions included in Ωc . Considering that M is built as a Hilbert space, we introduce a scalar product c from M × M onto R. With these definitions we formulate the velocity coupling operator as: ∀μ ∗ ∈ M , c μ ∗ , Π u˙ − Π d˙ =< μ ∗ , Π u˙ − Π d˙ >M (12.16) with c the classical scalar product on M . The global equations are coupled via Lagrange multipliers and can subsequently be written as: ∀ (v∗ , w∗ , μ ∗ ) ∈ U˙ ad,0 × D˙ ad,0 × M , given the initial conditions u (x, 0) , u˙ (x, 0) , d (0) , d˙ (0) , find (u, d, λ ) ∈ U ad × D ad × M such that: aα ,M (u, v∗ ) + aβ ,m (d, w∗ ) + c (λ , Π v∗ − Π w∗) + c μ ∗ , Π u˙ − Π d˙ = lα ,M (v∗ ) (12.17) The modified forms aα ,M , aβ ,m and lα ,M take into account the weighting functions α (x) and β (x), see [45] for details. In the atomistic subdomain the modified form aβ ,m that takes into account the distribution of the energy reads: aβ ,m (d, w∗ ) = w∗ · mβ d¨ − w∗ · fβ with:

βi = β (di )

mβ = [βi δi, j mi ] ,

,

fβ = [fβ ,i ]

(12.18) (12.19)

12.4.2 Discretized Problem In a manner which is by now standard the interpolation of each component of the displacement field is enriched with discontinuous functions in order to properly capture the traction-free discontinuity in the wake of the crack tip: ∀x ∈ ΩM

,

uh (x) =

∑

i∈NM

Ni (x)ui +

∑

Ni (x)HΓd uˆi

(12.20)

i∈Ncut

where Ni are standard finite element shape functions supported by the set of nodes NM included in the discretized domain ΩM . Nodes in Ncut have their support cut by the discontinuity. They hold additional degrees of freedom uˆi corresponding to the discontinuous function HΓd defined by:

12 A Multiscale Method for Dynamic Fracture

HΓd (x) =

x · nΓd x · nΓd

219

(12.21)

with nΓd the normal to the discontinuity Γ . Symbolically, eq. (12.20) can be written as ∀x ∈ ΩM , uh = NT U (12.22) where the matrix N contains the standard interpolation polynomials Ni (x) as well as the discontinuous function HΓd , and the array U contains the displacement degreesof-freedom ui and uˆi . The transition within the domain ΩM between the subdomain where the nodes are “enriched” and the part which has just the standard formulation does not affect the Molecular Dynamics computation other than through the coupling matrices. With the latter symbolic notation the bilinear form aα ,M and the linear form lα ,M become: ¨ + V∗T Kα U aα ,M (uh , v∗h ) = V∗T Mα U

(12.23)

lα ,M (v∗h ) = V∗T Fα

(12.24)

where the term that represents the body forces has been omitted for simplicity, and M=

K=

ρ NT NdΩ

(12.25)

∇NT K∇NdΩ

(12.26)

ΩM

ΩM

the mass and stiffness matrices, respectively. With the standard definition of the scalar product, the coupling term in the continuum can be discretized as follows: c (λ , Π v∗h ) = V∗T CMΛ = V∗T FLM

(12.27)

with CM the continuum coupling matrix. The vector Λ contains the Lagrange multipliers and its size equals the Ω c subset cardinal times the dimension of the space considered. FLM can be regarded as a fictitious force due to the coupling via the Lagrange multipliers. This force has a non-zero value only in the coupling zone Ωc . Using the Lagrange multipliers in the atomistic domain similar to that in the continuum domain: fLm = CmΛ (12.28) the weighted and coupled system (12.17) can be cast in a matrix-vector format: ¨ + Kα U + CMΛ + W∗T mβ d¨ − fβ − CmΛ + V∗T Mα U ˙ − CTm d˙ = V∗T Fα μ ∗T CTM U (12.29) Since this set must hold for any admissible (V∗ , W∗ , μ ∗ ) we finally obtain:

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⎧ ¨ + K α U = F α − CM Λ Mα U ⎪ ⎪ ⎪ ⎪ ⎨ mβ d¨ = fβ + CmΛ ⎪ ⎪ ⎪ ⎪ ⎩ T ˙ CM U = CTm d˙

(12.30)

with (U, d, Λ ) the set of unknowns. Details on the time integration scheme associated with this set of coupled ordinary differential equations are given in Ref. [45].

12.4.3 Time Integration Scheme The time integration scheme relies on discretization with a time step Δ t and has five stages: • Given the quantities at step n, compute the displacements Un+1 and dn+1 , ¨ ∗ and d¨ ∗ , neglecting the Lagrange • Compute the predictive accelerations U n+1 n+1 forces, ˙ ∗ and d˙ ∗ , • Compute the predictive velocities U n+1 n+1 ˙ n+1 and d˙ n+1 by taking into • Adjust these velocities to give the final velocities U account the coupling terms and Lagrange multipliers Λ n+1 . ¨ n+1 and d¨ n+1 . • Adjust the predictive accelerations to give the final accelerations U Below we specify the different steps of this predictor-corrector scheme: • With the displacements, velocities and accelerations at step n, we compute the displacements at step n + 1 as follows: 2 ˙n Δt+ 1 U ¨ Un+1 = Un + U 2 n (Δ t) (12.31) 2 dn+1 = dn + d˙ n Δ t + 1 d¨ n (Δ t) 2

• The predictive accelerations at step n + 1 are computed with the system (12.30) but without coupling terms: ¨∗ ˜ −1 Un+1 = M α (Fα ,n+1 − Kα Un+1 ) (12.32) d¨ ∗n+1 = m−1 β fβ ,n+1 ˜ α , which is stanNote that we use, for the continuum, a lumped mass matrix M dard for explicit time integration. • The predictive velocities are computed with the Newmark scheme: ∗ 1 ¨ ˙ ˙ ¨∗ U Δt n+1 = Un + 2 Un + Un+1 (12.33) d˙ ∗n+1 = d˙ n + 12 d¨ n + d¨ ∗n+1 Δ t

12 A Multiscale Method for Dynamic Fracture

221

• We next adjust the velocities by introducing the coupling terms: $ L ˙∗ − 1 M ˜ −1 ˙ n+1 = U U α FM,n+1 Δ t n+1 2 ˙dn+1 = d˙ ∗ + 1 m−1 fL n+1 2 β m,n+1 Δ t

(12.34)

• Finally we compute the accelerations: $ L ¨ ∗ −M ˜ −1 ¨ n+1 = U U α FM,n+1 n+1 −1 ∗ L +m f d¨ n+1 = d¨ β

n+1

(12.35)

m,n+1

The last steps enforce the coupling condition (12.16). From Eqs (12.16) and (12.30), the coupling condition becomes: T ˙∗ −1 ˙ ∗n+1 − M ˜ −1 CTM U α CMΛ n+1 Δ t = Cm dn+1 + mβ CmΛ n+1 Δ t

(12.36)

The new values of the Lagrange multipliers Λ n+1 are subsequently computed by solving: AΛ n+1 = bn+1 (12.37)

$

with:

=

A

bn+1 =

Δt T ˜ −1 T −1 2 CM Mα CM + Cm mβ Cm ˙ ∗ − CTm d˙ ∗ CTM U n+1 n+1

(12.38)

0.0014

0.0014

0.0012

0.0012

0.0010

0.0010 Displacement (nm)

Displacement (nm)

and bn+1 stands for the weak coupling condition on the predictive velocities. Thus, this term is a measure of the error compared to the solution that satisfies the system (12.30).

0.0008 0.0006 0.0004 0.0002

0.0008 0.0006 0.0004 0.0002

0.0000

0.0000 0

5

10

15

20 X (nm)

25

30

35

0

5

10

15

20

25

30

35

X (nm)

(a) Prediction step with weighted stiffness (b) Prediction step with full stiffness and and mass matrices mass matrices

Fig. 12.3 Wave propagation in a one-dimensional bar where the left and right sides are continuum domains, and the inside zone represents the atomistic region. Displacements at t = 150 × 10−15 s.

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0.014

Mechanical energy (zJ)

0.012 0.010 0.008 0.006 0.004 0.002 0.000 −0.002 0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

Time (ps)

Fig. 12.4 Energy transfer for the finite element - Molecular Dynamics coupling in case of prediction step with the weighted matrices. The drawn line is the energy in the left continuum domain. The dashed line is the energy in the right continuum domain, and the dash-dotted line represents the energy in the atomistic zone. The bold drawn line is the total mechanical energy.

0.020

0.015

Energy (zJ)

0.010

0.005

0.000

−0.005

−0.010 0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.11

Time (ps)

Fig. 12.5 Mechanical energy (drawn line) and the work stored in the Lagrange multipliers (dashed line) when the prediction step is with the full stiffness and mass matrices.

12.4.4 Energy Transfer between the Atomistics and Continuum Domains The energy transfer for the coupling between the continuum and atomistic domains is studied by means of a longitudinal bar discretized with finite elements, and containing an atomistic region, with a coupling zone on both sides. The bar is submitted to a traction wave, which is enforced by displacing the left-most

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20 elements in the initial configuration. The right-hand end is free. The whole domain is 59.142528 × 10−9 m long and 100 atoms have been put in the atomistic domain. The interatomic distance is re = 0.1234708 × 10−9 m, and the finite element size is h = re . We use a Lennard-Jones potential as constitutive model for the atoms, with a = 32.043529 × 10−21 J and a mass m = 0.0016599 × 10−24 g. The elastic material properties for the finite element model have been derived from the atomic properties [52]. The computation continues for 2000 time steps with Δ t = 1 × 10−15 s, which amounts to 95% of the critical time step. We first analyse the problem using weighted matrices for the predictive part of the solution strategy. We compute the lumped mass matrix, cf. Eq.(12.32), as follows: ˜ α ,I = α (XI ) M ˜I M

(12.39)

˜ is the classical lumped mass matrix. where XI is the position of the Ith node and M Similarly, the stiffness matrix is computed by weighting the elementary terms, thus taking into account the influence of the weighting function α [40]: Kα ,I,J =

Ω

α (X ) ∇u (NI (X )) : K : ∇u (NJ (X )) dΩ

(12.40)

The displacements shown in Figure 12.3(a) are for a coupling length Lc that includes 5 elements. When the wave passes through the coupling domain, we observe an amplification of the amplitude and a non-negligible reflection. The energy plots of Figure 12.4 show that, even though the total energy is conserved, the energy transfer between the domains is poor. Many reflections occur, causing information to be lost completely in the end. An explanation of this phenomenon resides in the discretization of the initial value problem. The introduction of the weighting in the elementary terms of the internal forces in the continuum is tantamount to considering a porous material where the porosity changes progressively with the distance. At the right-hand side, i.e. near the end of the coupling zone, the left continuum bar has a near-zero Young’s modulus and a near-zero mass. This construction does not cause any spurious reflections in the case of a coupling between two continuum domains, but it introduces a nonsymmetric problem between a continuum and a domain that is composed of atoms. The atomistic internal forces have been weighted with the weight function, and to solve the problem the continuum internal forces must be constructed such that full symmetry exists between both domains. The original continuum internal forces were expressed as: Fα = Kα U

(12.41)

In order to follow the same procedure as the atomistic one, we write: Fα ,I = α (XI ) KU

(12.42)

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By writing the internal forces in this manner, symmetry has been restored. Moreover, we have obtained a simpler solution scheme. Indeed, by writing the continuum internal forces in this manner, the second step of the procedure is equivalent to solving the “free” problems separately, since: $ ¨ ∗ = −M ˜ −1 ˜ −1 −1 U α Kα Un+1 = −M α α KUn+1 n+1 (12.43) ¨d∗ = m−1 fβ ,n+1 = m−1β −1β fn+1 n+1 β where α and β are the diagonal weighting matrices (e.g., α I = α (XI )). Finally we obtain: ∗ ¨ ˜ −1 U n+1 = −M KUn+1 (12.44) ¨d∗ = m−1 fn+1 n+1 The problem is therefore highly simplified. We first solve the “free” uncoupled problems separately, and then, as a second step, we couple them using the Lagrange multipliers. For this step we use the weighted mass matrices. We now apply this procedure to the example. The displacements are shown for different times in Figure 12.3(b). The simulations do not reveal spurious reflections anymore when the wave passes through the coupling zone, and the formation at t = 0 is preserved during the computation. Figure 12.5 shows the total mechanical eergy during the computation. We observe some fluctuations each time the wave crosses a coupling zone. In fact, work is stored by the Lagrange multipliers, and subsequently put back in the mechanical system when the wave exits the coupling zone. We observe that the work of the Lagrange multipliers is complementary to the mechanical energy, and the energy balance is therefore satisfied. Considering the energy plots of Figure 12.6 we observe that for different coupling lengths (i) the total energy is preserved, and that (ii) the energy correctly passes from one domain to the other when the wave traverses the coupling zones.

12.5 Dynamic Fracture 12.5.1 Mechanical Quantities in the Atomistic Domain In order to extract mechanical quantities from the atomistic domain, we adopt a continuum mechanics point of view to derive classical stress quantities. The atomistic stress tensor at an atom i is a measurement of the interatomic interactions of the atom with its neighbours. A widely used stress quantity defined on the atomistic domain is the virial stress, which takes into account the interactions and a kinetic energy contribution. Many formulations have been derived from this virial stress [47, 48, 49], but, as pointed out by Zhou [50], these definitions, even perfectly correct in a statistical and thermodynamical sense, do not correspond to the Cauchy stress or to any other mechanical stress. However, it can be shown that

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the interatomic interactions part of the virial stress reduces to the Cauchy stress with a physical meaning. We therefore adopt this definition for the stress tensor:

σi =

1 f ji ⊗ ri j 2Vi ri∑ j
(12.45)

where Vi is the volume of the atom i, ri j = ri − r j , and ri j = |ri j |. Subsequently, the average of this atomistic stress tensor is computed over the volume around i within the cut-off radius rc . The average atomistic stress at the atom i thus reads: σ i avg = Nn,i 1 Nn,i ∑ j=1 σ j . Finally, the Von Mises atomistic stress σc at an atom i is defined as follows: 2 avg 2 avg avg avg 2 2(σc,i )2 = σxx − σyy + σyy − σzzavg + σxx − σzzavg avg 2 avg 2 avg 2 (12.46) +6 σxy + σyz + σxz

12.5.2 Examples: Crack Propagation under Velocity Loading We now proceed with a two-dimensional simulation. A copper single crystal is considered in its (111) plane, so that the two-dimensional lattice is hexagonal. The Lennard-Jones potential is used in the molecular dynamics simulation with parameters [51]: a = 0.415 eV and b = 0.2277 nm. The Young’s modulus E and Poisson’s ratio ν for the continuum then become: E = 79.334 GPa and ν = 0.25 [52]. The copper atomic mass is taken as m = 0.105520602596 × 10−24 kg (mCu = 63.546 NA g with NA the Avogadro number: NA = 6.02214179 × 1023 mol −1 ), which corresponds to a mass density ρ = 1865.250812586 × 103 kgm−3 . In the present study, the temperature has not been taken into account, since the focus is on the coupling of Molecular Dynamics to an (extended) finite element method for crack propagation. When extending the methodology to explicitly include the temperature a “thermal equilibrium” has to be achieved in addition to the mechanical equilibrium. This can for instance be done using the Nose-Hoover thermostat method [53]. The domain of interest is 100 nm long and 77.5 nm wide with an initial crack of 10 nm. A large MD-Box is considered in order to properly trace the crack propagation during dynamic loading. The finite element mesh consists of 1221 quadrilateral elements and 4868 nodes. The element size is about 10 times the interatomic distance. 8875 atoms are put in the initial MD-Box. The width of the coupling domain is approximatively 3 nm and 33% of the atoms in this region hold Lagrange multipliers.

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Total energy drift

0.012

3.0e−11 2.5e−11

0.010 2.0e−11 0.008

1.5e−11 1.0e−11

0.006 5.0e−12 0.0e+00

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−1.5e−11 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

Time (ps)

Time (ps)

(a) Energy transfer for Lc consisting of 2 el- (b) Energy drift for Lc consisting of 2 eleements ments

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(c) Energy transfer for Lc consisting of 5 el- (d) Energy drift for Lc consisting of 5 eleements ments

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Total energy drift

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(e) Energy transfer for Lc consisting of 10 (f) Energy drift for Lc consisting of 10 eleelements ments

Fig. 12.6 Energy plots for the finite element - Molecular Dynamics coupling. In subfigures (a), (c) and (e) the drawn line is the energy in the first (left) continuum, the dashed line is the energy in the second (right) continuum, and the dash-dotted line represents the energy stored in the atoms. The bold drawn line is the total mechanical energy.

12 A Multiscale Method for Dynamic Fracture

(a) Cracked specimen 1 (Vp = 3.16 ms−1 ) at t = 140.744 ps.

(c) Cracked specimen 3 (Vp = 47.4 ms−1 ) at t = 32.102 ps

227

(b) Cracked specimen 2 (Vp = 31.6 ms−1 ) at t = 39.535 ps

(d) Cracked specimen 4 (Vp = 63.2 ms−1 ) at t = 29.256 ps

(e) Cracked specimen 5 (Vp = 126.5 ms−1 ) at t = 16.605 ps

Fig. 12.7 Cracked specimens at the end of the simulations

For computational reasons, the results that are presented from now on have been obtained by only including the first neighbours in the atomistic interactions. Indeed,

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the equilibration techniques and the updates are expensive when we take into account many neighbours. However, simulations on a smaller scale have indicated that at least qualitatively, the results are rather similar to those obtained with more neighbours. In order to simulate defects in the lattice, 0.5% of the atoms are removed in a random manner. Example calculations have been carried out for five different loading rates, applied to the top and bottom edges of the specimen: • • • • •

Specimen 1: Vp = 3.16 ms−1 ; Specimen 2: Vp = 31.6 ms−1 ; Specimen 3: Vp = 47.4 ms−1 ; Specimen 4: Vp = 63.2 ms−1 ; Specimen 5: Vp = 126.5 ms−1 .

For the lowest loading rate we observe from Figure 12.7 that there is only one main crack, which propagates in a horizontal sense. For the two next higher loading rates secondary branches develop, which suggest the appearance of evolving dislocations. For the two highest loading rates multiple crack branching is observed and the main crack no longer runs horizonally, but shows a tortuous pattern. Figure 12.8 shows the Von Mises stress just before and after the crack initiation for the specimen that is subjected to the medium loading rate. The right plots in this figure give the the stress distributions along the crack – along a horizontal line. The stress concentration clearly localizes at the crack tip before opening. Upon exceeding a threshold value, propagation occurs and the stress decreases rapidly which is accompanied by stress waves that are emitted radially. These results correspond to those obtained in classical fracture mechanics. Indeed, Figure 12.9 shows the stress components at the onset of crack initiation for specimen 4. The shape of the σxx , σxy and σyy components also corresponds to those predicted by classical fracture mechanics and the highest value is that of σyy , which corresponds to the fact that we have a Mode-I loading in the vertical direction. Considering the atomistic structure during crack propagation, a distinction can be made between cracks and dislocations. Figure 12.10 shows the atomic bonds at different time steps for specimen 2. While crack propagation is caused by atomic debonding, dislocations are characterized by a local re-arrangement of the lattice, rather than debonding. Furthermore, one can observe that, even if the main crack propagates in a horizontal sense, locally its path is governed by lattice defects. Finally, it is noted that the crack does not propagate in a smooth manner. Instead, void nucleations are observed ahead of the crack tip. The local process of crack propagation thus seems a succession of void nucleation, debonding and crack tip propagation. This phenomenon is known at the atomistic level, and has also been observed in macroscopic experiments (usually for high velocities, e.g. [54]). Furthermore, the main crack tip has been tracked during propagation for specimens 2, 3, and 4, in order to plot the curvilinear coordinate of the crack tip as a function of time. This enables the computation of the crack propagation velocity. In Figures 11(a) – 11(c), the evolution of the crack length has been plotted. For specimen 2 a crack velocity can straightforwardly be identified as Vcrack,2 ∼ 2815 ms−1 . 1 For specimen 3 one observes different regimes. First, the velocity equals Vcrack,3 ∼

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(a) Von Mises stress at the crack tip at t = 9.219 ps

(b) Stress distribution along the crack line at t = 9.219 ps

(c) Von Mises stress at the crack tip at t = 9.805 ps

(d) Stress distribution along the crack line at t = 9.805 ps

Fig. 12.8 Crack initiation for specimen 3: Vp = 47.4 ms−1

1967 ms−1 . Then, there is an acceleration at t ∼ 22 ps, resulting in a crack prop2 agation velocity Vcrack,3 ∼ 2581 ms−1 . Considering the displacement fields from t ∼ 22 ps onwards, e.g. Figure 11(d)), the main crack seems to be surrounded by two zones of evolving dislocations. This causes wave reflections, which can affect the crack propagation and its velocity. Moreover, as the velocity increases, more dislocations seem to arise. Finally, for specimen 4, three regimes can be identified: 1 • Initiation: Vcrack,4 ∼ 1892 ms−1 around t = 8 ps, 2 • First acceleration: Vcrack,4 ∼ 2447 ms−1 around t = 15 ps, 3 • Second acceleration: Vcrack,4 ∼ 2841 ms−1 around t = 24 ps.

From the displacement field in Figure 12.12 we observe that around these moments, similar changes occur as observed for specimen 3. For t ∼ 15 ps, Figures 12(a) and 12(b), dislocations emerge, and the main direction of the crack path changes. Around t ∼ 24 ps, Figures 12(c) and 12(d), a crack branching occurs. New free surfaces appear and the local configuration changes. This can explain the observed acceleration in crack propagation.

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(a) σxx at the crack tip at t = 7.986 ps

(b) σxy at the crack tip at t = 7.986 ps

(c) σyy at the crack tip at t = 7.986 ps.

Fig. 12.9 Stress components before crack initiation. Specimen 4: Vp = 63.2 ms−1

From the elastic material properties computed above, the following wave speeds can be deduced: 3 μ • Longitudinal wave speed: cL = λ +2 = 7144 ms−1 , ρ 3 • Shear wave speed: cS = μρ = 4124 ms−1 , • Rayleigh wave speed: cR ∼ 3785 ms−1 . Comparing these values with the numerically obtained velocities, the final crack speed appears to be around 75% of the Rayleigh wave speed, and for specimens 3 and 4, the initiatial velocity, i.e. before it accelerates, is around 50% of the Rayleigh wave speed.

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(a) Displacement field at t = 19.767 ps.

(b) Displacement field at t = 20.558 ps.

(c) Displacement field at t = 21.349 ps.

(d) Displacement field at t = 22.139 ps.

(e) Displacement field at t = 22.930 ps.

(f) Displacement field at t = 23.721 ps.

Fig. 12.10 Debonding and displacement around the crack tip. Specimen 2.

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7 Cr ace( r kt ehe7 - CuTvTl mr CeannCoTl r kme l i s

Crack path - Curvilinear coordinate (nm)

70 60 50 40 30 y = 2,81549x − 46,3283

20

R² = 0,999411

10 0

0

10

20

30

40

50

Time (ps)

)0

40 deye1=, 684) 9e− 1x=x6x2 R²eye0=, , 86, 4

30

deye2=) x1889e− 32=xxx8 R²eye0=, , , 163

20

10

0

0

10

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30

40

5 Ti me ( ps

(a) Specimen 2: Vp = 31.6 ms−1 .

(b) Specimen 3: Vp = 47.6 ms−1 .

7 Cr acmr kt m hm 7 - Cu5v5l i r Cm annCo5l r ki m el T p

60

)0

dm ym 2=s s ² s 28m − 2s =300s 9 xm ym 0=RR, 63²

s0 dm ym 2=, s 10) 8m − 3s =122s

30

9 xm ym 0=RR3) s

dm ym 1=, R2) s 8m − 1) =3RR3 9 xm ym 0=RR² 13s

20

10

0

0

10

20

30

4 5T i m e (p

(c) Specimen 4: Vp = 63.2 ms−1 .

(d) Displacement field at t = 23.721 ps.

Fig. 12.11 Crack tip coordinate as a function of time and displacement field of specimen 3 at t = 23.721 ps

We finally examine the evolution of the dislocation in specimen 3, Figure 13(a). It arises fom the crack at t = 26.251 ps and propagates in one of the main lattice directions (−60◦), see Figure 13(b). A propagation velocity Vshear,3 ∼ 5050 ms−1 can be identified, which, compared with the shear wave velocity cS seems somewhat high, but it is in the same order of magnitude. The graph also shows that the band propagation is influenced by defects along its path. Indeed, the global velocity is the result of a interaction between propagations, accelerations and local bifurcations.

12.6 Concluding Remarks A numerical approach has been proposed for combining a molecular dynamics method and a finite element method that exploits the partition-of-unity property of finite element shape functions (extended finite element method). The aim is to simulate dynamic fracture in an efficient manner on basis of elementary physical principles. To this end the zone around the crack tip is modelled using molecular

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(a) Displacement field at t = 15.814 ps.

(b) Displacement field at t = 18.977 ps.

(c) Displacement field at t = 24.512 ps.

(d) Displacement field at t = 26.884 ps.

Fig. 12.12 Main crack surrounded by crack branching and dislocations. Specimen 4.

dynamics. Around this so-called Molecular Dynamics Box a continuum mechanics approach is adopted, with the finite element method used for discretization. The partition-of-unity property of the finite element shape functions is exploited to model the crack in the wake of its tip as a traction-free discontinuity. The coupling between the continuum and molecular dynamics zones has a zonal character where the energy is partitioned over both models and a weak velocity coupling is enforced. In this manner, spurious reflections are avoided and energy is conserved when a wave travels from one zone into another [45]. The computation is limited by the size of the MD-Box. When a crack tip reaches the boundary of the atomistic domain and intersects the coupling zone, the crack is arrested and cannot propagate further. For this reason an adaptive MD-Box has been constructed, the atomistic domain is automatically resized and/or moved when the crack evolves during the computation. In this manner discontinuities always remain inside the atomistic domain. Two-dimensional examples are shown for different loading rates. Depending on the loading rate, different failure modes can be distinguished, ranging from stable propagation of a single crack to multiple branching, while phenomena like evolving dislocations and local mode mixity can also be identified.

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(a) Evolving dislocation band starting from the crack at t = 31.786 ps

Shear band path - Curvilinear coordinate (nm)

40

30 y = 5,05009x − 133,424 R² = 0,998223 20

10

0

26

27

28

29

30

31

32

33

Time (ps)

(b) Evolving dislocation: Length as function of time

Fig. 12.13 Evolving dislocation band for specimen 3: Vp = 47.4 ms−1

References [1] Griffith, A.A.: The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society of London A221, 163–198 (1921) [2] Irwin, G.R.: Analysis of stresses and strains near the end of a crack traversing a plate. Journal of Applied Mechanics 24, 361–364 (1957)

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[3] Rice, J.R.: A path independent integral and the approximate analysis of strain concentration by notches and cracks. Journal of Applied Mechanics 35, 379–386 (1968) [4] Henshell, R.D., Shaw, K.G.: Crack tip finite elements are unnecessary. International Journal for Numerical Methods in Engineering 9, 495–507 (1975) [5] Barsoum, R.S.: On the use of isoparametric finite elements in linear fracture mechanics. International Journal for Numerical Methods in Engineering 10, 25–37 (1976) [6] Fleming, M., Chu, Y.A., Moran, B., Belytschko, T.: Enriched element-free Galerkin methods for crack tip fields. International Journal for Numerical Methods in Engineering 40, 1483–1504 (1997) [7] Krysl, P., Belytschko, T.: The Element-free Galerkin Method fod dynamic propagation of arbitrary 3-D cracks. International Journal for Numerical Methods in Engineering 44, 767–780 (1999) [8] Belytschko, T., Black, T.: Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering 45, 601–620 (1999) [9] Moës, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 46, 131–150 (1999) [10] Réthoré, J., de Borst, R., Abellan, M.A.: A two-scale approach for fluid flow in fractured porous media. International Journal for Numerical Methods in Engineering 71, 780–800 (2007) [11] Albuquerque, E.L., Sollero, P., Aliabadi, M.H.: Dual boundary element method for anisotropic dynamic fracture mechanics. International Journal for Numerical Methods in Engineering 59, 1187–1205 (2004) [12] Barenblatt, G.I.: The mathematical theory of equilibrium cracks in brittle fracture. Advances in Applied Mechanics 7, 55–129 (1962) [13] Dugdale, D.S.: Yielding of steel sheets containing slits. Journal of the Mechanics and Physics of Solids 8, 100–108 (1960) [14] Hillerborg, A., Modeer, M., Petersson, P.E.: Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research 6, 773–782 (1976) [15] Needleman, A.: A continuum model for void nucleation by inclusion debonding. Journal of Applied Mechanics 54, 525–531 (1987) [16] Rots, J.G.: Smeared and discrete representations of localized fracture. International Journal of Fracture 51, 45–59 (1991) [17] Xu, X.P., Needleman, A.: Numerical simulations of fast crack growth in brittle solids. Journal of the Mechanics and Physics of Solids 42, 1397–1434 (1994) [18] Wells, G.N., Sluys, L.J.: Discontinuous analysis of softening solids under impact loading. International Journal for Numerical and Analytical Methods in Geomechanics 25, 691–709 (2001) [19] Wells, G.N., de Borst, R., Sluys, L.J.: A consistent geometrically non–linear approach for delamination. International Journal for Numerical Methods in Engineering 54, 1333–1355 (2002) [20] de Borst, R.: Numerical aspects of cohesive zone models. Engineering Fracture Mechanics 70, 1743–1757 (2003) [21] Duarte, C.A., Hamzeh, L.T.J., Tworzydlo, W.W.: A generalized finite element method for the simulation of three-dimensional crack propagation. Computer Methods in Applied Mechanics and Engineering 190, 2227–2262 (2001)

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[22] Réthoré, J., Gravouil, A., Combescure, A.: An energy conserving scheme for dynamic crack growth with the extended finite element method. International Journal for Numerical Methods in Engineering 63, 631–659 (2005) [23] Menouillard, T., Réthoré, J., Combescure, A.: Efficient explicit time stepping for the extended finite element method. International Journal for Numerical Methods in Engineering 68, 911–939 (2006) [24] Remmers, J.J.C., de Borst, R., Needleman, A.: A cohesive segments method for the simulation of crack growth. Computational Mechanics 31, 69–77 (2003) [25] Remmers, J.J.C., de Borst, R., Needleman, A.: The simulation of dynamic crack propagation using the cohesive segments method. Journal of the Mechanics and Physics of Solids 56, 70–92 (2008) [26] Réthoré, J., de Borst, R., Abellan, M.A.: A two-scale model for fluid flow in an unsaturated porous medium with cohesive cracks. Computational Mechanics 42, 227–238 (2008) [27] Abraham, F.F., Walkup, R., Gao, H., Duchaineau, M., Diaz De La Rubia, T., Seager, M.: Simulating materials failure by using up to one billion atoms and the world’s fastest computer: work-hardening. Proceedings of the National Academy of Sciences 99, 5783–5787 (2002) [28] Zhou, S.J., Lomdahl, P.S., Voter, A.F., Holian, B.L.: Three-dimensional fracture via large-scale molecular dynamics. Engineering Fracture Mechanics 61, 173–187 (1998) [29] Miller, R., Ortiz, M., Phillips, R., Shenoy, V., Tadmor, E.B.: Quasicontinuum models of fracture and plasticity. Engineering Fracture Mechanics 61, 427–444 (1998) [30] Parr, R.G., Gadre, S.R., Bartolotti, L.J.: Local density functional theory of atoms and molecules. Proceedings of the National Academy of Sciences 76, 2522–2526 (1979) [31] Car, R., Parrinello, M.: Unified approach for molecular dynamics and density-functional theory. Physical Review Letters 55, 2471–2474 (1985) [32] Kolhoff, S., Gumbsch, P., Frischmeister, H.F.: Crack propagation in bcc crystals studied with a combined finite element and atomistic model. Philosophical Magazine A 64, 851–878 (1991) [33] Shilkrot, L.E., Miller, R.E., Curtin, W.A.: Coupled atomistic and discrete dislocation plasticity. Physical Review Letters 89, 255011–255014 (2002) [34] Gracie, R., Belytschko, T.: Concurrently coupled atomistic and XFEM models for dislocations and cracks. International Journal of Numerical Methods in Engineering 78, 354–378 (2009) [35] Ben Dhia, H., Rateau, G.: The Arlequin method as a flexible engineering design tool. International Journal of Numerical Methods in Engineering 62, 1442–1462 (2005) [36] Prudhomme, S., Ben Dhia, H., Baumann, P.T., Elkhodja, N., Oden, J.T.: Computational analysis of modeling error for the coupling of particle and continuum models by the Arlequin method. Computer Methods in Applied Mechanics and Engineering 197, 3399– 3409 (2008) [37] Xiao, S.P., Belytschko, T.: A bridging domain method for coupling continua with molecular dynamics. Computer Methods in Applied Mechanics and Engineering 193, 1645–1669 (2004) [38] Zhang, S.L., Khare, R., Lu, Q., Belytschko, T.: A bridging domain and strain computation method for coupled atomistic-continuum modelling of solids. International Journal of Numerical Methods in Engineering 70, 913–933 (2007) [39] Guidault, P.A., Belytschko, T.: On the L2 and the H1 couplings for an overlapping domain decomposition method using Lagrange multipliers. International Journal of Numerical Methods in Engineering 70, 322–350 (2007)

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Chapter 13

Nonlinear Finite Element and Atomistic Modelling of Dislocations in Heterostructures Paweł Dłuz˙ ewski, Toby D. Young, George P. Dimitrakopulos, Joseph Kioseoglou, and Philomela Komninou

Abstract. A continuum and atomistic approach to the modelling of dislocations observed by the High Resolution Transmission Electron Microscopy (HRTEM) is discussed. We present a methodology for the analysis of dislocations, stacking faults and interfacial regions in crystal heterostructures by using experimental measurements, and atomistic/continuum models. The derived method is capable of handling dense arrays of dislocations as well as large areas of the interface between two crystal structures. As an example, we consider stacking faults in bulk GaN and a misfit dislocation network in a GaN/sapphire interfacial region through relaxation of strain and associated residual stresses; where the quantification of the latter is beyond the reach of experiment alone.

13.1 Introduction This chapter achieves a reciprocal relation between atomistic and continuum models of the extended crystal defects observed in semiconductor structures. In particular, a focus is applied to regions containing dislocations and/or stacking faults in bulk crystals, and to misfit dislocation networks in the interfacial region of a film/substrate heterostructure. The connection between the continuum and atomistic models is divided into two parts and illustrated by following two examples: (a) The atomistic models of dislocations, typically found in semiconductor materials, are constructed by means of a set of equations that describe the distortion Paweł Dłu˙zewski and Toby D. Young Institute of Fundamental Technological Research PAS, ul. Pawi´nskiego 5 B, 02-106 Warsaw, Poland e-mail: [email protected] and [email protected] George P. Dimitrakopulos, Joseph Kioseoglou and Philomela Komninou Department of Physics, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece e-mail: [email protected], [email protected], [email protected] M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 239–253. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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field of the straight–mixed dislocation introduced by a slip/climb plane (stacking fault) to the crystal lattice. Such partial dislocations and stacking faults as observed on HRTEM images are discussed in this chapter. (b) The misfit dislocations arising at a planar heterojunction formed between semiconductor film and substrate are reconstructed in the atomistic and 3D finite element models by taking advantage of experimentally obtained strain maps. In order to assure the fulfillment of a proper reciprocal relation between the continuous and discrete configurations of these systems, the theoretical predictions are referred back to observations obtained by experiment. From this, it may then be possible to investigate such a system in terms of its effectiveness in technological applications and devices. In particular, as a practical implementation of this scheme, we analyse structural defects introduced by the mismatch between GaN and sapphire lattices in three conjoined steps of modelling. In the first step the reconstruction of the lattice distortion fields was performed starting from observations of the GaN/sapphire interface by HRTEM. Having as input the experimental values of the lattice constants, 3D crystallographic models of the stacking faults and interface were constructed. Geometrical Phase Analysis [9] was then applied to the experimental and the corresponding simulated images in order to obtain initial lattice displacement and strain fields in the interfacial region. This data was taken as input for Finite Element Analysis (FEA) thereby overcoming the obstacle of a lack of empirical potentials suitable for describing heterostructures within molecular dynamics simulations. The 3D model, corresponding to a few parallel misfit dislocations in the HRTEM image, was constructed with a fundamental motif structure and multiplied in such a way as to obtain a sample of reasonable size with a number of dislocation nodes trapped inside thus forming a complex network of misfit dislocations. The remainder of this chapter is organised as follows: In section 13.2, the mathematical foundations of the nonlinear continuum theory are discussed and mutual tensor relations between the lattice and plastic distortions, Burgers vector and elasticplastic deformation tensors are discussed. Section 13.3 is devoted to dislocations and stacking faults reconstructed in the bulk crystals. Section 13.4 is devoted to the HRTEM analysis and reconstruction of the misfit dislocation net in the interfacial region of henerostructure GaN/Al2 O3 . In that section, we apply our method for reconstructing 3D source distortion maps from 2D and consider the relaxation process of the strain energy of the heterostructure by a discrete space formalism within FEA. Conclusions are drawn in Section 13.5.

13.2 Continuum Theory of Discrete Dislocations In the linear theory of dislocations the gradient of crystal deformation is decomposed additively into the lattice and plastic distortion tensors, according to

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gradutot = β + β pl ,

(13.1)

cf. [15]. By assumption, the lattice distortion β is composed of the antisymmetric tensor of crystal lattice rotation and symmetric tensor of elastic strain, respectively

β = w+ε .

(13.2)

The plastic distortion β pl is identified with an asymmetric strain tensor being isoclinic with the lattice rotation. This nomenclature comes from the theory of elastoplastic Cosserat continua where the particle rotation χ and elastic/plastic strains are identified respectively with: χ ≡ w, ε ≡ ε and ε pl ≡ β pl ; which gives grad utot = χ + ε + ε pl .

(13.3)

where the elastic strain tensor ε is no longer symmetric. Contrary to the Cosserat continua, in the continuum theory of dislocations it is assumed that the material particle cannot rotate independently of the displacement field, and therefore a simplified notation gathering together the elastic strain and crystal lattice rotation into a common lattice distortion tensor β is more convenient.

13.2.1 Burgers Vector According to the state–of–the–art of the nonlinear continuum theory of dislocations, the total deformation gradient can be decomposed multiplicatively into elastic and plastic deformation parts and rewritten in the form Ftot = F Fpl .

(13.4)

The integration of lattice distortions over Burgers circuits gives

b=

F d& x=

c

−& b=

dx = O

d& x= C

o

(1 + β& ) d& x,

(13.5a)

O

F−1 dx =

(1 − β ) dx ,

(13.5b)

o

& are the metric tensor of the Euclidean space, and the spatial and where 1, b and b true Burgers vectors, respectively. The symbols c, C, o, O denote the open and closed Burgers circuits situated respectively in the current (Eulerian) and reference (intermediate) configurations. It is worth emphasising here that the lattice distortion β is related to lattice spacings in the current configuration. Alternatively, the distortions can be referred back to the spacings in a perfect lattice, which gives the following reversible transformation rule

β& = (1 − β )−1 − 1 .

(13.6)

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Usually, in the linear theory the difference between differentiation over the current, reference and intermediate configurations are neglected, i.e. it is assumed that ∂u ∂u & , β ≈ β , O ≡ o, etc. Given such a simplified approach, the differential equa∂ x ≈ ∂& x tion sets obtained reduce to linear sets, and thanks to analytical methods available for linear systems many very useful analytical formulae have be derived for dislocations.

13.2.2 Analytical Equations for Mixed Straight Dislocation The total displacement field around a mixed straight-line dislocation in an isotropic elastic material is defined by the classical formulae coming back to works [18] and [22]. Here the equations are written in a more general form as xy y bx y − bx H arctan − ϕ , arctan + 2 2 2π x 2(1 − ν )(x + y ) x

2 − y2 1 − 2ν bx x ln (x2 + y2 ) + , uy = − 2π 4(1 − ν ) 4(1 − ν )(x2 + y2) bz y bz uz = arctan − . 2π x 2

ux =

(13.7)

where the edge and screw components of the Burgers vector, bx and bz , are parallel to the x and z axes respectively. The symbol ϕ denotes the angle composed by edge component of the Burgers vector with the crystal plane the partial dislocation is introduced into the crystal; the Heaviside step function H(x) is defined according to the following convention: H(x) = 1 for x ≥ 0, and H(x) = 0 for x < 0. For partial dislocations the plane mentioned above can be identified with a stacking fault plane, see Fig. 13.1. Fig. 13.1 Partial dislocaNon–vanishing components of lattice distortions are

βxx =

tion bounding a stacking fault.

−bx (3 − 2ν )x2y + (1 − 2ν )y3 bx (3 − 2ν )x3 + (1 − 2ν )xy2 , β = , xy 2π 2(1 − ν )(x2 + y2 )2 2π 2(1 − ν )(x2 + y2 )2

−bx (1 − 2ν )x3 + (3 − 2ν )xy2 bx (1 + 2ν )x2 y − (1 − 2ν )y3 , βyy = , 2 2 2 2π 2(1 − ν )(x + y ) 2π 2(1 − ν )(x2 + y2)2 y x −bz bz βzx = , βzy = , 2 2 2 2π x + y 2 π x + y2 (13.8)

βyx =

It is worth emphasizing that (13.7) does not describe the total displacement field but only its elastic part in a singly connected domain arctan yx = ϕ . The plastic

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distortions can be recovered here from the compatibility condition, which gives

βplxx =

x x2 + y2

y δ arctan − ϕ , x

βplxy = −

y x2 + y2

o dutot

= 0,

y δ arctan − ϕ , x (13.9)

where δ (·) denotes the Dirac delta function.

13.3 Atomistic Reconstruction of Dislocations In many cases the analytical solutions obtained by means of the linear theory of dislocations are used to generate the input files for atomistic modelling of the physical properties of dislocation cores. In case of the wurtzite structure of GaN such a method has been used in many papers starting from [1]. The structure of such obtained cores depends on which position the dislocation center has taken in the reference unit crystal cell as shown in figure 13.2.

Fig. 13.2 Dislocation core structures generated by means of (13.7) for b = 13 2110 in GaN, on the left: the reference unit cell with depicted positions of dislocation centers assumed to generate the 5:7 and 8-atom ring cores shown on the right, respectively.

The dislocation cores generated by the use of (13.7) do not hold symmetry of atomic bonds distribution. To avoid the asymmetry some modification into (13.7) was considered by [2], cf. [19]. Nevertheless, to obtain a symmetric distribution we should apply finite deformation theory in which the difference between differentiation over the reference and spatial configurations are taken into account. The problem is that the strict analytical solutions based on the nonlinear theory have not been obtained for most of the typical problems of the dislocation theory as yet.

13.3.1 HRTEM Investigations of Partial Dislocations in GaN GaN-based wide band-gap semiconductor technology has offered important device applications in optoelectronic and high power-frequency-temperature microelectronics. Extensive studies of dislocation core structures are being performed in

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order to assess their influence on a number of material properties and phenomena, e.g. [20, 1, 25, 17, 16, 7]. We have recently derived the core structures of the Shockley partial dislocations that delimit the I1 type of intrinsic stacking fault [11] which is a low energy fault of frequent occurrence. These defects can be constructed from a perfect crystal by removal or addition a disc of atoms in the (0001) basal plane (vacancy or interstitial disc) followed by a shear. Assuming that the line direction of the partial dislocation is along [1210], the dislocation may exhibit either the edge or mixed character. Let xz be the (0001) plane, with z along the dislocation line (i.e. [1210]) and x along [1010]. Consequently, the y axis is along [0001]. In the case of edge-type partials bedge = bx + by where bx = ± 13 [1010] and by = ± 12 [0001]. In the case of mixed-type partials, bmixed can be decomposed into a screw bz = 16 [1210] component and two edge components, bx = ± 16 [1010] and by = ± 12 [0001]. Using an empirical interatomic potential, 6 stable configurations for each of 4 variants of Shockley partials, 16 [2023], 16 [2023], 16 [2203], 16 [2203], have been obtained, and their energies and atomic configurations were given by [11]. A 57 -atom ring core in which the atoms are tetrahedrally coordinated was found energetically favourable among the edge partial dislocation configurations. The 57 and 12 atom rings were obtained as low energy cores for the mixed-type partials of the a and b cases, respectively, although none of these was found to comprise only tetrahedrally coordinated atoms. The HRTEM observations of edge and mixed 16 2023 partial dislocations in GaN have also been presented previously. Through GPA of strain fields, as well as peak finding on white dots, comparison between experimental images and simulated images of relaxed structures was performed. The 57 or 12-atom rings were clearly identified for the experimental image of figure 13.3a and an 8-atom ring was identified for the HRTEM image of figure 13.3b [13]. For the mixed partial dislocation of figure 13.3c, the peak-finding analysis has indicated the 12- or 10-atom rings [10].

Fig. 13.3 HRTEM images of edge partial dislocations 16 [2023] bounding a stacking fault I1 . The intensity peaks corresponding to the positions of atomic columns have been marked with red dots, the atom rings has been identified by both GPA and peak finding [12].

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13.4 Misfit Dislocations in Heterostructure GaN/Al2 O3 HRTEM provides a suitable method with which to quantitatively probe the details of lattice displacements with sub–angstrom accuracy [8]. The HRTEM simulated images of the interface model were generated using the multi–slice algorithm [24] of the EMS software. Prior to comparison with the simulations, the experimental conditions of thickness and defocus were determined by calculating the map of through focus thickness images of perfect GaN. In the HRTEM micrograph of figure 13.4a the GaN/Al2 O3 interface is depicted along the [1120] (GaN) and [1010] (Al2 O3 ) directions. Using Fourier filtering it is possible to visualise the edge component of one set of misfit dislocations as terminating fringes of the corresponding substrate lattice planes; figures 13.4b and 13.4c.

Fig. 13.4 In the left panel (a) HRTEM micrograph of the film/substrate system. In the right panel, measurements taken from the white bounding box in (a) are the Bragg images from Fourier filtering along (b) [0002] and (c) [1010]; and images of lattice distortions (d) εxx , (e) εxy and (f) wxy (thickness=2.2nm, defocus=59nm and white spots represent the atomic columns projection along [1210]).

13.4.1 Crystallographic Model The [1010] d-spacing of GaN was determined experimentally from figure 13.4a to be d = 2.725Å yielding a lattice parameter a = 3.1466Å. By assuming pure bi-axial strain, and with the use of the elastic constants C13 and C33 , the lattice c parameter was calculated to be ccal = 5.2025Å. The same parameter was determined experimentally to be cexp = 5.19Å. From the difference between ccal and cexp it is clear

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that in addition to bi–axial strain the GaN film also exhibits hydrostatic or uniaxial strain.

Fig. 13.5 (a) Schematic illustration along [1210] GaN/Al2 O3 heterostructure. (b) Plan-view along [0001] of two relatively rotated by π /6 hexagonal lattices showing the translation periodicity at the interface (dichromatic pattern). (c) Plan view along [0001] of the relatively rotated crystal structures (wurtzite GaN and Al2 O3 ) thus forming a dichromatic complex [21]. The translation periodicity and moiré pattern are shown as well as the trigonal-hexagonal primitive unit-cell employed here. (d) Enlarged view of the primitive unit-cell. Large and small circles denote distinct atomic species, shading of atoms denote levels along the projection direction.

The lattice mismatch between crystal layers is defined by the simple relation f = (aepi − are f )/are f ; where aepi and are f denote the epitaxial and reference lattice constants respectively. Taking GaN as reference the lattice mismatch was calculated to be f = 0.5124, and f = 0.3388 was found by the use of Al2 O3 as a reference. A lateral rotation of π /6 radians of the GaN basal plane significantly reduces the lattice mismatch to f = 0.1268 with GaN as the reference and f = 0.1452 with Al2 O3 as the reference. The latter configuration is investigated here. Figure 13.5a illustrates the crystallographic model of the bi–crystal along [1210] (i.e. the same projection direction as figure 13.4a). A 3D atomic configuration of the interface was duly constructed using crystallographic data and HRTEM image simulations were calculated. GPA was then applied to the simulated images in order to obtain the displacement and strain fields in the interface region. This data was used as input for FEA [14, 5].

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In figures 13.5b and 13.5c the unit–cell is enlarged. In figure 13.5d the coincident translational periodicity and primitive unit-cell are illustrated. The bi-crystal layer group is p3. Each unit-cell contains one dislocation; thereby its symmetry is reduced to p1 and there are three possible variants of 2π /3 radians (i.e. the co–set decomposition {p3} = {p1} ∪ 3+{p1} ∪ 3−{p1}). Figure 13.6 illustrates schematically the unit cell given in figures 13.5c and 13.5d. Rotation around the point of symmetry (black closed triangles) of the sub-cell (dashdot green lines) in figure 13.6a give rise variations of the primitive unit-cell (solid blue lines). The unit-cell is sequentially translated by the coincidence periodicity to obtain a honeycomb network of misfit dislocations crossing at 2π /3 radians depicted in figure 13.6b. The full structure is then taken into FEA described in Section 13.4.2.

Fig. 13.6 Schematic representation of (a) co-set decomposition of the primitive unit-cell employed here (c.f. figure 13.5c) and (b) construction of the resultant initial honeycomb misfit dislocation pattern through translation of the primitive unit-cell by the coincidence periodicity. Solid blue lines give the boundary of a unit-cell and dashed red lines denote misfit dislocations. The green dash-dot lines mark out boundaries of the sub-cells with point rotational symmetry.

13.4.2 Finite Element Modelling of Misfit Dislocations Linear elasticity predicts equal volumes of compressed and extended regions around dislocations while experimental evidence shows that extended regions occupy a larger volume than compressed regions [23]. This asymmetry is widely assumed to be quantified by the third-order elastic constants which play a crucial rôle in the elastic behaviour of real crystal structures. In the last decade a number of differing methods based on a theory of linear elasticity have been employed for the simulation and analysis of lattice dislocations. In this chapter a different approach is used that has emerged in more recent years and is based on a theory of nonlinear elasticity [5, 3]. In this way we aim to properly take into account the nonlinear elastic behaviour observed by experiment. We distinguish between three principal components that together contribute to the total deformation of the system. First, the source deformation field assumed to determine the lattice distortion field for the entire heterostructure; secondly, a chemical deformation gradient describing the transition between perfect lattices of

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Al2 O3 and GaN; and thirdly, the source deformation gradient determined by GPA in relation to some region of Al2 O3 chosen to be the reference region to measure the deformation map for the whole structure visible on the HRTEM image. The total deformation gradient F is decomposed therefore into three components: Ftot = F Fch F−1 ◦ ,

(13.10)

where F, Fch and F◦ are the elastic, chemical and lattice deformation gradient, respectively. The last gradient is extracted from the HRTEM image by computer image processing, cf. [5]. Alternatively, the gradient can be generated by using the analytical formulae of the continuum theory of dislocations, see [6]. The chemical deformation tensor in Al2 O3 region is assumed to be the unit tensor of the Euclidean space. In the GaN and transit regions it was assumed that ⎡ aGaN ⎤ 0 0 ⎢ aAl2 O3 ⎥ aGaN ⎢ ⎥ ⎢ ⎥ n(x) 0 0 Fch = ⎢ ⎥ aAl2 O3 ⎣ cGaN ⎦ 0 0 cAl2 O3

(13.11)

where n(x) is the molar fraction of GaN spanned on eight corner nodes of elements situated in the transit FE layer. The source deformation and distortion tensors extracted from HRTEM image satisfy the following relation F−1 ◦ = 1 −β◦ .

(13.12)

The continuum model used here is based on the integration of the equilibrium condition on the Cauchy stress tensor, div σ = 0 ,

(13.13)

In terms finite strains, where the stress measure conjugate by work with the logarithmic strain is referred to the undeformed (perfect) crystal lattice and satisfies the following linear constitutive equation

σ& = & c : ε& ,

(13.14)

where a hat denotes quantities in the reference lattice configuration. Here & ε is the Lagrangian logarithmic strain tensor defined as ε& = ln U, where U is the right stretch tensor of elastic deformation from the polar decomposition F = RU, where R is the orthogonal tensor of rotation. The fourth–order proper–symmetric tensor of elastic stiffness & c is assumed here to be a tensor of material constants. Due to nonlinear geometric changes between the reference and current lattice configurations, it can be shown that such constitutive equation leads to a nonlinear relation for the Cauchy stress

13 Nonlinear Finite Element and Atomistic Modelling

σ = c :ε ,

249

(13.15)

where

ε = A−T : ε& ,

AT det F−1 , c = A :& c :A

(13.16)

& KL IJ . The geometric meaning of the fourth–order proper– where Ai j IJ = Ri K R j L A & as an isotropic tensor function of ε& is discussed in [3]. symmetric tensor A Using the virtual work principle the following nonlinear matrix equation is found P(a) = f , where the vector P is a nonlinear function of nodal variables a, i.e. ⎡ ⎤ u T ⎣ P = grad W σ dv , a= n ⎦, Wσ d s. f= v ∂v β& ◦

(13.17)

(13.18)

and β& ◦ = (1 − β ◦ )−1 − 1. W denotes the weighting function determined in relation to the current (iterated) configuration. Using the Newton–Raphson method, the nonsymmetric equation set (13.17) was solved for displacements under the fixed fields β& ◦ and n. The components of distortion were cropped to represent the periodicity of parallel misfit dislocations on the interface and rescaled to a resolution of 9 × 11 pixels. The strain value at each pixel represented by a single corner node of the 27-node Lagrangian element [3, 4]. For an extension from the 2D mapping of figure 13.4 to a 3D continuum model of the strain energy similar operations are applied on the distortion tensor β as as were applied on the spatial coordinate system as described in subsection 13.4.1. For a single misfit dislocation the reconstruction of missing 3D source distortion components from 2D HRTEM images, cf. figure 13.5, was done by using the following relation [5] ⎡ ⎤ β&◦ xx 0 β&◦ xz ⎢√ ⎥ √ ⎢ ⎥ (13.19) β& ◦ = ⎢ 22 β&◦ xz 0 22 β&◦ zz ⎥ . ⎣ ⎦ β&◦ zx 0 β&◦ zz

13.4.3 Reconstruction of 3D Structure into FEA The GaN/Al2 O3 system FEA in the resultant configuration of relaxed strain is given in figure 13.6. In the left panel the complete structure is given where the lines of the finite element mesh are displayed. It can be seen that our model corresponds to a Al2 O3 substrate on which has been deposited a finite film of GaN. The right

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Al 2O3

βyy

0.00E+00 8.33E−02 1.67E−01 2.50E−01 3.33E−01 4.17E−01 5.00E−01 5.83E−01 6.67E−01 7.50E−01 8.33E−01 9.17E−01 1.00E+00

−2.30E−01 −1.92E−01 −1.55E−01 −1.18E−01 −8.05E−02 −4.31E−02 −5.83E−03 3.15E−02 6.88E−02 1.06E−01 1.43E−01 1.81E−01 2.18E−01

βxx

βzz

−5.17E−01 −4.61E−01 −4.06E−01 −3.50E−01 −2.94E−01 −2.39E−01 −1.83E−01 −1.27E−01 −7.17E−02 −1.60E−02 3.97E−02 9.53E−02 1.51E−01 −2.56E−01 −2.20E−01 −1.83E−01 −1.47E−01 −1.10E−01 −7.39E−02 −3.75E−02 −1.08E−03 3.53E−02 7.18E−02 1.08E−01 1.45E−01 1.81E−01

Fig. 13.7 Chemical composition and the lattice distortions βxx , βyy , βzz in the spring-back configuration induced by stress relaxation of the 27-node FE mesh reflecting the global honey-comb structure through a co-set decomposition of the primitive unit-cell described in Section 13.4.1 (c.f. figure 13.6).

panel shows the same as the left, but as a slice through the interface region in the [0001] direction revealing the GaN surface and network of three crossing line dislocations c.f. figures 13.5b and 13.6c. Residual stresses arising in the interface region are given in figure 13.8. These results indicate that a honeycomb network of misfit dislocations arising from the lattice mismatch between GaN and Al2 O3 is energetically stable.

σxx

σxx

σxy

σxy

Fig. 13.8 A cross-section through the [0001] direction revealing relaxed honeycomb Cauchy stress field [Pa] on (left) Al2 O3 and right GaN surface.

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13.5 Conclusions The aim of this paper was to achieve modelling of dislocations by means of the atomistic and continuum models including: (a) The reconstruction (preprocessing) of the dislocations and stacking faults by means of the use of a modified analytical formulae to input the mixed dislocation into the atomistic model, see (13.7) and Fig. 13.2. The reconstruction of analogical models based on the analysis of the HRTEM image is also discussed, see and Fig. 13.3. (b) The use of the computer image processing based on GPA as the input data to the FEA of interfacial region, see Fig. 13.4. The modelling takes into account the residual stresses and lattice distortion distribution induced by both: (a) the chemical mismatch of different type crystal lattices (b) misfit dislocations and (c) elastic residual stress/strain field accommodating the remaining mismatch. The continuum models used in this work have the significant advantage of treating large volumes without utilising millions of atoms in the formalism while providing an understanding of defect interactions. As a realistic example of this we have investigated the strain response affected by misfit dislocations in the heterostructure GaN/Al2 O3 . This was achieved starting from observations of HRTEM images, and through its computer image processing resulting 2D lattice distortion field, reconstruction of missing distortion field in the 3rd direction, and finally by the 3D FE modelling of the misfit dislocation net. The results from our method of modelling have been shown to be consistent with experiment. The method we have employed here is not limited to a traditional film/substrate system and can be easily extended to include more complex structures; such as for example, sequential multi-layers intersected by treading dislocations. Furthermore the computational process presented here can be used hand-in-hand with other techniques of HRTEM interpretation.

Acknowledgements This research was carried out in the framework of the European Commission PARSEM project MRTN-CT-2004-005583 additionally subsided by the Ministry of Science and Higher Education (MSHE) in Poland grant 131/6PREU/2005/7 in the period 1 Mar 2006 – 28 Feb 2009. Since 1 Mar 2009 the investigations are continued in the framework of the grant 0634/R/T02/2007/02 founded by MSHE.

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References [1] Béré, A., Serra, A.: Atomic structure of dislocation cores in GaN. Phys. Rev. B 65(20), 205323 (2002) [2] de Wit, R.: Theory of dislocations II, continuous and discrete dislocations in anisotropic elasticity. J. Research of Nat. Bur. Stand (USA) — A. Phys. Chem. 77A, 608 (1973) [3] Dłu˙zewski, P.: Anisotropic hyperelasticity based upon general strain measures. Journal of Elasticity 60(2), 119–129 (2000) [4] Dłu˙zewski, P., Jurczak, G., Antúnez, H.: Logarithmic measure of strains in finite element modelling of anisotropic deformations of elastic solids. Computer Assisted Mechanics and Engineering Science 10, 69–79 (2003) [5] Dłu˙zewski, P., Maciejewski, G., Jurczak, G., Kret, S., Laval, J.-Y.: Nonlinear FE analysis of residual stresses induced by misfit dislocations in epitaxial layers. Computational Materials Science 29(3), 379–395 (2004) [6] Dłu˙zewski, P., Young, T., Dimitrakopoulos, G., Komninou, P.: Continuum and atomistic modelling of the mixed straight dislocation. Int. J. Multiscale Computational Engineering (2009) (accepted for publication) [7] Fall, C.J., Jones, R., Briddon, P.R., Blumenau, A.T., Frauenheim, T., Heggie, M.I.: Influence of dislocations on electron energy-loss spectra in gallium nitride. Phys. Rev. B 65(24), 245304 (2002) [8] Hÿtch, M., Putaux, J.-L., Pénisson, J.-M.: Measurement of the displacement field of dislocations to 0.03Å by electron microscopy. Nature 423, 270–273 (2003) [9] Hÿtch, M.J., Snoeck, E., Kilaas, R.: Quantitative measurement of displacement and strain fields from HTEM micrographs. Ultramicroscopy 74, 131–146 (1998) [10] Kioseoglou, J., Dimitrakopulos, G., Komninou, P., Kehagias, T., Karakostas, T.: Phys. Status Solidi A 203, 2156 (2006) [11] Kioseoglou, J., Dimitrakopulos, G.P., Komninou, P., Karakostas, T.: Atomic structures and energies of partial dislocations in wurtzite GaN. Phys. Rev. B 70(3), 035309 (2004) [12] Kioseoglou, J., Dimitrakopulos, G.P., Komninou, P., Karakostas, T., Aifantis, E.: Dislocation core investigation by geometric phase analysis and the dislocation density tensor. J. Phys. D: Appl. Phys. 41, 035408 (2007) [13] Komninou, P., Kioseoglou, J., Dimitrakopulos, G., Kehagias, T., Karakostas, T.: Phys. Status Solidi A 202, 2888 (2005) [14] Kret, S., Dłu˙zewski, P., Dłu˙zewski, P., Laval, J.-Y.: On the measurement of dislocation cores distribution in GaAs/ZnTe/CdTe heterostructure by transmission electron microscopy. Philosophical Magazine A 83, 231–244 (2003) [15] Kröner, E.: Continuum theory of defects. In: Balian, R., Kleman, M., Poiries, J.-P. (eds.) Physics of Defects, pp. 215–315. North-Holland, Amsterdam (1981) [16] Lee, S.M., Belkhir, M.A., Zhu, X.Y., Lee, Y.H., Hwang, Y.G., Frauenheim, T.: Electronic structures of gan edge dislocations. Phys. Rev. B 61(23), 16033–16039 (2000) [17] Leung, K., Wright, A., Stechel, E.: Charge accumulation at a threading edge dislocation in gallium nitride. Applied Physics Letters 74, 2495 (1999) [18] Love, A.E.H.: Mathematical Theory of Elasticity. Cambridge University Press, Cambridge (1927) [19] Lymperakis, L.: Ab-Initio Based Multiscale Calculations of Extended Defects in and on Group III – Nitrides. PhD thesis, Department Physik der Fakultät für Naturwissenschaften an der Universität Paderborn (2005)

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[20] Lymperakis, L., Neugebauer, J., Albrecht, M., Remmele, T., Strunk, H.P.: Strain induced deep electronic states around threading dislocations in GaN. Phys. Rev. Lett. 93(19), 196401 (2004) [21] Potin, V., Ruterana, P., Nouet, G., Pond, R.C., Morkoç, H.: Mosaic growth of GaN on (0001) sapphire: A high-resolution electron microscopy and crystallographic study of threading dislocations from low-angle to high-angle grain boundaries. Phys. Rev. B 61(8), 5587–5599 (2000) [22] Read Jr., W.T.: Dislocations in Crystals. McGraw-Hill, London (1953) [23] Spaepen, F.: Interfaces and stresses in thin films. Acta Materialia 48, 31–42 (2000) [24] Stadelmann, P.: EMS - A software package for electron diffraction analysis and hrem image simulation in materials science. Ultramicroscopy 21, 131 (1987) [25] Wright, A., Grossner, U.: The effect of doping and growth stoichiometry on the core structure of a threading edge dislocation in GaN. Applied Physics Letters 73, 2751 (1998)

Chapter 14

Accuracy and Robustness of a 3-D Brick Cosserat Point Element (CPE) for Finite Elasticity M. Jabareen and M.B. Rubin

Abstract. The theory of a Cosserat point has been used to develop a 3-D brick Cosserat Point Element (CPE) for the numerical solution of problems in finite elasticity. In standard finite element procedures, an approximation of the kinematics (with possible strain enhancements) is assumed to be valid at each point in the element. Then, the stiffness of the element is obtained by integration over the element region. In contrast, the Cosserat approach for an elastic element connects the kinetic quantities to derivatives of a strain energy function. Once the strain energy of the CPE has been specified, the procedure needs no integration over the element region and it ensures that the response of the CPE is hyperelastic. Moreover, since the CPE is a structure, the strain energy function depends on both the reference element geometry and on the three-dimensional strain energy function of the elastic material. Restrictions on the strain energy function of the CPE have been developed which ensure that the CPE reproduces the exact constitutive response to all nonlinear homogeneous deformations. Consequently, the constitutive equations of the CPE analytically satisfy a nonlinear form of the patch test. It is convenient to separate the strain energy function into two parts: one controlling homogeneous deformations and the other controlling inhomogeneous deformations like: bending, torsion and higher-order hourlgassing modes of deformation. Developing a functional form for the strain energy of inhomogeneous deformations has proven to be challenging. Nevertheless, a functional form has been proposed which causes the CPE to be a truly user friendly element that can be used with confidence for problems of finite Mahmood Jabareen Faculty of Civil and Environmental Engineering, Technion - Israel Institute of Technology 32000 Haifa, Israel e-mail: [email protected] M.B. Rubin Faculty of Mechanical Engineering, Technion - Israel Institute of Technology 32000 Haifa, Israel e-mail: [email protected] This research was partially supported by MB Rubin’s Gerard Swope Chair in Mechanics and by the fund for the promotion of research at the Technion. M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 255–269. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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elasticity including: three-dimensional bodies, thin shells and thin rods. Examples are discussed which compare the response of the CPE with that of other elements based on enhanced strain and incompatible mode methods.

14.1 Introduction It is well known [34] that the standard eight node 3-D brick element for a nonlinear hyperelastic material based on the full integration Bubnov-Galerkin method using tri-linear shape functions exhibits unphysical locking for thin structures with poor element aspect ratios and for nearly incompressible materials. Special methods based on enhanced strains, incompatible modes and reduced integration with hourglass control (e.g. [6]-[8], [13], [23]-[25], [30]-[32]) have been developed to overcome these problems. However, it is also known (e.g.[14], [15], [20], [23], [24])) that these improved formulations can exhibit unphysical hourglassing in regions experiencing combined high compression with bending. This paper summarizes recent research on an eight node 3-D brick Cosserat Point Element (CPE) for finite deformations of a hyperelastic material. The CPE represents a new element technology that is based on the theory of a Cosserat point. Cosserat theories have been developed to model structures like shells (e.g. [22]) which are "thin" in one of their dimensions and structures like rods (e.g. [4], [5], [11], [12]) which are "thin" in two of their dimensions. For a Cosserat shell the position vector to material points on the reference surface of the shell is supplemented by a single director vector which models deformations of a line element through the shell’s thickness. For a Cosserat rod the position vector to material points on the reference curve is supplemented by two director vectors which model line elements in the rod’s cross-section. Following this line of thought it is natural to model a structure like a finite element which is "thin" in all three of its dimensions. The theory of a Cosserat Point is a continuum theory that has been developed to model the deformation of such structures. In its earliest form ([26], [27]) attention was limited to one-dimensional deformations and it was shown that the theory can be used to formulate the numerical solution of problems of finite elasticity in continuum mechanics. Later [28] the theory was generalized for the numerical solution of two- and three-dimensional thermomechanical problems. A discussion of these Cosserat theories of structures can be found in [29]. The theory of a Cosserat point is a fully nonlinear continuum theory which is developed in the spirit of three-dimensional continuum mechanics. Within the context of the direct approach the theory is proposed as a continuum model by introducing kinematics characterized by director vectors and kinetic balance laws which are used to determine these director vectors. Also, expressions for rate of work, kinetic energy, strain energy and rate of dissipation are used to develop restrictions on the constitutive equations in the theory. For an elastic CPE these equations determine the kinetic quantities in terms of derivatives of the strain energy function, just as

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the stress is related to derivatives of the strain energy function for three-dimensional hyperelastic materials. Alternatively, the kinematics and kinetics in the CPE can be related to standard expressions which are obtained by the Bubnov-Galerkin approach with shape functions based on a tri-linear kinematic approximation. In Section 14.2 the basic equations of the CPE will be developed using the standard Bubnov-Galerkin approach. However, in Section 14.3 it will be emphasized that for the CPE the constitutive equations are necessarily developed by the direct approach which does not assume that the kinematic approximation used in the Bubnov-Galerkin approach is valid pointwise. The strain energy of the CPE characterizes the response of the CPE as a structure and thus it depends on both the material and geometric properties of the CPE. In Section 14.4 restrictions on the strain energy function of the CPE are discussed which ensure that the CPE satisfies a nonlinear form of the patch test. Section 14.5 presents a functional form for the strain energy of the CPE which depends on both the material and geometric properties of the CPE. This coupling between material and geometric properties of the CPE is similar to what occurs in theories of structures like beam theory where the moment is related to the curvature of the beam by a coefficient EI with E being a material property and with I being a geometric property. Section 14.5 also includes a brief discussion of the procedure for determining constitutive coefficients which depend on the reference geometry of the CPE. Section 14.6 discusses examples which demonstrate the accuracy of the CPE and Section 14.7 discusses examples which demonstrate its robustness. Finally, Section 14.8 presents the main conclusion based on the results of numerous example problems.

14.2 Basic Equations of the CPE Using the Bubnov-Galerkin Approach

Fig. 14.1 Sketch of a general brick CPE showing the numbering of the nodes.

Figure 14.1 shows a sketch of an eight node 3-D brick CPE. Here, and throughout the text a superposed ( ∗ ) is used to denote some quantities related to the threedimensional theory which have counterparts in the CPE. Material points in the

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unstressed reference configuration are characterized by the convected coordinates θ i (i = 1, 2, 3) and are located by the position vector X∗ defined by X∗ =

7

∑ N j (θ i )D j

(14.1)

j=0

where D j are constant element director vectors and N j are tri-linear shape functions N 0 = 1, N 1 = θ 1 , N 2 = θ 2 , N 3 = θ 3 , N 4 = θ 1θ 2 , N 5 = θ 1 θ 3 , N 6 = θ 2 θ 3, N 7 = θ 1 θ 2θ 3

(14.2)

In contrast with standard finite element methods, the convected coordinates θ i have units of length and are restricted by |θ 1 | ≤

H1 H2 H3 , |θ 2 | ≤ , |θ 3 | ≤ 2 2 2

(14.3)

where Hi are characteristic lengths of the CPE determined by the conditions that |D1 | = 1, |D2 | = 1, |D3 | = 1

(14.4)

Within the context of the Bubnov-Galerkin approach the position vector x∗ to material points in the deformed present configuration of the CPE is expressed in the form x∗ =

7

∑ N jdj

(14.5)

j=0

where the present values d j (t) of the element directors are functions of time only. Also, the triads Di and di (i = 1, 2, 3) are restricted to be linearly independent with D1/2 = D1 × D2 • D3 > 0, d 1/2 = d1 × d2 • d3 > 0

(14.6)

Furthermore, the reciprocal vectors Di and di are defined by Di • D j = δ ji , di • d j = δ ji

(14.7)

where δ ji is the Kronecker delta symbol. Using these representations it is possible to determine the reference base vectors Gi , their reciprocal vectors Gi , the present base vectors gi , their reciprocal vectors gi and the velocity field v∗ , such that

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Gi = X∗,i , Gi • G j = δ ji , (i, j = 1, 2, 3) G1/2 = G1 × G2 • G3 , gi = x∗,i , gi • g j = δ ji , (i, j = 1, 2, 3) g1/2 = g1 × g2 • g3 v∗ = x˙ ∗ =

7

∑ N jw j,

w j = d˙ j

(14.8)

j=0

where a comma denotes partial differentiation with respect to θ i , a superposed (˙) denotes material time differentiation holding θ i fixed and w j are the director velocities. For the CPE it is convenient to define a deformation tensor F associated with homogeneous deformations and deformation vectors β i associated with inhomogeneous deformations by 3

F = ∑ di ⊗ Di , i=1

β i = F−1 di+3 − Di+3, (i = 1, 2, 3, 4)

(14.9)

It then can be shown that the three-dimensional deformation gradient F∗ associated with the kinematic assumption (14.5) is given by

3 4 F∗ = F I + ∑ ∑ N,ij+3 β j ⊗ Gi

(14.10)

i=1 j=1

From this expression it can be deduced that β i are pure measures of inhomogeneous deformations since when they vanish F∗ = F is independent of the coordinates θ i . It is well known (e.g. [10]) that the conservation of mass and the balance of linear momentum can be expressed in the forms

ρ ∗ g1/2 = ρ0∗ G1/2 , ρ ∗ v˙ ∗ = ρ ∗ b∗ + div∗T∗

(14.11)

where ρ0∗ and ρ ∗ are the reference and current values of the mass density, respectively, b∗ is the body force per unit mass, T∗ is the Cauchy stress tensor. Moreover, using the identity 1/2 j g g ,j = 0 (14.12) it follows that the divergence operator div∗ can be expressed in the form g1/2div∗ T∗ =

3

∑ t∗, jj ,

j=1

t∗ j = g1/2 T∗ g j , ( j = 1, 2, 3)

(14.13)

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Now, multiplying (14.11b) by g1/2 and defining the mass quantity m∗ , conservation of mass and balance of linear momentum can be written in the alternative forms 3

m∗ = ρ ∗ g1/2 = m∗ (θ i ), m∗ v˙ ∗ = m∗ b∗ + ∑ t∗, jj

(14.14)

j=1

Next, the Bubnov-Galerkin weak form of the balance of linear momentum can be obtained by multiplying (14.14b) by the shape functions N i to obtain 3 3 N i m∗ v˙ ∗ = N i m∗ b∗ − ∑ N,ij t∗ j + ∑ N i t∗ j , j j=1

(14.15)

j=1

Then, integration over the material region P∗ with closed smooth boundary ∂ P∗ yields 7

∑ myi j w˙ j = mbi + mi − ti,

(i = 0, 1, ..., 7)

(14.16)

j=0

These equations are the same as the director momentum equations of the CPE where the mass m, director inertia quantities yi j , external director couples bi due to body forces and external director couples mi due to surface tractions are defined by m=

P∗

ρ ∗ dv∗ , myi j =

mbi =

P∗

P∗

N i N j ρ ∗ dv∗ = my ji

N i ρ ∗ b∗ dv∗ , mi =

∂ P∗

N i t∗ dv∗

(14.17)

In these expressions dv∗ is the current element of volume, da∗ is the current element of area, t∗ is the traction vector applied to the surface ∂ P∗ and use has been made of conservation of mass in the forms m˙ = 0, y˙i j = 0

(14.18)

Also, the intrinsic director couples ti , which require constitutive equations, are defined by ti =

3

∑ N,ij t∗ j g−1/2dv∗ =

P∗ j=1

3

∑ N,ij gi T∗ dv∗

P∗ j=1

(14.19)

Then, using the definition 7

d 1/2 T = ∑ ti ⊗ di i=1

it can be shown the T is related to the volume averaged Cauchy stress [20]

(14.20)

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d 1/2 T =

P∗

T∗ dv∗

(14.21)

and that the reduced form of the balance of angular momentum requires T to be symmetric TT = T

(14.22)

For a three-dimensional hyperelastic solid a strain energy function Σ ∗ is proposed of the form

Σ ∗ = Σ&(C∗ ), C∗ = F∗T F∗

(14.23)

and the Cauchy stress is obtained by the expression T∗ = 2ρ ∗ F∗

∂ Σ& (C∗ ) ∗T F ∂ C∗

(14.24)

Thus, within the context of the Bubnov-Galerkin approach with full integration the constitutive equations for the intrinsic director couples ti can be determined by numerically evaluating the integral (14.19) which weights the Cauchy stress T∗ by gradients of the shape functions N i . Specifically, for a given strain energy function Σ ∗ , it is assumed that the kinematic approximation (14.1) for F∗ is valid pointwise and that T∗ is determined by the expressions (14.24).

14.3 Constitutive Equations of the CPE Using the Direct Approach Within the context of the direct approach, the kinematics of the CPE are characterized by introducing reference element director vectors Di and their present values di {Di , di (t)}, (i = 0, 1, ..., 7)

(14.25)

which satisfy the restrictions (14.7). Then, the conservation of mass and the balances of director momentum are proposed in the forms (14.18) and (14.16), respectively. Constitutive equations for the intrinsic director couples ti of a hyperelastic CPE are determined by procedures similar to those in the three-dimensional theory. Specifically, the CPE is considered to be a structure and the resistance to all deformational modes of the structure is characterized by a strain energy Σ (per unit mass) of the structure of the form

Σ = Σ (C, β i ), C = FT F

(14.26)

Moreover, it is convenient to introduce the rate of dissipation D by the expression

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˙ − mΣ˙ ≥ 0 D = W−K

(14.27)

where the rate of external work W done on the CPE and its kinetic energy K are defined by 7

W = ∑ (mbi + mi ) • wi i=0 7 7

K=

1 ∑ ∑ myi j wi • w j 2 i=0 j=0

(14.28)

Then, with the help of (14.16), (14.18), (14.20) and (14.22) the rate of dissipation (14.27) can be rewritten in the form 4

D = d 1/2 T • D + ∑ FT ti+3 • β˙ i − mΣ˙ ≥ 0

(14.29)

i=1

where D is the symmetric part of the rate tensor L defined by 3

˙ −1 = ∑ wi ⊗ di , D = L = FF i=1

1 L + LT 2

(14.30)

For an elastic CPE the strain energy function Σ has the general form

Σ = Σ (C, β i , Di )

(14.31)

and the rate of dissipation vanishes for all processes so that standard methods can be used to deduce that the kinetic quantities are related to derivatives of Σ , such that

∂Σ T ∂Σ F , ti+3 = F−T , (i = 1, ..., 4) ∂C ∂β i

7 0 i 1/2 j t = 0, t = d T − ∑ t ⊗ d j • di , (i = 1, 2, 3)

d 1/2 T = 2mF

(14.32)

j=4

The explicit dependence of Σ on Di will be discussed in Section 14.5. Furthermore, it can be shown that if the approximation (14.10) is assumed to be valid pointwise in the element, and if full integration is used, then the expressions (14.19) for the intrinsic director couples ti are consistent with the results (14.32) when the strain energy function Σ is specified by mΣ =

P∗

ρ ∗ Σ&∗ (C∗ )dv∗

(14.33)

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14.4 Restrictions Associated with a Nonlinear form of the Patch Test In [21] restrictions on the strain energy Σ of the CPE were developed which ensure that the CPE will reproduce exact solutions for all homogeneous deformations of a uniform homogeneous anisotropic elastic material for all reference element shapes. Specifically, it was shown there that these restrictions will be satisfied if Σ is related to the strain energy function Σ ∗ of the three-dimensional material such that mΣ (C, β i ) = Σ ∗ (C) + Ψ (β i , Di )

(14.34)

where the auxiliary deformation tensor F and C are defined by

4 T j F = F I+ ∑ β j ⊗V , C = F F

(14.35)

j=1

and the vectors Vi are determined by the reference geometry of the CPE 3

D1/2 V j = ∑

∗ i=1 P0

j+3

N,i

Gi dV ∗ , ( j = 1, 2, 3, 4)

(14.36)

Here, Ψ represents the strain energy per unit mass of inhomogeneous deformations which for the CPE includes bending, torsional and higher-order hourglass modes of deformation. Also, Ψ must satisfy the restrictions that

∂Ψ =0 ∂β i

f or

βi =0

(14.37)

which ensure that a nonlinear form of the patch test is satisfied by the CPE. In the remaining sections of the paper attention is limited to an isotropic compressible Neo-Hookean material which is characterized by the strain energy function 2 1 2 1 ρ0∗ Σ ∗ = K J − 1 + μ α 1 − 1 2 2 J = det(F), α 1 = J −2/3 C • I

(14.38)

where K and μ are the small deformation bulk and shear moduli, respectively, and use has been made of the work of Flory [9] to define α 1 as a scalar pure measure of distortional deformation. In particular, for examples of compressible material response the quantities {K, μ } and the small deformation Poisson’s ratio ν are specified by K = 1 GPa, μ = 0.6 GPa, ν = 0.25

(14.39)

whereas for nearly incompressible response these parameters are specified by

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K = 1000 GPa, μ = 0.6 GPa, ν = 0.4997

(14.40)

14.5 Determination of the Constitutive Coefficients In [14]-[18] the strain energy of inhomogeneous deformation Ψ was taken to be a quadratic function of β i which can be written in the form 2mΨ =

D1/2V μ 12 12 ∑ ∑ Bi j b i b j 6(1 − ν ) i=1 j=1

(14.41)

where the inhomogeneous strains are defined by b1 = H2 β 1 • D1 , b2 = H2 β 3 • D3 , b3 = H1 β 1 • D2 b4 = H1 β 2 • D3 , b5 = H3 β 2 • D1 , b6 = H3 β 3 • D2 b7 = H3 β 1 • D3 , b8 = H2 β 2 • D2 , b9 = H1 β 3 • D1 b10 = H2 H3 β 4 • D1 , b11 = H1 H3 β 4 • D2 b12 = H1 H2 β 4 • D3

(14.42)

By comparing exact solutions of pure bending and torsion of three right cylindrical elements with rhombic cross-sections like that shown in Figure 14.2, values of Bi j where determined in [17] and [18] which depend on Poisson’s ratio ν , the lengths Hi and the metric Di j = D i • D j

(14.43)

These values of Bi j fully couple bending and torsional modes of deformation and cause the CPE to respond well for elements which have distorted reference shapes.

Fig. 14.2 Sketch of the rhombic cross-section of a right cylindrical element.

14.6 Examples Demonstrating Accuracy of the CPE In [18] a number of examples of small deformations of thin structures were considered to examine the accuracy of predictions of the CPE. Specific attention was

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focused on investigating the accuracy of the CPE for irregular shaped elements. For example, solutions were obtained for a thin beam with irregular shaped elements by specifying values of the coefficients ai in Figure 14.3 which distort the beam’s central cross-section. Comparisons of solutions were also made for out-of-plane bending of the thin slanted cantilever beam and for a point load on the corner of a thin partially clamped rhombic plate shown in Figure 14.4. Furthermore, comparisons were made for a point load on the center of a thin fully clamped square plate with an irregular element mesh controlled by the parameters ai shown in Figure 14.5.

Fig. 14.3 Shear load on a thin cantilever beam with an irregular element mesh.

Fig. 14.4 Point load on the corner of a thin partially clamped rhombic plate.

Fig. 14.5 Point load on the center of a thin fully clamped square plate with an irregular element mesh.

These comparisons indicated that the CPE produces results which are as accurate as those predicted by enhanced strain/incompatible mode elements that can be found in the commercial computer codes ABAQUS [1], ADINA [2], ANSYS [3] and in the academic code FEAP [33].

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14.7 Examples Demonstrating Robustness of the CPE In [14], [16], [18] a number of examples of large deformations of thin structures and plane strain problems of nearly incompressible materials were considered to examine the robustness of predictions of the CPE. For example, in [14] a problem of large deformation of a single brick element was investigated which showed that although finite elements may be designed based on a hyperelastic material with a strain energy, the resulting numerical formulation may not be hyperelastic. Specifically, it was shown that the enhanced strain/incompatible modes elements in ABAQUS and ANSYS are hypoelastic and that a reduced integration element with hourglass control in ABAQUS is Cauchy elastic. Other elements in ADINA and FEAP were shown to be hyperelastic for this problem. In this regard, by design the CPE is analytically hyperelastic for all deformation paths. Figure 14.6 shows large deformations caused by a point load on a partially clamped thin rhombic plate and Figure 14.7 shows large deformations of thin circular cylindrical shell subjected to a pair of opposing point loads P. In [16] the problem of plane strain compression of a block was considered for a nearly incompressible material. There it was shown that the CPE exhibits physical shear buckling (see Figure 14.8) whereas the enhanced strain/incompatible mode elements in ABAQUS, ADINA and ANSYS exhibit unphysical hourglassing. Also, Jabareen and Rubin [18] considered plane strain indentation of a rigid perfectly bonded plate into a nearly incompressible block as shown in Figure 14.9. Figure 14.10 shows that the CPE allows flow near the corner of the plate whereas the Q1P0 mixed element in FEAP tends to lock there. The results of the CPE were validated by using the mixed higher order 9 node quadrilateral in FEAP. Recently, Jabareen and Rubin [19] have demonstrated that the CPE passes a nonlinear patch test in uniaxial strain which is failed by some enhanced strain/incompatible mode elements due to an unphysical instability.

Fig. 14.6 Point load on a partially clamped rhombic plate.

14 Cosserat Point Element

Fig. 14.7 Deformed shape of an eighth of a thin circular cylindrical shell subjected to a pair of opposing point loads P.

Fig. 14.8 Physical shear buckling predicted by the CPE for plane strain compression of a nearly incompressible material.

Fig. 14.9 Plane strain indentation of a rigid plate into a block.

Fig. 14.10 Deformed shapes for plane strain indentation of a rigid plate into a block.

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14.8 Conclusions The CPE can be used to model nonlinear elastic deformations of 3-D bodies, thin shells and rods and nearly incompressible materials. In addition, examples have been shown that indicate the CPE is free of hourglass instabilities that are observed in other element formulations in regions experiencing combined high compression with bending. Consequently, the CPE is truly a robust user friendly element that can be used with confidence to model problems in nonlinear elasticity.

References [1] [2] [3] [4] [5] [6] [7] [8]

[9] [10] [11] [12] [13] [14]

[15] [16] [17]

ABAQUS. Inc., Version 6.5-1, Providence RI. 02909-2499 ADINA. Inc., Version 8.3.1, Watertown MA 02472 ANSYS. Inc., University Advanced Version 9 Canonsburg, PA 15317 Antman, S.S.: The Theory of Rods. In: Handbuch der Physik, vol. VIa/2. Springer, Heidelberg (1972) Antman, S.S.: Nonlinear Problems of Elasticity. Springer, New York (1995) Belytschko, T., Ong, J.S.J., Liu, W.K., Kennedy, J.M.: Hourglass control in linear and nonlinear problems. Comput. Methods. Appl. Mech. Engrg. 43, 251–276 (1984) Belytschko, T., Bindeman, L.P.: Assumed strain stabilization of the eight node hexahedral element. Comput. Methods. Appl. Mech. Engrg. 105, 225–260 (1993) Bonet, J., Bhargava, P.: A uniform deformation gradient hexahedron element with artificial hourglass control. International Journal of Numerical Methods in Engineering 38, 2809–2828 (1995) Flory, P.: Thermodynamic relations for high elastic materials. Trans. Faraday Soc. 57, 829–838 (1961) Green, A.E., Adkins, J.E.: Large elastic deformations and non-linear continuum mechanics, Oxford (1961) Green, A.E., Naghdi, P.M., Wenner, M.L.: On The Theory Of Rods I: Derivations From The Three-dimensional Equations. Proc. Royal Soc. London A337, 451–483 (1974) Green, A.E., Naghdi, P.M., Wenner, M.L.: On The Theory Of Rods II: Developments By Direct Approach. Proc. Royal Soc. London A337, 485–507 (1974) Hutter, R., Hora, P., Niederer, P.: Total hourglass control for hyperelastic materials. Comput. Methods. Appl. Mech. Engrg. 189, 991–1010 (2000) Jabareen, M., Rubin, M.B.: Hyperelasticity and physical shear buckling of a block predicted by the Cosserat point element compared with inelasticity and hourglassing predicted by other element formulations. Computational Mechanics 40, 447–459 (2007) Jabareen, M., Rubin, M.B.: Modified torsion coefficients for a 3-D brick Cosserat point element. Computational Mechanics 41, 517–525 (2007) Jabareen, M., Rubin, M.B.: An improved 3-D Cosserat brick element for irregular shaped elements. Computational Mechanics 40, 979–1004 (2007) Jabareen, M., Rubin, M.B.: A generalized Cosserat point element (CPE) for isotropic nonlinear elastic materials including irregular 3-D brick and thin structures. United States Patent and Trademark Office, Serial No. 61/038,xxx (2008)

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[18] Jabareen, M., Rubin, M.B.: A generalized Cosserat point element (CPE) for isotropic nonlinear elastic materials including irregular 3-D brick and thin structures. Journal of Mechanics of Materials and Structures 3, 1465–1498 (2008) [19] Jabareen, M., Rubin, M.B.: Failures of the three-dimensional patch test for large elastic deformations. Communications in Numerical Methods in Engineering (2009) [20] Loehnert, S., Boerner, E.F.I., Rubin, M.B., Wriggers, P.: Response of a nonlinear elastic general Cosserat brick element in simulations typically exhibiting locking and hourglassing. Computational Mechanics 36, 255–265 (2005) [21] Nadler, B., Rubin, M.B.: A new 3-D finite element for nonlinear elasticity using the theory of a Cosserat point. Int. J. Solids and Structures 40, 4585–4614 (2003) [22] Naghdi, P.M.: The theory of shells and plates. In: Truesdell, C. (ed.) S. Flugge’s Handbuch der Physik, vol. VIa/2, pp. 425–640. Springer, Heidelberg (1972) [23] Reese, S., Wriggers, P.: Finite element calculation of the stability behaviour of hyperelastic solids with the enhanced strain methods. Zeitschrift fur angewandte Mathematik und Mechanik 76, 415–416 (1996) [24] Reese, S., Wriggers, P.: A stabilization technique to avoid hourglassing in finite elasticity. Int. J. Numer. Meth. Engng. 48, 79–109 (2000) [25] Reese, S., Wriggers, P., Reddy, B.D.: A new locking free brick element technique for large deformation problems in elasticity. Computers and Structures 75, 291–304 (2000) [26] Rubin, M.B.: On the theory of a Cosserat point and its application to the numerical solution of continuum problems. J. Appl. Mech. 52, 368–372 (1985) [27] Rubin, M.B.: On the numerical solution of one-dimensional continuum problems using the theory of a Cosserat point. J. Appl. Mech. 52, 373–378 (1985) [28] Rubin, M.B.: Numerical solution of two- and three-dimensional thermomechanical problems using the theory of a Cosserat point. J. of Math. and Physics (ZAMP) 46(Special Issue), S308–S334 (1995); Casey, J., Crochet, M.J. (eds.): Theoretical, Experimental, and Numerical Contributions to the Mechanics of Fluids and Solids. Brikhauser, Basel (1995) [29] Rubin, M.B.: Cosserat Theories: Shells, Rods and Points. Solid Mechanics and its Applications, vol. 79. Kluwer, The Netherlands (2000) [30] Simo, J.C., Rifai, M.S.: A class of mixed assumed strain methods and the method of incompatible modes. Int. J. Numer. Meth. Engng. 29, 1595–1638 (1990) [31] Simo, J.C., Armero, F.: Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. Int. J. Numer. Meth. Engng. 33, 1413–1449 (1992) [32] Simo, J.C., Armero, F., Taylor, R.L.: Improved versions of assumed enhanced strain tri-linear elements for 3D finite deformation problems. Comp. Meth. Appl. Mech. Engrg. 110, 359–386 (1993) [33] Taylor, R.L.: FEAP - A Finite Element Analysis Program, Version 7.5. University of California, Berkeley (2005) [34] Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method for Solid and Structural Mechanics, 6th edn. Elsevier, Amsterdam (2005)

Chapter 15

Possibilities of the Particle Finite Element Method in Computational Mechanics Eugenio Oñate, Sergio R. Idelsohn, Miguel Angel Celigueta, Riccardo Rossi, and Salvador Latorre

Abstract. We present some developments in the formulation of the Particle Finite Element Method (PFEM) for analysis of complex coupled problems in fluid and solid mechanics accounting for fluid-structure interaction and coupled thermal effects. The PFEM uses an updated Lagrangian description to model the motion of nodes (particles) in both the fluid and the structure domains. Nodes are viewed as material points which can freely move and even separate from the main analysis domain representing, for instance, the effect of water drops. A mesh connects the nodes defining the discretized domain where the governing equations are solved as in the standard FEM. The necessary stabilization for dealing with the incompressibility of the fluid is introduced via the finite calculus (FIC) method. An incremental iterative scheme for the solution of the non linear transient coupled fluid-structure problem is described. Extensions of the PFEM to allow for frictional contact conditions at fluid-solid and solid-solid interfaces via mesh generation are described. A simple algorithm to treat erosion in the fluid bed is presented. Examples of application of the PFEM to solve a number of coupled problems such as the effect of large Eugenio Oñate, International Center for Numerical Methods in Engineering (CIMNE), Technical University of Catalonia, Campus Norte UPC, 08034 Barcelona, Spain e-mail: [email protected] Sergio R. Idelsohn ICREA Research Professor at International Center for Numerical Methods in Engineering (CIMNE) e-mail: [email protected] Miguel Angel Celigueta, International Center for Numerical Methods in Engineering (CIMNE) e-mail: [email protected] Riccardo Rossi, International Center for Numerical Methods in Engineering (CIMNE) e-mail: [email protected] Salvador Latorre, International Center for Numerical Methods in Engineering (CIMNE) e-mail: [email protected] M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 271–310. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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wave on structures, the large motions of floating and submerged bodies, bed erosion situations and melting and dripping of polymers under the effect of fire are given.

15.1 Introduction The analysis of problems involving the interaction of fluids and structures accounting for large motions of the fluid free surface and the existence of fully or partially submerged bodies which interact among themselves is of big relevance in many areas of engineering. Examples are common in ship hydrodynamics, off-shore and harbour structures, spill-ways in dams, free surface channel flows, environmental flows, liquid containers, stirring reactors, mould filling processes, etc. Typical difficulties of fluid-multibody interaction analysis in free surface flows using the FEM with both the Eulerian and ALE formulation include the treatment of the convective terms and the incompressibility constraint in the fluid equations, the modelling and tracking of the free surface in the fluid, the transfer of information between the fluid and the moving solid domains via the contact interfaces, the modeling of wave splashing, the possibility to deal with large motions of the bodies within the fluid domain, the efficient updating of the finite element meshes for both the structure and the fluid, etc. For a comprehensive list of references in FEM for fluid flow problems see [7, 37] and the references there included. A survey of recent works in fluid-structure interaction analysis can be found in [18], [27] and [35]. Most of the above problems disappear if a Lagrangian description is used to formulate the governing equations of both the solid and the fluid domains. In the Lagrangian formulation the motion of the individual particles are followed and, consequently, nodes in a finite element mesh can be viewed as moving material points (hereforth called “particles”). Hence, the motion of the mesh discretizing the total domain (including both the fluid and solid parts) is followed during the transient solution. The authors have successfully developed in previous works a particular class of Lagrangian formulation for solving problems involving complex interaction between fluids and solids. The method, called the particle finite element method (PFEM, www.cimne.com/pfem), treats the mesh nodes in the fluid and solid domains as particles which can freely move and even separate from the main fluid domain representing, for instance, the effect of water drops. A mesh connects the nodes discretizing the domain where the governing equations are solved using a stabilized FEM. The FEM solution of the variables in the (incompressible) fluid domain implies solving the momentum and incompressibility equations. This is not such as simple problem as the incompressibility condition limits the choice of the FE approximations for the velocity and pressure to overcome the well known div-stability condition [7, 37]. In our work we use a stabilized mixed FEM based on the Finite Calculus (FIC) approach which allows for a linear approximation for the velocity and pressure variables.

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An advantage of the Lagrangian formulation is that the convective terms disappear from the fluid equations. The difficulty is however transferred to the problem of adequately (and efficiently) moving the mesh nodes. We use a mesh regeneration procedure blending elements of different shapes using an extended Delaunay tesselation with special shape functions [11, 13]. The theory and applications of the PFEM are reported in [2, 6, 11, 12, 14, 15, 26, 27, 28, 30, 31, 32]. The aim of this paper is to describe recent advances of the PFEM for a) the analysis of the interaction between a collection of bodies which are floating and/or submerged in the fluid, b) the modeling of bed erosion in open channel flows and c) the analysis of melting and dripping of polymer objects in fire situations. These problems are of great relevance in many areas of civil, marine and naval engineering, among others. It is shown that the PFEM provides a general analysis methodology for treat such a complex problems in a simple and efficient manner. The layout of the paper is the following. In the next section the key ideas of the PFEM are outlined. Next the basic equations for an incompressible thermal flow using a Lagrangian description and the FIC formulation are presented. Then an algorithm for the transient solution is briefly described. The treatment of the coupled FSI problem and the methods for mesh generation and for identification of the free surface nodes are outlined. The procedure for treating at mesh generation level the contact conditions at fluid-wall interfaces and the frictional contact interaction between moving solids is explained. A methodology for modeling bed erosion due to fluid forces is described. Finally, the potencial of the PFEM is shown in its application to problems involving large flow motions, surface waves, moving bodies in water, bed erosion and melting and dripping of polymers in fire situations.

15.2 The Basis of the Particle Finite Element Method Let us consider a domain containing both fluid and solid subdomains. The moving fluid particles interact with the solid boundaries thereby inducing the deformation of the solid which in turn affects the flow motion and, therefore, the problem is fully coupled. In the PFEM both the fluid and the solid domains are modelled using an updated Lagrangian formulation. That is, all variables in the fluid and solid domains are assumed to be known in the current configuration at time t. The new set of variables in both domains are sought for in the next or updated configuration at time t + Δ t (Figure 15.1). The finite element method (FEM) is used to solve the continuum equations in both domains. Hence a mesh discretizing these domains must be generated in order to solve the governing equations for both the fluid and solid problems in the standard FEM fashion. Recall that the nodes discretizing the fluid and solid domains are treated as material particles which motion is tracked during the transient solution. This is useful to model the separation of fluid particles from the main fluid domain in a splashing wave, or soil particles in a bed erosion problem, and to follow their subsequent motion as individual particles with a known density, an initial acceleration

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t

Current configuration t

t

V

t

t

u

Γv

V

Γt

t+

t +

0

V

0

t 0

0

Γv

V

Solid F

S

t +

2

0

0

V

Δt t

Δt Γt

S

x ,u 2

Δt V

t +

F t

t

Δt V

t +

V t

t +

Δu

Γt

ΔtV

Δt Γv

t +

Δt u

F

Next (updated) configuration

≡u

S

Fluid

Initial configuration

x ,u 1

1

We seek for equilibrium at t + Δt

Fig. 15.1 Updated lagrangian description for a continuum containing a fluid and a solid domain

and velocity and subject to gravity forces. The mass of a given domain is obtained by integrating the density at the different material points over the domain. The quality of the numerical solution depends on the discretization chosen as in the standard FEM. Adaptive mesh refinement techniques can be used to improve the solution in zones where large motions of the fluid or the structure occur.

15.2.1 Basic Steps of the PFEM For clarity purposes we will define the collection or cloud of nodes (C) pertaining to the fluid and solid domains, the volume (V) defining the analysis domain for the fluid and the solid and the mesh (M) discretizing both domains. A typical solution with the PFEM involves the following steps. (a) The starting point at each time step is the cloud of points in the fluid and solid domains. For instance nC denotes the cloud at time t = tn (Figure 15.2). (b) Identify the boundaries for both the fluid and solid domains defining the analysis domain nV in the fluid and the solid. This is an essential step as some boundaries (such as the free surface in fluids) may be severely distorted during the solution, including separation and re-entering of nodes. The Alpha Shape method [8] is used for the boundary definition (Section 15.5). (c) Discretize the fluid and solid domains with a finite element mesh n M. In our work we use an innovative mesh generation scheme based on the extended Delaunay tesselation (Section 15.4) [11, 12, 14]. (d) Solve the coupled Lagrangian equations of motion for the fluid and the solid domains. Compute the state variables in both domains at the next (updated)

15 Possibilities of the Particle Finite Element Method n

Initial “cloud” of nodes

275 Solid node Fluid node Fixed boundary node

C

n

C

→

n

Flying Sub-domains

V

n

Γ V

→

M

→

n

Mesh

n

n

nx

nu

n

M

Fixed boundary

M

, . , nv, na ,nε , nε , nσ .

n+1

C

C

→

n+1

n+1

V

→

n+1

n+1

M

→

n+2

n+1

Domain

Cloud

n+1

n

V

C

V

nΓ

Fixed boundary

Domain

n+1

Mesh

M

n+1

M

V

C

n+1x

n+1u

, . , n+1v, n+1a ,n+1ε , n+1ε , n+1σ

etc… Cloud

n+2

C

Fig. 15.2 Sequence of steps to update a “cloud” of nodes representing a domain containing a fluid and a solid part from time n (t = tn ) to time n + 2 (t = tn + 2Δ t)

configuration for t + Δ t: velocities, pressure, viscous stresses and temperature in the fluid and displacements, stresses, strains and temperature in the solid. (e) Move the mesh nodes to a new position n+1C where n + 1 denotes the time tn + Δ t, in terms of the time increment size. This step is typically a consequence of the solution process of step 4. (f) Go back to step 1 and repeat the solution process for the next time step to obtain n+2C. The process is shown in Figure 15.2. Figure 15.3 shows another conceptual example of application of the PFEM to model the melting and dripping of a polymer object under a heat source acting at a boundary. Figure 15.4 shows a typical example of a PFEM solution in 2D. The pictures correspond to the analysis of the problem of breakage of a water column [14, 28]. Figure 15.4a shows the initial grid of four node rectangles discretizing the fluid domain and the solid walls. Figures 15.4b and 15.4c show the mesh for the solution at two later times.

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Fig. 15.3 Sequence of steps to update in time a “cloud” of nodes representing a polymer object under thermal loads represented by a prescribed boundary heat flux q using the PFEM. Crossed circles denote prescribed nodes at the boundary

15.3 FIC/FEM Formulation for a Lagrangian Incompressible Thermal Fluid 15.3.1 Governing Equations The key equations to be solved in the incompressible thermal flow problem, written in the Lagrangian frame of reference, are the following:

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(a)

(b)

(c) Fig. 15.4 Breakage of a water column. (a) Discretization of the fluid domain and the solid walls. Boundary nodes are marked with circles. (b) and (c) Mesh in the fluid domain at two different times

Momentum

ρ

∂ vi ∂ σi j + bi = ∂t ∂xj

in Ω

(15.1)

Mass balance

∂ vi =0 ∂ xi Heat transport

∂T ∂ = ρc ∂t ∂ xi

in Ω

∂T ki +Q ∂ xi

(15.2)

in Ω

(15.3)

In above equations vi is the velocity along the ith global (cartesian) axis, T is the temperature, ρ , c and ki are the density (assumed constant), the specific heat and the conductivity of the material along the ith coordinate direction, respectively, bi and

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Q are the body forces and the heat source per unit mass, respectively and σi j are the (Cauchy) stresses related to the velocities by the standard constitutive equation (for incompressible Newtonian material)

σi j = si j − pδi j

1 1 ∂ vi ∂ v j , ε˙i j = si j = 2μ ε˙i j − δi j ε˙ii + 3 2 ∂ x j ∂ xi

(15.4a) (15.4b)

In Eqs.(15.4), si j is the deviatoric stresses, p is the pressure (assumed to be positive in compression), ε˙i j is the rate of deformation, μ is the viscosity and δi j is the Kronecker delta. In the following we will assume the viscosity μ to be a known function of temperature, i.e μ = μ (T ). Indexes in Eqs.(15.1)–(15.4) range from i, j = 1, nd , where nd is the number of space dimensions of the problem (i.e. nd = 2 for two-dimensional problems). Eqs.(15.1)–(15.4) are completed with the standard boundary conditions of prescribed velocities and surface tractions in the mechanical problem and prescribed temperature and prescribed normal heat flux in the thermal problem [2, 7]. We note that Eqs.(15.1)–(15.3) are the standard ones for modeling the deformation of viscoplastic materials using the so called “flow approach” [38, 39]. In our work the dependence of the viscosity with the strain typical of viscoplastic flows has been simplified to the Newtonian form of Eq.(15.4b).

15.3.2 Discretization of the Equations A key problem in the numerical solution of Eqs.(15.1)–(15.4) is the satisfaction of the incompressibility condition (Eq.(15.2)). A number of procedures to solve his problem exist in the finite element literature [7, 37]. In our approach we use a stabilized formulation based in the so-called finite calculus procedure [19]–[21],[28, 30, 32]. The essence of this method is the solution of a modified mass balance equation which is written as 3 ∂ vi ∂ ∂p +∑τ + πi = 0 ∂ xi i=1 ∂ xi ∂ xi

(15.5)

where τ is a stabilization parameter given by [10]

τ=

2ρ |v| 8 μ + 2 h 3h

−1 (15.6)

In the above, h is a characteristic length of each finite element (such as [A(e) ]1/2 for 2D elements) and |v| is the modulus of the velocity vector. In Eq.(15.5) πi are auxiliary pressure projection variables chosen so as to ensure that the second term in Eq.(15.5) can be interpreted as weighted sum of the residuals of the momentum

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equations and therefore it vanishes for the exact solution. The set of governing equations for the velocities, the pressure and the πi variables is completed by adding the following constraint equation to the set of governing equations [28, 32]

∂p τ wi + πi dV = 0 i = 1, nd (15.7) ∂ xi V where wi are arbitrary weighting functions (no sum in i). The rest of the integral equations are obtained by applying the standard Galerkin technique to the governing equations (15.1), (15.2), (15.3) and (15.5) and the corresponding boundary conditions [28, 32]. We interpolate next in the standard finite element fashion the set of problem variables. For 3D problems these are the three velocities vi , the pressure p, the temperature T and the three pressure gradient projections πi . In our work we use equal order linear interpolation for all variables over meshes of 3-noded triangles (in 2D) and 4-noded tetrahedra (in 3D) [28, 32, 40]. The resulting set of discretized equations has the following form

Momentum Mv˙ + K(μ )v − Gp = f

(15.8)

Gv + Lp + Qπ = 0

(15.9)

ˆ π + QT p = 0 M

(15.10)

CT˙ + HT = q

(15.11)

Mass balance

Pressure gradient projection

Heat transport

˙ = ∂ (·). The different maIn Eqs.(15.8)–(15.11) (·) denotes nodal variables, (·) ∂t trices and vectors are given in the Appendix. The solution in time of Eqs.(15.8)–(15.11) can be performed using any time integration schemes typical of the updated Lagrangian finite element method. A basic algorithm following the conceptual process described in Section 15.2.1 is presented in Box I. n+1 (a) j+1 denotes the values of the nodal variables a at time n + 1 and

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1. LOOP OVER TIME STEPS, t = 1 , NTIME Known values x, v, p, π, T, μ , f , q, C, V , M 2. LOOP OVER NUMBER OF ITERATIONS, i = 1 , NITER • Compute the nodal velocities by solving Eq.(15.8) t

t

t

t

t

t

⎡ 1 ⎢ ⎣ Δt

M + K ⎤⎥ v

⎡ 1 ⎢ ⎣ Δt

C + H⎤⎥

t

+1

t

i

+1

⎦

=

t

t

+1

t

t

t

f +G p t

+1

i

+

1 Δt

Mv t

Compute nodal pressures from Eq.(15.9) T L t +1 p i +1 = −G t +1 v i +1 − Q t +1 πi • Compute nodal pressure gradient projections from Eq.(15.10) n +1 i +1 ˆ −1 ⎡QT ⎤ t +1 pi +1 ˆ ˆ , M π = −M D⎣ D = diag ⎡⎣M D ⎤⎦ ⎦ • Compute nodal temperatures from Eq.(15.11) •

• t

t

+ Δt

⎦

T

i

+1

=

t

+ Δt

q+ 1 C T

Update position of analysis domain nodes:

+ Δt

x +1 = x + i

t

i

t

+ Δt

v +1Δt i

Define new “cloud” of nodes

• t

t

Δt

t

+ Δt

C +1 i

Update viscosity values in terms of temperature

+ Δt

μ = μ(

t

+Δt

T +1 ) i

Check convergence → NO → Next iteration i → i + 1 ↓ YES Next time step t → t + 1 • Identify new analysis domain boundary: +1V +1 • Generate mesh: M Go to 1 t

t

Box I. Flow chart of basic PFEM algorithm for the fluid domain

the j + 1 iterations. We note the coupling of the flow and thermal equations via the dependence of the viscosity μ with the temperature.

15.4 Overview of the Coupled FSI Algoritm Figure 15.5 shows a typical domain V with external boundaries ΓV and Γt where the velocity and the surface tractions are prescribed, respectively. The domain V is formed by fluid (VF ) and solid (VS ) subdomains (i.e. V = VF ∪VS ). Both subdomains interact at a common boundary ΓFS where the surface tractions and the kinematic variables (displacements, velocities and acelerations) are the same for both subdomains. Note that both set of variables (the surface tractions and the kinematic variables) are equivalent in the equilibrium configuration. Let us define t S and t F the set of variables defining the kinematics and the stressstrain fields at the solid and fluid domains at time t, respectively, i.e.

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tV FLUID

tV F

Velocity prescribed boundary Γv

Γt Traction prescribed boundary

SOLID

tV S

tt

Solid Domain Fluid Domain

Γv

tV

F

vFS

tFS tV

Γt

ΓSF

S

ΓFS

Prescribed velocities V at ΓFS in the fluid domain

Prescribed tractions t at ΓSF in the solid domain

ΓFS = ΓSF Note:

tFS and vFS

are equivalent

Fig. 15.5 Split of the analysis domain V into fluid and solid subdomains. Equality of surface tractions and kinematic variables at the common interface

S := [t xs , t us , t vs , t as , t ε s , t σ s , t Ts ]T t F := [t xF , t uF , t vF , t aF , t ε˙ F , t σ F , t TF ]T t

(15.12) (15.13)

where x is the nodal coordinate vector, u, v and a are the vector of displacements, velocities and accelerations, respectively, ε , ε˙ and σ are the strain vector, the strainrate (or rate of deformation) vectors and the Cauchy stress vector, respectively, T is the temperature and subscripts F and S denote the variables in the fluid and solid domains, respectively. In the discretized problem, a bar over these variables denotes nodal values. The coupled fluid-structure interaction (FSI) problem of Figure 15.4 is solved, in this work, using the following strongly coupled staggered scheme: 0. We assume that the variables in the solid and fluid domains at time t (t S and t F) are known. (a) Solve for the variables at the solid domain at time t + Δ t (t+Δ t S) under prescribed surface tractions at the fluid-solid boundary ΓFS . The boundary conditions at the

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part of the external boundary intersecting the domain are the standard ones in solid mechanics. The variables at the solid domain t+Δ t S are found via the integration of the equations of dynamic motion in the solid written as [40] Ms as + gs − fs = 0

(15.14)

where as is the vector of nodal accelerations and Ms , gs and fs are the mass matrix, the internal node force vector and the external nodal force vector in the solid domain. The time integration of Eq. (15.14) is performed using a standard Newmark method. Solve for the variables at the fluid domain at time t + Δ t (t+Δ t F) under prescribed surface tractions at the external boundary Γt and prescribed velocities at the external and internal boundaries ΓV and ΓFS , respectively. An incremental iterative scheme is implemented within each time step to account for non linear geometrical and material effects. Iterate between 1 and 2 until convergence. The above FSI solution algorithm is shown schematically in Box II.

LOOP OVER TIME STEPS n = 1,...ntime n

S,

n

F

LOOP OVER STAGGERED SOLUTION j = 1,...nstag Solve for solid variables (prescribed tractions at

n +1

ΓFS )

LOOP OVER ITERATIONS i = 1,...niter Solve for

n +1

S ji

Integrate Eq.(15.3) using a Newmark scheme Check convergence. Yes: solve for fluid variables NO: Next iteration i ← i + 1 Solve for fluid variables (prescribed velocities at

LOOP OVER ITERATIONS i = 1,...niter Solve for n +1Fji using the scheme of Section 15.4 Check convergence. Yes: go to C Next iteration i ← i + 1 C Check convergence of surface tractions at Yes: Next time step Next staggered solution j ← j + 1, i ← i + 1 Next time step

n +1

S ←n +1 S ij ,

n +1

n +1

ΓFS

F ←n +1 Fji

Box II. Staggered scheme for the FSI problem (see also Figure 15.3)

n +1

ΓFS )

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2D

3D

Fig. 15.6 Generation of non standard meshes combining different polygons (in 2D) and polyhedra (in 3D) using the extended Delaunay technique.

15.5 Generation of a New Mesh One of the key points for the success of the PFEM is the fast regeneration of a mesh at every time step on the basis of the position of the nodes in the space domain. Indeed, any fast meshing algorithm can be used for this purpose. In our work the mesh is generated at each time step using the so called extended Delaunay tesselation (EDT) presented in [11, 13, 14]. The EDT allows one to generate non standard meshes combining elements of arbitrary polyhedrical shapes (triangles, quadrilaterals and other polygons in 2D and tetrahedra, hexahedra and arbitrary polyhedra in 3D) in a computing time of order n, where n is the total number of nodes in the mesh (Figure 15.6). The C◦ continuous shape functions of the elements can be simply obtained using the so called meshless finite element interpolation (MFEM). In our work the simpler linear C◦ interpolation has been chosen [11, 13, 14]. Figure 15.7 shows the evolution of the CPU time required for generating the mesh, for solving the system of equations and for assembling such a system in terms of the number of nodes. the numbers correspond to the solution of a 3D flow in an open channel with the PFEM [32]. The figure shows the CPU time in seconds for each time step of the algorithm of Section 15.3.2. We see that the CPU time required for meshing grows linearly with the number of nodes, as expected. Note also that the CPU time for solving the equations exceeds that required for meshing as the number of nodes increases. This situation has been found in all the problems solved with the PFEM. As a general rule for large 3D problems meshing consumes around 30% of the total CPU time for each time step, while the solution of the equations and the assembling of the system consume approximately 40% and 20% of the CPU time for each time step, respectively. These figures prove that the generation of the mesh has an acceptable cost in the PFEM.

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Fig. 15.7 3D flow problem solved with the PFEM. CPU time for meshing, assembling and solving the system of equations at each time step in terms of the number of nodes

15.6 Identification of Boundary Surfaces One of the main tasks in the PFEM is the correct definition of the boundary domain. Boundary nodes are sometimes explicitly identified. In other cases, the total set of nodes is the only information available and the algorithm must recognize the boundary nodes. In our work we use an extended Delaunay partition for recognizing boundary nodes. Considering that the nodes follow a variable h(x) distribution, where h(x) is typically the minimum distance between two nodes, the following criterion has been used. All nodes on an empty sphere with a radius greater than α h, are considered as boundary nodes. In practice α is a parameter close to, but greater than one. Values of α ranging between 1.3 and 1.5 have been found to be optimal in all examples analyzed. This criterion is coincident with the Alpha Shape concept [8]. Figure 15.8 shows an example of the boundary recognition using the Alpha Shape technique. Once a decision has been made concerning which nodes are on the boundaries, the boundary surface is defined by all the polyhedral surfaces (or polygons in 2D) having all their nodes on the boundary and belonging to just one polyhedron. The method described also allows one to identify isolated fluid particles outside the main fluid domain. These particles are treated as part of the external boundary where the pressure is fixed to the atmospheric value. We recall that each particle is a material point characterized by the density of the solid or fluid domain to which it belongs. The mass which is lost when a boundary element is eliminated due to departure of a node (a particle) from the main analysis domain is again regained

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Fig. 15.8 Identification of individual particles (or a group of particles) starting from a given collection of nodes.

when the “flying” node falls down and a new boundary element is created by the Alpha Shape algorithm (Figures 15.2 and 15.8). The boundary recognition method above described is also useful for detecting contact conditions between the fluid domain and a fixed boundary, as well as between different solids interacting with each other. The contact detection procedure is detailed in the next section. We note that the main difference between the PFEM and the classical FEM is just the remeshing technique and the identification of the domain boundary at each time step. The rest of the steps in the computation are coincident with those of the classical FEM.

15.7 Treatment of Contact Conditions in the PFEM 15.7.1 Contact between the Fluid and a Fixed Boundary The motion of the solid is governed by the action of the fluid flow forces induced by the pressure and the viscous stresses acting at the common boundary ΓFS , as mentioned above. The condition of prescribed velocities at the fixed boundaries in the PFEM are applied in strong form to the boundary nodes. These nodes might belong to fixed external boundaries or to moving boundaries linked to the interacting solids. Contact between the fluid particles and the fixed boundaries is accounted for by the incompressibility condition which naturally prevents the fluid nodes to penetrate into the solid boundaries (Figure 15.9). This simple way to treat the fluid-wall contact at mesh generation level is a distinct and attractive feature of the PFEM formulation.

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Contact between fluid and fixed boundary n

C n

V Fluid

n

Γ e

hb > hcrit

Air

Air

e

Fixed boundary

n+1

C

n+1

n+1

V Fluid

Γ e

h > hcrit

Air

e

e

Fluid fixed boundary element h < hcrit

n+1

V

Fluid

n+1

Γ ut

i

tVe

i

t+ΔtVe

he < hc

Air Fixed boundary Fixed boundary

tVe

= t+ΔtVe

Node i moves in tangential direction due to incompressibility Fluid fixed boundary element

This prevents the node to penetrate into the fixed boundary

Contact is detected during mesh generation There is no need for a contact search algorithm Fig. 15.9 Automatic treatment of contact conditions at the fluid-wall interface

15.7.2 Contact between Solid-Solid Interfaces The contact between two solid interfaces is simply treated by introducing a layer of contact elements between the two interacting solid interfaces. This layer is automatically created during the mesh generation step by prescribing a minimum distance

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(hc ) between two solid boundaries. If the distance exceeds the minimum value (hc ) then the generated elements are treated as fluid elements. Otherwise the elements are treated as contact elements where a relationship between the tangential and normal forces and the corresponding displacement is introduced so as to model elastic and frictional contact effects in the normal and tangential directions, respectively (Figure 15.10). This algorithm has proven to be very effective and it allows to identifying and modeling complex frictional contact conditions between two or more interacting bodies moving in water in an extremely simple manner. Of course the accuracy of this contact model depends on the critical distance above mentioned. This contact algorithm can also be used effectively to model frictional contact conditions between rigid or elastic solids in standard structural mechanics applications. Figures 15.11–15.14 show examples of application of the contact algorithm to the bumping of a ball falling in a container, the failure of an arch formed by a collection of stone blocks under a seismic loading and the motion of five tetrapods as they fall and slip over an inclined plane, respectively. The images in Figures 15.11 and 15.14 show explicitely the layer of contact elements which controls the accuracy of the contact algorithm.

Contact between solid boundaries Fluid domain

t+

Solid t

M

Δt M

Solid

h < hc Fixed boundary

Solid

Fti Fni

Contact elements at the fixed boundary

Contact elements are introduced between the solid-solid interfaces during mesh generation

Vni h < hc

Contact forces

Vti

i e

Contact interface

Fti = - β K1(hc - h) Sign(Vti) Fni = K1(hc - h) – K2 Vni Sign(Vni)

Fig. 15.10 Contact conditions at a solid-solid interface

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Fig. 15.11 Bumping of a ball within a container. The layer of contact elements is shown at each contact instant

15.8 Modeling of Bed Erosion Prediction of bed erosion and sediment transport in open channel flows are important tasks in many areas of river and environmental engineering. Bed erosion can lead to instabilities of the river basin slopes. It can also undermine the foundation of bridge piles thereby favouring structural failure. Modeling of bed erosion is also relevant for predicting the evolution of surface material dragged in earth dams in

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Fig. 15.12 Failure of an arch formed by stone blocks under seismic loading

overspill situations. Bed erosion is one of the main causes of environmental damage in floods. Bed erosion models are traditionally based on a relationship between the rate of erosion and the shear stress level [16, 36]. The effect of water velocity on soil erosion was studied in [34]. In a recent work we have proposed an extension of the PFEM to model bed erosion [31]. The erosion model is based on the frictional work at the bed surface originated by the shear stresses in the fluid. The resulting erosion model resembles Archard law typically used for modeling abrasive wear in surfaces under frictional contact conditions [1, 24].

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Fig. 15.13 Motion of five tetrapods on an inclined plane

The algorithm for modeling the erosion of soil/rock particles at the fluid bed is the following: (a) Compute at every point of the bed surface the resultant tangential stress τ induced by the fluid motion. In 3D problems τ = (τs2 + τt )2 where τs and τt are the tangential stresses in the plane defined by the normal direction N at the bed node. The value of τ for 2D problems can be estimated as follows:

τt = μγt with

(15.15a)

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Fig. 15.14 Detail of five tetrapods on an inclined plane. The layer of elements modeling the frictional contact conditions is shown

γt =

vk 1 ∂ vt = t 2 ∂n 2hk

(15.15b)

where vtk is the modulus of the tangential velocity at the node k and hk is a prescribed distance along the normal of the bed node k. Typically hk is of the order of magnitude of the smallest fluid element adjacent to node k (Figure 15.15). (b) Compute the frictional work originated by the tangential stresses at the bed surface as

2 t t μ vtk Wf = τt γt dt = dt (15.16) hk ◦ ◦ 4 Eq.(16) is integrated in time using a simple scheme as n

W f = n−1W f + τt γt Δ t

(15.17)

(c) The onset of erosion at a bed point occurs when nW f exceeds a critical threshold value Wc defined empirically according to the specific properties of the bed material. 4. If nW f > Wc at a bed node, then the node is detached from the bed region and it is allowed to move with the fluid flow, i.e. it becomes a fluid node. As a consequence, the mass of the patch of bed elements surrounding the bed node vanishes in the bed domain and it is transferred to the new fluid node. This mass is subsequently transported with the fluid. Conservation of mass of the bed particles within the fluid is guaranteed by changing the density of the new fluid node so that the mass of the suspended sediment traveling with the fluid equals the mass originally assigned to the bed node. Recall that the mass assigned to a node is computed by multiplying the node density by the tributary domain of the node.

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Bed erosion due to fluid forces

τ μγ = t

Fluid m k

n

l

i

j

h

Bed domain n

k

t

V

K

γt=

t

1 ∂V 2 ∂n

=

t

τt k V

k

K 2h k

Vt

t

k

k

Bed

∫

t

0

τ γ t

t

dt =

∫

t

0

μ 4

VK t

h

k

Fluid m k i

2 dt > W

Then release node

c

“Eroded” domain Wk k

l i

j

k

l j

Bed domain

Fig. 15.15 Modeling of bed erosion by dragging of bed material

5. Sediment deposition can be modeled by an inverse process to that described in the previous step. Hence, a suspended node adjacent to the bed surface with a velocity below a threshold value is assigned to the bed surface. This automatically leads to the generation of new bed elements adjacent to the boundary of the bed region. The original mass of the bed region is recovered by adjusting the density of the newly generated bed elements. Figure 15.15 shows an schematic view of the bed erosion algorithm proposed.

15.9 Examples The examples presented below show the applicability of the PFEM to solve problems involving large motions of the free surface, fluid-multibody interactions, bed erosion and melting and dripping of polymers in fire situations.

15.9.1 Rigid Objects Falling into Water The analysis of the motion of submerged or floating objects in water is of great interest in many areas of harbour and coastal engineering and naval architecture among others.

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Fig. 15.16 2D simulation of the penetration and evolution of a cube and a cylinder in a water container. The colours denote the different sizes of the elements at several times

Fig. 15.17 Detail of element sizes during the motion of a rigid cylinder within a water container

Figure 15.16 shows the penetration and evolution of a cube and a cylinder of rigid shape in a container with water. The colours denote the different sizes of the elements at several times. In order to increase the accuracy of the FSI problem smaller size elements have been generated in the vicinity of the moving bodies during their motion (Figure 15.17).

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Fig. 15.18 Evolution of a water column within a prismatic container including a vertical cylinder

15.9.2 Impact of Water Streams on Rigid Structures Figure 15.18 shows an example of a wave breaking within a prismatic container including a vertical cylinder. Figure 15.19 shows the impact of a wave on a vertical column sustained by four pillars. The objective of this example was to model the impact of a water stream on a bridge pier accounting for the foundation effects.

15.9.3 Dragging of Objects by Water Streams Figure 15.20 shows the effect of a wave impacting on a rigid cube representing a vehicle. This situation is typical in flooding and Tsunami situations. Note the layer of contact elements modeling the frictional contact conditions between the cube and the bottom surface.

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Fig. 15.19 Impact of a wave on a prismatic column on a slab sustained by four pillars.

15.9.4 Impact of Sea Waves on Piers and Breakwaters Figure 15.21 shows the 3D simulation of the interaction of a wave with a vertical pier formed by a collection of reinforced concrete cylinders. Figure 15.22 shows the simulation of the falling of two tetrapods in a water container. Figure 15.23 shows the motion of a collection of ten tetrapods placed in the slope of a breakwaters under an incident wave. Figure 15.24 shows a detail of the complex three-dimensional interactions between water particles and tetrapods and between the tetrapods themselves. Figures 15.25 and 15.26 show the analysis of the effect of breaking waves on two different sites of a breakwater containing reinforced concrete blocks (each one of 4 × 4 mts). The figures correspond to the study of Langosteira harbour in La Coruna, Spain using PFEM. Figure 15.27 displays the effect of an overtopping wave on a truck circulating by the perimetral road of the harbour adjacent to the breakwater.

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Fig. 15.20 Dragging of a cubic object by a water stream.

Fig. 15.21 Interaction of a wave with a vertical pier formed by reinforced concrete cylinders.

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Fig. 15.22 Motion of two tetrapods falling in a water container.

15.9.5 Erosion of a 3D Earth Dam Due to an Overspill Stream We present a simple, schematic, but very illustrative example showing the potential of the PFEM to model bed erosion in free surface flows. The example represents the erosion of an earth dam under a water stream running over the dam top. A schematic geometry of the dam has been chosen to simplify the computations. Sediment deposition is not considered in the solution. The images of Figure 15.28 show the progressive erosion of the dam until the whole dam is dragged out by the fluid flow. Other applications of the PFEM to bed erosion problems can be found in [31].

15.9.6 Melting and Spread of Polymer Objects in Fire In the next example shown the PFEM is used to simulate an experiment performed at the National Institute for Stanford and Technology (NIST) in which a slab of polymeric material is mounted vertically and exposed to uniform radiant heating on one face. It is assumed that the polymer melt flow is governed by the equations of an incompressible fluid with a temperature dependent viscosity. A quasi-rigid

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Fig. 15.23 Motion of ten tetrapods on a slope under an incident wave.

behaviour of the polymer object at room temperature is reproduced by using a very high value of the viscosity parameter. As temperature increases in the thermoplastic object due to heat exposure, the viscosity decreases in several orders of magnitude as a function of temperature and this induces the melt and flow of the particles in the heated zone. Polymer melt is captured by a pan below the sample. A schematic of the apparatus used in the experiments is shown in Figure 15.29. A rectangular polymeric sample of dimensions 10 cm high by 10 cm wide by 2.5 cm thick is mounted upright and exposed to uniform heating on one face from a radiant cone heater placed on its side. The sample is insulated on its lateral and rear faces. The melt flows down the heated face of the sample and drips onto a surface below. A load cell monitors the mass of polymer remaining in the sample, and a laboratory balance measures the mass of polymer falling onto the catch surface. Details of the experimental setup are given in [4]. Figure 15.29 shows all three curves of viscosity vs. temperature for the polypropylene type PP702N, a low viscosity commercial injection molding resin formulation. The relationship used in the model, as shown by the black line, connects the curve

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Fig. 15.24 Detail of the motion of ten tetrapods on a slope under an incident wave. The figure shows the complex interactions between the water particles and the tetrapods.

for the undegraded polymer to points A and B extrapolated from the viscosity curve for each melt sample to the temperature at which the sample was formed. The result is an empirical viscosity-temperature curve that implicitly accounts for molecular weight changes. The finite element mesh used for the analysis has 3098 nodes and 5832 triangular elements. No nodes are added during the course of the run. The addition of a catch pan to capture the dripping polymer melt tests the ability of the PFEM model to recover mass when a particle or set of particles reaches the catch surface. For this problem, heat flux is only applied to free surfaces above the midpoint between the catch pan and the base of the sample. However, every free surface is subject to radiative and convective heat losses. To keep the melt fluid, the catch pan is set to a temperature of 600 K. Figure 15.30 shows four snapshots of the time evolution of the melt flow into the catch pan.

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Fig. 15.25 Effect of breaking waves on a breakwater slope containing reinforced concrete blocks. Detail of the mesh of 4-noded tetrahedra near the slope at two different times

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Fig. 15.26 Study of breaking waves on the edge of a breakwater structure formed by reinforced concrete blocks

To test the ability of the PFEM to solve this type of problem in three dimensions, a 3D problem for flow from a heated sample was run. The same boundary conditions are used as in the 2D problem illustrated in Figure 15.29, but the initial dimensions of the sample are reduced to 10× 2.5 × 2.5 cm. The initial size of the model is 22475 nodes and 97600 four-noded tetrahedra. The shape of the surface and temperature field at different times after heating begins are shown in Figure 15.32. Although the resolution for this problem is not fine enough to achieve high accuracy, the qualitative agreement of the 3D model with 2D flow and the ability to carry out this problem in a reasonable amount of time suggest that the PFEM can be used to model melt flow and spread of complex 3D polymer geometry. Figure 15.31 shows results for the analysis of the melt flow of a triangular thermoplastic object into a catch pan. The material properties for the polymer are the same as for the previous example. The PFEM succeeds to predicting in a very realistic manner the progressive melting and slip of the polymer particles along the vertical wall separating the triangular object and the catch pan. The analysis follows until the whole object has fully melt and its mass is transferred to the catch pan.

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Fig. 15.27 Effect of an overtopping wave on a truck passing by the perimetral road of a harbour adjacent to the breakwater

Fig. 15.28 Erosion of a 3D earth dam due to an overspill stream.

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Fig. 15.29 Polymer melt experiment. Viscosity vs. temperature for PP702N polypropylene in its initial undegraded form and after exposure to 30 kW/m2 and 40 kW/m2 heat fluxes. The black curve follows the extrapolation of viscosity to high temperatures.

We note that the total mass was preserved with an accuracy of 0.5% in both these studies. Gasification, in-depth absorption or radiation were not taken into account in these analysis. More examples of application of PFEM to the melting and dripping of polymers are reported in [33].

15.10 Conclusions The particle finite element method (PFEM) is ideal to treat problems involving fluidstructure interaction, large motion of fluid or solid particles, surface waves, water splashing, separation of water drops, frictional contact situations between fluid-solid and solid-solid interfaces, bed erosion, coupled thermal effects, melting and dripping of objects, etc. The success of the PFEM lies in the accurate and efficient solution of the equations of an incompressible fluid and of solid dynamics using an

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Fig. 15.30 Evolution of the melt flow into the catch pan at t = 400s, 550s, 700s and 1000s

updated Lagrangian formulation and a stabilized finite element method, allowing the use of low order elements with equal order interpolation for all the variables. Other essential solution ingredients are the efficient regeneration of the finite element mesh using an extended Delaunay tesselation, the identification of the boundary nodes using the Alpha-Shape technique and the simple algorithm to treat frictional contact conditions at fluid-solid and solid-solid interfaces via mesh generation. The examples presented have shown the great potential of the PFEM for solving a wide class of practical FSI problems in engineering. Examples of validation of the PFEM results with data from experimental tests are reported in [17].

Acknowledgements Thanks are given to Mrs. M. de Mier for many useful suggestions. This research was partially supported by project SEDUREC of the Consolider Programme of the

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Fig. 15.31 Melt flow of a heated triangular object into a catch pan.

Ministerio de Educación y Ciencia (MEC) of Spain, project XPRES of the National I+D Programme of MEC (Spain) and project SAYOM of CDTI Spain. Thanks are also given to the Spanish construction company Dragados for financial support for the study of harbour engineering problems.

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Fig. 15.32 Solution of a 3D polymer melt problem with the PFEM. Melt flow from a heated prismatic sample at different times.

References [1] Archard, J.F.: Contact and rubbing of flat surfaces. J. Appl. Phys. 24(8), 981–988 (1953) [2] Aubry, R., Idelsohn, S.R., Oñate, E.: Particle finite element method in fluid mechanics including thermal convection-diffusion. Computer & Structures 83(17-18), 1459–1475 (2005) [3] Butler, K.M., Ohlemiller, T.J., Linteris, G.T.: A Progress Report on Numerical Modeling of Experimental Polymer Melt Flow Behavior. Interflam, 937–948 (2004) [4] Butler, K.M., Oñate, E., Idelsohn, S.R., Rossi, R.: Modeling and simulation of the melting of polymers under fire conditions using the particle finite element method. In: 11th Int. Fire Science & Engineering Conference, University of London, Royal Halbway College, UK, September 3-5 (2007) [5] Codina, R., Zienkiewicz, O.C.: CBS versus GLS stabilization of the incompressible Navier-Stokes equations and the role of the time step as stabilization parameter. Communications in Numerical Methods in Engineering 18(2), 99–112 (2002)

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[6] Del Pin, F., Idelsohn, S.R., Oñate, E., Aubry, R.: The ALE/Lagrangian particle finite element method: A new approach to computation of free-surface flows and fluid-object interactions. Computers & Fluids 36, 27–38 (2007) [7] Donea, J., Huerta, A.: Finite element method for flow problems. J. Wiley, Chichester (2003) [8] Edelsbrunner, H., Mucke, E.P.: Three dimensional alpha shapes. ACM Trans. Graphics 13, 43–72 (1999) [9] García, J., Oñate, E.: An unstructured finite element solver for ship hydrodynamic problems. J. Appl. Mech. 70, 18–26 (2003) [10] Idelsohn, S.R., Oñate, E., Del Pin, F., Calvo, N.: Lagrangian formulation: the only way to solve some free-surface fluid mechanics problems. In: Mang, H.A., Rammerstorfer, F.G., Eberhardsteiner, J. (eds.) Fith World Congress on Computational Mechanics, Viena, Austria, July 7–12 (2002) [11] Idelsohn, S.R., Oñate, E., Calvo, N., Del Pin, F.: The meshless finite element method. Int. J. Num. Meth. Engng. 58(6), 893–912 (2003a) [12] Idelsohn, S.R., Oñate, E., Del Pin, F.: A lagrangian meshless finite element method applied to fluid-structure interaction problems. Computer and Structures 81, 655–671 (2003b) [13] Idelsohn, S.R., Calvo, N., Oñate, E.: Polyhedrization of an arbitrary point set. Comput. Method Appl. Mech. Engng. 192(22-24), 2649–2668 (2003c) [14] Idelsohn, S.R., Oñate, E., Del Pin, F.: The particle finite element method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves. Int. J. Num. Meth. Engng. 61, 964–989 (2004) [15] Idelsohn, S.R., Oñate, E., Del Pin, F., Calvo, N.: Fluid-structure interaction using the particle finite element method. Comput. Meth. Appl. Mech. Engng. 195, 2100–2113 (2006) [16] Kovacs, A., Parker, G.: A new vectorial bedload formulation and its application to the time evolution of straight river channels. J. Fluid Mech. 267, 153–183 (1994) [17] Larese, A., Rossi, R., Oñate, E., Idelsohn, S.R.: Validation of the Particle Finite Element Method (PFEM) for simulation of free surface flows. Engineering Computations 25(4), 385–425 (2008) (submitted) [18] Ohayon, R.: Fluid-structure interaction problem. In: Stein, E., de Borst, R., Hugues, T.J.R. (eds.) Enciclopedia of Computatinal Mechanics, vol. 2, pp. 683–694. J. Wiley, Chichester (2004) [19] Oñate, E.: Derivation of stabilized equations for advective-diffusive transport and fluid flow problems. Comput. Meth. Appl. Mech. Engng. 151, 233–267 (1998) [20] Oñate, E.: A stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation. Comp. Meth. Appl. Mech. Engng. 182(1–2), 355–370 (2000) [21] Oñate, E.: Possibilities of finite calculus in computational mechanics. Int. J. Num. Meth. Engng. 60(1), 255–281 (2004) [22] Oñate, E., Idelsohn, S.R.: A mesh free finite point method for advective-diffusive transport and fluid flow problems. Computational Mechanics 21, 283–292 (1998) [23] Oñate, E., García, J.: A finite element method for fluid-structure interaction with surface waves using a finite calculus formulation. Comput. Meth. Appl. Mech. Engrg. 191, 635– 660 (2001) [24] Oñate, E., Rojek, J.: Combination of discrete element and finite element method for dynamic analysis of geomechanic problems. Comput. Meth. Appl. Mech. Engrg. 193, 3087–3128 (2004)

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[25] Oñate, E., Sacco, C., Idelsohn, S.R.: A finite point method for incompressible flow problems. Comput. Visual. in Science 2, 67–75 (2000) [26] Oñate, E., Idelsohn, S.R., Del Pin, F.: Lagrangian formulation for incompressible fluids using finite calculus and the finite element method. In: Kuznetsov, Y., Neittanmaki, P., Pironneau, O. (eds.) Numerical Methods for Scientific Computing Variational Problems and Applications. CIMNE, Barcelona (2003) [27] Oñate, E., García, J., Idelsohn, S.R.: Ship hydrodynamics. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 3, pp. 579–610. J. Wiley, Chichester (2004a) [28] Oñate, E., Idelsohn, S.R., Del Pin, F., Aubry, R.: The particle finite element method. An overview. Int. J. Comput. Methods 1(2), 267–307 (2004b) [29] Oñate, E., Valls, A., García, J.: FIC/FEM formulation with matrix stabilizing terms for incompressible flows at low and high Reynold’s numbers. Computational Mechanics 38(4-5), 440–455 (2006a) [30] Oñate, E., García, J., Idelsohn, S.R., Del Pin, F.: FIC formulations for finite element analysis of incompressible flows. Eulerian, ALE and Lagrangian approaches. Comput. Meth. Appl. Mech. Engng. 195(23-24), 3001–3037 (2006b) [31] Oñate, E., Celigueta, M.A., Idelsohn, S.R.: Modeling bed erosion in free surface flows by the Particle Finite Element Method. Acta Geotechnia 1(4), 237–252 (2006c) [32] Oñate, E., Idelsohn, S.R., Celigueta, M.A., Rossi, R.: Advances in the particle finite element method for the analysis of fluid-multibody interaction and bed erosion in free surface flows. Comput. Meth. Appl. Mech. Engng. 197(19-20), 1777–1800 (2008) [33] Oñate, E., Rossi, R., Idelsohn, S.R., Butler, K.: Melting and spread of polymers in fire with the particle finite element method. Int. J. Num. Meth. in Engng. (2009) (submitted) [34] Parker, D.B., Michel, T.G., Smith, J.L.: Compaction and water velocity effects on soil erosion in shallow flow. J. Irrigation and Drainage Engineering 121, 170–178 (1995) [35] Tezduyar, T.E.: Finite element method for fluid dynamics with moving boundaries and interface. In: Stein, E., de Borst, R., Hugues, T.J.R. (eds.) Enciclopedia of Computatinal Mechanics, vol. 3, pp. 545–578. J. Wiley, Chichester (2004) [36] Wan, C.F., Fell, R.: Investigation of erosion of soils in embankment dams, J. Geotechnical and Geoenvironmental Engineering 130, 373–380 (2004) [37] Zienkiewicz, O.C., Taylor, R.L., Nithiarasu, P.: The finite element method for fluid dynamics. Elsevier, Amsterdam (2006) [38] Zienkiewicz, O.C., Jain, P.C., Oñate, E.: Flow of solids during forming and extrusion: Some aspects of numerical solutions. Int. Journal of Solids and Structures 14, 15–38 (1978) [39] Zienkiewicz, O.C., Oñate, E., Heinrich, J.C.: A general formulation for the coupled thermal flow of metals using finite elements. Int. Journal for Numerical Methods in Engineering 17, 1497–1514 (1981) [40] Zienkiewicz, O.C., Taylor, R.L.: The finite element method for solid and structural mechanics. Elsevier, Amsterdam (2005)

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Appendix The matrices and vectors in Eqs.(8)-(11) for a 4-noded tetrahedron are: Mi j = Gi j =

Ve

Ve

ρ NTi N j dV

BTi mN j dV

,

,

Ki j =

fi =

Ve

BTi DB j dV

NTi bdV +

Ve

Γe

NTi td Γ

Li j =

∇ Ni τ ∇N j dV T

Ve

q = [q1 , q2 , q3 ] , ˆ i ]kl = ˆ 2, M ˆ 3 , [M ˆ = M ˆ 1, M M

,

∂ ∂ ∂ ∇= , , ∂ x1 ∂ x2 ∂ x3

∂ Ni N j dV V e ∂ xk τ Ni N j dV δkl k, l = 1, 2, 3 τ

[Qk ]i j =

Ve

⎤

⎡

∂ Ni ⎢ ∂x ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ∂ Ni B = [B1 , B2 , B3 , B4 ]; Bi = ⎢ ⎢ ∂y ⎢ ⎢ ⎢ ⎢ ∂ Ni ⎢ ⎢ ∂z ⎢ ⎢ ⎣ 0 ⎡ 20 ⎢0 2 ⎢ ⎢0 0 D=μ⎢ ⎢0 0 ⎢ ⎣0 0 00 N = [N1 , N2 , N3 , N4 ] ,

Ci j =

Ve

ρ cNi N j dV

,

Hi j =

000 000 200 010 001 000

Ni = Ni I3

Ve

T

,

0 ∂ Ni ∂y

⎤ 0 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎦ 1

0

⎥ ⎥ 0 ⎥ ⎥ ⎥ ∂ Ni ⎥ ⎥ 0 ∂z ⎥ ⎥ ⎥ ∂ Ni 0 ⎥ ⎥ ∂x ⎥ ⎥ ⎥ ∂ Ni ⎥ ⎥ 0 ∂x ⎥ ⎥ ⎥ ∂ Ni ∂ Ni ⎦ ∂z ∂y

I3

∇N j dV ∇ T Ni [k]∇

is the unit matrix

,

m = [1, 1, 1, 0, 0, 0]T

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⎡ ⎤ k1 0 0 [k] = ⎣ 0 k2 0 ⎦ 0 0 k3

,

qi =

Ve

Ni QdV −

Γqe

Ni qn dΓ

In above equations indexes i, j run from 1 to the number of element nodes (4 for a tetrahedron), qn is the heat flow prescribed at the external boundary Γq , t is the surface traction vector t = [tx ,ty ,tz ]T and V e and Γ e are the element volume and the element boundary, respectively.

Chapter 16

A Framework for the Two-Scale Homogenization of Electro-Mechanically Coupled Boundary Value Problems Jörg Schröder and Marc-André Keip

Abstract. The contribution addresses the derivation of a meso-macro transition procedure for electro-mechanically coupled materials in two and three dimensions. In this two-scale homogenization approach piezoelectric material behavior will be analysed. In this context, a mesoscopic material model will be presented and implemented into an FE2 -homogenization approach. The resulting model is able to capture macroscopic boundary value problems taking into account attached heterogeneous representative volume elements at each macroscopic point. The model is also applicable for the calculation of effective electro-mechanical material parameters, which are efficiently computed by means of the proposed direct homogenization procedure.

16.1 Introduction For the macroscopic modeling of micro-heterogeneous and polycrystalline materials it is necessary to predict their effective properties. The derivation of upper and lower bounds and the computation of estimates for the overall properties have to be distinguished. The estimates of such bounds are based on the fundamental works [11], and later [45], [17], [46], and more recently [10], [26]. These methods have been applied for the prediction of mechanical as well as non-mechanical properties. Exact results for the overall properties of piezoelectric composites have been established in [6]. Jörg Schröder Institute for Mechanics, Faculty of Engineering Sciences, Department of Civil Engineering, University of Duisburg-Essen, Universitätsstraße 15, 45141 Essen, Germany e-mail: [email protected] Marc-André Keip Institute for Mechanics, Faculty of Engineering Sciences, Department of Civil Engineering, University of Duisburg-Essen, Universitätsstraße 15, 45141 Essen, Germany e-mail: [email protected]

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Estimates for overall thermo-electro-elastic moduli of multiphase fibrous composites based on self-consistent and Mori-Tanaka methods are given in [5]. Effective quantities of two-phase composites have been evaluated in [8], [7] using e.g. dilute, self-consistent, and Mori-Tanaka-schemes. In this context we refer also to [2], [3], [1], [4] and [9]. Universal bounds for effective piezoelectric properties of heterogeneous materials have been derived in [15], utilizing generalized Hashin-Shtrikman variational principles. The overall properties of (periodic) composites depend on the morphology of their (unit cells) mesostructure and the properties of their individual constituents. Therefore, it is possible to improve the performance characteristics of piezoelectric materials by means of topology optimization and homogenization techniques, see e.g. [36] and [37]. Utilizing a unit-cell method, [19] investigated the relation between effective properties and different geometries of microvoids based on a 3-D finite element analysis. A multi-scale finite element modeling procedure for the macroscopic description of polycrystalline ferroelectrics has been proposed by [44], [43]; here a homogenization procedure based on asymptotic expansions of the displacements and the electric potential were utilized, for the mathematical background see e.g. [29]. An approximation of macroscopic polycrystals by discrete orientation distribution functions has been discussed in [34], [35], [18]. This procedure is associated with the well known Reuss- and Voigt-bounds. An algorithm for the description of micro-heterogeneous coupled thermo-electro-magnetic continua has been presented in [50]. General works on homogenization theory are [13], [14], [40] and [16]. In the following a general direct homogenization procedure is presented which couples the macroscopic to the mesoscopic scale, in this context see also [38], [25], [24], [22], [30], [41], [23], [42], [47], [49], [21], [39]. The procedure is as follows: (a) At each macroscopic point: Localize suitable macroscopic quantities (e.g. the strains and the electric field) to the mesoscale. To be more specific, apply constraint conditions or boundary conditions, e.g. driven by the macroscopic strains and the electric field, on a representative volume element, see e.g. [31] and [20]. (b) Next, solve the equations of balance of linear momentum and Gauss’ law on the mesoscale under the applied macroscopic loading in order to obtain the dual mesoscopic quantities (e.g. the stresses and the electric displacements). (c) Next, perform a homogenization step, i.e. compute the average values of the dual quantities on the mesoscale. These macroscopic variables have to be transferred to the associated points of the macroscale. (d) Finally, solve the electro-mechanically coupled boundary value problem on the macroscale and proceed with step 1 until convergence is obtained on both scales. The numerical solution is based on separate finite element analyses on each scale. The overall algorithmic moduli needed for the Newton-Raphson iteration scheme on the macroscale are efficiently computed during the standard solution procedure on the mesoscale.

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16.2 Boundary Value Problems on the Macro- and the Mesoscale In the following we describe the electro-mechanically coupled boundary value problems (BVP) on both scales. The material behavior for the individual constituents on the mesoscale are modeled within a coordinate invariant formulation. Here we restrict ourselves to transversely isotropic material as presented in [32], [28], and [27].

16.2.1 Macroscopic Electro-Mechanically Coupled BVP The body of interest B ⊂ IR3 on the macroscopic scale is parameterized in x. u and φ denote the macroscopic displacement field and the macroscopic electric potential, respectively. The basic kinematic and electric variables are the linear strain tensor ε (x) := sym[∇u(x)] and the electric field vector E(x) := −∇ φ (x), where ∇ denotes the gradient operator w.r.t. x.

t

Q

n

∂ Bφ x∈B

∂ Bu

x∈B

∂ BD

∂ Bσ

Fig. 16.1 Boundary decomposition of ∂ B into mechanical, i.e. ∂ Bu ∪ ∂ Bσ = ∂ B with ∂ Bu ∩ ∂ Bσ = 0, / and electrical parts: ∂ Bφ ∪ ∂ BD = ∂ B with ∂ Bφ ∩ ∂ BD = 0. /

The governing field equations for the quasi-static case are the balance of linear momentum and Gauss’ law divx [σ ] + f = 0

and

divx [D] = q

in

B,

(16.1)

where divx denotes the divergence operator with respect to x. σ represents the symmetric Cauchy stress tensor, f is the given body force, D denotes the vector of electric displacements and q is the given density of free charge carriers. The boundary conditions in terms of displacements and surface tractions t are u = ub

on ∂ Bu

and t = σ · n on ∂ Bσ ,

(16.2)

and in terms of the electric potential and the electric surface charge Q we write

φ = φb

on ∂ Bφ

and

− Q = D · n on ∂ BD ,

(16.3)

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J. Schröder and M.-A. Keip

where n is a unit normal vector pointing outwards from the surface of the body. In the following we do not postulate the existence of a thermodynamical potential on the macroscale. Instead, we attach a representative volume element (RV E ) at each macroscopic point x, that delivers the constitutive response, see Fig. 16.2. RV E

ε,σ

RV E E, D

u ∈ RV E

x

φ ∈ RV E

x

x ∈ RV E

x ∈ RV E

∂ RV E

∂ RV E

Fig. 16.2 Attached RV E at x, associated to the macroscopic mechanical and electrical quantities.

In order to link the macroscopic variables {ε , σ , E, D} with their microscopic counterparts {ε , σ , E, D}, we define in this two-scale approach the macroscopic variables in terms of some suitable surface integrals over the boundary of the RV E with volume V . It should be remarked, that a definition of macroscopic quantities in terms of surface integrals is necessary in general. Respective definitions of macroscopic values by means of simple volume averages could lead to physically unreasonable results and would not allow for reliable interpretations of simple experiments, see e.g. [30]. The macroscopic strains and stresses are given by

ε :=

1 V

sym[u ⊗ n] da

and σ :=

∂ RV E

1 V

sym[t ⊗ x] da ,

(16.4)

∂ RV E

where u and t are the displacement and traction vectors on the boundary of the RV E , respectively. Furthermore, the macroscopic electric field and electric displacements are defined by the surface integrals E :=

1 V

−φ n da ∂ RV E

and D :=

1 V

−Q x da ,

(16.5)

∂ RV E

which are governed by the electric potential φ and the electric charge density Q on ∂ RV E .

16.2.2 Mesoscopic Electro-Mechanically Coupled BVP On the mesoscopic scale we consider a BVP defined on the RV E ⊂ IR3 , which is parameterized in the mesoscopic cartesian coordinates x. The governing balance equations are the balance of linear momentum neglecting body forces, and the Gauss’ law neglecting the density of free charge carriers

16 Two-Scale Homogenization of Electro-Mechanically Coupled BVPs

div[σ ] = 0

div[D] = 0

and

in

RV E .

315

(16.6)

The mesoscopic strains and electric field vector are given by ε := sym[∇u(x)] and E := −∇φ (x), where ∇ denotes the gradient operator and div the divergence operator with respect to x. In order to complete the description of the mesoscopic BVP we define some appropriate boundary conditions on ∂ RV E or some constraint conditions in the whole RV E . Therefore, we apply a generalized macro-homogeneity condition 1 1 ˙ dv , σ : ε˙ + D · E˙ = σ : ε˙ dv + D·E (16.7) V V RV E

RV E

in this context see [12]. The generalized macro-homogeneity condition is fulfilled if P = 0 holds, where P :=

1 V

RV E

1 σ : ε˙ dv − σ : ε˙ + V

˙. ˙ dv − D · E D·E

(16.8)

RV E

The simplest assumption of the mesoscopic fields that automatically fulfill the condition P = 0 is achieved by setting

and

σ = σ = const. or ε˙ = ε˙ = const.

(16.9)

D = D = const. or E˙ = E˙ = const.

(16.10)

for all points of the mesoscale. Equations (16.9)1,2 lead to the well-known Reussand Voigt-bounds, respectively. In the following we denote (16.9) and (16.10) as constraint conditions. More sophisticated expressions for suitable boundary conditions can be derived from the equivalent expression to (16.8) P=

1 1 ˙ · x)da, (16.11) (t − σ · n) · (u˙ − ε˙ · x) da + (Q + D · n) (φ˙ + E V V ∂ RV E ∂ RV E 56 7 4 56 7 4 P1

P2

where we utilized the Gauss theorem, the balance of linear momentum (16.6), the Cauchy theorem t = σ · n, Gauss’ law and Q = −D · n. Evaluating P1 = 0 leads to the Neumann- or Dirichlet-boundary conditions t = σ · n on ∂ RV E

or

u˙ = ε˙ · x on ∂ RV E .

(16.12)

Possible periodic boundary conditions, satisfying P1 = 0, are t+ (x+ ) = −t− (x− )

2 + (x+ ) = w 2 − (x− ) on x± ∈ ∂ RV E ± , and w

(16.13)

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J. Schröder and M.-A. Keip

for an illustration see Fig. 16.3.

Fig. 16.3 Mesoscopic mechanical BVP: Periodic boundary conditions on ∂ RV E .

In analogy to the above mentioned boundary conditions for the mechanical part, we obtain possible boundary conditions for the electrical part of the BVP on the mesoscale by evalutating the expression P2 = 0. Possible Neumann- or Dirichletboundary conditions, obtained from P2 = 0, are Q = −D · n on ∂ RV E

or

˙ · x on ∂ RV E . φ˙ = −E

(16.14)

Periodic boundary conditions, satisfying P2 = 0, are given by the conditions Q+ (x+ ) = −Q− (x− ) and φ2+ (x+ ) = φ2− (x− ) on x± ∈ ∂ RV E ± , (16.15) for an illustration see Fig. 16.4.

Fig. 16.4 Mesoscopic electrical BVP: Periodic boundary conditions on ∂ RV E .

For a detailed derivation of the respective boundary conditions we refer to [31].

16.3 Effective Properties of Piezoelectric Materials The aim of this two-scale approach is to achieve quadratic convergence within the Newton-Raphson iteration scheme for the discretized boundary value problem on the macroscale. For this we have to perform the consistent linearization of the macroscopic stresses and electric displacements with respect to the macroscopic strains and electric field. In detail we need the macroscopic (overall) mechanical moduli C, piezoelectric moduli e, and dielectric moduli , which enter the incremental constitutive relations

Δσ =

C : Δ ε − eT Δ E ,

−Δ D = − e : Δ ε − Δ E .

(16.16)

16 Two-Scale Homogenization of Electro-Mechanically Coupled BVPs

317

Formally, we obtain the overall moduli by the partial derivatives of the volume averages of the mesoscopic stresses and electric displacements with respect to the macroscopic strains and electric field, i.e. ⎫ ⎫ ⎤ ⎧ ⎧ ⎡ ⎬ ⎬ ⎨ ⎨ ⎥8 ⎢ ∂ σ dv ∂ σ dv 8 9 9 E⎩ ⎢ ε⎩ ⎭ ⎭ ⎥ Δσ ⎥ Δε 1⎢ RV E RV E ⎫ ⎫⎥ ⎧ ⎧ = ⎢ . (16.17) V⎢ ⎬ ⎬⎥ ⎨ ⎨ −Δ D ⎥ ΔE ⎢ ⎣ −∂ε D dv −∂E D dv ⎦ ⎭ ⎭ ⎩ ⎩ RV E

RV E

For the analysis of the mesoscopic BVP we additively split the mesoscopic strains and electric field into a constant part and a fluctuating part, i.e.

ε = sym[∇u(x)] = ε + 2 ε with

ε2 dv = 0

(16.18)

RV E

and 2 with E = −∇φ = E + E

2 dv = 0. E

(16.19)

RV E

Exploiting these relations in (16.17) leads after application of the chain rule to ⎤ ⎞8 ⎡ ⎡ ⎛ ⎤ 9 9 8 T ·∂ E C −eT 2 2 ∂ ε −e C : Δσ Δε ε E 1⎝ ⎦dv⎠ ⎣ ⎣ ⎦dv + = . V −Δ D E Δ 2 2 −e : ∂ ε − · ∂ E −e − ε E RV E RV E (16.20) In order to compute the sensitivities of the fluctuation fields w.r.t. the macroscopic counterparts we first consider the coupled BVPs on the mesoscale. The weak forms of the balance of linear momentum and Gauss’ law are 2 dv = div σ · δ w

Gu = − RV E

ε : σ dv − δ2 RV E

4

56

7 4 ∂ RV E

2 · (σ · n) da δw 56

(16.21)

7

Gext u

Gint u

and Gφ = − RV E

div D δ φ2 dv =

2 · D dv − δE RV E

4

δ φ2(D · n) da,

56

7 4 ∂ RV E 56

Gint φ

Gext φ

(16.22)

7

respectively. The linearization, LinGu,φ = Gu,φ (•) + Δ Gu,φ , of the weak forms yields for conservative loadings the linear increments

318

J. Schröder and M.-A. Keip

δ ε2 : ∂ε σ : Δ ε2 dv +

Δ Gint u = RV E

2 dv δ ε2 : ∂E σ · Δ E

(16.23)

2 · ∂E D · Δ E 2 dv. δE

(16.24)

RV E

and 2 · ∂ε D : Δ ε2 dv + δE

Δ Gint φ = RV E

RV E

In the following we use the abbreviations C := ∂ε σ ,

e = − {∂E σ }T = ∂ε D,

= ∂E D

(16.25)

for the tangent moduli. Finite Element Approximations: For the discretization of the weak forms we only have to account for the fluctuation fields of the displacements and electric potential. Thus, the approximation of the fluctuation fields, virtual fluctuation fields and incremental fluctuation fields appear as 2= w

nnode

∑

NuI 2 duI ,

2= δw

I=1

nnode

∑

NuI δ 2 duI ,

2= Δw

I=1

nnode

∑ NuI Δ 2duI

(16.26)

I=1

and

φ2 =

nnode

∑

φ NφI d2I ,

δ φ2 =

I=1

nnode

∑

φ NφI δ d2I ,

I=1

Δ φ2 =

nnode

φ

∑ NφI Δ d2I ,

(16.27)

I=1

u,φ d the nodal degrees of freedom and nnode where Nu,φ denote the ansatz functions, 2 the number of nodes per element. Let Beu and Beφ characterize the B-Matrices associated to fluctuation strains and electric field. The finite element approximations of the actual, virtual and incremental strains are

ε2 = Beu 2 du ,

δ ε2 = Beu δ 2 du ,

Δ ε2 = Beu Δ 2 du .

Analogously, the fields associated to the electric potential appear as φ 2 = −Be 2 E φ d ,

φ 2 = Be δ 2 δE φ d ,

φ 2 = −Be Δ 2 ΔE φ d .

Inserting the latter expressions into the linearized weak forms yields the discrete counterparts ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ nele uT eT e u eT T e φ e 2 2 2 =0 δ B CB dv Δ + B e B dv Δ + r d d d (16.28) ∑ u u u u φ ⎪ ⎪ ⎪ ⎪ e=1 ⎪ ⎪ V V ⎪ ⎪ ⎪ ⎪ 56 7 56 7 4 ⎪ ⎪ ⎩4 ⎭ e e kuu

k uφ

16 Two-Scale Homogenization of Electro-Mechanically Coupled BVPs

and

nele

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

∑ δ 2dφ T ⎪

e=1

⎪ ⎪ V ⎪ ⎪ ⎪ ⎩4

e 2u BeT φ eBu dv Δ d −

56

keφ u

7

4

e e 2φ BeT φ Bφ dv Δ d + rφ V

56

7

keφ φ

319

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

= 0,

(16.29)

with the right hand sides reu and reφ ; nele denotes the number of finite elements. After the standard assembling procedure, ⎞ ⎛ ⎡ ⎤ ⎜⎡ ⎤⎡ ⎤ ⎡ ⎤⎟ nele δ 2 reu ⎟ Δ2 du ⎜ keuu keuφ du ⎜ ⎣ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎦⎟ (16.30) + ⎟ = 0, ∑ 2φ ⎜ e e e ⎜ rφ ⎟ e=1 δ d Δ2 dφ ⎠ ⎝ kφ u kφ φ 56 7 4 56 7 4 56 7 4 2e ΔD

Ke

Re

we formally get the solution by inversion 2 = −K−1 R, ΔD

nele

with

K=

A

nele

Ke

and R =

e=1

A R. e

(16.31)

e=1

Computation of Overall Moduli: Now we compute the sensitivities of the mesoscopic fluctuation fields w.r.t. their incremental macroscopic counterparts. Therefore we linearize the weak forms at an equilibrium state: 2 dv = 0, δ ε2 : eT · Δ E + Δ E

δ ε2 : C : Δ ε + Δ ε2 dv − RV E

RV E

2 ·e : Δε +Δ2 ε dv + δE

RV E

2 · · ΔE+ ΔE 2 dv = 0. δE

(16.32)

RV E

After inserting the finite element approximations of the displacements and electric potential we obtain ⎧ ⎛ ⎪ ⎪ nele ⎨ ⎜ eT eT e 2u ∑ ⎪δ 2duT ⎜ ⎝ Bu C dv Δ ε + Bu CB dv Δ d ⎪ e=1 ⎩ Be Be 56 7 4 56 7 4 leuu keuu ⎞⎫ (16.33) ⎪ ⎪ ⎬ ⎟ T T e 2φ ⎟ = 0 − BeT BeT u e dv Δ E + u e Bφ dv Δ d ⎠ ⎪ ⎪ ⎭ Be Be 56 7 56 7 4 4 leuφ

keuφ

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J. Schröder and M.-A. Keip

and

⎧ ⎪

⎛

⎨ nele ⎪

⎜

∑ ⎪δ 2dφ T ⎜ ⎝

e=1 ⎪ ⎩

e 2u BeT φ eBu dv Δ d

BeT φ · e dv Δ ε +

Be

56

4

Be

7

leφ u

4

BeT φ dv Δ E −

+ Be

4

56

7

leφ φ

4

56

7

keφ u

Be

⎞⎫ ⎪ ⎪ ⎬ ⎟ eT e φ 2 ⎟ Bφ B dv Δ d ⎠ = 0. ⎪ ⎪ ⎭ 56 7

(16.34)

keφ φ

In a more compact notation we write nele

=0 δ2 duT leuu Δ ε + keuu Δ 2 du + leuφ Δ E + keuφ Δ 2 dφ

(16.35)

= 0. δ2 dφ T leφ u Δ ε + keφ u Δ 2 du + leφ φ Δ E + keφ φ Δ 2 dφ

(16.36)

∑

e=1

and nele

∑

e=1

After the standard assembling procedure we obtain ⎡ ⎣

2u δD 2φ δD

⎤ T ⎛8 ⎦ ⎝

Kuu Kφ u

⎤⎞ 9 ⎡ 2u ⎤ ⎡ ΔD Luu Δ ε + Luφ Δ E ⎦⎠ = 0 ⎦+⎣ ⎣ 2φ Kφ φ Lφ u Δ ε + Lφ φ Δ E ΔD

Kuφ

(16.37)

with the abbreviations nele

Kuu =

A

nele

keuu ,

K uφ =

e=1

e=1

nele

Luu =

A e=1

A

nele

keuφ ,

Kφ u =

e=1

nele

leuu ,

Luφ =

A e=1

A

nele

keφ u ,

Kφ φ =

nele

leuφ ,

Lφ u =

A e=1

Ak e=1

e , φφ

nele

leφ u ,

Lφ φ =

Al e=1

e φφ .

(16.38) The solution, i.e. the incremental nodal fluctuations due to the incremental macroscopic strains and electric field, is formally given by ⎡ ⎤ ⎤ 9−1 ⎡ 8 2u ΔD Kuu Kuφ Luu Δ ε + Luφ Δ E ⎣ ⎦. ⎦=− ⎣ (16.39) 2φ Kφ u Kφ φ Lφ u Δ ε + Lφ φ Δ E ΔD Inserting the finite element approximations in (16.20) and exploiting the definitions of the L-Matrices (16.38) leads to

16 Two-Scale Homogenization of Electro-Mechanically Coupled BVPs

⎡ ⎣

C −eT −e −

⎤ ⎦= 1 V

⎡

RV E

⎤ ⎡ T T ⎤⎡ Luu Lφ u 2u ∂ε Δ D 1 ⎣ ⎦⎣ ⎦. ⎦ dv + ⎣ V φ T T 2 ∂E Δ D Luφ Lφ φ −e − C −eT

321

⎤

(16.40)

Obviously, we have to compute the sensitivities of the nodal fluctuations w.r.t. the macroscopic strains and electric field. Utilizing (16.39), we obtain the derivatives of the incremental solutions ⎡ ⎤ ⎡ ⎤−1 ⎡ ⎤ Kuu Kuφ Luu Luφ 2u ∂ε Δ D ⎣ ⎦ = −⎣ ⎦ ⎣ ⎦, 2φ ∂E Δ D Kφ u Kφ φ Lφ u Lφ φ which leads to the final algorithmic expression for the macroscopic (overall) electromechanical tangent moduli ⎤ ⎡ ⎤ ⎡ C −eT C −eT 1 ⎦= ⎣ ⎦ dv ⎣ V −e − −e − RV E ⎡ T T ⎤8 9−1 ⎡ L L ⎤ L L uu uφ 1 ⎣ uu φ u ⎦ Kuu Kuφ ⎣ ⎦. − V T T K K φu φφ L uφ L φ φ Lφ u Lφ φ The authors would like to emphasize that the presented solution procedure is general in the sense that it can be applied to linear as well as to nonlinear problems. However, in this context it should be noted that quadratic convergence behavior on the macroscale can only be achieved, if an algorithmic consistent linearization of the nonlinear weak forms on the mesoscale has been carried out, see e.g. [33].

16.4 Numerical Examples In the following we discuss a two-dimensional and a three-dimensional BVP with heterogeneous representative volume elements. In the two-dimensional case we consider a square-shaped piezoelectric mesoscopic structure with ellipsoidal piezoelectric inclusion. The three-dimensional example discusses the determination of effective material parameters for voided piezoelectric materials.

16.4.1 Material Parameters and Invariant Formulation The material parameters which are applied in this contribution are taken from [48] and fitted to the underlying transversely isotropic material model utilizing a

322

J. Schröder and M.-A. Keip

least-squares approximation. We assume the existence of a quadratic electric enthalpy function from which the stresses and electric displacements are derived. In the present case of linear piezoelectric material behavior we obtain the general representation σ = C : ε − eT · E σi j = Ci jkl εkl − eki j Ek and (16.41) D = e:ε + · E Di = eikl εkl + ik Ek in direct and tensorial notation, respectively. Here C denotes the fourth-order elasticity tensor, e the third-order tensor of piezoelectric moduli and the second-order tensor of dielectric moduli. Following [32] we use a coordiante-invariant representation in the framework of the invariant-theory and focus on transversely isotropic solids, where a (with ||a|| = 1) is the preferred direction of the transversely isotropic material. We obtain C = λ 1 ⊗ 1 + 2 μ I + α3[1 ⊗ m + m ⊗ 1] + 2α2 m ⊗ m + α1 Ξ ,

(16.42)

where m = a⊗ a denotes the second-order structural tensor, 1 the second-order unity tensor, I the fourth-order unity tensor and Ξi jkl := [ai δ jk al + ak δil a j ] . The secondorder tensor of dielectric moduli is given by = −2γ1 1 − 2γ2m .

(16.43)

Finally, the third-order tensor of the piezoelectric moduli appears in the form e := −β1 a ⊗ 1 − β2 a ⊗ m − β3e

(16.44)

with the abbreviation {e}ki j := 12 [ai δk j + a j δki ]. Let the preferred direction a coincide with the x3 -axis, then we obtain the relations

λ = C12 ,

μ = 12 (C11 − C12 ),

α1 = 2C44 + C12 − C11 ,

α2 = 12 (C11 + C33 ) − 2C44 − C13 ,

α3 = C13 − C12 ,

(16.45)

between the parameters of the coordiante-independent and coordiante-dependent representation. The fitted material parameters for the elastic stiffness tensor are C11 = 222 ,

C12 = 108 ,

C13 = 111 ,

C33 = 151 ,

in units of GPa. The components of the dielectric tensor are 11 = 19 ,

33 = 0.495824

in units of 10−9 C/V m, and the piezoelectric components are e31 = −0.7 , in units of C/m2 .

e33 = 6.7 ,

e15 = 34.2

C44 = 30.5

16 Two-Scale Homogenization of Electro-Mechanically Coupled BVPs

323

16.4.2 Numerical Investigation of Two-Dimensional Mesostructure As first numerical example we discuss a homogeneous macroscopic BVP with periodic heterogeneous mesostructure. The macroscopic BVP consists of a statically determined body that is loaded with an electric potential on its upper and lower edge as depicted in Fig. 16.5.

Fig. 16.5 Macroscopic BVP with attached RV E.

The academic mesostructure is discretized with linear quadrilateral finite elements and composed of a transversely isotropic matrix material and a transversely isotropic ellipsoidal inclusion, see Fig. 16.6. The preferred directions of the matrix and the inclusion are perpendicular with respect to each other as indicated in the same figure. The material parameters of both materials are those given in section 16.4.1.

Fig. 16.6 Discretizations of the mesostructure with dimensions 0.1 mm × 0.1 mm.

In the following, both linear and periodic boundary conditions on the mesolevel are investigated. The individual distributions of the mesoscopic fields for the finest discretization are depicted in Fig. 16.7 and 16.8 for the linear case, and in Fig. 16.9 and 16.10 for the case of periodic boudary conditions. In both cases pronounced fluctuations of the mesoscopic fields can be observed. Considerable in this context are the non-zero mesoscopic distributions of corresponding fields that disappear on the macro level, as e.g. the mechanical stresses and the electric field in horizontal direction. It is worth to mention, that the electric field and stress maxima for linear boundary conditions are considerably higher compared to the case with periodic boundaries.

324

Fig. 16.7 Distributions of individual fields over the RV E for linear boundary conditions; left: φ˜ in [kV ], right: E1 in [kV /m].

Fig. 16.8 Distributions of individual fields over the RV E for linear boundary conditions; left: σ11 in [N/mm2 ], right: σ22 in [N/mm2 ].

Fig. 16.9 Distributions of individual fields over the RV E for periodic boundary conditions; left: φ˜ in [kV ], right: E1 in [kV /m].

Fig. 16.10 Distributions of individual fields over the RV E for periodic boundary conditions; left: σ11 in [N/mm2 ], right: σ22 in [N/mm2 ].

J. Schröder and M.-A. Keip

-4.6E-01 -2.3E-01 .0E+00 2.3E-01 4.6E-01

-1.2E+03 -6.4E+02 -1.1E+02 4.2E+02 9.5E+02

-3.8E+01 -1.5E+01 7.3E+00 3.0E+01 5.2E+01

-2.1E+01 -9.3E+00 2.5E+00 1.4E+01 2.6E+01

-4.6E-01 -2.3E-01 .0E+00 2.3E-01 4.6E-01

-1.2E+03 -6.4E+02 -1.1E+02 4.2E+02 9.5E+02

-3.8E+01 -1.5E+01 7.3E+00 3.0E+01 5.2E+01

-2.1E+01 -9.3E+00 2.5E+00 1.4E+01 2.6E+01

16 Two-Scale Homogenization of Electro-Mechanically Coupled BVPs

325

16.4.3 Determination of Effective Electro-Mechanical Moduli of a Three-Dimensional Voided Mesostructure In our second example we consider a three-dimensional discretization of both macro- and mesostructure in order to compute effective material parameters of an electro-mechanically coupled solid. In detail, we analyse the effective moduli of a piezoelectric material with microvoids. In this example, the void ratio of the mesostructure is 20%. In order to assess the needed mesh density on the mesolevel, a convergence study for the mesostructural discretization is conducted. The respective meshes consisting of 1371, 8209, 28703, and 53184 linear tetrahedral finite elements are depicted in Fig. 16.11. Furthermore, we assume periodic boundary conditions on the mesolevel.

a)

b)

c)

d)

Fig. 16.11 Different discretizations of the mesostructure: a) 1371, b) 8209, c) 28703, and d) 53184 linear tetrahedral finite elements.

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The results of the convergence study are depicted in Fig. 16.12. As can be seen, the results for the effective macroscopic moduli converge with increasing number of elements on the mesostructure. For a detailed discussion of the analysis of electromechanical moduli the reader is referred to [33]. C [GPa]

e [C/m2 ]

160

6

C11

140

120

3

100

C33

1.5

80

ele

0

a)

e31

4.5

0

10000

20000

30000

40000

50000

b)

0

10000

20000

30000

40000

50000

−e33 ele

Fig. 16.12 Convergence of effective macroscopic moduli: a) mechanical moduli C11 and C33 , b) piezoelectric moduli e33 and e31 .

16.5 Conclusion We presented a framework for a general two-scale homogenization procedure in order to analyze electro-mechanically coupled boundary value problems. In this context, a meso-macro transition procedure for electro-mechanically coupled materials in two and three dimensions was derived. The meso-macro transition procedure was implemented into an FE2 -homogenization environment, which allows for the computation of macroscopic boundary value problems under consideration of attached mesoscopic representative volume elements. It was shown, that the presented homogenization formulation is also capable for the efficient and accurate determination of effective electro-mechanical moduli.

References [1] Benveniste, Y.: Exact results in the micromechanics of fibrous piezoelectric composites exhibiting pyroelectricity. Proceedings of the Royal Society London A 441, 59–81 (1911) [2] Benveniste, Y.: Universal relations in piezoelectric composites with eigenstress and polarization fields, part i: Binary media: Local fields and effective behavior. Journal of Applied Mechanics 60, 265–269 (1993)

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[3] Benveniste, Y.: Universal relations in piezoelectric composites with eigenstress and polarization fields, part ii: Multiphase media–effective behavior. Journal of Applied Mechanics 60, 270–275 (1993) [4] Benveniste, Y.: Piezoelectric inhomogeneity problems in anti-plane shear and in-plane electric fields – how to obtain the coupled fields from the uncoupled dielectric solution. Mechanics of Materials 25(1), 59–65 (1997) [5] Chen, T.: Piezoelectric properties of multiphase fibrous composites: Some theoretical results. Journal of the Mechanics and Physics of Solids 41(11), 1781–1794 (1993) [6] Chen, T.: Micromechanical estimates of the overall thermoelectroelastic moduli of multiphase fibrous composites. International Journal of Solids and Structures 31(22), 3099– 3111 (1994) [7] Dunn, M.L., Taya, M.: An analysis of piezoelectric composite materials containing ellipsoidal inhomogeneities. Proceedings of the Royal Society London A 443, 265–287 (1918) [8] Dunn, M.L., Taya, M.: Micromechanics predictions of the effective electroelastic moduli of piezoelectric composites. International Journal of Solids and Structures 30, 161– 175 (1993) [9] Fang, D.N., Jiang, B., Hwang, K.C.: A model for predicting effective properties of piezocomposites with non-piezoelectric inclusions. Journal of Elasticity 62(2), 95–118 (2001) [10] Francfort, G.A., Murat, F.: Homogenization and optimal bounds in linear elasticity. Archive for Rational Mechanics and Analysis 94, 307–334 (1986) [11] Hashin, Z., Shtrikman, S.: On some variational principles in anisotropic and nonhomogeneous elasticity. Journal of the Mechanics and Physics of Solids 10, 335–342 (1962) [12] Hill, R.: Elastic properties of reinforced solids: Some theoretical principles. Journal of the Mechanics and Physics of Solids 11, 357–372 (1963) [13] Hill, R.: On constitutive macro-variables for heterogeneous solids at finite strain. Proceedings of the Royal Society London A 326(1565), 131–147 (1972) [14] Hill, R.: On the micro-to-macro transition in constitutive analyses of elastoplastic response at finite strain. Mathematical Proceedings of the Cambridge Philosophical Society 98, 579–590 (1985) [15] Hori, M., Nemat-Nasser, S.: Universal bounds for effective piezoelectric moduli. Mechanics of Materials 30(1), 1–19 (1998) [16] Krawietz, A.: Materialtheorie: Mathematische Beschreibung des phänomenologischen thermomechanischen Verhaltens. Springer, Berlin (1986) [17] Kröner, E.: Bounds for effective elastic moduli of disordered materials. Journal of the Mechanics and Physics of Solids 25, 137–155 (1977) [18] Kurzhöfer, I.: Mehrskalen-Modellierung polykristalliner Ferroelektrika basierend auf diskreten Orientierungsverteilungsfunktionen. PhD thesis, Institut für Mechanik, Fakultät Ingenieurwissenschaften, Abteilung Bauwissenschaften, Universität DuisburgEssen (2007) [19] Li, Z., Wang, C., Chen, C.: Effective electromechanical properties of transversely isotropic piezoelectric ceramics with microvoids. Computational Materials Science 27(3), 381–392 (2003) [20] Lupascu, D.C., Schröder, J., Lynch, C.S., Kreher, W., Westram, I.: Mechanical Properties of Ferro-Piezoceramics. In: Handbook of Multifunctional Polycrystalline Ferroelectric Materials. Springer, Heidelberg (2009) (in print)

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[21] Markovic, D., Niekamp, R., Ibrahimbegovic, A., Matthies, H.G., Taylor, R.L.: Multiscale modeling of heterogeneous structures with inelastic constitutive behavior. International Journal for Computer-Aided Engineering and Software 22(5/6), 664–683 (2005) [22] Michel, J.C., Moulinec, H., Suquet, P.: Effective properties of composite materials with periodic microstructure: a computational approach. Computer Methods in Applied Mechanics and Engineering 172(1-4), 109–143 (1999) [23] Miehe, C., Koch, A.: Computational micro-to-macro transitions of discretized microstructures undergoing small strains. Archive of Applied Mechanics 72(4), 300–317 (2002) [24] Miehe, C., Schotte, J., Schröder, J.: Computational micro-macro transitions and overall moduli in the analysis of polycrystals at large strains. Computational Materials Science 16(1-4), 372–382 (1999) [25] Miehe, C., Schröder, J., Schotte, J.: Computational homogenization analysis in finite plasticity simulation of texture development in polycrystalline materials. Computer Methods in Applied Mechanics and Engineering 171(3-4), 387–418 (1999) [26] Nemat-Nasser, S., Hori, M.: Micromechanics: Overall Properties of Heterogeneous Materials. North-Holland, London (1993) [27] Romanowski, H.: Kontinuumsmechanische Modellierung ferroelektrischer Materialien im Rahmen der Invariantentheorie. PhD thesis, Institut für Mechanik, Fakultät Ingenieurwissenschaften, Abteilung Bauwissenschaften, Universität Duisburg-Essen (2006) [28] Romanowski, H., Schröder, J.: Coordinate invariant modelling of the ferroelectric hysteresis within a thermodynamically consistent framework. A mesoscopic approach. In: Wang, Y., Hutter, K. (eds.) Trends in Applications of Mathematics and Mechanics, pp. 419–428. Shaker Verlag, Aachen (2005) [29] Sanchez-Palencia, E.: Non-Homogeneous Media and Vibration. Lecture Notes in Physics, vol. 172. Springer, Berlin (1980) [30] Schröder, J.: Homogenisierungsmethoden der nichtlinearen Kontinuumsmechanik unter Beachtung von Instabilitäten. Bericht aus der Forschungsreihe des Instituts für Mechanik (Bauwesen), Lehrstuhl I, Universität Stuttgart (2000) [31] Schröder, J.: Derivation of the localization and homogenization conditions for electromechanically coupled problems. Computational Materials Science (Corrected Proof, 2009) (in press) [32] Schröder, J., Gross, D.: Invariant formulation of the electromechanical enthalpy function of transversely isotropic piezoelectric materials. Archive of Applied Mechanics 73, 533–552 (2004) [33] Schröder, J., Keip, M.-A.: Computation of the overall properties of microheterogeneous piezoelectric materials (in preparation, 2009) [34] Schröder, J., Romanowski, H., Kurzhöfer, I.: Meso-macro-modeling of nonlinear ferroelectric ceramics. In: Ramm, E., Wall, W.A., Bletzinger, K., Bischoff, M. (eds.) Online Proceedings of the 5th International Conference on Computation of Shell and Spatial Structures, Salzburg, Austria, June 1-4 (2005) [35] Schröder, J., Romanowski, H., Kurzhöfer, I.: A computational meso-macro transition procedure for electro-mechanical coupled ceramics. In: Schröder, J., Lupascu, D., Balzani, D. (eds.) First Seminar on the Mechanics of Multifunctional Materials, Bad Honnef, Germany, May 7 - 10, Institut für Mechanik, Fakultät Ingenieurwissenschaften, Abteilung Bauwissenschaften, Universität Duisburg-Essen (2007) [36] Silva, E.C.N., Fonseca, J.S.O., Kikuchi, N.: Optimal design of periodic piezocomposites. Computer Methods in Applied Mechanics and Engineering 159(1-2), 49–77 (1998)

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Part IV

Geomechanics

Chapter 17

Modeling Concrete at Early Age Using Percolation Lavinia Stefan, Farid Benboudjema, Jean Michel Torrenti, and Benoît Bissonette

Abstract. The prediction of early age behavior of cementitious materials is of particular importance when it comes to the prediction of the crack occurring risks. Amongst the most important parameters that define the hydrating material are its elastic properties and the changes in volume that arise due to the very reaction of hydration. On a discrete generated microstructure, a percolation – type approach is applied. A forest fire algorithm allows taking into account the binding role played by the hydrates, and it reveals a threshold of hydration below which the rigidity of the concrete is negligible. The evolution of elastic characteristics is obtained by using a homogenization method applied to the percolated microstructure. Autogenous shrinkage is assumed to be due to the rise of a capillary pressure, the latter itself being a consequence of the hydration reaction. The capillary pressure is obtained from a model for desorption isotherm and is applied to the deformable skeleton corresponding to the percolated microstructure. Using this approach and Biot’s theory, it is possible to compute the autogenous shrinkage and its evolution around the threshold of percolation. Lavinia Stefan LMT, ENS Cachan, 61 Avenue du président Wilson, 94230 CACHAN, France and Département de Génie Civil, Pavillon Adrien-Pouliot, local 2928B, Université Laval, Québec G1V 0A6, Canada e-mail: [email protected] Farid Benboudjema LMT, ENS Cachan, 61 Avenue du président Wilson, 94230 CACHAN, France e-mail: [email protected] Jean Michel Torrenti LCPC, 58 boulevard Lefebvre, 75732 Paris cedex 15, France e-mail: [email protected] Benoît Bissonette Département de Génie Civil, Pavillon Adrien-Pouliot, local 2928B, Université Laval, Québec G1V 0A6, Canada

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17.1 Introduction Concrete is a material likely to be subjected to shrinkage and therefore to cracking risks. At early age, the mechanical characteristics of the material follow a fast evolution. Its Young’s modulus rapidly evolves from 0, when the material is in a liquid state, to values close to the service value, after several days. An intrinsic property of the hydration reaction is that it leads to a volume reduction of the reaction products. Moreover, due to the hydration phenomenon, capillary pressure arises, pressure that leads to autogenous shrinkage after the set. In the case that deformations are restrained, the material is stiff enough to oppose to the volume reduction induced by the hydration reaction, so cracks may occur if its strength is not yet fully developed. Therefore, in order to have a predictive model, the evolution of the mechanical properties, as well as the evolution of the autogenous shrinkage, are needed. The prediction of early age behaviour of concrete is subject of an important number of studies in literature and several models were developed in order to predict the risk of cracking [1]. However, progress is still possible, especially at very early age when concrete evolves from a state close to that of a liquid to a solid state. This transition can be modelled by an mechanical percolation approach [2, 3, 4], given a discrete generated hydrating microstructure, used to model the evolution of the material from the very instant of water – cement contact, up to its hardened state. The advantage of such an approach is that it permits to predict the evolution of the mechanical properties and of the autogenous shrinkage of concrete at early age, function of its hydration degree, as it shall be shown further on. In order to apply the percolation algorithm, we first need to have a virtual microstructure. This requires the use of a hydration model that can predict at any step the volumetric fractions of each phase involved in the hydration reaction. Several models that are able to ensure a correct computation of the above mentioned phases can be found in the literature ([5] to [10]). In the present study the computations are based on different hydration models proposed by Powers [5] and Jennings and Tennis [6]. Using these models one can estimate the amounts of the phases in hydrating cement pastes. Powers model considers only three phases (anhydrous cement, hydrated cement and porosity). Jennings and Tennis model allows explicit computation of the evolution of anhydrous phases (C3 S, C2 S, C3 A and C4 AF) and of the hydrated phases (C-S-H, portlandite and A f m ). Therefore, it is straight forward to generate an evolving random microstructure that includes the mentioned phases. More physical hydration models ([8] to [10]) exist. The proposed approach can be applied to these models, given slight development of the method. Once the virtual microstructure is generated, a percolation algorithm is then applied [4] in order to capture the solidified part, which contributes to the support of mechanical loadings, for each value of hydration degree. A self consistent scheme on the percolated cluster is used to obtain evolution of the elastic properties. Such an approach is fundamental to predict a correct evolution of mechanical properties at early-age. Otherwise, decrease of mechanical properties and/or non-zero Young modulus before hydration can be predicted [4]. The autogenous shrinkage is then predicted

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after the computation of evolution of desorption isotherms with respect to hydration degree and using Biot’s theory.

17.2 Hydration Models 17.2.1 Powers Model [5] This model has been used by several researchers (see for example [19, 14]). The cement paste is composed of three phases: the anhydrous cement grains, the hydrates and the porosity. It is assumed that the hydrates occupy a volume 2.13 times larger than that of the reactants, leading to the following relationships: V (1 − α ) 1 + 3.2 w/c

(17.1)

1 + 1.13α − Vanh (α ) 1 + 3.2w/c

(17.2)

Vpore (α ) = V − Vhyd (α ) − Vanh (α )

(17.3)

Vanh(α ) = V hyd (α ) = V

where Vanh , Vhyd , Vpore and V represent respectively the volume of anhydrous cement, the volume of hydration products, the volume of pores and the total volume of the system. w/c is the water to cement ratio and α is the hydration degree. It is straightforward to calculate from this simple model the volume of each phase of a cement paste for a given water to cement ratio.

17.2.2 Jennings and Tennis Model [6] In this model one consider the evolution of contents of anhydrous cement phases (C3 S, C2 S, C3 A and C4 AF), of hydrated phases (C-S-H, portlandite and A f m ) and of capillary porosity with respect to time. The composition of the cement is obtained from the modified form of the Bogue calculation proposed by [10]. Independent hydration of C3 S, C2 S, C3 A and C4 AF is assumed. A f t is not considered. A set of Avrami-type equations is used for the approximation of the degree of hydration of each compound. The general form of the equation is: αi = 1 − exp(−ai (t − bi)ci ) (17.4) where αi = degree of hydration of reactant i (i = C3 S, C2 S, C3 A or C4 AF) and t is the age (in days). Note that we use this set of equations in isothermal conditions (if the temperature is not constant thermo-activation must be taken into account).

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The constants ai , bi and ci are those determined empirically by [11]. They are given in Table 17.1. Table 17.1 Constants used in Eq. 17.4 [11] C3 S C2 S C3 A C4 AF

ai 0.25 0.46 0.28 0.26

bi 0.9 0 0.9 0.9

ci 0.7 0.12 0.77 0.55

The overall degree of hydration α is given by a weighted average Wi of the degrees of hydration of the individual compounds i: i=4

α = ∑ αiWi

(17.5)

i=1

Using data (density and molecular weight) compiled from various authors, the following equations for the prediction of the volume (in cm3 ) V j of each phase j were obtained by [6] (anhydrous, portlandite, A f m and C-S-H, respectively, for 1 g of cement paste): 1 1−α 1 + w/c ρc 1 (0.189αC3 SWC3 S + 0.058αC2SWC2 S ) = 1 + w/c 1 (0.849αC3 AWC3 A + 0.472αC4AF WC4 AF ) = 1 + w/c 1 (0.278αC3 SWC3 S + 0.369αC2SWC2 S ) = 1 + w/c

Vanh =

(17.6)

VCH

(17.7)

VA f m VC−S−H

(17.8) (17.9)

where w/c = water to cement ratio and ρc = density of cement (3.15 g.cm−3). C-S-H may also be dissociated into LD (low density) and HD (high density) C-S-H [7]. The capillary porosity Vcp is obtained, knowing the volume change due to the difference in solid volume between the products and the reactants Δi (cf Tab. 17.2,[6]). 4 1 Vcp = 1 − 1 + ∑ αiWi Δi (17.10) 1 + w/c i=1

Table 17.2 Difference in solid volume Δ i [6]

Δi [cm3 /g]

C3 S 0.437

C2 S 0.503

C3 A 0.397

C4 AF 0.136

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17.2.3 Comparison of the Two Models The two models give the volumetric fractions of hydrated and anhydrous phases. Jennings and Tennis model has a richer description. So it could take into account differences between cements. Note also that for the same degree of hydration it gives a smaller quantity of hydrates (so it will give also a smaller Young’s modulus). In the following applications we will use mainly Jennings and Tennis model. Powers model will be used for some examples.

17.2.4 Generation of the Microstructure The volumetric fractions of initial anhydrous cement phases and water for α = 0 are computed and then randomly disposed within a 90 × 90 elements two-dimensional representative elementary volume (REV). The size of the REV was chosen after several simulations that were made for microstructures of different sizes (starting from 50 × 50 elements REV up to 300 × 300 elements REV). Beginning with a 90 × 90 REV, the threshold of the evolutions of the mechanical characteristics, such as the Young modulus, does not vary with the increase of the number of the elements in the microstructure [4]. The initial microstructure evolves with the degree of hydration, following a set of rules: for each increment of hydration degree, the volumetric fractions of anhydrous cement phases that reacted and the hydrates formed are computed (Eqs. (17.3) – (17.7)). The anhydrous grains that are surrounded by the greatest number of pixels corresponding to water (pore phase) are replaced by the hydrates. Then, the remaining hydrates shall replace the pores that are found in the vicinity of anhydrous cement grains. Figure 17.1B corresponds to a degree of hydration α = 0.60 and it represents an example of the evolution of the initial microstructure represented in Figure 17.1A, for a water to cement ratio of 0.45.

Fig. 17.1 A. Initial generated microstructure; B. Hydrating microstructure α = 0.6, w/c = 0.45, black = water, white = anhydrous, grey = hydrates. Powers model.

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Such a description of the microstructure is enough for the calculation of the volumetric phases and the evaluation of the poro-mechanics properties. Of course it will not be the same if one consider the non linear behaviour of a cementitious material.

17.3 Percolation Once the virtual microstructure is generated, we can apply the percolation algorithm. The need of such a calculus arises from the particularity of cementitious materials. Indeed, for very low water to cement ratios, anhydrous cement particles are in contact from the very beginning. Still, the material has no stiffness or strength while it founds itself in the viscous state. It is only after setting that it starts to develop its mechanical properties. In order to capture the effect of “glue” played by hydrated phases and therefore the development of cohesion in cement-based materials, a “burning” type algorithm is used [2]. It is adapted to match the specificity of cement – based materials. As hydrates are formed, connections between solid fractions are made throughout the matrix, connections that propagate with the advancement of the hydration, ensuring the development of the solid skeleton. Once such paths will cross from one end of our virtual microstructure to the other, we can consider that a “set – like” phenomenon took place. The percolated cluster is identified and isolated, so we can then proceed to mechanical computations. The different steps of the “burning algorithm” are: (a) Fire is set on one of the sides of the microstructure, and only solid parts can participate to the percolated path (hydrates or anhydrous elements next to at least one hydrate); (b) Depending on the nature of the lighted pixel, fire propagates only under certain imposed conditions: • If the lighted element is an anhydrous grain, fire propagates to other hydrated grains found in the neighbourhood, grains which have never been previously lighted. • If the lighted element is a hydrate, fire propagates to both hydrated and unhydrated neighbours, which have never been previously lighted. Thus, 2 anhydrous grains of cement that are in contact with each other cannot ensure the cohesion of the system, so they shall not be considered as part of the percolation cluster, but an anhydrous grain that is found between 2 hydrates will be included in the percolation cluster. This set of rules ensures that for very low water to cement ratios there is not percolation for a degree of hydration equal to zero, knowing that the grains are in contact (there is percolation if you consider the contacts between grains like in the case of a sand; in the present context we need a percolation that simulates the cohesion in the medium).

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(c) Step 2 is repeated until fire stops to propagate. Mechanical percolation occurs, only if pixels on the opposite side have been lighted. The percolated microstructure corresponds to the assembly of pixels which have been lighted. (d) Percolation is tested from left to right, right to left, up to down and down to up. Only the common percolated pixels in all of the four tests will be kept. This allows the removing of “ dead end ” parts. (e) Finally, mechanical calculations are performed on the final percolated cluster. A typical percolation test for a microstructure of w/c = 0.45 is shown in Fig. 17.2. The first percolating cluster appears for a hydration degree of 0.26. It corresponds to about 6 hours after the water to cement contact.

a)

b)

c)

d)

Fig. 17.2 Percolation algorithm applied to the hydrating microstructure; a) α = 0,22 b) α = 0,25 c) α = 0,26 c) α = 0,29

17.4 Evolution of Poro-Mechanical Properties Elastic properties of each phases of cement paste are now well known, due to the nano-indentation techniques developed in the last years ([12] to [16]). Mean values are presented in Table 17.3. Knowing the volume fractions of each phase of cement paste, the computation of elastic properties is usually achieved by using one of the following 2 methods. The first method is to make use of the finite element computations ([17], [18]). However, there are several disadvantages as it can be quite time-consuming and significant numerical problems (i.e. mesh dependencies) can occur due to stress concentrations between different phases.

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Table 17.3 Elastic properties of phases Phase C3 S C2 S C3 A C4 AF CH C-S-H Afm

Ei [GPa] 135 130 145 125 42 22.5 22.4

νi

references

0.3 0.3 0.3 0.3 0.31 0.24 0.25

[12] [12] [12] [12] [15] [14] [16]

The second method is to use homogenization schemes that are able to compute the elastic properties of composite materials, provided that the elastic properties of each of the phases are known ([18] to [21]). Results depend on the morphology of the heterogeneous material and the shape of the phases. The time of calculus is dramatically reduced, and all problems that are linked to the type of finite element used, or the mesh density are no longer encountered. Due to its simplicity and good results that it provides, in the present paper a homogenization method was applied to the percolated cluster. The self-consistent scheme (with a spherical morphology) is adopted among various other methods, since it is the one that fits the best to a hydrating matrix. A Mori – Tanaka scheme will not provide good results as it is difficult to isolate at early-age a matrix surrounded by inclusions. The equations for the prediction of elastic mechanical properties are as presented by [19]: i=n

kh =

∑ Vi ki

i=1

1 + αm

−1 8i=n 9−1 ki ki −1 (17.11) ∑ Vi 1 + αm kh − 1 kh i=1

−1 8i=n 9−1 gi gi −1 (17.12) gh = ∑ Vi gi 1 + βm ∑ Vi 1 + βm gh − 1 gh i=1 i=1 i=n

where kh and gh = homogenized bulk and shear moduli, ki and gi = bulk and shear moduli of phase i, Vi is the volumetric fraction of phase i, α m and β m are functions of the mechanical properties of phases [19]. The mechanical characteristics are thus estimated by using the self-consistent scheme that can be applied to the percolation cluster only. In this manner, the volume of solids that is not found in the percolation cluster shall be evaluated as a void in the total volume. In Figure 17.3A we can see the evolution of total solid volume function of the degree of hydration, for different water to cement ratios. When the simulation takes into account only the percolated cluster, there is a threshold below which the solid fraction is not taken into account (continuous line). Indeed, before set, even if connections are made between solids, they provide nor strength or stiffness to the viscous material. Computations made on the entire microstructure will take into account the total amount of solid phase found in the REV (dotted line).

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In this case, when mechanical computations are performed, and especially for low water to cement ratios, a Young modulus for a zero degree of hydration is predicted (Figure 17.3B). This represents the essential difference between the present approach and the classical application of the self-consistent scheme [19], [20]. The latter gives excellent results, provided that the computations are not performed in the neighbourhood of the percolation threshold. For more important degrees of hydration, when almost all of the solids belong to the percolation cluster, both methods can be successfully applied.

35

w/c = 0.25

A

Young's modulus (GPa)

1

Solid fraction

0.8 w/c = 0.65 0.6 0.4 0.2 0

0

0.2

0.4 Hydration degree

0.6

B

w/c = 0.25

30 25 20 15 w/c = 0.65

10 5 0

0.8

0

0.2

0.4 Hydration degree

0.6

0.8

Fig. 17.3 A. Evolution of solid fraction; B. Evolution of Young’s modulus with the degree of hydration. Entire microstructure (dotted line) and percolated cluster (continuous line).

The Biot’s coefficient bi is computed using the following relation [10]: bi = 1 −

kh ks

(17.13)

where ks is the bulk modulus of the solid skeleton only. In order to compute it, it is necessary to “ extract ” the porosity and the unpercolated part of the microstructure and thus to consider a virtual microstructure that contains only the anhydrous and hydrated phases. By making these assumptions, the solid bulk modulus can be calculated by applying the homogenisation scheme on the solid part of the percolated microstructure (equations (17.11) and (17.12)). We shall use as input in our computations the volumetric fractions of anhydrous grains and hydrates phases re and f ) which are found in the total solid volume of the percolated spectively ( fanh hyd cluster, represented by the sum fhyd + fanh : fanh =

fanh , fanh + fhyd

fhyd =

fhyd fanh + fhyd

(17.14)

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1

Biot's coefficient

0.9 0.8 0.7 0.6 0.5 0.4 0.1

0.2

0.3

0.4

0.5

0.6

Hydration degree

Fig. 17.4 Evolution of Biot’s coefficient for different w/c ratios.

17.5 Autogenous Shrinkage With the advancement of the hydration reaction, the anhydrous cement grains and water are consumed and, provided the water to cement ratio is sufficiently low (inferior to 0.6), the material undergoes a desaturation. The relative humidity is below 100%, so according to Kelvin’s law, the water that is found in the partially saturated pores is subjected to a capillary depression pc . This capillary depression will be the cause of the autogenous shrinkage (see [22] for example). By combining Kelvin’s law with the relation between the saturation degree S and the relative humidity given by Van Genuchten’s relation, one obtains: 1−1/b pc (S) = a S−b − 1

(17.15)

where pc is the capillary depression, and a and b are constants. The a and b coefficients can be obtained by performing a desorption test. This type of test can be performed on hydrated materials. However, for hydrating materials, this is no more possible since this type of test needs about 1 year to be performed. These parameters can be obtained by an inverse analysis from the pore size distribution (determined from low temperature calorimetry [23] or mercury intrusion [24]). We use here another approach, since all these measurements induce a modification of the pore microstructure (cracking) and give a wrong image of the porosity (due for instance to the “ink bottle” effect). As shown by [25] on mature cement paste for various w/c ratios, we assume that the mass water content w50 at 50% RH is proportional to the C-S-H amount V ’C−S−H (cm3 per g of dry hcp): w50 = kVC−S−H

(17.16)

From Eq. (17.16), the corresponding saturation degree S50 at 50 % RH is calculated. By assuming that b coefficient in Eq. (17.15) remains constant and equal to 0.5 during hydration, one can predict the evolution of desorption isotherm (after the

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calculation of the capillary pressure with the Kelvin law). For k = 0.1 and w/c = 0.25, one obtains the results given in Figure 17.5. 1 0,9

D

Saturation degree

0,8

D

0,7 0,6

D

0,5 0,4 0,3 0,2 0,1 0 0

0,2

0,4

0,6

0,8

1

Relative humidity

Fig. 17.5 Evolution of desorption isotherm for w/c = 0.25 with respect to hydration degree.

Assuming an isodeformation of the porosity subjected to a given capillary pressure in the case of an unsaturated isothermal poroelastic approach, [26] and [27] showed that we have the following relation: d σ = kh d ε − bi Sd pc

(17.17)

where kh is the bulk modulus, bi is the Biot coefficient (in our case it is directly obtained by applying the self-consistent scheme), σ and ε are respectively the mean spherical stress and the mean spherical strain. In the case of free shrinkage, the average stress is null and one thus has: dε =

1 bi Sd pc kh

(17.18)

In this relation, kh , bi , S and pc depend on the degree of hydration. From our results (equation (17.18)), one can finally deduce the evolution of the autogenous shrinkage. Figure 17.6 presents this evolution for the w/c = 0.25. These calculations lead to a qualitatively correct evolution of the autogenous shrinkage for this water to cement ratio. It is found almost proportional to the hydration degree, as measured experimentally by several authors. However, we have to improve our model for larger water to cement ratio and to take into account the viscous behaviour of the material at early age in order to be more predictive. Indeed, Eq. (17.17) assumed only elastic strain. Due to creep under the capillary pressure, predicted autogenous shrinkage amplitude will be higher. Besides, due to hydration, saturation degree is decreasing, which slows the hydration rate. This feature needs also to be addressed in the future.

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Autogenous shrinkage (10 )

344 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

Hydration degree

Fig. 17.6 Evolution of autogenous shrinkage for w/c = 0.25.

17.6 Conclusions The percolation approach of early age of cementitious materials behavior makes it possible to describe the evolutions of the mechanical characteristics of these materials: Young modulus, Poisson ratio and Biot’s coefficient. Assuming an isotherm of desorption independent of the degree of hydration, an isodeformation of porosity and an aging elastic behavior we can then deduce the evolution of the autogenous shrinkage due to the capillary depression. Calculations lead to a good order of magnitude for the predicted autogenous shrinkage. However, this approach needs to be supplemented by taking account the viscous behavior at early age.

References [1] Acker, P., Torrenti, J.M., Ulm, F.J.: Comportement du béton au jeune âge, Paris, Hermès (2004) [2] Torrenti, J.M., Benboudjema, F.: Mechanical threshold of concrete at year early age. Materials and Structures 38(277), 299–304 (2005) [3] Smilauer, V., Bittnar, Z.: Microstructure-based micromechanical prediction of elastic properties in hydrating cement paste. Cem. Conc. Res. 36 (2006) [4] Stefan, L., Benboudjema, F., Torrenti, J.M., Bissonnette, B.: Prediction of elastic modulus of cement pastes at early ages. Accepted for publication in Computational Mat. Sciences (2009) [5] Powers, T.C., Brownyard, T.L.: Studies of the physical properties of hardened portland cement paste (nine parts). Journal of the American Concrete Institution 43 (October 1946 to April 1947) [6] Jennings, H.M., Tennis, P.D.: Model for the developing microstructure in Portland cement pastes. J. Am. Ceram. Soc. 77(12), 3161–3172 (1994) [7] Tennis, P.D., Jennings, H.M.: A model for two types of calcium silicate hydrate in the microstructure of Portland cement pastes. Cem. Conc. Res. 30, 855–863 (2000)

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[8] Bentz, D.P.: CEMHYD3D: Three-dimensional cement hydration and microstructure a development modelling package, version 2.0., NISTIR 6485, U.S. Department of Commerce (2000), http://ciks.cbt.nist.gov/monograph [9] van Breugel, K.: Simulation of hydration and formation of structure in hardening cement-based materials. PhD Thesis, Technical University Delft, Netherlands (1997) [10] Bishnoi, S., Scrivener, K.: Microstructural Modelling of Cementitious Materials using Vector Approach. In: Proc. the 12th International Congress on the Chemistry of Cement, Montreal, Canada (July 2007) [11] Taylor, H.F.W.: Modification of the Bogue Calculation. Adv. Cem. Res. 2(6), 73–77 (1989) [12] Velez, K., Maximilien, S., Damidot, D., Fantozzi, G., Sorrentino, F.: Determination of elastic modulus and hardness of pure constituents of Portland cement clinker. Cem. Conc. Res. 31(4), 555–561 (2001) [13] Damidot, D., Velez, K., Sorrentino, F.: Characterisation of interstital transition zone (ITZ) of high performance cement by nanoindentation technique. In: Proc. 11th International Congress on Chemistry of Cement, Durban, South Africa, May 11-16 (2003) CD-ROM [14] Constantinides, G., Ulm, F.-J.: The effect of two types of C-S-H on the elasticity of cement-based materials: results from nanoindentation and micromechanical modeling. Cem. Conc. Res. 34(1), 67–80 (2004) [15] Monteiro, P.J.M., Chang, C.T.: The elastic moduli of calcium hydroxide. Cem. Conc. Res. 25(8), 1605–1609 (1995) [16] Kamali, S., Moranville, M., Garboczi, E.G., Prene, S., Gerard, B.: Hydrate Dissolution Influence on the Young’s Modulus of Cement Paste. In: Li, et al. (eds.) Fracture Mechanics of Concrete Structures (FraMCoS-V), Proc. intern. symp., Vail (2004) [17] Haecker, C.-J., Garboczi, E.J., Bullard, J.W., Bohn, R.B., Sun, Z., Shah, S.P., Voigt, T.: Modeling the linear elastic properties of Portland cement paste. Cem. Conc. Res. 35, 1948–1960 (2005) [18] Smilauer, V., Bittnar, Z.: Microstructure-based micromechanical prediction of elastic properties in hydrating cement paste. Cem. Conc. Res. 36, 1708–1718 (2006) [19] Bernard, O., Ulm, F.J., Lemarchand, E.: A multiscale micromechanics-hydration model for the early-age elastic properties of cement-based materials. Cem. Conc. Res. 33(9), 1293–1309 (2003) [20] Sanahuja, J., Dormieux, L., Chanvillard, G.: Modelling elasticity of a hydrating cement paste. Cem. Conc. Res. 37, 1427–1439 (2007) [21] Sun, Z., Garboczi, E.J., Shah, S.P.: Modeling the elastic properties of concrete composites: Experiment, differential effective medium theory and numerical simulation. Cem. Conc. Res. 29, 22–38 (2007) [22] Hua, C., Acker, P., Ehrlacher, A.: Analyses and models of the autogenous shrinkage of hardening cement paste: I Modelling at macroscopic scale. Cem. Conc. Res. 25(7), 1457–1468 (1995) [23] Michaud, P.M.: Vers une approche chimio-poro-viscoelastique du comportement au jeune age des betons. PhD thesis, INSA de Lyon, Lyon (2006) [24] Haouas, A.: Comportement au jeune âge des matériaux cimentaires – caractérisation et modélisation chimio-hydro-mécanique du retrait. PhD thesis, Ecole Normale Supérieure de Cachan, Cachan (2007)

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[25] Baroghel-Bouny, V.: Water vapour sorption experiments on hardened cementitious materials Part I: Essential tool for analysis of hygral behavior and its relation to pore structure. Cem. Conc. Res. 37, 414–437 (2007) [26] Chateau, X., Dormieux, L.: Micromechanics of saturated and unsaturated porous media. Int. J. Numer. Anal. Meth. Geomech. (2002) [27] Coussy, O.: Revisiting the constitutive equations of unsaturated porous solids using a Lagrangian saturation concept. Int. J. Numer. Anal. Meth. Geomech. 31(15), 1675– 1694 (2007)

Chapter 18

Simulation of Incompressible Problems in Geomechanics Dieter Stolle, Issam Jassim, and Pieter Vermeer

Abstract. This article presents techniques for solving problems involving incompressibility in the context of low-order linear elements. It begins with describing a weak formulation that applies to both finite element and material point methods. Iterative solution schemes, including relaxation and explicit time stepping, are summarized. A strain enhancement procedure that is useful when incompressibility is introduced through plasticity is presented next, followed by a slope stability example that compares the iteration characteristics and vertical crest displacement corresponding to explicit and implicit matrix-free algorithms. Two procedures for dealing with pore pressure generation related to incompressibility and undrained conditions are described and an example is presented to compare pore pressures.

18.1 Introduction Limitations of low-order elements became apparent during the early development of the finite element methodology for solving boundary-valued problems involving incompressibility. Let us briefly consider the hypothetical problem shown in Fig. 18.1, in which we allow an ice slope fully fixed at the bottom and horizontally along the left hand boundary to creep assuming linear visco-elasticity (Maxwell body) from an elastic state of stress to one corresponding to an incompressible flow field. Dieter Stolle Department of Civil Engineering, McMaster University, Hamilton, Ontario, Canada, L8S4L7 e-mail: [email protected] Issam Jassim Institute of Geotechnical Engineering, Stuttgart University, Pfaffenwaldring 35D, 70569 Stuttgart, Germany e-mail: [email protected] Pieter Vermeer Institute of Geotechnical Engineering, Stuttgart University, Pfaffenwaldring 35D, 70569 Stuttgart, Germany e-mail: [email protected] M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 347–361. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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The objective is not to trace the actual history, but rather to obtain the steady-state flow field using the method of successive approximation. The elastic modulus and Poisson’s ratio are assumed to be 1000 MPa and 0.3, respectively, with a viscocity of 333 kPa·year and unit weight equal to 10 kN/m3 .

Fig. 18.1 Geometry of slope creep problem

Fig. 18.2 compares the spherical stress in element A and the ’steady-state’ horizontal velocity at node B that one obtains using regular and enhanced algorithms. One observes that the regular approach, based on an initial strain algorithm [16] together with Cholesky decomposition, predicts a lower velocity, when compared to that of the enhanced solution, that is due to "locking". The unrealistic diverging spherical stress that develops is attributed to the spherical stress not being directly coupled to the steady-state flow field, in spite of the material possessing elasticity. The enhanced approach that incorporates the strain smoothening described in this paper, as well as an iterative solver and a small amount of compressibility in the flow field, mitigates these problems.

Fig. 18.2 Comparison of regular and enhanced solutions

The objective of this paper is to review techniques for analyzing geotechnical problems that involve incompressibility and low-order finite element (FEM) or material point (MPM) methods. The literature in this area is quite extensive and only that, which is most relevant, is cited. The focus in the first part of the paper is on mesh-free algorithms and an enhanced volumetric strain technique that relaxes the effect of incompressibility constraints. A mesh-free implicit scheme that suppresses nonphysical temporal oscillations, which are common in explicit formulations, is then introduced. In the second part, the emphasis is on a technique that suppresses the spatial oscillations that occur in excess pressure predictions when adopting equal order interpolation for pressure and displacement. The usefulness of the techniques

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is demonstrated for a large displacement slope problem, as well as a foundation problem where the emphasis is on undrained pore pressure development.

18.2 Weak Formulation for MPM and FEM We begin here by writing the weak form for momentum balance for a material of density ρ that occupies domain V surrounded by boundary S; i.e., ¨ + δ uT ρ udV V

δ ε T σ dV − V

δ uT ρ bdV − V

δ uT tdS = 0

(18.1)

S

As usual, the dots over the displacement u denote differentiation with respect to time and δ implies a virtual quantity. The tensor variables t, b, ε and σ take on the familiar meanings of surface traction, body force, strain and stress, respectively. With regard to numerical implementation, the first step is to introduce approximations for the primitive variables; e.g., the displacement field within a finite element is approximated in terms of interpolation functions N and nodal displacements a via u = Na, which in turn allows one to define the strain using ε = Ba, in which B = LN is the usual finite element kinematic matrix related to the linear differential operator L and N. Equation (18.1) can now be written as

ρ NT N¨adV + V

BT σ dV − V

NT ρ bdV − V

NT tdS = 0

(18.2)

S

For details the reader is referred to Ref. [16]. To compare finite and material points methods, let us consider the case of gravity loading. Within the context of the FEM, integration is often carried out using Gauss quadrature integration rules; i.e., given momentum M¨a = R we calculate the mass matrix M IP

M = ∑ (ρ NT NΔ V )i

(18.3)

i=1

and the out-of-balance force R = (Fe − Fi ) that is related to the external Fe and Fi internal forces, defined as IP

Fe = ∑ (ρ NT bΔ V )i i=1 IP

Fi = ∑ (BT σ Δ V )i

(18.4)

i=1

with the representative volume of integration point i given by Δ Vi = |J|i wi where |J|i denotes the determinant of the Jacobian matrix, wi is the corresponding weighting factor and IP corresponds to the number of integration points. Within the material point method it is advantageous to use lumped masses and the notion of average

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strain [11], which results in Δ Vi being the actual participating volume for i. Particle volumes can be adjusted during deformation by assuming that the mass of i stays constant. For most applications dealing with materials that are near incompressible, the influence of volume changes can be neglected. Whether one deals with FEM or MPM, variables such as strain, stress and displacement can be considered as properties of the material point. Whereas for FEM, the material points (integration volumes or particles) are tied to the elements, for the MPM the material points are allowed to move from one element to another such that the state properties remain with the particles. From an implementation point of view, this implies that the computational grid for MPM acts as a means for determining changes that the material points experience during motion and that special bookkeeping is required to transfer information contained by the particles to the computational grid points and back again to the particles. Although not necessary, the material point method is most often based on using regular structured computational grids to simplify the bookkeeping. It is important to realize that the mesh is independent of the material points and can be updated periodically to enhance calculations at critical locations and, if desired, it can move with boundaries where non-homogeneous boundary conditions are specified. A MPM calculation cycle goes through three phases: Initialization where information is transferred from particles to the computational grid nodes; Lagrangian phase that parallels a finite element calculation where the system equations are setup and then primary variables (velocity and displacement) are evaluated, as well as stresses and strains at material points; and a convection phase where the location of particles and state variables are totally updated and the computational grid is redefined while keeping the location of the particles constant. As far as the Lagrangian phase is concerned, the location of particles can be held constant when going through iteration cycles similar to what is done for small strain applications. For cases where the location of particles is continuously updated, the mesh is often allowed to distort within the Lagrangian phase. For details on the material point method, the reader is referred to Ref. [3], [5] and [15].

18.3 Explicit Solution Scheme We will now look at explicit time integration, assuming that no special consideration is required other than that for the critical time-step size. For a body of given mass M, the momentum balance during a time increment Δ t is given by M

v1 − v0 = R0 − D0 Δt

(18.5)

where v, R and D are velocity, out-of-balance force and damping force, respectively. Subscripts 0 and 1 denote the values for nodal displacement a and force R at the beginning and end of the interval, respectively. The velocity v1 , on the other hand, corresponds to an interval where a1 = a0 + Δ tv1 . This implies that the stress

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increment Δ σ depends on the strain increment Δ ε that is determined from Δ a = Δ tv1 , and the stress and strain at time t1 = t0 + Δ t are σ1 = σ0 + Δ σ and ε1 = ε0 + Δ ε . Assumptions must be made about the nature of the damping. Viscous damping is most often assumed with D0 = α M(v0 + v1 )/2 where α is a Rayleigh damping coefficient. To overcome some of the problems associated with viscous damping, an alternative local form suggested in Ref. [5] has been adopted in this study; i.e., D0 = β R0 v0 /|v0 | where β is a damping coefficient that typically has a value between 0 and 0.7. It is important to realize that β is not directly related to α . Although critical damping may occur for β as high as 0.7, if a material undergoes appreciable wide spread plasticity, critical damping may occur for much smaller values of β , say less than 0.15. The maximum time step must be below the critical value given by Δ tcr = 2/ωmax where ωmax is the maximum frequency of the system. Since this value is not readily available, the Courant-Friedrichs-Lewy (CFL) stability condition is often used instead where Δ tcr = hmin /cd in which cd is the speed of sound of a compression wave and hmin is a characteristic length, often taken as some minimum representative distance that the wave travels within an element. For a triangular element it can be estimated as twice the element area divide by the length of the longest side. The critical time step is given by the element that minimizes Δ tcr . A small element with a high constrained (P-wave) modulus E p has a profound influence on the numerical time stepping efficiency of a numerical model. Let us for the moment consider a body that is in static equilibrium for a given set of boundary conditions and forces. If it is brought out of equilibrium due to some perturbation, it will undergo a cyclical motion until the damping forces bring the body to a state of rest; in other words R → 0 implies that velocity v → 0.

18.4 Dynamic Relaxation Explicit dynamic schemes are often used to solve problems that can be modeled as quasi-static processes. If one is only interested in the quasi-static solution, the actual mass of the system is not important and the analysis can be carried out using critical damping together with a scaled mass that minimizes the number of steps to convergence; i.e., the norms R or v are less than some small value. An efficient algorithm can be developed by specifying Δ tcr , a priori, and using the CFL equation to determine density for each element that delivers the same critical time step. That is, when determining the mass matrix denoted by MDR , we use

ρ DR = E p (

Δ tcr 2 ) hmin

(18.6)

Experience has shown that if we select Δ t = 1 then a critical time step of Δ tcr ≈ 1.05 provides an efficient algorithm [10]. The stability of the algorithm increases as Δ tcr is increased. Adopting a Δ t = 1, Eqn. (18.5) becomes

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MDR v1 = R0 − D0 + MDR v0

(18.7)

in which the subscripts 0 and 1 refer to the initial and updated vectors, respectively. Equation (18.7) is used iteratively until convergence is achieved, at which point the components of velocity v1 are negligible. If one were to adopt mass proportional viscous damping, then the algorithm would require estimating the dominant frequency ω associated with loading, say by the Rayleigh quotient, to form a damping matrix C = 2ω MDR . The idea is to critically damp the dominant frequency due to the load. Drawing a parallel between solid and fluid dynamics, Ref. [9] proposed an efficient algorithm for quasi-static solutions, which we refer to as transient relaxation. In its modified form, this solution scheme uses the iterative scheme based on MT R Δ a = R0 to update the field variables a1 = a0 + Δ a, σ1 = σ0 + Δ σ and ε1 = ε0 + Δ ε . The density for each element that is required to form MT R is determined from ρ T R = ρ DR /Δ tcr . The advantage of using this form, when compared to dynamic relaxation, is that there is no need to introduce non-physical damping to obtain the static solution. The number of steps to convergence can however be considerably higher than for dynamic relaxation.

18.5 Numerical Challenges with MPM Experience with relaxation, whether dynamic or transient, indicates that it can be quite efficient, at least for FEM; see, e.g., [10]. On the other hand, convergence for MPM can be much slower, making alternative quasi-static approaches more attractive [1]. It is not unusual to require time steps 100 times smaller than that dictated by the CFL stability requirement. An important reason for this is attributed to the introduction of nonphysical internal forces associated with particles moving between √ elements. An important observation is that the residual load norm R = RT R can double due to particles changing elements. The impact of algorithm-based unbalanced forces is masked in the regular explicit procedure, which implies that it is not always clear if a prediction at time t is contaminated by large out-of-balanced forces. To mitigate the impact of the nonphysical forces on the solution and convergence, the authors propose that the explicit algorithm should be modified by: • introducing an implicit unconditionally stable mesh-free dynamic scheme • updating the location of particles only after convergence is achieved within an implicit time increment; i.e., at the end of the Lagrangian phase. As far as the last point is concerned, the algorithm-based unbalanced forces still appear, they are only eliminated during the iteration. A zero load updating step can be introduced immediately after the convective phase to reduce the impact of the nonphysical residual forces. It should be noted that introduction of algorithmic

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oscillations has also been reported for FEM analysis involving contact problems, where oscillations are attributed to nonphysical stress relaxation [4].

18.6 Implicit Mesh-Free Dynamic Analysis Computational efficiency can often be improved by employing implicit time stepping algorithms, particularly for problems where low frequency response dominates. For finite element applications, the use of implicit forms is well established [16]. On the other hand, the use of implicit schemes is not wide spread in the context of MPM; see, e.g., Ref. [7] and [12] for some applications with implicit algorithms. The implicit algorithm proposed here is based on a mesh-free explicit technique, where either dynamic or transient relaxation is used to as a vehicle to achieve momentum balance convergence within a predefined time increment. The goal is not necessarily to circumvent the time step restrictions associated with explicit schemes when carrying out the analysis, but rather to improve the stability of calculations and suppress oscillations that occur in the iterative procedure. This is most important when considering problems that involve a coupling between pore pressure and the displacement field as nonphysical oscillations can lead to premature yielding. The implicit structure adopted herein follows the same structure as the explicit algorithm. For the transient relaxation scheme, we write MT R an+1 = MT R an1 + Rn1 − Dn1 − M 1

vn1 − v0 Δt

(18.8)

an+1 − an0 1 (18.9) Δt where the time increment on the right hand side corresponds to the ’physical’ time step. The superscript n acts as an iteration counter with the subscript referring to the time-stepping stage. Displacements and velocities are updated until convergence is achieved on the right hand side; i.e., until the terms of the vector on the RHS become smaller than some tolerance. Dynamic relaxation can be used in place of transient relaxation. It should also be pointed out, rather than using the fully implicit form given by Eqn. (18.8) and (18.9), one could implement a semi-implicit scheme. Within the context of the material point method, it is not unusual that the mass of a node can become small if a single particle remains in a computational element and this particle is about to leave the element. It is tempting when using explicit time marching to eliminate the influence of such a node by dropping it from the system of equations in order to avoid division by a small number that can lead to particles moving outside the mesh. By doing this, the algorithm inadvertently treats this node as corresponding to a fixed boundary condition, which introduces a constraint that does not exist. When using the implicit algorithm, this does not become an issue as all physical information is on the RHS of the equation and mass matrix M T R is formed exactly as described previously, being a property of the computational mesh vn+1 = vn1 + 1

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that is independent of the location of particles, but depends on the elastic properties of the particles.

18.7 Overcoming Limitations of Low Order Locking and difficulties with predicting reasonable pressure distributions can be overcome by using a higher order element, mixed formation. Arguably the most successful element that avoids locking and spurious variations in pressure has quadratic variation in displacement and linear in pressure. This element satisfies the BabuskaBrezzi condition, which from a pragmatic point of view requires that the interpolation for pressure be of lower order than that for displacement when the pressure field is not directly coupled to the flow field through a constitutive equation. A common way to stabilize the pressures is to introduce artificial compressibility, which converts the saddle point problem to a parabolic problem, for which a definite minimum in energy exists [14]. For 3-D problems, the use of low order elements together with explicit iterative solution schemes has become popular. Furthermore, lower order interpolation is required to avoid unnatural deformation modes that develop due to kinematic constraints that arise with higher order interpolation when adopting lumped mass iteration techniques. Considerable research has been carried out to develop solution stabilization techniques for (near) incompressible problems, in which pressure and velocity have equal order interpolation. There are two separate problems that must addressed: spurious pressure modes, and locking. It has been shown that that equal order interpolation is possible for the solution of steady state incompressible flows by iterative techniques that involve a ’velocity correction method’ (also referred to as the ’fractional step method’), as well as by introducing, for example, an elementsize sensitive term derived from momentum considerations to the mass conservation equation. Reference [17] summarizes various approaches to overcome problems associated with using equal order interpolation and explores the possibilities of various alternative iterative techniques. Locking that develops with low order elements, particularly when the material is incompressible, cannot be overcome with simple techniques such as reduced integration. A common approach to deal with this is through the use of enhanced strain fields such as bubble functions [13] or by nodal averaging techniques. Reference [2] demonstrates that locking effects can be mitigated in explicit dynamic applications by using a penalty formulation and pressure averaging. An approach proposed in Ref. [6] and recommended here is based on nodal volumetric strain averaging. This technique involves first determining the strain rates for each element in the usual manner and partitioning them into volumetric ε˙v = mT ε˙ and deviatoric ε˙d = ε˙ − mε˙v /3 components. A nodal volumetric strain rate for node n is determined by averaging via ∑ (ε˙vV )k ε˙ vn = (18.10) ∑ V˙k

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where the sum is over all elements attached to the node. Once the volumetric strain rate is determine for each node, the average volumetric strain rate ε˙ v for the element is determined by averaging the values of those nodes attached to the element.

ε˙ v =

1 ˙ εv d∑

(18.11)

with d being the number of vertices. The working assumption is that deviatoric strain rates need not be enhanced, only the volumetric components. As a result, the final strain rate within an element is redefined as

ε˙ = ε˙d +

mε˙ v 3

(18.12)

where the deviatoric component is the same as for the uncorrected strain rate field. Given that volumetric strain is not continuous at interfaces between dissimilar materials, the averaging process should be performed for each material separately.

18.8 Example - Slope Stability In this section, we consider the quasi-static, large displacement behaviour of the hypothetical 1 m high 60o model slope shown in Fig. 18.3. Roller boundaries are placed along the left hand side with full friction developing at the base. The base has the same elastic properties, but an undrained shear strength of 1000 kPa. Initially, the material is considered to be stress free. The simulations were completed by increasing the gravity over 20 equal load steps of 4 s duration to a maximum of 14 kN/m3 . A high damping coefficient with β = 0.7 was assumed for most simulations. For simulations employing relaxation, the critical time step was set at a value of 1.05. The time increment for the explicit scheme was based on having 50000 substeps within each 4 s step, which was much smaller than that required by the CFL stability criterion. While the material is assumed to have some elastic compressibility, the plastic deformations are incompressible. Convergence was based on R/Δ Fe < 0.005, with Δ Fe corresponding to the change in gravity loading during a load step. Proper balance is required between the number of material points and the computational mesh. A coarse mesh with a high density of material points still suffers from overly stiff response as the gradients calculated to update stresses can be poor in areas of high gradients. Two sets of analyses were completed, one with 3600 material points for the slope and a 20 × 11 computational mesh (coarse), with the other having 8100 particles and a computational of 60 × 30 (fine). The location of the material points were generated by defining the geometry via 4-noded quadrilateral elements and using bilinear interpolation functions. Owing to space limitations, only some main points and observations are presented.

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Fig. 18.3 Initial computational mesh for idealized cohesive slope on base

Fig. 18.4 compares the reduction of out-of-balance force error as a function of iteration corresponding to geometry update step 10 for simulations completed using dynamic relaxation, explicit time marching and the proposed implicit scheme. For this step many of the particles had already reached a plastic state of stress. The initial large values for error (greater than 1) are due to the unphysical residual forces associated with particles having changed elements during the convective phase. One clearly observes that the convergence characteristics (R/Δ Fe approaching a horizontal asymptote) are sensitive to the numerical algorithm. Dynamic relaxation and the implicit scheme were found to convergence either faster or with less noise than the explicit scheme. The larger values of the unbalanced force for the implicit scheme are due to the inertial term and damping not being included in the definition for R.

Fig. 18.4 Normalized unbalanced force norm history for geometry update step 10

Recalling that the full load was applied by step 20, the continued increase in displacement after update step 20, shown in Fig 18.5, was due to inertia. Inertial effects largely disappeared by step 30. By reducing the damping coefficient, the time (step) to attain maximum displacement was reduced, as one might expect. Slope failure with regard to its initial configuration had already occurred by step 10, yet the analysis was able to accommodate additional gravity load due to the slope moving to a new equilibrium configuration involving large deformations. The final configuration for the fine mesh based on the implicit analysis is shown in Fig. 18.6. Essentially, after the slope had failed it encountered a frictionless wall where the material was forced to move up and eventually come to rest after the internal resistance was suf-

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Fig. 18.5 Vertical displacement of crest as function of geometry update step

ficient to balance the driving force due to gravity. Analyses with the course mesh provided similar results.

Fig. 18.6 Total displacement contours at end of step 40 (fine mesh)

18.9 Treatment of Pore Pressure Generation The emphasis up to this point has been on solution techniques and the modeling of large deformations involving incompressible plasticity. This section examines undrained soil response with emphasis on the development of excess pore pressures. Two explicit matrix-free algorithms that relax the (near) incompressibility constraint corresponding to undrained conditions are examined.

18.9.1 Scheme A We begin by considering the momentum balance for a soil-water mixture, which is given by Eqn. (18.5); i.e., MΔ v = Δ t(R0 − D0 ), where M is the lumped mass matrix of the mixture, and Δ v = v1 − v0 is the change in nodal velocity during time Δ t, with the displacement increment being calculated from Δ a = v1 Δ t. The stress σ = σ´+ mp is now decomposed with the help of the Kronecker delta m into

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effective stress σ´ and pore pressure p (positive for suction). As usual, the effective stress increment Δ σ´ = CΔ ε is related to strain increment Δ ε = LΔ u = BΔ a and the constitutive tensor C. The other terms are the same as defined previously. If one relates pore pressure directly to volumetric strain εv via p = Qεv with εv = mT Ba and Q being a volumetric stiffness, it is possible to time-step using Eqn. (18.5) alone. However, given the tight relation between volumetric strains and the displacement field within an element, locking develops for large Q. To relax this constraint, an additional equation corresponding to pore pressure, which is now interpolated, p = Np, is introduced p )dV = 0 Q

T

N (εv − V T

T

N εv dV = V

T

N un dS − S

(18.13) B udV

(18.14)

V

with B = mB and un representing the normal displacement flux through surface S, which can be for the entire domain or for a subset. There are two important observations to be made here. The first is that Eqn. (18.13) and Eqn. (18.14) can be written in rate form and the second is that p/Q can be replaced by an interpolated volumetric strain. The resulting scheme shows similarities with the strain smoothening described earlier. An examination of Eqn. (18.13) indicates that as Q → ∞ , incompressibility is satisfied globally, although it is relaxed locally. A finite value for Q is retained to accommodate explicit-explicit time stepping. It should be noted that if pressure is specified, it is necessary to incorporate Eqn. (18.14) for those elements attached to the node that is specified.

18.9.2 Scheme B With this scheme we follow Ref. [9] and assume unit density. The transient relaxation equation for equilibrium is given by MΔ a = Δ t(Fe −

BT σ0 dV )

(18.15)

V

N V

Δp dV = Δ t κ2

N(mT Lu1 − V

p0 )dV Q

(18.16)

where κ = (2K/h)min with K being an effective bulk modulus and h a minimum characteristic element length. The time step is chosen as Δ t = ξ (h/κ )min with ξ < 1. It is noteworthy that Δ t depends on K and not Q. Given that the equations are uncoupled in the lumped mass scheme and the time-stepping acts as an iterative procedure to obtain a quasi-static solution, one can evaluate optimum values for each node as discussed in Ref. [9]. Alternatively, we can employ the relaxation technique

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described earlier where Δ t is set to 1 and appropriate densities are calculated for a given critical time step. In the scheme that follows, Eqn.(18.15) is partitioned as, MΔ a∗ = Δ t(Fe −

BT σ0 dV )

(18.17)

BT mp0 dV

(18.18)

V

MΔ a = MΔ a∗ − Δ t V

in which the first equation provides a displacement increment due to effective stresses only and second is a correction step to account for pore pressure influences. An examination of Eqn.(18.16) reveals that the displacement field corresponds to the end of the time-step with u1 = u0 + Δ u. To achieve the required stabilization to suppress pressure oscillations, the differential form of Eqn. (18.18) is substituted into the right hand side of Eqn. (18.16) noting that Δ u∗ = NΔ a∗ . After manipulation of terms and recognizing that Δ u∗ → 0 as equilibrium is approached, Eqn. (18.16) is replaced by T

M p Δ p = Δ t[

N (εv − V

p )dV − Q

B Δ u∗ dV − η T

V

T

B BpdV ]0

(18.19)

V

with the factor η ≥ 2. The addition of the last term in Eqn. (18.19) is similar to what one has if consolidation is included.

18.10 Example - Pore Pressure Generation under a Footing The 30 m × 30 m planar elastic foundation problem [8] shown in Fig. 18.7 is considered here. A uniformly distributed 3 m wide, 350 kPa load is applied on a soil that has hydraulic conductivity of zero, unit weight of 17 kN/m3 , elastic modulus of 30000 kPa and a corresponding Poisson’s ratio of 0.2. The objective of the analysis is to compare the pore pressure distribution from each scheme. The value of Q is assumed to be 30K, ξ = 0.1, and η = 8 is adopted for Scheme B. The bottom is fully fixed with roller boundaries along both sides. Pressure was not specified anywhere. The surface is traction free except where the load is applied. No attempt was made to optimize the time stepping. The 3-node triangular FEM mesh for this problem is relatively course. Both schemes predict a maximum vertical displacement under the load of 0.057 m, compared to a displacement of 0.067 m predicted when using 6-noded element with the same number of elements. The maximum pressure for Schemes A and B are 334 and 286 kPa, respectively. Scheme A is similar to what is done for the volumetric strain enhancement, which was very effective in accommodating incompressibility due to plasticity. However, close scrutiny of the contours indicates that Scheme A displays some pressure oscillations associated with undrained saturated conditions, whereas Scheme B is able to completely suppress these oscillations. The algorithm displays

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the typical characteristics of parabolic equations, which involves a continuous decay to a steady state solution.

Fig. 18.7 Pore pressure contour evolution for undrained elastic foundation

18.11 Concluding Remarks A uniform framework for accommodating incompressibility when using low order elements together with iterative solution techniques has been presented, whether for FEM or MPM analyses. Within the context for MPM, iteration history is sensitive to mesh density and number of particles, as well as the assumed damping characteristics. Optimum damping coefficients, which depend on energy dissipation due to plasticity, have yet to be established. The experience with the explicit time-stepping for obtaining quasi-static solutions was found to be mixed, particularly with regard to convergence and a tendency for non-monotonic iteration history. It was not uncommon for particles to leave the mesh due to not updating geometry often enough. Dynamic relaxation was found to converge the fastest. Convergence rate of the more stable implicit dynamic scheme for the quasi-static analysis was slower than that of dynamic relaxation. Incompressibility due to water, can be accommodated using the fractional step method together with the transient relaxation. Referring back to the ice creep example, to achieve a good quality solution, the volumetric strain must be enhanced, an iterative solution technique should be employed to maximize strain enhancement, and a little compressibility in the nonelastic strain field is advisable.

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References [1] Beuth, L., Vermeer, P., Nemeth, A.: Extended quasi-static material point method. Report: 35, IGS Stuttgart University (2008) [2] Bonet, J., Burton, A.J.: A simple averaged nodal pressure tetrahedral element for nearly incompressible dynamic explicit applications. Communi. Numer. Methods Eng. 14, 437–449 (1998) [3] Chen, Z., Brannon, R.: An evaluation of the material point method. Report: SAND 2002-0482, Sandia National Laboratories, Albuquerque, NM 87185-0893 (2002) [4] Chung, W.J., Cho, J.W., Belytschko, T.: On the dynamic effects of explicit FEM in sheet metal forming analysis. Eng. Comput. 15, 750–776 (1998) [5] Coetzee, C.J.: The material point method: theory and background. PhD Dissertation, University of Stellenbosch, South Africa (2005) [6] Detournay, C., Dzik, E.: Nodal mixed discretization for tetrahedral elements. In: Hart, R., Varona, P. (eds.) 4th international FLAC symposium, numerical modeling in geomechanics. Minnesota Itasca Consulting Group, Inc. Paper No. 07-02 (2006) [7] Guilkey, J., Weiss, J.: Implicit time integration for the material point method: quantitative and algorithmic comparisons with the finite element method. Int. J. Num. Methods Eng. 57, 1323–1338 (2003) [8] Huang, M., Yue, Z.Q., Tham, L.G., Zienkiewicz, O.C.: On the stable finite element procedures for dynamic problems of saturated porous media. Int. J. Num. Methods Eng. 61, 1421–1450 (2004) [9] Nithiarasu, P.: A matrix free fractional step method for static and dynamic incompressible solid mechanics. Int. J. Comput. Meth. Eng. Sci. Mech. 7, 369–380 (2006) [10] Sauve, R.G., Metzger, D.R.: Advances in dynamic relaxation techniques for nonlinear finite element analysis. J. Press Vess. Tech. 117, 170–176 (1995) [11] Stolle, D.F.E., Smith, W.S.: Average strain strategy for finite elements. Finite Elements Anal. Des. 40, 2011–2024 (2004) [12] Sulsky, D., Kaul, A.: Implicit dynamics in the material-point method. Comput. Methods Appl. Mech. Eng. 193, 1137–1170 (2004) [13] Taylor, R.L.: A mixed-enhanced formulation for tetrahedral finite elements. Int. J. Num. Methods Eng. 47, 205–227 (2000) [14] Wan, J.: Stabilized finite element methods for coupled geomechanics and multiphase flow. PhD thesis, Stanford University, CA, USA (2002) [15] Wieckowski, Z., Youn, S.-K., Yeon, J.-H.: A Particle-in-cell solution to the silo discharge problem. Int. J. Num. Methods Eng. 45, 1203–1225 (1999) [16] Zienkiewicz, O.C., Taylor, R.L.: The finite element method, 5th edn. ButterworthHeinemann, Oxford (2000) [17] Zienkiewicz, O.C., Wu, J.: Incompressibility without tears - how to avoid restrictions of mixed formulation. Int. J. Num. Methods Eng. 32, 1189–1203 (1991)

Chapter 19

Effect of Boundary, Shear Rate and Grain Crushing on Shear Localization in Granular Materials within Micro-polar Hypoplasticity Jacek Tejchman

Abstract. The paper deals with three different phenomena concerning shear localization in granular materials using the finite element method. In the first case, the boundary effect on the behaviour of granular materials during plane strain compression was investigated with a micro-polar hypoplastic constitutive model. The numerical calculations were carried out with different initial densities and horizontal boundary conditions. Comparisons of the mobilized internal friction and dilatancy between global and local quantities were made. In the second case, plane strain shearing of an infinite long and narrow granular strip of initially dense sand between two very rough walls under conditions of free dilatancy was investigated within micro-polar hypoplasticity enhanced by viscous terms. The calculations were performed under dynamic conditions with different shear rates. Finally, grain crushing was investigated during plane strain shearing of an infinite long granular strip with a micro-polar hypoplastic model enriched by formulae of breakage mechanics.

19.1 Introduction Localization of deformation in the form of narrow zones of intense shearing is a fundamental phenomenon in granular materials and can appear under both drained and undrained conditions. Localization can appear as a spontaneous one inside of granular materials or as an induced one at interfaces between granulates and structure members. An understanding of the mechanism of the formation of shear zones is important since they usually act as a precursor to ultimate soils failure. The formation of shear zones depends upon many different factors as: initial density, mean grain diameter, pressure level, direction of deformation rate, shear rate, specimen size and wall roughness. Jacek Tejchman Gdansk University of Technology, Faculty of Civil Engineering, 80-952 Gdansk-Wrzeszcz, Narutowicza 11/12, Poland e-mail: [email protected]

M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 363–376. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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The intention of the paper is to investigate the effect of three different parameters on shear localization, namely: boundary, shear rate and grain crushing. The plane strain FE calculations were carried out with a micro-polar hypoplastic constitutive model.

19.2 Micro-polar Hypoplastic Model Non-polar hypoplastic constitutive models [2] describe the evolution of the effective stress tensor as a non-linear tensorial function of the current void ratio, stress state and rate of deformation. They are capable of describing some salient properties of granular materials, e.g. non-linear stress-strain relationship, dilatant and contractant behaviour, pressure. dependence, density dependence and material softening. A further feature of hypoplastic models is the inclusion of critical states, in which deformation may occur continuously at constant stress and volume. In contrast to elasto-plastic models, the following assumptions are not necessary: decomposition of deformation into elastic and plastic parts, yield surface, plastic potential, flow rule. Moreover, neither coaxiality, i.e. coincidence of principal axes of stress and principal plastic strain increments nor stress-dilatancy rule are not assumed in advance. The hallmark of the hypoplastic models is their simple formulation and procedure for determining the parameters with standard laboratory experiments. The material parameters are related to granulometric properties, viz. size distribution, shape, angularity and hardness of grain. A further advantage lies in the fact that one single set of material parameters is valid for wide range of pressure and density. Hypoplastic constitutive models without a characteristic length can describe realistically the onset of shear localization, but not its further evolution. In order to account for the post bifurcation behaviour, a characteristic length can be introduced into the hypoplastic model by means of micro-polar, non-local or second-gradient theories. In this paper, a micro-polar theory was adopted [5]. The micro-polar model makes use of rotations and couple stresses, which have clear physical meaning for granular materials. The constitutive relationship requires ten material parameters. The calibration procedure was given in detail in [3].

19.3 Boundary Effects on Behaviour of Granular Material Our knowledge on the mechanical behavior of granular materials is largely based on element tests with uniform stress and homogeneous strain. Some important parameters can be obtained from the element tests such as internal friction angle and dilatancy angle. However, the parameters are determined with the aid of measurements of global (exterior) quantities at specimen boundaries (force and displacement), which may differ from their local quantities inside the material (impossible to be measured). These differences are caused by localized deformation in shear zones

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and the imperfect boundaries of specimen in laboratory tests. The effect of localized deformation and boundary conditions is important to differentiate between material behavior from structure behavior. The FE-calculations of plane strain compression were performed with the specimen size of bo × ho = 40 × 140 mm2 (bo – initial width, ho – initial height) [6]. The specimen depth was equal to 1.0 m with plane strain condition. The width and height of the specimen were similar as in the experiments by Vardoulakis [9]. Quasi-static deformation in the specimen was imposed through a constant vertical displacement increment prescribed at nodes along the upper edge of the specimen. Two sets of boundary conditions were assumed. First, the top and bottom boundaries were assumed to be ideally smooth, i.e. there is no shear stress along these boundaries. To preserve the stability of the specimen against horizontal sliding, the node in the middle of the top edge was kept fixed. To simulate a movable roller bearing in the experiment [9], the horizontal displacements along the specimen bottom were constrained to move by the same amount from the beginning of compression. Second, comparative calculations were performed with a very rough top and bottom boundary. In this case, the horizontal displacement and Cosserat rotation along both horizontal boundaries were assumed to be equal to zero. The calculations were carried out with large deformations and curvatures using the so-called “updated Lagrangian” formulation by taking into account the Jaumann stress rate and Jaumann couple stress rate and the actual geometry and area of finite elements. Fig. 19.1 shows the deformed meshes together with the distribution of void ratio e (increase of e is indicated by darker shadow) for the different wall roughness and initial void ratio eo . In the case of smooth walls, only one shear zone inside the specimen was observed. For very rough boundaries, two shear zones were formed (two intersecting shear zones or two branching shear zones). The thickness of the shear zone measured at the mid-point of the specimen is about 15 × d50 (eo = 0.55), 20 × d50 (eo = 0.70) and 30 × d50 (eo = 0.90) for smooth and very rough walls (on the basis of shear deformation). Its inclination against the horizontal is about θ ≈ 52◦ (eo = 0.55), θ ≈ 48◦ (eo = 0.70) and θ ≈ 46◦ (eo = 0.90) for smooth walls, and θ ≈ 47◦ (eo = 0.55), θ ≈ 46◦ (eo = 0.70) and θ ≈ 45◦ (eo = 0.90) for very rough walls. Fig. 19.2 (upper row) and Fig. 19.3 show the evolution of the mobilized friction angle φ calculated with global and local principal stresses. The global principal stresses were calculated according to σ1 = P/bl (P – resultant vertical force, b – actual specimen width) and σ3 = σc . The local stresses were calculated at the different mid-points of shear zones (at x1 = bo /4, bo /2 and 3bo /4, respectively). For smooth walls, the mobilized global friction angle increases to reach a pronounced peak and drops gradually at large deformation (the residual state is not reached at ut2 /ho = 2.5%). The global peak friction angles are: φ p = 51.5◦ at ut2 /ho = 0.022 (eo = 0.55), φ p = 37.7◦ at ut2 /ho = 0.029 (eo = 0.70) and φ p = 30.3◦ at ut2 /ho = 0.08 (eo = 0.90), respectively. For comparison, the global friction angles at large deformation of ut2 /ho = 25% are about: φres = 29.0◦ (eo = 0.55), φres = 27.0◦ (eo = 0.70) and φres = 23.0◦ (eo = 0.90), respectively.

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Fig. 19.1 Deformed FE-meshes with the distribution of void ratio e at residual state: a) smooth boundaries, initially dense sand (eo = 0.55), ut2 /ho = 0.071, b) smooth boundaries, initially medium dense sand (eo = 0.70), ut2 /ho = 0.071, c) smooth boundaries, initially loose sand (eo = 0.90), ut2 /ho = 0.014, d) very rough boundaries, initially dense sand (eo = 0.55), ut2 /ho = 0.071, e) very rough boundaries, initially loose sand (eo = 0.70), ut2 /ho = 0.071, f) very rough boundaries, initially loose sand (eo = 0.90), ut2 /ho = 0.014 (ut2 – vertical displacement of the top)

As compared to very smooth boundaries, the evolution of the mobilized global friction angle is different. Along with deformation, the global friction angle increases to a pronounced peak, drops gradually to reach a residual state. The global peak friction angles are: φ p = 51.7◦ at ut2 /ho = 0.021 (eo = 0.55), φ p = 38.2◦ at ut2 /ho = 0.029 (eo = 0.70) and φ p = 30.9◦ at ut2 /ho = 0.06 (eo = 0.90), respectively. The

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(a)

(b)

(c)

(d)

Fig. 19.2 Evolution of mobilized global friction angle φ versus normalized vertical displacement of the upper edge ut2 /ho (upper row), evolution of mobilized global dilatancy angle ψ versus normalized vertical displacement of the upper edge ut2 /ho (lower row): a) smooth boundaries, eo = 0.55, b) smooth boundaries, eo = 0.70, c) smooth boundaries, eo = 0.90, d) very rough boundaries, eo = 0.55, e) very rough boundaries, eo = 0.70, f) very rough boundaries, eo = 0.90

global friction angles at large deformation of ut2 /ho = 25% are about: φres = 43◦ (eo = 0.55), φres = 38◦ (eo = 0.70) and φres = 30◦ (eo = 0.90), respectively. Thus, the peak friction angles are only slightly higher (by 0.2o − 0.7o ) and the friction angles at large deformation of ut2 /ho = 25% are significantly higher (by 7◦ − 14◦ ) than for smooth walls. Usually, the local friction angles in shear zones reach their asymptotic values (except of points where two shear zones intersect each other, i.e. in the cases with very rough boundaries and eo = 0.55 and eo = 0.70. The local peak friction angles are similar as the global values independently of the wall roughness. For smooth walls, the mean local residual friction angles are significantly higher than the global ones (by 5◦ − 7◦ ). In the case of very rough walls, the mean local residual friction angles are significantly lower than the global ones for eo = 0.55 and eo = 0.70 (by 4◦ − 12◦) or higher for eo = 0.90 (by 4◦ ). The evolution of the mobilized global and local (in the mid-point of the shear zone) dilatancy/contractancy angle ψ is demonstrated in Figs. 19.2 (lower row) and 19.4. The dilatancy/contractancy angle was calculated with the formula , where plastic strain rates were replaced by total strain rates (ν˙ - volumetric strain rate, γ˙ shear strain rate). All global dilatancy/contractancy curves are seen to be smooth.

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(a)

(b)

(c)

(d)

(e)

(f)

Fig. 19.3 Evolution of mobilized local friction angle φ in the mid-points of shear zones versus normalized vertical displacement of the upper edge ut2 /ho : (a) smooth boundaries, eo = 0.55, (b) smooth boundaries, eo = 0.70, (c) smooth boundaries, eo = 0.90, (d) very rough boundaries, eo = 0.55, (e) very rough boundaries, eo = 0.70, (f) very rough boundaries, eo = 0.90

They show (independently of eo ) initial contractancy (up to ut2 /ho = 0.025 − 0.10) followed by dilatancy for initially dense and medium dense sand or further contractancy for initially loose sand. The peak dilatancy angles are about: 22◦ − 24◦ for eo = 0.55 and 5◦ for eo = 0.70. At large deformation for smooth boundaries, the dilatancy angle for initially dense sand (eo = 0.55) decreases gradually to approach the zero value in the residual state. In the case of eo = 0.70, the dilatancy also decreases gradually reaching again the region contractancy. When the specimen is initially loose (eo = 0.90), contractancy continuously increases. In the case of very rough walls, the dilatancy angle for eo = 0.55 and eo = 0.70 decreases gradually and increases next again with increasing deformation. When

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the specimen is initially loose, contractancy continuously decreases reaching the asymptote at +5◦ . Usually, the local dilatancy angles in the shear zones reach their asymptotic values except the cases with very rough boundaries (initially dense and medium dense sand). The local peak dilatancy angles are similar as the global values independently of the wall roughness. For smooth walls, the mean local residual dilatancy angles are higher than the global ones at large deformation ut2 /ho = 0.25 (by 10◦ − 35◦). In the case of very rough walls, the mean local residual dilatancy angles can be in some points of shear zones significantly higher than the global ones (by 5◦ − 35◦ ). An abrupt change of contractancy to dilatancy during shear zone occurrence close to the peak (observed in experiments with initially dense sand [9]) was only observed with initially loose sand. In contrast to global dilatancy curves, the local dilatancy curves indicate pronounced oscillations in the shear zone for 3 cases: in initially loose sand with smooth boundaries at peak (ut2 /ho = 0.03 − 0.08) and in some points of initially dense and medium dense sand with very rough boundaries for large vertical deformation (ut2 /ho = 0.15 − 0.25). Oscillations of volume changes were observed in experiments by Vardoulakis with initially dense specimens [9]. Approximately, the evolution of local dilatancy/contractancy is similar to the evolution of local friction.

19.4 Effect of Shear Rate In granular flow, there exist three well-defined asymptotic regimes, namely rapid flow, quasi-static flow and transitional moderate flow. In rapid flow, the internal stress is mainly due to collisions between grains and the material behaves like a dissipative gas. In quasi-static flow, the stress is due to friction between particles and the material shows solid-like behaviour. In transitional moderate flow, the material shows fluid-like behaviour. The dominating mechanism in the solid-like regime is frictional, while the fluid-like behaviour is primarily viscous. The behaviour of granular materials exhibiting fluid-like or dynamic behaviour is of importance in numerous engineering problems, e.g. bulk solid handling, debris flow, bed transport and fluidization. To describe the behaviour in dynamic regime, the micropolar hypoplastic model was enhanced by viscous terms according to Stadler and Buggisch [4]. FE-calculations were performed for an infinitely long and narrow granular strip of the height of ho = 50 mm (ho = 100 × d50) between two rigid very rough walls under conditions of free dilatancy [7]. The study was performed with only one element column with a width of 50 mm, consisting of 20 quadrilateral horizontal elements composed of four diagonally crossed triangles. The behaviour of an infinite shear layer was modelled by lateral boundary conditions, i.e. displacements and rotations along both sides of the column were constrained by the same amount. Consequently, the evolution of the state variables was independent of the layer length. The integration was performed with three sampling points placed in the middle of

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(a)

(b)

(c)

(d)

(e)

(f)

Fig. 19.4 Evolution of mobilized dilatancy angle ψ in the mid-points of shear zones versus normalized vertical displacement of the upper edge ut2 /ho : (a) smooth boundaries, eo = 0.55, (b) smooth boundaries, eo = 0.70, (c) smooth boundaries, eo = 0.90, (d) very rough boundaries, eo = 0.55, (e) very rough boundaries, eo = 0.70, (f) very rough boundaries, eo = 0.90

each element side. The calculations were carried out with large deformations and o = −(γ x + 1.0 kPa), curvatures. The following initial stress state was assumed: σ22 2 o o o σ11 = σ33 = Ko σ22 , γ - initial volumetric weight (γ = 16.7 kN/m3 ), x2 - vertical coordinate calculated from the specimen top, Ko – earth pressure coefficient at rest o - vertical normal stress, σ o - horizontal normal stress, σ o - nor(Ko = 0.45), σ22 11 33 mal stress perpendicular to the plane of deformation. The initial void ratio of dense sand (eo = 0.60) was assumed to be homogeneous in the specimen. The sand specimen was subject to shearing with free dilatancy. Shear deformation was initiated through constant horizontal displacement increments prescribed only at the nodes along the top of the strip. In the dynamic calculations, the horizontal velocity of

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the top edge changed from v = 5 mm/s up to v = 5 m/s, which corresponded to the shear rate γ˙ = 0.1 1/s up to γ˙ = 100 1/s ( = v/ho). The walls were assumed to be very rough (neither sliding nor rotation). The boundary conditions along the bottom were: u1 = 0, u2 = 0 and ω c = 0, and along the top boundary: u1 = nδ u (in o (the static calculations), or u1 = v (in dynamic calculations) ω c = 0, and σ22 = σ22 parameter n denotes the number of time steps and Δ u is the constant displacement increment in one step). The volume moment was neglected due to its insignificant effect (Tejchman 1989). The volume inertia moment was assumed as for a cylinder with a diameter d50 and mass density ρ . The evolution of the mobilized wall friction angle ϕ versus shear strain ut1 /ho across the granular strip with a different shear rate is demonstrated in Fig. 19.5 [7]. The results show that the friction angle and their fluctuation increase with increasing shear rate.

(a)

(b)

(c)

Fig. 19.5 Evolution of mobilized wall friction angle ϕ versus ut1 /ho during shearing of initially dense sand between two very rough boundaries: a) x2 = −25 mm, b) x2 = −6.25 mm, c) x2 = −3.75 mm, d) x2 = −1.25 mm, (a) γ˙ = 1 1/s, (b) γ˙ = 10 1/s, (c) γ˙ = 100 1/s (vertical co-ordinate x2 is measured from the top edge) [7]

The distribution of the Cosserat rotation and void ratio across the granular layer are shown in Fig. 19.6. The width of the shear zone occurring along the top wall decreases with increasing shear rate. Thus, our FE-calculations show that the thickness of the wall shear zone depends also upon viscosity and shear rate.

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The dependence of the wall friction angle on the shear rate is summarized in Fig. 19.7, which shows a parabolic dependence for the peak friction angle and a linear one for the residual friction angle.

(a)

(b)

Fig. 19.6 Shearing of dense sand between two very rough boundaries at ut1 /h = 0.2: (a) distribution of lateral displacement u1 and (b) Cosserat rotation ω c across the layer height x2 (results with γ˙ = 1 1/s – continuous line, results with γ˙ = 10 1/s – dotted line) [7]

Fig. 19.7 Relationship between wall friction angle ϕ = arctan μ and shear rate from FE analyses: a) peak wall friction angle, b) residual wall friction angle [7]

19.5 Effect of Grain Crushing Grain crushing is one of the phenomena which strongly influence the stress-strain behavior of granular bodies. It is caused both during compression and confined shearing. Since it is an inherent characteristic of these loading processes, it should be taken into account in constitutive modeling. It is particularly important when describing shear localization since the thickness of shear zones depends on the mean grain diameter which may diminish during grain crushing. The experimental results show that the breakage of particles increases with increasing confining pressure and

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that larger particles are more vulnerable to degradation in terms of their strength, but more resistive for they tend to be more cushioned. Particle breakage increases with increasing shear strain at a decreasing rate. A constant grading is reached at very large strains, where the process depends upon the pressure, and the characteristics of the initial grain size distribution (gsd). Small amount of particle breakage occur even at low stress levels, but in lower rates. The gsd tends to evolve towards an ultimate fractal distribution. The calculations were carried out with a micro-polar hypoplastic constitutive model, where the change of the median grain diameter with varying pressure and void ratio was taken into account with the help of formulae from breakage mechanics proposed by Einav [1]. Contrasted to the hypoplastic modeling framework, the thermodynamics of rate-independent dissipative materials entail definitions of yield functions and flow rules. Within the yield surface, the behavior is elastic. The yield function and the flow rule are activated upon yielding, which is directly connected to hypothesized energy balanced. The energy balance postulate enables to describe how the gsd evolves via the property of breakage as a thermodynamics internal variable. The quasi-static FE-calculations were performed for an infinitely long and narrow granular strip of the height of ho = 10 mm between two rigid very rough walls under conditions of free dilatancy [8]. The study was performed with only one element column with a width of 100 mm, consisting of 20 quadrilateral horizontal elements composed of four diagonally crossed triangles. The behaviour of an infinite shear layer was modelled by lateral boundary conditions, i.e. displacements and rotations along both sides of the column were constrained by the same amount. The calculations were carried out with large deformations and curvatures. The sand specimen was subject to shearing with free dilatancy under constant vertical pressure of p = 2000 kPa. Shear deformation was initiated through constant horizontal displacement increments (directed to the left) prescribed only at the nodes along the top of the strip. The walls were assumed to be very rough (neither sliding nor rotation). Figure 8 presents the effect of grain crushing on the evolution of the mobilized wall friction angle ϕ = arctan(σ12 /σ22 ) versus the shear deformation ut1 /ho . The distribution of the Cosserat rotation ω c (the positive Cosserat rotation (+ω c ) is directed counter clockwise) and void ratio e along the layer height at the residual state are demonstrated in Fig. 19.9. Finally, the evolution of the median grain diameter in the granular strip during shearing is shown in Fig. 19.10. The mobilized wall friction angle is ϕw = 40◦ at peak, and ϕw = 31◦ at residual state in the case of FE studies without grain crushing and ϕw = 30◦ with grain crushing (Fig. 19.8). The Cosserat rotation and non-symmetry of the stress tensor (σ12 = σ21 ) are noticeable during shearing. A shear zone is formed in the middle of the layer, which is characterized by the appearance of the Cosserat rotation (Fig. 19.9a) and a strong increase of the void ratio (Fig. 19.9b). At the upper and lower boundaries of the dilatant shear zone, a strong jump of the horizontal displacement, curvature, stresses and couple stress can be observed. The thickness of the shear zone (as visible from

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Fig. 19.8 Shearing of initially dense sand between two very rough boundaries: evolution of wall friction angle ϕw = arctan(σ12 /σ22 ) versus shear deformation ut1 /ho : a) without grain crushing, b) with grain crushing (p = 2000 kPa)

(a)

(b)

Fig. 19.9 Shearing of initially dense sand between two very rough boundaries: distribution of Cosserat rotation ω c and void ratio e across the layer height x2 : a) without grain crushing, b) with grain crushing (p = 2000 kPa)

o (5 mm) without grain crushing and 6 × d o (3 the Cosserat rotation) is about 10× d50 50 0 mm) with grain crushing ( d50 - initial mean grain diameter). The void ratio increases in the middle of each shear zone at residual state up to ec = 0.69. The change of the median grain diameter across the layer height is non-uniform. It takes place only in the hardening regime. The median grain diameter decreases 0 = 0.5 mm down to d 0 = 0.23 mm (in the middle of the during shearing from d50 50 shear zone) and down to 0.29 mm outside the shear zone at the wall.

19.6 Conclusions The following conclusions can be drawn from FE calculations: The boundary conditions have considerable effect beyond the peak, where a shear zone is fully developed. This can be ascertained with respect to mobilized friction and dilatancy (i.e. the global quantities can significantly differ from the local ones in

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Fig. 19.10 Shearing of initially dense sand between two very rough boundaries: evolution of median grain diameter d50 versus shear deformation ut1 /ho (p = 2000 kPa): a) x2 = 5 mm (mid-point), b) x2 = 4 mm, c) x2 = 3 mm, d) x2 = 2 mm, e) x2 = 1 mm, f) x2 = 0 mm

the shear zones). The pattern of shear zones seems to depend on the boundary conditions, too, in particular on the roughness of top and bottom-plates. The numerical calculations with smooth plates show one single shear zone, whereas rough plates give rise to two intersecting shear zones. The thickness of the shear zone is found to increase with increasing initial void ratio and remains fairly independent of the wall roughness. The shear zone inclination increases with decreasing void ratio and decreasing wall roughness. Both the global and local internal friction angles at the peak increase with decreasing void ratio and remain fairly independent of the boundary roughness. However, the residual friction angle, both global and local, is found to depend on both void ratio and boundary roughness. Similar observations can also be made on the dilatancy angles. The augmented hypoplastic model is applicable to both the solid-like and fluidlike behaviour of granular materials in particular in the range of moderate shear rates up to 10 1/s. The mobilized friction strongly increases with increasing shear rate and viscosity. The thickness of the wall shear zone depends also upon viscosity and shear rate. The numerical results show that the degree of particle breakage affects significantly the sand behavior in particular at the residual state. Both the residual mobilized shear resistance and thickness of the shear zone decrease with grain crushing. The median grain diameter decreases more strongly in the shear zone than outside it.

References [1] Einav, I.: Breakage mechanics. Part I-theory. J. Mech. Phys. Solids 55(6), 1274–1297 (2007) [2] Gudehus, G.: Comprehensive equation of state of granular materials. Soils and Foundations 36(1), 1–12 (1996)

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[3] Herle, I., Gudehus, G.: Determination of parameters of a hypoplastic constitutive model from properties of grain assemblies. Mechanics of Cohesive-Frictional Materials 4(5), 461–486 (1999) [4] Stadler, R., Buggisch, H.W.: Influence of the deformation rate on shear stresses in bulk solids – theoretical aspects and experimental results. In: Proc. Conf. Reliable Flow of Particulate Solids, Bergen. EFCE Pub. Series, vol. 49 (1985) [5] Tejchman, J., Górski, J.: Computations of size effects in granular bodies within micropolar hypoplasticity during plane strain compression. Int. J. for Solids and Structures 45(6), 1546–1569 (2008) [6] Tejchman, J., Wu, W.: FE-investigations of non-coaxiality and stress-dilatancy rule in dilatant granular bodies within micro-polar hypoplasticity. Int. Journal for Numerical and Analytical Methods in Geomechanics 33(1), 117–142 (2009) [7] Tejchman, J., Wu, W.: FE-investigations of shear localization in granular bodies under high shear rate. Granular Matter 11(2), 115–128 (2009) [8] Tejchman, J., Einav, I.: Effect of grain crushing on shear localization using micro-polar hypoplasticity. In: Proc. Int. Conf. “Computational Geomechanics”, ComGeo I, pp. 204– 214 (2009) [9] Vardoulakis, I.: Shear band inclination and shear modulus in biaxial tests. Int. J. Num. Anal. Meth. Geomech. 4, 103–119 (1980)

Part V

Biomechanics

Chapter 20

Biomechanical Basis of Tissue–Implant Interactions Romuald Bedzinski and Krzysztof Scigala

Abstract. Tissue-implant interactions were analysed by investigations of bone adaptation processes, especially in the region surrounding the implant. In such cases all factors stimulating bone negative remodelling should be taken in consideration. Because of that, a three-step analysis of implant alignment to the surrounding tissues was carried out. Basic parameters of the bone-implant interaction were estimated (on the macro level: stiffness characteristics, shear strains distribution, and bone tissue density distribution; on the micro level: trabecular structures development, trabecular microcracks distribution, and bone cells strain distribution). Estimation of each parameter was carried out by development of numerical tools which enabled control of bone tissue changes caused by changes in the implant design. All steps of the analysis were carried out using FE models and own simulation procedures.

20.1 Introduction Clinical practice shows that there is still a significant percentage of postoperative complications connected with lack of alignment of the implant and the surrounding tissues. Typical symptoms in such case are aseptic loosening of implant parts, implant migration, microcracks, and bone fractures or even implant fractures. In fact, most of the above-mentioned processes originate from pathological adaptation of tissues (like stress shielding), lack of osteoinduction on the implant surface, mechanical degradation of tissues, or generation of pathological tissue structures [3], [4], [8], [23]. Nevertheless, due to the biological nature of most of those processes, Romuald Bedzinski Wroclaw University of Technology, Lukasiewicza 7/9, 50-371 Wroclaw, Poland e-mail: [email protected] Krzysztof Scigala Wroclaw University of Technology, Lukasiewicza 7/9, 50-371 Wroclaw, Poland e-mail: [email protected]

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the biomechanical parameters significantly control their progress or intensity [9], [11], [12], [18]. Specifically, alignment of the strain characteristics of the implant and the surrounding tissues plays a major role in the process of implant integration and affects its parameters, which should be monitored or verified during implant design [4], [16]. The biomechanical alignment of the implant can be controlled by proper selection of the implant shape and dimensions but also by selection of the implant material. Some of the new generation of alloys exhibit lower stiffness and higher biocompatibility (Fig.20.1).

Fig. 20.1 Modulus of elasticity for typical and new generation alloys used for implant manufacturing [7]

Taking into consideration new materials, the biomechanical characteristics of the implant, determined by its design, should be investigated on the macro level as well as on the micro level. That leads to the conclusion that the implant design should take into account such parameters as major stiffness characteristics but also the influence of the implant shape and dimensions on tissue growth, remodelling, and even bone cell activity. The main aim of this analysis is to establish a protocol and numerical tools for implant analysis, which can be applied in implant design and selection for an average patient or in analysis of the treatment progress.

20.2 Material and Method The proposed analysis consisted of three steps. The first step was an analysis of implant mechanical characteristics on the macro level. Investigations of implant stiffness characteristics were carried out using FE models of the implant and the implant-bone complex (Fig. 20.2). The implant models were designed by direct measurement of implant dimensions and geometry and transfer of geometrical data

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into the geometrical model. For discretisation of geometrical models tetrahedralshaped elements with 10 nodes and 3 degrees of freedom were used. Models of the implant-bone complex were created by alignment of the already existing implant models into geometrical models of the femur bone. The femur bone model was created on the basis of CT data. It was used in simple loading models (such as bending in coronal plane – Fig. 20.1c) to investigate distribution of stiffness along the implant’s main axis and for comparison of the stiffness characteristics of the intact bone and the bone with implant. Calculations were carried out for a wide range of implant materials, including new generation alloys. Analysis of shear strain on the tissue-implant contact surface was also carried out using the same models of the bone-implant complex. For this analysis the Maquet model of hip joint loading was used [16]. In each of the above-mentioned analyses the material properties for implant alloys and bone tissue were taken from the literature [1], [5], [10], [19], [21]. All materials were considered as elastic, with linear stress-strain characteristics. The second step of the analysis consisted of numerical simulations of bone adaptation processes. The second step of the analysis consisted of numerical simulations of

Fig. 20.2 Model of implant (a), model of implant - bone complex (b), model of load used in stiffness characteristic calculations (c) and model of loading used in shear strain distribution analysis (d)

bone adaptation processes. Changes in bone density were analysed after implantation using the Carter model of bone remodelling [6], [8] to estimate tissue behaviour on the macro level. The analysis used two-dimensional FE models of the bone with implant (Fig. 20.3a). There is a significant difference in stress and strain distribution in the femur models depending on whether two-dimensional or three-dimensional models are used. Because of this, in the above-described case we used as a side plate technique [6], [13]. In such case, 2D models can still be used, however, results of the calculations are close to those obtained using 3D models. In the case of finite

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elements we used mesh plane elements with 8 nodes and 3 degrees of freedom at each node. For connections of the main model with the side plate we used beam elements with 2 nodes and 3 degrees of freedom at each node. The models were loaded by forces calculated using the Maquet model of hip and knee joint loading [17]. In this analysis the mechanical properties of bone tissue were assumed to be the lowest values observed for trabecular bone tissue at the beginning of the simulation. In further simulation iterations the mechanical properties values were changed according to the Carter model [2], [8]. To assess bone adaptation processes also on the micro level the Tsubota model of bone remodelling [23], [24] was used, as well as 2D bone models including trabecular structure of the cancellous bone (Fig. 20.3b). Two-dimensional FE models were created by transfer of geometrical information about the main coronal plane of the bone (including information about the cortical layer thickness) into the model. Next, a pattern of randomly placed circular shapes was created. Each circular shape consisted of pixel-size finite elements. The pattern was created in such way that it was homogenous and isotropic. During the subsequent simulation iterations the arrangement of finite elements in this pattern was changed according to the Tsubota model of trabecular tissue remodelling [23], [24]. In the case of the femur bone, the load model proposed by Beaupre was used [2]. In the case of tibia bone the Hurwitz model of knee loading was used [15]. Material properties of tissues and implant alloys were taken from the literature. The model was extended with an additional simulation procedure allowing calculations of bone mechanical degradation. The process of generation of microcracks in the trabecular structure was included into the procedure by using the Zioupos bone microdamage accumulation mode [26]. The third step of the analysis was an

Fig. 20.3 Model used in calculations of bone remodelling on the macro (a) and micro (b) levels

analysis of bone cell strain distribution in order to estimate influence of the external loading on the bone cells activity. It was assumed that strains on the osteocyte surface were proportional to the mechanical stimulus signal, which controls bone tissue growth and remodelling [14], [19], [20], [25]. Hierarchical FE modelling was used, from a bone model in the macro scale, to a model of bone tissue section, a model

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of trabecular structure, a model of single trabecula, a poroelastic model of trabecula material section, a model of osteocyte lacunae (including interosseous fluid flow), and a model of osteocyte itself (Fig. 20.4). The bone model in the macro scale was created using CT data from measurement of the tibia bone SAWBONE model. Material properties of cancellous and cortical bone tissue were taken from the literature [1], [5], [10], [18], [21]. The model was loaded by forces calculated from the Maquet model of knee loading [17]. After calculation of displacement distribution a block model of trabecular tissue section was created. The model was loaded at each node by displacements with the previously estimated values. Initial pattern of trabeculae was created by randomly placed spherical shapes consisting of finite elements. The loaded pattern of trabeculae was changed in further iterations of the simulation according to the Tsubota model. The next model was a model of a single trabecula. The dimensions and shape of a single trabecula were estimated from the model of tissue section. Selected trabecula geometry was measured in the model of tissue section by automatic estimation of several cross-section dimensions and the placement. The trabeculae material was assumed to be poroelastic when the pores in the material simulated osteocyte lacunas and the network of their interconnecting canaliculi. We calculated bone mineralized matter strains as well as interosseous fluid pressures and flow velocity. In the next step of the analysis a section of trabecula material was selected, small enough to consist of just one osteocyte and its lacunae.

Fig. 20.4 Hierarchical modelling for estimation of bone cells strain distribution

Model of this section is block shaped and consists of simplified geometry of osteocyte in the lacunae and simplified geometry of canaliculi connecting osteocyte with neighbours. Next, coupled field analysis was carried out. The pressures and velocities of interosseous fluid flow were calculated. Displacements of osteocyte surface were calculated as well. A numerical procedure for automatic model generation at each step and step-to step-data transfer was developed.

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20.3 Results The first step of the analysis demonstrated the significant influence of implant design and material on the basic biomechanical characteristics, describing implant alignment to the surrounding tissues. Sample stiffness characteristics of an implant manufactured using various alloys are shown in Fig.20.5. In all cases the maximum values of the bending stiffness were found in the proximal part of the implant, which is understandable because of the large implant cross-section in that region. It should be noted that the shape of the characteristics in each case (each implant alloy) is very similar, however analysis also proved that the application of new types of alloy leads to changes in the maximum stiffness, which can be as high as 45%. An implant manufactured using a new Ti-12Mo-6Zr-2Fe alloy has almost half lower bending stiffness than an implant manufactured using a classic Co-Cr-Mo alloy. Analysis of shear stress distribution on the contact surface between an implant and tissue shows that the implant shape has significant influence on the level of shear stress. Especially in the proximal medial part of the implant-bone complex we can observe bone tissue overloading.

Fig. 20.5 Sample stiffness characteristics for various implant materials

The second step of the analysis allowed us to develop numerical tools which can be used for analysis of the influence of implant design on the negative and positive bone adaptation as well as bone mechanical degradation. In the first analysis a change in the bone density was estimated according to the Carter model [8]. Figure 20.6a shows sample results of density distribution calculations in the model of the femur proximal part and in the model of femur bone with stem of hip joint endoprosthesis. On the left-hand side we can see that simulation in the intact model leads to formation of cortical layers on the lateral and medial sides of the bone. Some bone formations of medium density can also be observed in the distal part of the femur neck. In the case of the femur model with implant we can observe several differences in the density distribution according to the region around implant. In the region marked by (1) on the Fig. 20.6a we can observe good connection between implant and bone tissue. There was a continuous layer of bone tissue, without symptoms of bone resorption. Connection between bone and implant existed on the whole length of the implant. In the region marked (2) on the Fig. 20.6a we observe exactly the same results as previously. However, the length of this region is much

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shorter than on the opposite side. The region marked by (3) on the Fig. 20.6a can be characterised as showing abnormally low values of bone density, which is a result of a non-physiological bone resorption caused by high implant stiffness, as proved in the first stage of the investigation. The region marked by (4) on the Fig. 20.6a can be characterised by a significant decrease in the cortical bone layer thickness. It is also a result of implant maladjustment to the surrounding tissues. Rapid change of the bending stiffness at the end of stem leads to a decrease of bone loading in the part below the end of stem and bone resorption. This assumption results from discovery of a more detailed image of implant maladjustment compared to the first stage of the investigation.

Fig. 20.6 Sample results of bone remodelling simulations on the macro (a) and micro (b) levels

Figure 20.6b shows results of simulation of trabecular bone on the micro level. The sample results in the image show distribution of trabecula trabeculae in the proximal part of tibia bone with implanted tibial part of knee endoprosthesis. We can observe formation of structures transferring load form the implant to the distal part of bone. In the medial distal part of model we can observe trabeculae with high values of trabecula length and diameter. The structure is well developed and directed. For the most part the trabeculae are directed more or less vertically and connect the end of the endoprosthesis stem with the lower edge of the model (where model was constrained). There are also inclined trabeculae, connecting the end of stem with the medial layer of the cortical bone . The number of horizontally directed trabeculae is very low in that region. On the lateral distal part we can observe structures which are not so well developed as on the medial side. It is the result of overloading of the medial bone part, which was not corrected by the implant. In the proximal part of the model (just below the tibial implant part plate, we can observe well-developed structures of inclined trabeculae. The trabeculae in the medial part are directed from medial side to the lateral one. In the lateral part of the model the

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trabeculae are directed in the opposite direction (from the lateral side to the medial one). The structure in the medial part of the model is more dense that in the lateral part. In the model part which extends from the middle of the stem length to the end of the stem we can seen an effect of trabecular structure resorption. On the medial side of the stem the trabecular structure is poorly developed, consisting mostly of short and thin trabeculae. The trabeculae are poorly connected with the stem surface. On the lateral side of the stem we can observe complete resorption of structures. Application of additional procedures allows us to calculate damage accumulation in trabecular structure according to Zioupos. For each finite element in the model, a ratio of damage accumulation to negative change of elastic modulus and density was calculated. Negative change of density was calculated for each iteration separately. Results for damaged trabeculae are shown in the Fig. 20.6a (zones described as A, B, C). Zone (A located below end of stem) can be characterized by the relatively long incline cracks. Most of them are located below the lower stem surface. There is significantly less cracks in the medial side of the stem end. Those cracks are oriented more horizontally. In the B zone (located near connection of stem and plate of endoprosthesis on the medial side) we can observe cracks connected into a complex pattern without one dominating direction. Damage of trabecular structure in this case leads to a break of connection between the main structure and the part directly connected with the implant surface. In the C zone (located on the lateral side of stem) we can observe mostly vertically-oriented cracks. It should be noted that in the C zone resorption of trabecular structure was relatively high so only few trabeculae were left. We can assume that because of lack of trabeculae in the part below the analysed zone, cracks started from the lower part and propagated vertically. Also, in that case

Fig. 20.7 Selected results of bone cells strain analysis

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it lead to separation of the main trabecular load carrying structures from the implant stem surface. The third step of the analysis showed significant influence of mechanical loading of bone on the strain distribution in the osteocyte cells. Using procedure of hierarchical modelling (Fig. 20.7) it is possible to estimate of strain distribution in osteocyte network and further analysis of mechanical stimulus on the biological processes involved in tissues adaptation around implant. The results of tibia model loading show that a calculated displacement distribution is a result of several loading components. Application of forces on the articular surfaces and the forces simulating main muscles group activity lead to bending of tibia bone in coronal plane, but also to bending in sagittal plane. Some results of tibia compression and torsion are also visible in the distribution of displacements. Displacement distribution was calculated for every of five load cases simulating the balance phase of gait. Displacements calculated for each node of the model were stored in an external file and transferred into models of trabecular bone tissue. Those displacements were used for loading in the simulation of trabecular remodelling carried out using models of trabecular tissue sections. The results of the simulations are trabecula structures. In all models of trabecular structure highly directed structures were developed. In most cases it is possible to mark the main load-carrying structures by the trabecula dimensions and their alignment to the loading direction. Each of the calculated trabecular structure was controlled in such way as to calculate the whole structure volume. The obtained calculations were used to estimate the trabecular bone density for each bone section model. Comparison to typical trabecular tissue densities shows that that the developed structures satisfactorily mimic real trabecular tissue. For each of the developed trabecular structures stress distributions was measured to estimate load distribution among trabeculae. The next step of the analysis concerned trabeculae subjected to the biggest load. The load was transferred to the single trabecula models. For each trabecula displacement distributions in the solid phase were calculated as well as distributions of pressures and flow velocities of the interosseous fluid. According to the estimated pressure distributions of the interosseous fluid, all analysed trabeculae can be divided into 3 groups: group 1 nonlinear pressure distribution with minimal value in the middle of trabecula , group 2 - non-linear pressure distribution with maximal value in the middle of trabecula, group 3 - linear pressure distribution with maximal value at the end of trabecula and minimal value at the opposite end. The trabecula parts from each of the groups in which maximal pressure was recorded provided the basis for models of trabecular material sections. Values of displacements of solid phase of trabecula model were transferred to the solid part of trabecular material section model. Pressure values and velocities of the interosseous fluid were transferred to the external surfaces of a model of trabecular material section (specifically, ends of each canaliculi). Pressures and velocities of the interosseous fluid flow in the lacunae of osteocyte were calculated for the solid phase movement and external pressure load at the ends of each canaliculi. Results shows that pressures and velocities of the interosseous fluid in the lacunae are significantly higher than in the canaliculi connecting osteocytes. The recorded velocity distribution shows that the flow is mostly tangential to the

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osteocyte surface. Some disturbances were recorded in the regions of connection of each canaliculi with osteocyte lacunae. In those regions velocity changed direction from the direction tangential to the osteocyte surface to the direction tangential to the axis of canaliculi. Along the axis of canaliculi velocity decreased rapidly and in the short distance from lacunae its velocity was significantly lower. Most often the recorded pressure distribution was the higher in the lacunae than in each of the canaliculi. Mostly, the highest pressure was recorded on the one side of osteocyte and the minimum pressure on the opposite side. It means that as long as one side of the osteocyte was under significant pressure, the opposite side was under significant underpressure. On both sides the highest absolute pressure was recorded in the regions between the ends of the neighbouring canaliculi. Analysis of displacement component distributions on the osteocyte surface shows that surface displacement concentrations most often exist in the regions of connection of the osteocyte body and inset close to one of the ends of the osteocyte body. Hierarchical analysis shows that load distribution can be considered at many levels of bone tissue organisation. There is clearly a direct connection between load distribution in whole bone and organisation of trabecular structures and development of efficient load carrying structures consisting mostly of thick trabeculae. Load distribution among those trabeculae affects the flow of the interosseous fluid. We estimated the most frequent pressures distributions . The solid phase movement and the interosseous pressure are high enough to change the shape of osteocyte body. Distribution of load from the whole bone to the trabeculae and next to a network of lacunas and canaliculi is the main factor affecting the values of displacements of the osteocyte surfaces. Taking into consideration the possible mechanosensation role of the osteocyte network during the process of bone adaptation, the mechanism of load transfer from the macro level to the micro level was tracked and described.

20.4 Conclusion Analysis of the implant influence on the bone tissue reaction was carried out on the many levels of bone organisation. First, the results of the analysis conducted on the whole bone level show that changes of implant stiffness characteristics are so significant that new-generation implant materials should be considered. Lack of implant maladjustment can be detected by analysis of the bone-implant complex stiffness characteristics and analysis of shear stress distribution. A more detailed analysis of implant mechanical alignment to the surrounding tissues focuses on bone adaptation. Negative bone adaptation can be estimated on the macro level and trabecular level. The next step of that process deals with trabecular structure mechanical degradation, which cannot be compensated by biological processes. On the micro level it can be estimated by simulation of trabecular development, including damage accumulation combined with analysis of strain distribution among osteocyte cells. To sum up, it should be noted that there are several tools for analysis of implant interactions with the surrounding tissues. Numerical analysis of implant influence on the

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whole bone, bone trabecular tissue, and even bone cells is possible. Results of those analyses can be used in the process of implant design, including optimisation of its shape and dimensions. On the other hand, only biomechanical factors were taken into considerations. Biological factors should also be investigated, however, preliminary experimental investigations are necessary. In order to obtain osteointegration with an implant, development of haematoma is required as well as its transformation into fibrous tissue. There are several conditions that must be met to obtain osteointegration, such as appropriate relations of deformation including the avoidance of stress shielding in bone tissue surrounding an implant and bioacceptability. The purpose of the research was to analyse and assess biomechanical and biochemical conditions for proper shaping of the outer implant level and structure in order to obtain the required osteointegration process. The main goal of contemporary research is to obtain a proper implant-bone interaction, so that the process of osteointegration is achieved. It has been proved that osteoconduction of the bone cells is surface-dependent and that is why implant coatings seem to be indispensable. But to obtain osteointegration some more special conditions must be met, such as avoiding of stress-shielding or the adequate implant-bone strain relations. Acknowledgements. This project is supported by Polish Ministry of Science and Higher Education. Project number N501 042 31/2800.

References [1] An, Y.H., Draughn, R.A.: Biomechanical testing of bone and the bone – implant interface. CRC Press, New York (2000) [2] Beaupre, G.S., Orr, T.E., Carter, D.R.: An approach for time-dependent bone remodeling and remodeling applications: a preliminary simulation. Journal of Orthop. Res. 8, 662–670 (1990) [3] Burr, D.B., Robling, A.G., Turner, C.H.: Effects of Biomechanical Stress on Bones in Animals. Bone 30, 781–786 (2002) [4] Bedzinski, R.: Biomechanical engineering - selected topics. Oficyna Wydawnicza Politechniki Wroclawskiej, Wroclaw (1997) (in Polish) [5] Bedzinski, R., Ostrowska, A., Scigala, K.: Investigations of mechanical structure of long bone. Machine Dynamics Problems 28, 45–50 (2004) [6] Bedzinski, R., Scigala, K.: FEM analysis of strain distribution in tibia bone and relationship between strains and adaptation of bone tissue. Computer Assisted Mechanics and Engineering Sciences 10, 353–368 (2003) [7] Bedzinski, R., Scigala, K.: Problems of bone – orthopedic implant interactions. Biomechanica Hungarica 1, 47–56 (2008) [8] Carter, D.: Skeletal function and form. Cambridge University Press, Cambridge (2001) [9] Cowin, S.C.: Bone mechanics handbook, 2nd edn. CRC Press, New York (2001) [10] Currey, J.D.: The effect of porosity and mineral content on the Young’s modulus of elasticity of compact bone. Journal of Biomechanics 21, 131–139 (1998)

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[11] Frost, H.M.: A determinant of bone architecture. The minimum effective strain. Clinical Orthopedic 175, 286–292 (1983) [12] Frost, H.M.: Bone mass and the mechanostat: a proposal. Anat. Rec. 219, 1–9 (1987) [13] Huiskes, R.: If bone is the answer, then what is the question? J. Anat. 197, 145–156 (2000) [14] Huiskes, R., Mullender, M.G.: Osteocytes and Bone Lining Cells: Which are the Best Candidates for Mechano-Sensors in Cancellous Bone? Bone 20, 527–532 (1997) [15] Hurwitz, D.E., Sumner, D.R., Andriacchi, T., Sugar, D.A.: Dynamic knee loads during gait predict tibial bone distribution. Journal of Biomechanics 27, 423–430 (1998) [16] Knets, I., et al.: Bond strength of implant to the bone tissue and the stress - strain state of bone. Acta of Bioengineering and Biomechanics 8 (2006) [17] Maquet, P.G.J.: Biomechanics of the knee, 2nd edn. Springer, New York (1977) [18] Martin, R.B., Burr, D.B.: Structure, Function and Adaptation of Compact Bone. Raven Press, New York (1989) [19] Piekarski, K.R.: Biomechanics of bone. In: Morecki, A., Fidelus, K., Kedzior, K., Witt, A. (eds.) VII-A International Series of Biomechanics, Warszawa – Baltimore (1981) [20] Pozowski, A., Bedzinski, R., Scigala, K.: Stress distribution in varus knee after operative correction of its mechanical axis. Acta of Bioengineering and Biomechanics 3, 31–40 (2001) [21] Rho, J.Y., Kuhun-Spearing, L., Zioupos, P.: Mechanical properties and hierarchical structure of bone. Med. Eng. Phys. 20, 92–104 (1998) [22] Taylor, M., Tanner, K.E., Freeman, M.A.R.: Finite element analysis of the implanted proximal tibia: a relationship between the initial cancellous bone stresses and implant migration. Journal of Biomechanics 31, 303–310 (1998) [23] Tsubota, K., Adachi, T., Tomita, Y.: Functional adaptation of cancellous bone in human proximal femur predicted by trabecular surface remodeling simulation toward uniform stress state. Journal of Biomechanics 35, 1541–1551 (2002) [24] Tsubota, K.: Spatial and temporal regulation of cancellous bone structure. Medical Engineering & Physics 27, 305–311 (2005) [25] Weinans, H., Huiskes, R., Grootenboer, H.J.: The Behavior of Adaptive BoneRemodeling Simulation Models. Journal of Biomechanics 25, 1425–1441 (1992) [26] Zioupos, P.: The accumulation of fatigue microdamage in human cortical bone. Clinical Biomechanics 11, 365–375 (1996)

Chapter 21

Tooth-Implant Life Cycle Design Tomasz Łodygowski, Marcin Wierszycki, Krzysztof Szajek, Wiesław He¸dzelek, and Rafał Zagalak

Abstract. Dental restorations with the application of implants are very effective and commonly used in dental treatment. However, for some percent of patients, diverse complications can be observed. These problems can be caused by mechanical reasons such as loosening of the retaining screws or fracture and cracking of the dental implant components. These problems suggest the need for permanent modernization and development of dental implants. This paper describes selected aspects of the life cycle design process of the tooth-implant system Osteoplant. The authors would like to present what they mean by implant life cycle design as one part of the whole Digital Product Development (DPD) process. The sequential stages of this process are described and the tools and methods are discussed. The attention is focused on numerical simulations the mechanical behavior of dental implants and genetically based optimization algorithms. The tools and methodology of FE simulations of implant behavior are described. The whole process of optimization of a dental implant system is explained, and a self-developed optimization tool based on a genetic algorithm is presented. These processes are crucial for modern design procedure beyond the life sciences industry. Tomasz Łodygowski, Marcin Wierszycki, Krzysztof Szajek Department of Structural Mechanics, Poznan, University of Technology, ul. Piotrowo 5, 65 246 Poznan, Poland e-mail: [email protected], [email protected], [email protected] Wiesław He¸dzelek University of Medical Sciences, ul. Fredry 10, 61-701 Poznan, Poland e-mail: [email protected] Rafal Zagalak Foundation of University of Medical Sciences, ul. Teczowa 3, 60-275 Poznan e-mail: [email protected]

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21.1 Introduction Dental implants are a commonly applied treatment method of dental restorations. They are used in prosthetic dentistry to support restorations that resemble a tooth or group of teeth. The current significant position of this treatment method is a result of 50 years of research and development. In the 19th and the first half of the 20th century, there were many examples of metal alloys and porcelain formulations for dental implants. However the long-term success rates have been very poor. The real milestone in implantology was the year 1952. For the first time, Dr. Per Ingvar Branemark’s team from the University of Lund in Sweden used titanium metal as implant material. They observed that an implant screwed into bone had fused to the bone after several months. He called this phenomenon osseointegration. This discovery was the foundation of great success with dental implants. Currant development of dental implants requires close collaboration between doctors and engineers and uses a wide range of engineering methods and tools, e.g. computer-aided techniques. Furthermore, this cooperation requires deep understanding of certain medical issues by the designers. From an engineering viewpoint, the crucial issues are: the dental implant placement procedure and the mechanical behavior of the implant. Dental implant placement is a two-stage procedure. The first stage is a surgical placement of the implant, which can be done with a local anesthetic. The second stage is mounting of the tooth prosthesis. In the first step the implant is screwed or taped into a surgically prepared site in the gum. Next, the gum tissue is closed over the implant. The second step is simply the wait for osseointegration. The implant remains under the gum for 3 to 6 months. While the surface of the implant is fused into bone as a result of the osseointegration process, the patient can continue to wear his denture. In the third step, the implant is exposed by removing a small amount of gum tissue. After this the custom abutment is attached to the implant by a screw. At this point the screw, connecting abutment and implants, are torqued to a pre-determined torque moment, in order to achieve the correct pre-load value of the screw. This step is very important and crucial from a mechanical viewpoint, because it prevents future loosening of the screw but also causes significant stresses in the screw and the other implant components. At the end of the dental implant placement procedure, a ceramic crown is cemented over the abutment. During all these stages, computer-based tool are used. A 3D digital image such as computer tomography provides data which can be used to create a customized crown, manufacture precision drilling guides and virtually plan the surgical procedures. There are several types of dental implants. This paper is focused on the modern implant system Osteoplant, which was created and has been developed over 10 years by Foundation of University of Medical Sciences in Poznan since. The dental implantology is efficient and common but at the same time demands complicated treatment methods. The key direction of Osteoplant system development has been to simplify surgery and prosthetic procedures by reducing the number of necessary tools and instruments for complex implant-prosthetic surgery. Due to the smooth cooperation of experienced dentists, dental and engineering research laboratories, and medical and technical universities, Osteoplant is an affordable, safe and innovative implant system.

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Osteoplant Hex is two-component implant system consisting of the abutment and root with a connecting titanium screw (Fig. 1). The root of the implant is installed in the prepared hole drilled into the jawbone as described above. Implants differ in diameter (3,5 mm, 4,0 mm, 4,5 mm) and in length (9-16 mm) but generally size is similar to natural tooth. The permanent crown is placed on the abutment but apart from its basic function, it may also be used as the basis fore a more complicated prosthetic structure. The implant system goes beyond the implant itself. The final success of a dentistry treatment procedure requires not only a well-considered and designed implant, but also tools and instruments like drills, insertion tools, countersinks, bone taps, implant drivers and sockets. The design, engineering and prototyping of dental implant system is supported with computer techniques, in particular computeraided design (CAD), computer-aided manufacturing (CAM), and computer-aided engineering (CAE). These computer and information techniques support engineers in tasks such as analysis, simulation, design, manufacture and planning, crucial aspects of the Digital Product Development (DPD) process. These tasks compromise an implant life cycle.

Fig. 21.1 The Osteoplant dental implant

21.2 Motivation Numerous clinical observations of patients, who received the dental restorations implants, drew our attention to the risk of both early and late complications (Goodacre et al., 2003). These complications have both biological (failure of osseointegration or bone breaking, perforation and infections) and mechanical aspects. In many cases these mechanical problems are caused by biological phenomena such as bone remodelling or complex dysfunction of occlusion and mastication (Genna, 2004;

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Khraisat et al., 2002; Koczorowski and Surdacka, 2006; Teoh, 2000). In the late phase, the most frequent complications are loosening of the retaining screw, fracture and cracking of the dental implant components. The loosening of the retaining screw causes patient discomfort in the implant usage but can also be a source of much more serious problems. The reason for this complication is a defective static work of the mechanical implant system. Deep understanding of the mechanical behavior of implants and the possibility of numerical simulation are a prerequisite for minimizing the threat of screw loosening. The cracking of dental implant components always leads to much more serious complications and makes further treatment extremely difficult (Fig. 2). In extreme cases, overloading of the implant structure is the likely cause of these complications, but in many cases the character of loading and fracture indicates material fatigue as the basic reason (Wierszycki et al. 2006a; Wierszycki 2007; Zagalak, 2005). Our previous studies based on numerical simulations of the dental implant system (K´zkol et al., 2002) confirmed this observation. Research on implants in their real conditions is practically impossible and laboratory tests are very expensive. FEA tools however, can identify the reason for mechanical damage of the implant (Zienkiewicz and Taylor, 2005). CAE tools are widely used in the medical industry. Their use during the Osteoplant implant life cycle design reduces product development costs and time and more importantly, helps to improve the safety, comfort and durability of the dental implant system. In particular, the predictive capability of FEA tools has led us to the point where much of the design verification can be done using computer simulations rather than physical prototype testing.

Fig. 21.2 Dental implant: after implantation (a) and damaged (b)

Numerical simulation of dental implants has been carried out using a series of different FE models. For analyses of strength and fatigue, simplified axisymmetric models were used. For full simulation of implant structure behavior, a geometrically complex three dimensional model was necessary. All FEM calculations were carried out by commercial code Abaqus/Standard and Abaqus/Explicit. For the fatigue calculations, the fe - safe module was used. The numerical models were created directly on the basis of CAD models. During development of a dental implant system, some small and clear modifications and validations can be proposed using CAD and CAE tools. However, the major modifications need qualitatively improved methods.

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An optimization method which can be used in any work on dental implant system improvement must fulfil several specific requirements. First, this method must be based on results of FEA; therefore it should operate directly on values of the objective function instead of derivatives or other knowledge. Second, it can be used in the case of strong nonlinear problems and it should be resistant to possible discontinuities. The algorithm should also permit defining any type of constraints and finally, it should be easy or possible to join it with the environment of the FEA tool. One such technique is an optimization algorithm based on a genetic algorithm (GA). The optimization procedure is done by using self-developed software and Abaqus/CAE environment. In this tool, Abaqus solvers are used to evaluate individuals. This approach can also be applied for any, nonlinear problems, such as realistic simulation of the mechanical behavior of implant system. The CAD, CAE, CAM, CMM software and GA were used simultaneously during the implant system design procedure. The CAD models of implants, tools and instruments were created using CAD software. These full parametric models were starting points for whole tooth-implant life cycle design tasks, such as FE analyses, optimizations with GA, manufacture and planning. The created geometry of the implant was then transferred from the CAD system to a CAE environment. After FE calculation and optimization procedures, the modified geometry was transferred back to CAD software. Finally, the instructions for Computer Numerical Control (CNC) were prepared using CAM software. Digital data were used for validation procedures using the Cordinate-Measuring Machine (CMM), as well as documentation and marketing materials. In this paper the attention is focused on numerical simulation of the mechanical behavior of dental implants using FEA software and optimization based on genetic algorithms.

21.3 Numerical Simulation of Dental Implantsn 21.3.1 Model The following chapters describe the key stages of the FE model preparation procedure: geometry creation, material definition, mechanical assembly simulation, load and a description of the boundary conditions which are the necessary steps to continue the fatigue analysis.

21.3.1.1 Geometry and Mesh The geometry of a dental implant, in particular the screw connection, is very complex. For full simulation of implant structure behavior, a geometrically complex three dimensional model is necessary (Fig. 3b). This model, which includes a spiral thread, takes into consideration several important aspects, such as full simulation of kinematics of an implant, a description of the multiaxial state of stress and, most

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Fig. 21.3 Axisymmetric (a) and 3D (b) model of dental implant (CB - cancellous bone, CC cortical bone, R - root of an implant, S - screw, A - abutment)

important, simulation of screw loosening. The most interesting result will be the relationship between torque moment, the friction coefficient and the loosening or fatigue life of the screw under cyclic loads. Unfortunately, a three-dimensional model is very large (ca. 600K DOF). The design process, especially if optimization procedures are taken into consideration, requires a huge number of analyses. The crucial aspect is the characteristic of the numerical model of the implant which is used in the optimization process. In practice, the time of calculation cannot exceed several dozen minutes. This limitation means that a fully three-dimensional model of implant and typical modeling approaches cannot be used (Wierszycki et al., 2006b). For this study a special approach was proposed. Due to the fact that most parts of an implant are axisymmetric, an axisymmetric concept of modelling is a good option for simplification. The Abaqus/Standard offers special types of solid elements (CCL), which are capable of modelling the nonlinear asymmetric deformation of an axisymmetric structure in a very efficient way. These elements use standard isoparametric interpolation in the radial - symmetry axis plane, combined with the Fourier interpolation with respect to the angle of revolution (Zienkiewicz and Taylor, 2005). We called this approach 2.5D (Fig. 3a). The asymmetric deformation is assumed to be symmetric with respect to the radial - symmetry axis plane at an angle equal to 0 or π (Abaqus Manuals, 2007). The creation of an implant model with these elements is a two stage procedure. In the first step an axisymmetric model of an implant was created (Fig. 4a.). This, 2D model definition has a fully parametric geometry description. The final model of the implant has been created automatically by revolving an axisymmetric model about its axis of symmetry (Fig. 4). The threads of the screw and implant body were simplified to axisymmetric, parallel rings. This

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approach reduced the geometry description of an implant model from three dimensional to two dimensional, and at the same time enabled automatic modification of the shape of implant components during the design loop.

Fig. 21.4 The automatic creation of a 2.5D (b) model based on an axisymmetric model (a)

The mesh of the axisymmetric model was generated automatically. In the model of an implant, a special type of axisymmetric solid elements was used. The number of these elements along the circumferential direction is the compromise between time of calculation and accuracy of results. The several series of test analyses were carried out to evaluate which number of these element is optimal. The models with only classic three-dimensional solid elements (C3D), mixed classic and axisymmetric, and only axisymmetric elements were evaluated and compared. The details of these models are shown in Tab. 1. The maximum values of the equivalent Mises stresses at characteristic notches and global bending stiffness of the whole implant structure were used to compare the results. The results of comparable studies are shown in Fig. 5. In the results of this evaluation, the model with two axisymmetric and eight classic three-dimensional solid elements was used in optimization process analyses (Fig. 4b.). Thanks to this concept, the size of the problem (ca. 100K DOF) and the cost of the calculation were significantly reduced. In the case of fatigue calculation, the implant model was fixed into the small part of the jaw bone (Fig. 3a). The geometry of bone was simplified to the regular quasi-cylindrical domain. This simplification is justified for the function of bone in this model. The bone constitutes boundary conditions of an implant. Both types of bone were taken into consideration: cortical and cancellous. The cancellous bone is modelled using axisymmetric solid elements. The cortical bone is modelled as a layer of axisymmetric shell elements which are defined on the nodes of solid elements.

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Table 21.1 Selected parameters of comparable simulation of 3D analyses with the use of cylindrical (CCL-x) and 3D solid elements (C3D-x)

CCL-1 CCL-2 CCL-4 CCL-16 CCL-32

Floating Minimum Required Number Number Number point oper- memory diskspace of equa- of incre- of SDI ations per required tions ments iteration

Number Wallclock of EI time

[-] 6.51E+009 2.81E+010 1.61E+011 9.48E+011 5.15E+012

[-] 7 13 15 20 27

[MB] 47.31 87.25 176.93 447.38 1157.12

[MB] 150.73 366.96 1034.24 2969.60 8427.52

[-] 56226 93588 168312 317760 616656

[-] 7 7 9 13 13

[-] 33 32 47 84 82

[sec] 160 401 1565 8730 47685

Fig. 21.5 Reduction of time calculation as the result of a semianalitycal approach (red color - chosen configuration)

21.3.1.2 Material Properties The implant is made of medical titanium alloy, the mechanical properties of which are nonlinear. The description was based on certificates of conformity (ASTM F13698, ISO 5832 PT 2-93) and literature data (Niinomi, 1998) (Table 2). Table 21.2 Material properties of implant model components Implant’s body (Tita- Screw (6AL-4V- Abutment (6ALnium Grade 4) ELI) 4V-ELI) Young’s modulus [MPa] Poisson ratio Yield [MPa] Tensile Yield [MPa]

105 200 0.19 615.2 742.4

105 200 0.19 832.3 1004.0

105 200 0.19 802.8 970.4

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The jaw is composed of two kinds of bones, cancellous and cortical (Fig. 3a). The problem of describing the constitutive law of a bone is very complex. The mechanical characteristics and internal microstructure of cortical and cancellous bones are nonhomogeneous, anisotropic and variable in time. Changes in bone characteristics are caused by the phenomenon of tissue remodelling. It is very difficult to take these aspects into consideration in implant models. In finite element analysis, many concepts of mechanical properties of bone description could be applied, starting with the very simple, linearly elastic isotropic, going through the more complicated, transversely isotropic or orthotropic, and ending with the very complex, nonlinear anisotropic. The assumed material characteristics of the jaw bones are linearly elastic, homogeneous and isotropic and are discussed later (Tab. 4). This simplification is justifiable due to the role which the bone plays in fatigue analysis of an implant. The most important here is the influence of bone loss around an implant as well as bone flexibility on boundary conditions and implant fatigue life (Snauwaert et al., 2000; Zagalak et al., 2005). The best sources of fatigue material characteristics are the experimental tests of smooth specimens for constant strain amplitudes between fixed strain limits. This sources need many technically advanced resources, are expensive and complex. Some fatigue material data can be found in the literature. Also manufacturer’s certificates which are published by titanium manufacturers and fatigue researchers provide basic information about fatigue strength for specific endurance (Akahori et alo., 2005; Baptista et al., 2004, Guilhermeet al., 2005). These data are useful for stress-life analysis but are insufficient for strain-life approach. For strain-life fatigue analysis, six additional material properties are required: cyclic strain hardening coefficient K , cyclic strain hardening exponent n , fatigue strength exponent b, fatigue ductility exponent c, fatigue strength coefficient σ f , and fatigue ductility coefficient ε f . There are several approximated relationships to obtain these data. All of them are based on some physical interpretations of fatigue properties or relationships between fatigue properties and other well known physical parameters of materials. In this study the so-called Seeger (Draper, 1999) method was used. The fatigue strength coefficient σ f and cyclic strain hardening coefficient K are approximated with the help of the rescaling conventional monotonic ultimate tensile stress σu (Eq. 1 and 2). For titanium alloys:

σ f = 1.67σu

K = 1.61σu

(21.1) (21.2)

Seeger’s method is a modification of the universal slopes method and assumes that the slopes of elastic and plastic asymptotes of strain-life curve are the same for specific kinds of alloys. For titanium alloys b = −0.095 and c = −0.69 (Draper, 1999). Similarly, the cyclic strain hardening exponent and fatigue ductility coefficient were assumed as constants, n = 0.11 and ε f = 0.35 = 0.11 (Draper, 1999). The fatigue data of implant titanium alloys are shown in Tab. 3.

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Table 21.3 Fatigue material data

K’[MPa] n’ b c σ f [MPa] ε f

Implant body

Abutment

Screw

1195.26 0.11 -0.095 -0.69 1239.81 0.35

1561.7 0.11 -0.095 -0.69 1619.9 0.35

1616.44 0.11 -0.095 -0.69 1676.68 0.35

21.3.1.3 Mechanical Assembly The implant system is seemingly simple, but in fact it is quite a complex mechanical system (Kakol et all., 2002; Merz et all., 2000; Sakaguchi et all., 1993; Bozkaya and Müftü, 2005). The most important aspect of the numerical simulation of implant mechanical behavior is modelling of tightening. For this purpose, it was necessary to define the multiple contact surfaces between a root, an abutment and a screw. The friction characteristic on these surfaces is one of the key parameters influencing preload axial force, reduction of implant components mobility, resistance to screw loosening, but also fatigue life of a whole implant (Kakol et al., 2002). For the friction coefficient, a value ranging between 0.1 (as in a special finished surface) and 0.5 (as in dry titanium to titanium friction) may be found in literature. In the present analyses the variant values of friction coefficients were considered. The influence of a friction coefficient on fatigue life and screw loosening is substantial and has been evaluated. In the case of the three-dimensional model, the simulation of tightening can be defined in the way that describes a real physical process. The analyses were performed using the displacement control technique. The screw was subjected to a rotational displacement equal to 1 radian, which in turn was applied to the reference point of a rigid body defined on the top surface nodes of the screw head. This approach creates a very complex, nonlinear and strong discontinuous contact problem. If this analysis is performed using the implicit code, as Abaqus/Standard, a lot of small increments and equilibrium iterations are generally required to reach a converged solution. In the case of the axisymmetric model, the tightening is simulated in a simplified way. The middle part of the screw was subjected to temperature loading (Figure 6). The orthotropic thermal expansion property of the screw material was defined in such a way, that shrinking occurred only in the direction of the screw axis. The value of the axial force in a tightened screw was calculated from the empirical formula (Eq. 3). This equation is based on the assumption that the tightening moment Md is balanced by the sum of moments involved by friction on the threads and surfaces of contact between the screw and abutment:

P + π μgd2 sec α d2 D2 Md = N + μf (21.3) π d2 − μg P sec α 2 2

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where d2 is the diameter of the thread, P is the pitch, D2 is the effective diameter of head bolts, α stands for the angle of the thread, and the μg and μ f are the coefficients of friction on surfaces of the threads and between screw and abutment, respectively (Korewa, 1969). As is clearly seen, the axial force in the screw depends on the friction coefficient and torque moment. Finally, the effect of the assumed force (200N) was replaced by an equivalent temperature field (Wierszycki, 2007). This formula was validated based on results of the full simulation of tightening carried out using a three-dimensional implant model.

21.3.1.4 Loads The external loads of an implant model were applied in the second step of the simulation (Fig. 6). The values and directions of forces were assumed according to a physiologically proven scheme. The real loading of an implant is never axial. The vertical component of it is estimated at 600 N and the horizontal at 100 N (MericskeStern et al., 1995; Mericske-Stern et al., 1996; Mericske-Stern, 1998; Morneburg and Pröschel, 2002; Morneburg and Pröschel, 2003). To estimate the most detrimental distribution of stresses, only the maximal realistic components of mastication force were taken into account. For the fatigue calculations, it is necessary to define the character of load changeability in the shape of a load-time curve, the socalled load signal. In an applied low-cycled scheme of 24-hour loads, the average values were 60 N (Bielicki et al., 1975; Hêdzelek et al., 2004).

Fig. 21.6 Scheme of implant loads (H - horizontal bending component of load, N - axial force in a tightened screw)

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21.3.1.5 Boundary Conditions The boundary conditions of the implant model are defined by bone conditions around the implant body. The first, most simple definition of boundary conditions assumed that all degrees of freedom at the bottom part of the implant body were fixed. This assumption seemed to have its explanation in dental practice, where no movements of implants under physiological load are acceptable. It is acceptable if simulation is focused only on the behavior of the screw. However, the difference between rigid fixing and even high stiffness can be significant, especially in cyclic loading and the context of fatigue damage. In this case the stiffness of bone should be taken into consideration. In this approach, the boundary conditions of implants are replaced by a model of a small part of the bone (Fig. 3a). The geometry of a small part of the jaw surrounding the implant is very simplified, but it enables us to take into consideration the finite stiffness of boundary conditions and the changes in implant fixing conditions as well. The changing stiffness of the bone and bone loss phenomena are very important from a mechanical viewpoint. The bone loss phenomenon especially has a significant influence on implant behavior, stress distribution and therefore fatigue damage. The degree of encasement and osseointegration of the implant may not be 100%. It depends on bone quality, microstrains developed during healing and function, and the location of the implant in the jaw. This percentage may decrease to as low as 50%, due to phenomena of bone remodelling. In these analyses three levels of osseointegration were considered (Fig. 7).

Fig. 21.7 Levels of dental implant osseointegration in the jaw bone: a) no. 1, b) no. 2, c) no. 3

In the case of the first level, the implant body is fully fixed in the jaw bone (Fig. 7a). In the next two, the degree of implant body embedding decreases to 75 and

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50%, respectively (Fig. 7b and Fig. 7c). The top level of cortical bone was assumed as: (a) level no. 1 - entirely embedded to a bone (100%), (b) level no. 2 - 3mm below the conical implant surface (75%), (c) level no. 3 - 6mm below the conical implant surface (50%). The choice of levels of osseointegration expressed in this assumed geometry is based on the following assumptions: (a) the first level is a target position after a restoration treatment and was taken into considerations in all simulations, (b) the second level corresponds to bone removal which uncovers the first outer thread and where the hexagonal abutment changes into the inner thread, (c) the third level corresponds to those boundary conditions of an implant body for which bending acts on the weakest cross-section.

Table 21.4 Fatigue material data Bone Cortical all schemes Cancellous 1st scheme (D1) 2nd scheme (D2) 3rd scheme (D3) 4th scheme (D4)

Young’s modulus [MPa] 13 000 9 500 5 500 1 600 690

Bone remodelling is a very complex phenomenon, a result of changing the strain state in bone microstructure. The characteristic of bone changes permanently. The microstructure of cancellous bone is adapted to variant loads. The cancellous bone of jaw and maxilla is rebuilt quite fast; bone tissue is replaced by new tissue in a few months (Milewski, 2002). In order to take into consideration the effects of bone remodeling, a simplified approach was used. The four different values of the Young modulus of cancellous bone according to the Lek-holma i Zarba (Zagalak, 2003) classification were used in FEA models of an implant system (Tab. 4). The highest value of the Young modulus corresponds to the case in which cancellous is completely replaced by cortical bone. All mechanical characteristics of both cancellous and cortical bones were taken from the literature (Milewski, 2002; Misch, 1999).

21.3.2 FE Analyses - Strength Analyses The strength analyses of the implant system were carried out for service and exceptional loading. The most adverse load conditions are assumed. The applied

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horizontal force H triggers bending of the screw. Significant plastic strains appear on the notches of the screw, specifically on its thread and its head, only in cases of the most detrimental values of friction coefficient and torque moments. Nevertheless, from the point of view of a static work of the system, the stresses do not exceed the safe level. These results shown that distribution of stresses in implant components should not cause cracking. Local accumulations of stress, concentrated in a small area, and achieving the plastic yield of titanium, cannot be the source of observed mechanical failures. The results obtained also show a negligible effect of the stiffness of cancellous bone on stress distribution in the whole implant. No significant differences in stress values or concentration zones can be observed. However, the significant differences can be observed for the various levels of implant encasement (Fig. 8). For level no. 1, the stress concentration zones are observed mainly in the upper part of the screw in the plane of action of the bending (Fig. 8a). For level no. 2 the stress concentration zone covers the whole screw. The highest values of stresses are observed in the lower part of the screw. The areas with high values stresses are extended to include the first coil of outer thread of the implant body too. The stresses in the screw significantly increased as well (Fig. 8b). For Level no. 3, the zone of high stress value covers virtually the entire length of the screw as well as all coils of the outer and inner thread of the implant body and the thread of the screw in the same plane (Fig. 8c). It is worth mentioning that the plastic strains which appeared in the notches of the screw, specifically on its thread and its head, are in all cases on a safe level from the point of view of the static work of the whole system (Kakol et all, 2002).

21.3.3 Fatigue Calculations - Strain-Life Approach In the fatigue life calculation, nine cases of loading, three cases of bone configurations and three cases of jaw bone stiffness were taken into consideration. The estimation of the fatigue is always based on results of the finite element analysis; therefore all fatigue calculations were preceded by the FE analyses of implant models (Wierszycki et al., 2006a)(Fig. 9). Because the significant plastic strain occurred in case of a specific combination of friction coefficient and torque moment the fatigue calculations were carried out using the strain-life fatigue approach. This approach uses algorithms which incorporate formulae based on both elastic and plastic strain to estimate the fatigue life (Wierszycki et al., 2006a). This approach is based on the strain-life equation (Eq. 4) which is a function of the maximum shear strain γm ax and strain normal to the maximum shear strain εn :

σ f b c Δ γmax Δ εmax 2N f + 1.75ε f 2N f + = 1.65 2 2 E

(21.4)

where σ f is the fatigue strength coefficient, ε f is the fatigue ductility coefficient and N f stands for the number of cycles. Used algorithms are based on the stress results

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Fig. 21.8 Huber-Mises equivalent stress distribution after bending for three levels of osseointegration: a) no. 1, b) no. 2, c) no. 3

obtained from the finite element analysis (Fig. 7), variations in loading, hysteresis loop cycle closure, and material properties. Elastic stresses from the finite element analysis model are transformed into elastic-plastic stresses by means of the so called biaxial Neuber’s rule (Eq. 5). This relationship is defined as a hyperbola and is expressed by the following equation:

Δ σ Δ ε = Kt2

Sk2 E

(21.5)

where Δ σ and Δ ε are the amplitudes of true stresses and strains. The nominal stress-strain product is defined by elastic stress concentration factor Kt , nominal elastic stress sk and Young modulus E. During fatigue calculation, this formulae (Eq. 3) is used to calculate the principal stresses and strains for each node separately. The rainflow cycle counting algorithm is used to extract fatigue cycles. For biaxial

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Fig. 21.9 Algorithm of fatigue calculations based on FEA results. s - nominal stress, e nominal strain, Pk = P(t) - load signal, σ Δ - true stress amplitude, εΔ - true strain amplitude, E - Young’s modulus, K’ - cyclic strain hardening coefficient, n’ - cyclic strain hardening exponent, b - fatigue strength exponent, c - fatigue duc-tility exponent, σ ’ f - fatigue strength coefficient, ε ’ f - fatigue ductility coefficient, N f - number of cycles

fatigue methods, a critical plane procedure is used to calculate the orientation of the most damaged planes at nodes. Based on these data, the fatigue life is calculated using strain-life formulation (Eq. 4) (fe-safe Manuals, 2005; Draper, 1999). In order to estimate service life of an implant, a designed life is defined. Fe - safe calculates the factor FOS (Factor Of Strength) by which the stresses at each node can be increased or reduced to give the required life (fe safe Manuals, 2005). This kind of result is the most interesting and useful in design and optimization processes. The required level of stresses can be treated as an objective function. Fatigue life of an implant screw was calculated for nine separate cases of loading, three cases of boundary condition schemes and three cases of jaw cancellous bone density. For all of these cases, the same cyclic scheme of loading was assumed. The following assumptions were made:

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Fig. 21.10 Cyclic scheme of loading - daily load signal

(a) (b) (c) (d) (e)

maximum value of load at the physiological process of grinding food - 200 N, average value of load during chewing - 20 N, number of occlusal contacts during meals - 20 1/min, average duration of three meals during a day: 15min, 20 min i 15 min, number of occlusal contacts, associated with the action of saliva swallowing in the periods between meals - 2 u˚ 105 1/year.

A twenty-four-hour changeability scheme was assumed as a signal (Fig. 10), while the number of days corresponding to four years was assumed as the number of cycles. In addition to detailed guidance for the overall design conclusions may be drawn based on fatigue life calculations. The FOS distribution (Fig. 11) analysis for particular cases indicates the axial forces in the screw and the changes in the scheme of boundary conditions which have the greatest influence on fatigue changes. For different bone density and at the same time divergent stiffness of boundary conditions, significant differences of stress distributions present in the screw are noticeable (Fig. 12, 13 and 14). This does not lead to serious fatigue changes. For axial forces above 600 N, there is a noticeable increase in the areas endangered by fatigue failure. The degree of required stress reduction reaches ca. 30%. In the most unfavorable load case, the maximal axial force value is the result of a high torque moment and a very small friction coefficient on a screw thread. It is important to pay attention to the danger of increasing the tightening forces in an implant screw. In two-part implants, this high tightening force is motivated by biological and medical aspects. However, the increase in torsion moments and decrease in friction coefficients pose a danger for fatigue life of implant components.

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Fig. 21.11 The scale of FOS distribution

Fig. 21.12 FOS distribution in the screw - level no. 1 and bone stiffness D1 (a), D2 (b), D3 (c), D4 (d)

Fig. 21.13 FOS distribution in the screw - level no. 2 and bone stiffness D1 (a), D2 (b), D3 (c), D4 (d)

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Fig. 21.14 FOS distribution in the screw - level no. 3 and bone stiffness D1 (a), D2 (b), D3 (c), D4 (d)

21.3.4 Screw Loosing Simulation The simulation of screw loosening can only be performed using a 3D implant model (Fig. 3b). This simulation needs a FE model which can reflect a real physical characteristic of screw connection. In the first step of this simulation, the abutment and implant body were tightened by the screw described previously. The screw was rotated until the expected torque moment and axial force in the screw were obtained. Four values of tightening (0, 15, 25 and 35 Ncm) and two friction coefficients (0.1 and 0.2) were considered. In the second step of simulation, the torsional cyclic load (asymmetric force) was applied to the top of the abutment to initiate a small movement of the implant system. This kind of analysis is strongly nonlinear and extremely discontinuous due to contact conditions and the quasi-static character of the implant response. In this case it is very difficult to obtain any converged solution using implicit code. For simulations of screw loosening, dynamic analysis using an explicit code, like Abaqus/Explicit was carried out. Unfortunately, in this case the time of simulation of the whole screw loosening process is unacceptable. Due to this fact the simplified approach was proposed. The four separated simulations for different values of tightening were treated as four consecutive stages of the same screw loosening process. The obtained results, e.g. permanent rotation of the screw, clearly showed that this approach can reflect screw loosening phenomena. To measure a resistance to screw loosening, the rate of the frictional dissipation energy accumulated over each stage of the same screw loosening process was chosen. The energy dissipated by contact friction forces between the contact surfaces is as follows:

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0

S

tT · g˙slT dS dt

(21.6)

where gslT is the velocity field, tT is the frictional traction and S stands for the boundary (contact surfaces). This energy is a measure of the work to be performed during the screw loosening process. In Fig. 15 the change of the frictional energy dissipation is plotted for three different stages of the screw loosening process. It is clearly seen that dissipation of this energy increases when tightening moment decreases at the same time as a result of screw loosening. During design and optimization processes, each proposed modification can be evaluated in this way for screw loosening resistance.

Fig. 21.15 Gradient of energy dissipated through frictional effects during screw loosening (torque moment M = 0/15/25/35Ncm and friction coeff. n = 0.2)

21.4 Optimization Using Genetic Based Algorithms The numerical analyses of dental implant provide a satisfying estimation of mechanical behaviour for many different services and unexpected cases. They enable verification of existing systems and validation of new ones. Some small and clear modifications of existing dental implants are often easy to develop and validate using FEA. However, major modification needs qualitatively new methods of optimization. The optimization of the implant system was divided into several stages. The various objectives are considered: strength and fatigue life of the implant, fixation in a bone and minimizing of the screw loosening phenomenon.

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21.4.1 Optimization Algorithm - Genetic Algorithm The optimization of a dental implant is very complex and relatively difficult. The model which is the basis (starting point) for the optimization is very developed, and estimating the implant behaviour requires consideration of material nonlinearity, complex contact definition and prestress of the assembly screw. The level of complexity and large number of design parameters result in many local minima in design space. Thus, the optimization algorithm has to search optimal solutions globally, based on a limited set solution in the design space. Moreover, problems with FE analysis can occur and should be take into account; most frequent are rebuilding and convergence problems. The complexity of the geometry and large number of design parameters combination make it impossible to predict all problems with the geometry. As a consequence, for some configurations of design parameters which are in the feasible range, proposed shapes could be incorrect. Additionally, problem with mesh generation can occur, which can also be treated as a rebuild problem. Reasons for no convergence in analyses are usually caused by too large plastic strain or by contact problems. Both can occur for points in the feasible region of design space and have to be expected during the optimization process. One of the optimization algorithms which can fulfil all these requirements is a genetic algorithm. The genetic algorithm has been inspired by evolutionary biology and uses techniques such as inheritance, mutation, selection and crossover to find a better solution. Despite being a powerful optimization tool, the genetic algorithm uses simple rules, which makes it easy to implement. In the module presented in this work, the galileo (GPL) library was used.

Fig. 21.16 Genetic algorithm processing scheme

GA operates on individuals, which are the abstract representations of a real solution. In this optimization approach, a chromosome was constructed over the alphabet 0,1 as a binary string. The main idea of GA processing consists in generation of a set of initial solutions and use of evolutionary operators to improve them in successive iterations, called generations. An initial set of individuals, called an initial

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population, is usually randomly created. Every new population is subjected to evaluation. The evaluation process consists of assigning a fitness value to each individual. This value describes solution quality and is calculated according to an objective function and results of FE analyses. Fitness values are the basis of the selection process. The selection mechanism is responsible for improvment of next population quality. During selection, statistically only the most fitted individuals are chosen in order to allow them to take part in the creation of the next population. In the next stage, the selected individuals are subjected to crossover and mutation. The operators use selected chromosomes in order to create new design parameter configurations. The primary function of crossover is mixing the parent individual chromosomes and creating new offspring. In a classic form, crossover consists of sampling a chromosome partition point, dividing parent chromosomes and swapping obtained parts between them. The mutation is responsible for random distortion of chromosomes, which keeps the diversity of population on a sufficient level and helps to prevent from a local minimum. In the binary encoded chromosomes, the mutation samples gene changes its value to the opposite. In the last stage the new population replaces the previous one. Usually, GA ends when either maximum fitness value for the best fitted individual is obtained or a maximum number of generations is achieved. The general chart-flow of the genetic algorithm is presented in Fig. 16.

21.4.2 Goal This study was expected to identify a new dental implant shape which would lead to lower principal stresses in comparison with the current values. In this FE model, geometry and material properties are the same as for real dental implants. The loads that were applied come from the literature (Zygalak, 2003). For the presented configuration, the maximum principal stresses are greater than 625 MPa and are localized in the screw corner (Fig. 17).

21.4.3 Design Parameters The geometry of this two-component implantology system is quite complex and the choice of correct design parameters is crucial. The thread was excluded from consideration at this stage of the work. Six geometrical parameters were defined as real variables (Fig. 18). All of them refer to the upper part of the implant. The design parameters were encoded into the binary string. There were determined for each design parameter range and number of bits for encoding. The ranges come from both the geometry limitations and manufacturer requirements and are listed in Tab. 5. The larger the number of bits for encoding, the better the accuracy; on the other hand, the total increase of optimization time is substantial. In this, for angle

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Fig. 21.17 Maximal principal stresses in the initial dental implant design Table 21.5 Ranges and number of bits for design parameter encoding Name

Symbol

Min value

Max value Range

Bits

Resolution

Screw head diameter Screw head conic surface opening angle Screw head height Screw diameter Hexagonal slot height Hexagonal slot conic surface opening angle

A

0.95

1.6

0.65

3

0.093

B

0.0

80.0

80.0

4

5.33

C D E

1.5 1.0 0.05

6.4 1.5 2.65

4.9 0.5 2.6

3 3 3

0.7 0.071 0.371

F

11.0

90.0

79.0

4

5.27

variables, four bits strings were constructed, whereas for all the rest only three bits were used. The choice of particular design parameters based on the range influence on final results. All the binary strings reference to the design variables are joined into a chromosome. In this work the twenty bit string is used as follows: D D E E E F F F F chi =[gA1 , gA2 , gA3 |, gB4 , gB5 , gB6 , gB7 , |gC8 , gC9 , gC10 |gD 11 , g12 , g13 |g14 , g15 , g16 |g17 , g18 , g19 , g20 ] (21.7) The superscripts denote the design parameters symbols in accordance with Tab. 5.

21.4.4 Constraints The material used in the FE model simulates the elastic - plastic behavior. To some extent it is allowed to exceed the plastic stress limit in the dental implant, but

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Fig. 21.18 Design parameters

the plastic strain cannot be too large. In FE analysis it is assumed that maximum equivalent plastic strain cannot be higher than 10 ( t 2 p pl pl ε = ε |0 + ε˙ l : ε˙ p ldt (21.8) 3 0 The ε pl |0 donates initial equivalent plastic strain, which was equal to zero in this case whereas ε˙ p l donates an equivalent plastic strain rate. Principal stress reduction, which was set as a goal, directs changes in individual in opposite direction to this constraint. Thus, no stress or plastic strain constraints were necessary. In case of a rebuild or convergence problem, the death penalty method is applied. The individual is eliminated from further processing. In the evaluation of a single individual, the FE model of dental implant, which is time-consuming significantly limits the population size. On the other hand, the population has to be large enough to provide sufficient diversity and represent design space properly. In the presented work, a population of forty individuals was used. The number of epochs was not limited a priori. The optimization has been stopped, based on monitored results. The signal to break the procedure was sent after a few epochs without any new proposal for a better solution.

21.4.5 Genetic Algorithm Settings A ranking based method of selection is used. In contrast to the classic method of selection, roulette - wheel, this method sets the probability of being selected according to the position in the ranking instead of fitness. As a consequence, the probability increases of choosing an individual with fitness much lower than the maximum and results in higher diversity in the late stage of the optimization process. The

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one - point crossover method is applied. The selected individuals are joined into couples; their chromosomes are divided into two substrings at random positions and swapped between parent individuals. The new couple of offspring represent combinations of both parent features. In order to keep sufficient diversity during the optimization process, not the all couples were subjected to crossover. The intensity of crossover is determined with the crossover rate, which is constant and equal to 0.75. In the last stage of recombination, random distortions are introduced to chromosomes. The probability of each gene distortion is controlled by the mutation rate. In this work, the constant value of 0.02 was kept during whole optimization process. When the gene to mutate is selected, its value is changed to the opposite, from 0 to 1 and vice versa. The classic form of population replacement was used. The whole previous population was completely replaced with the new one. This method allows for increasing the diversity and searches the design space more extensively in comparison with the alternative methods, e.g. steady-state replacement. Unfortunately, it also moderates the convergence and increases the number of necessary FE analyses. The small population size which was used limits the variation of design parameters, therefore diversity is an important factor.

21.4.6 Incorporation of the Genetic Algorithm into Abaqus/CAE Genetic algorithm processes chromosomes which represent various combinations of design parameters. Each new chromosome has to be evaluated in order to assess its quality. The great advantage of the genetic algorithm is that evaluation processes can be fully elaborated in an exterior procedure. The procedure returns fitness values based on chromosomes. The tool employed in the evaluation procedure has to provide the possibility to rebuild the FE model for any design parameter configuration. Moreover, some procedures like meshing have to be created automatically. Thus, the reliability and capacity of the rebuilding tool are crucial. In this work, the Abaqus/CAE environment in cooperation with Abaqus/Standard code are used. Abaqus/CAE provides an efficient environment for user scripting. An objectoriented scripting language, called python, is available, allowing the user to add new procedures into the kernel and graphical user interface (GUI). In order to carry out the dental implant optimization, a new module has been created with full functional graphical user interface (Fig. 19). The optimization driver was imported as an external library into the Abaqus/CAE environment. The open source library called Galileo was used. The implemented module is divided into two main sections. The first organizes genetic optimization and employs mostly Galileo library, whereas the second provides the tool for chromosome evaluation. Every chromosome created in the reproduction process is a starting point for the evaluation procedure. In the first stage each chromosome is divided into substrings, which reference design parameters and are encoded. Finally, a set of real design parameter values is obtained and provides the basis for model rebuilding. All design parameters refer to the geometry of the dental

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Fig. 21.19 Optimization module for Abaqus/CAE

implants. Abaqus/CAE kernel script is created and run in order to change them in the model. The obtained geometry is controlled and if the shape is invalid, rebuild error is raised. The weakest point of the procedure is the creation of a mesh. The mesh is generated automatically and due to large number of analyses, it is not possible to control them manually. It is assumed that any errors in mesh result in higher principal stresses and eliminate the solution during selection. Based on modified geometry, the FE analysis definition is created and sent to the Abaqus/Standard solver. In the case of presented work, additional analysis has to be carried out to provide results for the axisymmertic model. The spatial dental FE model was built based on results for axisymmetric models with additional data such as lateral force magnitude. The module waits for analysis termination until maximum time is not exceeded. If the analysis takes too long, the solver stops and no convergence error is raised. This limitation can also eliminate well fitted solutions but is necessary to precede optimization. The analysis results and all information about its proceeding are stored in an output database, which can be read with the use of python script. All necessary information is extracted from the output database and objective function is calculated. Finally, fitness value returns to the optimization driver. The whole flow-chart of the individual evaluation procedure is presented in Fig. 20.

Fig. 21.20 Individual evaluation procedure

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In case of the rebuild error, a zero fitness value is returned for the individual. Zero fitness value guarantees that invalid solutions will be eliminated during the selection process.

21.4.7 Results The optimization was performed using 8 cpus in computations (Fujitsu-Siemens PRIMERGY TX300 S3). During the whole optimization process over, 1600 FEA analyses were carried out. The total time of optimization was ca. 250 hours.

Fig. 21.21 Maximal principal stress value for successive generated individuals

The evolution of the optimization process is presented in Fig. 21. It shows the changes of the maximum principal stresses in the upper part of the implant screw for successively generated solutions. The average value of the principal stress is drawn by a solid (red) curve. It can be observed that subsequent designs are being improved - the maximum principal stress is decreasing. Starting with a value of 625 MPa, the principal stress is reduced to 150 MPa. The graph does not consist of the first stage when the initial population was established. More than 320 FE analyses were necessary to find the forty correct individuals. If starting with the first population of random individuals the better solution is founded. Moreover, the reduction of principal stress is meaningful and equals 475 MPa, more than 75% of initial value. In the next generations, the design parameters from the best solution (peaks) were promoted more strong and thus they strongly influenced the population improvement. This results in a

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constant decrease of both average and minimum value of the principal stresses. After the 20th generation, the best solution is established and no further essential reduction of stress is observed. Because of the chosen reproduction strategy, diversity within the population is sufficient and even in the last monitored population, the individuals represent wide range of fitness values.

Fig. 21.22 Initial (a) and new (b, c) shapes of dental implant

Two proposals for new shapes are shown in Fig. 22. The lowest principal stresses were obtained for shape b) and equal 150MPa. The next proposal (Fig. 22c) provides reduction to 180MPa but still has hexagonal slot instead of the most optimal solution. The haxagonal slot plays an important role in preventing screw loosening, thus the solution b) will not be considered in further analyses. There are a few clear tendencies easy to explore. In the first place, the screw head is wider than the initial one and the opening angle of the screw head conic surface is moderated. Together with modification of the opening angle of the hexagonal slot, the conic surface makes clamping of the abutment and the body more efficient. Moreover, the moderation of the opening angle of the screw head conic surface reduces the stress concentration in the interior corner of the screw. All these modifications make the joint stiffer and reduce relative rotation between the abutment and the body. The implant starts to behave as a cantilever (Fig. 23). The final implant incorporates the body to carry lateral load. As a result the displacement of the points of the upper edge are reduced almost twice from 0.085 mm to 0.045 mm. The change of dental implant behavior also results in more uniform stress distribution (Fig. 24). The contact pressure is realized on the whole contact surfaces,

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Fig. 21.23 Deformation of initial (a) and final (b) dental implant (the displacements are scaled 40 times)

Fig. 21.24 Mises stress contour of initial (a) and final (b) dental implant

in contrast to the initial implant. As a result, the screw is bent less. All proposals of the new shapes were verified with the use of full three dimensional FE models. The results confirmed that the FE model built with cylindrical and solid elements provides compatible values and can be used for further optimization.

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21.5 Conclusions The dental implant system life cycle design is supported with many computer techniques that support engineers in many tasks, such as drafting, 3D design, analyses, simulations, manufacture, planning, documentation creating and preparing marketing materials preparing as well. The FEA simulation of the behavior of a dental implant system was presented. The results of simulation showed that computational modelling and 3D simulation enables realistic prediction of implant mechanical behavior under restoration treatment as well as service loads. On the basis of the results of fatigue analysis, it can be claimed that material fatigue is the basic reason for the observed complications and must be taken into consideration during the dental implant design cycle. Furthermore the application as a measure of loosening resistance, the energy dissipated through frictional effects, can lead to practical recommendations in dentistry. For screw loosening simulation, the modelling of tightening is a crucial task that must be carried out in such a way that it describes a real physical process as much as possible. Only 3D modelling allows for a full simulation of the kinematics of the implant, and, in consequence, the possibility of screw loosening. The presented optimization approach enables us to successfully incorporate FEA and genetic optimization procedures into the design cycle. The OSTEOPLANT dental implant has been optimized. The new shape has been found and it provided the solution to both substantial principal stresses and displacement reductions. One of the crucial design goals was the minimization of fatigue damage of the implant. The obtained level of principal stress in key parts of the implant connection enables us to reduce or even to eliminate the risk of fatigue damage. The study and the final results give evidence that this method is efficient and can be used during the design process. These stages of dental implant design clearly showed that CAD, CAM and CAE tools are the foundation for efficient and robust engineering and prototyping activities for medical devices. Acknowledgements. Financial support for this research was provided by grant N R13 0020 06. This help is kindly acknowledged.

References [1] Abaqus Manuals, SIMULIA, Pawtucket (2007) [2] Akahori, T., Niinomi, M., Fukui, H., Ogawa, M., Toda, H.: Improvement in fatigue characteris-tics of newly developed beta type titanium alloy for biomedical applications by thermo-mechanical treatments. Materials Science and Engineering C25, 248–254 (2005) [3] Baptista, C.A.R.P., Schneider, S.G., Taddei, E.B., da Silva, H.M.: Fatigue behavior of arc melted Ti-13Nb-13Zr alloy. International Journal of Fatigue 26, 967–973 (2004) [4] Bielicki, J., Pro´sba-Mackiewicz, M., Bereznowski, Z.: Chewing forces and their measurement (in polish). Protetyka Stomatologiczna 25, 241–251 (1975)

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[5] Bozkaya, D., Müftü, S.: Mechanics of the taper integrated screwed-in (TIS) abutments used in dental implants. Journal of Biomechanics 38, 87–97 (2005) [6] Draper, J.: Modern metal fatigue analysis. HKS, Inc. Pawtucket (1999) [7] fe-safe Manuals, Safe Technology Limited, U.K. (2005) [8] Genna, F.: Shakedown, self-stresses, and unilateral contact in a dental implant problem. European Journal of Mechanics A/Solids 23, 485–498 (2004) [9] Goodacre, C.J., Bernal, G., Rungcharassaeng, K., Kan, J.Y.: Clinical complication with implants and implant prostheses. International Journal of Prosthodontics 90, 121–129 (2003) [10] Guilherme, A.S., Henriques, G.E.P., Zavanelli, R.A., Mesquita, M.F.: Surface roughness and fatigue performance of commercially pure titanium and Ti-6Al-4V alloy after different polishing protocols. Journal of Prosthetic Dentistry 93(4), 378–385 (2005) [11] He¸dzelek, W., Zagalak, R., Łodygowski, T., Wierszycki, M.: Biomechanical studies of the parts of prosthetic implants using finite element method (in polish). Protetyka Stomatologiczna 51(1), 23–29 (2004) [12] Ka¸kol, W., Łodygowski, T., Wierszycki, M.: Numerical analysis of dental implant fatigue. Acta of Bioengineering and Biomechanics 4(1), 795–796 (2002) [13] Khraisat, A., Stegaroiu, R., Nomura, S., Miyakawa, O.: Fatigue resistance of two implant/abutment joint designs. Journal of Prosthetic Dentistry 88(6), 604–610 (2002) [14] Koczorowski, R., Surdacka, A.: Evaluation of bone loss at single-stage and two-stage implant abutments of fixed partial dentures. Advances in Medical Sciences 51, 43–46 (2006) [15] Korewa, W., Zygmunt, K.: Basics of Machine Construction (in polish). Wydawnictwo Nauko-wo-Techniczne, Warszawa (1969) [16] Mericske-Stern, R., Assal, P., Mericske, E., Burgin, W.: Occlusal force and tactile sensibility measured an partially edentulous patients with ITI implants. The International Journal of Oral and Maxillofacial Implants 3, 345–353 (1995) [17] Mericske-Stern, R., Assal, P., Buergin, W.: Simultaneous force measurements in 3 dimensions on oral endosseous implants in vitro and in vivo. Clinical Oral Implants Research 7, 378–386 (1996) [18] Mericske-Stern, R.: Three-Dimensional Force Measurements With Mandibular Overdentures Connected to Implants by Ball-Shaped Retentive Anchors. A Clinical Study. The International Journal of Oral and Maxillofacial Implants 13, 36–43 (1998) [19] Merz, B.R., Hunenbart, S., Belser, U.C.: Mechanics of the implant-abutment connection: an 8-degree taper compared to a butt joint connection. The International Journal of Oral and Maxillofacial Implants 15(4), 519–526 (2000) [20] Milewski, G.: Strength aspects of biomechanical interaction bone tissue-implant in dental bio-mechanics (in polish). Zeszyty Naukowe Politechniki Krakowskiej seria Mechanika 89 (2002) [21] Misch, C.E.: Contemporary Implant Dentistry. Mosby St. Louis (1999) [22] Morneburg, T.R., Pröschel, P.A.: In vivo forces on implants influenced by occlusal scheme and food consistency. International Journal of Prosthodontics 16, 481–486 (2003) [23] Morneburg, T.R., Pröschel, P.A.: Measurment of masticatory forces and implant load: a methodologic clinical study. International Journal of Prosthodontics 15, 20–27 (2002) [24] Niinomi, M.: Mechanical properties of biomedical titanium alloys. Materials Science and Engineering 243, 231–236 (1998)

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[25] Sakaguchi, R.L., Borgersen, S.E.: Nonlinear finite element contact analysis of dental implant components. The International Journal of Oral and Maxillofacial Implants 7, 655–661 (1993) [26] Snauwaert, K., Duyck, J., van Steeberghe, D., Quirynen, M., Naert, I.: Time dependent failure rate and marginal bone loss of implant supported prosthese: a 15-year follow-up study. Clinical Oral Investigations 4, 13–20 (2000) [27] Wierszycki, M.: Numerical analysis of strength of the dental implants and the human spine motion segment (in polish). PhD Thesis, Poznan University of Technology, Poznan (2007) [28] Wierszycki, M., Ka¸kol, W., Łodygowski, T.: Fatigue algorithm for dental implant. Foundations of Civil and Environmental Engineering 7, 363–380 (2006) [29] Wierszycki, M., Ka¸kol, W., Łodygowski, T.: Numerical complexity of selected biomechanical problems. Journal of Theoretical and Applied Mechanics 44(4), 797–818 (2006) [30] Zagalak, R.: Evaluation of mechanical properties of two dental implants Osteoplant (in polish). PhD Thesis, University of Medical Sciences in Poznan, Poznan (2003) [31] Zagalak, R., He¸dzelek, W., Łodygowski, T., Wierszycki, M.: Influence of the loss of bone and the loss their density on risk of fracture of the implant - a study using finite element (in polish). Implantoprotetyka 6(1), 3–7 (2007) [32] Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method. Elsevier, Amsterdam (2005)

Chapter 22

Predictive Modelling in Mechanobiology: Combining Algorithms for Cell Activities in Response to Physical Stimuli Using a Lattice-Modelling Approach Sara Checa, Damien P. Byrne, and Patrick J. Prendergast

Abstract. Computer simulation is a cornerstone of engineering design of high performance mechanical engineering equipment. However computer simulation is not so much used in the design of medical devices. The reasons for this are multifactorial and have their origin in the scepticism that biologists (and therefore clinicians) have regarding the plausibility of actually including all relevant factors in a computational model. In this work we have set out to model the response of tissues to changes in their mechanical environment, which could be caused by the presence of an implant or by damage of the tissue. We have used a lattice modelling approach, where the domain under analysis is discretized using both a finite element grid to compute biophysical stimuli and a ‘lattice’ to model cell activities. We applied the approach to two problems: (i) healing after bone fracture, and (ii) tissue regeneration inside a scaffold implanted into the body.

22.1 Introduction Mechanobiology is the study of how mechanical forces influence biological processes. The main aim is to determine how external loads create mechanical stimuli within tissues, how the cells sense these stimuli and emit signals that are translated into the cascade of biochemical reactions that stimulate cells to form or adapt a Sara Checa Trinity Centre for Bioengineering, School of Engineering, Trinity College, Dublin, Ireland e-mail: [email protected] Damien P. Byrne Trinity Centre for Bioengineering, School of Engineering, Trinity College, Dublin, Ireland e-mail: [email protected] Patrick J. Prendergast Trinity Centre for Bioengineering, School of Engineering, Trinity College, Dublin, Ireland e-mail: [email protected]

M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 423–435. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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tissue or organ [1]. One domain of applicability of mechanobiology is in the development of new clinical therapies, for example in bone fracture healing, distraction osteogenesis or osteoporosis, as well as in the improvement of implant design, such that tissue reactions due to the mechanical disturbance created by the implant can be minimized. Another important domain of applicability is in Tissue Engineering and Regenerative Medicine where conditioning of tissues in bioreactors requires knowledge of the appropriate biophysical milieu to promote tissue differentiation. Tissue repair occurs through the process known as cell differentiation. This process can be described as a change on the cell phenotype as a response to the mechanobiological environment. During tissue repair or regeneration parent cells, known as stem cells, change into more specialized cells, for example bone forming cells (osteoblasts) or cartilage forming cells (chondrocytes) which over time synthesize new extracellular matrix leading to the healing of the tissue. Efforts towards the elucidation of the mechanical rules driving this process have led to the development of several mechanobiological theories during the last 50 years. First ideas were proposed by Pauwels [2] who proposed chondrogenic differentiation to be regulated by hydrostatic pressure while shear strain regulates fibroblastic differentiation. Osseous formation was postulated to occur only under low mechanical environments provided by the formation of soft tissues. Based on Pauwels’ theory, Carter et al. [3] postulated that tissue differentiation is regulated by the mechanical loading history with hydrostatic pressure promoting the chondrogenic phenotype and tensile strain promoting fibrous tissue formation. Claes and Heigele [4] described the mechanical regulation of tissue differentiation quantitatively. Their hypothesis was that the amount of strain and hydrostatic pressure determined the differentiation of the tissue. By describing tissues as biphasic materials composed of a fluid phase embedded in a solid matrix, Prendergast et al. [5] proposed a tissue differentiation theory where stem cell fate is determined by a mechanical stimulus combination of fluid flow and shear strain. Over the last years these theories have been implemented in computer models to simulate tissue repair and differentiation in fracture healing [6,7,8,9], osteochondral defect healing [10], distraction osteogenesis [11,12], tissue engineering [13,14] and bone growth at bone/implant interfaces [15,16,17]. Although these models have been able to predict, to some extent, some of the main aspects of the tissue differentiation process, many limitations still remain. Included here are the description of the different cellular processes that occur during tissue repair, such as cell migration and proliferation, and the deterministic behaviour of the models. Towards an improvement in the description of the individual cellular events and the incorporation to some degree of the stochastic nature of these processes, a lattice based mechanobiological model for the simulation of tissue differentiation has been developed. Here we present the description of the model together with its application in two different mechanoregulated processes: a fracture healing situation and the tissue formation inside a tissue engineered scaffold.

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22.2 A Lattice-Based Mechanobiological Model In order to simulate how a mechanical stimulus produces a biological signal for cells to differentiate or adapt a tissue, the relevant mechanical stimuli acting on the cells and the cell response processes need to be determined. Here we propose the use of finite element analysis to determine the local mechanical environment surrounding the cells and the use of a lattice model [18] to simulate individual cell activity. In this approach, the regenerating tissue is divided into a fine grid (called a lattice) in which each of the positions (lattice point) represents a possible position for a cell to occupy, with a distance between points of the order of the size of the cells. In this lattice, the different cell phenotypes (stem cells, osteoblasts, chondrocytes, endothelial cells, etc) migrate, proliferate, differentiate, apoptose and synthesize new extracellular matrix, following a set of mechanoregulation rules. Phenotype specific cellular activity is implemented as cell dependent cellular activity rates [19]. In what follows the implementation of the different cellular activities is described.

22.2.1 Simulation of Cellular Activity

Cell migration Cell migration is implemented using a random walk model [20]. In each iteration the program steps through each lattice point and if the position is occupied by a migrating cell, then a new position is chosen randomly from the surrounding locations (Fig. 22.1). Migration is restricted by contact inhibition, i.e. cells are not allowed to migrate to positions already occupied by another cell.

Fig. 22.1 Schematic representation of the migration of a cell in one dimension. The approach can be easily extended to three dimensions

Even in this simple example, however, complexity may arise in the algorithm: for example, if in an iteration an occupied lattice point is selected for the cell to migrate

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to, a choice must be made – will the iteration be allowed to pass assuming then that the cell has been ‘contact inhibited’, or will a search be made for a free lattice point in that iteration. If it is the former, will then the occupied lattice point be removed for consideration in the following iteration? In the simulations presented here the approach we take is to assume cell contact inhibition when the chosen position is already occupied, moving on to the next iteration. Cell proliferation Cells proliferate following similar rules as for migration. In this case, after proliferation two new positions are selected at random. In three dimensions and when diagonal movement is not considered, this results in 21 possible states for the cells to occupy. Angiogenesis Angiogenesis is defined as the formation of capillaries from preexisting blood vessels and plays a key role during bone repair. Capillaries supply oxygen and nutrients which are essential for cell survival and proliferation. It is a complex dynamic process characterized by a coordinated sequence of cellular interactions. Basically, upon angiogenic stimulation vascular endothelial cells lining the interior walls of the vessels are activated and begin to degrade their surrounding basement membrane. Then the endothelial cells migrate into the interstitium, resulting in the formation of capillary sprouts. Endothelial cells behind the migrating endothelium of the sprouts proliferate so that the newly developing blood vessel elongates. Here, capillaries are modelled as a sequence of endothelial cells whose path is determined by the path of the cells at the capillary tips. The following fundamental events are considered: the formation of capillary sprouts from pre-existing sprouts or vessels, the growth of the sprouts, the chemotactic response of the cells, the branching process and the fusion of a sprout tip to another sprout tip or another sprout (anastomosis). Following experimental observations [21], the growth of the capillaries is regulated by the local mechanical environment at the capillary tip, determined using finite element analysis, such that a high mechanical stimulus inhibits the growth of the vessels [22]. The chemotatic response of endothelial cells to angiogenic growth factors is modelled as a high probability for the capillaries to grow towards regions with high concentrations of angiogenic stimulators [22]. Cell differentiation Following Prendergast et al., 1997, [5] mesenchymal stem cell differentiation is determined by a biophysical stimulus, combination of shear strain (γ ) and fluid flow (ν ), according to the following equation:

γ ν + (22.1) a b where a = 0.0375, b = 3 μ m/s and S is the mechanoregulatory stimulus. According to this theory, based on the level of mechanical stimulation, stem cells differentiate S=

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into fibroblasts, chondrocytes or osteoblasts which synthesize fibrous tissue, cartilage and bone, respectively (Fig. 22.2). Fig. 22.2 Mechanoregulation of stem cell fate by the combined biophysical stimuli of tissue shear strain (depicted on the x-axis) and fluid flow (depicted on the y-axis). If the stimuli remain high, fibrous connective tissue (fibroblasts) is differentiated from the mesenchymal cell pool, intermediate stimuli cartilage, low stimuli bone, with very low stimuli causing resorption. (adapted from Prendergast et al. [5])

Considering that even under an appropriate mechanical stimulus bone can not form if there is not sufficient oxygen supply, instead cartilage would form [23], and that oxygen diffuses only around 100 micrometers from blood vessels [24], the following algorithm describes the cell differentiation process: IF S > 3 THEN fibroblasts IF 1 < S < 3 THEN chondrocytes IF 0.53 < S < 1 THEN IF distance to blood vessel < 100μ m THEN inmature osteoblasts ELSE chondrocytes IF 0.1 < S < 0.53 THEN IF distance to blood vessel < 100μ m THEN mature osteoblasts ELSE chondrocytes IF S < 0.1 THEN cell apoptosis Matrix synthesis The synthesis of extracellular matrix by the different cell phenotypes is ac-counted for as a change of material properties in the finite element. Since different cell phenotypes may coexist inside a finite element, a rule of mixtures is used to calculate the new materials properties of the element. For example, in the case of an element where 8% of the lattice points inside the element are occupied by chondrocytes, 21% by osteoblasts and 28% by fibroblasts, the Young’s modulus would be computed as: e = 0.28 × Econnective + 0.08 × Ecartilage + 0.21 × Ebone + 0.43 × Egranulation (22.2)

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assuming that all positions not occupied by fibroblasts, chondrocytes or osteoblasts contribute to granulation tissue properties. Fig. 22.3 shows an hexahedral element which contains a 1000 lattice points and whose stiffness would follow the above equation.

Fig. 22.3 An example of a single cubic finite element filled with 10 × 10 × 10 = 1000 lattice points with a tissue type at each lattice point. The mechanical properties of the finite element is determined by the rule of mixtures

22.2.2 Computational Implementation To simulate the cell differentiation process over time and the synthesis of the new tissues an iterative process is used (Fig. 22.4). Initially the regenerating region is assumed to be formed by granulation tissue while mesenchymal stem cells are seeded in the lattice. Using poroelastic finite element analysis the local mechanical stimulus surrounding each of the cells is then determined. In the lattice, stem cells migrate and proliferate while new blood vessels invade the tissue. After stem cells have reached certain maturation age, they differentiate based on the mechanical stimulus and their vicinity to the newly formed capillaries. Differentiated cells then synthesize new extracellular matrix with specific material properties [22] which are then updated in the finite element model and a new iteration starts.

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Fig. 22.4 Schematic representation of the computational algorithm to model tissue regeneration. The differentiation process is regulated by the local mechanical environment and the local vascularity

22.3 Applications 22.3.1 Bone Fracture Healing A model of fracture healing in an human tibia with an external fixator under physiological loading conditions has been developed. The model consisted of a human left tibia with a fracture gap of 3 mm and a callus index of 1.4. A unilateral external fixator with two pins was modelled and inserted in the anterior medial side of the tibia (Fig. 22.5a). Muscle and ligament attachment data, force magnitudes and orientations were derived from Duda et al., 2002 [26] (Fig. 22.5b). A body weight of 80 Kg was assumed. The total knee load was split 60% - 40% on the medial and lateral sides, respectively, while the loading profile is based on the general shape of the weightbearing achieved by patients on their fractured leg. Initially the callus was assumed to be filled with granulation tissue, while mesenchymal stem cells began to proliferate and migrate from the periosteum, endosteum and marrow region at the site of the damaged cortical bone.

430

Fig. 22.5 a) 3D model of a human tibia fracture with an external fixator b) Schematic representation of the applied muscle and joint contact forces (adapted from Byrne [25])

S. Checa, D.P. Byrne, and P.J. Prendergast

(a)

(b)

Simulations predict an initial formation of cartilaginous and fibrous tissues between the bone ends (iteration 10, Fig 6) while intramembranous ossification occurred along the tibial cortical bone outside the fracture gap. Bone formation persisted in the external callus through intramembranous ossification, while cartilage was replaced by bone through endochondral ossification in the medullary cavity (iteration 20, Fig. 22.6). Stability of the callus was achieved after (almost) complete bone regeneration (iteration 60), time at which the external fixator was removed. Next bone resorption in the external callus and in the medullary cavity occurred.

22.3.2 Tissue Regeneration Inside a Scaffold Vascularization remains one of the main problems in bone tissue engineering. After scaffold implantation, blood vessels generally invade the construct due, in part, to the bone healing response induced by the surgical procedure. However, capillary growth is often too slow to provide adequate nutrients to the cells in the core of the scaffold, which results in peripheral bone formation [27]. In this study the process of vascularisation and tissue formation inside a regular printed type scaffold was investigated. Initially the scaffold was assumed to be filled with granulation tissue, mesenchymal stem cells were uniformly seeded while blood vessels were allowed to penetrate from the host tissue, simulating vascular invasion in an in vivo situation. The scaffold dissolved at a rate of 0.5%/day and had an initial porosity of 50%.

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Fig. 22.6 Cross-sectional view of the predicted healing patterns in (a) the medial side of the callus including the tibia, (b) the lateral side of the callus, and (c) a transverse slice through the centre of the fracture callus, over time. (adapted from Byrne [25])

Simulations predicted a high influence of the number of initially seeded cells on scaffold vascularisation, and therefore the formation of bone. A high number of uniformly seeded cells resulted in a dense extracellular matrix which caused the partial sealing of the pores. This resulted in a dense capillary network at the edges of the scaffold with no capillary formation in the deeper regions (Fig. 22.7). Simulations

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agree with experimental observations of a fast initial vascularisation followed by a slow rate of capillary growth after the first week postimplantation; which may explain the critical role initial seeding conditions may have on the vascularisation and tissue formation processes inside tissue engineered scaffolds.

Fig. 22.7 Cell distribution in a cross section through the scaffold where each dot represents a cell occupying a lattice point. After 60 days, under initial high cell seeding conditions, blood vessels are not able to fully penetrate the scaffold and as a result cartilage forms in the core. A reduction on the number of seeded cells allows for a better vascularisation of the scaffold and a more uniform formation of bone tissue. Due to symmetry the model was reduced to 1/8

22.4 Discussion and Perspective In this chapter we set out to describe our work on computational modelling of tissue regeneration. Previous models were based on the implementation of mechanoregulation rules driving tissue differentiation where cellular events, if considered at all, were generally described using diffusion equations; which assume that cells attempt to reach an homogeneous distribution. The approach presented here allows for the implementation of individual cellular events and their interaction with the local mechanical and biological environments, where model parameters can be experimentally determined [28]. Unfortunately, these models still present several limitations being probably the most important the selection of model parameters. A model is as good as its inputs and although all model parameters were taken from the literature, they do not

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correspond to a single experiment but to different experiments carried out on different conditions, different animals and different parts of the skeleton. How these parameters match in the specific situations remains unknown. Future efforts are therefore directed towards the validation of these models. Simulations of well controlled in vivo animal experiments have shown to be a powerful tool in the verification of tissue differentiation algorithms [29,30]; however they still present the problem of parameter estimation, since most of model parameters are difficult to be determined in in vivo conditions. Isaksson et al., [19] compiled, in a thorough literature review, most of these parameters from different in vitro experiments. Therefore, a combination of in vivo and in vitro experiments will be essential to further the predictive capabilities of these models. Ultimately the vision is for a field where models can be integrated across different dimensional scales and scientific disciplines so that they can be applied in medical engineering design and clinical applications [31]. Interpatient variability response during tissue repair is another important factor that needs to be considered. Although the models presented here are to some degree non-deterministic, due to the stochastic implementation of the cellular events [22], they have not been able to capture the interspecimen variability as seen in in vivo bone chamber experiments [30]. New approaches should be towards the implementation of this variability in tissue differentiation models. Acknowledgements. The research reported here is funded by a Science Foundation Ireland Principal Investigator grant to P.J. Prendergast titled “Mechanobiological modelling for tissue engineering and medical device design”, and by a European Commission FP6 project called SmartCap (www.smartcap.eu).

References [1] van der Meulen, M.C.H., Huiskes, R.: Why mechanobiology? A survey article. Journal of Biomechanics 35, 401–414 (2002) [2] Pauwels, F.: A new theory on the influence of mechanical stimuli on the differentiation of supporting tissue. The tenth contribution to the functional anatomy and causal morphology of the supporting structure. Z. Anat. Entwickl. Gesch. 121, 478–515 (1960) [3] Carter, D., Blenman, P., Beaupré, G.: Correlations between mechanical stress history and tissue differentiation in initial fracture healing. J Ortho. Res. 6, 736–748 (1988) [4] Claes, L., Heigele, C.: Magnitudes of local stress and strain along bony surfaces predicts the course and type of fracture healing. Journal of Biomechanics 32, 255–266 (1999) [5] Prendergast, P.J., Huiskes, R., Søballe, K.: Biophysical stimuli on cells during tissue differentiation at implant interfaces. Journal of Biomechanics 30, 539–548 (1997) [6] Andreykiv, A., van Keulen, F., Prendergast, P.J.: Simulation of fracture healing incorporating mechano-regulation of tissue differentiation and dispersal/proliferation of cells. Biomech. Model. Mechanobiol. 7, 443–461 (2008) [7] Isaksson, H., Wilson, W., van Donkellaar, C.C., Huiskes, R., Ito, K.: Comparison of biophysical stimuli for mechano-regulation of tissue differentiation during fracture healing. Journal of Biomechanics 39, 1507–1516 (2006)

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[8] Lacroix, D., Prendergast, P.J.: A mechano-regulation model for tissue differentiation during fracture healing: analysis of gap size and loading. Journal of Biomechanics 35, 1163–1171 (2002) [9] Hayward, L.N.M., Morgan, E.F.: Assessment of a mechano-regulation theory of skeletal tissue differentiation in an in vivo model of mechanically induced cartilage formation. Biomech. Model Mechanobio (2009), doi:10.1007/s10237-009-0148-3 [10] Kelly, D.J., Prendergast, P.J.: Mechano-regulation of stem cell differentiation and tissue regeneration in osteochondral defects. Journal of Biomechanics 38, 1413–1422 (2005) [11] Isaksson, H., Comas, O., van Donkelaar, C.C., Mediavilla, J., Wilson, W., Huiskes, R., Ito, K.: Bone regeneration during distraction osteogenesis: mechano-regulation by shear strain and fluid velocity. Journal of Biomechanics 40, 2002–2011 (2006) [12] Morgan, E.F., Longaker, M.T., Carter, D.R.: Relationships between tissue dilatation and differentiation in distraction osteogenesis. Matrix Biol. 25(2), 94–103 (2006) [13] Byrne, D.P., Lacroix, D., Planell, J.A., Kelly, D.J., Prendergast, P.J.: Simulation of tissue differentiation in a scaffold as a function of porosity, Young’s modulus and dissolution rate: application of mechanobiological models in tissue engineering. Biomaterials 28, 5544–5554 (2007) [14] Kelly, D.J., Prendergast, P.J.: Prediction of optimal mechanical properties for a scaffold used in osteochondral defect repair. Tissue Engineering 12, 2509–2519 (2006) [15] Geris, L., Andreykiv, A., van Oosterwyck, H., van der Sloten, J., van Keulen, F., Duyck, J., Naert, I.: Numerical simulation of tissue differentiation around loaded titanium implants in a bone chamber. Journal of Biomechanics 37, 763–769 (2008) [16] Huiskes, R., van Driel, W.D., Prendergast, P.J., Søballe, K.: A biomechanical regulatory model for periprosthetic fibrous tissue differentiation. J. Mater. Sci.: Mater. Med. 8, 785–788 (1997) [17] Liu, X., Niebur, G.L.: Bone ingrowth into a porous coated implant predicted by a mechano-regulatory tissue differentiation algorithm. Biomech. Model. Mechanobiol. 7, 335–344 (2008) [18] Simpson, M.J., Merrifield, A., Landman, K.A., Hughes, B.D.: Simulating invasion with cellular automata: Connecting cell-scale and population-scale properties. Physical Review E76 (2007) [19] Isaksson, H., van Donkelaar, C., Huiskes, R., Ito, K.: A mechano-regulatory bonehealing model incorporating cell-phenotype specific activity. Journal of Theoretical Biology 252, 230–246 (2008) [20] Pérez, M., Prendergast, P.J.: Random-walk model of cell-dispersal included in mechanobiological simulation of tissue differentiation. Journal of Biomechanics 40, 2244–2253 (2007) [21] Claes, L., Eckert-Hubner, K., Augat, P.: The fracture gap size influences the local vascularisation and tissue differentiation in callus healing. Langenbecks Arch. Surg. 388, 316–322 (2003) [22] Checa, S., Prendergast, P.J.: A Mechanobiological Model for Tissue Differentiation that Includes Angiogenesis: A Lattice-Based Modeling Approach. Annals of Biomed Eng. 37, 129–145 (2009) [23] Kanichai, M., Ferguson, D., Prendergast, P.J., Campbell, V.A.: Hypoxia promotes chondrogenesis in rat mesenchymal stem cells: a role for AKT and Hypoxia-Inducible Factor (HIF)-1α . Journal of Cellular Physiology 216, 708–715 (2008) [24] Carmeliet, P., Jain, M.K.: Angiogenesis in cancer and other diseases. Nature 407, 249– 257 (2008)

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[25] Byrne, D.P.: Computational modelling of bone regeneration using a three-dimensional lattice approach, PhD thesis, University of Dublin (2008) [26] Duda, G.N., Mandruzzato, F., Heller, M., Kassi, J.P., Khodadadyan, C., Haas, N.P.: Mechanical conditions in the internal stabilization of proximal tibial defects. Clinical Biomechanics 17(1), 64–72 (2002) [27] Kneser, U., Stangenberg, L., Ohnolz, J., Buettner, O., Stern-Strater, J., Möbest, D., Horch, R.E., Stark, G.B., Schaefer, D.J.: Evaluation of processed bovine cancellous bone matrix seeded with syngenic osteoblasts in a critical size calvarial defect rat model. J. Cell. Mol. Med. 10, 695–707 (2006) [28] Prendergast, P.J., Checa, S., Lacroix, D.: Computational models for tissue differentiation. In: De, S., Guilak, F., Mofrad, M. (eds.) Computational Methods in Biomechanics, Springer, New York (in press, 2009) [29] Geris, L., Vandamme, K., Naert, I., van der Sloten, J., Duyck, J.: Application of mechanoregulatory models to simulate periimplant tissue formation in an in vivo bone chamber. Journal of Biomechanics 41, 145–154 (2008) [30] Khayyeri, H., Checa, S., Tagil, M., Prendergast, P.J.: Corroboration of mechanobiological simulations of tissue differentiation in an in vivo bone chamber using a latticemodeling approach. J. Orthop. Res. (in press) (2009) [31] Clapworthy, G., Viceconti, M., Coveney, P.V., Kohl, P.: The virtual physiological human: building a framework for computational biomedicine. Phil. Trans. R. Soc. 366, 2975–2978 (2008)

Part VI

Structural Mechanics

Chapter 23

The Beam-to-Beam Contact Smoothing with Bezier’s Curves and Hermite’s Polynomials Przemysław Litewka

Abstract. This contribution concerns a key issue in numerical treatment of contact by the FEM, i.e. ensuring the C1 -continuity of contacting facets. After a discussion of basic issues of the frictional beam-to-beam contact two methods of curve construction – inscribed curve and node-preserving ones, are presented. Each of them is combined with a curve representation by Hermite’s polynomials and Bezier’s curves. In this way four possibilities of determination of a position vector for an arbitrary point on the curve are obtained. The resulting four different beam contact finite elements are briefly described. Then, on a base of this derivation and results of representative numerical examples, the elements are compared. To this end several criteria as: accuracy of curve approximation, number of degrees of freedom, code length, computer time and smoothing effectiveness, are used. The final conclusion is that despite the complicated, long code the inscribed curve method combined with the Hermite’s polynomials should be preferred.

23.1 Introduction Beam-to-beam contact is a special case in a wide range of contact analysis. Its numerical treatment differs in many technical aspects from a general case of 2D or 3D solids. This problem was first considered in [8, 9] for the simplest possible variant with circular cross-sections and linear shape functions. Later some extra features like rectangular cross-sections and more complicated representation of beam axes geometry, necessary in a situation of large relative sliding, were introduced [5, 6]. Soon it became obvious that, like in the general case of 2D and 3D solids contact, smoothing of adjacent contacting facets was a necessity to ensure a problem-free convergence behaviour of commonly used Newton-Raphson scheme in solving of Przemysław Litewka Poznan University of Technology, Institute of Structural Engineering, Piotrowo 5, PL – 61-138 Pozna´n, Poland e-mail: [email protected]

M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 439–452. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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non-linear equations resulting in the case of large displacement frictional contact problems. The case of the point-wise beam-to-beam contact is especially vulnerable to disadvantages resulting from the lack of C1 -continuity between the adjacent segments in the case of large sliding. These problems are especially pronounced when the number of contact points is small, like in the contact between beams. So it is especially important to find an effective and simple procedure of 3D curve smoothing to be used in a combination with any possible beam finite element. The purpose of this paper is to provide a comparison between several possibilities of smooth 3D curve representation. Hermite’s polynomials and Bezier’s curves are arguably the most natural choice. Also two methods of geometrical construction of the curve are addressed. They yield the curves, either inscribed in the polygon formed by the nodes of parent beam elements or passing through these nodes. In Section 23.2 the most important aspects of the frictional beam-to-beamcontact are discussed. Section 23.3 includes a description of the above mentioned methods of 3D curve construction. In Section 23.4 some details of FE discretization, including those providing a quantitative comparison of smoothing methods, are presented. Section 23.5 features representative numerical examples, which enable further comparison of effectiveness and computer efficiency of smoothing. Final remarks are formulated in Section 23.6.

23.2 Frictional Beam-to-Beam Contact In the presented analysis beams with circular cross-sections made of a homogenous linear elastic material are considered. It is assumed that they undergo large displacements but strains remain small. So it is possible to conclude that the cross-sections preserve their shape, size and remain plane. This feature is very important in an analysis of contact geometry. A stiff normal contact with the Coulomb friction is assumed. Resulting normal contact constraints are incorporated using the penalty method. The same method together with the analogy to plasticity is used to treat the friction. A detailed description of treatment of the frictional beam-to-beam contact was presented in [9, 6], so here only the most important issues are pointed out. The weak form of the formulation for the mechanical contact between two bodies can be expressed as

δΠ = δΠm + δΠs + δΠN + δΠT = 0

(23.1)

where the first two components correspond independently to deformations of each of the beams, the third term includes the normal contact contribution and the last one corresponds to the friction. If the penalty method is used the normal contact term can be expressed by δΠN = εN gN δgN (23.2)

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where εN is the penalty parameter and gN – the gap function. To define the latter a particular analysis is necessary depending on the contact problem in hand. In the case of the point-wise contact between beams m and s (Fig. 23.1) the first issue to solve is finding contact candidate points. To this end the procedure of closest points location on two 3D curves is applied. Arbitrary points on the curves are defined in terms of local co-ordinates ξm and ξs . Their values ξmn and ξsn corresponding to the closest points Cmn and Csn and their respective position vectors xmn and xsn can be obtained by means of solving of a non-linear set of orthogonality equations: (xmn − xsn ) · xmn,m = 0 (23.3) (xmn − xsn ) · xsn,s = 0 where the commas denote the derivatives with respect to the local co-ordinates. These equations in general may have several or no solution at all. However, it may be assumed that, at least locally, in the vicinity of the real contact point the solution does exist and that it is unique. ξm Cmn

x3

xmn x2 xsn

x1

m axis

dN Csn

s axis

ξs Fig. 23.1 The closest points on two curves

With the closest points located one can finally define the basic variable for the normal contact, i.e. the normal gap gN = dN − rm − rs

(23.4)

where rm and rs are radii of beams circular cross-sections. The gap must be expressed in terms of displacements [9, 6]. Defining of friction contribution, it is necessary to introduce a so called tangential gap. If the beam-to-beam contact is analyzed there are two such gaps, each corresponding to a relative movement of a contact point along each of two contacting beams. They both enter the friction contribution to the weak form, cf. Eq. (23.1), depending on the current friction/sliding state. Here the analysis of friction is performed using the analogy to the rigid plasticity with a non-associated sliding rule [7]. The current values of tangential gaps are split into elastic and plastic parts gT m = geT m + gTp m ,

p gT s = geT s + gTs

(23.5)

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The elastic parts are used to define friction force components acting along each of the contacting beams FT m = εT geT mn ,

FT s = εT geT sn

(23.6)

FT = FT = FT m tm + FTs ts

(23.7)

The resultant friction force is defined by

with a help of unit tangent vectors tm =

xmn,m , xmn,m

ts =

xsn,s xsn,s

(23.8)

The plastic parts of the gap can be obtained from a non-associated sliding rule g˙T m = γ˙ p

∂f , ∂ FT m

g˙T s = γ˙ p

∂f ∂ FT s

(23.9)

which is integrated with respect to a pseudo-time by means of a step-wise analysis. At each step the total gap must be calculated. It is done by adding increments dgT mn and dgT sn . They are defined in the current configuration (Fig. 23.2) as straight line distances between the current contact point (Cmn and Csn ) and a mapping of the previous contact point from the previous configuration (Cmp ’ and Csp ’) to the current configuration (Cmp and Csp ). After an introduction of sliding control parameters sm and ss the current tangential gaps can be expressed by < < < < gT mn = gT mp + <xmn − xmp < sm , gT sn = gT sp + <xsn − xsp < ss (23.10) Then the current trial elastic gaps are obtained using the previous plastic gaps p

get T mn = gT mn − gT mp ,

p

get T sn = gT sn − gT sp

(23.11)

and with this in hand trial friction forces are calculated within the penalty method framework as FTt m = εT get FTt s = εT get (23.12) T mn , T sn Than using Eq. (23.7) the sliding criterion is checked. If < < f =
(23.13)

the stick state exists. In this case the real values of the friction force components are equal to their trial values and there is no increment of the plastic gaps. In the opposite situation, when < < f = 0 (23.14)

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x3

dgTmn

x2 xmn

x1

Cmn

xmp

Cmp'

ξmn

ξmp Cmp

ξmp

axis m current configuration

axis m previous configuration

Fig. 23.2 Tangential gap increment

the slip case is encountered. The Euler return to the sliding limit surface is performed, which in the case of the Coulomb friction law does not require any iterative scheme. The real friction force components and the plastic tangential gaps are obtained in the following form FT m = μ pm FN , gTp mn = gTp mp +

1

εT m

FTt m − FT m ,

FT s = μ ps FN gTp sn = gTp sp +

(23.15)

1 t FT s − FTs εT s

(23.16)

Because the sliding rule is checked for the resultant and the analysis requires separate components of the friction force there are proportion parameters involved in Eq. (23.15), which are defined as pm =

FTt m , FTt

ps =

FTt s FTt

(23.17)

Now the friction contribution to the weak form (1) can be defined as

δ ΠTe = εT geT mn δ gT mn + εT geT sn δ gT sn

for stick

δ ΠT p = μεN pm sm gN δ gT mn + μεN ps ss gN δ gT sn

for slip

(23.18) (23.19)

The final issue is to express the kinematic variables in this formulation in terms of displacements. For the details, see [9, 6].

23.3 3D Curve Smoothing Procedures The treatment of the frictional beam-to-beam contact described in Section 23.2 is related to the current configuration of two contacting beams. This configuration is

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generally obtained segment-wise as a function of current position vectors of nodes representing the beam axes. At this point the effective smoothing procedures are vital to provide the C1 -continuity. Here four methods of construction of a smooth curve are described. They involve two types of polynomial representation of a 3D curve, i.e. Hermite’s polynomials and Bezier’s curves, as well as two types of curve layout related to the beam nodes, i.e. inscribed curve method and node-preserving method.

ξ

1

2 C R

CL

C12

A C23

x2 x

x1

x3

x3 x1

3

x2

Fig. 23.3 Construction of an inscribed curve segment

In the case of the inscribed curve method a segment of a resulting C1 -continuous curve is constructed on the base of three adjacent nodes, Fig. 23.3. The segment is spanned between the mid-points C12 and C23 , which locations are defined by the position vectors x12 =

x1 + x2 = (x12i )T , 2

x23 =

x2 + x3 = (x23i )T 2

(23.20)

Furthermore, the curve is tangent to straight lines passing through nodes 1, 2 and 2, 3, respectively. This ensures the C1 -continuity between the adjacent segments. Such a concept was already successfully used in smoothing of 2D solids in contact [3]. If the Hermite’s polynomials are used to define the curve mathematically, then each of three components of the position vector x for an arbitrary point A on the curve can be expressed by xi = a i ξ 3 + b i ξ 2 + ci ξ + d i

(23.21)

The unknown coefficients of the polynomial are determined from the boundary conditions x2i + x1i ξ = −1 ⇒ xi = x12i = 2 x3i + x2i ξ = 1 ⇒ xi = x23i = 2 (23.22) ∂ xi x2i − x1i ξ = −1 ⇒ = ϕ12i = l123 ∂ξ 2l12 ∂ xi x3i − x2i ξ =1 ⇒ = ϕ23i = l123 ∂ξ 2l23

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which force the curve to fulfil the C1 -continuity conditions discussed above. In Eqs. (23.22) the notation l12 , l23 , and l123 for the straight line distances 1–2, 2–3 and C12 –C23 , respectively, and x ji for the i-th component of the position vector of the j-th node was used. After some algebra [4] the position vector x can be given in terms of the node position vectors ⎫ ⎤⎡ ⎡ 3 ⎤⎧ ⎡ ⎤ ⎡ ⎤⎪ 1 1 0 ξ ⎪ 1 −1 1 1 ⎪ x1i ⎪ ⎬ ⎢ ξ 2 ⎥⎨ 1 ⎢ 0 0 −1 1 ⎥⎢ 0= 1= 1 ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ x xi = ⎣ ⎦ 2i 0= ⎦ ξ ⎪ ⎪ 8 ⎣ −3 3 −1 −1 ⎦⎣ −l123 l12 l123 =l12 ⎪ x3i ⎪ ⎭ ⎩ 2 2 1 −1 0 −l123 l23 l123 l23 1 (23.23) For the Bezier’s curves representation it is necessary to construct a so called control polygon. Here this polygon is formed to yield the same C1 -continuity as above and also a provision is taken to keep its nodes at regular distances in order to get as regular curve as possible. Hence, it is proposed that the polygon is composed of the end nodes C12 , C23 and the intermediate nodes CL , CR . The latter nodes must lie on the straight lines 1–2 and 2–3, respectively, (Fig. 23.3) to yield the curve tangent to these lines. Their position is somewhat arbitrary and here it is adopted as 5 1 xL = x2 + x1 , 6 6

5 1 xR = x2 + x3 6 6

(23.24)

Then the Bezier’s curve can be expressed in terms of Bernstein’s polynomials [2] to give the position vector x of a point A in the form: 1 (1 − ξ )3 x12i + 3 (1 + ξ ) (1 − ξ )2 xLi + 3 (1 + ξ )2 (1 − ξ ) xRi + (1 + ξ )3 x23i 8 (23.25) Arguably, a curve, which fits the nodes better can be obtained if the segment of the curve terminates at the nodes. The price to pay for this is a necessity to include four adjacent nodes in the analysis. The construction method for this case is illustrated in Fig. 23.4. The segment ends coincide with nodes 2 and 3 and the curve at these points is tangent to the lines parallel to the lines 1–3 and 2–4, respectively. xi =

ξ

2

CL CR

A 3

x2

x

1

x

x3 x1

4 x1

x2

Fig. 23.4 Construction of a node-preserving curve segment

x4

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The boundary conditions necessary to find the coefficients of the Hermite’s polynomial, Eq. (23.21) expressing one component of the position vector x of a point A read ξ = −1 ⇒ xi = x2i ξ = 1 ⇒ xi = x3i ∂ xi x3i − x1i (23.26) ξ = −1 ⇒ = ϕ13i = l23 ∂ξ 2l13 ∂ xi x4i − x2i ξ =1 ⇒ = ϕ24i = l23 ∂ξ 2l24 Again, after some algebra, this position vector can be expressed in terms of the node position vectors in the following form ⎤⎡ ⎡ 3 ⎤⎧ ⎡ ⎤⎡ ⎤⎫ 0 2 0 0 ξ ⎪ x1i ⎪ 1 −1 1 1 ⎪ ⎪ ⎢ ξ 2 ⎥⎨1 ⎢ 0 0 −1 1 ⎥⎢ ⎥⎢ x2i ⎥⎬ 0 0 2 0 ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = = xi = ⎣ ⎦ ⎣ 0= l23 l13 = 0 ⎦⎣ x3i ⎦⎪ ξ ⎪ 8 −3 3 −1 −1 ⎦⎣ −l23 l13 ⎪ ⎪ ⎭ ⎩ 2 2 1 −1 0 −l23 l24 0 l23 l24 x4i 1 (23.27) For the Bezier’s curve representation the control polygon must be constructed in a similar way as previously. Here the positions of intermediate nodes CL , CR are defined as xL =

1 l23 (x3 − x1 ) + x2 , 3 l13

1 l23 xR = − (x4 − x2 ) + x3 3 l24

(23.28)

and the final form of the position vector x is identical to the one given by Eq. (23.25). It is important to note that three of the presented four curve representations are non-linear in the position vectors of nodes. This is due to the presence of the lengths li j in the formulas. Only the Bezier’s curve model for the inscribed curve method (Eqs. (23.24) and (23.25)) is free of this. So it should be expected that its substitution to the frictional beam-to-beam formulation involving partial derivatives of x with respect to nodal displacements [6] would yield the simplest computer code for calculation of a tangent stiffness matrix and a residual vector.

23.4 Finite Element Discretization FE discretization of two beams and modeling of contact between them leads to a notion of a beam-to-beam contact element. The layout of this element depends on the method of the curve representation discussed in the previous section. For the case of inscribed curve the contact element will involve two pairs of beam elements, one from each beam. In terms of nodes there will be three nodes per beam involved, Fig. 23.5. The corresponding vector of nodal displacement has therefore 18 entries, i.e. three displacements per each of six involved nodes

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m2

Cm12

Cm23

Cmn

m1

s1

Csn Cs12

s3

Cs23

m3

s2

Fig. 23.5 Inscribed curve beam contact finite element

T q[18×1] = uTM[9×1] , uTS [9×1]

(23.29)

Consequently, the tangent stiffness matrix and the residual vector for this element have the dimension of 18. In the case of node-preserving modeling of the curves the resulting beam contact finite element must involve three beam elements per each of two beams. It means that four nodes per beam are involved, Fig. 23.6.

m1

s4

m2

Cmn

Csn

m4 m3

s2

s3

s1

Fig. 23.6 Node-preserving beam contact finite element

The vector of nodal displacement has 24 entries T q[24×1] = uTM [12×1] , uTS [12×1]

(23.30)

and the dimension of the tangent stiffness matrix and the residual vector is 24. The detailed determination of the element matrices and vectors can be found in [6]. Here only one aspect is presented, which allows a qualitative comparison of presented smoothing models. One of key matrices necessary in the contact formulation is a matrix Gmn (or Gsn ) used to express the variations of displacements δ umn (or δ usn ) for and arbitrary point on a curve m (or s) (i.e. the point A in Figs. 23.3 and 23.4) in terms of respective variations of nodal displacements δ uM (or δ uS ) δ umn = Gmn δ uM = Gmn 0 δ q , δ usn = Gsn δ uS = 0 Gsn δ q (23.31) The similar matrices H and M are necessary to express the first and second derivatives of the displacement variations with respect to the local co-ordinate ξ . To calculate all these matrices the symbolic algebra program Maple V is used. The length of

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the automatically generated optimized Fortran code for the considered four contact elements is given in Table 23.1. It can be noted that the lack of non-linearity in the smooth curve representation for the inscribed curve method with Bezier’s curves results in an extremely short code. So at this stage one might expect that the corresponding beam contact finite element would be the cheapest in terms of analysis time. Table 23.1 Number of lines in the Fortran code for basic matrices in the beam contact formulation Element type

Matrix G

Matrix H

Matrix M

Inscribed curve, Hermite

2291

1416

703

Inscribed curve, Bezier

51

45

27

Node-preserving, Hermite

983

959

838

Node-preserving, Bezier

718

764

652

23.5 Numerical Examples In the first example contact between two beams shown in Fig. 23.7 is considered. The beams have circular cross-sections with radius R = 0.1, length L = 6.0 and are initially spaced at 0.001. They are made of a material with Young’s modulus E = 250·105, Poisson’s ratio ν = 0.3 and the friction coefficient μ = 0.8. The penalty parameters used in the analysis are ε N = 1·104 and ε N = 1·104. The imposed displacements shown as arrows in the Fig. 23.7 are applied in 60 equal increments. Each beam is divided into 10 co-rotational finite elements [1]. In Fig. 23.8 the deformed layout of beam axes is shown for the case of inscribed curve, Hermite element. These results corresponding to the other elements are almost impossible to distinguish. The effectiveness of smoothing can be observed in Fig. 23.9 where the evolution of elastic gaps during the deformation process is presented. Despite the fact that the large sliding forces the contact point to pass several times from one beam segment to an adjacent one, the elastic gap curves are smooth. Since these values are used to determine the friction forces in the contact it can be concluded that all four smoothing methods fulfil their main task to yield the C1 -continuous curves. The differences between results for the inscribed curve and the node-preserving smoothing are present but small. With the increasing number of beam elements used they are decreasing and for a reasonably large number the discrepancy is negligible. Actually, one might argue that 10 elements used here are enough to get the satisfactory accuracy. On the other hand, the difference between results for Hermite’s polynomial and Bezier’s curve within the two curve construction methods is virtually nonexistent.

23 The Beam-to-Beam Contact Smoothing with Bezier’s Curves Δ2 = 1.0

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Z Z

Y

Δ1 = 2.0

X

2

Y

Δ2 = 1.0

Δ3 = 0.5

2

Δ2 = 1.0

1

1

Δ1 = 2.0 Δ3 = 0.5

X

Δ2 = 1.0

Δ3 = 0.5

Δ3 = 0.5 Y

2

Z

Y

X

Δ1 = 2.0

2

Δ1 = 2.0

Δ2 = 1.0

Z

X

Δ1 = 2.0

1

1

Δ3 = 0.5

Fig. 23.7 Beam axes layout for the first example Z Y

X

1

1

2 2

Fig. 23.8 Deformed beam axes for the first example 0.12 0.08

Inscribed curve, Hermite Inscribed curve, Bezier

on beam 2

0

on beam 1

gTe

0.04

-0.04

Node-preserving, Hermite Node-preserving, Bezier

-0.08 Pseudo-time

Fig. 23.9 Development of elastic tangential gaps in the deformation process for the first example

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X

Δ2 = 3.0 Z

Δ1 = 1.0

2

1

Y

Δ2 = 3.0

Δ3 = 0.5

X

Δ1 = 1.0

2 Δ2 = 3.0 Δ3 = 0.5

Δ3 = 0.5

1

Δ3 = 0.5 Y

2 Δ1 = 1.0

Z

Δ2 = 3.0

Y

Z

X

X

2 1

1

Δ2 = 3.0

Δ3 = 0.5

Fig. 23.10 Beam axes layout for the second example Z Y

X

Fig. 23.11 Deformed beam axes for the second example

0.24 0.2

gTe

0.16 0.12

Inscribed curve, Hermite Inscribed curve, Bezier Node-preserving, Hermite Node-preserving, Bezier

0.08 on beam 1

0.04 0 Pseudo-time

Fig. 23.12 Development of elastic tangential gap in the deformation process for the second example

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Table 23.2 presents the comparison of relative CPU times required for the analysis of the displacement process. It is very surprising to note that the contact element with the simplest formulation, i.e. inscribed curve Bezier does not yield the shortest time. On the other hand the inscribed curve Hermite element with the longest code (see Table 23.1) gives the fastest results. The reason for this is a different number of iterations to equilibrium state. It was found that in almost all of 60 increments the inscribed curve Hermite element required at least one iteration less than the other elements. On the other hand, differences in CPU time for a single increment using the considered elements are negligible. Table 23.2 Comparison of relative CPU time of analysis Element type

Relative CPU time [%] First example Second example

Inscribed curve, Hermite

100

100

Inscribed curve, Bezier

115

115

Node-preserving, Hermite 130

106

Node-preserving, Bezier

144

144

In the second example contact between two beams shown in Fig. 23.10 is considered. The data used are the same as in the first example, except that the beam 2 has the length L = 10.0. In the Fig. 23.11 the deformed layout of beam axes is shown for the case of inscribed curve, Hermite element. As previously, differences in deformed shape for the other contact elements are invisible. The results of elastic gap evolution shown in Fig. 23.12 again prove the effectiveness of smoothing. Finally, the comparison of the CPU time required for the analysis, shown in Table 23.2 leads to similar conclusions as for the first example.

23.6 Conclusions All presented smoothing techniques lead to effective contact finite elements. There is no sign of contact force discontinuities when large sliding occurs. Surprisingly the best results are obtained using the inscribed curve Hermite element. Its disadvantage, i.e. lack of accuracy in representing the curve can be diminished if a reasonable number of beam elements is used. The presented examples show that 10 elements per beam are enough to get satisfactory results.

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References [1] Crisfield, M.A.: A consistent co-rotational formulation for non-linear, three-dimensional beam-elements. Comp. Meth. Appl. Mech. Engng. 81, 131–150 (1990) [2] Farin, G.: Curves and Surfaces for Computer Aided Geometric Design. Academic Press Inc., Harcourt Brace Jovanovich Publishers, Boston (1993) [3] Krstulovi´c-Opara, L.: A C1 -continuous formulation for finite deformation contact. Dr.Ing. Dissertation, Institut für Baumechanik und Numerische Mechanik, Universität Hannover (2001) [4] Litewka, P.: Hermite polynomial smoothing in beam-to-beam frictional contact problem. Comput. Mech. 40(6), 815–826 (2007) [5] Litewka, P., Wriggers, P.: Contact between 3D beams with rectangular cross-sections. Int. J. Numer. Meth. Engng. 53, 2019–2041 (2002) [6] Litewka, P., Wriggers, P.: Frictional contact between 3D beams. Comput. Mech. 28, 26– 39 (2002) [7] Michałowski, R., Mróz, Z.: Associated and non-associated sliding rules in contact friction problems. Arch. Mech. 30, 259–276 (1978) [8] Wriggers P., Zavarise, G.: On contact between three-dimensional beams undergoing large deflections. Comm. Num. Meth. Engng. 13, 429-438 (1997). [9] Zavarise, G., Wriggers, P.: Contact with friction between beams in 3-D space. Int. J. Numer. Meth. Engng. 49, 977–1006 (2000)

Chapter 24

Synergic Combinations of Computational Methods and Experiments for Structural Diagnoses Giulio Maier, Gabriella Bolzon, Vladimir Buljak, Tomasz Garbowski, and Bartosz Miller

Abstract. The mechanical characterization of materials and the non-destructive assessment of possible damages in industrial plant components and in civil engineering structures and infrastructures is a problem which at present arises more and more frequently and acquires growing importance in both experimental and computational mechanics. The survey presented here concerns some representative, practically meaningful typical problems of this kind recently or currently tackled by our research team. It is intended to evidence the central role played by computer methods and by procedures of numerical mathematics, including soft computing, for the practical solutions of inverse analysis problems in real-life situations. The engineering applications dealt with herein concern steel pipelines and metal industrial components typical of the oil industry and existing large concrete dams possibly deteriorated by physico-chemical processes like alkali-silica reactions. Giulio Maier Department of Structural Engineering, Politecnico (Technical University) di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy e-mail: [email protected] Gabriella Bolzon Department of Structural Engineering, Politecnico (Technical University) di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy e-mail: [email protected] Vladimir Buljak Department of Structural Engineering, Politecnico (Technical University) di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy e-mail: [email protected] Tomasz Garbowski Department of Structural Engineering, Politecnico (Technical University) di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy e-mail: [email protected] Bartosz Miller Department of Structural Mechanics, Rzeszów University of Technology, ul. Pozna´nska 2, 35-959 Rzeszów, Poland e-mail: [email protected]

M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 453–476. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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24.1 Introduction In most practical situations including the real-life problems considered herein, the following circumstances characterize parameter identifications: (i) a finite domain containing the sought parameters can a priori be specified in their space (some times by an “expert” in the field), at least through lower and upper bounds on each parameter; (ii) correlation generally holds among the vectors of measurable quantities belonging to the responses of the system with differences only among parameters within the above mentioned domain; such vectors are “correlated” in sense that they turn out to be “almost parallel” in their space. The mathematical and computational methods listed below (and described in referenced publications) have been employed to solve the practical parameter identification problems here outlined in Sect. 24.2 and Sect. 24.3 and can be regarded, at present or in the near future, as central in most inverse problems in structural engineering. (a) The correlation reasonably expected among response vectors, as above noticed by remark (ii), makes potentially fruitful in the present context the recourse to Proper Orthogonal Decomposition (POD). Such procedure of growing interest in mechanics, can be summarized as follows (for details see e.g. [10, 22, 26, 27]): starting from say M points (“nodes”) in the P-dimensional domain of the sought parameters, let experiment simulations (or, very seldom, true experiments) lead to the M corresponding vectors (“snapshots” in the POD jargon), each one of N measurable response quantities. As suggested by the correlation of the snapshots gathered in the N × M matrix U, new Cartesian reference axes are determined such that, sequentially, a norm of the snapshot projections on each of them is made maximal; thereafter a “truncation” is carried out, namely only N axes with non-negligible component norms are preserved. Such procedure computationally implies the calculation of eigenvalues and eigenvectors of the (symmetric, positive semidefinite) matrix UT U = D. After this central computing effort, the “truncation” based on a comparative assessment of the above eigenvalues leads to a N × N matrix Φ with N $ N, which represents new “truncated” Cartesian reference. Thereafter, the snapshot “amplitudes” in the new reference are easily computed and gathered in N × M matrix A. (b) Suppose that computer simulations of the system response have been carried out by conventional modeling (say by a finite element code) on the basis of parameter vectors belonging to a suitable node grid on the domain in the P-dimensional parameter space. Then the system response based on a new parameter P-vector p (namely the amplitude vector a which defines the relevant snapshot u) can be computed by interpolation of the available results much more economically than by a further direct analysis through the conventional forward operator. Such interpolation can be efficiently performed by expressing the sought vector a(p) as a linear combination of Radial Basis Functions (RBF) (see e.g. [8, 27]). In the present study the selected RBFs are axially symmetrical functions of the “distances” (Euclidean norm) between the point p and the grid nodes in the parameter space pi , i = 1, ..., M. The N × M interpolation coefficients of such linear combinations are solutions to

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the linear equation system arising from the obvious requirement that the amplitude vector function equals the known amplitude N-vector a(p)i in each node of the grid. (c) Among the several first-order procedures now available in the broad literature on Nonlinear Mathematical Programming (NMP), the Trust Region Algorithm (TRA) has been selected to the present purposes. Its main features can be briefly pointed out as follows: at each iteration a two-variable quadratic programming problem is solved over the (hyper) plane defined by the gradient and by the Newton vector of the discrepancy function to minimize; the quadratic approximation of the objective function is generated by an approximation of the Hessian computed through the Jacobian of the “errors” (errors mean here differences between measurable and computed quantities, the latter computed as functions of the sought parameters); repeated diverse initializations have fragmentarily to be performed in order to avoid end of the step sequence in a local minimum different from the global one, due to possible lack of convexity. On TRA and on other NMP algorithms of first order (and of diverse orders, in particular of zero order or “direct search” like Nelder-Mead or “complex” algorithms) details can be found e.g in [11, 16]. (d) The possible presence of local minima and the presence of a domain “a priori” predictable in the space of the unknowns, clearly make Genetic Algorithms (GA) pertinent and promising conceptual and computational tools in the present context. On the other hand, Artificial Neural Networks (ANN) turn out to be of obvious interest for the engineering applications pursued herein, in view of the significant advantages provided by structural diagnosis made repeatedly “in situ” by a portable computer endowed with an ANN “trained” and “tested” once for all originally in a computer laboratory. In the vast literature on GA and ANN and on soft computing in general, references [19, 29] are among the primary sources of information for what follows.

24.2 Local Diagnostic Analyses of Concrete Dams 24.2.1 Preliminary Remarks The possible deterioration of existing concrete dams gives rise at present, and probably even more in the future, to real challenges in structural engineering, especially in developed countries where a multiplicity of large dams have been designed and built up several decades ago. The main causes of most damages in concrete dams are extreme loadings, such as earthquakes and floods, slow orogenetic motions of geological surroundings and, the most dangerous one, Alkali Silica Reaction (ASR). The physico-chemical process ASR, not clearly understood before the end of the Eighties, is generally dormant during one decade or more after casting, and later gradually reduces, sometimes to a large extent, concrete stiffness and strengths and generates irreversible expansions and consequent self-equilibrated stresses, see e.g. [2, 12].

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Statical overall diagnostic analyses are based on measurements (by pendula and collimators or in the near future by radar) of displacements generated either by quasi-static loading due to reservoir level variations (“ad hoc” fast, see e.g. [15]; or seasonal in service, see e.g. [4]). Dynamical overall diagnoses are based on excitations by vibrodynes and measurements by accelerators. Clearly, only elastic moduli (Young modulus alone in practice), assumed as zone-wise uniform, can be assessed by overall, statical or dynamical, nondestructive testing and inverse analyses. Inelastic properties of concrete and stresses in existing dams can be assessed by local “nondestructive” tests. These are briefly considered in what follows in order to evidence the potential improvements due to applications of recent developments in computational inverse analysis methodologies mentioned in the Introduction.

24.2.2 Tests on Dam Surfaces by Flat-Jacks Flat-jack tests are frequently applied to performed masonry structures and concrete dams since several decades. The state of the art of this popular procedure can be outlined as follows, see e.g. [17]: (α ) two parallel circumferential slots are digged; (β ) the cuts alter the existing stress state and the consequent elongations between marked points in direction orthogonal to the slots are measured by extensometers; (γ ) a flat-jack is inserted and pressurized until the previously measured elongations are fully recovered and, hence, the jack pressure approximately equals (to within a corrective empirical factor accounting for the jack geometry and welds) the compressive stress pre-existing there; (δ ) flat-jacks are inserted in both slots and pressurized: the pressure related to consequent measured elongations leads, again through semi-empirical formulae, to an estimation of the Young modulus; (ε ) estimations of Young modulus and, usually assumed “a priori”, of Poisson ratio, can be employed for an assessment of the normal stress on the basis of the elongation measurements in phase (α ), as an alternative (or as a validation) of the stress estimation at phase (γ ). To comparison purposes, a brief description is presented here below of the novel flat-jack diagnostic procedure made possible by progress in both computational and experimental mechanics. This procedure is novel also in terms of extension to plasticity and fracture and of experimental equipment, with recent innovative proposals described in [13, 14]. (I) Two slots are digged, one orthogonal to the other (“T geometry”) without intersection. (II) A conventional three-dimensional FE model is generated “una tantum” over a suitable domain with constrained boundary conditions in unperturbed zone; (III) The consequent perturbation of the stress state gives rise to elastic displacements; these are measured on the surrounding surface by Digital Image Correlation

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(DIC) at the nodes of suitable designed grid, nodes which are coincident with nodes of the FE mesh. (IV) The jacks are inserted into the slots and moderately pressurized; the consequent elastic displacements are measured by DIC. (V) The pressure is increased in order to cause inelastic deformations, including fracture propagation, particularly between the two slots. Once again displacements are measured by DIC. (VI) Inverse analysis based on the experimental data gathered in phase (IV) identifies elastic moduli. (VII) The elastic moduli identified at the stage (VI) are employed for the estimation of stresses on the basis of DIC measurements in stage (III). (VIII) Finally, the inelastic and fracture behavior of concrete is characterized by means of inverse analysis by exploiting the experimental data collected through DIC during phase (V). The above outlined innovative procedure has been validated by a number of “pseudo-experimental” exercises, namely: numerical values are assigned to the sought parameters; on this basis a computer simulation of the test generates measurable quantities; starting from these quantities inverse analysis leads to parameter estimates which are compared to the originally assumed values. A representative validation exercise is briefly described below (details will be available in journal papers). a)

b)

Fig. 24.1 Inelastic parameters to identify in single cohesive crack a and Drucker-Prager b model for concrete

In the elastic range, orthotropic and horizontally isotropic behavior is attributed to concrete, in view of possible consequences of the casting process (and, in some cases, of roller compaction). Specifically, in the reference frame with axes x and y on the dam surface and x and z on horizontal plane, the elastic moduli which govern the Hooke’s law and the values assumed for them are: Ex = Ez = EH = 20 GPa; Ey = EV = 30 GPa ; Gxy = Gyz = GV = 9.6 GPa; νxz = νH = 0.2 ; νyx = νV H = 0.2 ; νxy = νHV = νV H EEHV = 0.3 . The shear modulus in the isotropy plane Gxz = GH = 8.33 MPa depends on νH and EH . The parameters to identify are EH , EV and GV , since the Poisson ratios νH and νV H are assumed a priori known in view of their marginal influence on structural responses.

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In the inelastic range the simplest cohesive crack model of Fig. 24.1a is adopted for the expected mode I quasi-brittle fracture propagation. The parameters to identify are tensile strength ft and fracture energy G f , originally assumed equal to 3 MPa and 150 N/m, respectively, for initial forward analyses in view of pseudoexperimental validation. The simplest plasticity model is considered for concrete in the plastic range, namely Drucker-Prager yield criterion and flow potential: F = t − p tan(β ) − d ≤ 0 ;

G = t − p tan(ψ )

(24.1)

where p is the mean normal stress and t the equivalent shear stress (Fig. 24.1b). The dilatancy angle is a priori assumed ψ = 40 deg. The internal friction angle β (Fig. 24.1b), originally assumed β = 68 deg, is the third parameter to identify, together with ft and G f , since the cohesion d and compression strength fc are not independent but related to the above parameters as follows: 1 + 13 tan(β ) ft d fc = = (24.2) 1 − 13 tan(β ) 1 − 13 tan(β ) It is worth noting that the simplicity here adopted in material models is consistent with the novelty of the proposed flat-jack procedure and with the uncertainties of the experimental data achievable in the engineering context of large concrete dams. As for stresses to estimate, plane state represents an obviously reasonably hypothesis. Therefore in the previously introduced reference axes σz = τyz = τzx = 0, and, hence, sought components are σH = σx , σV = σy , τ = τxy , with the following assumption for numerical pseudo-experimental checks: σH = −2 MPa, σV = −6 MPa, τ = 1 MPa. The adopted finite element model, visualized in Fig. 24.2, exhibits the following features: 60, 400 tetrahedral FEs; 47, 200 DOF; zero displacements on the boundary supposed to be not influenced by the test. Digital Image Correlation (DIC) is an optical technique developed for full-field measurements of surface displacements (see e.g. [28]). It is based on comparison between two images of specimen taken in the undeformed and deformed state. Correlation between material points (actually small zones) in the undeformed and deformed images is obtained over the “Zone Of Interest” (ZOI) by an image-matching procedure based on gray levels stored in “pixels”. With respect to the extensometers employed so far in flat-jack tests and to other optical measurement methods, the special merits of DIC are: non-contact measurements; simple optic setup; no special preparation of specimens; no special illumination; smaller measurement “noise”. Moreover, for the present inverse analysis method meaningful advantage is provided by the substantial increase of the experimental data number and consequent regularization (in Tikhonov sense [9]) of the discrepancy minimization problem. Figure 24.3 visualizes the operative experimental phases of the proposed flatjack method. Over the ZOI shown in Fig. 24.4 photographs for DIC are taken at

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Fig. 24.2 Finite element model for simulations of the flat-jack tests. Areas 1 and 2 are concerned by DIC measurements during the inelastic phases of the test

stage (a), with only marks of slots before digging, and at stage (b) in the presence of the still empty slots; then at phase (c) after jack insertion and pressurization. The ZOI for DIC are reduced in phase (d-e) as shown in Fig. 24.4, in order to concentrate measurements near the places where plastic strains and fracture propagations are expected (i.e. over the sub-domain where inelastic material models have been adopted, as shown in Fig. 24.2). In the present engineering context it is particularly desirable, or even necessary, that the diagnostic assessment of possible local material damage be carried out repeatedly and routinely, by using portable equipment (including computers) apt to provide all the sought estimates soon and “in situ” and in “nondestructive” fashion (in particular without specimen extraction). Such practical requirements, besides the increase of reachable information evidenced earlier by comparison with the state-of-the-art, can be attained at present by suitable applications of the computational methods mentioned earlier in Sect. 24.1. After a number of numerical exercises, the following computing tools have been evaluated as probably optimal for the novel flat-jack methodology proposed by our team and outlined in what precedes. All three identification procedures are performed by exploiting Proper Orthogonal Decomposition (POD) together with Artificial Neural Networks (ANN) or with Trust Region Algorithm (TRA) through RBF interpolations. Computationally, the proposed diagnostic method proceeds according to the guidelines which follow. (i) The three sets of numerous (N) displacements measurable by DIC are computed by FE model (see Fig. 24.2), once for all in a computing center, as “snapshots” corresponding to M nodes of the grid over the expected domain in each one of the three spaces of parameters. Specifically, in a typical numerical exercise of this study:

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for elastic moduli (phase b-c, Fig. 24.3) N = 2086, M = 1089; for stresses (phase a-c) N = 2086, M = 3375; for inelastic parameters (phase d-e) N = 238, M = 1620. Clearly, the elastic moduli, identified first, have to be inserted as data for the two subsequent identifications and, hence, they represent an increase of the number of dimensions in the space where to generate the grids of nodes. Therefore isotropy and hence only Young modulus E has been considered here in preliminary validations of procedures apt to estimate by inverse analysis inelastic parameters. The description which follows is focused on the identification of three elastic moduli, but only on it for brevity.

Fig. 24.3 Phase sequence in the new flat-jack testing procedure on concrete dams

Fig. 24.4 Areas monitored by digital correlation instruments in the parameter identification stages specified in Fig. 24.3

(ii) The “compression” (in other terms approximation) of the information contents of the three large snapshot N × M matrices U generated once for all at phase (i), is accomplished, again once for all, by POD. The relevant “truncation” was made at an eigenvalue with order of magnitude ten thousand times less then the maximum eigenvalue. (iii) Any new “snapshot” of truly experimental data provided in the future by DIC will be compressed to its “amplitudes” in the truncated reference system generated

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by POD in phase (ii). This prospect makes computationally useful to generate a multilayer feed-forward Artificial Neural Network (ANN) once for all, in order to perform fast and “in situ” the transition from the compressed experimental data to the sought parameters; these are, at this stage, the three elastic moduli. In the present context the optimized architecture of ANN to be calibrated (“trained”, “tested” and “validated”) through POD generated “patterns” turned out to consist of 3-20-3 neurons in input, hidden and output layer, respectively. Other features which characterize the ANN in point are as follows: tan-sigmoid and linear transform function in hidden and output layer, respectively; “training” by a usual back-propagation procedure, Levenberg-Marquardt algorithm; the pattern set is divided into training, testing and validation subsets in the following proportions 0.6-0.2-0.2. (iv) As an alternative to ANN, the POD procedure mentioned at (ii) has been associated also to Trust Region Algorithm (TRA), so that in its iterations requiring derivatives, direct analysis is replaced by interpolation through Radial Basis Functions (RBF). The here selected formulation of these functions reads (p being the vector of sought parameters): g j (p) = ||p − p j ||3

(24.3)

The estimations achievable by the above methods, briefly indicated as POD-ANN and POD-RBF-TRA, have been comparatively checked from various standpoints. For example, the influence of random noise in DIC data on consequent errors in the estimations turns out to be comparable, as visualized in Fig. 24.5: a random perturbation of DIC pseudo-experimental data with uniform probability density over the interval 10 μm leads to the perturbations of elastic parameter estimations shown in Fig. 24.5a with the former method (iii); in Fig. 24.5b with the latter procedure (iv). This comparative numerical exercise on the two different computational procedures (POD-ANN and POD-RBF-TRA) have almost equal average estimation errors (2.245% and 2.352%, respectively).

a)

b)

Fig. 24.5 Scattering (in percentage) of the elastic parameter estimates by POD-ANN a and POD-RBF-TRA b diagnostic procedure on the basis of some randomly perturbed pseudoexperimental data from DIC

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It is reasonably expected that the displacement measurements by DIC, even if affected by same experimental errors (here ±10 μm), lead to more stable and more robust estimations then those provided by extensometers. In fact, the number of the latter was reasonably assumed as limited to 16 (even less in the present practice). The above conjecture turns out to be quantified and confirmed by numerical results in Fig. 24.6a compared to the preceding Fig. 24.5a.

a)

b)

Fig. 24.6 In a results analogous to those in Fig. 24.5b but on basis of similarly perturbed data provided by extensometers instead of by DIC. In b scattering of stresses resulting from the same DIC data and POD-RBF-TRA procedure like in Fig. 24.5b, and from the averages of the elastic moduli estimates considered there

Figure 24.6b shows errors of an identification by POD-RBF-TRA based again on DIC measurements, but concerns the estimates of the three stress components after the assessment of the concrete elastic stiffness. Finally, several numerical tests concerning the above POD-RBF-TRA procedure demonstrated that promising are estimations of the inelastic parameters specified in Fig. 24.1 when DIC provides data from stages (d-e), Fig. 24.3. Figure 24.7 illustrates convergence on the reference values when only the cohesive crack model of Fig. 24.1a is considered in the simulation beyond the elastic range, account taken of the stiffness and stresses estimate in the preceding stages (primarily here EH = 20 MPa, σH = −3 MPa), Figure 24.8 shows the same convergence behavior when both cohesive and Drucker-Prager models are involved in the FE simulation of the novel flat-jack test.

24.2.3 Dilatometric Tests in Depth Deterioration of concrete by a diffusive physico-chemical process like ASR clearly may concern the whole volume of a large dam. Damage assessment in interior castings still represents a challenge, which is tackled so far primarily by extraction of specimens to be tested in labs, according to techniques traditionally used in rock

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Fig. 24.7 Convergence of the Trust Region Algorithm for the estimation of concrete tensile strength ft and fracture energy G f (normalized by the reference values or ”targets”) based on DIC measurements, according to Fig. 24.4 phases (d-e)

Fig. 24.8 Same as in Fig. 24.7 but by test simulation including both crack (Fig. 24.1a) and DruckerPrager (Fig. 24.1b) models

mechanics (see e.g. [3]). Inverse analysis and relevant computational methods pointed out in what precedes inspired the procedure outlined below. The proposed instrumentation (not yet available on the market) is schematically depicted in Fig. 24.9. It is inserted in depth after hole drilling. The experiment and relevant parameter identification are articulated in the following stages: (i) further drilling process by, say, one meter (Fig. 24.9); (ii) the consequent changes in the stress state, supposed originally uniform with prevailing transversal, horizontal σH and vertical σV , normal stresses, give rise to deformation with radial displacements which are measured by 12 gauges (called “dilatometers”) included in the experimental equipment (Fig. 24.9); (iii) two pairs of radial “wedges” (or “arches”), in horizontal and vertical directions, are activated by small internal jacks and operate on the hole surface (Fig. 24.9) with a relationship force-versus-advancement like

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Fig. 24.9 Schematic representation of the proposed equipment for dilatometric tests in concrete dams.

the one shown in Fig. 24.10, generated by a FE simulation; (iv) the data gathered and digitalized in phase (iii) feed a POD-ANN procedure similar to the one earlier described with reference to flat-jack tests and implemented in order to estimate “in situ” the elastic moduli; (v) now the displacements measured in stage (ii) are employed for the identification of dominant stresses σH and σV by a procedure similar to the one applied at the preceding stage (iv), but using the elastic estimates produced at (iv); (vi) finally, experimental data extracted from the inelastic parts of force-versus-advancement measured relations like in Fig. 24.10 are exploited for another inverse analysis leading to parameters governing a plastic constitutive model. This model in preliminary validation exercises has been chosen same as for the flat-jack superficial diagnosis, namely Drucker-Prager model; however, the elliptic “cup” implemented in ABAQUS and in other commercial codes is added here, in view of possible dominance of compressive stresses deep in a dam. Therefore, the parameters to estimate are here 5, namely E, β and d like earlier (Fig. 24.1) and also cap position pd and ellipse eccentricity R in the implemented Drucker-Prager model. Identification procedures are here not described for brevity, since they are conceptually similar to those combined with flat-jack tests and specified in the preceding subsection. Sensitivity analysis is here particularly useful in order to mitigate possible doubts on the identifiability of the present procedure. Figure 24.11 visualizes by columns the changes of the vertical wedge displacements from their maximum values (about 10 mm) when the 5 material parameters are separately varied by 5% from their reference values.

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Fig. 24.10 Force versus arch advancement generated by a computer simulation of the wedging phase in the proposed dilatometric testing procedure

As a conclusion of this Section, it can be stated that current progress in computational methods applicable to inverse analysis techniques, even if they are developed by a simple deterministic approach, are likely to become substantially beneficial to the engineering practice for extensive diagnoses of large concrete dams.

Fig. 24.11 A comparative assessment of sensitivity of dilatometric measurements (Fig. 24.9) with respect to the parameters which govern Drucker-Prager model with “cap” attributed to dam concrete

24.3 Diagnostic Analysis of Metallic Industrial Components 24.3.1 Assessments of Material Parameters by Indentation Networks of pipelines for oil or gas transportation, steel components of power plants and of offshore structures are typical examples of meaningful engineering systems

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which require inspection, monitoring and assessment of present mechanical properties in view of structural analyses apt to compute safety margins. In fact, material parameters to be input into computer modeling may be somewhere unknown or deteriorated by aging processes like corrosion. Traditional indentation tests originally devised for the assessment of hardness (which has a peculiar metallurgical rather than mechanical meaning), since years have been developed into a popular nondestructive technique for the calibration of material constitutive models either by semiempirical formulae (see e.g. [25]), or, in recent times, by test simulations and inverse analyses (see e.g. [6, 7, 18]). It was proposed in [7] that experimental data apt to feed inverse analysis be provided not only by an instrumented indenter in terms of force versus penetration curves, but also by a laser profilometer in terms of geometric features of the residual imprint. Such enlargement of experimental information availability gives rise to the following broader spectrum of alternatives, among which an optimal option may be selected in each real-life situation of engineering practice: (i) the loading-unloading indentation plots are generated in digital form by an instrumented indenter and exploited as the only experimental data for inverse analyses which are carried out in situ by a portable computer; (ii) same as in (i) but instead of indentation plots, the imprint geometry only is measured and digitalized by a profilometer and then exploited for in situ parameter identification; (iii) imprint measurements only as in (ii), except that a mould of the imprint geometry is cast and taken to a laboratory endowed with profilometer and computer; (iv) the inverse analysis rests on experimental data from both the indentation plots and the imprint profile and it is performed either in situ or in the laboratory. Clearly, all the above alternative procedures are “nondestructive”, in the sense that no specimen is extracted and the caused damage is confined at the microscale; moreover, in view of their repetitiveness, they substantially benefit, in terms of time and resource savings, if the inverse problems are solved fast on small computers, possibly in situ. Therefore the computational procedures, apt to exploit “snapshot” correlations, as pointed out in Sect. 24.1, become practically desirable and useful in the present technological context. Such statement has been corroborated by many pseudo-experimental numerical exercises, on material parameter identification based on indentation tests including the illustrative ones which follow. The constitutive model adopted here for the material is classical: isotropic, elastic-plastic, associative, with exponential hardening and Huber-Mises yield function. Its description for uniaxial stress state can be given as follows, according to the handbook format of the commercial FE code ABAQUS employed herein:

σ = E ε , with ε ≤

σ0 σ0 , σ = σ01−n E n ε n , with ε > E E

(24.4)

The reference values here attributed to the three parameters for pseudoexperimental numerical validations are typical of some steels: σ0 = 405 MPa; E = 198 GPa; n = 0.111. A conical indentation conforming with Rockwell test geometry is simulated by the axisymmetrical FE model shown in Fig. 24.12 and characterized by the

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following features: 3320 degrees of freedom; zero displacements on the low plane and cylindrical boundaries; isotropic elastic indenter with E = 1170 GPa, ν = 0.1 (typical of diamond); Coulomb friction with coefficient equal to 0.15; finite strain nonlinearities according to ABAQUS code implementation (not discussed here).

Fig. 24.12 Axi-symmetric finite element model employed for simulations of conical indentations on isotropic materials

The simulation of an indentation test by means of the above specified model leads to loading-unloading relations and imprint profile visualized in Fig. 24.13a and Fig. 24.13b respectively. The computing time was 147 seconds on the computer with the processor Intel CoreDuo 2.2 GHz with 2 GB of RAM. The following data have been extracted from the results of the above computation: from the indentation plots 100 points of corresponding applied load and penetration depth values, assuming imposed forces at equal intervals; from the imprint profile 82 points with their two-dimensional coordinates. The discrepancy function defined as Euclidean norm of the “errors” has been minimized first by the TRA (as mentioned in Sect. 24.1) by using at each iteration the forward operator based on the above specified FE model. Satisfactory convergence occurred, as shown e.g. in Fig. 24.14, with different initializations, which were adopted in view of possible local minima. To investigate the influence of the noise on the inverse analysis, some moderate experimental “noise” was considered, namely randomly distributed error within the range ±2 N in the indentation curve, and ±1 μm in the imprint profile. Five different pairs of curves and imprints have been generated. For each case, the inverse problem was solved using the TRA with 4 different initializations. The results are presented in the Table 24.1. In Fig. 24.15 these results are compared to the “target” (namely to a plot defined by the reference values of the sought parameters) by means of uniaxial stress-strain

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a)

b)

Fig. 24.13 Force on the indenter versus its penetration a and profile of the residual imprint b computed by finite element model in Fig. 24.12

Fig. 24.14 Convergence of TRA with 4 different initializations in the inverse analysis problem based on the data visualized in Fig. 24.13

diagrams. It is worth noticing that all these curves practically coincide from an engineering perspective.

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Table 24.1 Estimates and relevant standard deviations achieved through TRA from 5 diverse conical indentations on the same material modeled according to Eq. (24.4), on the basis of pseudo-experimental data randomly perturbed as specify in the text Different cases

Young modulus

Hardening exponent

Yield stress

1 2 3 4 5

197.428 ± 8 GPa 201.170 ± 9 GPa 198.608 ± 11 GPa 190.748 ± 8 GPa 198.58 ± 5 GPa

0.10809 ± 0.0028 0.11294 ± 0.0036 0.10928 ± 0.0033 0.1114 ± 0.0033 0.1125 ± 0.0022

412.08 ± 10 MPa 401.4 ± 9 MPa 408.65 ± 7 MPa 407.01 ± 6 MPa 401.57 ± 4 MPa

Fig. 24.15 Uniaxial stress-strain curves constructed with the estimates from the 5 initializations leading to results of Table 24.1

Also the influence of systematic modeling errors is worth being investigated beside that of random errors. To this purpose, the indentation curve and the imprint to be used as inputs in the inverse analysis are shifted upwards by increasing their ordinate values by quantities randomly generated within the range 1% and 5%. Results concerning the yield limit σ0 in Eq. (24.4) are visualized in Fig. 24.16; in abscissa the input data sets are given, namely different material parameter combinations, while ordinate is reading the error in estimates. The above exercises of inverse analysis by the TRA show that indentation tests do provide data suitable to estimate material parameters in isotropic elastoplasticity with hardening, according to a diagnostic procedure of kind (iv) in the test classification presented earlier. Similar numerical investigations have demonstrated that sufficient data to the same purpose can be extracted from imprint profile alone or only from loading-unloading plots. However, in all these procedures heavy computing efforts are required by repeated FE simulations within the iteration sequence of

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Fig. 24.16 Influence of systematic modeling error for 10 different material parameter sets

an efficient optimization algorithm like TRA or with soft-computing technique like GA (which have been tested in parallel investigations). Therefore, as motivated in Sect. 24.1, the potential advantages in the present context of the recourse to POD and RBF have been investigated extensively by numerical exercises like those summarized here below. First, in the three-dimensional space of the sought parameters the following grid has been generated: σ0 from 330 MPa to 460 MPa with the step of 10 MPa; E from 170 GPa to 218 GPa with the step of 6 GPa; n from 0 to 0.2 with the step of 0.005. For each one of those “nodes” FE direct analysis has generated a vector (“snapshot”) of measurable quantities in the structure response to indentation. After performing POD and computation of the “amplitudes” in the new reference system, such large information has been “compressed” by “truncation”, namely dropping eigenvalues smaller than 10−6 times the largest one. Interpolation by RBF leads to the snapshot (through its “amplitudes”) of a new parameter set: the computing time for this generation of the new response turns out to be smaller by 5 orders of magnitude, than that required to the same purpose by the conventional FE analysis. After such very encouraging numerical tests of the above kind, POD-RBF sequence was combined first with TR algorithm. Practically the same results mentioned earlier on parameter estimations have been achieved with computing times orders-ofmagnitude shorter, by means of “ad hoc” software employable in a portable computer as highly advantageous in the engineering practice referred to. The whole identification process turns out to end within couple of seconds, while it would require approximately 12 hours on the same computer if TRA is combined with FEM at each of its iterations. Alternatively but with the same advantages pointed out above on TRA, POD has been combined with an ANN and have been investigated by similar pseudoexperimental numerical exercises of parameter estimation. The main features of here

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adopted ANN is: number of neurons 12-23-3 in the input-hidden and output layers; tan-sigmoid transfer function in the hidden layer, linear in the output layer. The total number of generated patterns is the same as in previous case, 20% of them used for validation and 20% for testing; the number of data at the input of the network is 12, which represent amplitudes of the indentation curve and residual imprint in the new POD basis; output consist of the 3 sought parameters. When “clear” numerical data are used, the results of ANN exhibits exactly the same level of accuracy as those obtained by POD-RBF-TR. When noise introduced, the results turn out to be slightly worse, but from engineering point of view quite acceptable.

24.3.2 Assessment of Residual Stresses by Indentations Residual stresses may significantly influence damaging processes, such as fatigue, and, hence durability of structures and plants. Beneficial compressive self-stress states are frequently generated near the surface of metallic industrial products, e.g. by thermal and/or chemical and/or mechanical treatments (see e.g. [20, 24]). Various experimental techniques are now available for the quantitative assessment of residual stresses (see e.g. [5, 6, 24, 30]). When these stresses are expected to vary with depth and such variation has to be quantified, repeated indentations and inverse analysis can be advantageous in engineering practice. The proposed procedure is here pointed out by following pseudo-experimental exercise concerning a metallic shaft to which Huber-Mises model is attributed with Young modulus 200 GPa, Poisson ratio 0.3 and yield stress 1060 MPa. Fig. 24.17a visualizes the adopted parameterization of the dependence on depth of the radial stress σ , of the assumed axi-symmeteric distribution. Analytically, such parametrization reads:

σ (x) = σmax sin ( f xm + p) , 0 < x < d0 where

σsur π−p , f= m σmax d0 −1

π /2 − p dmax log m = log d0 π−p p = arcsin

(24.5)

(24.6) (24.7)

The parameters to identify are σmax , σsur , d0 and dmax listed in Table 24.2. Three indentation tests are supposed to be carried out with depths 20, 50 and 150 μm, and the force versus penetration relationships are computed by an axially symmetric FE model with 14, 566 DOF (Fig. 24.17b) in order to generate the pseudo-experimental data. The inverse problem is first solved by Trust Region optimization algorithm using as a direct operator FE simulation. Several inverse analyses have been performed with different initializations. Table 24.2 lists the obtained mean values together with

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Fig. 24.17 Dependence of residual axi-symmetric stress on depth and its parameterization in a. In b the finite element model here adopted for the parameter identification

the standard deviations of the four parameters which govern the parameterized depth dependence of residual stresses. Table 24.2 Stochastic assessment of the influence of initialization on the estimates of the residual stress parameters in Fig. 24.17a

Maximal stress σmax Stress at the surface σsur Depth at which stress changes the sign d0 Depth at which maximum occurs dmax

a)

Mean value

Reference value (Error)

948 ± 30 MPa 429 ± 32 MPa 209 ± 5 μm 68 ± 6 μm

1040 MPa (8.8 ± 2.8%) 400 MPa (7.25 ± 8%) 190 μm (10 ± 2.6%) 67 μm (1.5 ± 8.9%)

b)

Fig. 24.18 Estimation accuracy visualized comparatively for inverse analyses: by means of TRA a and by GA b, on the basis of the same set of data and from same models of Fig. 24.17

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The problem under consideration turns out to have a number of local minima, and this circumstance can explain the scattering of the numerical results presented in Table 24.2. The same identification problem has been tackled by employing a Genetic Algorithm (GA) as the tool for minimization of the discrepancy function. In view of the large number of direct analyses generally required by GA here again a POD-RBF procedure is used. The computations of snapshots are performed through the same FE model depicted in Fig. 24.17b, by a total of 768 analyses with: sur σmax from 500 MPa to 1060 MPa, step 80 MPa; σσmax from 0.2 to 0.8, step 0.2; d from 100 μm to 220 μm, step 40 μm; and m from 0.4 to 1.4, step 0.2. For each indentation test the two snapshot matrices are generated, U1 [162 × 768] collecting data from the residual imprint, and U2 [200 × 768] for the indentation curve. The POD bases are constructed for both snapshot matrices, by truncating the one for the residual imprint after 5th term, and the one for the indentation curve after 6th term. The inverse multiquadric RBF here selected reads: 1 gi (p) = > ||p − pi ||2 + r2

(24.8)

where r = 0.5 represents a smoothing coefficient. Results from three different GA optimizations are visualized in Fig. 24.18b. A comparison with those previously achieved, Fig. 24.18a, would suggest that GA is preferable with respect to TRA to the present purpose, in view of its insensitivity as for local minima. But both procedures for discrepancy function minimization turn out to require POD in practical applications.

24.4 Closing Remarks The two kinds of engineering situations dealt with in what precedes are representative of a growing area of inverse analysis applications where mechanical characterization of structural materials is important and may become crucial. Other issues of current research by our team concern parameter of free-foils (especially geo-membranes and paper) [1] and ceramics and coatings [23]. The advantages provided by recent developments in computational methodology (like POD-RBF and soft computing) combined with non-traditional experimental equipments and techniques, include enlargement of the identifiable parameters set, and saving in resources and time, like e.g. when the whole procedure becomes performable “in situ” rather than in laboratory. Such advantages turned out here to be reachable by the flat-jack and dilatometric techniques for local diagnosis on concrete dams, superficially and in depth (Sect. 24.2) and by the indentation techniques applicable to industrial plants in order to identify material constitutive parameters and residual stresses (Sect. 24.3). Deterministic approaches only have been adopted here in both sections. However, similar computational advantages due to techniques like POD and RBF exploiting

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response correlation (Sect. 24.1) are reasonably expected also with stochastic approaches (i.e. assessment not only of estimates, but also of their covariance matrix which quantifies their uncertainties due to the random noise on experimental data). Severe computing efforts are likely to be mitigated also, and even more, in practical applications of stochastic techniques, like Monte Carlo, Bayesian and Kalman parameter identifications. Sensitivity analysis (see e.g. [21]) is another meaningful subject for parameter estimation which was considered herein only in passing. The design of the experiment and in particular, the selection of the measurable quantities to be actually measured in the tests, rests primarily on numerical derivatives of them with respect to the sought parameters: hence, once again, interpolations instead of “ex novo” direct analyses are economically attractive, even if sensitivities are to be assessed once for all, rather then routinely. The two inverse analysis problems and the relevant methodological novelties herein considered in Sect. 24.2 and Sect. 24.3 are susceptible to the following future developments, which are subject of research in progress. The flat-jack method described in Sect. 24.2 might be advantageously applied to periodic masonry if combined with homogenization procedure. The indentation techniques dealt with in Sect. 24.3 will be investigated in dynamics as well; in fact a dynamical noninstrumented indenter (like for traditional hardness tests by impact) would be by far more economical than the quasi-static ones to use for nondestructive tests “in situ”, since no contrast would be required in order to generate an imprint to analyze. Finally, another promising prospect might be represented by dynamical indentation employed to mechanically characterize bricks and mortar separately in periodic masonry, in order to calibrate constitutive models for overall structural analysis by a multi-scale approach. Acknowledgements. Research by our team in structural engineering of large dams (Sect. 24.2) and on diagnostic analysis of pipelines (Sect. 24.3) are supported by the Italian Ministry of University and by the companies Venezia Tocnologie and ENI, respectively. The authors thank the students (in particularly Ms Ada Zirpoli) who contributed by thesis works several computational exercises useful for the validation of the methodology presented herein.

References [1] Ageno, M., Bolzon, G., Maier, G.: Mechanical characterisation of free-standing elastoplastic foils by means of membranometric measurements and inverse analysis. Structural and Multidisciplinary Optimization 38, 229–243 (2009) [2] Ahmed, T., Burley, E., Rigden, S., Abu-Tair, A.: The effect of alkali reactivity on the mechanical properties of concrete. Construction and Buliding Materials 17, 123–144 (2003) [3] Amadei, B., Stephansson, O.: Rock stress and its measurements. Champion and Hall, London (1997)

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[4] Ardito, R., Maier, G., Massalongo, G.: Diagnostic analysis of concrete dams based on seasonal hydrostatic loading. Engineering Structures 30, 3176–3185 (2008) [5] Bocciarelli, M., Maier, G.: Indentation and imprint mapping method for identification of residual stresses. Computational Materials Science 39, 381–392 (2007) [6] Bolzon, G., Bocciarelli, M., Chiarullo, E.J.: Mechanical characterisaion of materials by microindentation and AFM scanning. In: Bhushan, B., Fuchs, H. (eds.) Applied Scanning Probe Methods XII - Characterization. Springer, Berlin (2008) [7] Bolzon, G., Maier, G., Panico, M.: Material model calibration by indentation, imprint mapping and inverse analysis. International Journal of Solids and Structures 41, 2957– 2975 (2004) [8] Buhmann, M.D.: Radial Basis Functions. Cambridge University Press, Cambridge (2003) [9] Bui, H.D.: Inverse problems in the mechanics of materials: an introduction. CRC Press, London (1994) [10] Chatterjee, A.: An introduction to the proper orthogonal decomposition. Current Science 78(7), 808–817 (2000) [11] Coleman, T.F., Li, Y.: An interior trust region approach for nonlinear minimization subject to bounds. SIAM Journal of Optimization 6(2), 418–445 (1996) [12] Comi, C., Fedele, R., Perego, U.: A chemo-thermo-damage model for the analysis of concrete dams affected by alkali-silica reaction. Mechanics of Materials 41(3), 210–230 (2009) [13] Fedele, R., Maier, G.: Flat-jack test and inverse analysis for the identification of stress states and elastic properties in concrete dams. Meccanica 42, 387–402 (2007) [14] Fedele, R., Maier, G., Miller, B.: Identification of elastic stiffness and local stresses in concrete dams by in situ tests and neural networks. Structure and Infrastructure Engineering 1(3), 165–180 (2005) [15] Fedele, R., Maier, G., Miller, B.: Health assessment of concrete dams by overall inverse analyses and neural networks. Int. J. of Fracture 137, 151–172 (2006) [16] Giannessi, F.: Constrained optimization and image space. Springer, New York (2005) [17] Gregorczyk, R., Lourenco, P.B.: A review on flat-jack testing. Engenharia Civil 9, 39– 50 (2000) [18] Gu, Y., Nakamura, T., Prchlik, L., Sampath, S., Wallace, J.: Micro-indentation and inverse analysis to characterize elastic-plastic graded materials. Materials Science and Engineering 345, 223–233 (2003) [19] Haykin, S.: Neural networks: A Comprehensive Foundation, 2nd edn. Prentice Hall, Englewood Cliffs (1998) [20] James, M.N., Hughes, D.J., Chen, Z., Lombard, H., Hattingh, D.G., Asquith, D., Yates, J.R., Webster, P.J.: Residual stresses and fatigue performance. Engineering Failure Analysis 14, 384–395 (2007) [21] Kleiber, M., Antunez, H., Hien, T.D., Kowalczyk, P.: Parameter sensitivity in non-linear mechanics. John Wiley, Chichester (1997) [22] Ly, H.V., Tran, H.T.: Modeling and control of physical processes using proper orthogonal decomposition. Mathematical and Computer Modeling 33, 223–236 (2001) [23] Maier, G., Bocciarelli, M., Bolzon, G., Fedele, R.: Inverse analyses in fracture mechanics. Plenary Lecture, 11th International Conference on Fracture. Int. J. of Fracture 138, 47–73 (2006) [24] McClung, R.C.: A literature survey on the stability and significance of residual stresses during fatigue. Fatigue & Fracture of Engineering Materials and Structures 30(3), 137– 205 (2007)

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[25] Oliver, W.C., Pharr, G.M.: An improved techniques for determining hardness elastic modulus using load and displacement sensing indentation experiments. Journal of Materials Research 7, 176–181 (1992) [26] Ostrowski, Z., Białecki, R.A., Kassab, A.J.: Estimation of constant thermal conductivity by use of Proper Orthogonal Decomposition. Computational Mechanics 37, 52–59 (2005) [27] Ostrowski, Z., Białecki, R.A., Kassab, A.J.: Solving inverse heat conduction problems using trained POD-RBF network. Inverse Problems in Science and Engineering 16(1), 705–714 (2008) [28] Réthoté, J., Hild, F., Roux, S.: Extended digital image correlation with crack shape optimization. International Journal for Numerical Methods in Engineering 73, 248–272 (2008) [29] Waszczyszyn, Z.: Neural networks in the analysis and design of structure. Springer, Wien, New York (1999) [30] Zhao, F., Huang, Y., Hwan, K.C.: Determination of uniaxial residual stress and mechanical properties by instrumented indentation. Acta Materialia 54(10), 2823–2832 (2006)

Chapter 25

Optimization of Marine Propulsion System’s Alignment for Aged Ships Lech Murawski and Wieslaw Ostachowicz

Abstract. The paper presents a numerical method of identification parameters of marine power transmission system’s alignment in the case of the lack of producers’ data. Proper shaft line alignment is often a problem for repair shipyards, for aged ships without sufficient documentation. The authors have proposed a combined experimental - analytical method for identification of some existing parameters and checking (and eventually correcting) power transmission system’s foundation. The specialised software has been developed for shaft line alignment calculations with influence coefficients. The example of analysis has been performed for cargo ship with medium-speed main engine and the second one with slow-speed propulsion system. Multivariant computations supported by measurements of the ships’ shaft line have been investigated.

25.1 Introduction Propulsion system’s foundation might be changed dangerously during ship hull repairing (especially welding works) even if there is no work with power transmission system. What is more, improvement of the shaft line - crankshaft alignment is difficult because very often aged ships haven’t available documentations. Repair shipyard does not know if foundation parameters of propulsion system are acceptable or not. Some of them (like shafts’ diameters, intermediate bearings’ reactions) are easy to measured but the others are practically inaccessible (shaft line deformation, shaft line - crankshaft interaction, stern tube and main engine bearings’ reactions, Lech Murawski Institute of Fluid Flow Machinery, Polish Academy of Sciences, ul. Fiszera 14, 80-952 Gdansk, Poland e-mail: [email protected] Wieslaw Ostachowicz Institute of Fluid Flow Machinery, Polish Academy of Sciences, ul. Fiszera 14, 80-952 Gdansk, Poland e-mail: [email protected] M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 477–491. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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Fig. 25.1 Typical model of the power transmission system isolated from ship hull

stresses). The analysis becomes much more difficult if our software takes into account isolated power transmission system from ship hull (see fig. 1) without boundary conditions (e.g. without hull displacements and stiffness) [9]. Usually, similar parameters of shaft line alignment before and after repair is a shipyard’s target. There are no well known cheap methods for inspection and eventually improvement of propulsion system foundation after ship hull repairing [13]. Anormalities of the propulsion system’s working parameters are often detected after ship’s repairing, during sea trial or even after certain period of ship’s voyage. Then, the failure repairing is very costly. Sometimes, tube of stern bearing or engine’s main bearing must be replaced and the proper shaft line alignment must be performed. The authors have proposed a method of shaft line alignment identification based on easy to measured parameters and numerical analysis. Elaborated specialised software can calculate influence coefficients for each bearing. Shaft line alignment

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identification should be performed before and after repairing process. Analysis shows if correction is necessary (if shaft line alignment parameters have been changed significantly during repairing). If yes, analysis, based on influence coefficients, gives us advice how to improve easily propulsion system’s foundation.

25.2 Methodology of Shaft Line Alignment Proper shaft line alignment is one of the most important actions during the design of the propulsion system [15]. It consists of determining the location of the main engine driving axis, intermediate bearings axis and stern tube bearing axis. The propulsion system bearings are usually mutually moved in the vertical plane. An example of the typical shaft line alignment of a container ship is shown in fig. 2. The shaft line alignment in horizontal plane is applied quite rarely (only in special ships or navy ships). Appropriate loadings of the shaft line and crankshaft bearings are the main targets of the shaft line alignment. Firstly, reactions in stern tube and intermediate bearings do not have to be too high or too small in all the propulsion’s service conditions. In the case of low static bearings’ reactions, the possible influence on lateral vibrations

Fig. 25.2 Shaft line alignment of a container ship

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(dynamic reactions) should be considered. If the static reaction is similar to the dynamic one, the loading direction might be changeable. Impact loading might be the cause of a quick deterioration of the bearing. All the bearings should be equally loaded. The Sommerfeld number may be a good parameter to characterise the bearings load. After assuming that Sommerfeld’s numbers should be equal in all the power transmission systems bearings (as well as all the revolutions and viscosity being the same in all the bearings). A low bearings loading is dangerous especially for the intermediate bearings (ship hull deformation may be the cause of decreasing relatively small reactions). Therefore, it is reasonable that the Sommerfeld number of the intermediate bearings are sometimes 30-50% greater than other bearings. The loading distribution of the stern tube bearing is another important problem, which should be taken into consideration. The stern tube bearing is the heaviest loaded bearing as it is relatively long. What is more, loading is strongly asymmetrical (by the propeller’s forces) so part seizure of the bearings is a danger. Therefore, the reactions distribution should be checked, for instance by comparing reactions on both the bearing edges (both reactions should be positive even with the dynamic component). Checking the relative deflection between the line of the journals and tube axis is an alternative method. In this method deformation of the shaft line, as well as the ship hull has to be determined. The crankshaft’s loadings by shear force and the bending moment, coming from the shaft line, must be appropriate. All the world producers of marine engines, define the allowable area of the crankshaft’s loading. An example of Sulzer’s allowable forces and moments is presented in fig. 3. The crankshaft’s shear force and bending moment have to be included in the allowable area in all the propulsion’s service conditions. Recently this requirement has changed to another: reactions, coming from shaft line’s loading, in the engine main bearings are given. In this case the model of the crankshaft must be more detailed. Calculation with the propeller thrust includes the moment inducted by the eccentricity of the thrust bearing. Calculations with maximum propeller thrust provide low or zero load for aft most, main bearing no. 1 (driving end) at alignment condition (i.e. ship hull deformation not considered). The thermal rise of the engine main bearing is not sufficient to avoid the a.m. substantial load reduction for main bearing no. 1. This is acceptable because the draught related ship hull bending, which is involved until the full thrust is available, will have shifted the static load from main bearing no. 2 to aft most no. 1. Finally, the bending stresses of the shaft line in the given final alignment, should be checked. Usually (for most typical marine power transmissions systems) these quasi-static stresses are not very high for all service conditions. Relatively large shaft line diameters are determined by the torsional vibration requirements. The most popular method used for shaft line alignment calculations is the Finite Element Method. The shaft line is modelled by linear beam elements. According to world producers of marine engines, the model of a crankshaft is simplified: there are fewer beams in the line arrangement (in the model there is no geometry of the cranks). The weight of the propeller and shafts, hydrodynamic forces acting on the propeller and the other forces coming from the propulsion system are modelled by

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Fig. 25.3 Sulzer’s allowable area of crankshaft loading

static, nodal forces and moments. Thermal expansion of the main engine body is modelled as a vertical movement of the main bearings. The value of this movement is specified by the engine’s manufacturers. Calculations have to be done for all the typical ships propulsion systems service conditions. A power transmission system working in nominal conditions (with hydrodynamic forces in a nominal propulsion speed and a "hot" main engine) is the most important variant. Parameters of a non-working propulsion system with a "hot" and a "cold" main engine must also be checked. Calculations must also be prepared for a disconnected power transmission system (during instalment). A shift of the shafts necks (SAG and GAP) has to be defined. As was already mentioned, beam model of the power transmission system is isolated from the ship hull. Therefore, determining the correctness of the boundary conditions is one of the most important, difficult and debated, during the marine power transmission systems static and dynamic calculations. Shaft line alignment and lateral vibration analysis are especially sensitive to proper boundary conditions’ determination. In the author opinion, stiffness and damping characteristics the of journal bearings oil film, ship hull and bearings frame should be taken into account

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[7]. Ship hull deformations, under different load conditions and regular sea waves, should also be analysed.

25.3 Assumption Influence on Analysis Results The influence of different boundary conditions on the shaft line alignment analysis has been discussed. The power transmission system (working in nominal conditions) of the container 2000 TEU was analysed. The following five variants of calculations have been done: 1. a classical model - all bearings are modelled as a pointwise, ideal stiff support; 2. a model similar to classical - the stern tube bearing is continuous but still ideally stiff; 3. all bearings are modelled as a pointwise with supports’ elasticity (uncoupled stiffness of a hull structures and oil film); 4. a stern tube bearing is modelled as a continuous support, the rest assumptions are the same as in variant no. 3; 5. a substitute stiffness of the hull structure is used, the rest of the assumptions are the same as in variant no. 4. Shaft line vertical absolute deformation and bearings’ static reactions, for the variants mentioned above, are shown in fig. 4 and 5. No. 1 and 2 mark stern tube bearing (stern and aft edge), no. 3 - intermediate bearing, no. 4, 5 and 6 - first of the main engine bearings. An especially large analysis assumption effect may be observed on the crankshaft flange’s loading. The crankshaft’s bending moment changes from 119 kNm (for an ideal stiff pointwise support) to 30 kNm. The shear force changes from 72 kN to 83 kN. All these effect should be taken into account during shaft line alignment identification.

Fig. 25.4 Shaft line deformation in different analysis assumptions

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Fig. 25.5 Shaft line deformation in different analysis assumptions

25.4 Identification Method Just before starting hull repairing, some typical measurements [1,5,6] should be performed. First of all, the jack-up test should be realized for intermediate bearings’ reaction identification. Also identification of shafts’ diameters is necessary. In the same time, the base of location of bearings body and shaft line position should be done. Position changes of the shaft line can be controlled by bending stresses measurement (fig. 6 - e.g. by strain gauge technique) [2,3]. Bending stresses should be recorded during full rotation of the shaft (with usage of turning gear) just before and after ship repair process. Changes of the stress level in the horizontal and vertical plane may be simulated by numerical analysis and then changes of the shaft line alignment may be identified.

Fig. 25.6 Bending stress measurement of the shaft line

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The relative position of bearing body might be based on measuring stress level [8] on foundation pads or on the other sensitive element of the bearing foundation (fig. 7). Stress level should be recorded during jack-up test, before ship repair process. The simultaneous measurements can be treated as calibration process. After measurements the dependence between bearing reaction and stress level of the foundation pads (or foundation knee) is recognised. The same measurements (even without jack-up test) performed after ship repairing, give us changes of the bearing reaction which are induced during overhaul.

Fig. 25.7 Stress measurement of bearing’s foundation pads and knee

Whole (before and after repairing process) measurements investigation gives us necessary data for shaft line alignment’s changes identification. The next step during expertises must be an assessment if identified changes are acceptable or not. The following limitations should be checked: loadings of the stern tube bearing, intermediate bearings and main bearings of the engine; stresses of the shaft line; interaction between shaft line and crankshaft. The shaft line alignment correction should be proposed if some limits are exceeded. Correction should be practicable and cheap. Generally, it should be restricted to displacements of the intermediate bearings in horizontal and vertical plane. Authors made a specialised computer program which can be useful for all this tasks (identification, assessment and correction) with limited (without design documentation) data availability.

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25.5 Software Description An assessment of the shaft line alignment for aged ships requires specialised software. The data for the software might be based on the measurements [5,6] before and after ship’s repairing in case of insufficient design data availability. During ship hull repairing power transmission system’s axis might be unacceptable displaced. Assessment of the shaft line alignment before ship repairing process as well as determination of propulsion system’s sensitivity on stable hull deformation is very important from technical and economical point of view. The authors performed specialised software based on Finite Element Method for bearing reactions’ identification and shaft line alignment’s influence coefficients determination. The identification of shaft line alignment with knowledge only of bearings’ reaction and shafts’ geometry is an inverse mathematical problem [4]. This problem is often ambiguous with multi solutions. For that reason the motion matrix equation (with taking into account geometric stiffness matrix) [14] is solved during iteration process. The modified Gauss algorithm is used for solving the equations. The shaft line is modeled by 2-node (with 6 degree of freedom for each node) linear beam elements with nonlinear boundary conditions. The beams can be cylindrical or conical and drilled. The bearings can be modelled by point wise or continuous support with taking into account bearing’s clearance and stiffness. Stiffness of the continuous bearing (e.g. stern tube bearing) is modelled by polynomial of 10th order. In the program there are procedures for field solution’s searching. Finding out variants fulfilling measured values is a target of the algorithm. All variables are defined in the program as dynamic one so there are no limitations for number of elements and bearings. An example of multi bearings shaft line is presented in Fig. 8.

Fig. 25.8 Shaft line model of two shaft’s universal supply ship

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25.6 Analysis Example for the Container Ship As a first step, the analysis has been performed for the container ship with typical propulsion system - constant pitch propeller driven directly by slow speed main engine. An example analysis has been made for container ship 4500 TEU. The hull stiffness characteristics on the base of FEM ship model analyses [10] like presented in fig. 9. This kind of the model contains about 50 000 œ 150 000 degrees of freedom. The methods of the characteristics determination of ship hull and main engine have been presented in the other authors’ papers [11, 12]. Fig. 25.9 FEM model of container ship

Analysis is performed on the basis of standard shipyard’s shaft line alignment. This alignment is correct but their parameters might be better. The aim of the calculations is showing software possibilities. Standard shaft line reactions are presented in fig. 10.

Fig. 25.10 Container’s shaft line reactions determined by the software for standard alignment

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The standard shaft line alignment might be improved by equalization of the aft stern tube bearing’s reaction and by reduction of the aft engine’s main bearing. Loading enlargement of the fore stern tube bearing (and aft stern tube bearing’s loading equalization) is a consequence of reaction reduction of the aft intermediate bearing (to 100 kN, 24 %). Resultant skew of the propeller shaft in the bearing tube is then decreased of about 8 %. Loading reduction of the main bearings (of about 9 %) is an effect of reaction increasing (to 300 kN) of fore intermediate bearing. The algorithm find out automatically new (for given intermediate bearings’ reactions) shaft line alignment. The modified parameters of the new shaft line alignment are presented in fig. 11.

Fig. 25.11 Modified shaft line alignment

Influence coefficients are also determined by the program. The coefficients are useful for designers because they show the relative changing of the shaft line parameters (shaft line deformation, reactions, bending moments, shear forces, stresses) if chosen bearing is moved vertically or longitudinally. An example of influence coefficients for vertical movement of aft intermediate bearing are presented on fig. 12. Influence coefficients of the bearings reaction under the influence of bearings’ longitudinal and vertical movements are also determined. The analysed shaft line is relatively stiff - it is short with big diameter and the propulsion system is equipped in many lateral bearings. For that reason, displacement of one bearing have an influence on only one shaft segment limited by adjacent bearings. Longitudinal movement of the intermediate bearings has hardly perceptible influence on the analysed shaft line alignment parameters.

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Fig. 25.12 Influence coefficients of shaft line deformation and stresses under the influence of vertical movement of aft intermediate bearing

25.7 Identification of the Middle-Speed Propulsion System The identification has been performed for aged ship destined for heavy repair process. It is universal supply ship with double-shaft, middle-speed propulsion system (with gear box). The shaft line is relatively flexible: 282 is a diameter of intermediate shaft and 400 is a diameter of propeller shaft. In the ship’s documentation, only design bearings’ reactions are available. Real reactions are known after measurements. Design and real shaft line alignment have been identified by the discussed software. Identified shaft line reactions are presented in fig. 13 and 14. After first step of identification, significant difference between design and real reactions are observed. The most important differences (which needed improvements) are as follows: unloaded aft edge and overloaded fore part of stern tube bearing and insufficiently loading of stern intermediate bearing.

Fig. 25.13 Design shaft line alignment and reactions

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Fig. 25.14 Real shaft line alignment and reactions

The authors’ software determine given shaft line alignment as a first step, and searching the field of optimal solutions in the second step. In the third step the influence coefficients are determined by the program. The coefficients are useful for fast and precisely determination of proper bearings position in vertical and longitudinal plane. Target of the shaft line alignment is determining proper and uniform bearings reaction, acceptable stress level and good interaction between shaft line and crankshaft or gearbox [15]. Proper shaft line alignment has been determined by software lift of the intermediate bearings. Improved bearings reaction distribution is presented in Figure 15. Comparison of the shaft line alignment’s parameters, before and after improvements, is shown in Figure 16. The shaft line alignment after correction is characterised by smoother axis deformation and uniform bearings reaction. Intermediate bearing no 1 is the heaviest loaded by the others because of desirable loading of stern tube bearing. After correction loading of the fore edge of stern tube bearing is reduced two times, while loading of aft edge become acceptable. Before correction the aft edge of stern tube bearing was unloaded and was threaten by hammering phenomenon.

25.8 Summary Changes of shaft line alignment of aged ship with insufficient data availability have been difficult for proper realisation with standard methodology. Shipyards are looking for well experienced workers because shaft line alignment has been performed by tests and errors method. Usually, it is very costly and the cost of the process is depended on engineer’s experience. Sometimes, repair shipyards make an assumption that bearing reaction deviation 50% is acceptable.

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Fig. 25.15 Shaft line alignment after correction

Fig. 25.16 Shaft line deformations before and after correction

In the paper the authors have proposed a quite simple and relatively cheap method for identification of the shaft line alignment parameters in the case of the lack of producers’ data. It is proposed combined experimental-analytical method for identification and correcting some existing parameters of power transmission system’s foundation. The specialised software has been developed and verified for shaft line alignment

References [1] Charchalis, A., Grza¸dziela, A.: Diagnosing the shafting of alignment by means vibration measurement. ICSV Congress, Garmisch-Partenkirchen (2000) [2] Cowper, B., DaCosta, A., Bobyn, S.: Shaft alignment using strain gauges. Marine Technology 36(2), 77–91 (1999)

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[3] Keshava Rao, M.N., Dharaneepathy, M.V., Gomathinayagam, S., Ramaraju, K., Charkravorty, P.K.: Computer-aided alignment of ship propulsion shafts by strain-gage methods. Marine Technology 28, 84–90 (1991) [4] Kiełbasi´nski, A., Schwetlick, H.: Numeryczna Algebra Liniowa, Wydawnictwo Naukowo-Techniczne, Warszawa (1992) [5] Murawski, L.: Analysis of measurement methods and influance of propulsion plant working parameters on ship shafting alignment - part I. Polish Maritime Research No 2(12), 4, 19–22 (1997) [6] Murawski, L.: Analysis of measurement methods and influance of propulsion plant working parameters on ship shafting alignment - part II. Polish Maritime Research No 3(13), 4, 15–20 (1997) [7] Murawski, L.: Influence of journal bearing modelling method on shaft line alignment and whirling vibrations. In: PRADS - Eighth International Symposium on Practical Design of Ships and Other Floating Structures, Shanghai-China, pp. 1205–1212. Elsevier, Amsterdam (2001) [8] Murawski, L.: Static and Dynamic Analyses of Marine Propulsion Systems. Oficyna Wydawnicza Politechniki Warszawskiej, Warszawa (2003) [9] Murawski, L.: Shaft line alignment analysis taking ship construction flexibility and deformations into consideration. Marine Structures No 1, 18, 62 (2005) [10] Murawski, L.: Statyczno-dynamiczne charakterystyki pracy okrêtowych uk¸sadów napêdowych i ich wp¸syw na drgania konstrukcji kad¸subów i nadbudówek statków. Zeszyty Naukowe Instytutu Maszyn Przep¸sywowych Polskiej Akademii Nauk, Gdansk (2006) [11] Murawski, L.: Ship structure dynamic analysis - effects of made assumptions on computation results. Polish Maritime Research No 2(48), 13, 3–7 (2006) [12] Murawski, L., Szmyt, M.: Stifness characteristics and thermal deformations of the frame of high power marine engine. Polish Maritime Research No 1(51) 14, 16–22 (2007) [13] Pressicaud, J.P.: Correlation between theory and reality in alignment of line shafting. Bureau Veritas, Paris (1986) [14] Zienkiewicz, O.C., Taylor, I.R.L.: The Finite Element Method. McGraw-Hill, London (1992) [15] Shafting alignment for direct coupled low-speed diesel propulsion plants. MAN B& W Diesel A/S, Copenhagen (1995)

Chapter 26

Experimental-Numerical Assessment of Impact-Induced Damage in Cross-Ply Laminates Gigliola Salerno, Stefano Mariani, Alberto Corigliano, Francesco Caimmi, Luca Andena, and Roberto Frassine

Abstract. In this work we provide an experimental-numerical investigation of impact-induced failure of layered composites. An experimental campaign, consisting of multiple drop tests, was performed to determine the delamination threshold (in the low-velocity regime) and to assess how, beyond this limit, failure mode is affected by the impact energy. A three-dimensional finite element model of the laminated composite was built; in the numerical model complex delamination events are simulated by means of a rate-dependent damage interface law. Numerical simulations of the whole impact event are directly compared to experimental results and then critically discussed.

26.1 Introduction Impacts on layered composites can be a major source of failure [1, 18], especially in aeronautical and aerospace structures subjected to debris strikes. In view of the complex propagation paths of stress waves arising close to the impact location, both inter-laminar and intra-laminar damaging phenomena may show up. In case of lowvelocity (and, therefore, low-energy) impacts, strength reduction and eventual failures are dominated by inter-laminar debonding processes, customarily termed delamination; in case of high-velocity (and high-energy) impacts, local perforation and spalling may affect the failure mode. Gigliola Salerno, Stefano Mariani, Alberto Corigliano Politecnico di Milano, Dipartimento di Ingegneria Strutturale, Piazza L. Da Vinci 32, 20133 Milano, Italy e-mail: [email protected], [email protected], [email protected] Luca Andena, Francesco Caimmi, Roberto Frassine Politecnico di Milano, Dipartimento di Chimica, Materiali e Ingegneria Chimica "Giulio Natta", Piazza L. Da Vinci 32, 20133 Milano, Italy e-mail: [email protected], [email protected], [email protected]

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Focusing on carbon-epoxy cross-ply laminates, in this work we provide an experimental investigation and a numerical characterization of some of the aforementioned local failure phenomena. Impact tests were performed in a drop-weight tester: mass and initial velocity of a spherical projectile were initially tuned so as to capture threshold values corresponding to the inception of delamination. To observe delamination growth as well as inception and propagation of intra-laminar damage up to failure, mass and/or velocity of the spherical projectile were subsequently increased. To model this kind of progressive failure and in order to fully account for the inception and growth of delamination, a softening interface constitutive law was adopted in the delamination-prone region to link tractions to displacement jumps along all the inter-laminar surfaces [2, 8, 9, 11, 17]. The interface model was identified on the basis of ad-hoc performed double cantilever beam (DCB) and end notched flexure (ENF) tests, and adopted in a fully three-dimensional finite element model to simulate the low-velocity impact tests. The present Chapter is organized as follows: in Section 26.2 the experimental results concerning DCB, ENF and low velocity impact tests are presented and discussed; Section 26.3 is devoted to the description of the adopted numerical model. The results of numerical simulations and the comparison with experimental findings are discussed in Section 26.4; some closing remarks are contained in the closing Section 26.5.

26.2 Experimental Campaign A preliminary campaign of standard tension and torsion tests has been first conducted on thin (length/height = 70) specimens to assess the elastic properties of a single lamina [12]. The relevant results, along with the mass density of the composite layer, are collected in Table 26.1. Table 26.1 Average elastic properties of the studied carbon/epoxy composite E11 (MPa)

E22 (MPa)

G12 (MPa)

G23 (MPa)

ν12

ρ (kg/m3 )

171430

7523

3560

3277

0.3

1626.5

Then, DCB and ENF tests have been performed to characterize the mode I and mode II inter-laminar fracture toughnesses, in case of either 0◦ /0◦ or 0◦ /90◦ interfaces1 . The compliance method [16, 19] has been adopted for data reduction, so as 1

A proper characterization of the mode-mixity in the neighbourhood of the tip of a crack laying along the 0◦ /90◦ interface would require the specification of the argument of the (complex) stress intensity factor, and the use of terms like mode I and mode II is slightly improper for such an interfacial crack. For easiness and consistency of notation we would nevertheless adopt them to define the mixity in the performed DCB and ENF tests.

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to compute the critical energy release rates according to ISO15024, ASTMD790, ASTMD5528 standards [3, 4, 13]. Rectangular specimens were produced, featuring a [0◦ /0◦ /90◦ /0◦ /0◦ /90◦ /0◦ /90◦ /90◦ /0◦ /0◦ /0◦ ] stacking sequence. The resulting samples were 4 mm thick and 210 mm long. A 50 μ m thick Teflon film was inserted on the midplane to create an initial delamination. Test results, in terms of toughness at the inception of delamination, are reported in Table 26.2. DCB results were obtained from a mode I pre-crack, while the ENF results were obtained directly from the inserted film as the unstable behavior of this test hampered the obtainment of mode II pre-cracks. For the DCB specimens, a significant increase of fracture toughness as a function of the delamination growth was observed; Figure 26.1 shows the results of DCB tests on 0◦ /90◦ interfaces. Post-mortem fractography of the delaminated area in DCB specimens (Figure 26.2) shows that the 90◦ layer acts as an obstacle to the delamination growth, with remarkable bridging between crack faces, see Figure 26.3. As to the ENF tests with 0◦ /90◦ interfaces (Figure 26.4) typical hackle marks, characteristic of shear failure in polymer composites [20], can be seen in both the 0◦ and the 90◦ laminae. In the latter, however, the fracture surfaces also show different fracture mechanisms involving extended debond and little matrix permanent deformation. This can help in explaining the strong differences in mode II toughness observed when moving from the 0◦ /0◦ interface to the 0◦ /90◦ interface. Table 26.2 Average fracture toughness values Interface

Toughness (J/m2 )

Value

0◦ /0◦

GIc GIIc GIc GIIc

239 2944 687 837

0◦ /90◦

2500

Toughness (N/m)

2000

Fig. 26.1 DCB tests, 0◦ /90◦ interface: G−resistance curves of the tested specimens.

1500

1000

500

0 70

90

110 130 150 Delamination length (mm)

170

190

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A series of low-velocity impact tests has been finally performed by striking circular laminated plates, fully constrained along their border, with a dart possessing an hemispherical tup (diameter 20 mm). To get a picture of the effects of impact energy on the delaminated area, both the striker mass and the free fall height were varied. Two cross-ply specimen configurations were tested, either 4 mm thick (with a stacking sequence [90◦,0◦ ]3S ) or 2.5 mm thick (with a stacking sequence [90◦ ,0◦ ]2S ). Examples of the recorded evolution in time of the impact force for both specimen thicknesses are shown in Figures 26.5 and 26.6, along with pictures showing the damage pattern on the outer laminae. In these plots, the delamination threshold load

Fig. 26.2 DCB tests, 0◦ /90◦ interface: typical fractography of the delaminated area (CPD represents the crack propagation direction).

Fig. 26.3 DCB tests, 0◦ /90◦ interface: lateral view of the specimen showing bridging between crack faces.

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is represented by an appreciable drop in the load graph, which testifies the inception of delamination growth [18].

26.3 Numerical Model Assuming intra-laminar damage to be negligible, impact-induced strength degradation processes mainly consist in delamination, i.e. inter-laminar micro-cracking eventually leading to debonding between laminae. Within a meso-mechanical framework, laminae are assumed to behave as linear elastic thin plates subject to coupled bending-shear deformation modes. These laminae are all bonded together in the initial configuration by the inter-laminar phases; since the thickness of such latter phases is small if compared to the whole thickness of the laminate, they are treated as lumped, zero-thickness surfaces. According to standard constitutive modeling for interfaces, in-plane membrane-like deformation modes are assumed not to affect the out-of-plane response of the resin-enriched phases; therefore, relationships linking tractions t, acting upon interface sides, to displacement discontinuities [u], occurring across the interface itself, need to be established. In order to deal with debonding, an elastic-damage constitutive law for the interlaminar surfaces is here adopted. To allow for finite strain effects in the post-impact simulations, an updated Lagrangian formulation is used; tractions t are thus to be

Fig. 26.4 ENF tests, 0◦ /90◦ interface: typical fractography of the delaminated area (CPD represents the crack propagation direction).

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considered as components of the Cauchy stress tensor in the current configuration, while displacement discontinuities [u] are to be considered as small jumps in

8000 7000

Force (N)

6000 5000 4000 3000 2000 1000 0 0

1

2

3

4

5

6

7

8

9

10

Time (ms)

Fig. 26.5 Low-velocity impact test on a [90◦ ,0◦ ]3S cross-ply laminate, impactor mass 14 kg, impactor velocity 1.5 m/s: damage pattern and evolution in time of the impact force.

12000 10000

Force (N)

8000 6000 4000 2000 0 0

1

2

3

4

5

Time (ms)

Fig. 26.6 Low-velocity impact test on a [90◦ ,0◦ ]2S cross-ply laminate, impactor mass 3 kg, impactor velocity 8.6 m/s: damage pattern and evolution in time of the impact force.

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the displacement field occurring across the interface along the whole deformation process. Considering a single interface, let us introduce a free energy ψ per unit surface according to [6, 7, 10]:

ψ=

1 (1 − d) K1 [u1 ]2 + K2 [u2 ]2 + K3 [u3 ] 2+ + K3 [u3 ] 2− 2

(26.1)

Here: [ui ], i=1,2,3, are the components of the displacement discontinuity vector aligned with the axes of a local interface reference frame, featuring x3 aligned with the out-of-plane direction; Ki are the interface stiffnesses; d is the damage variable; 2 + and 2 − respectively denote the positive and negative part of 2. In case of cross-ply laminates, symmetry leads to K1 = K2 . Through state equations, the components ti of the traction vector and the damage energy release rate Y are obtained as: ⎧ ⎫ (1 − d)K1[u1 ] ⎨ ⎬ ∂ψ (1 − d)K2[u2 ] = t= (26.2) ∂ [u] ⎩(1 − d)K [u ] + K [u ] ⎭ 3 3 + 3 3 − ∂ψ 1 = K1 [u1 ]2 + K2 [u2 ]2 + K3 [u3 ] 2+ (26.3) ∂d 2 For later use, we further define the three (single mode) components of the damage energy release rate Y as: Y =−

1 Y1 = K1 [u1 ]2 2

1 Y2 = K2 [u2 ]2 2

1 Y3 = K3 [u3 ] 2+ 2

(26.4)

A rate-dependent evolution law of the damage variable d is assumed and reads [6, 7, 10]: 8 @ 9n ? n Y − Y0 d= (26.5) n + 1 Yc − Y0 where:

1 Y = maxτ ≤τ (a1Y1 )α + (a2Y2 )α + (a3Y3 )α α

(26.6)

τ being time. In the above equations: α , ai and n are constitutive parameters; Y is the maximum achieved value of expression in (26.6), up to current time; Y0 is the elastic threshold value of Y ; Yc is the critical fracture energy. Parameters a1 , a2 and a3 define the ratios between critical fracture energies in case of single mode loadings, whereas α defines the shape of the elastic and critical domains in case of mixed-mode loadings. When damage is growing (d˙ > 0), the local constitutive law can be expressed ˙ where the superposed dot stands for time rate (namely in rate form as ˙t = EΓ [u], ˙t = ∂ t/∂ τ ) and the unsymmetric tangent stiffness matrix EΓ reads:

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EΓ = (1 − d)K − n

n 1 n + 1 Yc

n Y

n−1

K[u] ⊗ h

(26.7)

where ⊗ represents the outer product, K = diag{K1 K2 K3 }, and: 1 −1 (a1Y1 )α (a2Y2 )α (a3Y3 )α h = 2 (a1Y1 )α + (a2Y2 )α + (a3Y3 )α α [u1 ] [u2 ] [u3 ] (26.8) This model has been tested with data concerning low-velocity impacts on crossply laminates reported in [1], leading to close agreement with experimental outcomes.

26.4 Results Outcomes of the numerical simulations are here reported for the impact test on the [90, 0]3S laminate discussed in Section 26.2. The elastic moduli of each lamina have been already reported in Table 26.1, while the constitutive parameters adopted for the elastic-damage modeling of the inter-laminar surfaces are collected in Table 26.3. 8-node hexaedral elements have been used to discretize the laminae, whereas 4+4-node quadrilateral interface elements have been adopted for each inter-laminar surface. To assess the effects of the characteristic size of interface elements on the overall response of the laminate, analyses based on the three unstructured meshes shown in Figure 26.7 were run. Table 26.3 Elastic-damage model parameter values for a 0◦ /90◦ interface, see also [5, 14, 15]. K1 = K2 (N/m3 )

K3 (N/m3 )

a1 = a2

a3

n

Yc (J/m2 )

α

19845

1211

0.8

1.0

0.5

687

1.0

Figures 26.8 and 26.9 respectively depict the evolution in time of the contact force, namely of the force exerted by the striker on the specimen, and of the impactor kinetic energy. Both plots show a negligible effect of the mesh on the results. When compared to the experimental data, the force-time graphs display higher peak values, the absence of a signature of the delamination threshold load, and a reduced duration of the contact between striker and specimen, as testified by a premature reduction to zero of the contact force. This reduced duration of the contact can be seen also in Figure 26.9, with an early saturation to the eventual value of the impactor kinetic energy. All these results show that an accurate simulation of the ongoing impact-induced degradation process is in need to account for the intra-laminar damaging phenomena too.

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Figure 26.10 shows lateral views of the specimen, along with level sets of the out-of-plane displacement. The state of inter-laminar damage can be here inferred by the jumps in the contours beyond the completion of the test, i.e. at time τ = 11.5

Fig. 26.7 Impact test on a [90, 0]3S laminate: adopted unstructured meshes, featuring characteristic element size (A) 1.2 mm, (B) 0.65 mm, (C) 0.5 mm.

12500

Experimental data IDM - A 10000

IDM - B

Impact Force (N)

IDM - C 7500

5000

2500

0 0

0.002

0.004

0.006

0.008

0.01

Time (s)

Fig. 26.8 Impact test on a [90, 0]3S laminate: time evolution of the contact force (characteristic element size (A) 1.2 mm, (B) 0.65 mm, (C) 0.5 mm).

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ms; it can be observed that not all the surfaces have been completely damaged, since jumps occur among blocks of two/three laminae in the through the thickness direction.

26.5 Closing Remarks In this work an experimental-numerical investigation concerning impact induced failure in layered composites was discussed. Experimental results concerning DCB, ENF and low velocity impact tests were described. An elastic-damage interface model for the simulation of delamination was implemented. While the delamination behavior turned out to be correctly predicted by means of the proposed model, the correct representation of the whole behavior of laminates subject to impact loadings deserves the introduction of damage models for the layers. Acknowledgements. The contribution of MIUR Prin07 project 2007YZ3B24, Multi-scale problems with complex interactions in Structural Engineering is gratefully acknowledged. G. Salerno wishes to thank Prof. O. Allix for fruitful discussions during her permanence at LMT-Cachan, France.

18

Experimental data

16

IDM - A IDM - B

14

Impactor Kinetic Energy (J)

IDM - C 12 10 8 6 4 2 0 0

0.002

0.004

0.006

0.008

0.01

Time (s)

Fig. 26.9 Impact test on a [90, 0]3S laminate: time evolution of the kinetic impactor energy (characteristic element size (A) 1.2 mm, (B) 0.65 mm, (C) 0.5 mm).

26 Impact-Induced Failure in Layered Composites

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(a)

(b)

(c)

(d)

(e)

(f)

Fig. 26.10 Impact test on a [90, 0]3S laminate: level sets of out-of-plane displacement at time (a) τ =0.04 ms, (b) τ =0.52 ms, (c) τ =1 ms, (d) τ =2 ms, (e) τ =3 ms, (f) τ =11.5 ms.

References [1] Abrate, S.: Impact on Composite Structures. Cambridge University Press, Cambridge (1998) [2] Alfano, G., Crisfield, M.A.: Finite element interface models for the delamination analysis of laminated composites: Mechanical and computational issues. Int. J. Numer. Methods Engrg. 50, 1701–1736 (2001) [3] American Society for Testing and Materials (ASTM). Standard test method Flexural properties of unreinforced and reinforced plastics and electrical insulating materials. Standard D, 790-07 edition (2007) [4] American Society for Testing and Materials (ASTM). Standard test method for Mode I interlaminar fracture toughness of unidirectional fiber-reinforced polymer matrix composites. Standard D, 5528-01 edition (2007)

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[5] Allix, O.: A composite damage mesomodel for impact problems. Compos. Sci. Technol. 61, 2193–2205 (2001) [6] Allix, O., Corigliano, A.: Modeling and simulation of crack propagation in mixedmodes interlaminar fracture specimens. Intl. J. of Fracture 77, 111–140 (1996) [7] Allix, O., Ladevèze, P.: Interlaminar interface modelling for the prediction of delamination. Compos. Struct. 22, 235–242 (1992) [8] Corigliano, A.: Formulation, identification and use of interface elements in the numerical analysis of composite delamination. Int. J. Solids Struct. 30, 2779–2811 (1993) [9] Corigliano, A.: Damage and fracture mechanics techniques for composite structures. In: Milne, I., Ritchie, R.O., Karihaloo, B. (eds.) Comprehensive Structural Integrity, ch. 9, vol. 3, pp. 459–539. Elsevier Science, Amsterdam (2003) [10] Corigliano, A., Allix, O.: Some aspects of interlaminar degradation in composites. Comput. Methods Appl. Mech. Engrg. 185, 203–224 (2000) [11] Corigliano, A., Mariani, S.: Simulation of damage in composites by means of interface models: parameter identification. Compos. Sci. Technol. 61, 2299–2315 (2001) [12] Herakovich, C.T.: Mechanics of fibrous composites. John Wiley and Sons Inc., New York (1998) [13] International Organization for Standardization (ISO). Fibre-reinforced plastic composites - Determination of mode I interlaminar fracture toughness, GIc , for unidirectionally reinforced materials. Standard E, 15024:2001(e) edition (2001) [14] Ladevèze, P.: A damage computational approach for composites: basic aspects and micromechanical relations. Computational Mechanics 17, 142–150 (1995) [15] Ladevèze, P., Allix, O., Deü, J., Lévêque, D.: A mesomodel for localisation and damage computation in laminates. Comput. Methods Appl. Mech. Engrg. 183, 105–122 (2000) [16] Laksimi, A., Benzeggagh, M.L., Jing, G., Hecini, M., Roelandt, J.M.: Mode I interlaminar fracture of symmetrical cross-ply composites. Compos. Sci. Technol. 41, 147–164 (1991) [17] Mariani, S., Corigliano, A.: Impact induced composite delamination: state and parameter identification via joint and dual extended Kalman filters. Comput. Methods Appl. Mech. Engrg. 194, 5242–5272 (2005) [18] Schoeppner, G.A., Abrate, S.: Delamination threshold loads for low velocity impact on composite laminates. Composites A 31, 903–915 (2000) [19] Yang, Z., Sun, T.: Interlaminar fracture toughness of a graphite/epoxy multidirectional composite. J. Engrg. Materials Technol. 122, 428–433 (2000) [20] O’Brien, T.K.: Composite interlaminar shear fracture toughness, GIIc : shear measurement or sheer myth? NASA technical memorandum 110280 (1997)

Chapter 27

Finite Element Modeling of Stringer-Stiffened Fiber Reinforced Polymer Structures Werner Wagner and Claudio Balzani

Abstract. Within this contribution an approach for the simulation of stringerstiffened composite panels based on the finite element method is developed. Layered shear-elastic shell elements allowing for an arbitrary stacking sequence and a variable location of the reference plane along the thickness are used to model the preand postbuckling response of the composite. A ply discount model with constant knock-down factors is employed to account for ply failure which can occur as fiber fracture, matrix cracking, or fiber-matrix shear failure. Delamination, and particularly debonding of the stiffener from the skin, is modeled via interface elements in which the cohesive zone approach is implemented. Numerical validation examples with experimental evidence highlight the performance and applicability of the proposed model.

27.1 Introduction Due to their extraordinary high specific stiffness and strength fiber reinforced composite laminates are extremely suitable for high-performance lightweight engineering applications. This is exceptionally true for carbon fiber / epoxy matrix composites. The weight-saving effect stems from the fact that a structure can be designed with the same stiffness and strength with a weight which is at least one order of magnitude lower than that of e.g. metallic counterparts. Thus, in the aircraft industry metallic parts have successively been replaced by composite laminates over the past decades, and this trend is ongoing. Werner Wagner KIT – Karlsruhe Institute of Technology, Universität Karlsruhe (TH), Institute for Structural Analysis, Kaiserstr. 12, D – 76131 Karlsruhe, Germany e-mail: [email protected] Claudio Balzani KIT – Karlsruhe Institute of Technology, Universität Karlsruhe (TH), Institute for Structural Analysis, Kaiserstr. 12, D – 76131 Karlsruhe, Germany e-mail: [email protected] M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 505–523. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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It is evident that the most considerable weight-saving effect can be obtained if large structures are made of advanced composites. Thus, the aircraft industry aims for lightweight designs in which the entire fuselage and wings are made of light and safe composite panels since this will result in less fuel consumption and larger aircraft capacities. In the design phase of stringer-stiffened composite panels a massive experimental effort is required which is very expensive and time-consuming. Thus, there is a need for reliable simulation and certification procedures to minimize these efforts. Besides progressive damage scenarios such procedures should take into account the postbuckling behavior since fuselage panels tend to buckle under axial compression. Damage in composite laminates can occur as intralaminar damage in terms of matrix cracking, fiber fracture, or fiber-matrix shear failure, or interlaminar damage, namely delamination or skin-stringer debonding. All these damage scenarios are taken into account in the present paper. The postbuckling response is captured by geometrical nonlinearity which is included in a quadrilateral shell element formulation allowing for arbitrary stacking sequences and a variable location of the reference plane across the thickness. However, the formulation of the shell element is not subject of this contribution.

27.2 Intralaminar Damage The intralaminar (ply) damage model utilized in this work is based on a ply-byply failure analysis. After determining the state of deformation, i.e. the calculation of the Green strains E, it is subdivided into three steps within each ply, namely i) calculation of the second Piola-Kirchhoff stresses S, ii) evaluation of suitable failure criteria to check if the material is damaged1, and iii) application of a suitable degradation model in order to provide post-damaged solutions2 . In this work this methodology is described only for one ply. As a matter of course it has to be applied ply-by-ply throughout the laminate in layered structures. In the following the index 1 means the fiber direction, the index 2 is the in-plane direction transverse to the fibers, and the index 3 denotes the through-thickness direction. Thus the second Piola-Kirchhoff stresses and the corresponding Green strains, both written in vector notation, are defined by S = [S11 , S22 , S12 , S13 , S23 ]T

and E = [E11 , E22 , 2E12 , 2E13 , 2E23 ]T .

(27.1)

It should be noticed that the underlying shell theory omits both the normal strain and stress across the thickness, i.e. E33 = S33 = 0. 1 2

This issue is related to damage initiation. This issue is related to damage propagation.

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27.2.1 Transverse Isotropy The undamaged second Piola-Kirchhoff stresses are computed via a linear elastic transversely isotropic material law. The preferred orientation is the fiber direction. With a purely elastic free Helmholtz energy function given by

ψ0 =

1 T 0 EC l E, 2ρ

(27.2)

where ρ is the mass density, and starting from the second law of thermodynamics, one can easily derive the constitutive law S0 = ρ

∂ ψ0 =C l 0 E. ∂E

(27.3)

The symbol C l 0 denotes the linear elastic undamaged material matrix in terms of engineering constants which has the well-known form [2]: ⎤ ⎡ ⎡ 0 0 ⎤ C l 11 C l 12 0 0 0 E1∗ E2∗ ν12 0 0 0 ⎥ ⎢ 0 0 ⎢C l C l 0 0 0 ⎥ ⎢ 0 0 0 ⎥ E2∗ ν12 E2∗ ⎥ ⎥ ⎢ ⎢ 12 22 0 0 (27.4) C l =⎢ 0 0 C 0 0 ⎥ 0 0 G l 44 0 0 ⎥ = ⎢ 12 ⎢ ⎥. ⎥ ⎣ ⎢ ⎦ 0 ⎦ ⎣ 0 0 0 C κ G 0 0 0 0 l 55 0 12 0 0 0 0 κ G23 l 066 0 0 0 0 C Herein, κ is a suitable shear correction factor, ν12 is the in-plane Poisson’s ratio, G12 is the in-plane shear modulus, and G23 is the transverse/through-thickness shear modulus. One has to respect the following set of dependent parameters: E1∗ = E1 /Δ ,

E2∗ = E2 /Δ ,

2 and Δ = 1 − ν12

E2 , E1

(27.5)

where E1 is the Young’s modulus in fiber direction and E2 is the in-plane Young’s modulus transverse to the fibers. The above material law is related to the material coordinates and has to be transformed to a global coordinate system which is straightforward and is not shown here.

27.2.2 Damage Initiation The prediction of damage initiation requires suitable failure criteria. These should provide the ability to distinguish between matrix cracking (M), fiber fracture (F), and fiber-matrix shear failure (FM). This work thus relies on the Hashin failure criteria [16] and the extensions made in [15]. These criteria are established in terms of stress-strength relationships. The notation of uniaxial strength parameters is defined in Table 27.1.

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Table 27.1 Notation of the uniaxial strength parameters Symbol

Description

Rt11 Rc11 Rt22 Rc22 R12 R13 = R23

Tensile strength in fiber direction Compressive strength in fiber direction Tensile strength transverse to the fibers Compressive strength transverse to the fibers In-plane shear strength Through-thickness shear strength

An advantage of the enhanced Hashin criteria is that only uniaxial strength parameters are employed. Utilizing the Macauley brackets defined by x = (|x|+x)/2, where |x| is the norm of x, the fiber fracture failure index reads

fF =

S11

Rt11

2

−S11

+ Rc11

2 .

(27.6)

The matrix cracking failure index under transverse tension (S22 ≥ 0) is defined by

t fM =

S22 Rt22

2

+

S12 R12

2

+

S13 R13

2

+

S23 R23

2 ,

(27.7)

while that under transverse compression (S22 < 0) reads 8

9

c 2 R S S22 2 S12 2 S13 2 S23 2 22 c 22 fM = c −1 + + + + . (27.8) R22 2R23 2R23 R12 R13 R23 The failure index for fiber-matrix shear failure, i.e. debonding between fibers and matrix, is given by the expression

fFM =

−S11

Rc11

2

S12 + R12

2

S13 + R13

2 .

(27.9)

It may be mentioned that (27.6) is equal to the well-known maximum stress criteria for tension and compression in fiber direction merged together into one equation, and that only compressive normal stresses in fiber direction are accounted for in Eq. (27.9).

27.2.3 Damage Propagation The degradation model derived in the following is formulated in a thermodynamically consistent manner. It is very similar to continuum damage mechanics (CDM), cf. e.g. [17, 19, 21, 22, 23, 33]. However, since critical strain energy release rates

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are not available to the authors, no gradual softening is introduced but degradation occurs ideally brittle. This at first sight may trigger numerical instabilities. However, such effects could not be observed by the authors in simulations carried out by now. It further reduces the effort to derive the tangent operator required for a full Newton-Raphson iteration to a minimum. Consider a Helmholtz free energy function of the form: 5

ψ = ∑ (1 − di) ψi0 .

(27.10)

i=1

Herein, di are scalar-valued damage parameters accounting for the damage-induced material degradation and ψi0 are undamaged and purely elastic contributions which are defined by

ψ10 =

1 0 2 l E , C 2ρ 11 11

ψ40 =

ψ20 =

1 0 2 l E , C 2ρ 22 22

1 0 l E11 E22 , C ρ 12 1 0 2 l E . ψ50 = C ρ 66 23

ψ30 =

1 0 2 2 , C l 44 E12 + C l 055 E13 ρ

(27.11)

l 0kl are given in Eq. (27.4). It can be seen that ψ10 is an uncoupled normal Recall that C contribution in fiber direction, ψ20 is an uncoupled normal contribution transverse to the fibers, ψ30 serves to account for the in-plane normal coupling, i.e. transverse contraction, and ψ40 and ψ50 account for shear effects. The second law of thermodynamics, also known as dissipation inequality, in vector—respective matrix—notation for iso-thermal conditions reads D := ST E˙ − ρ ψ˙ ≥ 0 .

(27.12)

The time derivative (x) ˙ = dx/dt of the Helmholtz free energy function in an explicit form is given by the expression 5

ψ˙ = ∑ (1 − di) i=1

5 ∂ ψi0 ˙ E − ∑ ψi0 d˙i . T ∂E i=1

(27.13)

Substitution of Eq. (27.13) into Eq. (27.12) yields an explicit representation of the dissipation inequality: 9 8 5 5 ∂ ψi0 ˙ (27.14) D := S − ρ ∑ (1 − di) E + ρ ∑ ψi0 d˙i ≥ 0 . T ∂E i=1 i=1 Following the principles of rational continuum mechanics this expression must hold for every admissible process, i.e. for every E˙ and d˙i . Thus the term in square brackets in Eq. (27.14) must be equal to zero. From this it follows that

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S = ρ ∑ (1 − di) i=1

∂ ψi0 , ∂E

(27.15)

thus the stresses in the damaged material are determined. The explicit representation of the dissipation inequality, cf. Eq. (27.14), provides additional information concerning the damage parameters di , that is 5

ρ ∑ ψi0 d˙i ≥ 0 .

(27.16)

i=1

By definition it holds ρψi0 ≥ 0 so that it can be concluded that the evolution of the damage parameters has to be positive, i.e. d˙i ≥ 0. From this it follows that historydependence is required in order to avoid stiffness recreation when the material is damaged. For this purpose the following set of internal state variables αi is introduced:

α1n = max {α1n−1 , fF } , α3n = max {α3n−1 , fF , fM } ,

α2n = max {α2n−1 , fF , fM } , α4n = max {α4n−1 , fF , fFM } ,

α5n = max {α5n−1 , fF } ,

(27.17)

with the initial values of αi0 = 0. In Eq. (27.17) the superscript n denotes the current load step. This superscript will be omitted in the sequel. In order to obtain a perfectly brittle fracture behavior the damage parameters are defined as di := di∗

αi − 1

, αi − 1

(27.18)

where di∗ are constant knock-down factors. For a mathematically correct representation it may be defined that αi − 1 /(αi − 1) = 0 if αi − 1 = 0. In the most general case the knock-down factors can be different for each failure mode, but for simplicity we use d1∗ = d2∗ = d3∗ = d4∗ = d5∗ = 0.99. Thus we apply a one-percent-rule which means that the elasticity matrix entries are reduced to 1 % of their initial values in case of damage. A reduction to zero should be avoided since this may lead to zero entries on the main diagonal of the stiffness matrix if the entire laminate fails. In this event the stiffness matrix will loose its positive finiteness. The constitutive law including damage reads S =C l E, (27.19) with the material matrix given by ⎤ ⎡ (1 − d1)C l 011 (1 − d3)C l 012 0 0 0 ⎥ ⎢ ⎥ ⎢ (1 − d3)C l 012 (1 − d2)C l 022 0 0 0 ⎥ ⎢ 0 C l := ⎢ ⎥ . (27.20) l 0 0 0 0 (1 − d4)C 44 ⎥ ⎢ ⎦ ⎣ l 055 0 0 0 0 (1 − d4)C 0 l 66 0 0 0 0 (1 − d5)C

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It should be mentioned that the material matrix C l =C l 0 in case no damage occurred in the loading history since di = 0 in this case and thus S = S0 .

27.2.4 Validation Example The validation example presented in the sequel is a mono-stringer panel buckling test made of UD IM7/8552 carbon/epoxy prepregs. Specimens with a geometry as depicted in Fig. 27.1 are loaded in longitudinal direction (0◦ direction in Fig. 27.1). One specimen consists of a T-shaped stringer which is adhesively bonded to a flat skin. In the grey-colored areas the specimens are clamped in 50 mm long pottings which serve as load-introducers. The respective experiments have been carried out in the course of the COCOMAT project [12] and are reported in [37]. The material parameters utilized in this work are listed in Table 27.2.

Fig. 27.1 Geometry of the mono-stringer panels (grey-colored areas mark the location of pottings; lay-up skin: [90/-45/+45/0]s ; lay-up stringer blade: [(-45/+45)3 /06 ]s )

In the experiments no degradation prior to final collapse could be observed. Although after collapse all failure modes have been present (fiber rupture, matrix cracking, delamination) we assume that intralaminar damage was the major contributor which subsequently triggered delamination. Thus, it does not seem worthwhile to account for delamination in this case. Final failure was due to massive damage in the stiffener near the pottings. The finite element mesh in longitudinal direction consists of 10 elements in the potting regions and 60 elements along the free length. In transverse direction 4 elements in each stringer flange and 16 elements in the skin are used generating almost quadratic elements. The stringer blade consists of 4 elements in its height direction. The shell reference planes are located on the interior surface of the skin.

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Table 27.2 Engineering constants and strength parameters of IM7/8552 Symbol

Unit

Magnitude

Symbol

Unit

Magnitude

E1 E2 ν12 G12 G23

N/mm2 N/mm2 – N/mm2 N/mm2

147 300 11 800 0.3 6 000 6 000

Rt11 Rc11 Rt22 Rc22 R12 R13 = R23

N/mm2 N/mm2 N/mm2 N/mm2 N/mm2 N/mm2

2 379 1 365 39 170 102 78

Fig. 27.2 Load-shortening curves of the mono-stringer panel buckling test: Experiment vs. simulation

Concerning the boundary conditions all degrees of freedom are fixed in the pottings except the longitudinal displacement which is only fixed on the bottom edge. The load is introduced via a displacement-controlled arclength method. Therein a uniform compression is applied to the top edge, i.e. all nodes located on the top edge have the same longitudinal displacement in direction of the bottom edge. Graphs of the applied load plotted against the axial shortening for both the experiments and the simulation are provided in Fig. 27.2. In the experiments linear elasticity can be observed up to a shortening of about 1 mm. This is the point where local skin buckling starts which is expressed by a very mild kink in the load-shortening graph. Global stringer-based buckling starts at a shortening of about 1.8 mm. At the same time final failure occurs. In the simulation very good agreement with the experiments is obtained. The curves almost coincide concerning stiffness, onset of local skin buckling, global load-shortening behavior, and collapse load. Only the shortening at final failure is slightly overestimated, but the error is negligibly small. Fig. 27.3 shows contour plots of relative failure indices (RFI) for the three intralaminar failure modes after collapse, plotted on the deformed configuration. A relative failure index means the fraction between accumulated thickness of damaged plies and the overall thickness of the laminate. It can thus have a magnitude between 0 (no damage) and 1 (complete failure). An RFI provides a clear visualization of the

27 Finite Element Modeling

0.00

0.89

513

0.00

M

0.50 FM

0.00

0.72 F

Fig. 27.3 Mono-stringer panel buckling test: RFIs plotted on the deformed configuration at a shortening of 2.1 mm

damage state in shell elements where one cannot observe the damage across the thickness in one picture. It can be seen in Fig. 27.3 that the most critical damage is located in the stringer at the potting ends, as has been found also in the experiments. The most severe damage occurs as matrix cracking, but fiber fracture leads to collapse of the panel. The buckling shape, which is a mixture of local skin buckling and global stringer-based buckling, can also be observed in Fig. 27.3.

27.3 Interlaminar Damage Interlaminar damage in the course of this work denotes debonding of two adjacent layers within a laminate. This phenomenon is known as delamination [6, 14]. A special case of interlaminar damage occurs when a stringer debonds from the skin

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in aircraft fuselage panels. This effect will be titled skin-stringer separation in the following. Several methods have been proposed to model such effects with the finite element method. Two of these gained special attention in the past years, on the one hand purely fracture–mechanics–based methods such as the virtual crack closure technique (VCCT) [20, 28], and on the other hand a methodology relying on a combination of damage mechanics and fracture mechanics where the cohesive zone model [5, 13] is included in so-called interface elements. The second of the latter two is subject of this section.

27.3.1 Interface Element In this section the finite element formulation of an interface element is outlined. In this contribution a hexahedral element type is utilized which relates interfacial tractions t = [t1 ,t2 ,t3 ]T to a discontinuous displacement jump across the interface u = [u1 , u2 , u3 ]T . The displacement jump is defined as the relative displacement between the top and the bottom surface of the element and is also called separation. Let a superscript + denote the top surface and a superscript − denote the bottom surface. The nodes 1–4 are located on the bottom surface and the nodes 5–8 on the top surface. It should be mentioned that the initial thickness of the element is zero. A similar zero-thickness type has been used by e.g. [9, 11, 24, 25, 32, 36, 38, 39] among others. However, other element formulations are possible [4, 7, 35]. The indices 1, 2, 3 in the entries of t and u denote the directions of local element coordinate vectors A1 , A2 , A3 of which the origin is located in the element midpoint Xgc . The superscript g denotes that the element midpoint is related to the global coordinate system, namely the Euclidean space, which is spanned by the vectors e1 , e2 , e3 . The vectors A1 and A2 are the in-plane directions corresponding to mode II and III shearing and A3 is the through-thickness direction related to mode I opening. The methodology employed for the calculation of the local element basis vectors is adopted from [31] but utilizes the diagonal vectors of the element midplane. Details can be found in [34]. A transformation matrix T := Ai · e j = [A1 , A2 , A3 ]T

(27.21)

serves to transform the global position vectors of material points located on the top surface (X+ ) and those located on the bottom surface (X− ) from the global representation into the local element representation according to ⎡ +⎤ ⎡ −⎤ X1 X1 X+ = TT (Xg+ − Xgc ) = ⎣ X2+ ⎦ , X− = TT (Xg− − Xgc ) = ⎣ X2− ⎦ . (27.22) X3+ X3− Utilizing standard bi-linear Lagrangian shape functions NI = 14 (1 + ξI ξ )(1 + ηI η ), with ξ ∈ [−1, +1] and η ∈ [−1, +1] denoting natural coordinates and ξI ∈ {−1, +1}

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515

and ηI ∈ {−1, +1} their nodal values, the position vectors are approximated by X+ =

8

∑ NI XI

and

4

∑ NI XI .

X− =

(27.23)

I=1

I=5

The real, virtual, and incremental nodal displacement vectors are denoted by vI = [v1I , v2I , v3I ]T , δ vI = [δ v1I , δ v2I , δ v3I ]T , and Δ vI = [Δ v1I , Δ v2I , Δ v3I ]T . With these the real, virtual, and incremental displacement vectors of material points located on the top and bottom surface of the interface are approximated by the relationships u+ = u− =

8

∑ NI vI ,

δ u+ =

∑ NI vI ,

δ u− =

I=5 4

I=1

8

∑ NI δ vI ,

Δ u+ =

∑ NI δ vI ,

Δ u− =

I=5 4

I=1

8

∑ NI Δ vI ,

I=5 4

∑ NI Δ vI .

(27.24)

I=1

The discontinuous displacement jump defined as the relative displacements between the top and the bottom surface of the interface is now defined by the expression u = u+ − u− =

8

4

8

I=5

I=1

I=1

∑ NI vI − ∑ NI vI = ∑ BI vI .

(27.25)

Herein, operator matrices BI are introduced which serve to correctly interconnect the top and the bottom surface. They are defined by −1 ← 1 ≤ I ≤ 4 BI := I NI 1 , with I := (27.26) 1 ← 5≤I≤8 an interconnection parameter which distinguishes nodes on the top and the bottom surface and 1 a 3×3 identity matrix. The virtual and incremental displacement jump vectors can be obtained analogously:

δu =

8

∑ BI δ vI ,

Δu =

I=1

8

∑ BI Δ vI .

(27.27)

I=1

Compared with standard continuum elements, the separation vector u is a strain-like measure while the traction vector t is a stress-like measure which have to be related to each other. This relation is established in terms of a cohesive law. A straightforward finite element formulation yields the element residual vector reI and element tangential stiffness matrix KeIK : BTI t dA ,

reI = Γ0e

BTI C l coh BK dA .

KeIK =

(27.28)

Γ0e

Herein, I and K denote a specific node number, Γ0e is the area of the element midplane in the reference configuration, and C l coh is the traction linearization matrix

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which is obtained via a cohesive law. The integrals in (27.28) are solved via a 3×3 Newton-Cotes integration which is superior to a Gauss quadrature [1, 8, 29, 30].

27.3.2 Cohesive Zone Model The cohesive zone model goes back to Dugdale [13] and Barenblatt [5]. While Dugdale considers a plastic zone ahead of a crack tip, Barenblatt introduced his concept of molecular forces of cohesion acting at a crack tip which avoid infinite stresses at the crack tip and provide smooth closing of the crack faces. This model can be considered in a finite element code by simply implementing a suitable cohesive law into an interface element. It is generally accepted that a cohesive law should be able to predict the point of damage initiation and damage propagation. Damage initiation is predicted via a traction-strength based criterion, namely a quadratic power law which is given by

fD =

t1 RILS

2

+

t2 RILS

2

+

t3

RI

2 ,

(27.29)

see also [10]. In (27.29), RILS is the interlaminar shear strength and RI is the mode I strength. By including the Macauley brackets only mode I tension enters this criterion, so contact forces do not trigger delamination. In order to accurately predict damage propagation a fracture–mechanics–based method is employed, thus the separation energy release rate is compared with an allowable threshold, namely the fracture toughness Gc . Several cohesive laws have been proposed in the past. A very convenient way to account for a delamination mode coupling is to formulate the cohesive law in terms of a scalar-valued effective traction-separation law. The effective separation is then defined by 3 > ε := β 2 (u21 + u22 ) + u3 2 = uT C u ,

⎡

with

⎤ β2 0 0 2 C := ⎣ 0 β 0 ⎦ (27.30) 0 0 uu33

and β = RILS /RI . The most frequently used such model is a bi-linear one [10], see Fig. 27.4. However, it has been demonstrated [4] that a kink-free exponential model based on the universal binding law of Rose et al. [27], see also Fig. 27.4 and [3, 26], performs numerically more efficient since a coarser mesh provides the same quality of results. Hence, the latter one is employed in this work and relies on a cohesive free energy function of the form

ψcoh = 1 − (1 +

ε −ε / ε 0 K )e + −u3 2 . ε0 2

(27.31)

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Fig. 27.4 Two types of effective traction-separation laws: Bi-linear model from [10] (left) and smooth exponential model from [4, 34] (right) (σ : Effective traction; ε : Effective separation; σ 0 : Effective strength; ε 0 : Effective separation at delamination onset; ε u : Effective separation at complete decohesion; K: Initial stiffness; d: Damage parameter)

Herein, ε 0 is the effective separation at delamination onset, e is the Eulerian number, and K is a penalty stiffness accounting for contact effects. It can easily be shown that the second law of thermodynamics postulates that the tractions are obtained as the partial derivatives of the cohesive free energy function with respect to the separations. Partial derivation of the tractions with respect to the separations yields the traction linearization matrix C l coh which is required for a full Newton-Raphson iteration scheme. However, a detailed derivation of the cohesive law can be found elsewhere [4, 34] and is not repeated here. It may be mentioned that the cohesive law is made history–dependent via an internal variable which tracks the maximum effective separation observed in the loading history. This means that no stiffness recreation is possible. Thus the unloading path is different from the loading path, see Fig. 27.4.

27.3.3 Validation Example The aim of this example is to show the capability of the interface model to predict skin-stringer debonding under mixed mode loading conditions. A three point bending test performed within the COCOMAT project [12] which is reported in [18] is the basis for the numerical simulation. A photograph of the test is given in Fig. 27.5. The intralaminar elastic properties are those from Tab. 27.2. The interlaminar properties are given in Tab. 27.3. The fracture toughness has been reported in [18]. The specimen is composed of a curved skin and a T-shaped stringer. The test setup introduces a mixed mode loading state into the skin-stringer interface. On the right– hand side of Fig. 27.5 the opened specimen after failure is depicted. As can be seen transverse matrix cracking in the 0◦ ply of the skin next to the stiffener occurred first followed by delamination between the adjoining ±45◦ plies. However, we assume that the global response can be approximated with sufficient accuracy if debonding between skin and stringer is simulated.

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Table 27.3 Interfacial properties Symbol

Unit

Magnitude

RI RILS Gc

N/mm2 N/mm2 kJ/m2

51.0 115.7 0.609

Fig. 27.5 Photographs of the mixed mode debonding test: During the test (left) and opened specimen after the test (right), from [18]

Fig. 27.6 shows the geometry of the specimens. Ply drop-offs are located in the stringer flanges forming a graded thickness. The fiber directions in the stringer flanges follow from the lay-up of the stringer blade. The supports are located 7.0 mm from the free edges of the skin, thus their distance is 86.0 mm. Fig. 27.7 illustrates the finite element mesh. The reference planes of the skin and stringer flange shells are located on the interior surface of the skin. The displacement in longitudinal direction is fixed at the center node of the specimen. The left support is fixed in radial direction while the right support is fixed in both circumferential and radial direction. In longitudinal direction, 20 elements are used. In circumferential direction, 16 (from the free edge up to the left support) + 32 (from the left support up to the left stringer flange) + 120 (from the left stringer flange up to the stringer blade) + 96 (from the stringer blade up to the end of the right stringer flange) + 32 (from the end of the right stringer flange up to the right support) + 16 (from the right support to the free edge) elements are utilized. Interface elements are only inserted in the left stringer flange. Along the height of the stringer blade 16 shell elements are used. In the intersection line of the stringer blade and the stringer flange the corresponding nodes are linked pairwise via MPCs as depicted in the zoom view of Fig. 27.7. The MPCs assign the same nodal displacements and rotations representing a perfect bond. The loaded skin nodes are additionally linked in radial direction for a correct load introduction. A displacement-controlled arc-length method is employed where the loaded nodes are controlled in radial direction. Some deformed configurations at different load levels are plotted in Fig. 27.8. Comparing these with Fig. 27.5 we can conclude that they agree very well with the experiment. Fig. 27.9 presents load-deflection curves where the resultant of the

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519

Fig. 27.6 Mixed mode debonding test: Specimen geometry (Lay-up skin: [0/+45/-45/90]s ; lay-up stringer blade: [+45/-45/0/0]3s )

Fig. 27.7 Sketch of the finite element mesh and linking of nodes: Thickness of interface elements is virtually introduced in the zoom view (IF: Interface elements; the plotted mesh is coarser than that used in the analysis)

a)

b)

c)

d)

Fig. 27.8 Plots of some deformed configurations at different load levels: a) w = 0.0 mm, b) w = 5.2 mm (maximum load), c) w = 5.3 mm (after the load drop), and d) w = 8.0 mm

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Fig. 27.9 Load-deflection curves of the mixed mode debonding test

applied load (F) is plotted against the deflection of the center node of the skin (w). There is a sudden load drop at the maximum load which expresses unstable crack growth up to the first ply drop-off. Then there is stable crack propagation. This effect can be observed in both the experiment and the prediction. The simulation slightly underestimates the initial stiffness. This is very likely to be due to inaccurate modeling of the boundary conditions. In the FE analysis no friction effects are taken into account. In order to show the influence of friction alternative boundary conditions are applied in which the left support is also fixed in circumferential direction. For this analysis no interface layer is inserted. The corresponding load-deflection curve is included in Fig. 27.9. One can see that the initial slope increases dramatically. As expected, reality is in between the two predictions. Concerning the limit load and the subsequent damage growth excellent agreement between the prediction and the experiment can be observed.

27.4 Conclusions In this contribution, an intralaminar as well as an interlaminar damage model for composite laminates has been presented. The intralaminar damage model bases on a ply-by-ply failure analysis. It utilizes layered quadrilateral shell elements in which a material model is implemented which accounts for damage initiation and growth in terms of fiber fracture, matrix cracking, and fiber-matrix shear failure. The degradation model is thermodynamically consistent and ideally brittle. The interlaminar damage model utilizes interface elements in which the cohesive zone approach is implemented. The cohesive law is governed by a smooth and kink-free exponential response. Mode I compression, i.e. contact forces, do not contribute to the evolution of damage. A penalty contact algorithm avoids the interpenetration of the crack

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faces. The models are history-dependent, thus stiffness recreation is not possible. For both the intralaminar and the interlaminar damage model validation examples with experimental evidence have been presented which highlight the applicability of the proposed models.

References [1] Alfano, G., Crisfield, M.A.: Finite element interface models for the delamination analysis of laminated composites: mechanical and computational issues. International Journal for Numerical Methods in Engineering 50, 1701–1736 (2001) [2] Altenbach, H., Altenbach, J., Kissing, W.: Mechanics of Composite Structural Elements. Springer, Heidelberg (2004) [3] de Andrés, A., Pérez, J.L., Ortiz, M.: Elastoplastic finite element analysis of threedimensional fatigue crack growth in aluminum shafts subjected to axial loading. International Journal of Solids and Structures 36(15), 2231–2258 (1999) [4] Balzani, C., Wagner, W.: An interface element for the simulation of delamination in unidirectional fiber-reinforced composite laminates. Engineering Fracture Mechanics 75, 2597–2615 (2008) [5] Barenblatt, G.I.: The mathematical theory of equilibrium cracks in brittle fracture. Advances in Applied Mechanics 7, 55–129 (1962) [6] Bolotin, V. V.: Delaminations in composite structures: Its origin, buckling, growth and stability. Composites Part B: Engineering 27(2), 129–145 (1996) [7] Borg, R., Nilsson, L., Simonsson, K.: Modeling of delamination using a discretized cohesive zone and damage formulation. Composites Science and Technology 62(1011), 1299–1314 (2002), doi:10.1016/S0266-3538(02)00070-2 [8] de Borst, R., Rots, J.G.: Occurance of spurious mechanisms in computation of strainsoftening solids. Engineering Computations 6, 272–280 (1989) [9] de Borst, R., Schipperen, J.H.A.: Computational methods for delamination and fracture in composites. In: Allix, O., Hild, F. (eds.) Continuum damage mechanics of materials and structures, pp. 325–352. Elsevier Science Ltd., Amsterdam (2002) [10] Camanho, P.P., Dávila, C.G.: Mixed-mode decohesion finite elements for the simulation of delamination in composite materials. Technical Memorandum TM-2002-211737, NASA, Langley Research Center, Hampton, Virginia 23681-2199, USA (2002) [11] Crisfield, M., Mi, Y., Davies, G.A.O., Hellweg, H.B.: Finite element methods and the progressive failure-modelling of composite structures. In: Owen, D.R.J., Oñate, E., Hinton, E. (eds.) Computational Plasticity – Fundamentals and Applications, Proceedings of the 5th International Conference on Computational Plasticity, CIMNE, Barcelona, Spain, pp. 239–254 (1997) [12] Degenhardt, R., Rolfes, R., Zimmermann, R., Rohwer, K.: COCOMAT – Improved material exploitation of composite airframe structures by accurate simulation of postbuckling and collapse. Composite Structures 73, 175–178 (2006) [13] Dugdale, D.S.: Yielding of steel sheets containing slits. Journal of the Mechanics and Physics of Solids 8(2), 100–104 (1960) [14] Garg, A.C.: Delamination – a damage mode in composite structures. Engineering Fracture Mechanics 29(5), 557–584 (1988)

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[15] Goyal, V.K., Jaunky, N.R., Johnson, E.R., Ambur, D.R.: Intralaminar and interlaminar progressive failure analyses of composite panels with circular cutouts. Composite Structures 64(1), 91–105 (2004) [16] Hashin, Z.: Failure criteria for unidirectional fiber composites. Journal of Applied Mechanics – Transactions of the ASME 47(2), 329–334 (1980) [17] Kachanov, L.M.: On the creep fracture time. Izv Akad, Nauk USSR Otd. Tech. 8, 26–31 (1958) (in Russian) [18] Korjakins, A., Ozolinsh, O., Rikards, R.: Investigation of degradation by tests and development of degradation models – contribution to deliverable D07. COCOMAT Technical report, Riga Technical University, Riga, Latvia (2005) [19] Krajcinovic, D.: Constitutive equations for damaging materials. Journal of Applied Mechanics 50(4), 355–360 (1983) [20] Krüger, R.: Virtual crack closure technique: History, approach, and applications. Applied Mechanics Reviews 57(2), 109–143 (2004) [21] Lemaitre, J.: A Course on Damage Mechanics. Springer, Heidelberg (1996) [22] Maimí, P., Camanho, P.P., Mayugo, J.A., Dávila, C.G.: A continuum damage model for composite laminates: Part I – constitutive model. Mechanics of Materials 39(10), 897–908 (2007) [23] Maimí, P., Camanho, P.P., Mayugo, J.A., Dávila, C.G.: A continuum damage model for composite laminates: Part II – computational implementation and validation. Mechanics of Materials 39(10), 909–919 (2007) [24] Needleman, A.: An analysis of decohesion along an imperfect interface. International Journal of Fracture 42(1), 21–40 (1990) [25] Needleman, A.: An analysis of tensile decohesion along an interface. Journal of the Mechanics and Physics of Solids 38(3), 289–324 (1990) [26] Ortiz, M., Pandolfi, A.: Finite-deformation irreversible cohesive elements for threedimensional crack-propagation analysis. International Journal for Numerical Methods in Engineering 44(9), 1267–1282 (1999) [27] Rose, J.H., Ferrante, J., Smith, J.R.: Universal binding energy curves for metals and bimetallic interfaces. Physical Review Letters 47(9), 675–678 (1981) [28] Rybicki, E.F., Kanninen, M.F.: A finite element calculation of stress intensity factors by a modified crack closure integral. Engineering Fracture Mechanics 9(4), 931–938 (1977) [29] Schellekens, J.C.J.: Computational strategies for composite structures. Phd thesis, Technical University Delft (1992) [30] Schellekens, J.C.J., de Borst, R.: On the numerical integration of interface elements. International Journal for Numerical Methods in Engineering 36, 43–66 (1993) [31] Taylor, R.L.: Finite element analysis of linear shell problems. In: Whiteman, J.R. (ed.) The Mathematics of Finite Elements and Applications VI (MAFELAP 1987), pp. 191– 204. Academic Press, London (1988) [32] Tvergaard, V., Hutchinson, J.W.: The relation between crack growth resistance and fracture process parameters in elastic-plastic solids. Journal of the Mechanics and Physics of Solids 40(6), 1377–1397 (1992) [33] Voyiadjis, G.Z., Kattan, P.I., Taqieddin, Z.N.: Continuum approach to damage mechanics of composite materials with fabric tensor. Internation Journal of Damage Mechanics 16(3), 301–329 (2007) [34] Wagner, W., Balzani, C.: Simulation of delamination in stringer stiffened fiberreinforced composite shells. Computers and Structures 86(9), 930–939 (2008)

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[35] Wagner, W., Gruttmann, F., Sprenger, W.: A finite element formulation for the simulation of propagating delaminations in layered composite structures. International Journal for Numerical Methods in Engineering 51(11), 1337–1359 (2001) [36] Xu, X.P., Needleman, A.: Void nucleation by inclusion debonding in a crystal matrix. Modelling Simul. Mater. Sci. Eng. 1, 111–132 (1993) [37] de Zarate Alberdi, I.O., Rebollo, F.: Small degradation specimen test results – contribution to deliverable d07 (data base of degradation models). COCOMAT Technical report, AERnnova, Spain (2006) [38] Zou, Z., Reid, S.R., Li, S., Soden, P.D.: Modelling interlaminar and intralaminar damage in filament-wound pipes under quasi-static indentation. Journal of Composite Materials 36(4), 477–499 (2002) [39] Zou, Z., Reid, S.R., Li, S.: A continuum damage model for delaminations in laminated composites. Journal of the Mechanics and Physics of Solids 51(2), 333–356 (2003)

Index

hp-adaptive analysis 111 hp-adaptivity 19, 35 hpq-interpolation shape function “ghost” atom 215

113

acoustics problem 27 acoustic wave equation 4 adaptive analysis 117 adaptive modelling 117 adaptive strategy 141 algorithm 168, 171, 172, 186, 187, 189, 195, 201, 203 alkali silica reaction (ASR) 455 angiogenesis 426 approach 2.5D 396 Arlequin method 213 artiﬁcial immune system 166, 178, 183 artiﬁcial neural network (ANN) 155, 455 associated shape functions 61 atomistic/continuum models 239 autogenous shrinkage 342 automatic hp-adaptive strategy 126 Babuska-Brezzi condition 354 beam-to-beam contact 439 Bezier’s curves 445 biocompatibility 380 Bloch wave property 5 bone fracture healing 424, 429 boundary layer phenomenon 139 bridging domain method 213 bridging scale decomposition 213

Bubnov-Galerkin method Burgers vector 241, 242 burning algorithm 338

256

cancellous bone 397 capillary depression 342 carbon-epoxy cross-ply laminate 494 cell diﬀerentiation 424, 426 cell migration 425 cell proliferation 426 chondrocyte 424 cohesive-zone model 212 cohesive zone model 516 complex geometry 116 complex mechanical descriptions 117 computational bioengineering 168 consistent linearization 316 constitutive equations for intrinsic director couples 261 constitutive law 44, 45, 48, 50, 51 constrained approximation 34 contact boundary condition 52 contact inhibition 425 continuous dispersion relation 6 cortical bone 397 Cosserat point 256 Cosserat point element 256 Cosserat rod 256 Cosserat rotation 373 Cosserat shell 256 Coulomb friction 52, 440, 467 coupled multiphysics problem 20, 36 coupling element 217 coupling zone 217

526 crack 79 cracking of dental implant 394 crankshaft bearing 479 crystallographic model 245 damage initiation 507 damage propagation 508 DCB test 494 debonding 497 deformation gradient 259 delamination 495 delamination growth 494 dental implant 392 dental implant placement procedure 392 de Rham diagram 32 digital image correlation (DIC) 457, 458 Digital Product Development 393 dilatometric test in concrete dam 463 director momentum equations 260 discontinuous enrichment method 213 discrete-to-continuum bridging 213 discrete Bloch wave property 8 discrete dispersion relation 7 discretized eigenproblem 114 dislocations 239, 240 dispersive properties 4 dissipation inequality 509 distraction osteogenesis 424 distributed parameter systems (DPSs) 193 domain decomposition 85 drop-weight tester 494 dual-mixed formulation 24 dual-mixed formulation with weakly imposed symmetry 24 dual-mixed problem 36 dual formulation 23, 80 dynamic fracture 224 dynamic relaxation 215 Eﬀective Properties of Piezoelectric Material 316 elastic-damage interface model 502 elasticity problem 20 electro-mechanically coupled boundary value problem 313

Index endothelial cell 426 ENF test 494 enhanced volumetric strain technique 348 equilibrated residual method 117, 123 equilibrium problem of elastic body 112 evolution inclusion 44, 49, 51 excess pore pressures 357 explicit time integration 350 ﬁber-matrix shear failure 508 ﬁber fracture 508 ﬁber reinforced composite laminates 505 First Analysis 60, 61 ﬂat-jack test 456 forced vibration problem 114 fracture toughness 495 free vibration problem 113 frictional beam-to-beam contact 440 frictional contact problem 44, 45, 50, 52 friction force 442 fuzzy number 150, 158, 160 Gauss-Lobatto quadrature rule 10 generalized gradient 46 generalized macro-homogeneity condition 315 genetic algorithm 411 genetic algorithm (GA) 455 global friction angle 366 goal-oriented adaptivity 126 grain crushing 364 granular arithmetic crossover operator 155 granular character 150 granular evolutionary algorithm 154 granular Gaussian mutation operator 155 hanging node 129 Helmholtz equation 5 Helmholtz free energy function 509 hemivariational inequality 44, 45, 49 Hermite’s polynomials 444 hierarchical hpq-approximations 122 hierarchical approximation 117

Index

527

hierarchy of numerical models 118 homogenization procedure 312 hybrid multilayered laminate 153 hydration models 335 hyperelastic material 256 identiﬁcation for aged ship 488 identiﬁcation problem 149, 150, 157 immune optimization 178 immune system 166 implant design 380, 424 implicit mesh-free dynamic analysis 353 improper solution limit 136 inception of intra-laminar damage 494 interface condition 31 interface element 514 interlaminar damage 506 interval number 150, 157, 158, 161 intrinsic director couples 260 inverse problem 150, 485 irregular mesh 130 lattice model 425 Lennard-Jones potential locking 136, 348, 354

225

macroscopic electric displacements 314 macroscopic electric ﬁeld 314 macroscopic strains 314 macroscopic stresses 314 material point method 350 matrix cracking 508 matrix synthesis 427 mechanical behavior of the implant 392 mechanobiology 423 mechanoregulatory stimulus 426 mediator space 218 mesh-free algorithm 348 meshless method 59, 212 micro-polar hypoplastic model 364 microstructure 337 modiﬁcation step 125 Molodensky problem 99 momentum equation 27

multi-scale approach 213 multiscale model 183 non-associated sliding rule 442 non-penetration condition 79 non-polar hypoplastic constitutive model 364 non-reﬂecting boundary conditions 14 non-smooth perturbation 83 nonlinear mathematical programming (NMP) 455 nonmonotone subdiﬀerential boundary condition 43 Nose-Hoover thermostat method 225 optimal design 198, 199 optimally blending scheme 12 optimization problem 174, 196, 198 orthogonality equation 441 osseointegration 392 osteoblast 424 osteoplant 392 particle swarm optimization 188 partition-of-unity based ﬁnite element method 212 partition of unity 217 percolation 338 phase lag 4 phase lead 4 poroelasticity 29 pre-crack 215, 495 propagation of intra-laminar damage 494 proper orthogonal decomposition (POD) 454 prosthesis approache 129 quarter-point element

212

radial basis functions (RBF) 454 radiation condition 28 random walk model 425 Rayleigh dumping coeﬃcients 115 Rayleigh quotient 352 reproducing kernel particle 63 reproducing polynomial property 64 Rivara’s reﬁnement 129 row sum condition 60, 64

528 scaﬀold implantation 430 screw loosening 394 Second Analysis 60, 67 Seeger method 399 sensor density 197 sensor location problem 196 shaft line 479 shaft line alignment 479 shape derivative 83 shape functional 86 shape optimization 76 shear localization 364 shear rate 364 shell geometry 116 simplicial decomposition (SD) 200 singularity 38 singular limit 76 smoothing procedure 444 softening interface constitutive law 494 solid geometry 116 Sommerfeld number 480 spectral element method 4 spurious pressure modes 354 starter crack 215 Steklov-Poincar´e operator 76, 85 stern tube bearing 480 stringer-stiﬀened composite panels 506

Index Texas three-step strategy 125 topological asymptotic expansion 84 topological derivative 83, 87, 90 topology optimization 76 transition approximation 129 transition geometry 116 trust region algorithm (TRA) 455 two–stage granular strategy 154 two-mesh paradigm 126 two-scale approach 314 uniformly distributed particles

61

variational formulation 5, 23, 25, 28, 61, 79, 81, 114 variational inequality 79, 80, 82, 95 virial stress 224 viscous damping 351 void ratio 365 wall roughness 365 weak formulation 48 weighted averaging 4 Wendland function 103, 105 window function 63 zero row sum condition

60