Numerical Simulation Of Mechatronic Sensors And Actuators

Manfred Kaltenbacher Numerical Simulation of Mechatronic Sensors and Actuators

Manfred Kaltenbacher

Numerical Simulation of Mechatronic Sensors and Actuators With 286 Figures and 41 Tables

123

Dr. Manfred Kaltenbacher Universität Erlangen LS Sensorik Paul-Gordon-Str. 3/5 91052 Erlangen, Germany [email protected]

Library of Congress Control Number: 2007924154

ISBN 978-3-540-71359-3 Springer Berlin Heidelberg New York 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 for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2007 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. Typesetting: Digital data supplied by the author Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover: eStudio Calamar S.L., F. Steinen-Broo, Girona, Spain SPIN 11801375

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Printed on acid-free paper

Preface to the second edition

The second edition of this book fully preserves the character of the first edition to combine the detailed physical modelling of mechatronic systems and their precise numerical simulation using the Finite Element (FE) method. Most of the text and general appearance of the previous edition were retained, while the coverage was extended and the presentation improved. Starting with Chap. 2, which discusses the theoretical basics and computer implementation of the FE method, we have added a section describing the FE method for one-dimensional cases, especially to provide a easier understanding of this important numerical method for solving partial differential equations. In addition, we provide a section about a priori error estimates. In Chap. 3, which deals with mechanical fields, we now additionally discuss locking effects as occurring in the numerical computation of thin structures, and describe two well established methods (method of incompatible modes and of enhanced assumed strain) as well as a recently newly developed scheme based on balanced reduced and selective integration. The physical discussion of acoustic sound generation and propagation (see Chap. 5) has been strongly improved, including now also a description of plane and spherical waves as well as a section about quantitative measures of sound. The treatment of open domain problems has been extended and include a recently developed Perfectly Matched Layer (PML) technique, which allows to limit the computational domain to within a fraction of the wavelength without any spurious reflections. Recently developed flexible discretization techniques based on the framework of mortar FE methods for the numerical solution of coupled wave propagation problems allow for the use of different fine meshes within each computational subdomain. This technique has been applied to pure wave propagation problems (see Sec. 5.4.3) as well as coupled mechanical-acoustic field problems (see Sec. 8.3.2), where the computational grids of the mechanical region and the acoustic region can be independently generated and therefore do not match at the interface. Furthermore, we have investigated in the piezoelectric effect and provide in Chap. 9 an extended discussion on the modelling and numerical computation of nonlinear effects including hysteresis. In the last three years, we have established a research group on computational aeroacoustics to study the complex phenomenon of flow induced noise. Therewith, the totally new Chap. 10 contains a description of computational

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aeroacoustics with a main focus on a recently developed FE method for efficiently solving Lighthill’s acoustic analogy. Within Chap. 12, which deals with industrial applications, we have rewritten Sec. 12.5 to discuss latest computational results on micromachined capacitive ultrasound transducers, and have added a section on high power ultrasound sources as used for lithotripsy as well as a section on noise generation by turbulent flows. Most of the formulations described in this book have been implemented in the software NACS (see www.simetris.de).

Acknowledgment The author wishes to acknowledge the many contributions that colleagues and collaborators have made to this second edition. First of all I would like to express my gratitude to the members of the Department of Sensor Technology and its head Prof. Reinhard Lerch for the pleasant and stimulating working atmosphere. Amongst many, I wish to specially thank M.Sc. Max Escobar, M.Sc. Andreas Hauck, M.Sc. Gerhard Link, Dipl.-Ing. Thomas Hegewald and Dipl.-Ing. Luwig Bahr for fruitful discussions and proof reading. Much is owned by many intensive discussions with my wife Prof. Barbara Kaltenbacher with whom I work on hysteresis models and parameter identification for electromagnetics and piezoelectrics. Special thank is dedicated to Dr. Stefan Becker and his co-workers M.Sc. Irfan Ali and Dr. Frank Sch¨ afer for the contribution on computational aeroacoustics and the intensive cooperation within the current research project Fluid-Structure-Noise founded by the Bavarian science foundation BFS. Furthermore, the author would like to thank Dr. Bernd Flemisch and Prof. Barbara Wohlmuth for the fruitful cooperation on nonmatching grids. A common research project on Numerical Simulation of Acoustic-Acoustic- and Mechanical-Acoustic-Couplings on Nonmatching Grids founded by German Research Foundation DFG has just started. Moreover, the author wants to acknowledge the excellent working environment at the Johann Radon Institute for Computational and Applied Mathematics in Linz, Austria, where the author stayed for one semester in 2005/06 as an invited lecturer for coupled field problems within a special semester on computational mechanics. Special thank is dedicated to Prof. Ulrich Langer, who organized this event, and who did a great job in bringing together different researchers from all over the world. During this time, I also started the cooperation with Prof. Dietrich Braess on enhanced softening techniques to avoid locking in thin mechanical structures, to whom I would like to express my gratitude for revealing new and interesting perspectives to me. Manfred Kaltenbacher February 2007

Preface to the first edition

The focus of this book is concerned with the modelling and precise numerical simulation of mechatronic sensors and actuators. These sensors, actuators, and sensor-actuator systems are based on the mutual interaction of the mechanical field with a magnetic, an electrostatic, or an electromagnetic field. In many cases, the transducer is immersed in an acoustic fluid and the solid–fluid coupling has to be taken into account. Examples are piezoelectric stack actuators for common-rail injection systems, micromachined electrostatic gyro sensors used in stabilizing systems of automobiles or ultrasonic imaging systems for medical diagnostics. The modelling of mechatronic sensors and actuators leads to so-called multifield problems, which are described by a system of non-linear partial differential equations. Such systems cannot be solved analytically and thus a numerical calculation scheme has to be applied. The schemes discussed in this book are based on the finite element (FE) method, which is capable of efficiently solving the partial differential equations. The complexity of the simulation of multifield problems consists of the simultaneous computation of the involved single fields as well as in the coupling terms, which introduce additional non-linearities. Examples are moving conductive (electrically charged) body within a magnetic (an electric) field, electromagnetic and/or electrostatic forces. The goal of this book is to present a comprehensive survey of the main physical phenomena of multifield problems and, in addition, to discuss calculation schemes for the efficient solution of coupled partial differential equations applying the FE method. We will concentrate on electromagnetic, mechanical, and acoustic fields with the following mutual interactions: •

•

Coupling Electric Field – Mechanical Field This coupling is either based on the piezoelectric effect or results from the force on an electrically charged structure in an electric field (electrostatic force). Coupling Magnetic Field – Mechanical Field This coupling is two-fold. First, we have the electromotive force (emf), which describes the generation of an electric field (electric voltage re-

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spectively current) when a conductor is moved in a magnetic field, and secondly, the electromagnetic force. Coupling Mechanical Field – Acoustic Field Very often a transducer is surrounded by a fluid or a gaseous medium in which an acoustic wave is launched (actuator) or is impinging from an outside source towards the receiving transducer.

In Chap. 2, we give an introduction to the finite element (FE) method. Starting from the strong form of a general partial differential equation, we describe all the steps concerning spatial as well as time discretization to arrive at an algebraic system of equations. Both nodal and edge finite elements are introduced. Special emphasis is put on an explanation of all the important steps necessary for the computer implementation. A detailed discussion on electromagnetic, mechanical, and acoustic fields including their numerical computation using the FE method can be found in Chap. 3 – 5. Each of these chapters starts with the description of the relevant physical equations and quantities characterizing the according physical field. Special care is taken with the constitutive laws and the resultant non-linearities relevant for mechatronic sensors and actuators. In addition, the numerical computation using the FE method is studied for the linear as well as the non-linear case. In Chap. 4, where the electromagnetic field is discussed, we explain the difficulties arising at interfaces of jumping material parameters (electric conductivity and magnetic permeability), and introduce two correct formulations adequate for the FE method. At the end of each of these chapters, we present an example for the numerical simulation of a practical device. In Chap. 6, we study the interaction between electrostatic and mechanical fields and concentrate on micromechanical applications. After the derivation of a general expression for the electrostatic force, applying the principle of virtual work, we focus on the numerical calculation scheme. The simulation of a simple electrostatic driven bar will demonstrate the complexity of such problems, and will show the necessity of taking into account mechanical nonlinearities. The physical modelling and numerical solution of magnetomechanical systems is presented in Chap. 7. In this chapter, we first discuss the correct physical description of moving and/or deforming bodies in a magnetic field. Later, we derive a general expression for the electromagnetic force, again (as for the electrostatic force) by using the principle of virtual work. The discussion on numerical computation will contain a calculation scheme for the efficient solution of magnetomechanical systems and, in addition, electric circuit coupling as arise for voltage-driven coils. Especially for the latter case, we give a very comprehensive description of its numerical computation. Chapter 8 deals with coupled mechanical-acoustic systems and explains the physical coupling terms and the numerical computation of such systems. The simulation of the sound emission of a car engine will illustrate different

Preface to the first edition

IX

approaches concerning time-discretization schemes and solvers for the algebraic system. A special coupling between the mechanical and electrostatic field occurs in piezoelectric systems, which are studied in Chap. 9. After explaining the piezoelectric effect and its physical modelling, we concentrate on the efficient numerical computation of such systems. Whereas for sensor applications a linear model can be usually used, in many actuator applications non-linear effects play a crucial role, which we here account for by applying an appropriate hysteresis model. Since the efficiency of the solution (both with respect to elapsed CPU time and computer memory resources) is of great importance, Chap. 11 deals with geometric and algebraic multigrid solvers. These methods achieve an optimal complexity, that is, the computational effort as well as memory requirement grows only linearly with the problem size. We present new especially adapted multigrid solvers for Maxwell’s equation in the eddy current case and demonstrate their efficiency by means of TEAM (Testing Electromagnetic Analysis Methods) workshop problem 20 established by the Compumag Society [223]. After these rigorous derivations of methods for coupled field computation, Chap. 12 demonstrates the applicability to real-life problems arising in industry. This includes the following topics: • • • • •

Analysis and optimization of car loudspeakers Acoustic emission of electrical power transformers Simulation-based improvements of electromagnetic valves Piezoelectric stack actuators such as used, e.g., in common-rail diesel injection systems Ultrasonic imaging system based on capacitive micromachined ultrasound transducers

The appendices provide an introduction to vector analysis, functional spaces, and the solutions of non-linear equations. The structure of this book has been designed in such a way that in each of Chaps. 3 – 9 we first discuss the physical modelling of the corresponding single or coupled field, then the numerical simulation, followed by a simple computational example. If the reader has no previous knowledge of vector analysis, she/he should start with the first section of the Appendix. Chapter 2 can be omitted if the reader is only interested in the physical modelling of mechatronic sensors and actuators. The three chapters concerning the single field problems (mechanical, electromagnetic, acoustic) are written independently, so that the reader can start with any of them. Clearly, for the coupled field problems, the reader should have a knowledge of the involved physical fields or should have read the corresponding preceding chapters. Chapter 11 presents the latest topics on multigrid methods for electromagnetic fields and requires some knowledge on this topic. For a basic introduction of multigrid methods we refer to classical books [36, 185, 186]. Chapter 12 demonstrates

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the use of numerical simulation for industrial applications. For each of them, we first discuss the problem to be solved, followed by an analysis study applying numerical simulation to allow a better understanding of the different physical effects. For most applications, we also demonstrate measurements of the CAE-optimized prototype. Most in this book described formulations for solving multifield problems have been implemented in the software CAPA (see www.wissoft.de).

Acknowledgements The author wishes to express his gratitude to all the people who have inspired, increased, and sustained this work. Of course we feel obliged towards the Department of Sensor Technology in Erlangen and the previous Institute of Measurement Technology in Linz under the competent and generous leadership of Prof. Reinhard Lerch. The dynamic and stimulating atmosphere at the institute was certainly essential for this work; the author therefore thanks all his present and former colleagues among whom especially Dr. Reinhard Simkovics, Dr. Martin Rausch, Dr. Johann Hoffelner, Dr. Manfred Hofer, and Dipl.-Ing. Michael Ertl have to be mentioned. Much is owed to the long and fruitful cooperation with Dipl.-Math Hermann Landes and his company WisSoft. The author also thanks Dr. Stefan Reitzinger for many intensive and productive hours of work and discussion, and my wife Dozent Dr. Barbara Kaltenbacher for her assistance on mathematical problems and the cooperation on precise material parameter determination applying inverse methods. Moreover, we acknowledge the constructive working environment within the special research programs SFB 013 Numerical and Symbolic Scientific Computing in Linz (Prof. Ulrich Langer, Dr. Joachim Sch¨ oberl, Dr. Michael Schinnerl) funded by the Austrian science foundation FWF, and SFB 603 Modellbasierte Analyse und Visualisierung komplexer Szenen und Sensordaten in Erlangen (Dipl.-Math. Elena Zhelezina, Dr. Roberto Grosso, Dipl.-Inf. Frank Reck) funded by the DFG (Deutsche Forschungsgemeinschaft) (German Research Foundation) as well as the BMBF project Entwurf komplexer SensorAktor-Systeme (Prof. Peter Schwarz, Dipl.-Ing. Rainer Peipp). Additionally, the author thanks the industrial partners involved in this work for the opportunity of doing research on real-life problems. Finally, I would like to thank my copyeditor Dr. Peter Capper for reading the book very carefully and pointing out many errors and misspellings.

Manfred Kaltenbacher December 2003

Notation

Mathematical symbols e n t I IR r
C  C
Γ  Γ


ds

unit vector unit normal vector unit tangential vector unit matrix set of real numbers position vector contour integral

ds

closed contour integral

dΓ

surface integral

dΓ

closed surface integral

dΩ

volume integral

Ω

∇

nabla operator

curl , ∇× curl div , ∇·

divergence

grad , ∇ gradient ∂/∂x

partial derivative

∂/∂n

partial derivative in normal direction

d/ dx

total derivative

Finite Element Method u, a, etc. nodal vectors of displ., acceleration, etc. C damping matrix K stiffness matrix M mass matrix number of nodes nn nen number of nodes per finite element number of elements ne neq number of equations nd space dimension J Jacobi matrix |J | Jacobi determinant x, y, z global coordinates ¯ Ω whole simulation domain Ω simulation domain without boundary Γ boundary of simulation domain Dirichlet boundary Γe Γn Neumann boundary γP integration parameter (parabolic PDE) βH , γH integration parameters (hyperbolic PDE) ξ, η, ζ local coordinates

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Notation

Acoustics b diffusivity of sound B/A parameter of non-linearity c speed of sound sound-field intensity Ia k wave number adiabatic bulk modulus Ks KT isothermal bulk modulus Lp , SP L sound-pressure level LIa sound-intensity level sound-power level LPa p
acoustic pressure Pa acoustic power acoustic particle velocity v
wa acoustic energy density Za acoustic impedance acoustic density ρ
ψ scalar acoustic potential κ adiabatic exponent λ wavelength bulk viscosity ζv shear viscosity µv xs shock formation distance

Electromagnetics A B D E Fel Fmag H I, i J Ji M qe Qe P R uind Ve wel

magnetic vector potential magnetic flux density electric flux density electric field intensity electric force magnetic force magnetic field intensity electric current current density impressed current density magnetization electric charge density total electric charge electric polarization ohmic resistor induced voltage scalar electric potential electric energy density

Wel wmag Wmag ρe γ µ ν ε σe φ ψm Ψ δ

total electric energy magnetic energy density total magnetic energy specific electric resistance electrical conductivity magnetic permeability magnetic reluctivity electric permittivity electric surface charge magnetic flux reduced magnetic scalar potential total magnetic flux skin depth

Mechanics a [c] cL cT Em fV [Fd ] [Hd ] G m Pmech S [S] T [T] u v V [V] αM , αK ρ νp σ [σ] µL , λL

acceleration tensor of mechanical modulus velocity of longitudinal wave velocity of shear wave elasticity module volume force deformation gradient displacement gradient shear modulus mass mechanical power vector of linear strains tensor of linear strains Piola-Kirchhoff stress vector 2nd Piola-Kirchhoff stress tensor mechanical displacement velocity Green-Lagrangian strain vector Green-Lagrangian strain tensor damping coefficients density Poisson ratio Cauchy stress vector Cauchy stress tensor Lam´e-parameters

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

The Finite Element (FE) Method . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Finite Element Method for a 1D Problem . . . . . . . . . . . . . . . . . . . 2.3 Nodal (Lagrangian) Finite Elements . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Quadrilateral Element in IR2 . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Triangular Element in IR2 . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Tetrahedron Element in IR3 . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Hexahedron Element in IR3 . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.6 Global/Local Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.7 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Finite Element Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Time Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Parabolic Differential Equation . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Hyperbolic Differential Equation . . . . . . . . . . . . . . . . . . . . 2.6 Integration over Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Edge (N´ ed´ elec) Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Discretization Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 9 13 20 21 23 26 26 27 29 30 32 35 36 40 43 43 45

3

Mechanical Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Navier’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Deformation and Displacement Gradient . . . . . . . . . . . . . . . . . . . 3.3 Mechanical Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Plane Strain State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Plane Stress State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Axisymmetric Stress–Strain Relations . . . . . . . . . . . . . . . . 3.5 Waves in Solid Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 55 56 60 61 62 63 63 65

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3.7 Numerical Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Damping Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.3 Geometric Non-linear Case . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Locking and Efficient Solution Approaches . . . . . . . . . . . . . . . . . . 3.8.1 Incompatible Modes Method . . . . . . . . . . . . . . . . . . . . . . . . 3.8.2 Enhanced Assumed Strain Method . . . . . . . . . . . . . . . . . . 3.8.3 Balanced Reduced and Selective Integration . . . . . . . . . .

66 66 69 70 77 78 81 83 85

4

Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.2 Quasistatic Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.2.1 Magnetic Vector Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.2.2 Skin Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.3 Electrostatic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.4 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.4.1 Magnetic Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.4.2 Electrical Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.4.3 Dielectric Permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.5 Electromagnetic Interface Conditions . . . . . . . . . . . . . . . . . . . . . . 109 4.5.1 Continuity Relations for Magnetic Field . . . . . . . . . . . . . . 109 4.5.2 Continuity Relations for Electric Field . . . . . . . . . . . . . . . 111 4.5.3 Continuity Relations for Electric Current Density . . . . . 113 4.6 Numerical Computation: Electrostatics . . . . . . . . . . . . . . . . . . . . . 113 4.7 Numerical Computation: Electromagnetics . . . . . . . . . . . . . . . . . . 114 4.7.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.7.2 Discretization with Edge Elements . . . . . . . . . . . . . . . . . . . 121 4.7.3 Discretization with Nodal Finite Elements . . . . . . . . . . . . 122 4.7.4 Newton’s Method for the Non-linear Case . . . . . . . . . . . . 125 4.7.5 Approximation of B–H Curve . . . . . . . . . . . . . . . . . . . . . . . 129 4.7.6 Modelling of Current-loaded Coil . . . . . . . . . . . . . . . . . . . . 131 4.7.7 Computation of Global Quantities . . . . . . . . . . . . . . . . . . . 131 4.7.8 Induced Electric Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.8 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.8.1 Ferromagnetic Cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.8.2 Thin Iron Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5

Acoustic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.1 Wave Theory of Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.1.1 Conservation of Mass (Continuity Equation) . . . . . . . . . . 140 5.1.2 Conservation of Momentum (Euler Equation) . . . . . . . . . 141 5.1.3 Pressure-Density Relation (State Equation) . . . . . . . . . . . 143 5.1.4 Linear Acoustic Wave Equation . . . . . . . . . . . . . . . . . . . . . 145 5.1.5 Acoustic Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

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5.1.6 Plane and Spherical Waves . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.2 Quantitative Measure of Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.3 Non-linear Acoustic Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . 156 5.4 Numerical Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5.4.1 Linear Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.4.2 Non-linear Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.4.3 Non-matching Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5.4.4 Discretization Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 5.5 Treatment of Open Domain Problems . . . . . . . . . . . . . . . . . . . . . . 173 5.5.1 Absorbing Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 175 5.5.2 Perfectly Matched Layer (PML) Technique . . . . . . . . . . . 176 5.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 5.6.1 Transient Wave Propagation in Unbounded Domain . . . 184 5.6.2 Harmonic Wave Propagation in Unbounded Domain . . . 186 5.6.3 Acoustic Pulse Propagation over a Non-Matching Grid . 188 5.6.4 Non-linear Wave Propagation in a Channel . . . . . . . . . . . 190 6

Coupled Electrostatic-Mechanical Systems . . . . . . . . . . . . . . . . . 195 6.1 Electrostatic Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 6.2 Numerical Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 6.2.1 Calculation Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 6.2.2 Voltage-driven Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

7

Coupled Magnetomechanical Systems . . . . . . . . . . . . . . . . . . . . . . 207 7.1 General Moving/Deforming Body . . . . . . . . . . . . . . . . . . . . . . . . . . 207 7.2 Electromagnetic Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 7.3 Numerical Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 7.3.1 Electric Circuit Coupling: Voltage-loaded Coil . . . . . . . . . 211 7.3.2 Force Computation via the Principle of Virtual Work . . 213 7.3.3 Calculation Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 7.3.4 Moving Current/Voltage-loaded Coil . . . . . . . . . . . . . . . . . 218

8

Coupled Mechanical-Acoustic Systems . . . . . . . . . . . . . . . . . . . . . 229 8.1 Solid–Fluid Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 8.2 Coupled Field Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 8.3 Numerical Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 8.3.1 Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 233 8.3.2 Non-matching Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 8.3.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

9

Piezoelectric Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 9.1 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 9.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 9.3 Piezoelectric Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 245 9.4 Models for Non-linear Piezoelectricity . . . . . . . . . . . . . . . . . . . . . . 249

XVI

Contents

9.5 Hysteresis Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 9.6 Numerical Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 9.6.1 Linear Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 9.6.2 Non-linear Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 9.7 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 9.7.1 Computation of Impedance Curve . . . . . . . . . . . . . . . . . . . 261 9.7.2 Non-linear Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 10 Computational Aeroacoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 10.1 Requirements for Numerical Schemes . . . . . . . . . . . . . . . . . . . . . . 267 10.2 Lighthill’s Analogy and its Extension . . . . . . . . . . . . . . . . . . . . . . 270 10.3 Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 10.4 Validation: Co-Rotating Vortex Pair . . . . . . . . . . . . . . . . . . . . . . . 279 11 Algebraic Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 11.1 Preconditioned Conjugate Gradient (PCG) Method . . . . . . . . . . 283 11.2 Multigrid (MG) Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 11.3 Geometric MG Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 11.3.1 Geometric MG for Edge Elements . . . . . . . . . . . . . . . . . . . 288 11.3.2 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 11.4 Algebraic MG Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 11.4.1 Auxiliary Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 11.4.2 Coarsening Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 11.4.3 Prolongation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 11.4.4 Smoother and Coarse-grid Operator . . . . . . . . . . . . . . . . . 299 11.4.5 AMG for Nodal Elements . . . . . . . . . . . . . . . . . . . . . . . . . . 300 11.4.6 AMG for Edge Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 11.4.7 AMG for Time-harmonic Case . . . . . . . . . . . . . . . . . . . . . . 304 11.4.8 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 12 Industrial Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 12.1 Electrodynamic Loudspeaker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 12.1.1 Finite Element Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 12.1.2 Verification of Computer Models . . . . . . . . . . . . . . . . . . . . 316 12.1.3 Numerical Analysis of the Non-linear Loudspeaker Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 12.1.4 Computer Optimization of the Non-linear Loudspeaker Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 12.2 Noise Computation of Power Transformers . . . . . . . . . . . . . . . . . . 320 12.2.1 Finite Element Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 12.2.2 Verification of the Computer Models . . . . . . . . . . . . . . . . . 325 12.2.3 Verification of the Calculated Winding and Tank-surface Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 12.2.4 Verification of the Sound-field Calculations . . . . . . . . . . . 328 12.2.5 Influence of Tap-changer Position . . . . . . . . . . . . . . . . . . . . 328

Contents

XVII

12.2.6 Influence of Stiffness of Winding Supports . . . . . . . . . . . . 330 12.3 Fast-switching Electromagnetic Valves . . . . . . . . . . . . . . . . . . . . . 330 12.3.1 Modelling and Solution Strategy . . . . . . . . . . . . . . . . . . . . 331 12.3.2 Actuator Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 12.3.3 Actuator Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 12.3.4 Dynamics Optimization I: Electrical Premagnetization . 336 12.3.5 Dynamics Optimization II: Overexcitation . . . . . . . . . . . . 337 12.3.6 Switching Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 12.4 Cofired Piezoceramic Multilayer Actuators . . . . . . . . . . . . . . . . . . 340 12.4.1 Setup of Multilayer Stack Actuators . . . . . . . . . . . . . . . . . 341 12.4.2 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 12.4.3 Measured and Simulated Results . . . . . . . . . . . . . . . . . . . . 343 12.5 Capacitive Micromachined Ultrasound Transducers . . . . . . . . . . 345 12.5.1 Requirements to Numerical Simulation Scheme . . . . . . . . 346 12.5.2 Single CMUT Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 12.5.3 CMUT Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 12.5.4 Controlled CMUT Array . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 12.6 High-Intensity Focused Ultrasound . . . . . . . . . . . . . . . . . . . . . . . . 357 12.6.1 Piezoelectric Transducer and Input Impedance . . . . . . . . 357 12.6.2 Pressure Pulse Computation . . . . . . . . . . . . . . . . . . . . . . . . 357 12.6.3 High-Power Pulse Sources for Lithotripsy . . . . . . . . . . . . . 361 12.7 Noise Generated from a Flow around a Square Cylinder . . . . . . 366 12.7.1 3D Flow and 2D Acoustic Computations . . . . . . . . . . . . . 366 12.7.2 3D Flow and 3D Acoustic Computations . . . . . . . . . . . . . 377 A

Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 A.1 Vector Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 A.2 Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

B

Scalar and Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 B.1 The Nabla (∇) Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 B.2 Definition of Gradient, Divergence, and Curl . . . . . . . . . . . . . . . . 386 B.3 The Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 B.4 The Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 B.5 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 B.6 Divergence Theorem (Gauss Theorem) . . . . . . . . . . . . . . . . . . . . . 392 B.7 The Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 B.8 The Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 B.9 Stoke’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 B.10 Green’s Integral Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 B.11 Application of the Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 B.12 Irrotational Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 B.13 Solenoidal Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

C

Appropriate Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

XVIII Contents

D

Solution of Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 D.1 Fixed-point Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 D.2 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

1 Introduction

Each modern industrial process environment needs sensors to detect the physical quantities involved (e.g., electric current, mechanical torque, temperature, etc.), a signal-conditioning circuit, and an interface to computers, where the process parameters are controlled. According to the controlling signals, power

Fig. 1.1. Industrial process

electronic circuits supply the actuators, which will steer the process (see Fig. 1.1). As indicated by the gray boxes in Fig. 1.1, the main topic of this book is concerned with sensors and actuators, especially their sophisticated design. Therefore, we provide a comprehensive discussion on the precise physical modelling of sensors and actuators and, furthermore, on the numerical solution of the governing partial differential equations. To be precise, we concentrate our investigation on mechatronic sensors and actuators. Mechatronic sensors, actuators and sensor-actuator systems are based on the mutual interaction of

2

1 Introduction

(a) Electromagnetic valve

(b) Capacitive micromachined ultrasound transducers (CMUT) Fig. 1.2. Examples of mechatronic sensors and actuators

the mechanical field and an electrostatic, an electromagnetic, or an acoustic field. Typical examples are: electromagnetic valves for injection systems in vehicles, capacitive micromachined ultrasound transducers (CMUTs) for medical imaging systems, electrodynamic loudspeakers, surface acoustic wave (SAW) devices for telecommunications (e.g., in a mobile phone), piezoelectric or magnetostrictive actuators for ultrasound cleaning, etc. (see Fig. 1.2). The main transducing mechanisms for mechatronic sensor and actuators are displayed in Fig. 1.3 (those in boldface will be discussed within this book). In most cases, the fabrication of prototypes within the design process of modern mechatronic transducers is a lengthy and costly task. The still widely used experimental-based design, as shown in Fig. 1.4(a) exhibits many disadvantages. First, it is very time-consuming, since for each change in the design a new prototype has to be fabricated and the relevant parameters have to be measured. In particular, if a design failure is recognized late, high costs result.

1 Introduction

3

Fig. 1.3. Transducing mechanisms of mechatronic sensors and actuators

Furthermore, not all parameters of interest can be measured (e.g., magnetic field or mechanical stresses inside a solid body) and the measurement setup may influence the (dynamical) behavior of the prototype.

(a) Experimental-based design

(b) CAE-based design

Fig. 1.4. Design process

Since for the development of modern mechatronic transducers all the different coupling mechanisms of the involved physical fields have to be considered, the design process is a very complex task. Therefore, an increasing need for reliable and usable computer modelling tools capable of precisely simulating the multifield interactions arises. Such appropriate computer-aided

4

1 Introduction

engineering (CAE) tools offer many possibilities to the design engineer. Arbitrary modification of transducer geometry and selective variation of material parameters are easily performed and the influence on the transducer behavior can be studied immediately. In addition, the simulation provides access to physical quantities that cannot be measured, and simulations strongly support the insight into physical phenomena. Thus, a CAE-based design as displayed in Fig. 1.4(b) can tremendously reduce the number of necessary prototypes within the design process. However, we want to emphasize that a direct physical control of the transducer design is possible only with the help of experiments, whereas the computer simulation is always based on a model of reality. Therefore, the quality of the results depends on the suitability of the physical model as well as the material parameters. Moreover, numerical effects may spoil the results unless proper computational methods are used. For these reasons, one should also be aware of the risk of trusting every result the CAE environment computes. These facts make it very important that the user of such tools has both a deep physical understanding of the ongoing processes and mathematical knowledge of the simulation algorithms. Summarizing, one can say that an appropriate use of CAE tools for the design of mechatronic transducers can tremendously reduce the number of necessary prototypes within the design process, but the user should always critically question each result. The use of these CAE tools in the design of mechatronic transducers started only some years ago. The main reason for the lack of CAE environments, capable of performing multifield simulations that arise in the design of mechatronic sensors and actuators, is the complexity of such problems. We will demonstrate this by discussing the requirements on a CAE tool for the numerical simulation of an electrodynamic loudspeaker. Figure 1.5 shows an electrodynamic loudspeaker with all its components. A cylindrical, small light voice coil is suspended freely in a strong radial magnetic field, generated by a permanent magnet. The magnet assembly, consisting of pole, back plate and top plate, helps to concentrate most of the magnetic flux within the magnet structure and, therefore, within the narrow radial gap. When the coil is loaded by an electric voltage, the interaction between the magnetic field of the permanent magnet and the current in the voice coil results in an axial Lorentz force. The voice coil is wound onto an aluminum holder, which is attached to the rigid, light cone diaphragm in order to couple the forces more effectively to the air and, hence, to permit acoustic power to be radiated from the assembly. The dust cap does not usually fulfill any acoustic function, but prevents the penetration of dust into the air gap of the magnetic assembly. The main function of the suspension and the surround is to allow free axial movement of the moving coil driver, while nonaxial movements are suppressed almost completely. To obtain a precise description of an electrodynamic loudspeaker we have to model the electromagnetic, mechanical, and acoustic fields, and in addition, the following coupling mechanisms:

1 Introduction

5

Fig. 1.5. Electrodynamic loudspeaker

•

Mechanical Field – Acoustic Field The normal component of the surface velocity of the solid must meet the normal component of the particle velocity of the fluid.

•

Mechanical Field – Magnetic Field The interaction between these two physical fields is given by the Lorentz force and the motional electromotive force (emf) term leads to additional eddy currents in the coil.

•

Electric Circuit – Magnetic Field Due to the fact that the coil is voltage driven, the magnetic field equation has to be solved together with the electric circuit equation.

For the design of electrodynamic loudspeakers, the frequency dependence of the axial pressure response at 1 m distance and the electrical input impedance of the voice coil are the two most important parameters. Since the overall physical model consists of a system of coupled, non-linear partial differential equations a transient analysis has to be performed and then the results are transformed to the frequency domain. To obtain a good frequency resolution within the frequency range of interest (0 − 20 kHz) around 10 000 time steps have to be computed. Since the coupled system of partial differential equations cannot be solved analytically, a numerical scheme has to be applied. The finite element (FE) method has been established as the standard method for numerically solving the partial differential equations describing the physical fields including their couplings. Thus, a static, transient and time-harmonic analysis including non-linearities (e.g., material non-linearities, geometric non-linearities, etc.) can be performed very efficiently. This method is currently used in most commercial computer codes. Of course it has to be mentioned that for different physical field problems (e.g., fluid dynamics, high-frequency electromagnetics, etc.) different numerical methods (e.g., finite

6

1 Introduction

volume, finite difference, finite integration, boundary element) might be the methods of choice [22, 33, 64, 66, 213].

2 The Finite Element (FE) Method

The finite element (FE) method has become the standard numerical calculation scheme for the computer simulation of physical systems [16, 104, 226]. The advantages of this method can be summarized as follows: •

Numerical efficiency: The discretization of the calculation domain with finite elements yields matrices that are in most cases sparse and symmetric. Therefore, the system matrix, which is obtained after spatial and time discretization, is sparse and symmetric, too. Both the storage of the system matrix and the solution of the algebraic system of equations can be performed in a very efficient way.

•

Treatment of non-linearities: The modelling of non-linear material behavior is well established for the FE method (e.g., non-linear curves, hysteresis).

•

Complex geometry: By the use of the FE method, any complex domain can be discretized by triangular elements in 2D and by tetrahedra in 3D.

•

Analysis possibilities: The FE method is suited for static, transient, harmonic as well as eigenfrequency analysis.

The two essential disadvantages of the FE method are given by •

Discretization: The effort for the discretization of the simulation domain is quite high, since in 2D the whole cross section and in 3D the whole volume has to be subdivided into finite elements.

•

Open domain problems: Models that need the treatment of an open boundary, e.g., the simulation of radiation characteristics of an ultrasound array, lead in the general case to errors due to the limitation of the simulation domain. One of several approaches to overcome this problem is

8

2 The Finite Element (FE) Method

the use of absorbing boundary conditions, perfectly matched layer (PML) techniques or so-called infinite elements (see Sect. 5.5).

Fig. 2.1. From the strong formulation to the algebraic system of equation

The general approach of the FE method is shown in Fig. 2.1. Starting from the partial differential equation (PDE) with given boundary conditions, we multiply it by appropriate test functions and integrate over the whole simulation domain. Performing a partial integration, we arrive at the variational formulation, also called the weak formulation. Applying Galerkin’s approximation method using finite elements (FE) results in the algebraic system of equations. As already mentioned, the use of the FE method requires the discretization of the whole domain (see Fig. 2.2). For the discretization triangular as well as quadrilateral finite elements are used in 2D and tetrahedral as well as hexahedron finite elements in 3D. The physical quantity of interest (e.g., temperature, mechanical displacement, etc.) is approximated by so-called shape functions and the solution of the algebraic equation yields the physical quantity in the discretization points, the so-called finite element nodes, for Lagrangian finite elements and along the edges for N´ed´elec finite elements.

Fig. 2.2. FE method: Discretization of the domain with quadrilateral finite elements

2.1 Finite Element Formulation

9

2.1 Finite Element Formulation In the following, all steps—from the strong formulation of the partial differential equation (PDE) to the algebraic equation—will be briefly described by means of the following simple PDE with the searched for quantity u(r, t), the known source term f (r, t) at each time t of the interval (0, T ), and the corresponding initial and boundary conditions. Given: f : Ω × (0, T ) → IR u0 : Ω → IR ¯ × [0, T ] → IR Find: u : Ω ∂u = ∇ · ∇u + f ∂t

(2.1)

u = ue on Γe × (0, T ) ∂u = un on Γn × (0, T ) ∂n u(r, 0) = u0 , r ∈ Ω . In this so-called strong formulation of the initial-boundary value problem IR ¯ the simulation domain, Ω the simulation denotes the set of real numbers, Ω domain without the boundary Γ = Γe ∪ Γn , Γe the boundary with prescribed Dirichlet boundary condition, and Γn the boundary with prescribed Neumann boundary condition. Now, let us introduce for any t ∈ [0, T ] the space Tt Tt = {u(·, t) | u(·, t) ∈ H 1 (Ω), u(r, t) = ue (r, t) on Γe } ,

(2.2)

and G, the space of so-called test functions, as G = {w | w ∈ H 1 (Ω), w = 0 on Γe } ,

(2.3)

with H 1 the standard Sobolev space (see Appendix C). It has to be noted that the spaces Tt , t ∈ [0, T ] vary with time, whereas the space G is timeindependent. In the first step, we multiply the partial differential equation with an arbitrary test function w and perform an integration over the whole domain Ω    ∂u − ∇ · ∇u − f dΩ = 0 . w ∂t Ω Applying Green’s first integration theorem to the above equation results in     ∂u ∂u dΩ + (∇w) · (∇u) dΩ = dΓ . (2.4) w wf dΩ + w ∂t ∂n Ω Ω Ω Γn

10

2 The Finite Element (FE) Method

Thus, the weak formulation (often also called variational formulation) for the initial-boundary problem is as follows: Given: f u0

: Ω × (0, T ) → IR : Ω → IR

Find: u(t) ∈ Tt such that for all w ∈ G and t ∈ [0, T ]  w Ω

∂u dΩ + ∂t



 (∇w) · (∇u) dΩ =

Ω

 wf dΩ +

Ω



wun dΓ

(2.5)

Γn

u = ue on Γe × (0, T )  wu(0) dΩ = wu0 dΩ .

Ω

Ω

Since the Neumann boundary condition is now incorporated into the equation of the weak form, it is also called natural. The Dirichlet boundary condition on u still has to be explicitely forced, and is therefore called essential. Formally it can be proven that the two formulations according to (2.1) and (2.5) are mathematically equivalent, provided u is sufficiently smooth [99]. To discretize (2.5), which is still infinite dimensional, we now apply the approximation according to Galerkin’s method. Let us define the finite dimensional spaces Tht and Gh according to Tht ⊂ T

Gh ⊂ G .

Therefore, we perform the domain discretization (see Fig. 2.2), and approximate the searched for quantity u(t) as well as the test function w by u(t) ≈ uh (t)

w ≈ wh ,

(2.6)

with h the discretization parameter (defining the mesh size). Furthermore, we decompose uh (t) into the searched for value v h (t) and the known Dirichlet values uhe (t). For v h , wh , and uhe we choose the following ansatz v h (t) =

neq 

Na (r)va (t)

(2.7)

Na (r)ca

(2.8)

Na (r)uea (t) ,

(2.9)

a=1 neq

wh = uhe (t) =



a=1 ne  a=1

where Na (r) denotes appropriate shape functions (often also called interpolation or basis functions), neq the number of unknowns, which is equal to the

2.1 Finite Element Formulation

11

number of finite element nodes with no Dirichlet boundary condition, and ne the number of finite element nodes with Dirichlet boundary condition. Substituting (2.7) – (2.9) into (2.5) results in  neq    neq     neq neq   ∂  N a ca Nb vb dΩ + ∇ N a ca · ∇ Nb vb dΩ ∂t Ω a=1 Ω a=1 b=1 b=1  neq    n     neq ne e   ∂  + N a ca Nb ueb dΩ + ∇ N a ca · ∇ Nb ueb dΩ ∂t Ω a=1 Ω a=1 b=1 b=1   neq   neq Na ca f (ra ) dΩ + Na ca un (ra ) dΓ . (2.10) = Ω a=1

Γn a=1

Now, since we can put the sums before the integrals and having in mind that ∇ just operates on the shape functions N (r) (ca as well as ub are constants with respect to the space variables), we may write (2.10) for the 2D plane case as follows  neq neq     ∂vb ca Na Nb dΩ ∂t Ω a=1 b=1

 

∂Na ∂Nb ∂Na ∂Nb + + ∂x ∂x ∂y ∂y Ω   − Na f dΩ − Na un dΓ Ω

+

+

Ω

ne   b=1





dΩ vb

Γn

ne    b=1



Ω

∂Na ∂Nb ∂Na ∂Nb + ∂x ∂x ∂y ∂y 

Na Nb dΩ

∂ueb ∂t



 dΩ ueb

= 0.

Since the equation has to be fulfilled for all coefficients ca , we obtain the defining equations for the searched for finite element node values vb : for each a (a = 1, .., neq ) we have to solve an equation as follows

12

2 The Finite Element (FE) Method

b=1

 



neq   

∂vb + Na Nb dΩ ∂t Ω Na f Ω ne  

dΩ −





dΩ vb



Na un dΓ Γn

∂Na ∂Nb ∂Na ∂Nb + ∂x ∂x ∂y ∂y Ω b=1

  ne  ∂ueb + Na Nb dΩ = 0. ∂t Ω +





 −

Ω

∂Na ∂Nb ∂Na ∂Nb + ∂x ∂x ∂y ∂y



 dΩ ueb (2.11)

b=1

Thus, the semidiscrete Galerkin formulation can be written in matrix form as follows (2.12) Mv˙ + Kv = f , with v˙ = ∂v/∂t, v the nodal unknowns and f the right-hand side vector. •

Mass matrix M: M = [Mab ]  Mab = Na Nb dΩ

(2.13)

Ω

1 ≤ a, b ≤ neq •

Stiffness Matrix K: K = [Kab ]    ∂Na ∂Nb ∂Na ∂Nb Kab = + dΩ ∂x ∂x ∂y ∂y Ω

(2.14)

1 ≤ a, b ≤ neq •

Right-hand side f : f = [fa ]   fa = Na f dΩ + Ω ne  



Na un dΓ

Γn

∂Na ∂Nb ∂Na ∂Nb − + ∂x ∂x ∂y ∂y Ω b=1  ne   ∂ueb − Na Nb dΩ ∂t Ω



 dΩ ueb (2.15)

b=1

1 ≤ a ≤ neq 1 ≤ b ≤ ne

(2.16)

2.2 Finite Element Method for a 1D Problem

13

This example was supposed to illustrate the main steps of the FE method. Note that the mass and stiffness matrix may take different forms depending on the physical phenomena they model and on the material parameters (see Sects. 3, 4, and 5). The resulting equation (2.12) is still infinite dimensional due to the time dependence. Therefore, in Sect. 2.5 we will discuss timediscretization schemes, in order to arrive at the algebraic system of equations. Before, we will provide a detailed description of the FE method by means of applying it to a 1D problem, followed by discussing all steps necessary for computer implementation.

2.2 Finite Element Method for a 1D Problem In order to illustrate the main idea of the FE method, we will consider the following 1D differential equation −

∂2u + c u = f (x) ∂x2 u(a) = ua

(2.17)

u(b) = ub , where [a, b] defines the computational domain. As described in Sec. 2.1, the first step is to derive the weak form of (2.17). For this purpose, we choose an appropriate test function v, multiply (2.17) by this test function and integrate over the whole domain b

 2  ∂ u v − 2 + c u − f (x) dx . ∂x

(2.18)

a

For the first term in (2.18) we perform an integration by parts b a

b b ∂v ∂u ∂2u ∂u

dx . v 2 dx = v −

∂x ∂x a ∂x ∂x a

Provided that the test function v(x) vanishes on the Dirichlet boundary (first restriction on the test function, second one will be the existence of a first-order derivative in the weak sense, see Sec. C), we obtain for (2.18) b  a

∂v ∂u + cvu ∂x ∂x



b dx =

vf dx . a

Therewith, the weak (variational) formulation reads as follows:

(2.19)

14

2 The Finite Element (FE) Method

Given: f, c : [a, b] → IR Find: u ∈ V = {u ∈ H 1 (a, b)|u(a) = ua , u(b) = ub } such that for all v ∈ W = {v ∈ H 1 (a, b)|v(a) = v(b) = 0} a(u, v) =< f, v >

(2.20)

with b  a(u, v) =

∂v ∂u + cvu ∂x ∂x

 dx

a

b < f, v > =

vf dx . a

In (2.20) a(u, v) is called a bilinear form and < f, v > an inner product in the specified functional space. In the next step, we divide the computational domain into cells, so-called finite elements. Therewith, in our case, we divide the interval [a, b] into a set of smaller intervals [xi−1 , xi ], i = 1, ..., M such that the following properties are fulfilled •

Ascending order of node positions xi−1 < xi for i = 1, ..., M

•

Complete covering of the domain [a, b] =

M 

[xi−1 , xi ]

x0 = a, xM = b

i=1

•

No intersection of intervals [xi−1 , xi ] ∩ [xj−1 , xj ] = 0 for i = j

For simplicity, we choose an equidistant discretization, so that we obtain (see Fig. 2.3) a−b xi = a + ih h = i = 0, ..., M . M The unknown quantity u(x) is now approximated by a linear combination of finite functions with local support, which means that these functions are just different from zero in a ’small’ interval (see Fig. 2.4). Such a choice is given e.g., by piecewise linear hat-functions, defined as follows (see Fig. 2.5)

2.2 Finite Element Method for a 1D Problem

15

Fig. 2.3. Subdivision of the computational domain into finite elements

Fig. 2.4. Finite element function with local support: suppN (x) = [α, β]

Ni (x) =

⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x−xi−1

a ≤ x ≤ xi−1

⎪ ⎪ ⎪ ⎪ ⎪ ⎩

xi < x ≤ xi+1

xi−1 < x ≤ xi

h xi+1 −x h

(2.21)

xi+1 < x ≤ b

0

Our chosen ansatz (shape) functions fulfill the requirement

Fig. 2.5. Piecewise, linear hat-functions

Ni (xj ) = δij =

1

i=j

0

i = j

and the approximation of the unknown u(x) is given by

(2.22)

16

2 The Finite Element (FE) Method

u(x) ≈ uh (x) =

M−1 

Ni (x)ui + N0 (x)ua + NM (x)ub

(2.23)

i=1

uh (x = xi ) = ui .

(2.24)

The discretized weak (variational) formulation reads as Given: f, c : [a, b] → IR Find: uh ∈ Vh = {uh (x) = v h ∈ Wh = {v h (x) =

M−1 

Ni (x)ui + N0 ua + NM ub } such that for all

i=1 M−1 

Ni (x)vi }

i=1

a(uh , v h ) = < f, v h >

(2.25)

with b  a(uh , v h ) =

∂v h ∂uh + cv h uh ∂x ∂x

 dx

a

b < f, v h > =

v h f dx . a

For the finite dimensional functional spaces V h , W h we have the property V h ⊂ V , Wh ⊂ W. Now, we have to set up the algebraic system of equations in order to obtain the unknowns ui at all the finite element nodes within our grid. Using the approximation according to (2.23) for u as well as the test function v results in ⎛ ⎞ M−1  b M−1  ∂ ∂ ⎝ Ni (x)vi Nj (x)uj + N0 ua + NM ub ⎠ dx ∂x i=1 ∂x j=1 a ⎞  ⎛M−1 b M−1   + c Na (x)va ⎝ Nj (x)uj + N0 ua + NM ub ⎠ dx a

i=1

j=1

−

b M−1  a

 Ni (x)vi

f dx = 0 .

i=1

Considering that we can interchange the integrals and the sums, and that all vi as well as uj are constants (no function of x), we obtain

2.2 Finite Element Method for a 1D Problem M−1  i=1

⎛ vi ⎝

M−1 

b  uj

j=1

∂Ni ∂Nj + cNi Nj ∂x ∂x

17

 dx

a

b  +

∂Ni ∂x

a

b −



∂NM ∂N0 ua + ub ∂x ∂x ⎞



 + cNi (N0 ua + NM ub )

dx

Ni f dx ⎠ = 0 .

a

Letting vi with i = 1, .., M − 1 run through all unit vectors in IRM , we obtain for each i an equation M−1 

Sij uj = fi

i = 1, ..., M − 1

j=1

Su = f

(2.26)

with b  Sij =

∂Ni ∂Nj + cNi Nj ∂x ∂x

 dx

(2.27)

a

b

b Ni f dx −

fi = a

∂Ni ∂x



∂N0 ∂NM ua + ub ∂x ∂x

 dx

a

b −

cNi (N0 ua + NM ub ) dx .

(2.28)

a

According to the properties of our chosen ansatz functions (see (2.22)), we get the following pattern for our system matrix S (see Fig. 2.6) ⎛ ⎞ ∗ ∗ 0 0 ··· 0 ··· 0 ⎜ ⎟ ⎜∗ ∗ ∗ 0 ··· 0 ··· 0⎟ ⎜ ⎟ 0 j ∈ {i − 1, i, i + 1} ⎜ ⎟ ⎜0 ∗ ∗ ∗ ··· 0 ··· 0⎟ Sij = ⎜ ⎟ ∗ j ∈ {i − 1, i, i + 1} ⎟ S=⎜ ⎜ .. .. .. .. . . .. .. .. ⎟ ⎜. . . . . . . .⎟ ⎜ ⎟ ∗...nonzero entry ⎜ ⎟ ⎜0 · · · ··· ∗ ∗ ∗⎟ ⎝ ⎠ 0 · · · ··· 0 ∗ ∗ Now, let us compute the nonzero entries of S by evaluating (2.27) and using (2.21)

18

2 The Finite Element (FE) Method

Fig. 2.6. Shape functions for nodes xi−1 , xi and xi+1

xi  Si,i−1 =

∂Ni ∂Ni−1 + cNi Ni−1 ∂x ∂x

 dx

xi−1

=

     xi    1 −1 x − xi−1 xi − x +c dx h h h h xi−1

=

−1 ch + h 6 x i+1 

Si,i =

∂Ni ∂Ni + cNi Ni ∂x ∂x

 dx

xi−1

   xi     1 1 x − xi−1 x − xi−1 = +c dx h h h h xi−1 x i+1 

1 − h

+



1 − h



 +c

xi+1 − x h



xi+1 − x h

 dx

xi

2 2ch = + h 3 Si−1,i = Si,i−1 =

−1 ch + . h 6

For simplicity, we set our source term f (x) equal to 1 over the whole computational domain and assume ua = ub = 0, which results in

2.2 Finite Element Method for a 1D Problem

19

x i+1

fi =

Ni (x) dx xi−1

xi =

x − xi−1 dx + h

xi−1

x i+1

xi+1 − x dx = h . h

xi

Let us choose L = 10 and M = 5, so that our mesh size h is 2. For the parameter c we choose once the value 0 and once 0.5, and we set the boundary values ua as well as ub to zero. Substituting these values, results in ⎛ ⎛ ⎞ ⎞ 1 − 21 0 0 2 ⎜ 1 ⎜ ⎟ ⎟ 1 ⎜− 1 − 0 ⎟ ⎜2⎟ ⎜ 2 ⎜ ⎟ ⎟ 2 f Kc=0 = ⎜ = ⎜ ⎟ ⎟ ⎜ 0 −1 1 −1 ⎟ ⎜2⎟ 2 2⎠ ⎝ ⎝ ⎠ 0 ⎛

5 3 ⎜ 1 ⎜−3 ⎜

Kc=0.5 = ⎜ ⎜ 0 ⎝ 0

0 − 12 1 − 31 0

0

2 ⎞

⎟ − 13 0 ⎟ ⎟ 1⎟ 5 − 3 3⎟ ⎠ 1 5 0 −3 3

5 3 − 31

Solving the two algebraic systems of equations results in the solutions displayed in Fig. 2.7. It is worth mentioning that the FE solution corresponding

Fig. 2.7. Solutions for the cases c = 0 and c = 0.5

to c = 0, which solves the 1D Poisson equation (Laplace operator, see (2.17)), is exact at the FE nodes. However, the case c = 0.5 already shows an error at the FE nodes. An a priori error estimate for the discretization error will be discussed in Sec. 2.8.

20

2 The Finite Element (FE) Method

2.3 Nodal (Lagrangian) Finite Elements As previously discussed, the FE method subdivides the simulation domain into small elements (e.g., triangles, tetrahedra, etc.) and the unknown quantities are approximated by interpolation functions that have local support. After spatial and time discretization, we end up with an algebraic system of equations. Now our task is to discuss the computation of the matrices (stiffness, mass, etc.) as well as the right-hand side suitable for a computer implementation. The first important step is to rewrite the integration over the whole domain (see e.g., (2.14)) as a sum of integrations over the element domains, e.g., for the stiffness matrix    ∂Na ∂Nb ∂Na ∂Nb + Kab = dΩ ∂x ∂x ∂y ∂y Ω  ne    ∂Na ∂Nb ∂Na ∂Nb + dΩ , = ∂x ∂x ∂y ∂y e=1 Ωe

with ne the number of finite elements within the mesh. Therefore, we can introduce the so-called element stiffness (mass, etc.) matrix and obtain K=

ne 

ke

ke = [kpq ]

e=1

 

kpq =

∂Np ∂Nq ∂Np ∂Nq + ∂x ∂x ∂y ∂y

 dΩ

Ωe

1 ≤ p, q ≤ nen ,

 the assembly operator (for the with nen the number of element nodes and assembling procedure see Sect. 2.4). In the second step, we shall briefly discuss the computation of the element matrices and the right-hand side. For this task, we need to compute the interpolation functions, their derivatives, and perform numerical integration. The easiest and most general strategy is to introduce transfer functions for the different geometric elements (quadrilateral, tetrahedral, etc.) to their parent elements (see Sect. 2.3). For these parent elements, we have to develop appropriate shape functions as well as numerical integration schemes. In addition, since we need the derivatives of the interpolation functions with respect to the global coordinates x, y, z (also called global derivatives), we have to develop a procedure for performing this task with the help of the transformation as well as local shape functions of the geometric elements (see Sect. 2.3.6). In the case that we choose for the transformation from the local to the global coordinate system nen  Ni (ξ)xi x(ξ) = i=1

2.3 Nodal (Lagrangian) Finite Elements

21

the same interpolation functions Ni as for the unknown quantity uh (ξ) =

nen 

Ni (ξ)uhi

i=1

we call these finite elements isoparametric. 2.3.1 Basic Properties Let us assume that we want to solve a partial differential equation with the scalar unknown u on the domain Ω with Dirichlet boundary Γe and Neumann boundary Γn as displayed in Fig. 2.8. After performing the domain

Fig. 2.8. Domain to be discretized

discretization—in this case with triangular finite elements—we obtain 12 finite elements and 11 finite element nodes (see Fig. 2.9). The numbers in the parenthesis are the equation numbers, and as can be seen, for Dirichlet nodes this number is zero (compare with the decomposition of u into the unknown v and known values ue , see Sect. 2.1). Figure 2.10 displays the interpolation

Fig. 2.9. Discretization of the domain with finite elements

function N5 (r) for Eq. (5) (finite element node 9). As can be seen, the interpolation function has a local support, which means that it has the value 1

22

2 The Finite Element (FE) Method

at the node, decreases to zero approaching the neighboring nodes and is zero outside the neighboring elements. Therefore, the ansatz according to (2.7) – (2.9) is allowed, since, e.g., at any node b it exhibits exactly the value vb v h (t)|r=rb =

neq 

Na (rb )va (t) = vb (t) .

a=1

In addition, it is now clear that the mass, stiffness as well as effective sys-

Fig. 2.10. Shape (basis) function for the unknown in FE node 9

tem matrices show a sparse profile, since the integrals defining their entries (see (2.13) and (2.14)) are nonzero only if the supports of the interpolation functions Na and Nb overlap, which only happens if the functions belong to neighboring nodes. For our simple example we obtain the following matrix structure ⎛ ⎞ a11 a12 0 0 0 0 ⎜ a21 a22 a23 0 0 a26 ⎟ ⎜ ⎟ ⎜ 0 a32 a33 a34 a35 a36 ⎟ ⎜ ⎟ ⎜ 0 0 a43 a44 a45 0 ⎟ . ⎜ ⎟ ⎝ 0 0 a53 a54 a55 a56 ⎠ 0 a62 a63 0 a65 a66 The main properties to be fulfilled by any nodal finite element are: 1) Smoothness on each element interior Ω e 2) Continuity across each element boundary Γ e 3) Completeness Let us assume that we have to evaluate finite element matrices with partial derivatives of order m in the weak formulation. Then, properties 1 and 2 demand for any nodal finite element basis functions, which are m times differentiable in Ω e and (m − 1) times differentiable over the boundary Γ e . For example, for m = 1 any finite element with linear interpolation functions

2.3 Nodal (Lagrangian) Finite Elements

23

(C 0 finite element) fulfills property 1 and 2. In general we call finite elements satisfying property 1 and 2 conforming, or compatible. Now, let us illustrate the property completeness. We assume the following ansatz for the unknown quantity u uh (x) =

nen 

Na uea ,

a=1

with nen the number of nodes for the finite element e, and the property uh (xea ) = uea . The finite element is said to be complete if uea = c0 + c1 xea + c2 yae + c3 zae

(2.29)

uh (x) = c0 + c1 x + c2 y + c3 z ,

(2.30)

implies with arbitrary constants c0 ... c3 . This means the element interpolation functions are capable of exactly representing an arbitrary linear polynomial. Therewith, it is guaranteed that by reducing the mesh size he , the approximated solution converges towards the exact solution. 2.3.2 Quadrilateral Element in IR2 For the computation of the element matrices, it is convenient to transform each finite element to its reference element, where the numerical integration can be performed easily (see Sect. 2.3.7). Let us investigate this transformation for the bilinear quadrilateral element in two space dimension as displayed in Fig. 2.11. The local coordinates x

 2 (-1,1)

1

1 (1,1)

2


y 3 x

4

3 (-1,-1) x

Fig. 2.11. Bilinear quadrilateral element

4 (1,-1)

24

2 The Finite Element (FE) Method

  ξ ξ= η

(2.31)

are related to the global coordinates x=

  x y

via the following transformation ⎧ 4  ⎪ ⎪ Ni (ξ, η)xei ⎨ x(ξ, η) = i=1 x(ξ) = 4  ⎪ ⎪ Ni (ξ, η)yie ⎩ y(ξ, η) =

(2.32)

(2.33)

i=1

Now, in order to compute explicitly the basis functions Ni , we choose the following bilinear ansatz x(ξ, η) = α0 + α1 ξ + α2 η + α3 ξη

(2.34)

y(ξ, η) = β0 + β1 ξ + β2 η + β3 ξη .

(2.35)

The shape functions have to be constructed in such a way that the relations x(ξi , ηi ) = xei y(ξi , ηi ) = yie are fulfilled. Since the local coordinates take only the following values node i ξi ηi 1 2 3 4

1 -1 -1 1

1 1 -1 -1

we obtain 8 equations for the eight unknowns α0 .. β3 . Using the solution for αi and βi in (2.34) and (2.35) and comparing the coefficients with (2.33), we arrive at the following explicit form of the shape function for node i (see Fig. 2.12) 1 (2.36) Ni (ξ) = Ni (ξ, η) = (1 + ξi ξ)(1 + ηi η) . 4 Now, let us investigate the three mentioned properties for the quadrilateral element. 1) Smoothness on each element interior Ω e : The shape functions Ni define smooth functions, if each interior angle of the quadrilateral is less then 180o.

2.3 Nodal (Lagrangian) Finite Elements

25

Fig. 2.12. Shape function of node i for a quadrilateral element

2) Continuity across each element boundary Γ e : Figure 2.12 displays the shape function Ni for node i defined by (ξi , ηi ), and it is easy to see that (2.37) Ni (ξi , ηi ) = δij is fulfilled. Along the boundary, e.g., η = −1, we obtain Ni (ξ, −1) =

1 + ξi ξ , 2

which is exactly the shape function for the 1D case. Since this shape function is typically the same for all edges, the quadrilateral element fulfills the continuity condition. 3) Completeness: uh =

nen 

Ni (ξ, η)uei

i=1

=

nen 

Ni (ξ, η)(c0 + c1 xei + c2 yie )

i=1

=

n en 

 Ni (ξ, η) c0 +

i=1

n en  

i=1

 Ni (ξ, η)xei 

c1 + 

x(ξ,η)

n en  

 Ni (ξ, η)yie

i=1

 y(ξ,η)

Summing up all four shape functions results in nen  i=1

Ni (ξ, η) =

1 [(1 − ξ)(1 − η) + (1 + ξ)(1 − η) 4 +(1 + ξ)(1 + η) + (1 − ξ)(1 + η)]

= 1, which proves the completeness.

c2 

26

2 The Finite Element (FE) Method

2.3.3 Triangular Element in IR2

Fig. 2.13. Transformation from global to local domain

The linear triangular element is defined by its three nodes as displayed in Fig. 2.13. The local coordinates are as follows node i ξi ηi 1 2 3

0 0 1 0 0 1

Similar to the quadrilateral element (see above) we obtain the local shape functions given by N1 = 1 − ξ − η N2 = ξ N3 = η . 2.3.4 Tetrahedron Element in IR3 The linear tetrahedron element is defined by its four coordinates as shown in Fig. 2.14. node i ξi ηi ζi 1 2 3 4

0 1 0 0

0 0 1 0

0 0 0 1

Let us compute the transformation that maps any arbitrary tetrahedral element in the global (x, y, z)-domain to a parent tetrahedron in the local (ξ, η, ζ)-domain by choosing the following linear ansatz

2.3 Nodal (Lagrangian) Finite Elements

27

xi = α0 + α1 ξi + α2 ηi + α3 ζi

(2.38)

yi = β0 + β1 ξi + β2 ηi + β3 ζi .

(2.39)

We know that the transformation has to satisfy the following relations at the four nodes of a tetrahedron element x(ξ) = xea

a = 1, ... , 4 .

Therefore, we obtain (see Fig. 2.14) x1 = α0

y 1 = β0

(2.40)

x2 = α0 + α1 x3 = α0 + α2

y 2 = β0 + β 1 y 3 = β0 + β 2

(2.41) (2.42)

x4 = α0 + α3

y 4 = β0 + β 3 .

(2.43)

Solving the above system of equations, we arrive at a general expression for the shape function Ni as a function of the local coordinates N1 = 1 − ξ − η − ζ N2 = ξ N3 = η N4 = ζ .

Fig. 2.14. Transformation from global to local domain

2.3.5 Hexahedron Element in IR3 In many 3D applications hexahedron elements are used for the domain discretization, due to their good approximation property. Figure 2.15 displays

28

2 The Finite Element (FE) Method

the hexahedron element in its global and local coordinate system. node i ξi ηi ζi 1 2 3 4 5 6 7 8

1 -1 -1 1 1 -1 -1 1

1 1 -1 -1 1 1 -1 -1

-1 -1 -1 -1 1 1 1 1

For the element a trilinear mapping is applied between the global (defined by x) and the local (defined by ξ) element domain [99] x(ξ) = α0 + α1 ξ + α2 η + α3 ζ + α4 ξη + α5 ηζ + α6 ξζ + α7 ξηζ .

(2.44)

The coefficients αi are determined by the relations (see Fig. 2.15)

Fig. 2.15. Hexahedron element: notation in the global and local domain

x(ξ a ) = xea

a = 1, .., 8 ,

(2.45)

which results in Na (ξ) =

1 (1 + ξa ξ)(1 + ηa η)(1 + ζa ζ) . 8

(2.46)

Using a simple degeneration technique [99], one can obtain a pyramid, a wedge as well as a tetrahedron element from a hexahedron one.

2.3 Nodal (Lagrangian) Finite Elements

29

2.3.6 Global/Local Derivatives For the computation we need to evaluate derivatives of the shape functions with respect to the global coordinate system (see e.g., (2.14)). Since the shape functions Ni depend on the local coordinates (ξ, η, ζ), we may write Na,x = Na,ξ ξ,x + Na,η η,x + Na,ζ ζ,x Na,y = Na,ξ ξ,y + Na,η η,y + Na,ζ ζ,y Na,z = Na,ξ ξ,z + Na,η η,z + Na,ζ ζ,z , with the notation, e.g., ξ,x = ∂ξ/∂x. In matrix form, we obtain ⎞ ⎡ ⎤⎛ ⎞ ⎛ ξ,x η,x ζ,x Na,ξ Na,x ⎟ ⎢ ⎥⎜ ⎟ ⎜ ⎜ Na,y ⎟ = ⎢ ξ,y η,y η,y ⎥ ⎜ Na,η ⎟ . ⎠ ⎣ ⎦⎝ ⎠ ⎝ Na,z ξ,z η,z ζ,z Na,ζ

(2.47)

Now, we do not have the explicit form of the derivatives of the local coordinates with respect to the global coordinates. However, we can express this relation as follows ⎞ ⎡ ⎤⎛ ⎞ ⎛ x,ξ y,ξ z,ξ Na,x Na,ξ ⎟ ⎢ ⎥⎜ ⎟ ⎜ ⎜ Na,η ⎟ = ⎢ x,η y,η z,η ⎥ ⎜ Na,y ⎟ . (2.48) ⎠ ⎣ ⎦⎝ ⎠ ⎝ Na,ζ x,ζ y,ζ z,ζ Na,z Comparing (2.47) and (2.48) we arrive at ⎡

ξ,x η,x ζ,x

⎤

⎡

x,ξ y,ξ z,ξ

⎤−1

⎢ ⎥ ⎢ ⎥ ⎢ ξy η,y η,y ⎥ = ⎢ x,η y,η z,η ⎥ , ⎣ ⎦ ⎣ ⎦ ξ,z η,z ζ,z x,ζ y,ζ z,ζ   

(2.49)

(J T )−1

with J the Jacobi matrix. The computation of J can be performed by the transformation between the global and local coordinate systems x(ξ, η, ζ) = y(ξ, η, ζ) = z(ξ, η, ζ) =

nen  a=1 nen  a=1 nen 

Na (ξ, η, ζ)xea Na (ξ, η, ζ)yae Na (ξ, η, ζ)zae .

a=1

Therefore, the explicit expression for the Jacobian reads as

30

2 The Finite Element (FE) Method

⎡ n en

Na,ξ xea

n en 

Na,η xea

n en 

Na,ζ xea

⎤

⎤ ⎢ a=1 ⎥ a=1 a=1 x,ξ x,η x,ζ ⎥ ⎢ n n n en en en   ⎢ Na,ξ yae Na,η yae Na,ζ yae ⎥ J = ⎣ y,ξ y,η y,ζ ⎦ = ⎢ ⎥. ⎥ ⎢ a=1 a=1 a=1 z,ξ z,η z,ζ ⎦ ⎣ n n n en en en   e e e Na,ξ za Na,η za Na,ζ za ⎡

a=1

a=1

(2.50)

a=1

The algorithm can be summarized as follows (nint denotes the number of integration points, see next section): for l := 1, nint Determine: Wl , ξ˜l , η˜l , ζ˜l for a := 1, nen Calculate: Na , Na,ξ , Na,η , Na,ζ at (ξ˜l , η˜l , ζ˜l ) end Compute Jacobi matrix, determinant and its inverse Compute global derivatives Na,x , Na,y , Na,z at (ξ˜l , η˜l , ζ˜l ) end 2.3.7 Numerical Integration For the computation of the element matrices as well as element right-hand sides we have to numerically evaluate an integral of the form  f (x) dΩ . (2.51) Ωe

Since we perform a transformation of each finite element to its parent element, (2.51) changes, e.g., for a hexahedron, to 1 1 1 f (x(ξ))|J | dξ dη dζ ,

(2.52)

−1 −1 −1

with |J | the Jacobi determinant (see Sect. 2.3.6). In the 1D case a Gaussian quadrature formula is optimal, since by using nint integration points, we achieve an accuracy of order 2nint (see e.g., [99]) 1 g(ξ) dξ = −1

nint 

g(ξ˜l )Wl + E

l=1

ξ˜l ... zero positions of Legendre polynomial with order nint Wl ... weighting factor for integration point l E ... error.

(2.53)

2.3 Nodal (Lagrangian) Finite Elements

31

For our 3D case, we can write 1 1 1

1

f (x(ξ))|J | dξ dη dζ =

2

3

nint nint nint  

g(ξ˜l1 , η˜l2 , ζ˜l3 )Wl1 Wl2 Wl3 + E

l1 =1 l2 =1 l3 =1

−1 −1 −1

=

nint 

g(ξ˜l , η˜l , ζ˜l ) Wl + E .

(2.54)

l=1

In the following, we give for each discussed geometric element the integration points as well as the weighting factors. •

Quadrilateral elements (Gaussian quadrature): l

ξl

ηl

Wl

1 −0.57735026919 −0.57735026919 1.0 2 0.57735026919 −0.57735026919 1.0 3 0.57735026919 0.57735026919 1.0 4 −0.57735026919 0.57735026919 1.0 •

Triangular elements (Gaussian quadrature): l

ξl

ηl

Wl

1 0.166 666 67 0.166 666 67 0.166 666 67 2 0.666 666 67 0.166 666 67 0.166 666 67 3 0.166 666 67 0.666 666 67 0.166 666 67 •

•

Tetrahedron elements (Gaussian quadrature): l

ξl

ηl

ζl

Wl

1 2 3 4

0.585 410 0.138 196 0.138 196 0.138 196

0.138 196 0.585 410 0.138 196 0.138 196

0.138 196 0.138 196 0.585 410 0.138 196

0.041 666 7 0.041 666 7 0.041 666 7 0.041 666 7

Hexahedron elements (Gaussian quadrature): l

ξl

ηl

ζl

Wl

1 2 3 4 5 6 7 8

-0.57735026919 0.57735026919 0.57735026919 -0.57735026919 -0.57735026919 0.57735026919 0.57735026919 -0.57735026919

-0.57735026919 -0.57735026919 0.57735026919 0.57735026919 -0.57735026919 -0.57735026919 0.57735026919 0.57735026919

-0.57735026919 -0.57735026919 -0.57735026919 -0.57735026919 0.57735026919 0.57735026919 0.57735026919 0.57735026919

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

32

2 The Finite Element (FE) Method

2.4 Finite Element Procedure In the previous section we discussed the computation of the element matrices as well as right-hand side. The still-open question of the assembly procedure will be addressed here. In the first step we introduce the nodal equation array N E, which relates the global equation number P to the global node number A. ⎧ ⎨ P : if the quantity is unknown at A N E(A) = 0 : if the quantity is known at A ⎩ (e.g., Dirichlet boundary condition) A ... global node number P ... global equation number.

Fig. 2.16. Example: global node numbers and in parenthesis the local node numbers

Since the whole simulation domain is discretized with finite elements, and we first compute the element matrices (right-hand side) and then assemble it to the global system matrix (right-hand side), we need information given by the following information element node array IEN IEN (a, e) = A a ... local element node number e ... element number A ... global node number. Assuming that we solve a scalar PDE, we have just one unknown per FE node, and the local node number coincides with the local equation number. Combining the N E array with the IEN array results in the equation array EQ EQ(a, e) = N E (IEN (a, e)) = P . The EQ array connects the element node number (element equation number) a of element e with the global equation number P . The following simple example, displayed in Fig. 2.16, will demonstrate all the steps that have to

2.4 Finite Element Procedure

33

be performed for the assembly process. The FE mesh consists of two finite elements with given Dirichlet boundary conditions u1 , u2 at node 1 and 2. Let us write the algebraic system of equations for some discretized PDE on this domain (e.g., the Poisson equation) in the following general form ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ u1 f1 K11 K12 K13 K14 K15 K16 ⎢ K21 K22 K23 K24 K25 K26 ⎥ ⎢ u2 ⎥ ⎢ f2 ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ K31 K32 K33 K34 K35 K36 ⎥ ⎢ u3 ⎥ ⎢ f3 ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (2.55) ⎢ K41 K42 K43 K44 K45 K46 ⎥ ⎢ u4 ⎥ = ⎢ f4 ⎥ . ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎣ K51 K52 K53 K54 K55 K56 ⎦ ⎣ u5 ⎦ ⎣ f5 ⎦ K61 K62 K63 K64 K65 K66 u6 f6 Since we know u1 and u2 we ⎤ ⎡ K33 K34 K35 K36 ⎢ K43 K44 K45 K46 ⎥ ⎥ ⎢ ⎣ K53 K54 K55 K56 ⎦ K63 K64 K65 K66

can rewrite (2.55) as ⎡ ⎤ ⎡ ⎤ u3 f3 − K31 u1 − K32 u2 ⎢ u4 ⎥ ⎢ f4 − K41 u1 − K42 u2 ⎥ ⎢ ⎥=⎢ ⎥ ⎣ u5 ⎦ ⎣ f5 − K51 u1 − K52 u2 ⎦ . u6 f6 − K61 u1 − K62 u2

(2.56)

The nodal equation array N E is given by 1 2 3456 0∗ 0∗ 1 2 3 4 ∗: at this node we already know the quantity (Dirichlet boundary condition) and using it for setting up the algebraic system, we recognize that we obtain a similar system as in (2.56) with four unknowns. Since there is no connection between the nodes 1, 2 and 5, 6 (see Fig. 2.16), the values of K51 , K52 , K61 , and K62 are zero. The number of nodes for the linear quadrilateral element is nen = 4 and the unknown quantity is a scalar. Therefore, we obtain the following element matrices as well as right-hand sides for the two finite elements ⎡ 1 ⎡ 1 ⎤ ⎤ 1 1 1 k11 k12 f1 k13 k14 1 1 1 1 1 ⎢ {k21 ⎢ ⎥ } (k22 ) (k23 ) {k24 }⎥ ⎥ f 1 = ⎢ < f21 > ⎥ k1 = ⎢ 1 1 1 1 ⎣ {k31 ⎣ ⎦ } (k32 ) (k33 ) {k34 } < f3 > ⎦ 1 1 1 1 k41 k42 k43 k44 f41 ⎡

⎡ ⎤ 2 2 2 2 ) (k12 ) (k13 ) (k14 ) (k11 < f12 2 2 2 2 ⎢ (k21 ) (k22 ) (k23 ) (k24 ) ⎥ 2 ⎢ < f22 ⎢ ⎥ k2 = ⎢ 2 2 2 2 ⎦ f =⎣ ⎣ (k31 < f32 ) (k32 ) (k33 ) (k34 ) 2 2 2 2 (k41 ) (k42 ) (k43 ) (k44 ) < f42 ()

... contributes to the global stiffness matrix K

< > ... contributes to the global right-hand side f { } ... contributes to the local right-hand side f e .

⎤ > >⎥ ⎥ >⎦ >

34

2 The Finite Element (FE) Method

The entries of the right-hand side f 1 for element 1 compute as ⎤ f11 1 1 ⎢ f21 − k21 u1 − k24 u2 ⎥ ⎥. f1 = ⎢ 1 1 ⎣ f31 − k31 u1 − k34 u2 ⎦ f41 ⎡

Combining the N E array with the element node array IEN , we arrive at the EQ array IEN local 1 node 2 number 3 4

el.nr. 12 13 35 46 24

EQ local 1 node 2 number 3 4

el.nr. 12 01 13 24 02

Assembling: EQ(1, 1) = 0

&

EQ(2, 1) = 1 & EQ(3, 1) = 2

1 K11 ← K11 + k22 1 K12 ← K12 + k23

f1 ← f21

1 K22 ← K22 + k33 1 K21 ← K21 + k32

f2 ← f31

EQ(4, 1) = 0

⎧ K11 ⎪ ⎪ ⎨ K13 EQ(1, 2) = 1 ⎪ K14 ⎪ ⎩ K12 ⎧ K33 ⎪ ⎪ ⎨ K31 EQ(2, 2) = 3 K34 ⎪ ⎪ ⎩ K32 ⎧ K44 ⎪ ⎪ ⎨ K41 EQ(3, 2) = 4 K43 ⎪ ⎪ ⎩ K42 ⎧ K22 ⎪ ⎪ ⎨ K21 EQ(4, 2) = 2 K23 ⎪ ⎪ ⎩ K24

← ← ← ←

2 K11 + k11 2 K13 + k12 2 K14 + k13 2 K12 + k14

f1 ← f12

← ← ← ←

2 K33 + k22 2 K31 + k21 2 K34 + k23 2 K32 + k24

f3 ← f22

← ← ← ←

2 K44 + k33 2 K41 + k31 2 K43 + k32 2 K42 + k34

f4 ← f32

← ← ← ←

2 K22 + k44 2 K21 + k41 2 K23 + k42 2 K24 + k43

f2 ← f42 .

2.5 Time Discretization

35

Thus, we obtain the following algebraic system of equations ⎤ ⎤ ⎡ 1 ⎡ 1 2 1 2 2 2 f2 + f12 k22 + k11 k23 + k14 k12 k13 ⎥ ⎥ ⎢ 1 ⎢ 1 2 1 2 2 2 ⎥ ⎢ f3 + f42 ⎥ ⎢ k32 + k41 k33 + k44 k42 k43 ⎥ ⎥ ⎢ ⎢ K=⎢ ⎥ f =⎢ ⎥. 2 2 2 2 ⎥ 2⎥ ⎢ ⎢ k21 k k k f 24 22 23 2 ⎦ ⎦ ⎣ ⎣ 2 2 2 2 k31 k34 k32 k33 f23 Comparing the result with Fig. 2.16, we can set up the following rules •

Diagonal entries: K11 is the entry of K due to node 3. Since this node belongs to element 1 as well as 2, the entry in K will be a sum of the contributions from element 1 and element 2. The entry K33 is due to global node 5, and therefore, only element 2 will contribute to it.

•

Off-diagonal entries: The same situation as described for the diagonal entries occurs.

Figure 2.17 summarizes the whole assembly procedure, and the following pseudo-code demonstrates the computer implementation. //loop over all finite elements for e := 1, ne //loop over all element nodes (equations) for a := 1, nen eq1 := EQ(a, e) if ( eq1 > 0 ) //loop over all element nodes (equations) for b := 1, nen eq2 := EQ(b, e) if ( eq2 > 0 ) K(eq1, eq2) := K(eq1, eq2) + ke (a, b) end end end end end

2.5 Time Discretization In the following we will describe single time-step discretization algorithms for parabolic (e.g., electromagnetic field in the eddy current case) as well as for hyperbolic (mechanical as well as acoustic field) partial differential equations.

36

2 The Finite Element (FE) Method

Fig. 2.17. FE procedure

2.5.1 Parabolic Differential Equation Let us consider the following semidiscrete Galerkin formulation (discrete in space and continuous in time) Mu(t) ˙ + Ku(t) = f (t) ,

(2.57)

with M the mass matrix, K the stiffness matrix, f the right-hand side, u the vector of unknowns at the finite element nodes and u˙ its derivative with respect to time. In a first step, let us subdivide the time interval [0, T ] into M subintervals, which are defined as follows [0, T ] =

M  n=1

[tn−1 , tn ] ,

t0 = 0 < t1 < t2 < .... < tn < .. < tM = T .

2.5 Time Discretization

37

For simplicity, we assume the time step size ∆t = tn − tn−1 = T /M to be constant over the whole time interval of interest. In a second step, we approximate the time derivative of the unknown u(t) by the forward time difference scheme ≈ u(t) ˙

u − un u(tn+1 ) − u(tn ) = n+1 . ∆t ∆t

(2.58)

In the third step, we have to decide at which point in time we evaluate the elliptic part Ku(t) and the right-hand side f (t). Applying a convex linear combination with the time integration parameter γP to these terms, will result in the following system of equations M

u(tn+1 ) − u(tn ) +γP Kun+1 +(1−γP)Kun = γP f n+1 +(1−γP)f n . (2.59) ∆t

Table 2.1 is listing the different time discretization methods depending on the value of γP . Table 2.1. Finite difference schemes Method

γP 0.0

forward difference; forward Euler

0.5

trapezoidal rule; Crank-Nicolson

1.0

backward difference; backward Euler

Rearranging the terms in (2.59), we arrive at ( ' (M + γP ∆tK) un+1 = ∆t γP f n+1 + (1 − γP )f n + (M − (1 − γP )∆tK) un . (2.60) We obtain the same system of algebraic equations, when we apply the general trapezoidal difference scheme [99], which is defined as follows * ) un+1 = un + ∆t (1 − γP )u˙ n + γP u˙ n+1 . (2.61) From (2.61) we can compute u˙ n+1 u˙ n+1 =

un+1 − un 1 − γP − u˙ n γP ∆t γP

and substituting it into the time discrete version of (2.57) at t = tn+1 leads to (M + γP ∆tK) un+1 = γP ∆tf n+1 + M (un + (1 − γP )∆tu˙ n ) .

(2.62)

38

2 The Finite Element (FE) Method

To see that (2.62) and (2.60) are equivalent, we use the relation Mu˙ n + Kun = f n and rewrite the right-hand side of (2.60) as follows ( ' ∆t γP f n+1 + (1 − γP )f n + (M − (1 − γP )∆tK) un ( ' = γP ∆tf n+1 + Mun + (1 − γP )∆t f n − Kun    Mu˙ n

= γP ∆tf n+1 + M (un + (1 − γP )∆tu˙ n ) . Applying the general trapezoidal difference method, we can write the solution process as a predictor-corrector algorithm, where we distinguish between an effective mass and effective stiffness formulation. 1. Effective Mass Formulation • Perform predictor step: u ˜ = un + (1 − γP )∆t u˙ n . •

(2.63)

Solve algebraic system of equations: M∗ u˙ n+1 = f n+1 − K˜ u

(2.64)

∗

M = M + γP ∆t K . •

Perform corrector step: un+1 = u ˜ + γP ∆t u˙ n+1 .

(2.65)

2. Effective Stiffness Formulation • Perform predictor step: u ˜ = un + (1 − γP )∆t u˙ n . •

Solve algebraic system of equations: 1 M˜ u γP ∆t 1 M. K∗ = K + γP ∆t

K∗ un+1 = f n+1 +

•

(2.66)

(2.67)

Perform corrector step: u˙ n+1 =

˜ un+1 − u . γP ∆t

(2.68)

2.5 Time Discretization

39

According to the choice of the integration parameter γP , one distinguishes between an explicit and implicit algorithm. 1. Explicit Algorithm: γP = 0 For this case, M∗ = M and lumping of the mass matrix might be considered (see e.g., [99]). Thus, the system matrix of the algebraic system is a diagonal matrix, and the only time-consuming part of the solution process are the matrix vector multiplications on the right-hand side. The disadvantage of this algorithm is clearly the lack of stability, which implies restrictions on the time step value ∆t, depending on the material parameters and the quality of the mesh. In addition, one has to consider that the mass matrix must be regular. This is, for example, not the case within electromagnetic computations, where the entries in the mass matrix are zero for regions with zero electrical conductivity (e.g., air). A solution to this problem may be the application of a mixed explicit/implicit time discretization [99]. 2. Implicit Algorithm: 0 < γP ≤ 1 In this case, M∗ computes as the sum of the mass matrix M and the stiffness matrix K multiplied by γP ∆t. Now M∗ is a sparse matrix and the algebraic system of equations has to be solved by any direct or iterative solver. For γP = 0.5 the time discretization is called the Crank-Nicolson scheme (second-order accurate), and for γP ≥ 0.5 the algorithm is absolute stable (independently of ∆t). In order to investigate in the accuracy of the time discretization scheme, we will expand u(t) at time t = tk+γP in a Taylor series u(t) = u(tk+γP ) + u(t ˙ k+γP ) (t − tk+γP ) + u ¨ (tk+γP ) ) * +O (t − tk+γP )3 .

(t − tk+γP )2 2 (2.69)

By considering the relation tk+γP = tk + γP ∆t we obtain for the evaluation of (2.69) at t = tk u(tk ) = u(tk+γP ) + u(t ˙ k+γP ) (−γP ∆t) + u ¨ (tk+γP ) ) * +O (−γP ∆t)3

(−γP ∆t)2 2

as well as at t = tk+1 u(tk+1 ) = u(tk+γP ) + u(t ˙ k+γP ) ((1 − γP )∆t) + u ¨(tk+γP ) ) * +O ((1 − γP )∆t)3 .

((1 − γP )∆t)2 2

Therewith, we obtain for the difference of u(tk+1 ) and u(tk ), which is used for approximating the time derivative (see (2.58)), the following expression

40

2 The Finite Element (FE) Method

u(tk+1 ) − u(tk ) = u˙ ∆t + u ¨

∆t2 − 2γP ∆t2 + O(∆t3 ) . 2

(2.70)

Now, (2.70) clearly shows that only for γP = 0.5 (Crank-Nicolson), we arrive at a second order time discretization scheme, and for all other choices the scheme is first-order accurate. 2.5.2 Hyperbolic Differential Equation For the hyperbolic case, we arrive after the spatial discretization at the following system of second-order ordinary differential equations (still continuous in time) (2.71) M¨ u(t) + Ku(t) = f (t) . Now, we will apply a finite difference scheme of second order to approximate u ¨(t) u ¨(t) ≈

− 2un + un−1 u u(tn+1 ) − 2u(tn ) + u(tn−1 ) = n+1 2 ∆t ∆t2

(2.72)

with ∆t the time step size. Similar to the parabolic case, we substitute the elliptic part Ku(t) and the right-hand side f (t) by a convex, linear combination at t = tn−1 , tn and tn+1 weighted by the integration parameter αH . Therewith, we obtain the following algebraic system of equations M

un+1 − 2un + un−1 + αH Kun+1 + (1 − 2αH )Kun + αH Kun−1 ∆t2 = αH f n+1 + (1 − 2αH )f n + αH f n−1 , (2.73)

which can be rewritten as (M + αH ∆t2 K)un+1 = αH ∆t2 f n+1 + (1 − 2αH )∆t2 f n + αH ∆t2 f n−1 + (2M − (1 − 2αH )∆t2 K)un

(2.74)

− (M + αH ∆t K)un−1 . 2

Choosing αH = 0 and transforming M to a diagonal matrix (mass lumping, see e.g., [99]), results in an explicit time discretization scheme, which does not need any algebraic solver. However, such schemes will not be unconditionally stable. A stability analysis will exhibit the so-called CFL (Courant-FriedrichLevi) condition [110] 2 . (2.75) ∆t < + λmax (M−1 K) In (2.75) λmax denotes the largest eigenvalue of the matrix (M−1 K). Since this value is of the order O(h−2 ), we should choose the time step size ∆t and the mesh size h of the same order. For αH = 1/4 the above scheme is unconditional stable.

2.5 Time Discretization

41

For practical applications, especially if an additional damping matrix C is present, the Newmark schemes are mainly used. Let us start at the semidiscrete Galerkin formulation M¨ un+1 + Cu˙ n+1 + Kun+1 = f n+1 ,

(2.76)

with M the mass matrix, C the damping matrix (e.g., C = αM M + αK K, see Sect. 3.7.2), K the stiffness matrix, f the right-hand side, u the vector of unknowns at the finite element nodes and u˙ as well as u ¨ its first and second derivative with respect to time. Thus, we have (see e.g., [99]) * ∆t2 ) (1 − 2βH )¨ un + 2βH u¨n+1 un+1 = un + ∆t u˙ n + 2 * ) u˙ n+1 = u˙ n + ∆t (1 − γH )¨ un + γH u ¨n+1 .

(2.77) (2.78)

In (2.77) and (2.78) n denotes the time-step counter, ∆t the time-step value and βH , γH the integration parameters. Substituting un+1 and u˙ n+1 according to (2.77) and (2.78) into (2.76) leads to the following algebraic system of equations M∗ u¨n+1 = f n+1 − C (u˙ n + (1 − γH )∆t u¨n )   ∆t2 −K un + ∆t u˙ n + (1 − 2βH )¨ un 2

(2.79)

M∗ = M + γH ∆t C + βH ∆t2 K . According to the choice of the integration parameters βH and γH , one distinguishes similarly to the parabolic case between an explicit and implicit algorithm. 1. Explicit Algorithm: βH = 0 and C = αM M Choosing for γH the value 0.5 we achieve a second-order accurate scheme. Similar to the parabolic case, it makes sense to lump the system matrix, so that again the time-consuming part of the solution process is the matrix vector multiplications on the right-hand side. Of course, the stability depends on the time step value ∆t, the material parameters and the quality of the mesh. However, this kind of algorithm is used quite often in acoustic computations, especially when the discretization is performed by a mapped mesh. 2. Implicit Algorithm: βH = 0 , γH = 0 In this case, M∗ computes as the sum of the mass matrix M, the damping matrix C and the stiffness matrix K with appropriate integration factors. The matrix M∗ is now sparse and the algebraic system of equations has to be solved by any direct or iterative solver. For βH = 0.25 and γH = 0.5 the time discretization is second-order accurate with respect to time, if C

42

2 The Finite Element (FE) Method

vanishes. To keep the second-order accuracy even in the damped case, one has to extend the Newmark scheme to the Hilbert-Hughes-Taylor scheme (see [99]). Writing the solution process for one time step as a predictor-corrector algorithm we arrive at the effective mass as well as effective stiffness formulations. 1. Effective Mass Formulation • Perform predictor step: u ˜ = un + ∆t u˙ n + (1 − 2βH )

∆t2 u ¨ 2 n

˜˙ = u˙ n + ∆t (1 − γH )¨ un . u •

(2.80) (2.81)

Solve algebraic system of equations: ˜˙ M∗ u ¨n+1 = f n+1 − K˜ u − Cu

(2.82)

M∗ = M + γH ∆t C + β∆t2 K . •

Perform corrector step: un+1 = u ˜ + βH ∆t2 u ¨n+1 ˜˙ + γH ∆t u u˙ n+1 = u ¨ n+1 .

(2.83) (2.84)

2. Effective Stiffness Formulation According to (2.77) and (2.78) we can express u¨n+1 and u˙ n+1 as follows ˜ un+1 − u βH ∆t2 ˜˙ + γH ∆t u ˜˙ + γH (un+1 − u =u ¨n+1 = u ˜) . βH ∆t

u¨n+1 =

(2.85)

u˙ n+1

(2.86)

Therefore, we obtain • Perform predictor step: u ˜ = un + ∆t u˙ n + (1 − 2βH )

∆t2 u ¨ 2 n

˜˙ = u˙ n + ∆t (1 − γH )¨ un . u •

•

Solve algebraic system of equations:   1 γH ∗ ˜ C u ˜ M+ K un+1 = f n+1 − Cu˙ + βH ∆t2 βH ∆t 1 γH C+ K∗ = K + M. βH ∆t βH ∆t2

(2.87) (2.88)

(2.89)

Perform corrector step: ˜ un+1 − u βH ∆t2 = u˜˙ + γH ∆t¨ un+1 .

u ¨n+1 =

(2.90)

u˙ n+1

(2.91)

2.7 Edge (N´ ed´ elec) Finite Elements

43

2.6 Integration over Surfaces Very often one has to evaluate an integral over a 3D surface or along a contour in 2D. In the first case the integration is performed within a 1D finite element in a 2D space and in the second case within a 2D finite element in a 3D space. Let us assume a scalar function f (x, y, z) as the integrand of a surface integral. According to [157], we can write    f (x, y, z) dΓ = f (x(ξ, η), y(ξ, η), z(ξ, η)) |xξ (ξ, η) × xη (ξ, η)| dξ dη Γ

(2.92) ⎛ ∂x ⎞

with

⎛ ∂x ⎞

∂ξ

∂η

⎜ ⎟ xξ = ⎝ ∂y ∂ξ ⎠

⎜ ∂y ⎟ xη = ⎝ ∂η ⎠.

∂z ∂ξ

∂z ∂η

For the second case of a contour integral over a scalar function f (x, y), we obtain   f (x, y) ds = f (x(ξ), y(ξ)) |xξ (ξ)| dξ , (2.93) C



with xξ =

∂x ∂ξ ∂y ∂ξ

 .

Therefore, with slight modifications we can evaluate such integrals by performing an integration in the local domain and instead of the Jacobian determinant we have to compute the expressions given in (2.92) and (2.93).

2.7 Edge (N´ ed´ elec) Finite Elements Edge finite elements belong to the family of vector finite elements (shape functions are vectors) and assign the degrees of freedom to the edges rather than to the nodes of the element. These types of elements were first introduced by Whitney (see e.g., [79]). Their importance in electromagnetics were realized by N´ed´elec (see e.g., [163]), who constructed edge elements on quadrilateral and tetrahedron elements. Many important studies followed for the further development of different electromagnetic field problems (see e.g., [14, 26, 121, 160, 214]). Within edge elements, a physical vector quantity A (e.g., the magnetic vector potential) is approximated by the following ansatz A ≈ Ah =

ne  k=1

Aek Ek .

(2.94)

44

2 The Finite Element (FE) Method

In (2.94) ne defines the number of edges in the finite element mesh, Ek the edge shape function associated with the k-th edge, and Aek the corresponding degree of freedom, namely the line integral of the physical vector quantity along the k-th edge  A · ds .

Aek =

(2.95)

k

For edge shape functions Ek of lowest order, the following conditions have to be fulfilled [197]: 1. The tangential component of Ek along the edge k has to be constant. 2. The tangential component of Ek along edges l = k is zero. 3. The divergence of Ek is zero inside the element. Since, in this work, N´ed´elec finite elements are exclusively used for 3D magnetic field computation, we will restrict ourselves to this case using tetrahedron elements. Let us consider the following vector ansatz function E1 (∇

Fig. 2.18. Tetrahedron element: nodes are denoted by 1, .. , 4 and edges by 1, .. , 6

defines here the derivatives with respect to the global coordinates x) E1 = N1 ∇N2 − N2 ∇N1

(2.96)

along edge 1 defined by the nodes 1 and 2 (see Fig. 2.18) and N1 and N2 the nodal interpolation functions (see Sect. 2.3.4) in node 1 and 2, respectively. To check condition 1, we compute, e.g., the tangential component of E1 along edge 1 E1 · t1 = N1 t1 · ∇N2 − N2 t1 · ∇N1 . It has to be mentioned that we have to compute the derivatives with respect to the global coordinate system (see Sect. 2.3.6). Since t1 · ∇N1 = −1/(2l1) and t1 · ∇N2 = 1/(2l1 ) (l1 denotes the length of edge 1), we obtain E1 · t1 =

1 . l1

2.8 Discretization Error

45

Condition 2 is also fulfilled, since N1 vanishes along the edges (4,5,6), N2 along the edges (2,3,6) and t2 · ∇N1 , t3 · ∇N1 , t4 · ∇N2 , t5 · ∇N2 are zero. Therewith, shape function E1 has no tangential component along the edges (2,3,4,5,6). Condition number 3 states that the divergence of E1 has to vanish inside the element. Applying the divergence to E1 results in ∇ · (N1 ∇N2 − N2 ∇N1 ) = N1 ∆N2 + ∇N1 · ∇N2 − N2 ∆N1 − ∇N2 · ∇N1 = N1 ∆N2 − N2 ∆N1 . Since the interpolation functions N1 as well as N2 are linear functions, the value of ∇ · E1 is zero. To obtain for the tangential components of Ek the value 1, we have to scale the vector function with the length of the corresponding edge length, which results in the following edge interpolation functions for a linear tetrahedron E1 = (N1 ∇N2 − N2 ∇N1 ) l1 E2 = (N1 ∇N3 − N3 ∇N1 ) l2 E3 = (N1 ∇N4 − N4 ∇N1 ) l3 E4 = (N2 ∇N3 − N3 ∇N2 ) l4 E5 = (N4 ∇N2 − N2 ∇N4 ) l5 E6 = (N3 ∇N4 − N4 ∇N3 ) l6 .

2.8 Discretization Error When applying the FE method, it is of great importance to have some knowledge of the discretization error. Since the error analysis is a quite sophisticated task, we will just provide a short overview containing important results. For a detailed discussion on this topic we refer to [110]. For our purpose of error analysis let us consider the following variational form: Given: f, c : Ω → IR Find: u ∈ Vg such that for all v ∈ V0 a(u, v) = < f, v > with

(2.97)

46

2 The Finite Element (FE) Method



 ∇v · ∇u dΩ +

a(u, v) = Ω

c v u dΩ Ω



< f, v > =

vf dΩ Ω

Vg = {u ∈ H 1 |u = g on Γ } V0 = {v ∈ H 1 |v = 0 on Γ } . Applying the FE method to our problem, results in Given: f, c : Ω → IR Find: uh ∈ Vgh such that for all v h ∈ V0h a(uh , v h ) = < f, v h > with



(2.98) 

∇v h · ∇uh dΩ +

a(uh , v h ) = Ω

Ω



h

c v h uh dΩ

h

< f, v > =

v f dΩ Ω

with Vgh ⊂ Vg and V0h ⊂ V0 . In general, we distinguish between a priori and a posteriori error estimates: •

The a priori error estimate is expressed as ||u − uh || ≤ C1 (u) hα

(2.99)

with α > 0 and C1 (u) being a positive constant. In (2.99) u denotes the exact solution, uh our solution obtained by the FE method, h the discretization parameter (e.g., longest edge in the FE mesh) and || · || an adequate norm. The constant α depends on the smoothness of the solution u and the polynomial degree of the chosen FE basis (shape) functions. In general, this constant is not known. •

The a posteriori error estimate reads as ||u − uh || ≤ C2 (uh , h) .

(2.100)

As for the a priori error estimate, u denotes the exact solution, uh our solution obtained by the FE method, h the discretization parameter and || · || an adequate norm. The constant C2 depends on the FE solution and the discretization parameter h. Providing an optimal error estimator, it is possible to obtain C2 or at least sharp bounds for C2 (see e.g., [4]).

2.8 Discretization Error

47

In the following, we will concentrate on a priori estimates, and we assume that our bilinear form as defined in (2.98) is V0 -elliptic and V0 -bounded (for a proof see e.g., [110]). V0 -ellipticity means that there exists a constant C1 > 0 so that a(v, v) ≥ C1 ||v||2H 1 for all v ∈ V0 . V0 -boundedness states that there exists a constant C2 > 0, such that |a(u, v)| ≤ C2 ||u||H 1 ||v||H 1 for all u, v ∈ V0 . In addition to the norms defined in App. C, we introduce the energy norm of the error e = (u − uh ) associated with the bilinear form a(u, v) , + ||u − uh ||a = a(u − uh , u − uh ) = a(e, e) . Since V0h ⊂ V0 , we may rewrite (2.97) by a(u, v h ) =< f, v h > for all v h ∈ V0h . Now, let us subtract from this equation the discrete weak form (see (2.98)). This operation results in a(u − uh , v h ) = 0 for all v h ∈ V0h , which is known as the Galerkin orthogonality of the error. This result implies that in a certain sense uh is the best approximation of u in Vgh . The most important theorems for the a priori estimation of the discretization error in H 1 (Ω) are C´ ea’s lemma (see theorem 1 below) and the BrambleHilbert lemma [78]. The lemma of C´ ea allows us to transform the estimation of the discretization error to an estimation of the approximation error. Theorem 1: Let the bilinear form a(·, ·) be V0 -elliptic and V0 -bounded. Then, the error estimation reads as ||u − uh ||H 1 ≤

C1 C2

inf ||u − wh ||H 1 .

w h ∈Vgh

Since the approximation error inf ||u − wh ||H 1

w h ∈Vgh

can be estimated from above by the interpolant (see Fig. 2.19) Π h (u) =

M  i=0

Ni (x) ui ⊂ Vgh ,

(2.101)

48

2 The Finite Element (FE) Method

we may rewrite (2.101) by ||u − uh ||H 1 ≤

C1 C2

inf ||u − wh ||H 1 ≤

w h ∈Vgh

C1 ||u − Π h (u)||H 1 . C2

u(x) u uh h  (u)

x0=a

xM=b h

Fig. 2.19. Exact solution u, interpolant Π (u) and the FE solution uh

According to this important result and using the Bramble-Hilbert lemma, we will obtain practical a priori error estimates. For our FE discretization we will use piecewise, continuous polynomial basis functions of order m, and can state the following theorem [78]: Theorem 2: Let a(·, ·) be a V0 -elliptic and V0 -bounded bilinear form. Furthermore, there exists derivatives in the weak sense of the exact solution u up to order (k + 1) with k ≤ m, so that u ∈ Vg ∩ H k+1 . Then, we estimate the a priori error by ||u − uh ||H 1 ≤ C hm |u|H m+1 (2.102) with C independent of the discretization parameter h and m the polynomial order of the basis functions defined on the reference element. In (2.102) |u|H m+1 stands for the semi norm (see C.11). For our considered bilinear form the exact solution has derivatives in the weak sense up to m+1 = 2, so that the error will be of order O(h). It has to be noticed, that an increase of the polynomial order of our basis functions makes just sense for sufficiently smooth u. If u ∈ H k with 2 ≤ k ≤ m + 1, and we use polynomials of order m for the basis functions, then the following estimate holds

2.8 Discretization Error

49

||u − uh ||H 1 ≤ C˜ hk−1 |u|H k . This means that for such a case an increase of our assumed polynomial order m of the basis functions, will not result in an decrease of the error. Therefore, a reduction of the discretization error can just be achieved by a refinement of the FE mesh (decreasing h). If we now consider the case for • • • •

c = 0 in our bilinear form (i.e. the bilinear form corresponds to the Laplace operator) u ∈ Vg ∩ H 2 linear basis functions a quasi uniform discretization of our computational domain,

then we arrive at the following practical estimates ||u − uh ||H 1 ≤ C h|u|H 2 = O(h) ||u − uh ||L2 ≤ C˜ h2 |u|H 2 = O(h2 )

(2.103) (2.104)

with C, C˜ being constants and independent of the discretization. However, problems with discontinuous jumps in material parameters (e.g., magnetic permeability at an iron–air interface) usually do not permit solutions that are smooth enough to be in H 2 . In such a case, the bounds deteriorate to ||u − uh ||H 1 ≤ C hα |u|H 2 = O(hα ) ||u − uh ||L2 ≤ C˜ h2α |u|H 2 = O(h2α )

(2.105) (2.106)

with 0 < α < 1. The value of α depends on the smoothness (regularity) of the solution u.

3 Mechanical Field

3.1 Navier’s Equation Let us consider a solid body with prescribed volume force fV and support at equilibrium, which means that the sum of all forces as well as the sum of all moments are zero. In the first step we cut out a small part of this solid body so that the faces of this small body are parallel to the Cartesian coordinate system (see Fig. 3.1a). Now, we have to apply forces at the cutting planes to still guarantee equilibrium (see Fig. 3.1b). These forces correspond to inner

Fig. 3.1. (a) Cutting out a small body; (b) Mechanical stress σ on a surface of the small body

forces acting within the solid body. Since the applied forces are distributed all over the cutting planes, we describe them by mechanical stresses (force per unit area). The stress state of each face is defined by its stress vectors σ x , σ y

52

3 Mechanical Field

and σ z , where the index denotes the associated face. Thus, we can describe the stress state as follows σ x = σxx ex + σxy ey + σxz ez

(3.1)

σ y = σyx ex + σyy ey + σyz ez

(3.2)

σ z = σzx ex + σzy ey + σzz ez

(3.3)

with (σxx , σyy , σzz ) the normal stresses and (σxy , σxz , σyx , σyz , σzx , σzy ) the shear stresses and ei defining the unit vector in direction i. Now in a second case, we consider a body with an oblique cutting face and investigate on the mechanical stress σ α on the face dΓα (see Fig. 3.2). We  x z y

y

x

z Fig. 3.2. Mechanical stresses on the surface of a small body with an oblique face

define the normal vector nα by nα = nx ex + ny ey + nz ez , with the property |nα | = n2x + n2y + n2z = 1. According to Fig. 3.2, the equilibrium of all forces results in dΓα σ α − dΓx σ x − dΓy σ y − dΓz σ z = 0 .

(3.4)

In (3.4) dΓα denotes the infinite small surface of the oblique cutting face, and the surfaces of the three other faces can be expressed as dΓx = dΓα nx

dΓy = dΓα ny

dΓz = dΓα nz .

Therewith, we can rewrite (3.4) by σ α = nx σ x + ny σ y + nz σ z , which results by using (3.1) - (3.3) in

3.1 Navier’s Equation

53

σ α = (σxx nx + σyx ny + σzx nz ) ex + (σxy nx + σyy ny + σzy nz ) ey + (σxz nx + σyz ny + σzz nz ) ez . To obtain a more compact expression, we introduce the mechanical stress tensor [σ], called the Cauchy stress tensor, as ⎡ ⎤ σxx σxy σxz [σ] = ⎣ σyx σyy σyz ⎦ . (3.5) σzx σzy σzz The stress tensor now allows us to express any physical stress vector σ α acting on a face defined by its normal vector nα as follows σ α = [σ]T nα . After having defined the stress state, we can now investigate in the equation for translation  fV dΩ + [σ]T dΓ = 0 (3.6) Ω

Γ

and the equation for rotation  (r × fV ) dΩ + (r × [σ]) dΓ = 0 , Ω

(3.7)

Γ

which have to be fulfilled, if the body is at rest. First, let us consider the equation for translation in x-direction given by  ) T * [σ] ex · dΓ = 0 . (3.8) fV · ex dΩ + Ω

Γ

Rewriting [σ]T ex by σ x and applying the divergence theorem to the second term of (3.8), we obtain   fx dΩ + ∇ · σ x dΩ = 0 , (3.9) Ω

Ω

with fV = (fx fy fz )T . Since this result has to hold for each volume Ω, we may write (3.9) in the following form fx + ∇ · σ x = 0 .

(3.10)

Similar expressions are obtained for the y- and z-directions fy + ∇ · σ y = 0 fz + ∇ · σ z = 0 .

(3.11)

The equilibrium condition for the rotation around the x-axis with the position vector r = (x y z)T reads as

54

3 Mechanical Field

-

 (yfz − zfy ) dΩ + Ω

(yσ z − zσy ) · n dΓ = 0 .

(3.12)

Γ

Applying the divergence theorem and omitting the volume integral results in the following differential form yfz − zfy + ∇ · (yσ z ) − ∇ · (zσ y ) = 0 .

(3.13)

Using the vector identity ∇ · (ξu) = ξ∇ · u + u · ∇ξ

(3.14)

we obtain yfz − zfy + y∇ · σ z + σ z · ∇y − z∇ · σ y − σ y · ∇z = 0 .

(3.15)

With the result of (3.11) we can simplify (3.15) to σzy = σyz .

(3.16)

The relations for the remaining two axes of rotation yield σzx = σxz and σyx = σxy .

(3.17)

Therefore, the equilibrium equation for a body at rest can be expressed as follows (3.18) fV + ∇[σ] = 0 , where [σ] denotes the Cauchy stress tensor. Since the stress tensor [σ] is symmetric, it is convenient to write it as a vector of six components using Voigt notation [18] ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ σ11 σ1 σxx ⎜ σyy ⎟ ⎜ σ22 ⎟ ⎜ σ2 ⎟ ⎡ ⎤ ⎡ ⎤ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ σxx σxy σxz σ11 σ12 σ13 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎣ σyx σyy σyz ⎦ = ⎣ σ21 σ22 σ23 ⎦ ; ⎜ σzz ⎟ = ⎜ σ33 ⎟ = ⎜ σ3 ⎟ . (3.19) ⎜ σyz ⎟ ⎜ σ23 ⎟ ⎜ σ4 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ σzx σzy σzz σ31 σ32 σ33 ⎝ σxz ⎠ ⎝ σ13 ⎠ ⎝ σ5 ⎠ σxy σ12 σ6 By introducing the differential operator B ⎛ ∂ ⎞T ∂ ∂ ∂x 0 0 0 ∂z ∂y ⎜ ⎟ ∂ ∂ ∂ ⎟ B=⎜ ⎝ 0 ∂y 0 ∂z 0 ∂x ⎠ , ∂ ∂ ∂ 0 0 ∂z ∂y ∂x 0

(3.20)

(3.18) takes on the following form B T σ + fV = 0 .

(3.21)

In the dynamic case the sum of all forces is equal to the inertia force and we arrive at Navier’s equations describing the dynamical behavior of mechanical systems (3.22) fV + B T σ = ρa , with ρ denoting the density of the medium and a the acceleration of the body.

3.2 Deformation and Displacement Gradient

55

3.2 Deformation and Displacement Gradient Figure 3.3 displays the relation between an arbitrary point P0 in Ω0 , the initial configuration of a body, and the corresponding point P in the deformed configuration defined by Ω. Furthermore, X defines the position of any material point in Ω0 (often called Lagrangian coordinates), and x the position of the corresponding point in Ω (often called Eulerian coordinates). The motion

Fig. 3.3. Body with volume Ω0 in the initial configuration and with volume Ω in the deformed configuration

of the body shall be described by the unique map Φ x = Φ(X, t) .

(3.23)

To locally compute the deformation, we introduce the deformation gradient [Fd ], which maps the differential line element dX in Ω0 to the corresponding differential line element dx in Ω dx = [Fd ] dX ∂x = ∇X Φ [Fd ] = ∂X ⎡ ∂x ∂x ∂x ⎤ ∂X ∂Y

⎢ ∂y =⎢ ⎣ ∂X

∂Z

∂y ∂y ∂Y ∂Z

⎥ ⎥. ⎦

(3.24)

∂z ∂z ∂z ∂X ∂Y ∂Z

Since the map Φ is bijective, the Jacobi determinant |J | of [Fd ] is different from zero, and in order to exclude intersections, it has to be greater than zero. By introducing the displacement vector u according to u(X, t) = x − X = Φ(X, t) − X ,

(3.25)

56

3 Mechanical Field

we obtain Fd = ∇X (X + u) = I + ∇X u = I + [Hd ] ⎡ ∂ux ∂ux ∂X

⎢ ∂u y ∇X u = ⎢ ⎣ ∂X

∂Y

∂ux ∂Z

∂uy ∂uy ∂Y ∂Z

⎤ ⎥ ⎥, ⎦

(3.26)

(3.27)

∂uz ∂uz ∂uz ∂X ∂Y ∂Z

with [Hd ] the displacement gradient. The knowledge of Fd allows the definition of transformations for differential quantities. In particular, the transformation between the differential surface element dΓ0 in Ω0 to dΓ in Ω is given by [220] dΓ = n dΓ = |J |[Fd ]−T n0 dΓ0 = |J |[Fd ]−T dΓ0 ,

(3.28)

and for a differential volume element by dΩ0 = |J | dΩ .

(3.29)

3.3 Mechanical Strain In order to first provide a basic physical understanding of the mechanical strain, we will derive the relation between mechanical displacement and strain in the linear case for the configuration shown in Fig. 3.4. Due to a given

Fig. 3.4. Initial state and deformed state for a infinite small rectangle

load case, a general point P0 (x, y) of the elastic body at the initial state will

3.3 Mechanical Strain

57

undergo a deformation according to the displacement components ux (x, y) and uy (x, y). We assume that the infinite small rectangle with side-length ∆x and ∆y will deform in a parallelogram with small angles α and β. Therewith, the side-length in x-direction computes for the deformed body as ux (x + ∆x, y) − ux (x, y) ≈ ux (x + ∆x, y) − ux (x, y) = ∆ux . cos α

(3.30)

Now, the elongation in x-direction computes for the limit ∆x → 0 lim

∆x→0

ux (x + ∆x, y) − ux (x, y) . ∆x

Expanding the term ux (x + ∆x, y) in a Taylor series ux (x + ∆x, y) = ux (x, y) +

∂ux ∆x + ... higher oder terms ∂x

and neglecting the higher order terms, results in sxx =

∂ux . ∂x

(3.31)

In (3.31) sxx is the unit elongation of the elastic body in x-direction and we call it the normal strain in x-direction. The same derivation can be applied to the unit elongation in y-direction, which will define the normal strain in y-direction ∂uy . syy = ∂y Now, let us compute the shearing of the body, which means to obtain a relation between α, β and the displacements ux , uy . According to Fig.3.4, we obtain u (x+∆x,y)−u (x,y)

tan α =

y y uy (x + ∆x, y) − uy (x, y) ∆x = . x (x,y) ∆x + ux (x + ∆x, y) − ux (x, y) 1 + ux (x+∆x,y)−u ∆x

(3.32)

Expanding the terms ux (x + ∆x, y) and uy (x + ∆x, y) in a Taylor series up to the linear term, substituting these expressions into (3.32) and performing the limit ∆x → 0, results in tan α =

1

∂uy ∂x x + ∂u ∂x

.

(3.33)

Assuming that the terms ∂uy /∂x and ∂ux /∂x are small compared to 1, the angle α will also be small, and we obtain the following approximation of (3.33) α=

∂uy . ∂x

Applying the same steps for the computation of β results in

58

3 Mechanical Field

β=

∂ux . ∂y

The total shearing of our elastic body computes as the sum of α and β α+β =

∂ux ∂uy + = 2sxy , ∂x ∂y

and we call sxy the shear strain. For deriving the general relation between the mechanical strain and displacement, we consider the case as displayed in Fig. 3.5. The displacement u maps the initial configuration into the deformed one. The deformation state of a body is defined by considering the change of a line element between two neighboring points (P0 (X, Y, Z), Q0 (X, Y, Z)) in the initial configuration and (P (x, y, z), Q(x, y, z)) in the deformed configuration (see Fig. 3.5). Since the

Fig. 3.5. Strain measurement

metric of a space—the measure of the length and angle of the deformation—is defined by the square of the line element, we obtain for the differential element dl0 in the initial configuration dl02 = dX 2 + dY 2 + dZ 2 = dXT dX

(3.34)

and for dl in the deformed configuration dl2 = dx2 + dy 2 + dz 2 = dxT dx .

(3.35)

With the help of (3.24), we can express the difference as follows dl2 − dl02 = dxT dx − dXT dX = dXT [Fd ]T [Fd ] dX − dXT dX ) * = dXT [Fd ]T [Fd ] − I dX = dXT 2[V] dX ,

(3.36)

3.3 Mechanical Strain

59

where [V] denotes the Green–Lagrangian strain tensor. Thus, V measures the difference between the square length of an infinitesimal segment in the deformed configuration and in the initial configuration. Since we can express [Fd ] by (I + ∇X u) the Green–Lagrangian strain tensor [V] takes the form ( 1' (I + ∇X u)T (I + ∇X u) − I [V] = 2 * 1) * 1) ∇X u + (∇X u)T + (∇X u)T ∇X u . = (3.37) 2 2 As can be easily seen, the first part in the above equation defines the linear strain, whereas the addition of the second part allows the description of large deflections. The explicit form of the Green–Lagrangian strain tensor written in vector notation is given as follows  ⎤ ⎡ ) (2 ) ' * * ∂uy ∂ux 2 ∂uz 2 1 + ∂X + ∂X ⎢ 2  ∂X  ⎥ ⎢ ⎢ ⎥ ⎥ ( ' ∂u 2 y * ) ⎢ ⎥ ⎥ ⎢ 1 ) ∂ux *2 2 ∂uy ∂uz ∂Y ⎢ ⎥ ⎥ ⎢2 + + ∂Y ∂Y ∂Y ⎢ ⎢ ⎥ ∂uz ⎥ ⎢ ⎥ ⎢ ) ( ' 2 ) ∂u *2 ⎥ *2 ∂Z ⎢ ⎥ ⎥ ⎢ ∂uy x z ' ( ⎥ + ⎢ 12 ( ∂u ⎥, + + V=⎢ ∂Z ∂Z ∂Z ⎢ ∂uy + ∂uz ⎥ ⎢ ⎥ ⎢ ∂Z ⎥ ⎥ ⎢ ∂ux ∂ux ∂Y ∂u ∂u y y ∂uz ∂uz ⎢) ⎢ ⎥ ⎥ * ⎢ ∂uz ⎢ ∂Y ∂Z + ∂Y ∂Z + ∂Y ∂Z ⎥ ∂ux ⎥ ⎢ ∂X + ∂Z ⎥ ⎢ ⎥ ∂uy ∂uy ∂uz ∂uz ∂ux ∂ux ⎥ ⎣' (⎦ ⎢ + ∂Z ∂X + ∂Z ∂X ⎣ ⎦ ∂Z ∂X ∂uy ∂ux + ∂Y ∂X ∂uy ∂uy ∂uz ∂uz ∂ux ∂ux    ∂X ∂Y + ∂X ∂Y + ∂X ∂Y ⎡

∂ux ∂X

⎤

(3.38)

S

with [S] the tensor of the linear strains in vector notation. The linear strain tensor [S] in vector form according to Voigt notation reads as ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ s11 s1 sxx ⎜ syy ⎟ ⎜ s22 ⎟ ⎜ s2 ⎟ ⎡ ⎤ ⎡ ⎤ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ sxx sxy sxz s11 s12 s13 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎣ syx syy syz ⎦ = ⎣ s21 s22 s23 ⎦ ; ⎜ szz ⎟ = ⎜ s33 ⎟ = ⎜ s3 ⎟ . (3.39) ⎜ 2syz ⎟ ⎜ 2s23 ⎟ ⎜ s4 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ szx szy szz s31 s32 s33 ⎝ 2sxz ⎠ ⎝ 2s13 ⎠ ⎝ s5 ⎠ 2sxy 2s12 s6 The factor of 2 on the shear strains results from the requirement that the computed energy in Voigt notation and index notation must be the same. It should be noted that the Green–Lagrangian strain tensor [V] defines the strains in the initial configuration, whereas the Cauchy stress tensor [σ] defines the stress for the deformed configuration. In the non-linear case, we will introduce the second Piola–Kirchhoff stress tensor, which is also defined for the initial configuration, and therefore fits to the Green–Lagrangian strain tensor. However, in the linear case we do not have to distinguish between the initial and the deformed configuration and can relate the Cauchy stress tensor [σ] to the linear strain tensor [S].

60

3 Mechanical Field

3.4 Constitutive Equations The simplest and most frequently used relation between the stress and strain is the linear law of elasticity known as Hook’s law [168, 225]. Assuming an isotropic material, it can be expressed by knowledge of the shear modulus G and the Poisson ratio νp only, as follows (sij denotes the components of the linear strain tensor [S])   νp (sxx + syy + szz ) ; σxy = 2Gsxy σxx = 2G sxx + 1 − 2νp   νp σyy = 2G syy + (sxx + syy + szz ) ; σyz = 2Gsyz (3.40) 1 − 2νp   νp (sxx + syy + szz ) ; σzx = 2Gszx . σzz = 2G szz + 1 − 2νp The often-used elasticity modulus Em is computed via G and νp by G=

Em . 2(1 + νp )

(3.41)

By introducing the so-called Lam´e parameters λL and µL νp Em (1 + νp )(1 − 2νp ) Em , µL = 2(1 + νp )

(3.42)

λL =

(3.43)

we obtain the following explicit form of the stress–strain relation for isotropic materials ⎤ ⎛ ⎞ ⎡ λL + 2µL ⎞ ⎛ λL λL 0 0 0 sxx σxx ⎥ ⎟ ⎢ ⎟ ⎜ λL λL + 2µL λL 0 0 0 ⎥ ⎜ s ⎟ ⎜ σyy ⎟ ⎢ ⎢ ⎥ ⎜ ⎜ yy ⎟ ⎟ ⎢ ⎜ ⎥ ⎟ ⎟ ⎜ λL λL λL + 2µL 0 0 0 ⎥ ⎜ szz ⎟ ⎜ σzz ⎟ ⎢ ⎥ ⎜ ⎜ ⎟=⎢ ⎟ . (3.44) ⎜ ⎥ ⎟ ⎢ ⎟ ⎜ 0 0 0 µL 0 0 ⎥ ⎜ 2s ⎜ ⎟ ⎜ σyz ⎟ ⎢ yz ⎥ ⎜ ⎟ ⎢ ⎟ ⎜ ⎢ ⎥ ⎜ ⎜σ ⎟ ⎢ 0 0 0 0 µL 0 ⎥ ⎝ 2sxz ⎟ ⎠ ⎝ xz ⎠ ⎣ ⎦ 0 0 0 0 0 µ L σxy 2sxy Therefore, Navier’s equation can be expressed by µL ∇ · ∇u + (λL + µL )∇ ∇ · u + fV = ρ

∂2u . ∂t2

(3.45)

For the general anisotropic case, we introduce the tensor of elasticity moduli [c] in the form

3.4 Constitutive Equations

61

[σ] = [c][S] σij = cijkl skl ,

(3.46)

which has the following properties cijkl = cijlk cijkl = cjikl cijkl = cklij .

(3.47)

Due to symmetry, we can write the stress as well as the strain tensors as vectors. Thus, we can combine the four indices of the cijkl to two indices cIK as follows ij / kl I / K 11 22 33 23 13 12

1 2 3 4 5 6

Expressing the linear strain vector S by Bu and combining (3.22) with (3.46), results in ∂2u B T [c]Bu + fV = ρ 2 . (3.48) ∂t 3.4.1 Plane Strain State

Fig. 3.6. Plane strain case

62

3 Mechanical Field

The plane strain case as depicted in Fig. 3.6 is a simplification of the general case and can be used when the third dimension (assumed as the zdirection) is very large and within each cross section (in our case the xy-plane) the same boundary conditions as well as forces act on the body. Therefore, the dependence of the mechanical displacements ux and uy on the z-coordinate can be neglected szy = szx = 0 szz = 0 . The stress–strain relation for the isotropic case simplifies to ⎛ ⎞ ⎡ ⎤ ⎛ ⎞ sxx σxx λL + 2µL λL 0 ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ ⎜ ⎜ σyy ⎟ ⎢ ⎟ λL λL + 2µL 0 ⎥ ⎜ ⎟=⎢ ⎥ ⎜ syy ⎟ . ⎝ ⎠ ⎣ ⎦ ⎝ ⎠ 0 0 µL σxy 2sxy

(3.49) (3.50)

(3.51)

3.4.2 Plane Stress State A demonstrative example for the plane stress case is shown in Fig. 3.7, where a thin sheet is loaded by mechanical forces at the boundary and the forces act within the defined plane. By cutting an infinitely small piece out of the whole

Fig. 3.7. Plane stress case

thin sheet, we can assume to have stresses on its surface as displayed in Fig. 3.7. Therefore, the following relations are fulfilled σzx = σzy = 0 σzz = 0 .

(3.52) (3.53)

By using (3.52) and (3.53), we immediately obtain for the isotropic case (see (3.44))

3.5 Waves in Solid Bodies

szx = szy = 0 szz

63

(3.54)

λL =− (sxx + syy ) , λL + 2µL

which leads to the following simplifications for the plane an isotropic material ⎛ ⎞ ⎡ 2λL µL ⎤ 2λL µL σxx 0 λL +2µL + 2µL λL +2µL ⎜ ⎟ ⎢ ⎥ 2λL µL 2λL µL ⎜ σyy ⎟ ⎢ ⎥ ⎜ ⎟=⎢ λL +2µL λL +2µL + 2µL 0 ⎥ ⎝ ⎠ ⎣ ⎦ σxy 0 0 µL

(3.55) stress case assuming ⎛

sxx

⎞

⎜ ⎟ ⎜ syy ⎟ ⎜ ⎟. ⎝ ⎠ 2sxy

(3.56)

3.4.3 Axisymmetric Stress–Strain Relations In a cylindrical-coordinate system, the displacement components read as ur displacement in radial direction (r-direction) uz displacement in axial direction (z-direction) uθ displacement in circumferential direction (θ-direction) Since in the axisymmetric case the mechanical displacements do not depend on the θ-coordinate, the following equations must hold uθ = 0

(3.57)

srθ = szθ = 0 .

(3.58)

Thus, the stress–strain relation for the isotropic case is given by ⎛

σrr

⎞

⎜ ⎟ ⎜ σzz ⎟ ⎜ ⎟ ⎜ ⎟= ⎜ σrz ⎟ ⎝ ⎠ σθθ

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

λL + 2µL λL

λL

0

λL

λL + 2µL 0

λL

0

0

λL

λL

µL

0

0 λL + 2µL

⎤ ⎛ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

srr

⎞

⎜ ⎟ ⎜ szz ⎟ ⎜ ⎟ ⎜ ⎟. ⎜ 2srz ⎟ ⎝ ⎠ sθθ

(3.59)

3.5 Waves in Solid Bodies In this section we want to investigate different wave types that can propagate within solids. In the general case, the mechanical displacement u can be decomposed into an irrotational part (curl u = 0) and a solenoidal part (div u = 0) (Helmholtz decomposition) u = grad ϕ + curl ψ .

(3.60)

By using this decomposition for (3.45) and setting fV to zero, we arrive at

64

3 Mechanical Field

µL ∇ · ∇grad ϕ + (λL + µL )∇ ∇ · grad ϕ + µL ∇ · ∇curl ψ ∂ 2 grad ϕ ∂ 2 curl ψ +(λL + µL )∇ ∇ · curl ψ = ρ +ρ 2 ∂t ∂t2 (3.61)

grad

  2 ∂ ϕ ρ 2 − (λL + 2µL )∇ · ∇ϕ ∂t  2  ∂ ψ +curl ρ 2 − µL ∇ · ∇ψ = 0 . ∂t

(3.62)

This relation is fulfilled if ϕ and ψ solve the two following equations λL + 2µL ∂2ϕ ∇ · ∇ϕ = 2 ∂t ρ ∂2ψ µL ∇ · ∇ψ . = 2 ∂t ρ

(3.63) (3.64)

First, we will choose for (3.63) an ansatz expressed by ϕ = f (ξ) = f (k · r − ωt) ,

(3.65)

which defines a wave propagation in direction of k with velocity c. By using this ansatz for (3.63), we obtain with     2 ∂2ϕ ∂ ∂ϕ ∂ξ ∂ ∂ϕ 2∂ ϕ = (3.66) = −ω = ω ∂t2 ∂t ∂ξ ∂t ∂t ∂ξ ∂ξ 2     2 ∂ϕ ∂ξ ∂ϕ ∂ ∂2ϕ ∂ 2 ∂ ϕ = (3.67) = k = k i i ∂x2i ∂xi ∂ξ ∂xi ∂t ∂ξ ∂ξ 2 (3.68) the following result ω2

∂2ϕ = ∂ξ 2

 

=

λL + 2µL ρ λL + 2µL ρ

  3 

i=1

ki2

∂2ϕ ∂ξ 2

3 ∂2ϕ  2 k . ∂ξ 2 i=1 i   

(3.69)

k2

Now, since the relation c2 = ω 2 /k 2 holds, (3.63) is fulfilled, if c2 takes on the value (λL + 2µL )/ρ. The mechanical displacement for the scalar component ϕ computes

3.6 Material Properties

∂ϕ ∂ϕ ∂ϕ ex + ey + ez ∂x ∂y ∂z ∂ϕ ∂ξ ∂ϕ ∂ξ ∂ϕ ∂ξ ex + ey + ez = ∂ξ ∂x ∂ξ ∂y ∂ξ ∂z ∂ϕ . =k ∂ξ

65

u = grad ϕ =

(3.70)

Thus, (3.70) clearly shows that the mechanical displacements are in the direction of the wave propagation, which defines a longitudinal wave with velocity cL . . λL + 2µL (1 − νp )Em = . (3.71) cL = ρ (1 + νp )(1 − 2νp )ρ By choosing the ansatz F = F(ξ) = F(k · r − ωt)

(3.72)

we can solve (3.64). In this case the mechanical displacement is computed by u = curl ψ ∂ψ = k× , ∂ξ

(3.73)

and we obtain a mechanical displacement that is perpendicular to the direction of propagation. The type of wave is called a shear wave, which propagates with velocity cT . . / µL Em G = = . (3.74) cT = ρ 2(1 + νp )ρ ρ The ratio of the two velocities . cL = cT

λL + 2µL µL

(3.75)

leads to the following inequality cL >

√ 2cT .

(3.76)

3.6 Material Properties The mechanical material properties are defined by the density ρ and the tensor of mechanical moduli [c]. In the general case, the entries of [c] depend on the mechanical stress σ. Figure 3.8 displays a typical stress–strain curve of a metallic material obtained by a tensile test. For the region defined by stresses up to σp , we find a strict proportionality between the stress and strain as

66

3 Mechanical Field

used in linear elasticity computations. For stresses larger than σp , we can find a super-proportional increase of the strain till the stress reaches σy , the socalled yield stress. By a further increase of the applied force, the stress again strongly increases due to stiffening effects of the material until the sample breaks at σb . For a more detailed discussion, especially on material models

Fig. 3.8. Stress–strain curve for a metallic material

(e.g., viscoelastic, viscoplastic, etc.) we refer to [18, 220]. Furthermore, heating up a solid body will also result in a mechanical deformation. The resulting thermal strain sth ij can be modeled as follows sth ij = αi (T − T0 ) ,

(3.77)

with αi the so-called thermal expansion coefficient in direction i and T0 the reference temperature. For a homogenous and isotropic material, the value of α is the same for all directions, so that the shear strains are zero. Therefore, Hook’s law can be written as * ) (3.78) [σ] = [c] [S] − [Sth ] & α(T − T0 ) for i = j sth . (3.79) ij = 0 for i = j In Table 3.1 the mechanical properties of some materials are summarized.

3.7 Numerical Computation 3.7.1 Linear Elasticity The strong formulation for linear elasticity problems reads as follows:

3.7 Numerical Computation

67

Table 3.1. Mechanical properties of some materials ρ (kg/m3 )

Material aluminum iron copper PVC

2.7 × 103 7.7 × 103 8.9 × 103 1.1 × 103

Em (N/m2 ) 7.20 × 1010 21.6 × 1010 12.5 × 1010 0.30 × 1010

νp 0.34 0.29 0.35 0.48

α (1/T)

cL (m/s)

24 × 10−6 12 × 10−6 12 × 10−6 11 × 10−6

6.3 × 103 5.9 × 103 4.7 × 103 2.2 × 103

cT (m/s) 3.13 × 103 3.20 × 103 2.26 × 103 1.10 × 103

Given: u0 u˙ 0 ρ, cij fV

:Ω :Ω :Ω :Ω

→ IRd → IRd → IR → IRd .

¯ × [0, T ] → IRd Find: u(t) : Ω B T [c]Bu + fV = ρ¨ u.

(3.80)

Boundary conditions u = ue on Γe × (0, T ) T

[σ] n = σ n on Γn × (0, T ) . Initial conditions u(r, 0) = u0 , r ∈ Ω ˙ 0) = u˙ 0 , r ∈ Ω . u(r, For simplicity, we will set the boundary conditions to zero (ue = 0, σ n = 0). Multiplying (3.80) by an appropriate test function u
and performing a partial integration will transform (3.80) to its variational formulation, which reads as follows: Find u ∈ H10 such that   

T ¨ dΩ + (Bu ) [c]Bu dΩ = ρu · u u
· fV dΩ (3.81) Ω


Ω

Ω

for any u ∈ Let us perform the spatial discretization with standard nodal finite elements, which approximate the continuous displacement u as follows ⎞ ⎛ n
n n
n nd  Na 0 0   u ≈ uh = Na uia ei = Na ua ; Na = ⎝ 0 Na 0 ⎠ , (3.82) a=1 i=1 a=1 0 0 Na H10 .

with nd the space dimension and n
n the number of finite element nodes with no Dirichlet boundary condition. Applying the same approximation to the

68

3 Mechanical Field

test function u
, we have the following semidiscrete Galerkin formulation for linear elasticity ⎛ n
n n
n     ⎝ ρNTa Nb dΩ u ¨ b + (Bau )T [c]Bbu dΩ ub a=1 b=1

Ω

Ω

 −

⎞

NTa fV (ra ) dΩ ⎠ = 0 ,

(3.83)

Ω

with

⎛ ∂N

0 ∂Na ∂y

0 0

0

∂Na ∂z

a

∂x

⎜ Bau = ⎝ 0 0

0 ∂Na ∂z ∂Na ∂y

∂Na ∂Na ∂z ∂y a 0 ∂N ∂x ∂Na 0 ∂x

⎞T ⎟ ⎠ .

(3.84)

In matrix form, we may write (3.83) as Mu u ¨ + Ku u = f ,

(3.85)

with Mu =

ne 

 meu

;

meu

Ku =

ne 

f =

(3.86)

(Bpu )T [c]Bqu dΩ

(3.87)

Ωe

 keu ; keu = [kpq ] ; kpq =

e=1 ne 

ρNTp Nq dΩ

= [mpq ] ; mpq =

e=1

Ωe

 f

e

e

NTp fV (rp ) dΩ .

; f = [f p ] ; f p =

e=1

(3.88)

Ωe

The time discretization is performed by a standard Newmark method as explained in Sect. 2.5.2. Thus, we arrive at the following time-stepping scheme for an effective mass formulation: •

Perform predictor step: ∆t2 (1 − 2βH ) u ¨n 2 ˜˙ = u˙ n + (1 − γH )∆t u ¨n . u

u ˜ = un + ∆t u˙ n +

•

(3.89) (3.90)

Solve algebraic system of equations: ˜˙ M∗u u ¨n+1 = f n+1 − Ku u ˜ − Cu u M∗u

(3.91) 2

= Mu + γH ∆t Cu + βH ∆t Ku .

(3.92)

In (3.91) we have introduced a damping matrix Cu according to a standard Rayleigh model (see Sect. 3.7.2).

3.7 Numerical Computation

•

69

Perform corrector step: ˜ + βH ∆t2 u ¨n+1 un+1 = u ˜˙ + γH ∆t u u˙ n+1 = u ¨ n+1 .

(3.93) (3.94)

3.7.2 Damping Model In general, vibrating mechanical systems will always show a damped behavior. The reason for the damping is mainly to friction within the material and its mathematical model is usually an added velocity proportional damping term. Therefore, within the FE method a constant damping matrix Cu is introduced and the term Cu u˙ is added to the semidiscrete Galerkin formulation given in (3.85). In many FE formulations, the Rayleigh damping model is applied, so that Cu is computed via a combination of the mass matrix Mu and the linear stiffness matrix Ku Cu = αM Mu + αK Ku . (3.95) In (3.95) αM denotes the mass proportional and αK the stiffness proportional damping coefficients. As shown in [16], a mode superposition analysis includ-

Fig. 3.9. Damped sine curve

ing damping according to (3.95) leads to the following relation αM + αK ωi2 = 2ωi ξi ,

(3.96)

with ωi the i-th eigenfrequency (in rad/s) and ξi the modal damping for the i-th eigenmode. The modal damping ξi corresponds to the loss factor tan δi for ωi , so that we obtain

70

3 Mechanical Field

tan δi = 2ξi =

αM + αK ωi2 . ωi

(3.97)

˙ can be deIn addition, it can be shown (see [16]) that (3.85) (with Cu u) composed in a system of non-coupled single degree of freedom differential equations with unit mass as well as stiffness x¨i (t) + 2ξi ωi x˙ i (t) + ωi2 xi (t) = fi (t) ,

(3.98)

with generalized displacements xi and forces fi . The technically relevant solution of (3.98) will be an exponentially (with amplitude ξi ) damped sine curve as displayed in Fig. 3.9. The relation between the logarithmic decrement Di and the modal damping factor ξi computes as Di = ln . ξi =

xn 2πξi = + xn+1 1 − ξi2 Di2 . Di2 + 4π 2

(3.99)

(3.100)

Therefore, if we measure e.g., the decay of a mechanical wave excited with frequency ωi propagating within a solid body, we can compute the damping factor ξi and thus the loss factor tan δi . The computation of αM and αK for this ξi can then be performed using (3.97) as follows αM + αK (ωi + ∆ω)2 = 2(ωi + ∆ω) ξi αM + αK (ωi − ∆ω)2 = 2(ωi − ∆ω) ξi .

(3.101) (3.102)

The value ∆ωi shall be kept small, so that we meet the prescribed ξi at ωi . However, if we have to model a wide frequency range by fixed αM and αK , which is, e.g., the case within a transient analysis, the actual ξ will differ from ξi according to (3.97). Let us suppose we perform a transient analysis of a thickness mode piezoelectric transducer with resonance frequency 1 MHz, and we set the damping coefficient ξ at resonance frequency to 0.005. Then, we will compute the Rayleigh damping coefficients in order to meet this damping at resonance frequency, which will exhibit αM = 3.1 × 104 and αK = 7.9 × 10−10 . For all other frequencies within the excitation signal, the damping ξ will be according to Fig. 3.10. 3.7.3 Geometric Non-linear Case We have derived the partial differential equation for the mechanical field by considering the equilibrium equations (translation as well as rotation) for an elastic body (for simplicity we just consider the static case) ∇ [σ] + fV = 0 u = ue T

[σ[ n = σ n

on Γe

(3.103)

on Γn .

(3.104)

3.7 Numerical Computation

71

Fig. 3.10. Damping factor ξ as a function of frequency (ξi = 0.005 at 1 MHz; computed αM = 3.1 × 104 and αK = 7.9 × 10−10 )

In (3.103) [σ] denotes the mechanical stress tensor, fV any volume force, and u the mechanical displacement. However, (3.103) is just applicable for linear mechanics, since we are mixing up quantities defined in the deformed configuration (e.g., Cauchy stress tensor [σ]) and quantities defined in the initial configuration (e.g., mechanical volume force fV ). Now, if we perform any computation, we always start at the initial configuration and aim at calculating the deformation of the body due to any prescribed boundary conditions and volume forces given for the initial configuration. Thus, we have to transform the Cauchy stress tensor [σ] from the deformed to the initial configuration using (3.28)    [σ] dΓ = |J |[σ][Fd ]−T dΓ0 = [τ ] dΓ0 . (3.105) Γ

Γ0

Γ0

In (3.105), [τ ] stands for the 1st Piola–Kirchhoff tensor, which is a nonsymmetric stress tensor. Therefore, we introduce the 2nd Piola–Kirchhoff tensor [T], which represents no physical stresses, but is symmetric and computes as [T] = [Fd ]−1 [τ ] = |J |[Fd ]−1 [σ] [Fd ]−T .

(3.106)

Thus, we can rewrite (3.103) as ∇X ([Fd ] [T]) + fV = 0 ,

(3.107)

including only quantities defined on the initial configuration (no mixing of quantities defined in the deformed and initial state as in (3.103)). According

72

3 Mechanical Field

to (3.107), we define the non-linear operator F as a function of the mechanical displacement u F (u) = ∇X ([Fd ] [T]) + fV = 0 . (3.108) Now, a Newton step can be written as (see Appendix D.2) uk+1 = uk + s

with F
(uk )[s] = −F(uk ) .

(3.109)

First, let us derive the weak formulation of (3.107). For an arbitrary test function u
∈ H10 and assuming homogeneous Neumann boundary condition σ n = 0, we obtain  u
(∇X · ([Fd ] [T]) + fV ) dΩ = 0 (3.110) Ω0



[T] · [Fd ]T ∇X u
dΩ =

Ω0



u
· fV dΩ ,

(3.111)

Ω0

⎛ ∂u


with

∂u
x ∂u
x ∂X ∂Y ∂Z x

⎜
⎜ ∇X u
= ⎜ ∂uy ⎝ ∂X

∂u
y ∂u
y ∂Y ∂Z

∂u
z

∂u
z

∂u
z

∂X

∂Y

∂Z

⎞ ⎟ ⎟ ⎟ ⎠

(3.112)

being in general a nonsymmetric tensor. Since [T] is a symmetric tensor, the scalar product with [Fd ] will always result in a symmetric tensor. Therefore, we rewrite the first term in (3.110) as   ( 1' [Fd ]T ∇X u
+ ∇X T u
[Fd ] dΩ . [T] · [Fd ]T ∇X u
dΩ = [T] · 2 Ω0

Ω0

(3.113) According to (3.109), we need the Frech´et derivative F
, which can be approximated by F (uk + s) − F(uk ) (see Appendix D.2). In its weak form, we have to compute  ( 1' [Fd (uk + s)]T ∇X u
+ ∇X T u
[Fd (uk + s)] dΩ [T(uk + s)] · 2 Ω0    1 − [T(uk )] · [Fd (uk )]T ∇X u
+ ∇X T u
[Fd (uk )] dΩ . (3.114) 2 Ω0

Now, let us remember the following relations * 1 ) [c] [Fd ]T [Fd ] − I 2 [Fd ] = I + ∇X u , [T] = [c][V] =

(3.115) (3.116)

3.7 Numerical Computation

73

with [c] the tensor of the mechanical elasticity coefficients (in our case we assume constant entries). The evaluation of the term [T(uk + s)] leads to [T(uk + s)] = [c][V(uk + s)] * 1) [V(uk + s)] = [Fd (uk + s)]T [Fd (uk + s)] − I 2 ( *T ) * 1 ') I + ∇X (uk + s) = I + ∇X (uk + s) − I 2 ( *T ) * 1 ') I + ∇X uk + ∇X s = I + ∇X uk + ∇X s − I . (3.117) 2 Neglecting all second-order terms, we arrive at   1 [Fd (uk )]T [Fd (uk )] − I [V(uk + s)] ≈ 2   *T * ) 1 ) + I + ∇X uk ∇X s + ∇X T s I + ∇X uk 2   1 = [V(uk )] + [Fd (uk )]T ∇X s + ∇X T s [Fd (uk )] . (3.118) 2 In addition, the term [Fd (uk + s)] can be expressed as follows [Fd (uk + s)] = I + ∇X (uk + s) = [Fd (uk )] + ∇X s .

(3.119)

With the help of these expressions for [V(uk + s)] and [Fd (uk + s)], we can rewrite the first term in (3.114) as     1 (3.120) [T(uk )] + [c] [Fd (uk )]T ∇X s + ∇X T s [Fd (uk )] 2 Ω0     1 T T
k T
k · dΩ . [Fd (u )] + ∇X s ∇X u + ∇X u [Fd (u )] + ∇X s 2 Setting all second-order terms to zero, we arrive at    1 [T(uk )] · [Fd (uk )]T ∇X u
+ ∇X T u
[Fd (uk )] dΩ 2 Ω0    1 T T
k
+ [T(u )] · ∇X s ∇X u + ∇X u ∇X s dΩ 2 Ω0    1 T k T k [c] [Fd (u )] ∇X s + ∇X s [Fd (u )] + 2 Ω0   1 T
k T
k · [Fd (u )] ∇X u + ∇X u [Fd (u )] dΩ . 2

(3.121)

74

3 Mechanical Field

Substituting this result into (3.114), we get    1 T k T k [c] [Fd (u )] ∇X s + ∇X s [Fd (u )] 2 Ω0   1 T
k T
k · [Fd (u )] ∇X u + ∇X u [Fd (u )] dΩ 2    1 T T
k
+ [T(u )] · ∇X s ∇X u + ∇X u ∇X s dΩ . 2

(3.122)

Ω0

Therefore, the Newton step can be evaluated as follows    1 [c] [Fd (uk )]T ∇X s + ∇X T s [Fd (uk )] 2 Ω0   1 T
k T
k · [Fd (u )] ∇X u + ∇X u [Fd (u )] dΩ 2    1 T T
k
+ [T(u )] · ∇X s ∇X u + ∇X u ∇X s dΩ 2 Ω0  = u
· fV dΩ Ω0

 −

[T(uk ] ·

  1 [Fd (uk )]T ∇X u
+ ∇X T u
[Fd (uk )] dΩ (3.123) 2

Ω0

uk+1 = uk + s . Before we go over to the discretized version of (3.123), let us apply some helpful transformations. Using the relation [Fd (uk )] = [I + ∇X (uk )], we obtain   1 [Fd (uk )]T ∇X s + ∇X T s [Fd (uk )] 2 1 1 [I + ∇X (uk )]T ∇X s + ∇X T s [I + ∇X (uk )] = 2 2 ( ( 1 ' 1 ' T ∇X s + ∇X s + ∇X T uk ∇X s + ∇X T s∇X uk . =   2   2 Bs

Bnl (uk ) s

The differential operator B has already been defined (see (3.20)) and B nl computes as follows

3.7 Numerical Computation

⎛

B nl

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ =⎜ k ⎜ ∂ux ⎜ ∂Y ⎜ ⎜ k ⎜ ∂ux ⎜ ∂X ⎝

∂uk ∂ x ∂X ∂X

∂uk y ∂ ∂X ∂X

∂uk ∂ z ∂X ∂X

∂uk ∂ x ∂Y ∂Y

∂uk y ∂ ∂Y ∂Y

∂uk ∂ z ∂Y ∂Y

∂uk ∂ x ∂Z ∂Z

∂uk y ∂ ∂Z ∂Z

∂uk ∂ z ∂Z ∂Z

∂ ∂Z

+

∂uk ∂ x ∂Z ∂Y

∂uk y ∂ ∂Y ∂Z

+

∂uk y ∂ ∂Z ∂Y

∂uk ∂ z ∂Y ∂Z

+

∂uk ∂ z ∂Z ∂Y

∂ ∂Z

+

k ∂uk ∂ ∂uy ∂ x ∂Z ∂X ∂X ∂Z

+

k ∂uk y ∂ ∂uz ∂ ∂Z ∂X ∂X ∂Z

+

∂uk ∂ z ∂Z ∂X

∂uk ∂ x ∂X ∂Y

+

k ∂uk ∂ ∂uy ∂ x ∂Y ∂X ∂X ∂Y

+

k ∂uk y ∂ ∂uz ∂ ∂Y ∂X ∂X ∂Y

+

∂uk ∂ z ∂Y ∂X

75

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ . (3.124) ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

The second term in (3.123) contains a scalar product between two tensors, which in general for two tensors [A] and [B] computes as [A] · [B] = A11 B11 + A12 B12 + A13 B13 + A21 B21 + A22 B22 + A23 B23 + A31 B31 + A32 B32 + A33 B33 . Using this relation, we can rewrite this term as follows ( 1' ˜ T [T]B ˜ ∇X T s ∇X u
+ ∇X T u
∇X s = B [T(uk )] · x 2  T ∂ ∂ ∂ ˜ = B . ∂X ∂Y ∂Z

(3.125) (3.126)

Approximating s and u
by nodal finite elements (see (3.82)) will result in the following discrete Galerkin formulation k KNL u (u )s = f a − f i .

(3.127)

In (3.127) KNL u , f a (external applied forces) as well as f i (internal forces due to stresses) are calculated as follows KNL u =

ne 

keu ; keu = [kpq ]

e=1



kpq = Ωe



(Bpu )T [c]Bqu

dΩ +

 '

(

+

(Bpnl )T [c]Bqnl dΩ Ωe

B˜pT [T]B˜q I dΩ

(3.128)

Ωe

fa =

ne 

f e ; f e = [f p ]

(3.129)

NTp fV (rp ) dΩ

(3.130)

e=1



fp = Ωe

76

3 Mechanical Field

fi =

ne 

f e ; f ei = [f p ]

(3.131)

) u *T Bp + Bpnl [T(uk )] dΩ ,

(3.132)

e=1



fp = Ωe

with Bpu as given in (3.84), I the identity matrix and T the second Piola– Kirchhoff tensor in vector notation. The operator Bpnl depends on uk and computes as follows ⎞ ⎛ k k k

Bpnl

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ∂ukx ⎜ ∂Y ⎜ ⎜ k ⎜ ∂ux ⎜ ∂X ⎜ ⎝ ∂ukx ∂X

∂ux ∂Np ∂X ∂X

∂uy ∂Np ∂X ∂X

∂uz ∂Np ∂X ∂X

∂uk x ∂Np ∂Y ∂Y

∂uk y ∂Np ∂Y ∂Y

∂uk z ∂Np ∂Y ∂Y

∂uk x ∂Np ∂Z ∂Z

∂uk y ∂Np ∂Z ∂Z

∂uk z ∂Np ∂Z ∂Z

∂Np ∂Z

+

∂uk ∂uk y ∂Np x ∂Np ∂Z ∂Y ∂Y ∂Z

+

k ∂uk y ∂Np ∂uz ∂Np ∂Z ∂Y ∂Y ∂Z

+

∂uk z ∂Np ∂Z ∂Y

∂Np ∂Z

+

∂uk ∂uk y ∂Np x ∂Np ∂Z ∂X ∂X ∂Z

+

k ∂uk y ∂Np ∂uz ∂Np ∂Z ∂X ∂X ∂Z

+

∂uk z ∂Np ∂Z ∂X

∂Np ∂Y

+

∂uk ∂uk y ∂Np x ∂Np ∂Y ∂X ∂X ∂Y

+

k ∂uk y ∂Np ∂uz ∂Np ∂Y ∂X ∂X ∂Y

+

∂uk z ∂Np ∂Y ∂X

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (3.133)

with ∂ukα /∂β computed within each integration point by e ∂ukα  ∂Na k = u ∂β ∂β a α a=1

n

α , β ∈ {X, Y, Z} .

(3.134)

The operator B˜p is given by  B˜p =

∂Np ∂Np ∂Np ∂X ∂Y ∂Z

 .

(3.135)

The iterative solution process is stopped if the incremental error as well as the residual error fulfill ||uk+1 − uk ||2 < δa ||uk+1 ||2

||f k+1 − fa ||2 i ||f a ||2

< δr ,

(3.136)

with appropriate δa and δr . To guarantee that the Newton method converges to the correct solution, a line search algorithm should be applied to obtain an optimal relaxation parameter η in each Newton step (see Appendix D) uk+1 = uk + ηs .

(3.137)

3.7 Numerical Computation

77

3.7.4 Numerical Example In order to show the influence of the discretization as well as the order of the shape functions used for the approximation, we will compute the deflection of a beam due to a mechanical load. Figure 3.11 displays the setup and since the beam is fully supported at both sides, we exploit the symmetry. The beam has

Fig. 3.11. Setup

a length of L = 1000 µm and a thickness of d = 4 µm, and we perform a plane strain analysis. On the left side (x = 0), where the beam is fully supported, we set ux = uy = 0 and on the right side (symmetry) we set ux = 0. The nodal force has a value of 1 N and is supplied at x = L/2 and y = d. In the first step we will perform a linear analysis using once linear and once quadratic shape functions for the approximation. The discretization in the y-direction is done just by one finite element and the discretization in the x-direction is varied. As displayed in Table 3.2, the obtained tip displacement for the linear shape functions converges very slowly to the correct value when increasing the number of finite elements ne . The reason for this effect is that linear shape functions exhibit a very poor approximation of the true solution when applied to thin walled structures. This effect is called locking (see e.g., [16, 99]), and will be discussed in detail in the subsequent section. Table 3.2. Tip displacement uy at x = L/2 and y = 0

ne 20 40 80 160 320

Type of basis function Linear Quadratic –0.145 µm –4.436 µm –0.947 µm –4.457 µm –2.182 µm –4.465 µm –3.238 µm –4.468 µm –3.683 µm –4.468 µm

In the second step we perform a non-linear analysis and compare the results to the linear one (see Fig. 3.12). For the discretization, quadrilateral elements

78

3 Mechanical Field

with quadratic shape functions have been used with one finite element in the y-direction and 40 finite elements in the x-direction. Figure 3.12 clearly demonstrates that due to the large deflection, which is of the order of the thickness of the beam, a non-linear analysis has to be performed for our setup.

Fig. 3.12. Displacement along the beam obtained by linear and non-linear analysis

3.8 Locking and Efficient Solution Approaches For some problems in computational mechanics, we recognize that the straight forward application of the displacement-based finite element method (as described in Sec. 3.7.1) will provide inaccurate results and show slow convergence when reducing the mesh size. Being specific, the finite element solution results in too small displacements (see example in the previous section), and we will refer to this effect as locking. In general, the three main locking phenomena are as follows: •

Shear locking: The shear locking effect occurs by the application of the displacement-based finite element method to thin walled structures, where especially the relation t << h holds (t denotes the thickness and h the mesh size).

•

Volumetric (Poisson) locking: For nearly incompressible materials (Poisson ration νP → 0.5, which means for the Lam´e parameters λL >> µL )

3.8 Locking and Efficient Solution Approaches

79

volumetric locking will occur when applying the displacement-based finite element method. •

Membrane locking: This type of locking just occurs in curved beam and shell elements.

We will concentrate ourself on the shear locking and start with a mathematical investigation (see e.g., [29]). We consider the following weak formulation for the unknown u in an appropriate Hilbert space X a0 (u, v) +

1 (Bu, Bv) =< f, v > . t2

(3.138)

In (3.138) t denotes a parameter with 0 < t ≤ 1, a0 : X ×X → IR a continuous, symmetric and coercitive bilinear form, B : X → L2 a continuous mapping and f a given function in the computational domain Ω. Now, one can show that for a given f , there exists a v0 such that < f, v0 >= 0 and Bv0 = 0. Then, the solution u is bounded from below by ||u|| ≥ C||f || with a constant C independent of t. However, in the FE setting, we obtain [30] ||uh || ≤ t2 C(h−2 )||f || . Therewith, for small values of the parameter t the standard finite element method will result in too small displacements, and we call the phenomena locking. Let us demonstrate the shear locking effect by considering a thin Timoshenko beam. There, the assumption is made that a plane section original perpendicular to the neutral axis remains plane, but according to the shear deformations rotates as displayed in Fig. 3.13. The shear angle γ and curvature 

 Neutral axis

w

Beam cross-section

Fig. 3.13. Timoshenko beam assumptions

80

3 Mechanical Field

κ can be expressed as follows γ=

∂w −β ∂x

κ=

∂β . ∂x

Therewith, the total potential energy Epot computes as (see e.g., [16])

Epot =

Em I 2

L 

∂β ∂x

2

GAk 2

dx+

0

L 

∂w −β ∂x

2

L dx+

0

pw dx . (3.139) 0

In (3.139), Em denotes the elasticity modulus, G the shear modulus, A = bt the cross sectional area with t the thickness and b the width of the beam, I the moment of inertia (I = bt3 /12), k the shear correction factor and p the external load. Therein, the first term defines the bending energy, the second term the shear energy and the last one the potential of the load. By normalizing Epot and neglecting for the further considerations the load term, we obtain 2 L  2 L  ∂β ∂w 1 ˜ Epot = −β dx + 2 dx . (3.140) ∂x t ∂x 0

0

˜pot , we arrive at the weak By applying the variation of the potential energy E formulation for the Timoshenko beam, which will be similar to (3.138) with B(β, w) =

∂w −β. ∂x

From (3.140) we can conclude that for the case of a very thin beam (t << 1), the term B(β, w) has to approach zero. This is actual the case, since the shear deformation will approach zero, e.g., ∂w/∂x = β with γ = 0. For the finite element method, we subdivide our computational domain [0, L] into finite elements of equal mesh size h. Choosing linear basis functions, e.g., c1 x + c0 , it is easily shown that the following inequality holds x0 +h

h3 3 h2 c1 = (c1 x + c0 ) dx ≥ 3 3

x0 +h

2

x0

c21 dx . x0

With this result, we obtain for the terms in (3.140) the following relation x0 +h 

x0

∂wh − βh ∂x

2

h2 dx ≥ 3

x0 +h 

∂β h ∂x

2 dx .

x0

This clearly shows us that with linear basis functions a strong locking will occur, and the FE solution will result in too small displacements.

3.8 Locking and Efficient Solution Approaches

81

In the following, we will investigate efficient solution approaches for treating the locking effect. Therewith, we will discuss two very popular methods, namely the method of incompatible modes and the method of enhanced assumed strain. Furthermore, we will show that locking still occurs for second-order basis functions and we will apply a balanced reduced and selective integration technique. 3.8.1 Incompatible Modes Method The method of incompatible modes (IM) has been first introduced in [216] and was enhanced for general shaped bilinear quadrilateral elements in [207]. It is one of the most popular methods in computational mechanics to avoid shear, as well as volumetric locking. The key idea is based on extending the shape functions of quadrilateral (hexahedron) elements to account for higherorder modes. The additional degrees of freedom are kept internal and no compatibility of these displacements are taken into account. Therewith, these incompatible displacements can be eliminated at the element level, so that the global number of degrees remain the same compared to standard quadrilateral (hexahedron) elements in structural mechanics. The mechanical displacement u for the IM method is extended by incompatible displacements α u ≈ uh + αh =

nen 

Na ua +

a=1

nIM 

NIM a αa .

(3.141)

a=1

For a bilinear quadrilateral element, the number of additional shape functions nIM is two. They are defined by N1IM = 1 − ξ 2

N2IM = 1 − η 2 ,

and for a hexahedron we extend the standard shape functions by N1IM = 1 − ξ 2

N2IM = 1 − η 2

N3IM = 1 − ζ 2 .

Therewith, in addition to the standard differential operator Bau (see (3.84)), we define the differential operator BaIM containing the incompatible modes ⎛ ⎜ BaIM = ⎜ ⎝

∂NaIM ∂x

0 0

0

0

∂NaIM ∂y

0

0

∂NaIM ∂z

0 ∂NaIM ∂z ∂NaIM ∂y

∂NaIM ∂NaIM ∂z ∂y ∂NaIM 0 ∂x ∂NaIM 0 ∂x

⎞T ⎟ ⎟ . ⎠

(3.142)

This will lead to the following system of equations on the element level   e  e  e u f kuu keuα (3.143) = e e e kαu kαα α 0

82

3 Mechanical Field

with



 T

(B u ) [c] B u dΩ

keuu = Ωe



keαα

=

T

(B u ) [c] B IM dΩ

keuα = Ωe

) IM *T B [c] B IM dΩ .

Ωe

From (3.143) we can compute the incompatible displacements as −1

αe = − (keαα )

keαu ue

(3.144)

and obtain ˜ e ue = f e k ˜ e = ke − ke (ke )−1 ke . k uu uα αα αu

(3.145) (3.146)

Now, as pointed out in [207], the straightforward application of the IM method as described above does not pass the so-called patch test. This test is a necessary requirement for displacement-type elements in structural mechanics to assess convergence. Shortly speaking, the patch test requires αe to be zero, whenever ue corresponds to a rigid body motion or constant strain condition (for a detailed discussion we refer to [226]). Now, let us define ue0 the set of displacements corresponding to one of the above cases, then we obtain from (3.144) −1 αe0 = − (keαα ) keαu ue0 . Since keαα is positive definite, we arrive for αe0 to be zero at the following relation  ) IM *T keαu u0 = B [c] B u u0 dΩ = 0 . (3.147) Ωe

The assumption on u0 results for the term [c] B u u0 in some constant stress. Consequently, the requirement (3.147) reduces to  B IM dΩ = 0 . Ωe

Investigating this relation for the bilinear quadrilateral element, we obtain (see Sec. 2.3.6)

3.8 Locking and Efficient Solution Approaches



1 1

 B2IM Ωe

⎜ ⎜ ⎝

−ξy,η

83

⎞

0

⎟ ξx,η ⎟ ⎠ dξ dη −1 −1 ξx,η −ξy,η ⎛ ⎞ ηy,ξ 0 1 1 ⎜ ⎟ ⎜ 0 −ηx,ξ ⎟ dξ dη . dΩ = 2 ⎝ ⎠ −1 −1 −ηx,ξ ηy,ξ

B1IM dΩ = 2 Ωe

⎛

0

(3.148)

(3.149)

The derivatives of the global coordinates x = (x, y)T w.r.t. to the local coordinates ξ = (ξ, η)T compute as x,ξ = x,η =

nen  a=1 nen 

ξa (1 + ηa η) xa

(3.150)

ηa (1 + ξa ξ) xa .

(3.151)

a=1

Therewith, whenever the geometric shape of the quadrilateral element is a rectangle or parallelogram, (3.150) and (3.151) are constant, and (3.148) will integrate to zero. The simple but effective idea proposed in [207] to fulfill the condition even in the general case is to evaluate (3.150) and (3.151) at ξ = η = 0. A generalization of this idea can be found in [102] and [24]. 3.8.2 Enhanced Assumed Strain Method Perhaps the most popular method to overcome locking effects in structural mechanics is the method of enhanced assumed strain introduced in [199]. The method is based on the Hu-Washizu principle, eliminating there the mechanical stress and resulting in a nonconforming method. Within the Hu-Washizu principle, all three mechanical fields (displacement, strain and stress) remain in the weak (variational) formulation, which reads as follows (already the discrete weak form): Find uh , Sh and σ h for given a volume force f h , such that  ) h *T ) * η [c] Sh − σ h dΩ = 0 (material law) (3.152)  Ω

Ω

) h *T h Bv σ dΩ − 

 vh · f h dΩ = 0

(force balance law) (3.153)

Ω

) h *T ) h * τ Bu − Sh dΩ = 0 ,

(kinematics)

(3.154)

Ω

for all test functions η h , vh and τ h (equivalent to Sect. 3.7.1, we have set possible prescribed boundary stresses σ n to zero). By introducing the enhanced strain ShEAS as follows

84

3 Mechanical Field

Sh = Buh + ShEAS and substituting it into (3.154), we obtain the following important relation  ) h *T h τ SEAS dΩ = 0 . Ω

Moreover, we set

* ) σ h = [c] Buh + ShEAS

and choose η h from the same function space (space of enhanced gradients) as ShEAS , so that  ) h *T h η σ dΩ = 0 . Ω

Therewith, the discrete stresses σ h and enhanced strains ShEAS are energy orthogonal. This allows us to rewrite the weak formulation of the Hu-Washizu principle as    ) h *T ) h *T h h [c] Bu dΩ + [c] SEAS dΩ = vh · f h dΩ Bv Bv Ω



Ω

) h *T η [c] Buh dΩ +

Ω



Ω

) h *T η [c] ShEAS dΩ = 0 .

(3.155)

Ω

The above discrete weak formulation of the assumed enhanced strain method is fully equivalent to a mixed finite element formulation (see e.g., [29]). In order to satisfy the above mentioned orthogonality condition, we choose the following ansatz for ShEAS ShEAS =

|J0 | EAS B α = B˜EAS α |J |

with |J | the determinant of the Jacobian and |J0 | = |J |ξ=η=0 . In [199], where the assumed enhanced strain has been originally introduced, B EAS computes for a bilinear quadrilateral as ⎛ ⎞ ξ 0 ⎜ ⎟ ⎟ B EAS = ⎜ ⎝0 η⎠ . ξ η For a detailed discussion on the choice of B EAS for quadrilateral as well as hexahedron elements we refer to [7] and [39]. Using the same ansatz for η h , we may rewrite (3.155) on the element level in matrix form

3.8 Locking and Efficient Solution Approaches



keuu keuα keαu keαα

with



u

 e

 =

αe

 Ωe

keαα =

(3.156)

0 

(B u )T [c] B u dΩ

keuu =

f

85

 e

 '

B˜EAS

(T

(B u )T [c] B˜EAS dΩ

keuα = Ωe

[c] B˜EAS dΩ .

Ωe

Similar to the method of incompatible modes, we can perform a static condensation on the element level, and arrive at the modified element stiffness matrix ˜ e = ke − ke (ke )−1 ke . k uu uα αα αu 3.8.3 Balanced Reduced and Selective Integration The approach of a balanced reduced and selective integration technique is motivated by developments for the Mindlin–Reissner plate by [10] as well as [40]; see also [28]. We apply selective reduced integration only to a portion of the shear term (for the Timoshenko beam this shear term is the second term in (3.140)). The amount of the portion is derived by a scaling argument. In this way we obtain a robust finite element treatment and do not encounter zero-energy modes or checkerboard modes as in a pure reduced and selective integration technique (see e.g., [99]), while carrying for the elimination of locking. We consider a 3D body with thickness 2t. Since t is small in applications of interest, we take special care for avoiding locking effects. To this end we will compare the trial functions later with those of a well-established Mindlin– Reissner plate model. The finite element functions with quadratic polynomials in z are written with orthogonal polynomials in z as ui (x, y) = Ui (x, y) + zθi (x, y) + (z 2 −

t2 ) φi (x, y), 3

i = 1, 2, 3. (3.157)

For convenience, we also write x1 = x, x2 = y and x3 = z when evaluating the strains Sik = 12 (∂ui /∂xk + ∂uk /∂xi ). In particular, Si3

1 = 2

      ∂φ3 ∂U3 ∂θ3 1 1 2 t2 + z 2φi + + (z − ) , θi + ∂xi 2 ∂xi 2 3 ∂xi

i = 1, 2. (3.158)

By integrating over the thickness, we obtain

86

3 Mechanical Field



t Sii2 dz = 2t −t

∂Ui ∂xi

2

 2  2 ∂θi ∂φi 2 8 + t3 + t5 , 3 ∂xi 45 ∂xi   

i = 1, 2,

MR-plate

t 2 S12 dz −t

1 = t 2



∂U1 ∂U2 + ∂x2 ∂x1

2

1 + t3 6 



2  2 ∂θ2 2 5 ∂φ1 ∂φ2 ∂θ1 + + t + , ∂x2 ∂x1 45 ∂x2 ∂x1   MR-plate

t −t

8 2 S33 dz = 2t θ32 + t3 φ23 , 3

t 2 Si3 dz = −t

 2  2  2 ∂φ3 1 ∂U3 1 ∂θ3 2 t θi + + t3 2φi + + t5 2 ∂xi 6 ∂xi 45 ∂xi   

i = 1, 2.

MR-plate

The terms that are encountered in the Mindlin–Reissner plate theory are emphasized by the mark ’MR-plate’. The internal stored energy of the solution is proportional to t3 . We avoid locking if the terms with a factor proportional to t do not spoil the finite element solution. It follows from Korn’s inequality that the gradients of U1 and U2 contribute to the energy if they are not small. Therefore, U1 and U2 must be small of order t. The same holds for θ3 . This is consistent with the fact that these terms are set to zero in plate theory. The critical terms are the shear terms with Si3 , where the first term in the integral is relaxed in finite element computations unless polynomials of high order are used. We first consider the case that only polynomials are chosen that are linear in z, i.e., we have φi = 0, i = 1, 2, 3, in (3.157). In this case the shear term is softened as follows. The full integral is incorporated with a factor α=

t2 , h 2 + t2

(3.159)

while selective reduced integration is applied to the rest of the shear term, i.e., to the portion h2 β =1−α= 2 . (3.160) h + t2 Specifically, quadrature formulas with 1 point in the x, y-plane are used. This procedure is equivalent (on parallelograms) to incorporating only the integral of the squared mean-value of θi + ∂U3 /∂xi on each element. The idea to apply a softening only to a portion of the shear term in order to save coercitivity was

3.8 Locking and Efficient Solution Approaches

87

first suggested by [10]. The factors α and β are fixed following [40]. The shear term is a singular perturbation in comparison to the other terms of the internal stored energy. The importance of keeping the coercivity and not reducing the full term was discussed in [29] in the framework of saddle point problems with penalty terms. So our approach differs from the typical application of the EAS concept; see e.g., [176]. The correct softening is more if quadratic polynomials in z are
involved 2 is to be reduced, but the second term present. Only the first term in Si3 must not. This can be realized by different implementations.
2 separated during The most direct way is to keep the three terms in Si3 the assembling and the whole finite element computation. So it is possible to apply the selective reduced integration only to the critical term, i.e., the first one. An alternative is a separation of the even and the odd functions in z, and it goes with φ3 = 0. Recalling (3.158), we see that a softening of the even part of Si3 is required. Note that +t 2 Si3 (x, y, z)dz −t

1 = 2

+t 2 ( Si3 (x, y, z) + Si3 (x, y, −z) ) dz −t

1 + 2

+t 2 ( Si3 (x, y, z) − Si3 (x, y, −z) ) dz. (3.161) −t

A separation of the first and the second term is performed here via a symmetry argument. Only a portion of the even part (with the weight β) is incorporated into the selective reduced integration. A third possibility provides the EAS method. We replace the standard trial functions for Si3 by Si3 + Sˆi3 with the enhanced strain being in each element of the form Sˆi3 = a(x − xe ) + b(y − ye ), where xe and ye are the (local) coordinates of the center of the element. The mean value of the enhanced strain in each element is zero and yields an appropriate softening. Since it does not depend on z, it acts only on the even part of the shear term. Although one finds in literature investigations with the EAS method applied to the complete shear term, the extension to a portion with a given factor is straightforward. The shear components are proportional to the thickness t of the structure, whereas the bending terms scale with t3 . We decompose the material tensor [cE ] with the scaling from (3.159) and (3.160) as follows

88

3 Mechanical Field

[cE ] = [cE ]α + [cE ]β ⎛ 0 c11 c12 c13 0 ⎜ c21 c22 c23 0 0 ⎜ ⎜ c31 c32 c33 0 0 E [c ]α = ⎜ ⎜ 0 0 0 αc44 0 ⎜ ⎝ 0 0 0 0 αc44 0 0 0 0 0 ⎛ ⎞ 000 0 0 0 ⎜0 0 0 0 0 0⎟ ⎜ ⎟ ⎜ 0 0 0 0 0 0⎟ E ⎜ ⎟, [c ]β = ⎜ ⎟ ⎜ 0 0 0 βc44 0 0 ⎟ ⎝ 0 0 0 0 βc44 0 ⎠ 000 0 0 0

(3.162) ⎞

0 0 ⎟ ⎟ 0 ⎟ ⎟, 0 ⎟ ⎟ 0 ⎠ c66

α=

β=

h2 . h 2 + t2

h2

t2 , + t2

(3.163)

(3.164)

In (3.163) and (3.164) h denotes the largest side length and t the smallest side length (for thin structures t corresponds to the thickness) of the elements. Therewith, the element stiffness matrix keu of the mechanical part computes as keu =

14 

BiT [cE ]α B |Ji | wistd

i=1

+

9 1  T (Bi+5 − Bi ) [cE ]β (Bi+5 − Bi ) |Ji | wistd 2 i=5

1 T (B3 + B1 ) [cE ]β (B3 + B1 ) |J1 | w1red 2 + B2T [cE ]β B2 |J2 | w2red , +

(3.165)

where |Ji | denotes the Jacobian determinant at integration point i. As can be seen from (3.165), the first two terms are integrated by the standard scheme (see Tab. 3.3) and the last two terms by the reduced scheme (see Tab. 3.4). In particular, only the symmetrical part of the critical shear term is delt with by reduced integration, and the decomposition (3.161) is used for this purpose. To test the developed method, we will compare our results with those of the Mindlin–Reissner plate bending element of Hughes and Tezduyar [100] (referred to as MR HT) and the Mindlin–Reissner plate bending element of Simo and Rifai (referred to as MT SR). The latter are based on the method of enhanced assumed strains [199]. Contrary to these two formulations, we perform a full 3D-analysis of the structures with hexahedron elements of 2nd order. Specifically, we apply our new approach with balanced reduced integration (referred as present) as well as standard finite elements that do not contain precautions to locking (referred as standard).

3.8 Locking and Efficient Solution Approaches

89

Table 3.3. Points and weights for the standard integration ξi 0.7958 -0.7958 0.0 0.0 0.0 0.7588 0.7588 -0.7588 -0.7588 0.0 0.7588 0.7588 -0.7588 -0.7588

ηi 0.0 0.0 0.7958 -0.7958 0.0 0.7588 -0.7588 0.7588 -0.7588 0.0 0.7588 -0.7588 0.7588 -0.7588

ζi 0.0 0.0 0.0 0.0 -0.7958 -0.7588 -0.7588 -0.7588 -0.7588 0.7958 0.7588 0.7588 0.7588 0.7588

wistd 0.8864 0.8864 0.8864 0.8864 0.8864 0.3352 0.3352 0.3352 0.3352 0.8864 0.3352 0.3352 0.3352 0.3352

Table 3.4. Points and weights for the reduced integration ξi ηi

ζi √ 0 0 −1/ 3 0 0 0√ 0 0 1/ 3

wired 8/3 8/3 8/3

The three structures of investigation are a square, a rhombic, and a circular plate. The data is chosen as described in [199]. In these examples, a simply supported boundary condition is used on the lateral boundary. This means in our displacement formulation that we set the mechanical displacement in thickness direction for all nodes along the bottom line of the outer boundary to zero (see e.g. Fig. 3.14). If a symmetry of the plate is exploited, the computation needs only be performed on a half or a quarter of the plate and the displacement in the direction of the normal is set to zero on the ’symmetry boundaries’. In a first numerical example, we consider the bending of a square plate. The plate has a length of L = 10 m, a thickness of 0.1 m and is simply supported. The modulus of elasticity Em and Poisson ratio νP are chosen to be 10.92 N/m2 and 0.3, and the plate is loaded by a uniform pressure p of 1 N/m2 . We use symmetry boundary conditions in order to just model a quadrant. Figure 3.14 displays the mesh using 4 by 4 elements in the plane and one element in the thickness direction. Table 3.5 shows the numerical results. [A value of 4.0644 · 104 is reported in [199] for the result of a series.] Moreover the stiffness of the 3D model is 0.5−

90

3 Mechanical Field

Fig. 3.14. Mesh for rectangular plate (due to symmetry, only one fourth of the structure is meshed) Table 3.5. Bending of a square plate (all meshes just have one finite element in thickness direction) center displacement × 104 mesh MR HT MT SR standard present 2 4 8 16 32 64

x x x x x x

2 4 8 16 32 64

3.9712 4.0439 4.0593 4.0632

3.9712 4.0436 4.0593 4.0632

3.6688 4.0438 4.0711 4.0831 4.0913 4.0947

4.0036 4.1766 4.1497 4.1173 4.1035 4.1025

1% smaller than that of the plate. This is explained by the additional degrees of freedom of the 3D model. It is not an effect of the reduced integration since the standard finite elements show the same effect asymptotically. The 30o skew plate, as displayed in Fig. 3.15, has a side length of 100 m, thickness of 1 m and is loaded by a uniform pressure of 1 N/m2 . The modulus of elasticity Em is chosen to be 103 N/m2 and the Poisson ratio νP to 0.3. All along the boundary the plate is simple supported, which means for a

Fig. 3.15. Mesh for rhombic plate

3.8 Locking and Efficient Solution Approaches

91

displacement formulation setting the mechanical displacement in thickness direction to zero. The results of our computations as well as the one reported in [199] are listed in Tab. 3.6. Table 3.6. Bending of a rhombic plate (all meshes just have one finite element in thickness direction)

mesh 4x4 8x8 16 x 16 32 x 32 64 x 64 128 x 128

center displacement × 10−2 MR HT MT SR standard present 3.8803 3.9841 4.1565 4.2727 4.3883 4.4668

1.9628 3.0536 3.8283 4.3270 4.5474 4.6460

3.0288 3.8631 4.3217 4.5258 4.6593 4.6965

There is the well-known complication that we have not only the danger of locking, but also less regularity at the 120o corner. Therefore, the convergence of the 3D finite element computations is slower. [Note that the FE results of the plate models do not reflect the reported result of a series which is 4.455 · 10−2 .] For the circular plate example we use the same material parameters as for the square plate above. The radius is chosen to be 5 m, the thickness 0.1 m, and the plate is loaded by a uniform pressure of 1 N/m2 . The plate is simply

Fig. 3.16. Mesh for circular plate

supported on its boundary. Again, we model just one quadrant of the plate for symmetry reasons. The thickness of the plate will be discretized for the computation by one element, and we will use isoparametric elements in the plane as shown in Fig. 3.16. Since isoparametric elements of second degree are used, the curved boundary is approximated by quadratic polynomials. – Table 3.7 contains the numerical results.

92

3 Mechanical Field

Table 3.7. Bending of a circular plate (all meshes just have one finite element in thickness direction)

mesh 12 48 192 768

elements elements elements elements

center displacement × 104 MR HT MT SR standard present 3.9070 3.9649 3.9789 3.9822

3.6966 3.9140 3.9664 3.9791

3.7595 3.8366 3.8640 3.8696

3.8788 3.8784 3.8741 3.8726

4 Electromagnetic Field

4.1 Maxwell’s Equations The full system of partial differential equations describing the electromagnetic field was published for the first time by James Clerk Maxwell in his work A Treatise on Electricity and Magnetism, Vol. I,II [153, 154] in the year 1862. He based his theory on the work and experiments of Amp` e re, Gauss, and Faraday. His great contribution lies in the unification of the different equations to a set of partial differential equations. The introduction of the displacement current, which generalizes Amp`ere’s law, allowed him to foresee the physical phenomena of the propagation of electromagnetic waves. In general, we distinguish two domains in electromagnetism, both are of course included in Maxwell’s equations: •

•

The high-frequency domain, which includes the study of electromagnetic waves and propagation of energy through matter. In general, we will define high-frequency domains, as domains, where the displacement currents cannot be neglected. The low-frequency domain includes the major part of electromagnetic devices like motors, relays or transformers. These are all applications at frequencies below a few tens of kHz. Strictly speaking, any application in which displacement currents can be neglected is a low-frequency application. In these domains, corresponding to the quasistatic case, we can, in general, study electric and magnetic fields as separate quantities. At high frequencies, the electric and magnetic fields are interdependent.

According to Fig. 4.1 we can divide electromagnetism into different cases, where each case represents a particular aspect of Maxwell’s equations. These equations are a set of partial differential equations, linear in space and time, applied to electromagnetic quantities. When electromagnetic fields interact with materials, the equations can assume non-linear forms. The electromagnetic quantities involved in Maxwell’s equations are:

94

4 Electromagnetic Field

Fig. 4.1. Classification of Maxwell’s equations [104]

Notation E D H B J qe M P

Unit

Description

(V/m) (As/m2 ) (A/m) (T) (A/m2 ) (As/m3 ) (T) (As/m2 )

electric field intensity electric flux density (electric induction) magnetic field intensity magnetic flux density (magnetic induction) current density charge density magnetization electric polarization

In addition, we define the following material parameters • • • •

Magnetic permeability µ (Vs/Am) Magnetic reluctivity ν = 1/µ (Am/Vs) Electric permittivity ε (As/Vm) Electric conductivity γ (1/Ωm = S/m)

The four partial differential equations (PDEs), stated as Maxwell’s equations in differential form, fully describe all phenomena in electromagnetic fields ∂D ∂t ∂B ∇×E = − ∂t ∇ · D = qe

∇×H = J+

∇·B = 0.

(4.1) (4.2) (4.3) (4.4)

In addition, to get solvability of the four PDEs, the following constitutive equations are introduced

4.1 Maxwell’s Equations

95

J = γ(E + v × B)

(4.5)

D = εE = ε0 E + P B = µH = µ0 H + M .

(4.6) (4.7)

It has to be stated that (4.5) is a direct consequence of the force relation defined by Lorentz (4.8) fV = qe (E + v × B) . Thus, the total electromagnetic force acting on the electric volume charge qe is given by a term defined by the product of the electric field E and a volume charge qe at rest and a term defined by the magnetic induction B and a volume charge qe at velocity v. As already mentioned, Maxwell’s equations are based on experiments and concluding empirical laws stated by Amp`ere, Faraday, and Gauss. Law of Amp` ere With the help of experiments, Amp`ere was able to prove in the year 1820 that an electric current (dc or ac) generates a magnetic field (Fig. 4.2). This

Fig. 4.2. Experiments of Amp`ere

empirical law states that the line integral of the magnetic field intensity along a closed contour is equal to the electric current I H · ds = I . (4.9) C(I)

The direction of the magnetic field intensity H is related to the direction of the electric current I via the right-hand rule. For general distributed currents within a cross section Γ with boundary curve C, the law of Amp`ere reads as  H · ds = J · dΓ , (4.10) C

Γ

96

4 Electromagnetic Field

and by applying Stokes’ theorem (see Appendix B.9), we obtain   (∇ × H) · dΓ = J · dΓ . Γ

(4.11)

Γ

Since this relation has to be independent of the surface, the law results in the differential form, which reads as follows ∇ × H = J.

(4.12)

Maxwell found that (4.12) does not fully describe the general case and added the so-called displacement current density ∂D/∂t in order to obtain a description for the general case (see e.g., [205]). Since we are mainly concerned in the quasistatic case, we will not discuss this term in detail. Law of Faraday According to his experiments, Faraday postulated that the relation between the induced voltage uind in an open conductive loop and the magnetic flux φ through it (Fig. 4.3), is given by uind = −

dφ . dt

(4.13)

The magnetic flux φ is defined by  B · dΓ ,

φ=

(4.14)

Γ

with Γ the cross section passed by the magnetic induction B. A change of the

Fig. 4.3. Loop within a time-varying magnetic field

magnetic flux with respect to time can be obtained in two different cases. First, the loop is at rest and the magnetic field is time dependent, and secondly,

4.1 Maxwell’s Equations

97

the loop changes its position as a function of time and the magnetic field is constant. Let us consider a surface Γ on which E and B are defined, then we can rewrite (4.2) as   ∂B · dΓ . (∇ × E) · dΓ = − ∂t Γ

Γ

Through the use of Stoke’s theorem, we obtain  ∂B · dΓ , E · ds = − ∂t C(Γ )

Γ

where C is the contour that limits the surface Γ. We now look at the application of this expression in an example as depicted in Fig. 4.4. Let us assume that the magnetic field is concentrated in the high-permeable cylinder and the magnitude of B is constant at each cross section of the cylindrical core. Furthermore, an electrical conductive loop, electrically isolated from the core, encloses the core such that it forms a contour C(Γ ).

Fig. 4.4. Induced voltage in an open conductive loop, caused by a time-varying magnetic field

First, note that assuming the direction of dΓ as in Fig. 4.4b, the direction of ds must be as shown (right-hand rule). With the direction of ∂B/∂t as shown in Fig. 4.4b, we obtain −

∂B · dΓ < 0 ; E · ds < 0 , ∂t

which makes E point in the direction opposite to ds.

98

4 Electromagnetic Field

The circulation of E along C leads to an electromotive force (emf), detectable by a voltmeter as an electric potential difference. The electromotive force computes as E · ds .

uind =

(4.15)

C

It has to be noted that the electric field due to the time variation of the magnetic field is solenoidal, and thus, the measured voltage depends on the way the electric voltmeter is connected to the open loop [95]. Because of the fact that B only depends on time and not on its position, we conclude that ∂B/∂t = dB/ dt. Since the integration over the surface Γ and the time derivative are independent operations, we can write   dφ d ∂B B · dΓ = − · dΓ = − , − ∂t dt dt Γ

Γ

and obtain the expression uind = −

dφ . dt

(4.16)

Now let us consider the case of a conductive slab positioned in a uniform and time-independent magnetic field B (see Fig. 4.5). As already mentioned

Fig. 4.5. Electrically conductive slab moving with velocity v in a time-constant magnetic field B

in this section, the total electromagnetic force on a charge is given due to an electrostatic field acting on a charge in rest, and a magnetic field acting on a moving charge. Therefore, an electromagnetic force Fmag will act on the electric charges Qe within the conductive slab (see Fig. 4.5) Fmag = Qe (v × B) ,

(4.17)

4.1 Maxwell’s Equations

99

and the charges will distribute on the surface of the slab as depicted in Fig. 4.5. Due to this separation of positive and negative charges, an electrostatic field E arises, which will result in the restoring force Fel = Qe E .

(4.18)

By assuming the equilibrium state, we obtain Fmag + Fel = 0 E = −v × B .

(4.19) (4.20)

If we now connect a resistor R to the slab in parallel by using a flexible cable (so that the resistor is at rest), then the electric field E will result in an electric current i. Therefore, the term v × B is called the motional electromotive force. By neglecting the resistance of the connections, the potential difference across the resistor, which we define by the voltage u, is given by u = Ri = E · ds = l · (v × B) , (4.21) with l the length of the conductive slab (direction points into the direction of the current). Hence, the mechanical power Pmech we apply to the slab (since we move it with velocity v) is converted into electric power Pel Pmech = Fmech · v = Ri2 = Pel .

(4.22)

Simply speaking, we have the case of an electric generator with an ohmic load.

Law of Gauss By means of experiments, Gauss postulated that the amount of electric flux density D crossing a closed surface Γ is equal to the total electric charges qe within the volume Ω (Fig. 4.6)  D · dΓ = qe dΩ . (4.23) Γ

Ω

Applying the divergence theorem (Appendix B.6), the above equation reads as   (∇ · D) dΩ = qe dΩ (4.24) Ω

Ω

and, since the law has to hold for any volume Ω, we end up at the differential form ∇ · D = qe . (4.25) Therefore, the sources of the electric field are given by the electric charges, and according to this relation the electric field is irrotational.

100

4 Electromagnetic Field

Fig. 4.6. Sources of the electric field

Solenoidal magnetic field Since magnetic materials cannot be split into a piece only having a north pole or a south pole; no magnetic charges exist, and the magnetic field is always solenoidal (closed field lines). This property of the magnetic field can be stated by B · dΓ = 0 , (4.26) Γ (Ω)

which means that the magnetic flux is conservative. Applying the divergence theorem  (∇ · B) dΩ = 0 , Ω

and generalizing it for any volume Ω, we obtain the differential form ∇·B = 0.

(4.27)

It is very interesting to note that this equation, which has been introduced by Maxwell, is necessary to guarantee that no magnetic charges exist. If we apply the operator ∇· to (4.2) ∇ · (∇ × E) = −∇ ·

∂B ∂t

(4.28)

we achieve the following relation ∂ (∇ · B) = 0 . (4.29) ∂t The solution of this equation without considering the relation ∇·B = 0 would result in ∇ · B = const. = 0, and therefore, in the existence of magnetic charges.

4.2 Quasistatic Electromagnetic Fields

101

4.2 Quasistatic Electromagnetic Fields The most important case for electromagnetic sensors and actuators is the quasistatic case often referred to as the eddy current case. For quasistatic electromagnetic fields we can neglect the displacement current density term ∂D/∂t, which transforms Maxwell’s equations including the constitutive equations to the following subset ∇×H = J

(4.30)

∂B ∇ × Es = − ∂t ∇·B = 0

(4.31) (4.32)

J = γ(Etotal + v × B)

(4.33)

B = µH = µ0 H + M .

(4.34)

In (4.33) Etotal denotes the total electric field intensity given as a sum of any irrotational part Ei (defined by (4.3)) and of any solenoidal part Es Etotal = Ei + Es .

(4.35)

Now, we can rewrite (4.33) as J = Ji + γ(Es + v × B) ,

(4.36)

with Ji an impressed current density due to a given electric potential difference (current- or voltage-loaded coil). 4.2.1 Magnetic Vector Potential According to (4.32) the magnetic field is solenoidal and therefore can be described by the curl of a vector B = ∇ × A.

(4.37)

The vector A is called the magnetic vector potential. This ansatz results for (4.2) in the following relation ∂ ∇ × Es = − (∇ × A) ∂t   ∂A ∇ × Es + = 0. ∂t

(4.38)

Since we are concerned with eddy current problems, the electric field is pure solenoidal. Thus, we set Es = −∂A/∂t and arrive at ∇ × ν∇ × A = Ji − γ

∂A + γ(v × ∇ × A) . ∂t

(4.39)

102

4 Electromagnetic Field

It has to be noted that the introduction of the magnetic vector potential A leads to non-uniqueness of the solution. This can be seen very easily by adding the gradient of a scalar function to A, which leads to a new vector A∗ A∗ = A + ∇ψ .

(4.40)

Since the curl of any grad is zero, we arrive at B = ∇ × (A∗ − ∇ψ) = ∇ × A∗ .

(4.41)

To obtain a unique solution, we have to gauge the magnetic vector potential A, which is, e.g., achieved by a Coulomb gauge [22] ∇· A = 0.

(4.42)

In the 2D case the magnetic vector potential is given by A = A(x, y)ez

(4.43)

and therefore (4.42) is automatically guaranteed. The same holds for the axisymmetric case A = A(r, z)eϕ . (4.44) Furthermore, it has to be noted that we are interested in the quantity B computed via ∇ × A, and therefore, no physical need exists to gauge the magnetic vector potential A. 4.2.2 Skin Effect Supplying an electric conductor with cross section ΓC by an alternating current with frequency fC , we will find that the higher the frequency, the more the current will be concentrated near the surface of the conductor and tend to zero towards the center of the conductor. To investigate this effect, let us simplify the setup to an infinitely extended half-plane as depicted in Fig. 4.7. Therefore, the current density J and, respectively, the electric field intensity E only depend on the coordinate z and point towards the x-direction. Using (4.12), we obtain −

∂Hy = Jx ∂z ∂By ∂Ex =− . ∂z ∂t

(4.45) (4.46)

By applying the operation ∂/∂z to (4.45) and using (4.46) as well as the constitutive laws (4.5) and (4.7), we arrive at −

∂ 2 Hy ∂Ex ∂Hy ∂Jx =γ = −γµ . = ∂z 2 ∂z ∂z ∂t

(4.47)

4.2 Quasistatic Electromagnetic Fields

103

Fig. 4.7. Infinite extended half-plane

Equation (4.47) is called the diffusion equation, because it models the diffusion of electromagnetic energy into the conductor. For the time-harmonic case with angular frequency ω, (4.47) gets the form ˆ y (z) ejωt Hy = H ˆy ∂2H ˆ , = jωγµH y ∂z 2

(4.48) (4.49)

which has the general solution ˆ y (z) = C1 e H

√

jωγµz

+ C2 e−

√ jωγµz

.

(4.50)

Since for physical reasons, the magnetic field decreases with distance z, the constant C1 has to be zero. By using the relation + 1+j j= √ 2 ˆ (z = 0) = H0 , we arrive at the solution for and the boundary condition H y the magnetic field inside the infinite half-plane (using (4.48)) H y (z) = H0 e−z/δ ej(ωt−z/δ) Hy (z) = Re(H y ) = H0 e

−z/δ

(4.51) cos(ωt − z/δ) .

(4.52)

In (4.51) δ denotes the so-called skin (penetration) depth and as can be seen from (4.50), it is a function of frequency and material properties δ=√

1 . πf γµ

(4.53)

Additionally, we obtain solutions for B, J, and E By (z) = B0 e−z/δ cos(ωt − z/δ) −z/δ

cos(ωt − z/δ) Jx (z) = J0 e −z/δ cos(ωt − z/δ) . Ex (z) = E0 e

(4.54) (4.55) (4.56)

104

4 Electromagnetic Field

Thus, the magnitudes of the electromagnetic quantities not only decay exponentially but also their phase changes with z/δ, so that at a certain depth, the field quantities are even in opposite directions. Concluding, it has to be noted that the expression for the skin depth δ is not restricted to this simplified example but can be considered as a general relation for eddy current problems (see e.g., [142, 201]).

4.3 Electrostatic Field For the static case, the physical quantities do not depend on time, and Maxwell’s equations with constitutive laws can be separated into a magnetic and an electric subsystem. For the electric subsystem, called the electrostatic case, we obtain the following system of partial differential equations ∇×E = 0 ∇ · D = qe D = εE .

(4.57) (4.58) (4.59)

Since the curl of the electric field intensity E is zero, we can express it by the gradient of a scalar potential Ve , which is called the electric scalar potential E = −∇Ve .

(4.60)

Thus, by combining (4.58), (4.59), and (4.60), we arrive at −∇ · ε∇Ve = qe ,

(4.61)

which describes the electrostatic field within any media characterized by the material quantity ε.

4.4 Material Properties 4.4.1 Magnetic Permeability The constitutive law within the magnetic field is given by (4.7), which relates the magnetic induction B and the magnetic field intensity H via the magnetic permeability µ. In general, the magnetic permeability is a tensor of rank 2, but for most cases we can assume a scalar value, which can be decomposed as follows µ = µ0 µr µ0 = 4π 10−7 (Vs/Am) .

(4.62) (4.63)

4.4 Material Properties

105

Therein, µr is called the relative permeability and characterizes the magnetic material. Often, instead of the magnetic permeability µ the magnetic reluctivity ν given by 1 ν= (4.64) µ is used. In general, we differentiate between two types of magnetic materials, namely •

Soft magnetic materials, which can be classified into diamagnetic, paramagnetic, and ferromagnetic materials

•

Hard magnetic materials of which permanent magnets are fabricated

By placing a diamagnetic material within an external magnetic field, the internal magnetization M will be opposite to the external field. Therefore, the total magnetic field within such a material will be smaller than the external applied field (see Fig. 4.8(a). Within paramagnetic materials the internal

Fig. 4.8. Cube within an external magnetic field (a) diamagnetic material; (b) paramagnetic material; (c) ferromagnetic material

magnetization will be directed already in the direction of the external magnetic field. However, the effect is small, as can be seen in Fig. 4.8(b). Only for ferromagnetic materials will the resulting internal magnetization be large and, as shown in Fig. 4.8(c), the magnetic field concentrates inside the cube. By switching off the external field, the magnetization almost disappears, and we call it a reversible process. However, if the amplitude of the external field is large, irreversible processes will take place, and after switching off the external field a remanent magnetic field can be detected. Therefore, the following definitions of permeabilities exists: •

Reversible permeability: µrev = lim

∆H→0

∆B . ∆H

(4.65)

This case is given if a large constant magnetic field is superimposed on a small alternating field.

106

•

4 Electromagnetic Field

Differential permeability: dB . (4.66) dH This permeability is always larger than the reversible permeability, since it will also contain irreversible processes. µdiff =

•

Starting permeability

dB |B=0,H=0 . (4.67) dH By first applying an external magnetic field to a ferromagnetic material, the magnetic induction will rise strongly until the material is saturated. If we now decrease the amplitude of the external applied field, we will recognize that the magnetic induction inside the material will decrease along a curve different from the initial magnetization curve and at external field zero, we will have a remanent magnetic induction Br of the material. This effect is due to the already discussed irreversible processes inside the body. Finally, by driving the external magnetic field to a negative maximum amplitude and back to the positive amplitude, we will measure a similar hysteresis curve as displayed in Fig. 4.9. The quantity Hc is called the coercitive magnetic µs =

Fig. 4.9. Hysteresis and initially magnetization curve for a soft magnetic material

field intensity (force), since we have to externally apply this value in order to obtain zero magnetic induction inside the material. In addition to the magnetic field dependency of the magnetic permeability, magnetic materials completely lose magnetization, if the temperature exceeds the so-called Curie temperature TCurie , which can be modelled as follows B = µ(H)H µ = f (T ) =

&

(4.68) µ = µ(H) : T < TCurie . µ = µ0 : T > TCurie

(4.69)

4.4 Material Properties

107

In general, Br is small for soft magnetic materials, which also means that the magnetization follows the external field almost with no essential retardation effect. This physical phenomenon is quite contrary for hard magnetic

Fig. 4.10. Typical magnetization curve of hard magnetic material

materials, which exhibit a large remanent magnetic field Br . A typical B–H curve is given in Fig. 4.10 and Table 4.1 summarizes the data of important permanent magnets. Table 4.1. Properties of permanent magnets Material Br (T) Hc (kA/m) (BH)max (kWs/m3 ) Alnico 1.25 60 50 0.38 240 25 Ferrite 800 230 Ne-Fe-B 1.15

Therefore, if we consider a magnetic assembly including permanent magnets, we can describe the magnetic field via the magnetic vector potential by the following partial differential equation (assuming a known magnetization M and no motional emf) ∇×

∂A 1 ∇ × A = Ji − γ +∇×M. µ ∂t

(4.70)

If we just consider a static field problem, and decompose the magnetic field intensity H into HJ generated by a current density J and HM describing the magnetic material, we can introduce the so-called reduced magnetic scalar potential ψm according to H = HJ + HM HM = −∇ψm . Therefore, by using (4.4) we obtain the describing equation for ψm

(4.71) (4.72)

108

4 Electromagnetic Field

∇ · µ(HJ + HM ) = 0

(4.73)

∇ · µ∇ψm = ∇ · (µHJ ) ,

(4.74)

and HJ is computed via Biot–Savart’s law (see e.g., [95])  1 J × er


dΩ , H(x , y , z ) = 4π r2

(4.75)

Ω

with (x
, y
, z
) the field (observation) point, Ω the volume of a current loaded structure and er the unit vector pointing from the volume Ω to the field point. 4.4.2 Electrical Conductivity In general, the conductivity is a tensor of rank 2, but for many applications it can be assumed to be a scalar value. Table 4.2 gives the values for some materials. The electrical conductivity γ mainly depends on the temperature T . Table 4.2. Electrical conductivity of some materials Iron

Aluminum Copper

Conductivity (S/m) 1 × 10

7

3.5 × 10

7

Steel Carbon

5.8 × 10 5 × 106 3 × 104 7

In most cases the dependency is available for the specific electrical resistivity ρe = 1/γ and takes the relation ) * ρe = ρ20 1 + α(T − T20 o C ) + β(T − T20 o C )2 , (4.76) with α the linear and β the quadratic temperature coefficient. In (4.76) ρ20 denotes the specific resistivity at T = T20 o C . Therefore, we obtain the electric conductivity (see Fig. 4.11) γ=

γ20 , (1 + α(T − T20 o C ) , +β(T − T20 o C )2 )

(4.77)

with γ20 = 1/ρ20 . 4.4.3 Dielectric Permittivity The modelling of dielectric materials is quite similar to that of magnetic materials. Instead of the magnetization M, we define the polarization P for characterizing dielectric materials [95] D = ε0 εr E = ε0 E + P .

(4.78)

4.5 Electromagnetic Interface Conditions

109

Fig. 4.11. Dependency of the electrical conductivity of steel on temperature

Table 4.3. Relative permittivity of dielectric materials Air Insulating Glass Water PZT 5A paper (3-direction) εr 1 3 6 80 3000

The scalar permittivity ε can be decomposed into the permittivity of vacuum ε0 = 8.854 × 10−12 (As/Vm) and the relative permittivity εr . The material can be described by a scalar value εr (see Table 4.3), and therefore D, E, and P have the same direction. However, materials with a large value of relative permittivity exhibit a strong anisotropy and are called ferroelectrics. This is the special case for piezoelectric materials. These materials also show a hysteresis as depicted in Fig. 4.12 with Dr denoting the dielectric remanent and Ec the coercitive electric field intensity.

4.5 Electromagnetic Interface Conditions 4.5.1 Continuity Relations for Magnetic Field According to (4.4) and by application of the divergence theorem, we obtain ∇·B = 0

 Ω

(∇ · B) dΩ = 0 B · dΓ = 0 . Γ

(4.79)

110

4 Electromagnetic Field

Fig. 4.12. Hysteresis and initial polarization curve

At the interface between two materials with different permeabilities, we have to evaluate the surface integral according to Fig. 4.13 for b → 0.

Fig. 4.13. Continuity of the normal component of B

B · dΓ = B1 · n1 + B2 · n2 = 0

lim

b→0

Γ

n1 · (B1 − B2 ) = 0 B1n = B2n µ1 H1n = µ2 H2n .

(4.80) (4.81)

Therefore, at an interface of changing permeability, the normal component of the magnetic induction B is continuous. By using (4.30) and assuming no current density at the interface (see Fig. 4.14), we obtain the continuity relation for the magnetic field intensity H

4.5 Electromagnetic Interface Conditions

111

(applying Stoke’s theorem)

Fig. 4.14. Continuity of the tangential component of H

 lim b→0

Γ

(∇ × H) · dΓ = 0 lim H · ds = 0 b→0

H1 · s − H2 · s = 0 st · (H1 − H2 ) = 0 H1t = H2t .

(4.82)

Therefore, at an interface of changing permeability, the tangential component of the magnetic field intensity H is continuous. By using the material relation between B and H we get the defining equation for the tangential component of the magnetic induction B1t B2t = . (4.83) µ1 µ2 4.5.2 Continuity Relations for Electric Field Using the integral form of (4.3)   (∇ · D) dΩ = qe dΩ , Ω

(4.84)

Ω

with the relation dΩ = b dΓ and b → 0 (see Fig. 4.15) we obtain  lim

b→0 Ω

lim qe dΩ = σe dΓ  qe dΩ = σe dΓ ,

b→0

(4.85) (4.86)

Γ

with σe the surface charge along the interface. By applying the divergence theorem

112

4 Electromagnetic Field

Fig. 4.15. Continuity of the normal component of D



(∇ · D) dΩ =

Ω

D · dΓ ,

(4.87)

Γ

and performing the step for b → 0, we arrive at lim D · dΓ = D1 · n1 + D2 · n2 = n1 (D1 − D2 ) . b→0

(4.88)

Γ

Therefore, for the normal components of the electric field quantities, the following relations have to hold D1n = D2n + σe ε1 E1n = ε2 E2n + σe .

(4.89) (4.90)

Fig. 4.16. Continuity of the tangential component of E

For the tangential component of the field quantities at an interface between two materials with different permittivities and no time-varying magnetic field (Fig. 4.16), we have to start at (4.2) and perform similar steps as for the tangential component of the magnetic field, and obtain

4.6 Numerical Computation: Electrostatics

E1t = E2t D1t D2t = . ε1 ε2

113

(4.91) (4.92)

4.5.3 Continuity Relations for Electric Current Density Similar to (4.80), we can rewrite (4.30) after applying the divergence on both sides as ∇·J = 0, (4.93) which results in the following interface condition J1n = J2n .

(4.94)

Assuming a jump of the electrical conductivity γ at an interface, we obtain for the eddy current case (see Sect. 4.2.1 and v = 0) γ1

∂A1n ∂A2n = γ2 . ∂t ∂t

(4.95)

4.6 Numerical Computation: Electrostatics Let us consider the situation of a domain with given electric volume charges qe , where we want to compute the generated electrostatic field. The strong formulation of this problem will be stated as follows Given: qe : Ω → IR ε : Ω → IR . ¯ → IR Find: Ve : Ω −∇ · ε∇Ve = qe .

(4.96)

Boundary conditions Ve = 0 on Γ = Γe . To obtain the variational formulation, we multiply (4.96) with an appropriate test function ω ∈ H01 (see Appendix C) and perform a partial integration. Therefore, the weak form reads as: Find Ve ∈ H01 such that   ε ∇w · ∇Ve dΩ − wqe dΩ = 0 , (4.97) Ω

Ω

114

4 Electromagnetic Field

for any w ∈ H01 . Using standard nodal finite elements, we approximate the continuous electric scalar potential Ve as well as the test function w by Ve ≈ Veh =

neq 

Na Vea

(4.98)

Na wa ,

(4.99)

a=1 neq

w ≈ wh =



a=1

with neq the number of equations. Therefore, (4.97) is transformed into the following discrete formulation ⎛ ⎞  neq neq    ⎝ ε (∇Na )T ∇Nb dΩ Ve b − Na qe (ra ) dΩ ⎠ = 0 . (4.100) a=1 b=1

Ω

Ω

In matrix form, (4.100) may be written as KVe Ve = f V ,

(4.101)

e

with KVe =

ne 

 keVe

;

keVe

e=1

fV =

ne 

e

ε (∇Np )T ∇Nq dΩ (4.102)

= [kpq ] ; kpq = Ωe

 f eV

e

e=1

; f eV = [fp ] ; fp =

Np qe (rp ) dΩ .

e

(4.103)

Ωe

4.7 Numerical Computation: Electromagnetics The numerical computation of electromagnetic fields has been performed for more than 20 years. For the domain discretization nodal as well as edge finite elements have been used successfully. Nevertheless, in recent years inaccurate results at material parameter interfaces in the magnetostatic as well as in the eddy current case, and spurious modes in Maxwell’s eigenvalue problems have been reported. Therefore, we will perform a precise investigation in the formulation and further discretization of electromagnetic fields in the eddy current case. 4.7.1 Formulation Let us consider a domain Ω with boundary Γ consisting of two subdomains Ω1 and Ω2 with interface Σ12 as displayed in Fig. 4.17. The material relations are given by H = νB and J = γE with ν, γ the magnetic reluctivity (ν = 1/µ) and

4.7 Numerical Computation: Electromagnetics

115

Fig. 4.17. Domain with given material properties

electrical conductivity of the material. On the boundary Γ = ∂Ω the normal component of the magnetic induction (B·n = 0 on Γ ) shall be prescribed with n being the normal unit outward vector on Γ . In addition, we consider the continuity conditions that have to be fulfilled at an interface Σ12 of changing magnetic permeability as well as conductivity [B · n] = B1 · n − B2 · n = 0

(4.104)

[H × n] = H1 × n − H2 × n = 0 [J · n] = J1 · n − J2 · n = 0 .

(4.105) (4.106)

The partial differential equation to be solved has been derived in Sect. 4.2.1, and assuming no moving bodies, reads as γ

∂A + ∇ × ν∇ × A = Ji . ∂t

(4.107)

The boundary conditions change to n × A = 0 on Γ , and the interface conditions take on the form [A × n] = 0 [ν n × ∇ × A] = 0

 ∂A γ n· = 0. ∂t

(4.108) (4.109) (4.110)

Therefore, the strong formulation for the eddy current case reads as follows: Given: A0 : Ω → IRd γ, ν : Ω → IR . ¯ × [0, T ] → IRd Find: A(t) : Ω

116

4 Electromagnetic Field

γ

∂A + ∇ × ν∇ × A = Ji . ∂t

(4.111)

Boundary conditions n × A = 0 on Γ × (0, T ) . Interface conditions A1 × n = A2 × n on Σ12 × (0, T ) ν1 n × ∇ × A1 = ν2 n × ∇ × A2 on Σ12 × (0, T ) ∂A1 ∂A2 = γ2 n · on Σ12 × (0, T ) . γ1 n · ∂t ∂t Initial condition A(r, 0) = A0 , r ∈ Ω . Now, multiplying (4.111) by appropriate test functions A
and applying Green’s first integral theorem in vector form (Appendix B.10) will transform the PDE into its variational formulation, which reads as follows: Find A ∈ HΣ 0 (curl) such that    ∂A dΩ + ∇ × A
· ν ∇ × A dΩ = A
· Ji dΩ , (4.112) γA
· ∂t Ω

Ω

Ω

for any A
∈ HΣ 0 (curl) with the Sobolev space 2 3 2 3 HΣ 0 (curl) = {u ∈ (L (Ω)) | ∇ × u ∈ (L (Ω)) ,

u × n|Γ = 0, [n × u]|Σ = 0} .

(4.113)

If the conductivity γ is positive, the problem has a unique solution in H0 (curl), whereas for γ = 0 (globally, or only in some regions), the solution is unique only up to gradient fields in the case that Ω is simply connected. Since we are interested in the quantity B computed via ∇ × A, there is no need to gauge the vector potential A, though. The fact that our variational formulation (4.112) automatically fulfils the interface conditions (4.108) - (4.110) can be shown as follows. Starting at (4.112) and performing partial integration (Green’s first integral theorem in vector form, see Appendix B.10), we obtain

4.7 Numerical Computation: Electromagnetics



γA
·

∂A dΩ + ∂t

Ω



A
∇ × ν ∇ × A dΩ +

Ω1



+

Σ

A
∇ × ν ∇ × A dΩ +

Ω2



+





Σ

117

(A
× ν1 ∇ × A) · n1 dΓ    A
·(n×ν1 ∇×A1 ) (A
× ν2 ∇ × A) · n2 dΓ    A
·(n×ν2 ∇×A2 )

(A
× ν∇ × A) · n dΓ

Γ



=

A
· Ji dΩ

(4.114)

Ω

with n1 = −n2 = n. Now, let us choose a test function A
being concentrated near the interface Σ and zero elsewhere. Furthermore, we assume A
to point into normal direction, i.e., A
= wn with w an arbitrary scalar function. Since the term (n × ν∇ × A) results in a vector orthogonal to n, the scalar product with A
is zero. Moreover, due to (4.106) and the fact that the intersection of the support of A
with the interior of Ω1 and Ω2 is made arbitrary small, and the interface Σ is at distance from the boundary Γ , only the first term in (4.114) remains   ∂A1 ∂A2 dΓ = γ2 wn · dΓ . (4.115) γ1 wn · ∂t ∂t Σ

Σ

This holds for any w, so that

 ∂A γn· = 0. ∂t

(4.116)

On the other hand, when we choose A
arbitrarily tangential to Σ and the support of A
as concentrated to Σ, then the volume integrals in (4.114) are vanishing, while only the surface integrals over Σ can give nonzero values   A
· (n × ν1 ∇ × A1 ) dΓ = A
· (n × ν2 ∇ × A2 ) dΓ Σ

Σ

[ν n × ∇A] = 0 .

(4.117)

The different asymptotic behaviors of volume and surface integrals, as it was exploited here for deriving (4.117), can more easily be made obvious in a special situation without losing generality. Let us specify Σ as a plane orthogonal to the z-axis Σ = {(x, y, 0)|(x, y) ∈ [0, 1]2 } . Then, our proof amounts to choose

118

4 Electromagnetic Field

⎛

wx (x, y)

⎞

⎟ ε ⎜ ⎟ A
(x, y, z) = ⎜ ⎝ wy (x, y) ⎠ φ (z) 0 with φε (z) as a hat function having support [−ε, ε] and value 1 at z = 0 (see Fig. 4.18). Now, observe that φε equals 1 on Σ for all ε, while, since

Fig. 4.18. Hat function φε (z) with support [−ε, ε]


ε

φε (z) dz goes to zero as ε → 0, volume integrals containing A
in their integrand have to vanish as ε → 0. Note that this situation, although looking quite simple, is sufficiently significant, since any smooth interface can locally be smoothly transformed to a piece of a plane. The edge finite elements [162] for the spatial discretization of (4.112) are H0 (curl)-conform, and solving the resulting algebraic system of equations leads to correct results. Nevertheless, the solution of the algebraic system requires special care in order to obtain an optimal solver (see e.g., [9,92]). We suggest to add a fictive electric conductivity γ
to regions with zero electric conductivity to obtain a variational form, which is elliptic [191]. Of course, this fictive conductivity γ
has to be chosen small compared to the reluctivity of the material. The proof of convergence even in the case of γ
→ 0 is given in [180]. For the application of nodal finite elements, we have to perform additional steps. As shown in [74], the space HΣ 0 (curl) has the decomposition for any convex domain Ω ) 1 *3 HΣ ⊕ grad H01 (Ω) . (4.118) 0 (curl) = H0 (Ω) −ε

This is equivalent to splitting the magnetic vector potential A as follows A = w + ∇φ

∇·w = 0,

(4.119)

*3 ) with w ∈ H01 (Ω) and φ ∈ H01 (Ω). The same decomposition is done for the test function A
= v + ∇ψ. Since Ji is assumed to be divergence free, ψ does not enter into the right-hand side.

4.7 Numerical Computation: Electromagnetics

119


To guarantee ∇ · w = 0, we may add the term Ω ν ∇ · v ∇ · w dΩ to the variational formulation (4.112), which corresponds to a penalty formulation (see e.g., [78]). The variational formulation changes to: Find (w, φ) ∈ (H01 (Ω)3 , H01 (Ω)) such that   ν∇ × v · ∇ × w dΩ + ν ∇ · v ∇ · w dΩ Ω



+

Ω

∂ γ(v + ∇ψ) · (w + ∇φ) dΩ ∂t

Ω



Ji · v dΩ ,

=

(4.120)

Ω

for any (v, ψ) ∈ (H01 (Ω)3 , H01 (Ω)). Thus, the standard continuous nodal finite elements can be used to approximate both fields w and φ and we arrive at a conforming FE approximation. It should be noted that due to the decomposition of A into w and ∇φ the magnetic induction B as well as eddy current density Jeddy compute as B = ∇ × A = ∇ × (w + ∇φ) = ∇ × w ∂w ∂∇φ ∂A = −γ −γ . Jeddy = −γ ∂t ∂t ∂t

(4.121) (4.122)

As reported in many scientific contributions, the discretization of (4.120) with nodal finite elements leads to correct values in the eddy current case, where the permeability is constant all over the domain (see e.g., [143]). Inaccurate results have been demonstrated in the case of domains with materials of different magnetic reluctivities (e.g., iron–air interface). Therefore, we will concentrate on this problem and in the following investigate the magnetostatic case (∂/∂t = 0)   ν∇ × v · ∇ × w dΩ + ν ∇ · v ∇ · w dΩ Ω

Ω



Ji · v dΩ .

=

(4.123)

Ω

In the first step we will consider a domain as displayed in Fig. 4.17, where we have a jump of the reluctivity at the interface Σ between the subdomains Ω1 and Ω2 . If we perform similar steps as for the previous variational formulation (4.112), we will see that (4.123) fulfills the following interface conditions [ν n12 × ∇ × w] = 0 [ν n12 ∇ · w] = 0 .

(4.124) (4.125)

120

4 Electromagnetic Field

Of course, from the numerical point of view, the approximation of these interface conditions using linear interpolation functions will be poor, but applying quadratic interpolation functions will give acceptable results. So, the question arises, what is the reason for the poor results discretizing (4.123) with nodal finite elements? Let us consider the case of a ferromagnetic cube embedded in air (see Fig. 4.19). Assuming the case ν2 → 0 (permeability is very

Fig. 4.19. Ferrormagnetic cube in air

large in the iron), we arrive at a nonconvex domain. Now, according to [46], it is known that for nonconvex domains the discretization with nodal finite elements produces wrong solutions due to the nondensity of smooth fields. In [46] the authors could prove that by introducing a special weighting function inside the divergence integral, nodal finite elements can yet be used for the approximation. Therefore, the second term in the variational formulation (4.123) has to be changed to   ν ∇ · v ∇ · w dΩ → ν s ∇ · v ∇ · w dΩ , (4.126) Ω

Ω

with s=

0

raα .

(4.127)

a∈A

In (4.127) A denotes the set of all re-entrant corners, ra the distance to each re-entrant corner, and α an exponent. This means that w has to be in HY HY = {w ∈ H0 (curl) | ∇ · w ∈ Y} .

(4.128)

By a correct choice of the weighted Sobolev space Y, the space H10 is dense in HY , and thus, nodal finite elements do lead to correct results, and we call it the weighted regularization method. It has to be noted that this idea can be implemented in a simple way by setting the weighting function s to zero for finite elements near re-entrant corners [118]. This was exactly the treatment the authors in [172] employed, where a current vector potential formulation was used. Since the current vector potential is just defined in electric conductive regions, in most cases the boundary will exhibit re-entrant corners.

4.7 Numerical Computation: Electromagnetics

121

4.7.2 Discretization with Edge Elements Performing an edge finite element discretization of (4.112), we first define the approximation of the vector potential A as A≈

meq 

Eb Ab .

(4.129)

b=1

In (4.129) meq defines the number of edges with unknown magnetic vector potential in the finite element mesh, Eb the edge shape function associated with the b-th edge, and Ab the corresponding degree of freedom, namely the line integral of the magnetic vector potential along the b-th edge  Ab = A · ds . (4.130) b

Applying the same discretization to the test function A
, we arrive at (see (4.112)) ⎛  meq meq    ⎝ γEa · Eb dΩ A˙ b + ν(∇ × Ea ) · (∇ × Eb ) dΩ Ab a=1 b=1

Ω

Ω

 −

⎞ Ea · Ji dΩ ⎠ = 0 .

(4.131)

Ω

In matrix form, we obtain MA A˙ + KA A = f

(4.132)

with MA =

ne 

 m

e

; m = [mpq ] ; mpq =

e=1

KA =

ne 

 ke ; ke = [kpq ] ; kpq =

e=1

f =

ne  e=1

γEp · Eq dΩ

e



(4.133)

Ωe

*T ) ν Bpcurl Bqcurl dΩ (4.134)

Ωe

Ep · Ji dΩ .

f e ; f e = [fpe ] ; fpe =

(4.135)

Ωe

For the computation of Bpcurl we know, according to Sect. 2.7 that the edge shape function Ep for a tetrahedra is given by Ep = (Ni ∇Nj − Nj ∇Ni )lp ,

(4.136)

122

4 Electromagnetic Field

with p defining the edge and i, j the corresponding vertices. To find a more explicit form for Bpcurl , let us rewrite Ep as follows ⎛ ∂Nj ⎞

⎛ ∂Ni ⎞

∂x

∂x

⎜ ∂N ⎟ ⎜ ∂N ⎟ j ⎟ ⎜ i⎟ N Ep = lp N i ⎜ − l p j ⎝ ∂y ⎠ ⎝ ∂y ⎠ ∂Nj ∂z ⎛ ∂Nj Ni ∂x −

i Nj ∂N ∂x

⎞

(4.137)

∂Ni ∂z

⎜ ∂N ⎟ j ∂Ni ⎟ = lp ⎜ ⎝ Ni ∂y − Nj ∂y ⎠ . Ni

∂Nj ∂z

(4.138)

i − Nj ∂N ∂z

Therefore, we obtain ⎛ ∂Ez ∂y

−

⎜ ∂Ex Bpcurl = ∇ × Ep = ⎜ ⎝ ∂z − ∂Ey ∂x −

∂Ey ∂z ∂Ez ∂x ∂Ex ∂y

⎞

⎛

∂Ni ∂Nj ∂y ∂z

−

⎜ ⎟ i ∂Nj ⎟ = 2lp ⎜ − ⎜ ∂N ⎠ ⎝ ∂z ∂x ∂Ni ∂Nj ∂x ∂y

−

∂Nj ∂Ni ∂y ∂z ∂Nj ∂Ni ∂z ∂x ∂Nj ∂Ni ∂x ∂y

⎞ ⎟ ⎟ ⎟ . (4.139) ⎠

The time discretization is performed by applying the trapezoidal method (see Sect. 2.5.1) using, e.g., an effective mass formulation, which results in the following scheme: •

Perform predictor step: A˜ = An + (1 − γP )∆t A˙ n .

•

Solve algebraic system of equations: M∗A A˙ n+1 = f n+1 − KA A˜ M∗A

•

(4.140)

= MA + γP ∆t KA .

(4.141) (4.142)

Perform corrector step: An+1 = A˜ + γP ∆t A˙ n+1 .

(4.143)

4.7.3 Discretization with Nodal Finite Elements According to (4.120), we have to compute the unknowns of the vector w and the scalar φ at all finite element nodes. To obtain four equations for the three unknowns of w and the one of φ, we set once the test function ψ and once the test function v to zero, which results in

4.7 Numerical Computation: Electromagnetics



 ν∇ × v · ∇ × w dΩ + Ω



' ( ˙ + ∇φ˙ dΩ γv· w

ν ∇ · v ∇ · w dΩ + Ω

Ω



Ji · v dΩ

= 

123

(4.144)

Ω

'

( ˙ + ∇φ˙ dΩ = 0 , γ ∇ψ · w

(4.145)

Ω

˙ = ∂w/∂t and φ˙ = ∂φ/∂t. In the first step we will perform a spatial diswith w cretization of (4.144) and (4.145) by introducing the following approximations of the continuous functions w and φ ⎞ ⎛ n
n n
n nd  Na 0 0   w ≈ wh = Na wia ei = Na wa ; Na = ⎝ 0 Na 0 ⎠ (4.146) a=1 i=1 a=1 0 0 Na


φ≈φ = h

nn 

Na φa .

(4.147)

a=1

In (4.146) and (4.147) n
n denotes the number of nodes with no Dirichlet boundary condition, nd the space dimension and Na the interpolation function for node a. Applying the same approximations for the test functions A
and ψ, we arrive at ⎛  n
n n
n   ⎜ T ˙ b + γ NTa ∇Nb dΩ φ˙ b ⎝ γ Na Nb dΩ w       a=1 b=1

Ω



Ω

1 Bab

2 Bab



ν (∇ × Na )T (∇ × Nb ) dΩ wb +   

+ Ω

 −

ν (∇ · Na )T (∇ · Nb ) dΩ wb    Ω

curl )T (Ba

⎞

div )T (Ba

Na · Ji dΩ ⎠ = 0

(4.148)

Ω

⎛ ⎞    ⎝ γ(∇Na )T Nb dΩ w ˙ b + γ∇Na · ∇Nb dΩ φ˙ b ⎠ = 0 . n
n

n
n

a=1 b=1

Ω

Ω

The different operators B have the following explicit form

(4.149)

124

4 Electromagnetic Field

⎛

0

a − ∂N ∂z

∂Na ∂z

0

a − ∂N ∂y

∂Na ∂x

Na Nb

0

0

0

Na Nb

0

0

0

Na Nb

⎜ Bacurl = ⎜ ⎝ ⎛ ⎜ 1 =⎜ Bab ⎝

∂Na ∂y

⎛ ∂Na

⎞

⎟ a ⎟ − ∂N ∂x ⎠ 0 ⎞ ⎟ ⎟ ⎠

∂x

0

0

⎞

⎜ ⎟ ∂Na ⎟ Badiv = ⎜ ⎝ 0 ∂y 0 ⎠ (4.150) a 0 0 ∂N ∂z ⎞ ⎛ b Na ∂N ∂x ⎟ ⎜ 2 ∂Nb ⎟ =⎜ Bab (4.151) ⎝ Na ∂y ⎠ . b Na ∂N ∂z

Thus, we can write the spatially discretized matrix equation as 2  1 2    1 w˙ Mww Mwφ w Kww 0 f mag + = , MTwφ Mφφ φ 0 0 0 φ˙

(4.152)

with w, φ the unknowns at the nodes and the matrices as well as the righthand side vector compute as Mww =

ne 

 meww

;

meww

e=1

Mwφ =

ne 

(4.153)

Ωe

 2 γBpq dΩ

mewφ ; mewφ = [mpq ] ; mpq =

e=1

Mφφ =

1 γBpq dΩ

= [mpq ] ; mpq =

ne 

(4.154)

Ωe

meφφ ; meφφ = [mpq ] ;

e=1

(4.155)

 γ∇Na · ∇Nb dΩ

mpq = Ωe

Kww =

ne 

keww ; keww = [kpq ]

e=1



kpq =

* ) ν (Bpcurl )T Bqcurl + (Bpdiv )T Bqdiv dΩ

(4.156)

Ωe

f mag =

ne 

f emag ; f emag = [f a ] ;

e=1

 (Na Jix , Na Jiy , Na Jiz )T dΩ ,

fa = Ωe

with ne the number of finite elements.

(4.157)

4.7 Numerical Computation: Electromagnetics

125

The time discretization is performed by applying the trapezoidal method (see Sect. 2.5.1) both to w, φ using, e.g., an effective mass formulation, which results in the following scheme: •

Perform predictor step: w ˜ = w n + (1 − γP )∆t w˙ n ˜ = φ + (1 − γP )∆t φ˙ . φ n n

•

(4.158) (4.159)

Solve algebraic system of equations: 1

M∗ww Mwφ

2

w˙ n+1 φ˙ n+1

MTwφ Mφφ



 =

f n+1 0



1 −

Kww 0 0

0

M∗ww = Mww + γP ∆t Kww . •

2

w ˜ ˜ φ

 (4.160) (4.161)

Perform corrector step: ˜ + γP ∆t w˙ n+1 wn+1 = w ˜ + γP ∆t φ˙ φ =φ . n+1

n+1

(4.162) (4.163)

4.7.4 Newton’s Method for the Non-linear Case The main non-linearity within magnetic field computation is due to the dependence of the permeability µ on the magnetic field, which can be modelled as follows B = µ(H)H

(4.164)

H = ν(B)B ,

(4.165)

with µ denoting the magnetic permeability and ν the magnetic reluctivity. Since experimental setups for measuring the non-linear behavior of magnetic materials exhibit a functional relation between H and B, we can derive the following alternative formulation H(B) = ν(B)B H(B) ν(B) = B H
(B)B − H(B)
ν (B) = . B2

(4.166) (4.167) (4.168)

For the magnetostatic case, we obtain from (4.112) the following weak formulation   ν(|∇ × A|) (∇ × A) · (∇ × A
) dΩ = Ji · A
dΩ , (4.169) Ω

Ω

126

4 Electromagnetic Field

with the test function A
∈ H0 (curl). This equation can be expressed with the non-linear operator F as F (A) = 0 . Newton’s method is now defined by (see Appendix D.2) Ak+1 = Ak + S ,

where S solves F
(Ak )[S] = −F(Ak ) .

(4.170)

Therefore, in the first step we must derive the linearized form of F F
(Ak )[S] = F (Ak + S) − F(Ak ) + O(||S||2 ) .

(4.171)

The term F (Ak + S) − F(Ak ) for arbitrary test functions A
∈ H0 (curl) computes in its weak form as  ν(|∇ × (Ak + S)|) (∇ × (Ak + S)) · (∇ × A
) dΩ Ω

 −

ν(|∇ × Ak |) (∇ × Ak ) · (∇ × A
) dΩ .

(4.172)

Ω

By adding the term ν(|∇ × Ak |)(∇ × (Ak + S) · (∇ × A
) to (4.172) as well as at the same time subtracting it from (4.172) and combining similar terms leads to     ∇ × (Ak + S) · (∇ × A
) dΩ ν(|∇ × (Ak + S)|) − ν(|∇ × Ak |) Ω

 +

ν(|∇ × Ak |) (∇ × S) · (∇ × A
) dΩ .

(4.173)

Ω

Now we can perform the following approximations   ν(|∇ × (Ak + S)|) − ν(|∇ × Ak |) ≈ ν
(|∇ × Ak |) |∇ × (Ak + S)| − |∇ × Ak | |∇ × (Ak + S)|2 − |∇ × Ak |2 |∇ × (Ak + S)| + |∇ × Ak | 2 2 |∇ × Ak | + |∇ × S| + 2(∇ × Ak ) · (∇ × S) − |∇ × Ak |2 = |∇ × (Ak + S)| + |∇ × Ak | (∇ × S) · (∇ × Ak ) ≈ . |∇ × Ak |

|∇ × (Ak + S)| − |∇ × Ak | =

With the help of these approximations, (4.173) can be written as

(4.174)

4.7 Numerical Computation: Electromagnetics



ν
(|∇ × Ak |)

Ω

127

(∇ × S) · (∇ × Ak ) (∇ × (Ak + S) · (∇ × A
) dΩ |∇ × Ak |  . + ν(|∇ × Ak |) (∇ × S) · (∇ × A
) dΩ(4.175) Ω

Neglecting all terms of second order and retaining only terms linear in S, we arrive at  (∇ × S) · (∇ × Ak ) (∇ × Ak ) · (∇ × A
) dΩ ν
(|∇ × Ak |) |∇ × Ak | V  + ν(|∇ × Ak |) (∇ × S) · (∇ × A
) dΩ . (4.176) Ω

Therefore, the resulting Newton step computes as (see (4.170))  ν(Bk ) (∇ × S) · (∇ × A
) dΩ Ω



+

ν
(Bk ) (∇ × S) · eBk Bk eBk · (∇ × A
) dΩ

Ω



=

Ji · A
dΩ

Ω



−

ν(Bk )(∇ × Ak ) · (∇ × A
) dΩ

(4.177)

Ω

Ak+1 = Ak + S ,

(4.178)

with Bk = ∇ × Ak , Bk = |Bk | and eBk = Bk /Bk . Now, let us apply the FE method to (4.177) using, e.g., nodal finite elements. Therefore, we have to consider instead of (4.169) a weak formulation similar to (4.123). Since we have to
ensure ∇ · w = 0 (see (4.119)), we apply a penalty formulation by adding ν ∇ · v ∇ · w dΩ to (4.177) (having the Ω

decomposition A = w + ∇φ in mind)

128

4 Electromagnetic Field








ν(Bk ) (∇ × v ) · (∇ × S) dΩ + Ω



+

ν(Bk ) ∇ · v
∇ · S dΩ

Ω

ν
(Bk )Bk (∇ × v
) · eBk (∇ × S) · eBk dΩ

Ω



= Ω

Ji · v
dΩ 

−

  ν(Bk ) (∇ × wk ) · (∇ × v
) + (∇ · wk )(∇ · v
) dΩ . (4.179)

Ω

Performing the same procedure as in Sect. 4.7.3, we arrive at the following algebraic system of equations   (4.180) KL (w k ) + KNL (w k ) S = f mag − KL (w k )w k = f res . In (4.180) KL and f mag are calculated as in (4.156) and (4.157). The matrix KNL , also called the tangent stiffness matrix, is computed as follows KNL =

ne 

keNL ; keNL = [kNL pq ]

e=1

kNL pq

 =

) * ν
(Bk ) (Bpcurl · eBk ) Bk (Bqcurl · eBk )T dΩ , (4.181)

Ωe

with ne the number of finite elements and ν
(Bk ) as well as eBk to be evaluated at each integration point during the numerical integration. The iterative solution process is performed, until the following two stopping criteria are fulfilled ||f res ||2 ||w k+1 − w k ||2 < δa < δr , (4.182) ||w k+1 ||2 ||f mag ||2 with appropriate δa and δr . Since a standard Newton method is not globally convergent, it is necessary to apply a technique that controls the updating by wk+1 = wk + ηS ,

(4.183)

with some relaxation parameter η. The scalar parameter η > 0 is introduced to control the convergence during early steps of the iteration process, or in the presence of nonmonotonic material relations. A common algorithm to compute η is a line search (see Appendix D) defined by approximately minimizing |G(η)| = |S T f res (w k + ηS)| .

4.7 Numerical Computation: Electromagnetics

129

4.7.5 Approximation of B–H Curve As can be seen from Sect. 4.7.4, not only values of the magnetic reluctivity ν as a function of the magnetic induction B but also first derivatives of ν are required for the solution of the non-linear equation. Figure 4.20(a) shows a typical B–H curve obtained by linear interpolation of the measured B– H values. Since, from the physical point of view, the B–H curve is always 2 1.8 1.6 1.4

B (T)

1.2 1 0.8 0.6 0.4 0.2 0 0

5

10 H (kA)

15

20

(a) Typical B–H curve 12000

10000

Q (Am/Vs)

8000

6000

4000

2000

0 0

0.5

1 B (T)

1.5

2

(b) Corresponding ν(B) curve Fig. 4.20. Typical B–H curve and computed ν(B) curve

130

4 Electromagnetic Field

monotone, which is not the case for the ν(B) curve as displayed in Fig. 4.20(b), it makes sense to find a functional expression for H(B) and compute ν as well as ν
as given in (4.167) and (4.168). At this point, the question arises if one should apply an interpolation or an approximation scheme to obtain the function H(B). Since the measured B–H values will always just be accurate up to a measurement error, it is not appropriate to fix the curve to these values at the measurement points, as would be done by interpolation. Instead, one should do an approximation to the measurement points combined with some regularization to avoid oscillations. We will use so-called smoothing splines [49] and minimize the following regularized least squares functional B end α

J [H] =

(H

(B)) dB + α 2

N 

2

(H(Bk ) − Hk ) ,

(4.184)

k=1

0

with H

(B) the second derivative with respect to B and N the number of measured B–H values. Therefore, we obtain for small values of α a very smooth approximation and for large values of α a solution, which will be very close to the discrete points. For determining the regularization parameter α a posteriori, we use a discrepancy principle with bisection, i.e., set α equal to the largest member of the geometrically decreasing sequence 2−j such that for the minimizer B α of J α the residual 3 4N 4 5 (H(Bk ) − Hk )2 (4.185) k=1

is of the order of magnitude of the measurement noise. Note that—unlike ν—the B–H curve itself always has to be monotone for physical reasons, so in order to be able to monitor and preserve this property we prefer to apply the described smoothing spline technique directly to the collection of (Bk , Hk )N k=1 points. Practically, the monotonicity of the B–H curve approximation is gained by putting restrictions on the coefficients of the spline curve. These restrictions arise naturally, if we require H
(B) > 0. Consequently, the task is to find a monotone approximation that fulfills the discrepancy principle for a given data noise level. Note additionally that the noise level is given in terms of (Bk , Hk )N k=1 points. If we would deduce an estimate for the data noise in terms of the (νk , Bk )N i=k points, this would, due to the strong variation in scale of both function and derivative values of the curve under consideration, lead to a locally too pessimistic noise estimate and hence to a poor curve approximation. For a detailed discussion on this topic we refer to [181] and [169].

4.7 Numerical Computation: Electromagnetics

131

4.7.6 Modelling of Current-loaded Coil In this section we will investigate the numerical computation of electromagnetic fields excited by a current-loaded coil. Figure 4.21 displays a typical setup of a coil with a ferrite core.

(a) Structure of the winding around the ferrite core

(b) Hollow cylinder

Fig. 4.21. Coil with a ferrite core

To simplify the geometric modelling of the winding structure as well as the further meshing, we will substitute the complex structure by a hollow cylinder (see Fig. 4.21(b)). This simplification can be done since the change in the computed magnetic field can be neglected. The relation for the current density J in the coil is now defined by J = JeJ =

I INc eJ = eJ . Γw κΓc

(4.186)

In (4.186) eJ denotes the unit vector for the direction of J, I the current, Γw the cross section of one winding, Nc the number of turns, Γc the cross section of the hollow cylinder, and κ the filling factor. 4.7.7 Computation of Global Quantities In order to compare simulated results with measurements, we have to evaluate quantities derived from the computed field.

132

4 Electromagnetic Field

Magnetic Flux and Inductance In general, the inductance L for a magnetic assembly is defined by [95] L=

Ψ . I

(4.187)

In (4.187) Ψ denotes the total magnetic flux and I the electric current loading the coil. The total magnetic flux Ψ for a coil with Nc turns computes as Ψ = Nc φ , where φ is the flux defined by (see Fig. 4.22)  φ= B · dΓ .

(4.188)

Γ

Expressing the magnetic induction B via the magnetic vector potential A

Fig. 4.22. Magnetic field crossing Γ

and using Stoke’s theorem (Appendix B.9), we arrive at  φ = (∇ × A) · dΓ Γ

A · ds .

=

(4.189)

C(Γ )

It has to be noted that the orientation of ds depends on the normal vector n according to the right-hand rule. General 3D Case To obtain a very accurate numerical value for the magnetic inductance, we compute the magnetic flux over the whole coil volume and then normalize it to the cross section of the coil Γc (see Fig. 4.21). Thus, we obtain

4.7 Numerical Computation: Electromagnetics

⎛ 1 ⎜ ⎝ Γc i=1 n

φ=



133

⎞ ⎟ (A · eJ ) dx dy dz ⎠ ,

(4.190)

Ωie

with n the number of finite elements within the coil region (hollow cylinder) and eJ the unit vector defining the direction of the current density J in the coil. 2D Plane Case

y Cut

Gc

Gc

Region 1

Region 2

l

x

Gc

(a) Full 3D setup

(b) 2D model

Fig. 4.23. Example for a 2D plane case

According to the 2D plane model (see Fig. 4.23), we have A = Az (x, y)ez ds = dz ez . Thus, the evaluation of (4.189) gives us φ = −A1z l + A2z l = l(A2z − A1z ) . Since the magnetic vector potential can vary over the cross section, we compute the magnetic flux φ by the following averaging ⎛ ⎞   n n 1 2  l ⎜ ⎟ φ= A2z dx dy − A1z dx dy ⎠ , ⎝ Γc i=1 i=1 Ωie

Ωie

with n1 and n2 the number of finite elements in regions 1 and 2. Axisymmetric Case The situation for the axisymmetric case is displayed in Fig. 4.24. In the ax-

134

4 Electromagnetic Field

(a) Full 3D setup

(b) 2D model

Fig. 4.24. Example for a axisymmetric case

isymmetric case, just the circumferential component of the magnetic vector potential is different from zero, thus A = Aϕ (r, z)eϕ ds = dϕ eϕ . Here we also perform an averaging, which means to compute the integral over the volume of the coil and normalize it to the cross section Γc of the coil Ψ = Nc φ =

n  2πNc  rAϕ dr dz , Γc i=1

(4.191)

Ωie

with n the number of finite elements in Γc . 4.7.8 Induced Electric Voltage Let us consider an electrically open coil, and the task of computing the induced electric voltage due to a time-varying magnetic field. Then, according to Sect. 4.1, we have to evaluate dφ uind = −Nc , (4.192) dt with Nc the number of turns. Thus, we can simply use (4.190) (or its 2D ˙ and multiply the result by −Nc to arrive at (4.192). versions), replace A by A

4.8 Numerical Examples

135

4.8 Numerical Examples 4.8.1 Ferromagnetic Cube In a first study, we will consider the numerical computation of the magnetic induction B within a ferromagnetic cube surrounded with air (see Fig. 4.25). A parameter jump in the magnetic reluctivity across the ferromagnetic cube/air interface by a factor of 1000 is assumed. The boundary conditions for the

Fig. 4.25. Geometry setup and finite element mesh

magnetic vector potential A = (Ax , Ay , Az ) are applied as follows: z=0 z=H x=0 x=L y=0 y=L

Az Az Az Az Az Az

=0 =0 = 0 Ay = 0 Ay = 0 Ax = 0 Ax

= A0 = −A0 = −A0 = A0

The geometry of this cube is fixed by its side length L and height H. The used finite element mesh is displayed in Fig. 4.25. In the first step we computed the magnetic induction by discretizing (4.123) using nodal finite elements with linear interpolation functions (linear hexahedra). As shown in Fig. 4.26(a), instead of a constant magnetic induction over the ferromagnetic cube, we obtain very poor results near the corners and edges. In the next step, we have used nodal finite elements with quadratic interpolation functions. As can be seen from Fig. 4.26(b), the wrong solution at the corners has not changed much, but along the edges the solution has been improved considerably. By applying the weighted regularization method (see Sect. 4.7.1), we obtain for the case of linear interpolation functions the results displayed in Fig. 4.27(a), and, for the case of quadratic interpolation the results shown in Fig. 4.27(b). As can

136

4 Electromagnetic Field

(a) Nodal finite elements with linear in- (b) Nodal finite elements with terpolation functions quadratic interpolation functions Fig. 4.26. Magnetic induction: Standard formulation

(a) Nodal finite elements with linear in- (b) Nodal finite elements with terpolation functions quadratic interpolation functions Fig. 4.27. Magnetic induction: Weighted regularization

be seen, the magnetic induction is almost constant all over the ferromagnetic cube when using the weighted regularization. Of course, also in this case, the quadratic interpolation functions yield the better results. 4.8.2 Thin Iron Plate In a second example, we study the distribution of the magnetic induction within a thin iron plate (Fig. 4.28(a)). At the outer boundary the tangential component of the magnetic vector potential A has been set to zero, whereas at the symmetry planes the normal component has to vanish. The given current density generates the magnetic field, and within the iron plate the magnetic induction shall be constant in the x-direction and will decrease in the y-direction. The finite element discretization using linear hexahedra and tetrahedra for the iron plate and coil is shown in Fig. 4.28(b). The computational results by using the standard finite element formulation are displayed in Fig. 4.29(a), and show a strong variation of the magnetic induction in the x-direction. The formulation with the weighted regularization produces a magnetic induction distribution within the plate as shown in Fig. 4.29(b). If we

4.8 Numerical Examples

137

(a) Geometry setup: Iron plate and coil (b) Finite element mesh without ambient air Fig. 4.28. Iron plate surrounded by a coil

(a) Standard regularization

(b) Weighted regularization

Fig. 4.29. Magnetic induction in the iron plate

compare the evaluated inductances of the coil (global quantity)—computed with the standard formulation (regularization term all over the domain) and with weighted regularization—then the difference is less then 0.5 %. This result indicates that the error occurring at interfaces of jumping permeability is a local one for the standard formulation.

5 Acoustic Field

Acoustic waves can propagate in non-viscous media just in the form of longitudinal waves. Thus, the particles of the fluid move forwards and backwards in the direction (and opposite) of the propagation and produce locally a compression and expansion of the fluid (see Fig. 5.1). The repulsive force, which is essential for the propagation of the wave, is generated by the pressure change in the media.

Fig. 5.1. Compression and expansion of a fluid

5.1 Wave Theory of Sound The wavelength λ is uniquely defined by the frequency f and the speed of sound c via c (5.1) λ= . f Assuming a frequency of 1 kHz, the wavelength in air takes on the value of 0.343 m (c = 343 m/s) and in water 1.5 m (c = 1500 m/s). The propagation of

140

5 Acoustic Field

waves is described by the time and the spatial variation of the density ρ ρ=

mass (kg/m3 ) volume

the pressure p p=

force (N/m2 ) cross section

and the velocity v distance (m/s) . time These quantities can be decomposed in their mean and alternating part according to v=

ρ = ρ0 + ρ
p = p0 + p


(5.2)




v = v0 + v , where we denote by ρ
the acoustic density, p
the acoustic pressure and v
the acoustic particle velocity. For linear acoustics, we have the following relations between the mean and alternating quantities ρ
<< ρ0

p
<< p0

v
<< v0 .

(5.3)

The acoustic field is fully described by the equation of mass conservation (continuity equation), the equation of momentum conservation (Euler’s equation, Newton’s law for fluids) and the pressure-density relation (state equation). These basic physical relations will be discussed in the following section. 5.1.1 Conservation of Mass (Continuity Equation) The propagation of an acoustic wave through a fluid (gas) causes local changes in the density. Therefore, we have to mathematically model the fact that a net flow across a closed surface causes a mass change due to a change in density inside the volume in order to guarantee mass conservation. Let us consider a volume Ω enclosed by a surface Γ as displayed in Fig. 5.2. The mass m at time t and given density ρ(x, y, z, t) computes as  m(t) = ρ(x, y, z, t) dΩ . Ω

Now, if we model a positive net flow across the surface Γ during the time interval dt, then the mass inside the volume Ω will decrease. Thus, we obtain  ρv dt · dΓ = − (ρ(t + dt) − ρ(t)) dΩ , (5.4) Γ

Ω

5.1 Wave Theory of Sound

141

Fig. 5.2. Considered volume Ω with closed surface Γ

which states that a reduction of mass in the volume Ω with respect to time has to be compensated by a mass flow through the surface Γ enclosing the volume. Expanding the expression ρ(t + dt) in a Taylor series up to the linear term ∂ρ dt ρ(t + dt) ≈ ρ(t) + ∂t changes (5.4) to    ∂ρ dt dΩ . (5.5) ρv dt · dΓ = − ∂t Γ Ω Applying the divergence theorem (see Appendix B.6) to the term on the left side of (5.5) and cancelling dt results in     ∂ρ ∇ · (ρv) dΩ = − dΩ . (5.6) ∂t Ω Ω Since we integrate the same arbitrary volume, we can omit the integration and obtain the continuity equation in differential form ∇ · (ρv) = −

∂ρ . ∂t

(5.7)

5.1.2 Conservation of Momentum (Euler Equation) Let us consider an infinitely small volume, which moves with the fluid and has the mass dm = ρ dΩ. Then Newton’s law states a dm = df .

(5.8)

For non-viscous fluids, the repulsive force is caused by the change in pressure, and the pressure is understood to be (−pn) with n the outward normal vector on the surface. Thus, we obtain for the infinite small volume dΩ = ( dx dy dz) (see Fig. 5.3) − dfx = (p(x + dx/2, y, z) − p(x − dx/2, y, z)) dy dz − dfy = (p(x, y + dy/2, z) − p(x, y, − dy/2, z)) dx dz − dfz = (p(x, y, z + dz/2) − p(x, y, z − dz/2)) dx dy .

142

5 Acoustic Field

Fig. 5.3. Infinite small volume with pressure state

Now we can expand the terms in a Taylor series up to the linear term, e.g, p(x + dx/2, y, z) and p(x − dx/2, y, z) ∂p ∂x ∂p p(x − dx/2, y, z) ≈ p(x, y, z) − ∂x

p(x + dx/2, y, z) ≈ p(x, y, z) +

dx 2 dx . 2

This results in ∂p dΩ ∂x ∂p dΩ dfy = − ∂y ∂p dfz = − dΩ , ∂z or in compact form we may write dfx = −

df = −∇p dΩ .

(5.9) (5.10) (5.11)

(5.12)

Since we use a coordinate system fixed in space, the position of the fluid particles depends on time. Thus, the expression for the acceleration a as a function of the acoustic velocity v reads as ∂v ∂v ∂x ∂v ∂y ∂v ∂z dv = + + + dt ∂t ∂x ∂t ∂y ∂t ∂z ∂t ∂v = + (v · ∇)v . ∂t

a=

Using (5.8), (5.12), and (5.13) yields Euler’s equation   ∂v ρ + (v · ∇)v = −∇p . ∂t

(5.13)

(5.14)

5.1 Wave Theory of Sound

143

5.1.3 Pressure-Density Relation (State Equation) According to the quasistatic theory of thermodynamics, we can describe the local state of a fluid at any time by the density ρ and the specific internal energy Eint . Other state variables like the pressure p or the temperature T are determined from equations of state defined by ρ and Eint . In the context of acoustic wave propagation, we shall also consider the specific entropy s, which is related to heat transfer. In general, the pressure is regarded to be a function of the density and the specific entropy p = p(ρ, s) .

(5.15)

However, if the specific entropy can be considered as constant, which is expressed by ds = 0, (5.16) dt the state is adiabatic and the pressure is just a function of ρ. Now, the question of sound propagation being an adiabatic or isothermal process can be answered by considering the heat conduction process within a fluid. Actually, the temperature rises and falls according to the density and pressure fluctuations in a sound wave. Since the relation cp

∂T ds =T ∂t dt

(5.17)

holds (cp denotes the specific heat at constant pressure), we can rewrite the thermal diffusion equation by T

ds = γT ∆T . dt

(5.18)

with γT the coefficient of thermal conductivity. If conduction dominates the process, we can approximate (5.18) by ∆T = 0 and the sound is considered as isotherm, otherwise (5.18) is approximated by ds/dt = 0 and the sound is considered adiabatic. Both cases are approximations of reality, but for freely propagating acoustic waves both in gases as well as liquids at frequencies of interest, the state is adiabatic (for a detailed discussion see [170]). Therewith, no thermal exchange of energy between the particles exists. Physically we can express this behavior by (5.19) pΩ κ = const. , with κ the adiabatic exponent and Ω the considered volume. The adiabatic exponent κ is computed by the ratio of specific heat cp at constant pressure and the specific heat cΩ at constant volume κ=

cp . cΩ

(5.20)

Now, let us consider a fluid in rest with (Ω0 , p0 , ρ0 ) and a fluid, in which an acoustic wave propagates, defined by (Ω, p, ρ). According to (5.19) we obtain

144

5 Acoustic Field



Ω0 Ω

κ =

p . p0

(5.21)

By assuming a constant mass, we derive a relation between the volumes and densities m0 = ρ0 Ω0 = ρΩ ρ Ω0 = . Ω ρ0 This relation, in combination with (5.21), yields  κ ρ p = . p0 ρ0

(5.22)

(5.23)

For an ideal gas, the relation p = ρRT

(5.24)

holds (R denotes the gas constant, e.g., in air R = 297 Ws/kgK), and we may express the speed of sound c by [170] √ c = κRT (5.25) to obtain its dependency on the temperature T . Since for linear acoustics we may assume ρ
<< ρ0 , we can perform the following approximation  κ  κ ρ0 + ρ
ρ ρ
= ≈1+κ . (5.26) ρ0 ρ0 ρ0 This result combined with (5.23) yields the linearized state equation for acoustic waves p0 p
= κ = c2 . (5.27) ρ
ρ0 For liquids, such as water, the pressure-density relation is mainly expressed by the adiabatic bulk modulus Ks according to Ks ∂p(ρ, s) = ∂ρ ρ . Ks . c= ρ

(5.28) (5.29)

The reciprocal 1/Ks is known as the adiabatic compressibility. In the case of a liquid, a negligible error occurs, if we consider the state to be adiabatic or isotherm. The reason for this behavior is the little difference between the adiabatic bulk modulus Ks , which is quite difficult to measure, and the isothermal bulk modulus KT (for details see [170]).

5.1 Wave Theory of Sound

145

5.1.4 Linear Acoustic Wave Equation For linear acoustic wave propagation, the following relations hold v ∇ρ  ρ∇ · v ≈ ρ0 ∇ · v ∂v (v · ∇)v  . ∂t Thus, Euler’s equation (5.14) as well as the continuity equation (5.7) read as ∂v 1 = − ∇p ∂t ρ0 1 ∂ρ . ∇·v = − ρ0 ∂t

(5.30) (5.31)

Since any derivative with respect to time as well as space of the constant quantities (v0 , p0 , ρ0 ) are zero, we may substitute the total physical quantities (v, p, ρ) by their alternating ones (v
, p
, ρ
). Applying the time derivative to (5.31) and interchanging time and space derivatives yields ∇·

1 ∂ 2 ρ
∂v
=− . ∂t ρ0 ∂t2

(5.32)

Expressing the term ∂v/∂t by (5.30) results in a relation between pressure and density   1 1 ∂ 2 ρ
. (5.33) ∇ · − ∇p
= − ρ0 ρ0 ∂t2 Using the relation ρ
= p
/c2 according to the state equation (5.27), we obtain the acoustic wave equation for p
∆p
=

1 ∂ 2 p
. c2 ∂t2

(5.34)

By applying the curl operator to (5.30), it is easy to show that the acoustic velocity v
is irrotational   ∂v
+ ∇p
= 0 ∇ × ρ0 ∂t ∂ ρ0 (∇ × v
) + ∇ × ∇p
= 0 ∂t (5.35) ∇ × v
= 0 . Thus, the acoustic velocity v
can be expressed as the gradient of a scalar potential, the so-called acoustic velocity potential v
= −∇ψ .

(5.36)

146

5 Acoustic Field

By introducing the acoustic velocity potential in Euler’s equation, we obtain from (5.30) the relation between ψ and p
−ρ0

∂ (∇ψ) = −∇p
∂t ∂ψ , p
= ρ0 ∂t

(5.37)

where the integration constant can be set to zero, since (5.36) defines ψ only up to a constant. With (5.37) we may write the wave equation (5.34) for the acoustic velocity potential 1 ∂2ψ ∆ψ = 2 2 . (5.38) c ∂t 5.1.5 Acoustic Quantities Let us first consider the linear version of Euler’s equation (see (5.30)) and take the dot product with the acoustic velocity v
v
· ρ0

∂v
= −v
· ∇p
∂t = −∇ · (p
v
) + p
∇ · v
.

(5.39)

Using the linear version of the mass conservation and the linear pressuredensity relation ρ
= p
/c2 results in ∂v
1 ∂ρ
= −∇ · (p
v
) − p
∂t ρ0 ∂t     2 p
1 ∂ ∂ 1



ρ0 v · v = −∇ · (p v ) − . ∂t 2 2 ∂t ρ0 c2 v
· ρ0

(5.40)

Therewith, we obtain ∂wa + ∇ · Ia = 0 ∂t 1 wakin = ρ0 v
· v
2 2 p
wapot = 2ρ0 c2

(5.41) (5.42) (5.43)

with wa the total acoustic energy density and Ia the acoustic energy flux, also called the acoustic intensity. Furthermore, we refer to wakin as the acoustic kinetic energy density and wapot the acoustic potential energy density. Now let us assume that the acoustic source is driven with constant frequency f (angular frequency ω = 2πf ), so that we can express the acoustic field variables by

5.1 Wave Theory of Sound

147

p
(t) = pˆ cos(ωt) ˆ cos(ωt + ϕ) . v
(t) = v Therewith, we obtain the following expressions for the time averaged acoustic energy density wa as well as intensity Ia (T = 1/f is the period time) waav

1 = T

t 0 +T

t0

1 ρ0 vˆ2 cos2 (ωt + ϕ)    2

1 dt + T

1 2 (1+cos(2(ωt+ϕ))

t 0 +T

t0

1 pˆ2 2 ρ 0 c2

cos2 (ωt)   

dt

1 2 (1+cos(2ωt))

1 1 pˆ2 ρ0 vˆ2 + 4 4 ρ 0 c2 t 0 +T 1 ˆ cos(ωt) cos(ωt + ϕ) dt = pˆv    T =

Iav a

t0

=

1 2

cos ϕ+ 12 cos(2ωt+ϕ)

ˆ pˆv cos ϕ . 2

By using these results and substituting it in (5.41), we arrive at ∇ · Iav a = 0. This means that the spatial variation of Iav a has to be zero, or expressing it in integral form  ∇ · Iav dΩ = Iav a a · n dΓ = 0 . Ω

Γ (Ω)

However, if the considered volume encloses any acoustic sources, the above derivation does not apply, and we obtain Paav = Iav (5.44) a · n dΓ . Γ (Ω)

In (5.44) we denote by Paav the average acoustic power radiated by all sources enclosed by the surface Γ . Strictly speaking, each acoustic wave has to be considered as transient, having a beginning and an end. However, for some long duration sound, we speak of continuous wave (cw) propagation and we define for the acoustic pressure p
a mean square pressure (p
)2av as well as a root mean squared (rms) pressure p
rms 3 4 t0 +T 4 41 2 prms = 5 (p − p0 ) dt . (5.45) T t0

148

5 Acoustic Field

In (5.45) T denotes the period time of the signal or if we cannot strictly speak of a periodic signal, an interminable long time interval. Any acoustic media is defined by its acoustic impedance Za , which computes as the quotient of sound pressure p
and the magnitude of the acoustic particle velocity v
p
Za =
. (5.46) v If the particles of the fluid have the same oscillation in the plane normal to the direction of propagation, the corresponding acoustic wave is called a plane wave. For plane waves, the acoustic impedance is constant and is called the specific acoustic impedance Z0 =

p
= ρ0 c (Ns/m3 ) . v


(5.47)

In air the acoustic impedance Z0 has a value of 408 Ns/m3, in water a value of 1.5 × 106 Ns/m3 and in solids values between (3 − 110) × 106 Ns/m3 . For non-plane waves the sound field impedance is not constant, e.g., for a spherical wave Za computes as Za =

p
jρ0 ckr jkr = Z0 , = v
1 + jkr 1 + jkr

(5.48)

with k = ω/c = 2π/λ the wave number and r the distance from the source.

5.1.6 Plane and Spherical Waves In order to get some physical insight in the propagation of acoustic sound, we will consider two special cases plane and spherical travelling waves. Let’s start with the simpler case, the propagation of a plane wave as displayed in Fig. 5.4. Thus, we can express the acoustic pressure by p
= p
(x, t) and the velocity by

Fig. 5.4. Propagation of a plane wave

v
= v(x, t)ex . Using these relations together with the linear pressure-density law, we arrive at the following 1D linear wave equation

5.1 Wave Theory of Sound

∂ 2 p
1 ∂ 2 p
− = 0, ∂x2 c2 ∂t2

149

(5.49)

which can be rewritten in factorized version as    1 ∂ 1 ∂ ∂ ∂ − + p
= 0 . ∂x c ∂t ∂x c ∂t

(5.50)

This version of the linearized, 1D wave equation motivates us to introduce the following two functions ξ = t − x/c η = t + x/c with two main properties ∂ ∂ ∂ = + ∂t ∂ξ ∂η

1 ∂ = ∂x c



∂ ∂ − ∂η ∂ξ

 .

Therewith, we obtain for the factorized operator ∂ 1 ∂ 1 ∂ − =− ∂x c ∂t 2c ∂ξ

∂ 1 ∂ 1 ∂ + = ∂x c ∂t 2c ∂η

and the linear, 1D wave equations transfers to −

1 ∂ ∂
p = 0. 4c2 ∂ξ ∂η

The general solution computes as a superposition of arbitrary functions of ξ and η (5.51) p
= f (ξ) + f (η) = f (t − x/c) + g(t + x/c) . This solution describes waves moving with the speed of sound c in +x and −x direction, respectively. Now let us derive the relation between acoustic pressure p
and particle velocity v
. For this, we substitute the above relations in the linearized, 1D equations for momentum and mass conservation, and use the linear pressuredensity relation ∂v
∂p
+ ρ 0 c2 =0 ∂t ∂x   ∂v
∂p
∂v
∂p
+ + ρ0 c − =0 ∂ξ ∂η ∂η ∂ξ ∂
∂
(p − ρ0 cv
) + (p + ρ0 cv
) = 0 ∂ξ ∂η

∂p
∂v
+ =0 ∂t ∂x     ∂v
∂v
∂p
1 ∂p
+ − ρ0 + =0 ∂ξ ∂η c ∂η ∂ξ ∂ ∂ (ρ0 cv
− p
) + (ρ0 cv
+ p
) = 0 . ∂ξ ∂η ρ0

Once adding and once subtracting these two obtained equations will result in

150

5 Acoustic Field

∂ (ρ0 cv
+ p
) = 0 ∂η

∂ (ρ0 cv
− p
) = 0 ∂ξ

so that (ρ0 cv
+ p
) as well as (ρ0 cv
− p
) become functions of ξ and η, and we arrive at 1 p
v
= (f (t − x/c) + g(t + x/c)) = . (5.52) ρ0 c ρ0 c Therewith, the value of the acoustic pressure over acoustic particle velocity for a plane wave is constant and results in the specific acoustic impedance (see 5.47). Furthermore, the acoustic energy density wa and the acoustic intensity Ia simplifies to wa =

p
ρ0 v
p
+ = 2 2 2ρ0 c ρ 0 c2

(5.53)

Ia =

p
ex = cwa ex . ρ0 c

(5.54)

2

2

2

2

In (5.53) we realize that the acoustic kinetic and potential energy density are equal, and (5.54) demonstrates that the acoustic energy propagates with the speed of sound c. The second case of investigation will be a spherical wave, where we assume a point source located at the origin. In the first step, we rewrite the linearized wave equation in spherical coordinates and consider that the pressure p
will just depend on the radius r. Therewith, the Laplace-operator reads as ∆p
(r, t) = and we obtain

1 ∂ 2 rp
∂ 2 p
2 ∂p
= + ∂r2 r ∂r r ∂r2

1 ∂ 2 rp
1 ∂ 2 p
− = 0. 2 r ∂r2 c2 ∂t  

(5.55)

1 ∂ 2 rp
r ∂t2

A multiplication of (5.55) with r results in the same wave equations as obtained for the plane case (see (5.49)), just instead of p
we have rp
. Therefore, the solution of (5.55) reads as p
(r, t) =

1 (f (t − r/c) + g(t + r/c)) , r

(5.56)

which means that the pressure amplitude will decrease according to the distance r from the source. The assumed symmetry requires that all quantities will just exhibit a radial component. Therewith, we can express the time averaged acoustic intensity Iav in normal direction n by a scalar value just depending on r Iav · n = Irav and as a function of the time averaged acoustic power P av of our source

5.2 Quantitative Measure of Sound

Irav =

P av . 4πr2

151

(5.57)

According to (5.57), the acoustic intensity decreases with the squared distance from the source. This relation is known as the spherical spreading law. In order to obtain the acoustic velocity v
= v
(r, t)er as a function of the acoustic pressure p
, we substitute the general solution for p
(see (5.56), in which we set without loss of generality g = 0) into the linear Euler equation (see (5.30))   ∂v
1 ∂p
1 ∂ f (t − r/c) =− =− ∂t ρ0 ∂r ρ0 ∂r r   1 ∂ F (t − r/c) v
= − (5.58) ρ0 ∂r r with f (t) = ∂F (t)/∂t. Using the relation 1 ∂F (t − r/c) ∂F (t − r/c) =− ∂r c ∂t and performing the differentiation with respect to r results in 1 1 ∂F (t − r/c) F (t − r/c) + ρ0 r ∂r ρ0 r 2 1 1 ∂F (t − r/c) F (t − r/c) = + ρ0 c r ρ0 r 2 ∂t 

v
(r, t) = −

(5.59) (5.60)

f /r=p





=

p F (t − r/c) . + ρ0 c ρ0 r 2

(5.61)

Therewith, spherical waves show in the limit r → ∞ the same acoustic behavior as plane waves. Now with this acoustic velocity-pressure relation, we may rewrite the acoustic intensity for spherical waves as p
1 ∂F 2 (t − r/c) p
p
+ + , F (t − r/c) = 2 ρ0 c ρ0 r ρ0 c 2ρ0 r3 ∂t 2

Ir =

2

which results for the time averaged quantity (assuming F (t−r/c) is a periodic function) in the same expression as for the plane wave (p
)av . ρ0 c 2

Irav =

5.2 Quantitative Measure of Sound Sound is characterized by its pressure amplitude and its frequency spectra defining the tone color. For an average, young person (about 20 years) the

152

5 Acoustic Field

sensible frequency range is from 16 Hz to 16 kHz. It is of great interest that the human perception concerning the frequency of sound signals is a relative one. This means that frequency differences in two sound signals are sensed as equal, when the ratio of sound is the same (strictly speaking this is true for frequencies above 500 Hz, see [228]). E.g., a change in the frequency from 500 Hz to 600 Hz and from 5000 Hz to 6000 Hz will be sensed as equal. Therewith, in most analysis of acoustic signals, 1/3-octave filters, and for some technical reasons also octave filters are used. Their specifications are as follows: •

Filter specifications (see Fig. 5.5):

fm =

√

fl ... lower frequency limit fl fu

fu ... upper frequency limit fm ... center frequency

•

Octave filter: fu = 2fl Therewith, the center frequency computes by fm√= width of the filter is f = fu − fl = fu /2 = fm / 2.

•

1/3-Octave filter: fu =

√ 3 2fl

Therewith, the center√frequency calculates as fm = bandwidth is f = ( 3 2 − 1)fl ≈ 0.26fl.

√ 2fl and the band-

√ 6 2fl ≈ 1.12fl and the

Fig. 5.5. Filter function (band-pass)

The sound is often broadband and so makes sense to display the frequency spectra in so-called frequency bands by applying the above mentioned filters. Furthermore, if we consider e.g., traffic noise, then a too resolved frequency spectra will lead in most cases to non-reproducible results, so that adequate frequency bands are used, instead. Figure 5.7 displays the frequency spectra of an acoustic signal, which is generated by a turbulent flow (see Fig. 5.6)

5.2 Quantitative Measure of Sound

153

Fig. 5.6. Visualization of the flow structure near a forward-facing step

over a forward-facing step (flow-induced noise). As can be clearly seen, the levels for the octave filtered spectra are the largest, followed by the 1/3-octave filtered one and the original signal (fully resolved spectra).

Fig. 5.7. Frequency spectra of acoustic pressure level SPL (see (5.62): full resolution, 1/3-octave and octave filtered

Similar to the perception of the tone color is the perception of the human ear to pressure amplitude changes, e.g., a pressure amplitude change from p0 to 2p0 is sensed as equal with a pressure amplitude change from 5p0 to 10p0 (p0 has to be larger than 40 dB, see [228]). Now, it has to be mentioned that the threshold of hearing of an average human is at about 20 µPa and the threshold of pain at about 200 Pa, which corresponds to an interval of 107 . Thus, logarithmic scales are mainly used for acoustic quantities. The most common one is the decibel (dB), which expresses the quantity as a ratio

154

5 Acoustic Field

relative to a reference value. The sound pressure level Lp (SPL) is defined by Lp = 20 log10

prms pref

pref = 20 µPa .

(5.62)

The reference pressure pref corresponds to the sound at 1 kHz that an average person can just hear. The sound-intensity level LIa is then defined by LIa = 10 log10

Iaav Iref

Iref = 10−12 W/m , 2

(5.63)

with Iref the reference sound intensity corresponding to pref . Furthermore, the sound-power level LPa computes as LPa = 10 log10

Paav Paref

Paref = 10−12 W ,

(5.64)

with Paref the reference sound power corresponding to pref . In Tables 5.1 and 5.2 some typical sound pressure and sound power levels are listed. Table 5.1. Typical sound pressure levels SPL Threshold Voice of hearing at 5 m 0 dB

60 dB

Car at 20 m

Pneumatic hammer at 2 m

80 dB

Accelerating motor at 5 m

100 dB

110 dB

Jet at 3 m 140 dB

Table 5.2. Typical sound power levels and in parentheses the absolute acoustic power Pa Voice

Fan

Loudspeaker

Jet airliner

30 dB (25 µW)

110 dB (0.05 W)

128 dB (60 W)

170 dB (50 kW)

In general, an acoustic signal is not mono-frequent, but consists of an superposition of signals with different frequencies. For simplicity, let us analyze a pressure signal, which can be expressed as follows p
(t) = A1 cos(ω1 t) + A2 cos(w2 t) . Then, the mean square pressure computes as

5.2 Quantitative Measure of Sound 2 p
rms

1 = T

t o +T

155

) 2 * A1 cos2 (ω1 t) + 2A1 A2 cos(ω1 t) cos(ω2 t) + A22 cos2 (ω2 t) dt

t0 t o +T

=

t o +T

A21 (1 + cos(2ω1 t)) dt + 2

t0

A22 (1 + cos(2ω2 t)) dt 2

t0 t o +T

A1 A2 (cos((ω1 − ω2 )t) + cos((ω1 + ω2 )) dt .

+ t0

Assuming that T = n1 (2π/ω1 ) = n2 (2π/ω2 ) = n3 (2π/(ω1 +ω2 )) = n4 (2π/(ω1 − ω2 )) with ni ∈ IN, then all integrals of the form t o +T

Ai cos(ωi t)) dt t0

are zero, and we obtain

/

A2 A21 + 2. 2 2 Generalizing the above case, we can rewrite (5.63) by


p rms =

N 

Lp = 10 log10 Lav i = 10 log10



p2rms,k

k=1

p2ref p2rms,k p2ref

= 10 log10

N 

 10

Lav i /10

(5.65)

k=1

(5.66)

In (5.65) Lp is the overall sound pressure level OSPL, and can be interpreted as the formula for adding sound pressure levels generated by different, incoherent sound sources. Figure 5.8 displays the increase of the overall sound pressure level OSPL achieved by two sound signals in comparison to the sound pressure level SPL of the individual signals. Therewith, due to the logarithmic scaling, which corresponds to the perception of the human ear, the sum of two equal sound pressure levels just increase the overall sound pressure level OSPL by 3dB, when both sound signals have the same sound pressure level SPL. For the case that one sound signal has a SPL of about 20 dB smaller than the other, we can neglect the contribution of this sound signal to the overall sound pressure level OSPL. An interesting point we want to stress here is the fact that the above formula can be also applied to frequency bands, as e.g., obtained by an octavefiltering.

156

5 Acoustic Field

Fig. 5.8. Increase of the overall sound pressure level achieved by two sound signals as a function of the difference in SPL of the individual signals

5.3 Non-linear Acoustic Wave Equation In this section we will consider a general formulation for non-linear wave propagation in lossy and compressible fluid media. Furthermore, we assume zero mean velocity v0 , so that v = v
. Thus, we will derive Kuznetsov’s equation for non-linear acoustics [135], which is a second-order approximation for viscous heat-conducting fluids. For this case, the equation of continuity as given in (5.7) still holds. However, Euler’s equation changes to a more general equation for momentum conservation, known as Navier-Stokes equation (see [135] as well as [59])   ( 'µ ∂v v (5.67) + (v · ∇)v + ∇p = µv ∆v + + ζv ∇(∇ · v) . ρ ∂t 3 In (5.67) ζv denotes the bulk viscosity and µv the shear viscosity. Since the following vector identities are fulfilled (domain is assumed to be convex) ∇(∇ · v
) = ∆v
+ ∇ × ∇ × v
1 (v · ∇)v
= ∇(v
· v
) − v
× ∇ × v
, 2

(5.68) (5.69)

we can rewrite (5.67) as ρ

ρ ∂v
+ ∇(v
· v
) + ∇p = ∂t 2



 4µv + ζv ∆v
(5.70) 3 ( 'µ v + ζv ∇ × ∇ × v
+ ρ0 v
× ∇ × v
. − 3

5.3 Non-linear Acoustic Wave Equation

157

The last two terms in (5.70) describe acoustic streaming, and according to [84] decrease exponentially with the distance to solid walls. Since in our case we can still consider the acoustic velocity v as irrotational and set ρ = ρ
+ ρ0 as well as p = p
+ p0 (assume ∇p0 = 0), (5.70) reads as   4µv ρ0 ∂v
∂v
+ ρ
+ ∇(v
· v
) + ∇p
= + ζv ∆v
. ρ0 (5.71) ∂t ∂t 2 3 The state equation, defining the relation between the acoustic pressure p
and density ρ
within the fluid, can be expressed for the non-linear wave propagation as follows [84]   1 ∂p
p
1 B
2 κ 1
p . (5.72) − − ρ = 2− c ρ0 c4 2A ρ 0 c4 cΩ cp ∂t In (5.72) B/A denotes the parameter of non-linearity, κ the adiabatic exponent (see (5.20)) and cp , cΩ the specific heat capacitance at constant pressure as well as constant volume. Now, the goal is to combine the three equations (5.7), (5.71), and (5.72) into a single wave equation with the acoustic scalar potential ψ as the unknown. Therefore, we will substitute any physical quantity in a second-order term by its linearized one, since the resulting errors will be of third order. This means that we will express second-order terms in the continuity equation (5.7) and Euler equation (5.71) by the linearized ones obtained for the linear wave propagation (see (5.30) and (5.31)). Let us start with the continuity equation in the form ∂ρ
+ ρ0 ∇ · v
= −ρ
∇ · v
− v
· ∇ρ
, (5.73) ∂t where we have arranged all second-order terms on the right. By using ρ
≈

p
c2

∇ · v
≈ −

(linear state equation)

1 ∂ρ
ρ0 ∂t

(linear continuity equation)

and the relation

∂p
1 ∂p
2 = , ∂t 2 ∂t we derive the following approximation p


−ρ
∇ · v
≈

p
∂p
1 ∂p
2 ρ
∂ρ
≈ = . ρ0 ∂t ρ0 c4 ∂t 2ρ0 c4 ∂t

(5.74)

By using the linear state as well as the linear Euler equations, we transform the second term on the right-hand side of (5.73) to     1 ρ0 ∂v
ρ0 ∂(v
· v
)




−v · ∇ρ ≈ −v · . (5.75) ∇p · − = −v = c2 c2 ∂t 2c2 ∂t

158

5 Acoustic Field

Thus, our modified equation of continuity reads as ρ0 ∂(v
· v
) ∂ρ
1 ∂p
2 + ρ0 ∇ · v
= + . ∂t 2ρ0 c4 ∂t 2c2 ∂t

(5.76)

Furthermore, the use of (5.72) for ρ
leads to   2
1 ∂ p 1 B ∂p
2 κ 1 ∂p
1 − − − + ρ0 ∇ · v
2 4 4 c ∂t ρ0 c 2A ∂t ρ0 c cV cp ∂t2 =

ρ0 ∂(v
· v
) 1 ∂p
2 + . 2ρ0 c4 ∂t 2c2 ∂t

(5.77)

Now, let us apply a similar procedure to (5.71). For the term ρ
∂v
/∂t we find the following approximation ρ


p
∂v
p
1 ∂v
1 ≈ 2 ≈ − 2 ∇p
= − ∇p
2 . ∂t c ∂t c ρ0 2ρ0 c2

(5.78)

In addition, we can rewrite according to [1] the term on the right of (5.71) within the second-order approximation by     4µv 4µv 1 ∂p
+ ζv ∆v
≈ + ζ . (5.79) v ∇ 2 3 ρ0 c 3 ∂t Using (5.78) and (5.79) for (5.71), we arrive at our modified Euler equation   4µv ∂p
1 ρ0 1 ∂v


2

ρ0 ∇p − ∇(v · v ) − + ∇p = + ζv ∇ . (5.80) 2 2 ∂t 2ρ0 c 2 ρ0 c 3 ∂t Similar to the linear case, we apply the divergence operator to (5.80)     4µv 1 ρ0 1 ∂v
∂p


2

∆(v + ζ ρ0 ∇ · ∆p − · v ) − ∆ + ∆p = v ∂t 2ρ0 c2 2 ρ 0 c2 3 ∂t (5.81) and a time derivative to (5.77)   1 ∂ 1 B ∂ 2 p
2 κ 1 ∂ 3 p
1 ∂ 2 p
− − − + ρ0 (∇ · v
) = c2 ∂t2 ρ0 c4 2A ∂t2 ρ 0 c4 cV cp ∂t3 ∂t 1 ∂ 2 p
ρ0 ∂ 2 (v
· v
) + . 2ρ0 c4 ∂t2 2c2 ∂t2

(5.82)

Subtracting (5.82) from (5.81), allowing to interchange time and spatial derivative (means that the term ρ0 (∂/∂t(∇ · v
)) cancels out) yields   3
1 ∂ p 1 ∂ 2 p
1 B ∂ 2 p
2 κ 1
∆p − 2 =− − − c ∂t2 ρ0 c4 2A ∂t2 ρ 0 c4 cV cp ∂t3   4µv ρ0 1 ∂p
+ ζ − ∆(v
· v
) − ∆ v 2 ρ 0 c2 3 ∂t 2
2 2

ρ0 ∂ (v · v ) 1 1 ∂ p − 2 + ∆p
2 . (5.83) − 4 2 2 2ρ0 c ∂t 2c ∂t 2ρ0 c2

5.3 Non-linear Acoustic Wave Equation

159

Furthermore, we use the relations between time and spatial derivatives according to the linear wave equation for pressure and acoustic velocity 1 ∂ 2 p
c2 ∂t2 1 ∂ 2 v
∆v
= 2 . c ∂t2 ∆p
=

(5.84) (5.85)

Applying (5.84) and (5.85) to the right-hand side of (5.83) yields ∆p
−

1 B ∂ 2 p
1 ∂ 2 p
b ∂∆p
ρ0 ∂ 2 (v
· v
) − = − − , (5.86) c2 ∂t2 c2 ∂t ρ0 c4 2A ∂t2 c2 ∂t2

with b the diffusivity of sound     4µv 1 1 1 κ + ζv + b= − . ρ0 3 ρ 0 cV cp Since the relations between the pressure as well as velocity to the scalar velocity potential ψ still hold in the non-linear case v
= −∇ψ ∂ψ p
= ρ0 , ∂t we can rewrite (5.86) as follows       ∂ψ ∂ψ ∂ψ 1 ∂2 b ∂ ∆ ρ0 ∆ ρ0 − 2 2 ρ0 =− 2 ∂t c ∂t ∂t c ∂t ∂t  2 2 ∂ψ 1 B ∂ ρ0 − ρ0 c4 2A ∂t2 ∂t 2 ρ0 ∂ − 2 2 (∇ψ · ∇ψ) . c ∂t

(5.87)

Thus, the non-linear wave equation, also called Kuznetsov’s equation, has the form    2 2 1 ∂ψ ψ B ∂ ∂ c2 ∆ψ − 2 = − b∆ψ + 2 + ∇ψ · ∇ψ . (5.88) ∂t ∂t c 2A ∂t

Besides Kuznetsov’s equation, the following three partial differential equations modelling non-linear wave propagation are widely used (see e.g., [84]): •

Burger’s equation: ∂p
b ∂ 2 p
βa ∂p
− 3 , = p ∂x 2c ∂τ 2 ρ0 c3 ∂τ

(5.89)

160

5 Acoustic Field

with τ = (t − x/c) the retarded time, b the diffusivity of sound and βa = 1 + B/2A the coefficient of non-linearity. The Burger’s equation allows to study the combined effect of dissipation and non-linearity on progressive plane waves. •

Westervelt equation: ∇2 p
−

1 ∂ 2 p
b ∂ 3 p
βa ∂ 2 p
2 + = − , c2 ∂t2 c4 ∂t3 ρ0 c4 ∂t2

(5.90)

with b the diffusivity of sound. This partial differential equation can describe the propagation of plane waves including the non-linearity effects as well as dissipation. This means that we can use this equation, when cumulative non-linear effects dominate local non-linear effects. For example, once the propagation distance is greater than a wavelength, the waveform distortion is dominated by the cumulative effect. The approximation is not appropriate for the simulation of standing waves. •

Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation: c b ∂ 3 p
βa ∂ 2 p
2 ∂ 2 p
− ∇2⊥ p
− 3 3 = , ∂x∂t 2 2c ∂t 2ρ0 c3 ∂t2

(5.91)

with βa = 1 + B/2A the coefficient of non-linearity, b the diffusivity of sound and ∇⊥ the nabla operator including just partial derivatives with respect to the transversal coordinates (in our case y and z, since the wave propagates in the x-direction). The simplification (compared to Kuznetsov’s equation) is given by modelling directional wave propagation fulfilling k⊥  1, kx with k⊥ the wave number in the transversal direction and kx the wave number in the direction of propagation. This approximation is mainly used for investigation of diffraction, non-linearity and dissipative effects in directional sound beams, e.g., occurring in medical ultrasound, acoustic microscopy or non-destructive testing.

5.4 Numerical Computation Within this section we will discuss the FE formulation of the linear wave equation as well as Kuznetsov’s wave equation for the non-linear case. Furthermore, we will derive an FE formulation, which allows different mesh sizes for the computational subregions.

5.4 Numerical Computation

161

5.4.1 Linear Acoustics The strong formulation for linear acoustics expressed by the acoustic pressure reads as follows1 Given: f : Ω → IR c : Ω → IR . ¯ × [0, T ] → IR Find: p
(t) : Ω 1
p¨ − ∆ p
= f . c2

(5.92)

Boundary conditions p
= p
e on Γe × (0, T ) ∂p
= p
n on Γn × (0, T ) . ∂n Initial condition p
(r, 0) = p
0 , r ∈ Ω p˙
(r, 0) = p˙
0 , r ∈ Ω . In (5.92) f denotes any excitation function for generating the acoustic wave. For simplicity we set the boundary as well as initial conditions to zero. To obtain the variational formulation, we multiply (5.92) by an appropriate test function and perform a partial integration. Thus, the weak form reads as: Find p
∈ H01 such that    1

w p ¨ dΩ + ∇w · ∇p dΩ − wf dΩ = 0 (5.93) c2 Ω

Ω

Ω

for any w ∈ H01 . Using standard nodal finite elements, we approximate the continuous acoustic pressure p
as well as the test function w by p
≈ p
h =

neq 

Na p
a

(5.94)

Na wa .

(5.95)

a=1 neq

w ≈ wh =

 a=1

Thus, (5.93) is transformed to the following semidiscrete Galerkin formulation 1

same formulation holds for the acoustic velocity potential ψ

162

5 Acoustic Field neq neq  

⎛



⎝

a=1 b=1

1 Na Nb dΩ p¨
b c2

Ω

 +

(∇Na ) · ∇Nb dΩ p
b −

Ω



⎞ Na f (ra ) dΩ ⎠ = 0 . (5.96)

Ω

In matrix form, (5.96) reads as Mp¨
+ Kp
= f ,

(5.97)

with M=

ne 

 m

e

e

; m = [mpq ] ; mpq =

e=1

K=

ne 

Ωe



f =

(∇Np ) · ∇Nq dΩ

ke ; ke = [kpq ] ; kpq =

e=1 ne 

1 Np Nq dΩ c2

Ωe

 f

e

e

; f = [fp ] ; fp =

e=1

Np f (rp ) dΩ . Ωe

The time discretization is performed by applying a standard Newmark algorithm (see Sect. 2.5.2), which results in the following time-stepping scheme (effective mass matrix formulation): •

Perform predictor step: ∆t2 p˜
= p
n + ∆t p˙
n + (1 − 2βH ) p¨
n 2 ˜ p˙
= p˙
+ (1 − γH )∆t p¨
. n

•

n

(5.98) (5.99)

Solve algebraic system of equations: M∗ p¨
n+1 = f n+1 − Kp˜
n ∗

(5.100)

2

M = M + βH ∆t K . •

Perform corrector step: p
n+1 = p˜
+ βH ∆t2 p¨
n+1

(5.101)

˜ p˙
n+1 = p˙
+ γH ∆t p¨
n+1 .

(5.102)

The question if one should solve for the acoustic pressure or acoustic velocity potential is a question of the boundary conditions.

5.4 Numerical Computation

•

163

Acoustic pressure: p
= p
e on Γ Clearly, in this case we will solve for p
.

•

Normal component of acoustic particle velocity: n · v
= vn
on Γ Due to the relation v
= −∇ψ we obtain n · v
= vn
= −

∂ψ , ∂n

and we will solve for ψ and compute p
according to (5.37). •

Mixed boundary conditions: p
= p
e on Γe ; n · v
= vn
on Γn In an acoustic pressure formulation the boundary conditions will read ( by use of (5.30)) p
= p
e on Γe ∂p
∂vn = −ρ0 on Γn , ∂n ∂t whereas for the acoustic velocity potential ψ we obtain (using (5.37)) ∂ψ = p
e on Γe ∂t ∂ψ = vn on Γn . − ∂n

ρ0

For the time harmonic case, we will always apply an acoustic pressure formulation with boundary conditions defined by pˆ = pˆe on Γe ∂ pˆ = −jωρ0 vˆn on Γn . ∂n 5.4.2 Non-linear Acoustics The strong formulation for the non-linear wave equation is given as follows Given: f : Ω → IR c : Ω → IR B/A, b : Ω → IR . ¯ × [0, T ] → IR Find: ψ(t) : Ω    2 B/A ∂ψ 1 ∂ 1 ∂2ψ 2 b(∆ψ) + − ∆ψ = f + 2 + (∇ψ) . c2 ∂t2 c ∂t 2c2 ∂t

(5.103)

164

5 Acoustic Field

Boundary conditions ψ = ψe on Γe × (0, T ) ∂ψ = ψn on Γn × (0, T ) . ∂n Initial condition ψ(r, 0) = ψ0 , r ∈ Ω ˙ 0) = ψ˙ 0 , r ∈ Ω . ψ(r, For simplicity we set all boundary conditions as well as initial conditions to zero. Now, to obtain the variational formulation of (5.103), we multiply (5.103) by an appropriate test function and perform partial integration for the Laplace term on the left- and right-hand sides. Thus, the variational formulation reads as follows: Find ψ ∈ H01 such that    1 ¨ w ψ dΩ + (∇w) · (∇ψ) dΩ = ωf dΩ c2 Ω Ω Ω    b ˙ dΩ − ∇ 2 w · (∇ψ) c Ω  2 B wψ˙ ψ¨ dΩ + c4 2A Ω    2 ˙ dΩ (5.104) + w ∇ 2 ψ · (∇ψ) c Ω

for any w ∈ H01 . In the following, we will assume that c and b are constant all over the computational domain. Using the same ansatz for the approximations as in the linear case (see (5.94)), we arrive at the following semidiscrete Galerkin formulation  neq neq    1 − N N dΩ ψ¨b 2 a b c a=1 b=1 Ω  − (∇Na ) · (∇Nb ) dΩ ψb Ω



b (∇Na ) · (∇Nb ) dΩ ψ˙ b c2 Ω  neq    2 B Na Nb + Nc ψ˙ c dΩ ψ¨b c4 2A c=1 −

Ω

5.4 Numerical Computation





neq  2 + Na (∇Nb ) · ∇ Nc ψc c2 c=1 Ω   + Na f dΩ = 0.

165

 dΩ ψ˙ b

(5.105)

Ω

In matrix form, we may write (5.105) as ˙ ψ¨ − N2 (ψ)ψ˙ = f , Mψ ψ¨ + Kψ ψ + Cψ ψ˙ − N1ψ (ψ) ψ

(5.106)

with

Mψ =



ne  e=1

Kψ =

ne 

keψ ; keψ = [kpq ] ; kpq =

(5.108)

Ωe



ne 

b (∇Np ) · (∇Nq ) dΩ (5.109) c2

ceψ ; ceψ = [cpq ] ; cpq =

ne 

Ωe

(n1ψ )e ; (n1ψ )e = [n1pq ] ;

e=1

n1pq



2 B Np Nq c4 2A

= Ωe

N2ψ

(∇Np ) · (∇Nq ) dΩ

Ωe

e=1

N1ψ =

(5.107)



e=1

Cψ =

1 Np Nq dΩ c2

meψ ; meψ = [mpq ] ; mpq =

=

ne 

n en 

 Nc ψ˙ c

(5.110)

(n2ψ )e ; (n2ψ )e = [n2pq ] ;

e=1

n2pq

 =

2 Np (∇Nq ) · c2

 ∇

Ωe

f =

dΩ

c=1

ne  e=1

nen 

 Nc ψc

dΩ

(5.111)

c=1



f e ; f e = [fp ] ; fp =

Np f (rp ) dΩ .

(5.112)

Ωe

The time discretization is performed by a standard Newmark algorithm (see Sect. 2.5.2). Since (5.106) is a non-linear equation, we have to apply an iterative scheme. By shifting all non-linearities to the right-hand side of the equation system, we arrive at the following scheme:

166

5 Acoustic Field

1. Perform predictor step: 2 ¨ = ψk ˜ = ψ + ∆t ψ˙ + ∆t (1 − 2βH ) ψ ψ n+1 n n n 2 k ˜ ¨ = ψ˙ . ψ˙ = ψ˙ + (1 − γH )∆t ψ n

n

n+1

(5.113) (5.114)

2. Solve algebraic system of equations: k+1

M∗ψ ψ¨n+1 = f n+1

(5.115)

˜ − Cψ ψ˜˙ + N1 (ψ˙ k )ψ¨k + N2 (ψ k )ψ˙ k − Kψ ψ ψ ψ n+1 n+1 n+1 n+1 M∗ψ = Mψ + γH ∆t Cψ + βH ∆t2 Kψ .

(5.116)

3. Perform corrector step: ¨ k+1 ˜ + βH ∆t2 ψ ψ k+1 =ψ n+1 n+1

(5.117)

k+1 ˜ ¨ k+1 . ψ˙ n+1 = ψ˙ h + γH ∆t ψ n+1

(5.118)

4. Test convergence: k+1 k ψ¨n+1 − ψ¨n+1 2 k+1 ψ¨n+1 2

& ≤ δψ

fulfilled : perform next time step (5.119) not fulfilled : k := k + 1 goto step 2 .

5.4.3 Non-matching Grids Within this section, we will investigate flexible discretization techniques for the approximative solution of acoustic wave propagation problems. In particular, we will introduce non-matching grids, where we can allow totally different grids to exist at the interface of two subregions. This is in particular of great interest in computational aeroacoustics, where we need a much finer grid in the source region, where the turbulent flow generates the sound, than in the regions, where we just compute the propagation of sound (see Sect. 10). In order to keep as much flexibility as possible, we intend to use independently generated grids which are well suited for approximating the solution of decoupled local subproblems in each subdomain. Therefore, we have to deal with the situation of non-conforming grids appearing at the common interface of two subdomains. Special care has to be taken in order to define and implement the appropriate discrete coupling operators. For this type of problem, we can use the well-tested and well-studied framework of mortar methods (see e.g., [21, 217]). The global domain and its decomposition is displayed in Fig. 5.9. Thus, in each subdomain we have to solve the wave equation for the acoustic pressure p
i : Ωi × (0, T ) → IR,

5.4 Numerical Computation

167

Fig. 5.9. Acoustic domain with two subregions Ω1 and Ω2 with different discretizations

1 ¨
p − ∆ p
i = fi , c2 i

in Ωi × (0, T ), i = 1, 2.

(5.120)

completed by appropriate initial conditions at time t = 0 and boundary conditions on the global boundary Γf . Interface condition For simplicity, we use the same equation and primal variable in both subdomains, and the interface is just artificial, i.e., no material change occurs. We refer to [13, 136] for the treatment of more general situations. Therefore, in the strong setting, it is natural to impose continuity in the trace and flux of the acoustic pressure, i.e., p
1 = p
2 and

∂p
2 ∂p
1 = on Γ I . ∂n ∂n

The flux coupling condition will be enforced in a strong sense by introducing the Lagrange multiplier λ=−

∂p
∂p
1 =− 2. ∂n ∂n

(5.121)

However, the continuity in the trace will be understood in a weak sense  (p
1 − p
2 )µ dΓ = 0 (5.122) ΓI

for all test functions µ out of a suitable Lagrange multiplier space. We note that a correct functional framework is presented in [13]. Coupled formulation We proceed as in Sec. 5.4.1 and obtain from (5.120), ignoring for the moment the boundary condition on Γ f and the excitations fi ,

168

5 Acoustic Field

 Ωi

1 wi p¨
i dΩ + c2








∇wi · ∇p i dΩ − Ωi

wi ni ·∇p
i dΓ = 0

ΓI

for all test functions wi , i = 1, 2. Inserting the definition of the Lagrange multiplier (5.121) and summing up, we obtain the symmetric evolutionary saddle point problem of finding p
1 , p
2 , λ such that 2   i=1

Ωi

1 ¨
wi p i dΩ + c2

 

∇wi · ∇p
i dΩ



Ωi

+

(w1 − w2 )λ dΓ = 0

(5.123)

(p
1 − p
2 )µ dΓ = 0,

(5.124)

ΓI



ΓI

for all µ and wi , i = 1, 2. In contrast to the purely primal coupled formulation (8.9),(8.10), we now face a primal-dual problem where the coupling is realized in terms of Lagrange multipliers. Spatial discretization For the approximation of the acoustic pressures p
1 and p
2 , we may again use standard nodal finite elements on the triangulations T1 and T2 , respectively. Special attention has to be paid to the construction of a discrete Lagrange multiplier space Mh for approximating the interface flux λ. It is convenient to define Mh with respect to the (d−1)-dimensional grid inherited from one of the subdomains Ω1 , Ω2 . The interface grid Tn chosen in this way is referred to as non-mortar or slave side, and the other interface grid Tm as mortar or master side. There exist several possibilities for choosing the basis functions spanning Mh , most of them with equal mathematical stability and approximation properties. In Fig. 5.10, four types of basis functions for a 1D interface using linear or bilinear finite elements are presented: the standard ones coinciding with the trace space Wh of the finite element space on Tn , the dual

Fig. 5.10. Basis functions: standard, discontinuous dual, continuous dual, and constant Lagrange multipliers

ones spanned by piecewise linear discontinuous basis functions satisfying a biorthogonality relation with the nodal basis functions of Wh [217], the dual

5.4 Numerical Computation

169

continuous ones where the discontinuous dual basis functions are modified by cubic polynomials [218], and the piecewise constant ones spanned by basis functions, which are constant from one edge midpoint to the next. Again switching to the algebraic formulation and assuming that we have chosen the Lagrange multipliers with respect to Ω1 , the discretization of (5.123), (5.124) yields ⎛ ⎞⎛
⎞ ⎛ ⎞⎛
⎞ p¨ 1 M1 0 0 K1 0 D p1   fu ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜
⎟ ⎜ 0 M2 0 ⎟ ⎜ p¨
⎟ + ⎜ 0 K2 M ⎟ ⎜ p 2 ⎟ = . (5.125) ⎝ ⎠ ⎠⎝ 2⎠ ⎝ ⎠⎝ 0 T T ¨ 0 0 0 D M 0 λ λ The coupling matrices D, M are given by D=

nIe 

 De ; De = [Dpq ]; Dpq = Γ eI

e=1

Npm φnq dΓ,

(5.126)



Npm φnq dΓ,

M = [Mpq ]; Mpq =

(5.127)

ΓI

where Npn and Npm denote the finite element basis functions on Tn and Tm , respectively, and φnq denotes the Lagrange multiplier basis function associated with the slave node q. We note that the assembly of D poses no difficulty since all basis functions involved are defined with respect to the same grid Tn . Moreover, if the dual Lagrange multipliers are used, D becomes a diagonal matrix. However, the assembly of M is more involved, since Npm and φnq are defined with respect to different grids. We will discuss the necessary steps for a straight interfaces in 2D, and refer to [67] for a detailed description in 2D and 3D including curved interfaces. In order to determine whether two surface elements en and em intersect, m m m we loop over all vertices pm k , k = 1, . . . , nv , of e , where nv denotes the corresponding number of vertices. The two elements intersect, if we find at n least one vertex pm k which lies inside e . An easy way to justify this, is to use the transformation of global to local coordinates, which is usually available in any finite element code. If all local coordinates of pm k with respect to the element en are within the correct ranges, the elements en and em intersect. Otherwise, it may still be possible that the element en is completely covered by the element em . Therefore, one has to repeat the procedure interchanging the roles of en and em . If still no appropriate vertex is found, the two elements do not intersect. We consider the case that the interface Γ is part of the x-axis. The corner coordinates of en and em are denoted by xn and xm  with  = 1, 2, respectively. m We assume that xn1 < xn2 and xm 1 < x2 . The possible four situations and resulting intersections are illustrated in Fig. 5.11.

170

5 Acoustic Field

Fig. 5.11. Four situations of intersecting elements en and em in 2D.

5.4.4 Discretization Error In this section, we will investigate the discretization error of the FE method applied to the wave equation and will provide practical rules for choosing the mesh size h as a function of the wave number k. In the first step we will restrict to the time harmonic case to get rid off the error introduced by the time discretization. The wave equation in the time harmonic case, also called Helmholtz equation, reads as follows ∆ˆ p + k 2 pˆ = 0

(5.128)

with k = ω/c = 2πf /c the wave number. The comparison of the FE solution to the analytic solution of this kind of PDE has already been discussed in Sect. 2.2 (just substitute there the parameter c by k). There, we have observed that the FE solution for the parameter c = 0 is no longer exact at the FE nodes (see Fig. 2.7). In the context of the Helmholtz equation we decompose the total discretization error into a local error eint (interpolation error) defined by eint = pˆ − pˆI and into the pollution error epol computed by epol = pˆI − pˆh . This decomposition is illustrated in Fig. 5.12. A very important result concerning the error analysis has been reported in [106], which reads as follows ) * eh ≤ C1 (kh) + C2 k 3 h2 kh < 1 (5.129) eh =

|ˆ p − pˆh |H 1 |ˆ p|H 1

with | |H 1 the H 1 semi-norm (see Sect. C) and C1 , C2 constants independent of k as well as h. In (5.129) the first term represents the local error and the second term the pollution error. The main effect of the pollution error is that the wave number k h of the FE solution is different from the wave number k of the analytic solution. We call this dispersive and therefore speak from the dispersion error. The behavior of the dispersion error has been extensively studied in [3]. This analysis provides a concrete guideline for choosing the order q of the finite element basis functions and the mesh size h in order that the dispersion error is virtually eliminated. This relation is given by

5.4 Numerical Computation

171

^ p(x)

p^ p^ h p^ I epol eint x Fig. 5.12. Decomposition of the discretization error into the interpolation erroreint and pollution error epol

q+

kh 1 > + C(kh)1/3 2 2

(5.130)

with the constant C, which can in practice be set to one. In addition, we can precisely define the dispersion error in the small wave number limit kh << 1 and the high wave number limit kh >> 1  2 q! (kh)2q+2 1 h cos(k h) − cos(kh) = + O(kh)2q+4 kh << 1 2 (2q)! 2q + 1  2q+1 sin(kh) khe h kh >> 1 cos(k h) − cos(kh) = 2 2(q + 1) Let us consider the simple case of an acoustic wave in a channel with given Dirichlet boundary conditions on the left and right end (see Fig. 5.13). We

Fig. 5.13. Acoustic wave in a channel with given Dirichlet boundary conditions

keep the discretization fixed by 20 finite elements and change the wave number k (see Table 5.3) as well as the order q of the finite element basis functions.

172

5 Acoustic Field

Table 5.3. Choice of wave number k and resulting number of finite elements per wavelength

k λ/h

9π/2 19π/2 29π/2 8.9

4.2

2.7

For comparison we compute the accumulated error Ea 3 4 2 4 nn (ˆ ph − pˆ) 4 4 i=1 Ea = 4 nn  5 pˆ2 i=1

with nn the number of finite element nodes. The analytic solution pˆ computes by cos(kL) sin(kx) . pˆ = cos(kx) − sin(kL) Figure 5.14 clearly demonstrates that we effectively reduce the error by increasing the order q of the finite element basis functions. However, as shown by the case k = 29/2 π, we do not fulfill (5.130) at the mesh with q = 1 and so no reduction of the error is achieved by increasing the order of the finite element basis functions from q = 1 to q = 2. This clearly demonstrates, that (5.130) is a very important and practical guidline for acoustic computations.

Fig. 5.14. Accumulated error

In a second example, we consider a sine pulse travelling through a channel and record the wave signal at a distance of two times the fundamental wave-

5.5 Treatment of Open Domain Problems

173

length. Now, in addition to the correct choice of the mesh size h we have to deal with the time discretization. Figure 5.15 displays the setup. The choice of Propagation direction

Excitation: sine pulse Recording of wave signals

p´(t) t

Fig. 5.15. Setup of the transient case study

the time and spatial discretization is taken as a function of the fundamental exciting frequency f0 , which results in a fundamental wavelength λ0 = c/f0 . Table 5.4 lists the mesh size h as well as time step size ∆t for the three different computations. In Fig. 5.16 it can be clearly noticed that around the Table 5.4. Time and spatial discretization Discretization 1 Discretization 2 Discretization 3 h ∆t

1/(20f0 ) λ0 /20

1/(40f0 ) λ0 /40

1/(60f0 ) λ0 /60

expected pulse wave parasitic waves arise due to the numerical dispersion. In fact, the generated parasitic waves occur since the discrete speed of sound ch is a function of the wave number k and therefore of the frequency of the wave. Clearly, this is a pure numerical effect, since in a homogeneous media the continuous speed of sound c is constant. For a detailed discussion on this topic we refer to [43].

5.5 Treatment of Open Domain Problems Many applications within computational acoustics are open-domain problems. In order to correctly compute such problems with the FE method, we have to define appropriate boundary conditions. Using a simple homogeneous Dirichlet or Neumann boundary condition will result in a total reflection of the

174

5 Acoustic Field

Fig. 5.16. Recorded pulse signals at a distance of two times the fundamental wavelength

outgoing waves at the boundary. Therefore, special boundary conditions have to be developed for absorbing the waves impinging on the artificial boundary imposed on the acoustic domain. The three main approaches handling open-domain problems with the FE method can be briefly described as follows: •

Infinite finite elements: At the boundary an additional layer of finite elements, so-called infinite finite elements try to model the effect of waves in the outer region (noncomputational domain). For a basic discussion we refer to [166], and for an advanced formulation see [52].

•

Absorbing boundary conditions: The general idea of applying at the boundary a special condition, so-called absorbing boundary conditions, is that only outgoing waves can pass the boundary.

•

Perfectly matched layer (PML) technique: By this method the computational domain is surrounded by a damping layer, which is constructed in such a way that the acoustic impedance of this layer matches the impedance of the computational (wave propagation) domain and in which the waves are absorbed.

5.5 Treatment of Open Domain Problems

175

In the following, we will provide details about ideas and computer implementation of local absorbing boundary conditions of first order and of the perfectly matched layer technique. 5.5.1 Absorbing Boundary Conditions In [42,60] absorbing boundary conditions for general classes of wave equations are discussed. Subsequently, a hierarchy of highly absorbing local boundary conditions that approximate the theoretical non-local relation have been obtained. The locality of these conditions preserves the sparsity of the FE matrices. For an explanation of how absorbing boundary conditions work, let us consider a boundary located at x = 0. A wave with one degree of freedom

Fig. 5.17. Wave propagating in the direction of negative x values

propagating in the direction of negative x values can be expressed by u− (x, t) = u0 ej(ωt+kx)

(5.131)

with ω the angular frequency and k the wave number. By considering the relation for the speed of sound c = ω/k, it can be easily proved that the wave fulfills the following condition   ∂ ∂ −c u = 0. (5.132) ∂t ∂x Thus, the wave can totally pass (no reflection) the boundary. However, a wave, which is propagating towards positive x values and that can be mathematically modeled by (5.133) u+ (x, t) = u0 ej(ωt−kx) will not fulfill (5.132) and will be totally reflected. Indeed, (5.132) defines a locally absorbing boundary condition of first order (see [42,60]). An extension

176

5 Acoustic Field

of this method to waves in solids (especially piezoelectric materials) can be found in [93]. To derive the correct formulation including the absorbing boundary condition as given in (5.132), we start from the weak form (see (5.93)) without setting the boundary integral to zero (which would correspond to homogeneous Neumann boundary condition)     1 ∂p


dΓ − w p¨ dΩ + ∇w · ∇p dΩ − w wf dΩ = 0 . (5.134) c2 ∂n Ω

Ω

Γ

Ω

Now, in all directions we can express the 1D spatial derivative in (5.132) as the normal derivative in (5.134). By doing so, special attention has to be given to the direction of the normal vector relative to the positive direction of the x axis. In our case, the normal points out of the boundary towards the infinite domain that is in the same direction of the outgoing wave, whereas x is positive in the direction opposite to the propagation of the normal wave. Finally, changing the direction of the normal vector and substituting (5.132) into (5.134) results in     1 1 ∂p


w p¨ dΩ + ∇w · ∇p dΩ + w dΓ − wf dΩ = 0 . (5.135) c2 c ∂t Ω

Ω

Γ

Ω

The additional surface integral, including a first time derivative of the acoustic pressure p
, may be seen as a damping matrix C acting only on the surface of the computational domain. After performing the spatial discretization using finite elements, we obtain the following formula for the computation of C 

Γ

C=

ne  e=1

e

c

e

; c = [cpq ] ; cpq =

1 Np Nq dΓ , c

(5.136)

Γe

with nΓe the number of surface elements (for the evaluation of the surface integral see Sect. 2.6). The matrix C is almost empty, since only terms along the boundary Γ of the domain contribute to its non-zero entries. 5.5.2 Perfectly Matched Layer (PML) Technique There has been much research work on the PML-technique since the first introduction of this technique [20] (see e.g., [45,85,97,200]). In this section we will focus on the acoustic wave propagation in the frequency domain and invest in basic ideas concerning impedance matching, describe the PML-technique and prove the perfect matching. For linear acoustics in the 1D case the relation between the acoustic particle velocity v
= vξ eξ and the acoustic pressure p
is given by (conservation of mass and momentum, see Sect. 5.1)

5.5 Treatment of Open Domain Problems

∂vξ
∂p
= −ρ0 c2 ∂t ∂ξ

177

∂vξ
1 ∂p
=− , ∂t ρ0 ∂ξ

which results in the linear wave equation for p
1 ∂ 2 p
∂ 2 p
− = 0. c2 ∂t2 ∂ξ 2

(5.137)

For a plane wave the acoustic impedance Za is calculated by ρ0 c, and the reflection coefficient R between two media as displayed in Fig. 5.18 computes as Za2 − Za1 R= with Za1 = ρ0 c Za2 = ρ˜c˜ . (5.138) Za2 + Za1 Clearly, to make R equal to zero, Za2 has to match Za1 . However, we are free

Fig. 5.18. Plane wave impinging perpendicular (left) and at angle ϕ1 to the interface (right)

in choosing the two quantities ρ˜ and c˜ in the damping region in such a way that just their product equals ρ0 c. Therefore, by setting ρ˜ = ρ0 (1 − jσξ )

c˜ =

c 1 − jσξ

(5.139)

in the damping region, we obtain for the acoustic impedance Za2 = Za1 . Furthermore, the wave equation (5.137) reads in the time-harmonic case as ∂ 2 pˆ ω2 p ˆ + =0 c˜2 ∂ξ 2

(5.140)

with ω the pulsation of the wave and pˆ the acoustic pressure amplitude. Without any restrictions, we choose σξ = σ0 (constant within the damping region) and obtain ω2 ∂ 2 pˆ 2 (1 − jσ ) p ˆ + = 0. (5.141) 0 2 ∂ξ 2 c   ˜2 k

Therewith, we arrive at a wave equation with a complex wave number k˜

178

5 Acoustic Field

ω k˜ = (1 − jσ0 ) = k(1 − jσ0 ) . c

(5.142)

Since the general solution of (5.141) is pˆ = pˆ0 ej(ωt−kξ) = pˆ0 ej(ωt−kξ) e−σ0 ξ , ˜

we achieve our goal that we have impedance matching and damping of the wave at the same time. However, if we consider a plane wave impinging at an angle ϕ to the normal vector of the interface (see Fig. 5.18), then the acoustic impedances compute by (see e.g., [170]) Za1 =

ρ0 c cos ϕ1

Za2 =

ρ˜c˜ cos ϕ2

and our method will not be working. Therefore, as introduced in [20] we need a splitting of the acoustic pressure p
into p
x , p
y and p
z and introduce artificial damping. Therewith, the mass as well as momentum conservation equation for linear acoustics change to ∂v
∂p
x + σx p
x = −ρ0 c2 x ∂t ∂x
∂p
y ∂v y + σy p
y = −ρ0 c2 ∂t ∂y
∂p
z ∂v + σz p
z = −ρ0 c2 z ∂t ∂z

1 ∂p
∂vx
+ σx vx
= − ∂t ρ0 ∂x ∂vy
1 ∂p
+ σy vy
= − ∂t ρ0 ∂y ∂vz
1 ∂p
+ σz vz
= − ∂t ρ0 ∂z

(5.143) (5.144) (5.145)

In the above equations σx , σy and σz are damping functions, which are zero within the acoustic propagation domain and which are different from zero within the PML-layer enclosing the acoustic propagation domain (see Fig. 5.19). By applying a Fourier-transformation to (5.143) - (5.145) and rear-

Fig. 5.19. Computational setup: propagation region surrounded by a PML-region

ranging the involved terms, we arrive at the following equations

5.5 Treatment of Open Domain Problems

1 jω + σx 1 pˆy = −ρ0 c2 jω + σy 1 pˆz = −ρ0 c2 jω + σz

pˆx = −ρ0 c2

∂ˆ vx 1 ∂ pˆx 1 vˆx = − ∂x ρ0 jω + σx ∂x ∂ˆ vy 1 ∂ pˆy 1 vˆy = − ∂x ρ0 jω + σy ∂y ∂ˆ vz 1 ∂ pˆz 1 vˆz = − . ∂z ρ0 jω + σz ∂z

179

(5.146) (5.147) (5.148)

As can be seen from (5.146) - (5.148), we can directly compute vˆx , vˆy and vˆz as a function of pˆx , pˆy and pˆz . Therewith, we obtain for the total acoustic pressure pˆ = pˆx + pˆy + pˆz the following modified Helmholtz equation       ∂ ∂ ∂ 1 ∂ pˆ 1 ∂ pˆ 1 ∂ pˆ ηy ηz + ηx ηz + ηx ηy ∂x ηx ∂x ∂y ηy ∂y ∂z ηz ∂z + ηx ηy ηz k 2 pˆ = 0 .

(5.149)

In (5.149) k = ω/c denotes the acoustic wave number and the functions η1 , η2 and η3 compute as follows ηx = 1 − j

σx ω

ηy = 1 − j

σy ω

ηz = 1 − j

σz . ω

(5.150)

This decomposition of the acoustic pressure p
into its x-component, ycomponent and z-component is the key point of the PML-technique. For a physical interpretation we will consider an interface between a propagation and a PML region, which is parallel to the y-axis and has a normal vector pointing in x-direction (see Fig. 5.20). Now, the x-component p
x of the to-

Fig. 5.20. Physical interpretation of PML-technique

tal acoustic pressure p
can be regarded as a plane wave propagating just in x-direction and which will be damped in the PML-region with the damping coefficient σx . Therewith, we have the 1D case and achieve our goal of a reflectionless interface, if simply the impedances matches. This simply consideration already provides us with the information, where to choose the individual damping coefficients σx , σy and σz different from zero. Figures 5.21 and 5.22 display the PML-technique for the 2D and 3D case.

180

5 Acoustic Field

Fig. 5.21. Construction of PML layer in 2D

Fig. 5.22. Construction of PML layer in 3D

At this point we want to note that there is a second general method to derive (5.149). This is known as a mapping of the solution of Helmholtz equation in the real coordinate space to a complex coordinate space, e.g., an analytic continuation of the solution (see e.g., [45, 208]). Now, let us prove that the perfectly matched layer will have the same acoustic impedance as the propagation medium (a similar proof is given in [20] for electromagnetic waves). Since any general solution of the homogeneous wave equation can be expressed by the superposition of plane waves, we can restrict our investigation to plane waves impinging at an arbitrary angle ϕ on the interface, where the perfectly matched layer starts. Therewith, we consider a setup as displayed in Fig. 5.23. In this 2D consideration, the set of partial differential equations describing the plane wave in the layer are as follows

5.5 Treatment of Open Domain Problems

181

∂vy
∂vx
1 ∂
1 ∂
+ σx vx
= − (px + p
y ) + σy vy
= − (p + p
y ) (5.151) ∂t ρ0 ∂x ∂t ρ0 ∂y x ∂vy
∂v
∂p
y ∂p
x + σx p
x = −ρ0 c2 x + σy p
y = −ρ0 c2 . (5.152) ∂t ∂x ∂t ∂y Our goal is to compute the acoustic impedance in the perfectly matched layer. Since we assume a plane wave, we can perform the following ansatz for the

Fig. 5.23. Wave propagation in the PML region

general solutions of the components of the acoustic particle velocity and pressure vx
= v0 sin ϕ ej(ωt−kx x−ky y) vy
= v0 cos ϕ ej(ωt−kx x−ky y)

(5.153)

p
x = px0 e

(5.154)

˜

˜x x−k ˜y y) j(ωt−k

˜

˜

p
y = py0 e

˜x x−k ˜y y) j(ωt−k

˜

.

In the first step, we substitute these general solutions into (5.152) and obtain ρ0 c2 k˜x v0 sin ϕ ρ0 c2 v0 sin ϕ k˜x = jω + σx ηx ω 2˜ 2 ρ0 c ky v0 cos ϕ ρ0 c v0 cos ϕ k˜y . = = jω + σy ηy ω

px0 =

(5.155)

py0

(5.156)

Using this intermediate results in (5.151) will lead us to the following two relations   j k˜x c2 k˜x sin ϕ k˜y cos ϕ ηx sin ϕ = (5.157) + ω2 ηx ηy   j k˜y c2 k˜x sin ϕ k˜y cos ϕ ηy cos ϕ = + , (5.158) ω2 ηx ηy so that we can e.g., express k˜y by a function of k˜x ηy cos ϕ ˜ kx . k˜y = ηx sin ϕ

182

5 Acoustic Field

Substituting this result into (5.157) leads to a quadratic equation for k˜x ηx2 sin2 ϕ =

* k˜x2 c2 ) 2 sin ϕ + cos2 ϕ , 2 ω    =1

form which we just take the positive solution ' ω σx ( . k˜x = ηx sin ϕ = k sin ϕ 1 − j c ω

(5.159)

Performing similar steps results in the expression for k˜y ' ω σy ( . k˜y = ηy cos ϕ = k cos ϕ 1 − j c ω

(5.160)

Therewith, the substitutions of these terms into (5.155) and (5.156) lead to px0 = ρcv0 sin2 ϕ

py0 = ρcv0 cos2 ϕ ,

so that the total pressure computes as ) * px0 + py0 = ρcv0 sin2 ϕ + cos2 ϕ = ρcv0 . The acoustic impedance of the perfectly matched layer now results in the same expression as for a plane wave in the propagation media Za =

p = ρ0 c . v

The proper choice of the damping functions is of great importance, especially in order to obtain a very robust and efficient PML-technique. For this purpose, let us consider the case, in which a wave is propagating within the PML-layer in y-direction and having an angle of ϕ with respect to the y-axis. The total pressure p
computes as p
= p
x + p
y = px0 ej(ωt−kx x−ky y) + py0 ej(ωt−kx x−ky y) . ˜

˜

˜

˜

(5.161)

Substituting the values for k˜x , k˜y according to (5.159), (5.160) and considering that for the chosen case σx = 0, we obtain p
= (px0 + py0 ) ej(ωt−kx sin ϕ−ky cos ϕ) e−(σy /c) cos ϕ = p0 e−(σy /c) cos ϕ . (5.162) Assuming a layer thickness of L, the damped wave will be totally reflected at the outer boundary of the PML-region and this reflected wave at the interface between propagation and PML region takes the value p
r

= p0 e

−(2/c) cos ϕ

RL o

σy (y)dy

= p0 R .

(5.163)

5.5 Treatment of Open Domain Problems

183

A reasonable choice of the reflection factor R is 10−3 , since we have to take care that a too strong damping in a too small PML-region can strongly disturb the numerical solution. In addition, in order to get rid of the dependence of the overall damping on the speed of sound c, we will choose all damping functions σy direct proportional to c (see (5.163)). In a first case, we will assume a constant damping σy = σ0 . Therewith, we obtain from (5.163) the following relation for σ0 σ0 =

−c ln R . 2L cos ϕ

(5.164)

In a second case, we consider a quadratically increasing damping function, hence we set y2 σy = σ0q 2 , L and assume that y is equal to zero at the interface and is increasing within the PML-region. Again, exploiting (5.163) we arrive at a relation for the constant factor σ0q 3c ln R . (5.165) σ0q = 2L cos ϕ In the last step, we will introduce a singular function, given by σy =

c , L−y

(5.166)

, which means that we increase the damping inverse with the distance. We want to emphasize that the damping functions according to (5.164) as well as (5.166) are different from zero at the interface, and therefore will introduce a discontinuity at the interface. However, due to the properties of the PML-technique, no spurious reflections will occur. For the FE formulation of the wave equation within a PML region, we start at the strong setting, which reads as follows. Given: fˆ : Ω → C I c, ρ : Ω → C I. Find: pˆ ∂ ηy ηz ∂x



1 ∂ pˆ ηx ∂x



∂ + ηx ηz ∂y



¯ →C :Ω I

1 ∂ pˆ ηy ∂y



∂ + ηx ηy ∂z + ω2



1 ∂ pˆ ηz ∂z



ηx ηy ηz pˆ − fˆ = 0 . (5.167) c2

184

5 Acoustic Field

Boundary conditions ∂ pˆ = 0 on Γ ∂n

(5.168)

In (5.167) and (5.168) fˆ denotes any acoustic source term, C I the set of complex numbers, Γ the outer boundary of the computational domain and ηx , ηy , ηz the damping functions, which compute according to (5.150). In the first step, we multiply (5.167) by an appropriate test function v and integrate over the whole domain Ω         ∂ ∂ ∂ 1 ∂ pˆ 1 ∂ pˆ 1 ∂ pˆ v ηy ηz + ηx ηz + ηx ηy ∂x ηx ∂x ∂y ηy ∂y ∂z ηz ∂z Ω

+ ω2

( ηx ηy ηz ˆ dΩ = 0 . p ˆ − f c2

(5.169)

Now, we apply Green’s integral theorem to the second-order spatial derivatives and incorporate the homogeneous Neumann boundary condition (5.168). These steps lead to the following weak formulation of (5.168): Find pˆ
∈ H 1 such that   ηy ηz ∂v ∂ pˆ ηx ηz ∂v ∂ pˆ ηx ηy ∂v ∂ pˆ ηx ηy ηz + + dΩ − ω 2 v pˆ dΩ ηx ∂x ∂x ηy ∂y ∂y ηz ∂z ∂z c2 Ω  Ω (5.170) = v fˆ dΩ Ω

for any v ∈ H 1 . Using standard nodal finite elements, we arrive at the following discrete complex algebraic system of equations ) * K − ω 2 M pˆ = fˆ (5.171) with K the stiffness matrix, M the mass matrix, pˆ the nodal vector of complex acoustic pressure and fˆ the complex nodal vector of the right-hand side. For a stability investigation we refer, e.g., to [17, 31].

5.6 Numerical Examples 5.6.1 Transient Wave Propagation in Unbounded Domain Let us consider a 2D case, where a wave is excited by a 10-kHz sine-signal for a specific time (four periods) at the beginning of the computation (see Fig. 5.24). The configuration allows us to evaluate the performance of the proposed absorbing boundary method. Figure 5.25 shows simulation results for the model described above, as a sequence of snapshots of the acoustic

5.6 Numerical Examples

185

Fig. 5.24. 2D plane case consisting of a plate excited with a 10-kHz sinusoidal pressure load (r = 375 mm, r/λ ≈ 10)

Fig. 5.25. Acoustic pressure distribution at different times showing the effect of absorbing boundary conditions

pressure pulse for different times. In Fig. 5.25, the acoustic pressure obtained when using first-order absorbing boundary conditions is presented. At a time of t = 0.44 ms the plate at the bottom is still being excited, and the sinusoidal generated wave can be observed. At a time of t = 0.96 ms, the results start to show spurious reflections and at t = 0.104 ms, the effect of these spurious reflections has propagated back into the acoustic domain. However, the amplitudes of these reflected waves are small. For a single point fixed in space (located at (x, y) = (0 mm, 115 mm)), Fig. 5.26 displays the pressure signal as a function of time. At this point, we want to note, that with this proposed method, we can not limit the computational domain in the acoustic near field. For such cases, the order of the absorbing boundary condition has to be increased. For latest research on this topic, we refer to [76].

186

5 Acoustic Field

Fig. 5.26. Acoustic pressure as a function of time using first-order absorbing boundary conditions for a fixed point located at (x, y) = (0 mm, 115 mm)

5.6.2 Harmonic Wave Propagation in Unbounded Domain In order to evaluate the PML method, we perform numerical simulations of cylindrical waves and compare it with analytic results. As displayed in Fig. 5.27, we prescribe the acoustic pressure on the surface of a cylinder and compute the radiated sound field. Although the problem is 2D, we will perform

Fig. 5.27. Computational setup: propagation region surrounded by a PML-region

a full 3D computation. The dimension of the propagation domain is fixed by λ/2 and the PMLthickness L is varied from L = λ/2 over λ/4 to λ/8 (see Fig. 5.28). Further-

5.6 Numerical Examples

187

more, we perform a discretization, where we use 20, 40, 60 and 80 bilinear hexahedron elements per wavelength λ. The analytical solution is given by

L = λ/8

L = λ/4

L = λ/2

Fig. 5.28. Coarsest grids (h ≈ λ/20) of the three different setups concerning the thickness of the PML

(2)

pˆ = pˆ(R0 )

H0 (kr) (2)

(5.172)

H0 (kR0 )

(2)

with H0 the Hankel function, r = (x, y) the position vector, k the wave number, and R0 the radius, where we define the excitation pˆ(R0 ) = 1. We compute the relative accumulated error (sum over all finite element nodes N in the propagation region) as follows 3 4 N 4 h 4 pi − pˆi )2 4 i=1 (ˆ 4 Etotal = 4 100% . (5.173) N  5 2 pˆi i=1

In Fig. 5.29, we display the iso-lines for the three cases having different PML-thickness L and which was computed on the coarsest grid (mesh size h ≈ λ/20). Table 5.5, lists the computed error according to (5.173). It has to be noted that in addition to the inexact approximation of the free radiation condition by the PML, we measure with (5.173) the overall error (see e.g., [3,43,106,107]). However, since this is a practical case, we are interested in the overall error of our numerical scheme. As can be seen from Table 5.5, we achieve a very consistent reduction of the error, by decreasing the mesh size h as well as increasing the layer thickness. The great superiority of the inverse distance damping function compared to the constant as well as quadratically increasing one can be clearly seen, when studying the error Etotal in Table 5.5 as well as the iso-lines in Fig. 5.30. As a rule of thumb, we can summarize that one should use the same mesh size in the PML-region as used in the propagation

188

5 Acoustic Field

L = λ/2

L = λ/4

L = λ/8

Fig. 5.29. Iso-lines on the coarsest grid (h ≈ λ/20) of the three setups concerning the thickness of the PML (inverse distance weighted damping function) Table 5.5. Computational error Etotal for different mesh sizes h and different PMLthickness L

h

Damping constant L = λ/8 L = λ/4 L = λ/2

Damping quadratically increasing L = λ/8 L = λ/4 L = λ/2

λ/20

2.9712

1.3760

0.4919

2.8872

1.1259

0.3685

λ/40

1.3537

0.5378

0.1817

1.0470

0.2376

0.1238

λ/60

0.9302

0.3329

0.1065

0.3721

0.0776

0.0880

λ/80

0.6163

0.2338

0.0982

0.2152

0.0413

0.0787

h

Damping inverse with distance L = λ/8 L = λ/4 L = λ/2

λ/20

0.5576

0.4137

0.3174

λ/40

0.3139

0.1119

0.0773

λ/60

0.1580

0.0557

0.0409

λ/80

0.1159

0.0374

0.0232

region and the mesh should have at least 10 bilinear finite elements within the thickness L of the PML-layer. 5.6.3 Acoustic Pulse Propagation over a Non-Matching Grid To show the capability of non-matching grids, we will investigate the numerical simulation of a single acoustic spherical pulse of frequency 1000 Hz and magnitude 1, propagating over an interface with quite different spatial discretizations. The computational domain in the (r, z)-plane is shown in the left picture of Fig. 5.31. The pulse is imposed in form of an essential boundary condition on ΓD . We assume that the grid on Ω1 has to be substantially finer than that required by the acoustic wavelength. Therefore, we use

5.6 Numerical Examples

Constant damping

Quadratic damping

189

Inverse damping

Fig. 5.30. Iso-lines of the solution for a PML-thickness of λ/4 and the coarsest grid (h ≈ λ/20) using the three different damping functions

Fig. 5.31. Spherical pulse. Axisymmetric computational domain (dimensions in meters)

2·40·40 = 3200 elements on Ω1 , as depicted in the right picture of Fig. 5.31. In order to compare the conforming method with the non-conforming one, we take 6400 elements on Ω2 in both cases, choosing nx = 80, ny = 40 for the conforming and nx = 160, ny = 20 for the non-conforming case. In Fig. 5.32, the isolines for the velocity potential at time t = 1.6 ms are visualized. Whereas the conforming method exhibits numerical noise before and behind the pulse, the non-conforming approach is much closer to the expected solution. Inside Ω1 , the radial symmetry of the isolines from the conforming method is observably disturbed. The poor quality of the conforming method can be easily explained by the fact that in order to obtain matching interface grids, the mesh on Ω2 is simply too coarse to correctly resolve the solution. In order to examine the transient behavior more closely, we visualize the evolution of the acoustic potential at the point (0, 0.1 m) in Fig. 5.33. In addition to the comparison of both approaches, we employ a reference solution, obtained with a uniformly fine grid of 54500 elements. As expected from the observations above, the conforming approach exhibits quite strong oscillations, while the behavior of the non-conforming method is quite smooth and visually coinciding with the reference solution. We remark that even the reference solution is subject to small unphysical oscillations directly after the pulse.

190

5 Acoustic Field

Fig. 5.32. Isolines of the acoustic velocity potential at time t = 1.6 ms. Matching (left) and non-matching grids (right)

Fig. 5.33. Evolution of the acoustic potential at the point (0, 0.1)T .

5.6.4 Non-linear Wave Propagation in a Channel Let us consider a plane wave in a semi-infinite channel as displayed in Fig. 5.34. The excitation is performed by a rigid-body motion on the left side, so

5.6 Numerical Examples

191

Fig. 5.34. Simulation model

that we obtain a defined acoustic particle velocity. The prescribed mechanical motion follows a sine-burst signal as displayed in Fig. 5.35 with a frequency of 100 kHz and an amplitude of 100 µm. For the medium we use water with

Fig. 5.35. Signal form used for excitation (mechanical motion)

a parameter of non-linearity B/A equal to 5 and a diffusivity of sound value b = 6 · 10−9 m2 /s. Now, as a function of the distance x from the source (vibrating mechanical solid body), analytical solutions exist (see e.g., [25]). Introducing the dimensionless distance σ as σ = x/¯ x with x ¯ the shock formation distance, we obtain the following two formulas •

σ < 1.0: Fubini solution v(x, t) = v0

 ∞

 2 Jn (nσ) sin[n(ωt − kx)] , nσ n=0

(5.174)

with v(0, t) = v0 sin(ωt) .

(5.175)

192

5 Acoustic Field

In (5.174) v0 stands for the particle velocity, k = ω/c0 the wave number, c0 the speed of sound, Jn for the Bessel function of order n, and x the coordinate for the propagation direction. •

σ > 3.0: Fay solution v(x, t) = v0

∞

 n=0

 2/Γ sin[n(ωt − kx)] , sinh[n(1 + σ)/Γ ]

with Γ =

(5.176)

B/A . b

The shock formation distance x ¯ itself computes for a harmonic excitation with velocity amplitude v0 and the fundamental wavelength λ as [94] x ¯=

λ2 . 4π 2 v0 (1 + B/(2A))

(5.177)

For the acoustic part, we specify homogeneous Neumann boundary conditions on the whole boundary. For the mechanical solid body, we prescribe the mechanical displacement for all FE nodes with an amplitude of 100 µm, which varies with time according to the sine-burst signal. The FE mesh consists of linear quadrilateral elements with just one finite element for the width of the channel and 250 finite elements per fundamental wavelength for the length of the channel. In order to also resolve higher harmonics, which will arise due to the non-linearity, this fine mesh has been used. For the time discretization a time step value of 20 ns is specified, which corresponds to 500 time samples per fundamental time period. Again, as for the spatial discretization, this small time step is necessary to be able to resolve higher harmonics. In Fig. 5.36 it is clearly demonstrated how the number of higher harmonics grows with the distance from the source. The comparison of the Fay solution with the numerical data is displayed in Fig. 5.37 for different distances. A detailed discussion on parameter variations (B/A, b, spatial and time discretization), especially how they influence the results, is given in [94].

5.6 Numerical Examples

σ = 0.2

σ = 0.2

σ = 0.6

σ = 0.6

σ=1

σ=1

193

Fig. 5.36. Time signal and frequency spectra of Fubini solution and numerical computation for different distances

194

5 Acoustic Field

σ=3

σ=4

σ=5

σ=6

σ=7

σ=8

Fig. 5.37. Time signal of Fay solution for distances between σ = 3 and 8

6 Coupled Electrostatic-Mechanical Systems

In electrostatic-mechanical systems, the structure is subject both to rigid motions and elastic deformations by means of the electrostatic force. Thus, the change on the geometry in turn may strongly influence the electric field and thus the electrostatic force distribution. A typical application is a micromachined pump [224], shown in Fig. 6.1. If an electric voltage is applied to

Fig. 6.1. Schematic view of an electrostatically driven micropump [224]

the electrodes, the elastic pump diaphragm is deformed by the electrostatic force and bends towards the counterelectrode. Thereby, fluid will be sucked in through the inlet valve. When the supply voltage is switched off, the relaxation of the diaphragm will push the fluid through the outlet valve.

196

6 Coupled Electrostatic-Mechanical Systems

6.1 Electrostatic Force An elegant method for deriving a general formula for the electrostatic force Fel is obtained by applying the principle of virtual work (displacement). Thus, a virtual displacement δr of a body due to a force Fel will result in a variation of the electrostatic energy δWel δWel = Fel · δr .

(6.1)

The electrostatic energy density wel is computed by the scalar product of the electric field intensity E and the electric flux density D wel =

1 D · E. 2

(6.2)

By using the constitutive law D = εE (assuming constant ε), we obtain the following expression 1 ε (6.3) wel = E · E = D · D . 2 2ε The total electrostatic energy Wel is computed by integration of the energy density wel over the volume Ω  (6.4) Wel = wel dΩ . Ω

Now, the variation of the electrostatic energy δWel computes as ⎛ ⎞  δWel = δ ⎝ wel dΩ ⎠ Ω

  =

D · δD D2 δε − ε 2ε2

 dΩ ,

(6.5)

Ω

with D = |D|. By using the scalar electric potential Ve (E = −∇Ve ), the first term in (6.5) can be rewritten as   D · δD dΩ = − (∇Ve · δD) dΩ . (6.6) ε Ω

Ω

Furthermore, the vector identity ∇ · (Ve δD) = Ve ∇ · δD + δD · ∇Ve and ∇ · δD = δqe (variation of the free charges) lead to    − (∇Ve · δD) dΩ = (Ve δqe ) dΩ − ∇ · (Ve δD) dΩ . Ω

Ω

Ω

(6.7)

6.1 Electrostatic Force

197

The use of the divergence theorem (see Appendix B.6) transforms the third integral in a surface integral over Γ (Ω)   ∇ · (Ve δD) dΩ = Ve δD · dΓ . (6.8) Ω

Γ (Ω)

Now, we know that the electric potential Ve is proportional to 1/r (with r denoting the distance), and therefore, the electric flux density D is proportional to 1/r2 [221]. Since the surface only can grow proportional to r2 , the surface integral will tend to zero for increasing Ω. The variation of the volume charge δqe is equal to the scalar product of the virtual displacement δr and the gradient of qe δqe = δr · ∇qe . In addition, we can use the relation ∇(Ve qe ) = qe ∇Ve + Ve ∇qe in order to modify the second term in (6.7)   Ve δqe dΩ = δr · (Ve ∇qe ) dΩ Ω

Ω

   = δr · ∇(Ve qe ) − qe ∇Ve dΩ Ω

⎛

⎜ = δr · ⎝



-

Γ (Ω)

Ve qe dΓ −

⎞ ⎟ qe ∇Ve dΩ ⎠ .

(6.9)

Ω

The surface integral in (6.9) will tend to zero for increasing volume Ω for the same reason as discussed above. Therefore, the differential energy δWel computes by using (6.5), (6.9), and the relation δε = δr · ∇ε   1 D2 δWel = δr · qe E dΩ − δε dΩ 2 ε2 Ω Ω ⎞ ⎛    2 E ∇ε dΩ ⎠ . (6.10) = δr · ⎝ qe E − 2 Ω

Comparing the above expression with (6.1), we obtain the equation for the total electrostatic force Fel    E2 ∇ε dΩ , (6.11) Fel = qe E − 2 Ω

198

6 Coupled Electrostatic-Mechanical Systems

and for the electrostatic volume force fΩel fΩel = qe E −

E2 ∇ε . 2

(6.12)

The first term models the electric force acting on electric volume charges qe in an electric field E. The second term exhibits a force at surfaces of changing permittivities, and therefore, we are looking for a surface force expression. Let us assume a small volume ∆Ω in form of a cylinder with height h, which is located between the two materials with permittivities ε1 and ε2 (see Fig. 6.2). Having no volume charge qe within ∆Ω, we can compute the electrostatic

Fig. 6.2. Interface between two materials with different permittivity

force ∆F as follows  ∆F =

r2 fΩel dΩ = ∆Γ

fΩel dr r1

∆Ω

r2 ∆Γ =− (E 2 ∇ε) dr . 2

(6.13)

r1

The scalar product of the force ∆F with the normal vector n and the relation dε = dr · ∇ε = n · dr ∇ε, lead to ∆Γ n · ∆F = − 2 ∆F12 = −

∆Γ 2

r2 E2

dε dr dr

r1 ε2

E 2 dε .

(6.14)

ε1

Now, we know that at an interface of changing permittivity the normal component of electric flux density and the tangential component of electric field

6.1 Electrostatic Force

199

intensity are continuous (see Sect. 4.5.2). With the help of the normal vector n and the tangential vector t, we can decompose E and D as follows E = (E · t) t + (E · n) n = Et + En D = (D · t) t + (D · n) n = Dt + Dn . These relations allow us to write the term E 2 = E · E in the form     Dn Dn D2 + Et · + Et = 2n + Et2 . E·E= ε ε ε The above result in combination with (6.14) leads to the term for the surface force fΓ el in the direction of n (∆F12 → dF12 , ∆Γ → dΓ) ⎛ ⎞ ε2 ε2 dε 1 dF12 = − ⎝Dn2 fΓ el = + Et2 dε⎠ dΓ 2 ε2 ε1 ε1   1 Dn2 1 Et2 (ε1 − ε2 ) − (6.15) = + 2 ε2 ε1 2 fΓ el = fΓ el n . Now, within the FE computation we can evaluate (6.15) to obtain at each FE node on the boundary the corresponding electrostatic force. However, the more accurate and general approach is to use the principle of virtual work and apply it to the FE formulation. According to (6.1), we can calculate the force in a direction r by  1 δ Fr = E · D dΩ . (6.16) δr 2 Ω

Let us consider the setup as displayed in Fig. 6.3, where a virtual displacement of the micromechanical cantilever will just lead to a deformation of the surrounding finite elements in the air gap and thus in a change of the electrostatic energy just within these elements. Therewith, the force Fr can be expressed by Fr =

=

  ne   δ 0 E · E dΩ δr Ωe 2 e=1 ne   n=1

δE dΩ + 0 E δr Ωi

 Ωe

0 δ( dΩ) E·E 2 δr

 .

(6.17)

Here the first term takes account of the changing electric field within the domain, whereas the second term deals with the geometric distortion of the element.

200

6 Coupled Electrostatic-Mechanical Systems

Fig. 6.3. Micromechanical cantilever with FE discretization; nodes with a point may move during a virtual deformation and nodes with a cross are fixed

Now, since in the coupling to the mechanical field, we are interested in the local force, we may rewrite (6.17) for the total force acting on a node i located at the surface of the electromechanical cantilever ne   i

Fri

=

e=1

δE dΩ + 0 E δr Ωe

 Ωi

δ( dΩ) 0 E·E 2 δr

 .

(6.18)

In (6.18) nie is the number of finite elements to which node i belongs (see Fig. 6.3). In the next step, we will now transform the integration over the global space of the finite elements to an integration over the reference elements. Thus, we substitute dΩ by |J| dξ dη dζ with |J| the determinant of the Jacobi matrix J ne   i

Fri

=

e=1

δE |J| dξ dη dζ + 0 E · δr Ωe

 Ωi

δ|J| 0 E·E dξ dη dζ 2 δr

 . (6.19)

After the solution process of the electrostatic field, we obtain in all FE nodes the electric potential Ve , form which we can compute the electric field intensity E = −∇Ve = −

nen 

∇Na (x)Vea

(6.20)

a=1

with nen the number of element nodes per finite element and Na the shape function at node a. Performing the computation at the reference element, we may rewrite (6.20) by (see Sec. 2.3.6) E = −(J T )−1

nen  a=1

∇Na (ξ)Vea .

(6.21)

6.1 Electrostatic Force

201

According to (6.19), we have to evaluate the variation of E with respect to the direction r, which computes by using (6.21) as nen δE δ(J T )−1  =− ∇Na (ξ)Ve a . δr δr a=1

(6.22)

Applying the variation to (J T )−1 will lead to quite complex terms. In order to arrive at a more convenient expression, we utilize the following relation (I denotes the unit matrix) δ(J−1 )T δJT −1 T δI δ(J−1 · J) = (J ) + JT = =0 δr δr δr δr δ(J−1 )T δ(J)T = −(J−1 )T (J−1 )T . δr δr Therewith, the variation of the electric field intensity E in the direction r can be computed from the FE results as follows δE δ(J)T = −(J−1 )T E. δr δr

(6.23)

Substituting this result into (6.19) allows us to directly compute the force acting on a node i by ne  

 δJT = 0 E (J ) E |J| dξ dη dζ . δr Ωe Ωe e=1 (6.24) For any integration point, the transposed Jacobian matrix computes as ⎛ n ⎞ n n en en en   ⎞ ⎛ Na,ξ xa Na,ξ ya Na,ξ za xξ xη xζ ⎜ ⎟ a=1 a=1 ⎟ ⎟ ⎜ a=1 ⎜ n n en en en   ⎜ yξ yη yζ ⎟ ⎜ n T Na,η xa Na,η ya Na,η za ⎟ J =⎜ ⎟, ⎟=⎜ ⎟ ⎠ ⎜ a=1 ⎝ a=1 a=1 ⎝ n ⎠ n n en en en zξ zη zζ   Na,ζ xa Na,ζ ya Na,ζ za i

Fri

δ|J| 0 E·E dξ dη dζ − 2 δr

a=1

and its variation with respect to δr ⎛ n en N δxa ⎜ a=1 a,ξ δr ⎜ n δJ T ⎜ en N δxa =⎜ a,η δr ⎜ a=1 δr ⎝ n en a Na,ζ δx δr a=1



−1 T

a=1

a=1

as n en  a=1 n en  a=1 n en  a=1

a Na,ξ δy δr a Na,η δy δr a Na,ζ δy δr

n en  a=1 n en  a=1 n en  a=1

a Na,ξ δz δr a Na,η δz δr a Na,ζ δz δr

⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎠

The term δxa /δr (δya /δr, δza /δr) obtains the value 1, if and only if node a is equal with node i, which belongs to the surface and is virtually moved, and

202

6 Coupled Electrostatic-Mechanical Systems

the variation is in the x-direction (y-direction, z-direction). In all other cases, we apply 0 to those terms. The last term in (6.24) to be evaluated, is the variation of the Jacobian determinant, which computes as follows δ|J| = δr

δxξ δr (yη zζ

− zη yζ ) +

δxη δr (yξ zζ

− zξ yζ ) +

δxζ δr (yξ zη

− zξ yη ) +

δyξ δyη δyζ δr (xη zζ − zη xζ ) + δr (xξ zζ − zξ xζ ) + δr (xξ zη − zξ xη ) + δzξ δzζ δzη δr (xη yζ − yη xζ ) + δr (xξ yζ − yξ xζ ) + δr (xξ yη − yξ xη ) .

6.2 Numerical Computation The challenges in the simulation of electrostatic-mechanical transducers can be summarized as follows. First, one has to deal with at least two different physical fields, usually the electrostatic and mechanical field. In many cases the transducer is immersed in an acoustic fluid (e.g., in ultrasound applications), and therefore, the acoustic wave propagation also has to be computed. In addition, all non-linearities of the involved physical fields have to be considered (e.g., geometric non-linearity within the mechanical field due to large displacements and/or strains). Furthermore, the coupling mechanisms between the physical fields have to be taken into account, which are highly non-linear in nature. Therefore, even in the case when all single fields are governed by linear partial differential equations, the coupled system is non-linear and has to be solved by an iterative method. 6.2.1 Calculation Scheme The FE formulation of the electrostatic field and the mechanical field has already been derived in Sect. 4.6 as well as Sect. 3.7. Using predictor values for the electric potential in order to compute the electrostatic force and predictor values for the mechanical displacement to update the configuration for the computation of the electrostatic field, we can split the coupled system of partial differential equations into a mechanical part and an electric part. To guarantee a full coupling between the two fields, we perform an iterative solution process. The resultant decoupled discrete matrix system of equations reads as follows    n+1    ∗ f u (Ve n+1 ) uk+1 0 Mu k . (6.25) = ) f V (un+1 Ve n+1 ) 0 KVe (un+1 k k k+1 e

M∗u

In (6.25) denotes the effective mechanical mass matrix, f u (Ve ) the nodal vector of external mechanical and electrostatic forces, u the nodal vector of mechanical displacement, f V (u) the electric source vector and Ve the nodal e

6.2 Numerical Computation

203

vector of scalar electric potential, n the time step, and k the iteration counter within each time step. Since the system matrix KVe and the nodal vector f V are updated by the mechanical displacement ukn+1 , we use a moving-mesh e technique to avoid large mesh deformations in the FE grid. Therefore, the entries of the system matrix KVe change as functions of ukn+1 . The solution algorithm of the fully coupled system of equations is displayed in Fig. 6.4. The outer iteration loop controls the iterative coupling process

(a) Linear mechanics

(b) Non-linear mechanics

Fig. 6.4. Coupled electrostatic–mechanical simulation schemes

between the mechanical and electrostatic equation, which is performed in the simplest case by a fixed-point method. The convergence test is based on the following stopping criterion n+1 ||2 ||un+1 k+1 − uk

||un+1 k+1 ||2

< δo ,

(6.26)

with u the nodal vector of mechanical displacements, δo an adjustable accuracy, || ||2 the L2 -norm, k the iteration counter for the outer loop (electrostatic-mechanical iteration) and n the time step number.

204

6 Coupled Electrostatic-Mechanical Systems

Due to a movement and/or deformation of the mechanical structures in electrostatic devices, the finite elements computing the electrostatic field will be deformed (see Fig. 6.5 for a simple voltage-driven bar), and thus will lead to numerical inaccuracy. These deformations have to be controlled, and, before an intersection of finite elements occurs (see Fig. 6.5), a remeshing of the simulation domain has to be performed. To avoid this problem, we use a

Fig. 6.5. Intersection of finite elements by using the standard method

special kind of moving-mesh technique. Therefore, the finite elements for the electric field are modified in order to be able to handle mechanical degrees of freedom, too. Of course, the mechanical stiffness of these elements has to be chosen very small compared to the stiffness of the bar. Now, the deformation of the bar is not only acting on the first layer of finite elements around the bar, but on all finite elements between the bar and the counterelectrode (see Fig. 6.6).

Fig. 6.6. No intersection of finite elements by using moving-mesh technique

Since the deformations of the mechanical structures especially in micromachined capacitive transducers are often large compared to the geometric dimensions, a non-linear formulation for the mechanical field has to be utilized. This leads to the calculation scheme as shown in Fig. 6.4(b). An inner iteration loop is introduced for the computation of the mechanical field. The stopping criterion for this loop should be based on an error and on a residual principle n+1 ||2 ||r n+1 ||un+1 j+1 − uj j+1 ||2 < δ < δir . (6.27) ia n+1 ||uj+1 ||2 ||f n+1 ||2 u

6.2 Numerical Computation

205

In (6.27) r denotes the nodal residual vector of the mechanical system, f V e the nodal vector of external mechanical and electrostatic forces, δia as well as δir adjustable accuracy values and j the iteration counter for the inner (mechanical) iteration loop. 6.2.2 Voltage-driven Bar

Fig. 6.7. Setup of the voltage-loaded bar

To investigate the behavior of electrostatic transducers, we will perform simulations of a voltage-loaded bar. Figure 6.7 displays the setup of a twosided clamped bar (for symmetry reasons just half of the bar is shown). The goal is to compute the deflection of the bar as a function of the applied voltage till snap-in occurs. To account for typical geometries of micromachined electrostatic transducers, we set the thickness of the bar to 1 µm, the length to 100 µm and the air-gap distance to 3 µm. For the discretization, quadrilateral elements with quadratic interpolation functions have been used. The thickness of the electrodes has been neglected. Therefore, all nodes belonging to the upper electrode are assigned inhomogeneous Dirichlet boundary conditions with a value according to the applied voltage. Furthermore, homogeneous Dirichlet boundary conditions are applied to all nodes belonging to the lower electrode. At the left end of the bar ux as well uy are set to zero to account for the full support and ux is set to zero at the right end of the bar to account for the symmetry. In the first step, we perform the simulation of the coupled electrostaticmechanical system with linear mechanics. The obtained results are displayed in Fig. 6.8. For this setup, it can be clearly seen that the mechanical field has to be computed by the non-linear formulation. Due to the strong deflection of the bar, inner stresses arise, which increase the stiffness of the bar. In a second computation, we have also modelled a prestressing of the bar. Due to the fabrication of micro-electromechanical systems (MEMS), inner stresses will remain within the structure after the fabrication process. We have assumed a value of 10 MPa, which is a typical value for such structures, and performed

206

6 Coupled Electrostatic-Mechanical Systems

Fig. 6.8. Tip displacement as a function of the applied electric voltage (linear mechanics with dashed line and non-linear mechanics with solid line) 0 Ŧ0.2

Tip displacement (Pm)

Ŧ0.4 Ŧ0.6 Ŧ0.8 Ŧ1 Ŧ1.2 Ŧ1.4 Ŧ1.6 Ŧ1.8 0

2

4

6 Voltage (V)

8

10

12

Fig. 6.9. Tip displacement as a function of the applied electric voltage using a non-linear mechanical formulation with (dashed line) and without (solid line) prestressing

a non-linear analysis for the mechanical field. As displayed in Fig. 6.9, the influence of the prestressing is quite strong. Now the voltage can reach values of about 10 V until snap-in occurs.

7 Coupled Magnetomechanical Systems

Let us consider a magnetic solenoid valve as depicted in Fig. 7.1. The valve

Fig. 7.1. Principal setup of a magnetic solenoid valve

consists of a ferromagnetic pot with a copper coil and a ferromagnetic armature that is held in its position by a compression spring. If the copper coil is loaded by a current pulse, a magnetic force on the armature is generated that causes a motion of the armature, when its amplitude is larger than the spring force. Due to the time-varying magnetic field, eddy currents are generated in the magnet pot as well as armature. As a consequence, a time delay between the applied current and the magnetic force will occur. Milling cuts into the magnet pot will reduce these eddy currents, and therefore will lead to a shorter switching time.

7.1 General Moving/Deforming Body Due to the movement of an electric conductive body in a magnetic field, the so-called motional emf term (electromotive force term)

208

7 Coupled Magnetomechanical Systems

γv × B ,

(7.1)

with γ the electric conductivity and v the velocity of the body has to be taken into account. The magnetic induction B can be expressed by the magnetic vector potential via B = ∇ × A resulting in the expression for the eddy current density Jv induced in an electrically conductive body Jv = γv × B = γv × ∇ × A .

(7.2)

The total partial differential equation in the quasistatic case including moving bodies reads as ∇ × ν∇ × A = Ji − γ

∂A + γv × ∇ × A . ∂t

(7.3)

If the velocity is a priori known, the additional term remains linear, but will lead to a so-called convective term. Therefore, the numerical computation will need some upwind technique for stability reasons (see e.g., [22, 184]). A typical example, which can be modelled by (7.3), is the investigation of an eddy current sensor as displayed in Fig. 7.2, where the material under test can be assumed to be infinitely wide and its velocity to be known.

Fig. 7.2. Eddy current sensor used for NDE (nondestructive evaluation)

However, if the magnetic field significantly changes due to moving and/or deforming parts (e.g., moving parts within a magnetic valve or current/voltageloaded coils), we have to apply an updated Lagrangian formulation of the electromagnetic field equations [190]. Let us consider a general point P of the moving body at time t and t + ∆t as shown in Fig. 7.3. Now, we can express the change of the magnetic vector potential ∆A within ∆t according to the fixed coordinate system (x, y, z) A(x + ∆x, t + ∆t) − A(x, t) ∆A = . ∆t ∆t

(7.4)

By assuming small ∆x and ∆t, we can express the term A(x + ∆x, t + ∆t) by a Taylor series up to the linear term

7.2 Electromagnetic Force

209

Fig. 7.3. Moving point P at time t and t + ∆t

∂A ∆t ∂t ∂A ∂y ∂A ∂z ∂A ∂x ∆t + ∆t + ∆t + ∂x ∂t ∂y ∂t ∂z ∂t ∂A = A(x, t) + ∆t + (v · ∇) A ∆t , ∂t

A(x + ∆x, t + ∆t) ≈ A(x, t) +

(7.5)

with v = (∂x/∂t, ∂y/∂t, ∂z/∂t)T . Now by letting ∆t → 0, we obtain from (7.4) using (7.5) ∂A dA = + (v · ∇) A . (7.6) dt ∂t On the other hand, we can rewrite the term γv × ∇ × A as follows γv × ∇ × A = γ∇A (v · A) − γ(v · ∇)A ,

(7.7)

where ∇A denotes the operation of the nabla operator just on the magnetic vector potential A. By comparing (7.6) and (7.7), we can rewrite (7.3) for the updated Lagrangian formulation ∇ × ν∇ × A = Ji − γ

dA + γ∇A (v · A) . dt

(7.8)

It has to be noted that the term γ∇A (v · A) in (7.8) will always be zero in the 2D plane as well as axisymmetric case, since the vectors v and A are orthogonal. However, in the general 3D case, one has to consider this term.

7.2 Electromagnetic Force Applying the principle of virtual work (displacement) as used for deriving the electrostatic force will lead to a general expression for the magnetic force Fmag . The variation of the magnetic energy δWmag relates the virtual displacement δr and magnetic force Fmag as follows

210

7 Coupled Magnetomechanical Systems

δWmag = δr · Fmag .

(7.9)

Using the two magnetic quantities B and H, we can compute the electromagnetic energy density wmag by wmag =

1 B· H. 2

(7.10)

Assuming an isotropic body with magnetic permeability µ we obtain for (7.10) wmag =

B·B H·H =µ . 2µ 2

Performing an integration over the whole volume Ω gives us the total magnetic energy Wmag , stored within the body   B·B 1 1 dΩ . (7.11) B · H dΩ = Wmag = 2 2 µ Ω

Ω

Thereby, the variation of the magnetic energy leads to       B · δB B·B 1 δWmag = δµ dΩ . dΩ − µ 2 µ2 Ω

(7.12)

Ω

Since δµ can be expressed by δµ = δr · ∇µ we obtain for the second term in (7.12)       1 B·B B·B 1 δr · δµ dΩ = ∇µ dΩ 2 µ2 2 µ2 Ω Ω  1 (H · H ∇µ) dΩ . = δr · 2

(7.13)

Ω

According to [222], we may rewrite the first term in (7.12) as ⎞ ⎛     B · δB dΩ = δr · ⎝ (J × B) dΩ ⎠ . µ Ω

Ω

This result, in combination with (7.13) and (7.9), leads to the total electromagnetic force Fmag   1 Fmag = (J × B) dΩ − (H · H ∇µ) dΩ , (7.14) 2 Ω

Ω

7.3 Numerical Computation

211

and the expression for the magnetic volume force fmag 1 fmag = J × B − H · H ∇µ . 2

(7.15)

The first term describes a magnetic force that arises due to the current in a conductor (can also be a conductive body carrying eddy currents) placed in a magnetic field. The magnetic force described by the second term acts on all those interfaces where the magnetic permeability µ changes. Analogously to the electrostatic case, we can rewrite the second term in (7.15) in the form of a surface force fΓmag acting at the interface between two materials with permeabilities µ1 and µ2     1 1 1 Bn2 + Ht2 (µ1 − µ2 ) n . fΓmag = − (7.16) 2 µ2 µ1

7.3 Numerical Computation The precise numerical simulation of magnetomechanical systems involves the computation of the magnetic field, the mechanical field and the non-linear coupling terms. Since, in the general case, the single fields themselves are described by non-linear partial differential equations (magnetization curve in the magnetic field and geometric non-linearity in the mechanical field), the solution of large-scale problems demands for a very efficient calculation scheme. 7.3.1 Electric Circuit Coupling: Voltage-loaded Coil Let us assume the case as displayed in Fig. 7.4. The voltage source with amplitude u(t) is directly connected to the coil, which has an ohmic resistance defined by Rs . Therefore, the circuit equation reads as u(t) = Rs i(t) + Ψ˙ .

(7.17)

In (7.17) Ψ˙ = Nc φ˙ denotes the time derivative of the total magnetic flux, which is computed by the FE method using the magnetic vector potential A as follows  ˙ ˙ · dΓ ˙ Ψ = Nc φ = Nc B  ' = Nc Γf

Γf

( ˙ · dΓ = Nc ∇×A

˙ · ds , A

(7.18)

C(Γf )

with Γf the cross section of the ferrite core. The evaluation of the contour

212

7 Coupled Magnetomechanical Systems

Fig. 7.4. Voltage-loaded coil with electric circuit

integral is usually done by averaging the magnetic vector potential A over the whole volume Ωc of the coil (see Sect. 4.7.6)  ' ( Nc ˙ · eJ dΩ , Ψ˙ = A (7.19) κΓc Ωc

with eJ a unit vector pointing into the direction of the current density J. The FE formulation for computing the electromagnetic field in the eddy current case reads as n+1 n+1 + KA An+1 − f˜ A = 0 . (7.20) MA A˙ In (7.20) A and A˙ denote magnetic vector potential as well as its time derivative at the FE nodes, MA the magnetic mass matrix, KA the magnetic stiffness n+1 matrix, f˜ A the vector due to a current flowing in the coil and excited by a voltage u(t) (see (7.17)) and n the time step counter. Now, if we can assume that the skin effect in each winding can be neglected (see Sect. 4.2.2), we model the coil as in the current-loaded case, so that f˜ A computes as n+1 f˜ A = in+1 f A

fA =

ne 

f eA

f eA = [f p ]

e=1

fp =

(7.21)

Nc κΓc

 (Np eJx Np eJy Np eJz )T dΩ .

(7.22)

Ωe

Therefore, we can set up the following matrix equation system using (7.17) and (7.20) 

MA −f A ' (T fA Rs



n+1 A˙ in+1



 +

KA 0 0 0



An+1 in+1



 =

0 un+1

 .

(7.23)

n+1 For the time discretization of A˙ we will apply a general trapezoidal method (see Sect. 2.5.1)

7.3 Numerical Computation

An+1

( n n+1 . = An + ∆t (1 − γP )A˙ + γP A˙

213

'

(7.24)

Therefore, the following predictor/corrector algorithm results Predictor:

˜ = An + (1 − γP )∆t A˙ n . A

Algebraic system of equations:  ∗     n+1 M −f A ˜ −KA A A˙ ' A (T = un+1 in+1 fA Rs M∗A = MA + γP ∆tKA . Corrector:

˜ + γP ∆tA˙ n+1 . An+1 = A

(7.25)

(7.26) (7.27)

(7.28)

Since, in the above equation, the system matrix is indefinite, it is preferable to solve the algebraic system by constructing a Schur complement. First, we n+1 can express A˙ using (7.26) as follows n+1 −1 −1 ˜ . A˙ = (M∗A ) f A in+1 − (M∗A ) KA A      

(7.29)

d1

d0 −1

Instead of computing the inverse (M∗A ) , we will solve two algebraic systems, both having the same system matrix M∗A M∗A d0 = f A

(7.30)

M∗A d1

(7.31)

= −KA A˜ .

Now, since we know d0 and d1 , we can compute in+1 as follows Rs in+1 = un+1 + f TA d1 − f TA d0 in+1 i

n+1

=

un+1 + f TA d1 Rs + f TA d0

.

(7.32)

Therefore, instead of solving (7.26), we solve (7.30) and (7.31), compute in+1 n+1 via (7.32) and then calculate A˙ from (7.29). 7.3.2 Force Computation via the Principle of Virtual Work In this section we will demonstrate the use of the principle of virtual work for the local force calculation within the FE method [47]. The magnetic force is

214

7 Coupled Magnetomechanical Systems

obtained by the derivative of the magnetic energy while keeping the magnetic flux constant. Thus, the force in the direction of δr computes as ⎛ B ⎞   δ ⎝ H · dB⎠ dΩ . Fr = − (7.33) δr 0

Ω

The minus sign in (7.33) is explained as follows. We assume that the variation δr and the force Fr point in the same direction. The magnetic force always moves/deforms a body in such a way that the magnetic energy becomes a minimum. Therefore, the variation of the energy will be negative, and we have to apply a minus in (7.33).

Fig. 7.5. Movable body with discretization including the layer of virtually distorted finite elements

Now let us assume an FE mesh of the simulation domain as displayed in Fig. 7.5. The total force acting on the movable/deformable body computes as a sum over all finite elements ne surrounding the body ⎞ ⎛ B   ne  δ ⎝ H · dB⎠ dΩ . Fr = − (7.34) δr e=1 Ωe

0

Since the magnetic permeability within the virtually distorted elements is that of air, we may write B B·B H · dB dΩ = . 2µ0 0

The variation now leads to

7.3 Numerical Computation

⎛ ⎞   ne  B · B δ( dΩ) ⎠ ⎝ B · δB dΩ + Fr = − . µ δr 2µ0 δr 0 e=1 Ωe

215

(7.35)

Ωe

The local force Fri , which will act at the node i located on the surface of the body (see Fig. 7.5), computes as a sum over all elements surrounding the body and containing node i ⎛ ⎞   nie  B B · B δ( dΩ) δB i ⎝ ⎠. dΩ + (7.36) · Fr = − µ0 δr 2µ0 δr e=1 Ωe

Ωe

First, we will change the integration over the global elements to an integration over the reference elements, which means that we have to substitute dΩ by |J| dξ dη dζ with |J| the determinant of the Jacobi matrix J . Applying this substitution to (7.36), we arrive at ⎛ ⎞  nie   B · B δ|J | ⎝ B · δB |J | dξ dη dζ + Fri = − dξ dη dζ ⎠ . (7.37) µ δr 2µ0 δr 0 e=1 Ωe

Ωe

Since we will evaluate the integral numerically, we need the expression for δB/δr for each integration point. After the computation of the magnetic vector potential Ah , we can compute B as follows B=

nen nd  

∇ × (Na Ahai ) =

i=1 a=1

nen 

∇Na × Aha ,

(7.38)

a=1

with nen the number of nodes per finite element, Na the shape function at node a and Aha the calculated vector potential at node a (all three components). Performing the computation at the reference element, (7.38) changes to B=

nen 

(J T )−1 ∇ξ Na × Aha ,

a=1

with ∇ξ = (∂/∂ξ ∂/∂η ∂/∂ζ)T , and the variation of B leads to nen δB δ(J T )−1  = ∇ξ Na × Aha . δr δr a=1

Since the variation of the term δ((¸J T )−1 J T )/δr is zero, we obtain δJ T δ(J T )−1 T J + (J T )−1 =0 δr δr δ(J T )−1 δJ T = −(J T )−1 (J T )−1 . δr δr

(7.39)

216

7 Coupled Magnetomechanical Systems

Using this relation, we may write (7.39) as follows δJ T δB = −(J T )−1 B, δr δr

(7.40)

which changes (7.37) to ⎞ ⎛   nie T T  B δJ B · B δ|J | ⎝ B |J | dξ dη dζ − dξ dη dζ ⎠ . Fri = · (J T )−1 µ δr 2µ δr 0 0 e=1 Ωe

Ωe

(7.41) T The computation of ∂δJ T /δr and δ|J | /δr is analogous to the electrostatic case (see Sec. 6.1). 7.3.3 Calculation Scheme Applying the FE method to both partial differential equations, we achieve the following semidiscrete Galerkin formulation of the problem Mu u¨n+1 + Cu u˙ n+1 + Ku (un+1 )un+1 −f u (An+1 , A˙ n+1 ) = 0 MA (un+1 )A˙

(7.42)

n+1

+ KA (An+1 , un+1 )An+1 −f n+1 (An+1 , un+1 , u˙ n+1 ) = 0 . A

(7.43)

In (7.42) and (7.43) Mu denotes the mechanical mass matrix, Cu the mechanical damping matrix, Ku the mechanical stiffness matrix, f u the mechanical force vector, un+1 the nodal displacements, MA the magnetic mass matrix, KA the magnetic stiffness matrix, An+1 the nodal magnetic vector potentials, and f A the nodal magnetic source vector. For the time discretization, we apply the Newmark scheme for the mechanical equation and the general trapezoidal scheme for the magnetic equation (see Sect. 2.5). Using predictor values for the magnetic vector potential to calculate f u and predictor values for the mechanical displacement to compute MA , KA and f A a decoupling into a mechanical and a magnetic matrix equation is achieved. To still ensure a strong coupling between the magnetic and mechanical quantities, we have to solve these equations within each time step iteratively (see Fig. 7.6). The outer iteration loop controls the iterative coupling process between the magnetic and mechanical equation, which is performed in the simplest case by a fixed-point method. The convergence test is based on the following displacement stopping criterion n+1 ||2 ||un+1 k+1 − uk

||un+1 k+1 ||2

< δo ,

(7.44)

7.3 Numerical Computation

(a) Linear magnetics and mechanics

217

(b) Non-linear magnetics and mechanics

Fig. 7.6. Coupled magnetomechanical simulation

with u the nodal vector of mechanical displacements, δo an adjustable accuracy, || ||2 the L2 -norm, k the iteration counter for the outer loop (magnetomechanical iteration) and n the time-step number. When we have to consider a non-linear magnetization curve, we have an additional inner loop for the magnetic equation denoted by the iteration counter i, which has to fulfill the following convergence criteria n+1 ||An+1 ||2 i+1 − Ai

||An+1 i+1 ||2

mag < δia

||rn+1 i+1 ||2 ||f n+1 ||2 A

mag < δir .

(7.45)

In (7.45) r n+1 i+1 denotes the nodal residual vector of the magnetic equation, mag i the iteration counter for the inner (magnetic) iteration loop, and δia as mag well as δir adjustable accuracy values. The first convergence criterion in (7.45) is based on the magnitude of the magnetic vector potential increments n+1 ∆A = Ai+1 − An+1 , the second criterion on the magnitude of the residual i

218

7 Coupled Magnetomechanical Systems

vector r (right-hand side criterion). By applying also a non-linear formulation for the mechanical field, we need an additional inner loop for the mechanical equation. This loop will be denoted by the iteration counter j and has to fulfill the following stopping criteria n+1 ||un+1 ||2 j+1 − uj

||un+1 j+1 ||2

mech < δia

||r n+1 j+1 ||2 ||f n+1 ||2 u

mech < δir .

(7.46)

mech In (7.46) r n+1 j+1 is the nodal residual vector of the mechanical equation, δia mech as well as δir adjustable accuracy values, and j the iteration counter for the inner (mechanical) iteration loop.

7.3.4 Moving Current/Voltage-loaded Coil In this section we will consider the case of a current/voltage-loaded coil, and will study different approaches to overcome the problem of moving parts within the electromagnetic field. In particular, we will discuss the simulation of an electrodynamic loudspeaker as displayed in Fig. 7.7. The magnetic

Fig. 7.7. Design of an electrodynamic loudspeaker

field in the air gap is dominated by the constant radial magnetic field of the permanent magnet. When the coil is driven by a voltage signal, the interaction between the resulting electric current and the magnetic field in the air gap leads to an axial magnetic force (Lorentz force), acting on the coil. Therefore, the whole structure, consisting of the coil, former, suspension, surround and cone diaphragm, starts to move and generates the acoustic sound field. The computation of the magnetic field by applying the FE method requires the discretization of the whole simulation domain (permanent magnet, magnetic assembly, coil, former, diaphragm, suspension, surround, and ambient air). Therefore, in an updated Lagrangian formulation the moving coil of the electrodynamic loudspeaker causes mesh distortion of the surrounding finite elements (see Fig. 7.8). In addition, if the displacements exceed the element

7.3 Numerical Computation

219

Fig. 7.8. Mesh deformation due to movement

size, a new mesh of the simulation domain is required. In recent years several approaches have been proposed to overcome this problem: 1. Coupled Finite-Element–Boundary-Element Method [91,113,134, 164, 167]: Applying a finite-element–boundary-element method, where the solids are discretized by finite elements and the magnetic field in the surrounding air with boundary elements, results in a total separation of stationary and moving parts (see Fig. 7.9). This method requires no remeshing of the domain, as long as no contacts between the different parts occur. Since only the solid parts have to be discretized, the preprocessing, especially in 3D, is easier and open-domain problems can be computed exactly. Due to the updating of the boundary element matrices according to the mechanical displacements, the motional emf term is implicitly taken into account. However, the main disadvantage of this method is the large amount of memory and CPU time needed by the BEM. 2. Motional EMF-Term Method [44, 71, 83, 103, 109]: Since only nonmagnetic parts move within the electrodynamic loudspeaker, the motional emf term γ(v × B)

(7.47)

fully describes the movement of parts with conductivity γ and velocity v in a magnetic field with flux density B. Therefore, no remeshing of the calculation domain has to be performed. It has to be pointed out that the velocity of the moving parts in an electrodynamic loudspeaker is not a constant value, but changes from time step to time step resulting in a new setup of the system matrices. Furthermore, the motional emf term

220

7 Coupled Magnetomechanical Systems

Fig. 7.9. FE–BE discretization of stationary and moving part as well as of the surrounding air

leads to a nonsymmetric system matrix and, depending on the conductivity, the velocity as well as the mesh size, an upwind technique has to be utilized. In addition, the change of the overall magnetic field due to the different positions of the moving coil is not taken into account. 3. Coupling via Lagrange multiplier [77, 137, 152, 184]: This technique can be considered as a general method for coupling independent meshes. As displayed in Fig. 7.10 for the example of a rotating machine, the rotor-mesh and stator-mesh have a common interface Γc within the air gap. The two meshes are coupled by forcing the normal Stator

Air gap

Gc

Rotor

Fig. 7.10. Common interface in an air gap of a rotary electric machine

component of the magnetic induction [B · n] and the tangential component of the magnetic field intensity [H × n] at Γc of the rotor as well as stator to be equal. A common method for solving this problem is a La-

7.3 Numerical Computation

221

grange multiplier formulation. Thus, the velocity term is fully included, and no remeshing is necessary. The main disadvantage of this method is that the condition of the system matrix becomes poor. Furthermore, this method is restricted to problem cases in which the common interface Γc does not change its geometry, as is the case in rotary and translational electric machines. 4. Moving-Mesh Method [48, 73, 77, 108, 140, 188, 189]: For this method, the whole volume (moving as well as stationary parts) is discretized by finite elements. If any movements occur, the coordinates of the finite elements, discretizing the moving parts, are updated in such a form that the distortions of the elements remain small. This can be achieved by also solving the mechanical equation within the air gap (of course with a very low mechanical stiffness and no mass) together with that of the moving parts (in our example: coil, holder, suspension, cone diaphragm). Therefore, the structure of system matrices will remain and the motional emf term is implicitly taken into account. However, the number of unknowns within the mechanical equation increases, the entries of the magnetic system matrix change and, if the movement is too large a remeshing has to be performed. In addition, one has to be careful in choosing the mechanical stiffness for the surrounding air region, so that the reaction to the moving/deforming body is negligible. 5. Moving-Material Method [19, 202]: The whole simulation domain is discretized by finite elements and will be unchanged throughout the simulation. Any movements of parts within the simulation domain are considered by changing the material values within each finite element according to the motion (see Fig. 7.11). Thus, the structure of the finite element matrices will not change and no remeshing is required. However, the entries of the system matrices change, and the motional emf term is not implicitly taken into account. z

z

z

z

Dz Region of coil Surrounding air 0

1 f(z)|t

r

0

1 f(z)|t+Dt

r

Fig. 7.11. Moving-material method: Configuration at time step t and t + ∆t. The function f (z) defines a spatial weighting function for the current

222

7 Coupled Magnetomechanical Systems

In the following, we will introduce two new methods only, based on the FE method, which perfectly fit to a moving current/voltage-loaded coil. Therefore, the system matrices will remain constant throughout the whole simulation, and no problem with mesh distortion will arise. Modified Motional EMF-Term Method Due to the motional emf, an additional electric current iemf will be induced in the coil with opposite direction to coil current i excited by the applied voltage u. Thus, (7.17) changes to u(t) = Rs (i(t) − iemf (t)) + Ψ˙ .

(7.48)

The current iemf computes according to (7.2) iemf =

Nc  

γ(v × ∇ × A) · dΓ .

(7.49)

i=1 Γ

wi

Now, by modelling the complex winding structure as a hollow cylinder, (7.49) reads as  iemf = γ (v × ∇ × A) · eJ dΓ . (7.50) Γc

Since we can express the ohmic resistor Rs as Rs =

lNc lNc2 = , γΓw γ κΓc

(7.51)

with l the length of one winding, we can rewrite (7.50) in the form of a volume integral over the volume Ωc of the hollow cylinder suitable for the FE method (see Sect. 4.7.6)  N2 Rs iemf = c (v × ∇ × A) · eJ dΩ . (7.52) κΓc Ωc

Therefore, the coupled FE-network matrix equation results in (for comparison see (7.23))         MA −f A A 0 KA 0 A˙ = , (7.53) + cTA 0 i u i f TA Rs with cA including the FE discretization of (7.52). Applying the time discretization as shown in Sect. 7.3.1, we arrive at    n+1    ˜ MA + γP ∆tKA −f A −KA A A˙ = (7.54) ˜ . f TA Rs un+1 − cTA A in+1

7.3 Numerical Computation

223

The emf term has been treated explicitly, since we have used the predictor ˜ instead of the unknown vector potential A. This method has been value A successfully applied in the dynamic simulation of the electrodynamic loudspeaker within small-signal parameter studies (see Sect. 12.1). However, if the amplitude of the exciting voltage increases, as is the case for large-signal studies, the motion of the coil will not remain within the homogeneous field region. In this case, the change of the overall magnetic field has to be taken into account, which is not possible by this calculation scheme. Modified Moving-Material Method The restriction of the modified motional emf term method to small movements can be overcome by using the standard moving material method with the following changes [174] •

Same handling of the motional emf term as in the modified motional emf term method, which means, in an explicit way (using the predictor value ˜ instead of the unknown vector potential An+1 ). A

•

Evaluating the spatial weighting function f (z) (see Fig. 7.11) within each numerical integration point.

•

Applying the solution strategy for a voltage-loaded coil as shown in Sect. 7.3.1. Therefore, in each time step first of all the current in+1 is computed and then the magnetic vector potential An+1 .

The third point allows the voltage-loaded coil to be treated as if it were a current-loaded coil, since we first compute the current in the coil and then the magnetic vector potential (see Sect. 7.3.1). Since the permeability of the coil and of the surrounded air is the same (both have relative permeability of 1), the movement of the coil and therefore the change of the material within the finite elements will not change the entries of the system matrices. However, the vector f A has to be recomputed according to the motion of the coil. Verification In the following, we will set up a simple example of a moving voltage-loaded coil, and compare the moving-mesh, modified motional emf, and modified moving-material methods. Figure 7.12 displays the test case, which is a reduced setup of an electrodynamic loudspeaker. The coil is held by a spring and a permanent magnet generates the constant magnetic field Bconst in the air gap. The values have been chosen as follows:

224

7 Coupled Magnetomechanical Systems

Bconst Nc γ ρ Γc k c ˆ U f

= 0.75 T constant magnetic field in the air gap = 100 number of turns = 5.8 × 107 S/m conductivity of the winding wire = 8900 kg/m3 density of winding wire = 0.5 mm2 cross section of the coil = 249.6N/m spring constant = 0.131 Ns/m damping coefficient = 3 V amplitude of applied voltage = 131 Hz frequency of applied voltage

Fig. 7.12. Test case: voltage-loaded coil in a homogeneous static magnetic field (axisymmetric setup)

Due to the small frequency of the applied voltage, we can neglect the induction of the coil in a semianalytical approach. Furthermore, due to the axisymmetric setup, the magnetic induction B is orthogonal to the electric current density J, and we assume that within the coil the magnetic induction B and the velocity v are also orthogonal. The movement of the coil can be described by Newton’s law as m

dx d2 x + k x = Fmag (t) = Fˆmag sin(ω t) . +c 2 dt dt

(7.55)

In (7.55) m denotes the mass of the coil, Fmag the magnetic force and ω = 2 π f the angular frequency. Introducing the normalized coefficients δ and ω0 , we obtain

7.3 Numerical Computation

d2 x dx Fˆmag + ω02 x = sin(ω t) + 2δ 2 dt dt m c 1 δ= = 177.7 2m s k 2 ω0 = = (2 π 131 Hz)2 , m

225

(7.56) (7.57) (7.58)

with ω0 the angular frequency of the free, undamped oscillation, which is equal to the angular frequency ω = 2 π f . For t  1/δ, we obtain for the amplitude of the forced oscillation Fˆmag + m (ω 2 − ω02 )2 + 4δ 2 ω 2 ω = 2 π f = ω0 . xˆ =

(7.59) (7.60)

The amplitude of the electromagnetic force Fmag can be computed via Fˆmag = B l Iˆ l = 2 r π Nc ,

(7.61) (7.62)

with r the average radius of the coil (r = 13.225 mm) and Iˆ the amplitude of the coil current. In the first step, we will neglect the motional emf term in computing the electric current IˆnoEMF ˆ ˆ U U IˆnoEMF = + 2 ≈ = 104.70 mA RE R E + (2πf L E )2 RE =

2πrNc2 = 28.653 Ω, γΓc

(7.63) (7.64)

with R E denoting the ohmic resistance and L E the inductance of the coil. Using (7.57) and (7.60) – (7.63), we can compute the amplitude xˆ of the axial motion via (7.59) x ˆnoEMF =

B 2 r π Nc IˆnoEMF = 6.05 mm. c ω0

(7.65)

In a second step, let us also consider motional emf. Thus, the eddy current induced within the coil due to the motion, computes as ˆ B Γc /Nc , IˆEMF = γ vˆ B Γc /Nc = γ 2 π f x

(7.66)

with vˆ = 2 π f xˆ the amplitude of the velocity and x ˆ the amplitude of the axial motion. The total current in the coil changes to Iˆ = IˆnoEMF − IˆEMF .

(7.67)

226

7 Coupled Magnetomechanical Systems

In Table 7.1 we have summarized the results obtained by the semianalytical method and the numerical simulation for IˆnoEMF and xˆnoEMF . The differences in the values are due to the fact that within the semianalytical computation we have neglected the inductance of the coil (this will reduce the difference to 0.08% assuming an inductance of 3.9 mH, obtained by numerical computation) and any magnetic stray field. Table 7.1. Comparison of semianalytical and numerical computation without considering motional emf IˆnoEMF x ˆnoEMF

Semianalytical method Numerical simulation Relative difference 104.70 mA 104.10 mA −0.6% 6.05 mm 6.00 mm −0.8%

Table 7.2. Comparison of the different numerical calculation schemes

x ˆ

Motional emf term Moving material Moving mesh 0.534 mm 0.533 mm 0.532 mm

To see the influence of the motional emf, we have computed the motion of the coil with all three numerical methods and the results for the calculated amplitude of the axial motion are given in Table 7.2. Since the moving coil always remains within the homogeneous magnetic field, the three numerical methods show almost no difference. It can be clearly seen that the amplitude x ˆ due to the motional emf term has strongly decreased (e.g., the case of an electromagnetic brake). The comparison of the effective current in the coil including motional emf, is summarized in Table 7.3. The value of Iˆ for the semianalytical method has been obtained by using the numerically computed amplitude of the motion x ˆ and evaluating (7.67) with the help of (7.66). The Table 7.3. Comparison of semianalytical and numerical methods including motional emf

Iˆ rel. difference

Semianalytical Motional emf Moving material Moving mesh 9.28 mA 9.267 mA 9.265 mA 9.20 mA −0.14% −0.16% −0.9%

advantages and disadvantages of the three numerical calculation schemes are summarized in Table 7.4.

7.3 Numerical Computation

227

Table 7.4. Comparison of the numerical calculation schemes Motional emf Moving material Moving mesh Including the emf term Coil position considered

yes

yes

yes

no

yes

yes

Mesh deformation

no

no

yes

Matrix entries change

no

no

yes

Concluding, we can state that for the special application electrodynamic loudspeaker the modified moving-material method is the method of choice, since it includes all physical effects and no mesh-updating or even remeshing is necessary.

8 Coupled Mechanical-Acoustic Systems

In many technical applications, the sensor/actuator is immersed in an acoustic fluid. Therefore, mechanical vibrations will generate acoustic waves, which itself will act as a surface pressure load on the vibrating structure. In general, we distinguish between the following two situations concerning mechanicalacoustic systems: •

Strong Coupling: In this case, the mechanical and acoustic field equations including their couplings have to be solved simultaneously. A typical example is a piezoelectric ultrasound array immersed in water (see Fig. 8.1).

Fig. 8.1. Acoustic sound field of a piezoelectric ultrasound array antenna

•

Weak Coupling: If the pressure forces of the fluid on the solid are negligible, a sequential computation can be performed. For example, the acoustic sound field of an electric transformer as displayed in Fig. 8.2 can be obtained in this way.

230

8 Coupled Mechanical-Acoustic Systems

Thus, in a first simulation the mechanical surface vibrations are calculated, which are then used as the input for an acoustic field computation.

Fig. 8.2. Acoustic sound field of an electric transformer due to the Lorentz forces acting on its winding

8.1 Solid–Fluid Interface

Fig. 8.3. Solid–fluid interface

At a solid–fluid interface, the continuity requires that the normal component of the mechanical surface velocity of the solid must coincide with the normal component of the acoustic velocity of the fluid (see Fig. 8.3). Thus, the following relation between the velocity v of the solid expressed by the mechanical displacement u and the acoustic particle velocity v
expressed by the acoustic scalar potential ψ arises

8.2 Coupled Field Formulation

∂u v
= −∇ψ ∂t n · (v − v
) = 0 ∂ψ ∂u = −n · ∇ψ = − . n· ∂t ∂n

231

v=

(8.1)

In addition, one has to consider the fact that the ambient fluid causes on the surface a mechanical stress σ n σ n = −np
= −nρ0

∂ψ , ∂t

(8.2)

which acts like a pressure load on the solid. In (8.2) ρ0 denotes the mean density of the fluid. When modelling special wave phenomena, we often arrive at a partial differential equation for the acoustic pressure. Therewith, we will also derive the coupling conditions between the mechanical displacement and acoustic pressure at a solid–fluid interface. For the first coupling condition, the continuity of the velocities, we have to establish the relation between the acoustic particle velocity v
and the acoustic pressure p
. According to the linearized Euler equation (see 5.30), we can express the normal component of v
by n·

1 ∂p
∂v
=− . ∂t ρ0 ∂n

(8.3)

Therewith, since n · v = n · v
holds, we get the relation to the mechanical displacement by 1 ∂p
∂2u . (8.4) n· 2 = − ∂t ρ0 ∂n The second coupling condition as defined in (8.2) is already established for an acoustic pressure formulation.

8.2 Coupled Field Formulation Let us consider a setup of a coupled mechanical-acoustic problem as shown in Fig. 8.4, where at the interface Γ I we have to consider the solid–fluid coupling. Now, within the domain Ωs the partial differential equation for the mechanical field (see 3.80), within the domain Ωf the partial differential equation for the acoustic field (see 5.92) and along the interface Γ I the coupling conditions according to (8.1) and (8.2) have to be fulfilled. In a first step, let us transform the partial differential equations to their weak form without setting the boundary integral (obtained by using integration by parts) to zero. Therefore, we obtain for the mechanical system    

T
¨ dΩ (Bu ) [c]Bu) dΩ − ρu · u u · σ n dΓ = u
· fV dΩ , (8.5) Ωs

Ωs

ΓI

Ωs

232

8 Coupled Mechanical-Acoustic Systems

Fig. 8.4. Setup of a coupled mechanical-acoustic problem

and for the acoustic system (acoustic excitation f being zero)     1 ¨ dΩ + w ψ ∇w · ∇ψ dΩ + w n · ∇ψ dΓ − w n f · ∇ψ dΓ = 0 . c2 Ωf

Ωf

ΓI

Γf

(8.6) It should be noted that the plus sign in (8.6) in front of the the boundary integral over Γ I is due to the choice of n (see Fig. 8.4). In a second step, we will now incorporate the coupling conditions. With the help of (8.2), we can rewrite the boundary integral in (8.5) as follows    ∂ψ dΓ . (8.7) u
· σ n dΓ = − u
· n p dΓ = − u
· n ρ0 ∂t ΓI

ΓI

ΓI

Using (8.1), we obtain for the boundary integral along the interface Γ I (at the outer boundary Γ f we set for simplicity ∂ψ/∂n = 0) in (8.6) the following form   ∂ψ ∂u dΓ = − w n · dΓ . (8.8) w ∂n ∂t ΓI

ΓI

Thus, we arrive at the following coupled system of equations     ∂ψ ¨ dΩ + (Bu
)T [c]Bu) dΩ + dΓ = ρu
· u u
· n ρ0 u
· fV dΩ(8.9) ∂t Ωs Ωs ΓI Ωs    1 ∂u dΓ = 0 . (8.10) w ψ¨ dΩ + ∇w · ∇ψ dΩ − wn· c2 ∂t Ωf

Ωf

ΓI

8.3 Numerical Computation

233

8.3 Numerical Computation 8.3.1 Finite Element Formulation Before we perform a domain discretization, we multiply (8.10) by −ρ0 in order to obtain in (8.10) a boundary integral similar to the one in (8.9). Thus, the matrices, occurring from an FE discretization of these two boundary integrals, will be transposed to each other and hence symmetry of the overall system matrix will be obtained. Using nodal finite elements, we approximate the mechanical displacement u as well as the scalar acoustic potential ψ as follows ⎞ ⎛ nd  n1 n1 Na 0 0   Na uhia ei = Na uha ; Na = ⎝ 0 Na 0 ⎠ (8.11) u ≈ uh = a=1 i=1 a=1 0 0 Na n 2  ψ ≈ ψh = Na ψah , (8.12) a=1

with n1 the number of nodes with unknown mechanical displacement and n2 the number of nodes with unknown acoustic potential. Following the same procedure as described in Sect. 3.7.1 for the mechanical equation and in Sect. 5.4.1 for the acoustic equation we obtain a coupled system of equations          u˙ u ¨ 0 Cuψ u Mu 0 Ku 0 + + T ¨ ˙ Cuψ 0 ψ 0 −Mψ 0 −Kψ ψ ψ   fu = . (8.13) 0 The new matrix Cuψ computes as follows Cuψ =

nIe  e=1

 Ceuψ ; Ceuψ = [Cpq ] ; Cpq = Γe

⎛

⎞ N p N q nx ρ0 ⎝ Np Nq ny ⎠ dΓ , (8.14) N p N q nz

with nIe the number of finite elements along the interface. At this point, it should be emphasized that the coupled system of equations remains symmetric. This is not the case if instead of the acoustic velocity potential an acoustic pressure formulation is used. The time discretization is performed by a standard Newmark algorithm (see Sect. 2.5.2), which reads for the effective mass matrix formulation as follows

234

•

8 Coupled Mechanical-Acoustic Systems

Perform predictor step: ∆t2 (1 − 2βH ) u ¨n 2 ˜˙ = u˙ n + (1 − γH )∆t u ¨n u 2 ∆t ˜ = ψ + ∆t ψ˙ + ¨ (1 − 2βH ) ψ ψ n n n 2 ˜ ¨ . ψ˙ = ψ˙ n + (1 − γH )∆t ψ n u ˜ = un + ∆t u˙ n +

•

(8.16) (8.17) (8.18)

Solve algebraic system of equations: 

M∗u

C∗uψ

(C∗uψ )T −M∗ψ

•

(8.15)



u ¨n+1



ψ¨n+1

 =

fu 0



 −

Ku

0

  u ˜

˜ 0 −Kψ ψ  ⎛ ˜⎞ u˙ 0 Cuψ ⎝ ⎠ (8.19) − ˜ CTuψ 0 ψ˙

M∗u = Mu + βH ∆t2 Ku M∗ψ = Mψ + βH ∆t2 Kψ

(8.20) (8.21)

C∗uψ = γH ∆t Cuψ .

(8.22)

Perform predictor update: ˜ + βH ∆t2 u ¨ n+1 un+1 = u ˜˙ + γH ∆t u u˙ n+1 = u ¨n+1

(8.23)

˜ + βH ∆t2 ψ ¨ =ψ n+1

(8.25)

˜˙ + γ ∆t ψ ¨ ψ˙ n+1 = ψ . H n+1

(8.26)

ψ n+1

(8.24)

8.3.2 Non-matching Grids In many practical cases, we wish to perform the spatial discretization within the elastic body independent to the discretization of the surrounding fluid. A typical example is given in Fig. 8.5, where on the one hand we need a much finer grid within the mechanical structures (plates) and on the other hand wish to have a regular grid within the fluid domain. In particular, the advantages of using non-matching grids are demonstrated in Fig. 8.5, where the mechanical regions have to be resolved by a substantially finer grid than the fluid region. Therewith, we will investigate in such flexible discretization techniques for the approximative solution of coupled mechanical-acoustic problems. For the mechanical-acoustic coupling, the problem formulation for nonmatching grids remains essentially the same as for the matching situation. This

8.3 Numerical Computation

235

Fig. 8.5. Non-matching discretization of mechanical structure and surrounding fluid

means, in contrast to the problem setting considered in Sec. 5.4.3 (acousticacoustic coupling over non-matching grid), no additional Lagrange multiplier has to be introduced. For the spatial discretization, we use two independently generated triangulations Ts and Tf on Ωs and Ωf , respectively, and approximate the displacement u on Ts and the potential ψ on Tf by finite elements. The two triangulations inherit two (d−1)-dimensional grids ∂Ts and ∂Tf on Γ I . Due to the flexible construction of both grids, the finite element nodes on ∂Ts and ∂Tf will in general not coincide. On the contrary, motivated by different spatial scales required for the resolution of the local subproblems, the difference in the mesh sizes can become quite large. The discretized version of (8.9), (8.10) results in the same matrix-equations as for the matching grid approach (see 8.13). The coupling between the two grids is represented by the matrices Cuψ and CT uψ which realize the boundary integrals in (8.9) and (8.10). Their entries are given by  Cuψ = [Cpq ]; Cpq = ρf Nps Nqf n dΓ ∈ IRd , (8.27) ΓI

where Nps is the scalar basis function associated with the node p on ∂Ts , and Nqf is the one for node q on ∂Tf . For a detailed discussion of the element wise assembly of Cuψ we refer to [67]. 8.3.3 Numerical Examples Noise Radiation from an Oil Pan and a Car Engine A typical problem that is encountered in numerical acoustics of automotive applications consists of the calculation of the sound radiated from machine parts. In the standard approach, first the structural response due to dynamic loads is calculated based on the FE method. In a second step, the acoustic radiation is calculated, again applying the FE method. The surface grid is dis-

236

8 Coupled Mechanical-Acoustic Systems

Fig. 8.6. Surface grid of an oil pan

Fig. 8.7. Embedding of oil pan into finite element grid

played in Fig. 8.6, whereas a detail of the finite element model, showing the embedding of the oil pan structure in a sphere consisting of acoustic finite elements, is shown in Fig. 8.7. The total finite element model consisted of approx. 500 000 3D acoustic finite elements and the resulting sound field at a driving frequency of 600 Hz as computed by this model is shown in Fig. 8.8. Next,

Fig. 8.8. Sound field radiated at 600 Hz

this modelling scheme was applied in the simulation of a complete car engine. Here, the surface grid consists of 8 200 2D elements. This model was embedded in a sphere of radius 2 m and the mesh generator NETGEN [194] was used to mesh the unfilled volume with acoustic finite elements. The embedding of the engine in the finite element mesh is displayed in Fig. 8.9, whereas the innermost layer of acoustic elements (tetrahedra) is shown in Fig. 8.10. The results of the transient structural simulation were used as excitations for the acoustic calculations. In the simulations, two different models have been considered, consisting of 210 000 and 1 340 000 finite elements, respectively. For the numerical computation three different approaches have been performed: •

Direct, implicit: An implicit Newmark time-integration scheme has been used and the re-

8.3 Numerical Computation

237

Fig. 8.10. Innermost tetrahedra layer around car engine

Fig. 8.9. Embedding of car engine into finite element grid

sulting algebraic system of equation has been solved by a direct solver. •

Iterative, implicit: Again, an implicit Newmark time-integration scheme has been used, but now we have applied an iterative solver (GMRES with ILU as preconditioner, see [127]).

•

Explicit: An explicit Newmark time-integration scheme with lumped mass matrix has been used. Therewith, the system matrix is a pure diagonal and the algebraic solver just divides the right-hand side by the diagonal entries. In Table 8.1 and 8.2, the required computer resources are summarized. Table 8.1. Computer resources, small model Solver Direct, implicit Explicit Iterative (GMRES)

Time steps

Memory

2 000 364 MB 20 000 20 MB 2 000 38 MB

The direct, implicit solution was not available for the large model, since it required 10 GB of main memory. For the iterative solver, convergence was achieved for an accuracy of 10−8 (relative residual norm) within six iterations

238

8 Coupled Mechanical-Acoustic Systems Table 8.2. Computer resources, large model Solver Direct, implicit Explicit Iterative (GMRES)

Time steps – 20 000 2 000

Memory – 102 MB 206 MB

Fig. 8.11. Comparison of normalized acoustic pressure at evaluation point for small model

per time step. Of course, this convergence behavior strongly depends on the quality of the generated grid. For the small model, no difference was found between the solution for the direct and the iterative solver. However, as can be seen from Figs. 8.11 and 8.12, due to the diagonal mass matrix formulation for the explicit time scheme, the solution based on the explicit time stepping differs from both solutions obtained with implicit time stepping. This difference vanishes, when an even finer time discretization is used for the explicit scheme (for our examples we applied a time step of ten times smaller as used for the implicit scheme). Acoustic Wave Generation by a Multiple Plate Structure To demonstrate the applicability of the non-matching grid technique, we present the generation and radiation of acoustic waves by multiple structures, which admits the steering of the waves by exciting the structures in a specified chronological order. In particular, we use for the structure Ωs 25 cylindrical silicon plates with diameter 50 µm and height 1 µm. They are placed as a (5 × 5)-array, each plate having a distance of 50 µm to its nearest

8.3 Numerical Computation

239

Fig. 8.12. Comparison of normalized acoustic pressure at evaluation point for large model

neighbors. An excitation force with frequency f = 1 MHz is applied on their lower end. For the acoustic domain Ωf , which is assumed to be water, a cuboid of length and width 1200 µm and height 420 µm is chosen. Due to symmetry reasons, we use as computational domain one quarter of the original one. In

Fig. 8.13. Cylindrical plates attached to the fluid domain and isosurfaces of the acoustic potential, deformed plates

Fig. 8.13 a part of the finite element meshes is shown, for which a uniform grid of 40×40×28 cubes is used to discretize the acoustic domain and a grid of 768 hexahedra is employed for each plate. Thus, having a mesh width of ha = 600 µm /40 = 15 µm, we use ten elements per wavelength for Ωf . If one had to employ matching grids, it would be quite difficult to generate them,

240

8 Coupled Mechanical-Acoustic Systems

and if the mesh-width could not be very small over the whole domain, the resulting element shapes would possibly result in a poor approximation of the solution. The nonconforming approach admits to use the grid desired for each subdomain regardless of the grids for the other subdomains. Figure 8.14 shows snapshots, taken every ten time steps of 3.5 ns, of the evolution of the acoustic velocity potential ψ along with the deformation of the structures (magnified by a factor of 1000).

Fig. 8.14. Evolution of the acoustic velocity potential and of the deformed structures, successive excitation: snapshots after 10, 20, . . . , 80 time steps

8.3 Numerical Computation

241

For the results presented in Fig. 8.14, the plates are excited successively, and as can be seen, the waves emitting from the structures add up as expected to constitute the superposed global sound beam. Given a target point, it is possible to optimally steer the acoustic wave towards this point by appropriately adjusting the chronological order of excitation of the silicon chips. This principle is used in so-called capacitive micro-machined ultrasound transducers (CMUTs) (see Sec. 12.5). There, the deformation of the structure is induced by an electrostatic surface force acting on the loaded electrodes.

9 Piezoelectric Systems

The piezoelectric transducing mechanism is based on the interaction between the electric quantities, electric field intensity E and electric induction D, with the mechanical quantities, mechanical stress [σ] and strain [S]. By applying a mechanical load (force) to a piezoelectric transducer (e.g., piezoelectric material with top and bottom electrode), one can measure a change of the electric charge (sensor effect, see Fig. 9.1). This mechanism is called the direct

Fig. 9.1. Piezoelectric force sensor with a charge amplifier

piezoelectric effect, and is due to a change in the electric polarization of the material. The so-called inverse piezoelectric effect is observed when loading a piezoelectric transducer with an electric voltage, resulting in mechanical deformations (actuator effect). This setup can be used, e.g., in a positioning system (see Fig. 9.2).

9.1 Constitutive Equation Having the numerical computation of the dynamic behavior of piezoelectric transducers in mind, we choose D and σ as the physical quantities depending

244

9 Piezoelectric Systems

Fig. 9.2. Piezoelectric actuator (bimorph setup)

on the electric field intensity E and mechanical strain S. Therefore, after applying a Taylor series up to the linear term, we obtain ∂σij ∂σij 0 |E (Skl − Skl )+ |S (Ek − Ek0 ) ∂Skl ∂Ek ∂Di ∂Di 0 0 Di = Di (Sij , Ek0 ) + |E (Skl − Skl )+ |S (Ek − Ek0 ) . ∂Skl ∂Ek

0 σij = σij (Sij , Ek0 ) +

(9.1) (9.2)

0 Without loss of generality, we can set Skl = 0 and Ei0 = 0. As shown in [198] the partial derivatives in (9.1) yield the material tensors

cE ijkl =

∂σij ∂Skl

ekij = −

∂σij ∂Ek

εSik =

∂Di . ∂Ek

(9.3)

Therefore, we arrive at the material law describing the piezoelectric effect [σ] = [cE ][S] − [e]T E S

D = [e][S] + [ε ]E .

(9.4) (9.5)

The material tensors [cE ], [εS ], and [e] appearing in (9.4) and (9.5) are the tensor of elastic modulus, of dielectric constants, and of piezoelectric moduli. The superscripts E and S indicate that the corresponding material parameters have to be determined at constant electric field intensity E and at constant mechanical strain S, respectively.

9.2 Governing Equations For the linear case, we will use the Cauchy stress tensor [σ] and the linear strain tensor [S] as mechanical quantities. Starting at Navier’s equation (3.22) applying Voigt notation (9.6) fV + B T σ = ρa , and using (9.4) as well as (3.38) for the strain–displacement relation, we arrive at ) * (9.7) ρ¨ u − B T [cE ]Bu − [e]T E = fV .

9.3 Piezoelectric Material Properties

245

Since piezoelectric materials are insulating, i.e., do not contain free-volume charges, the electric field is determined by ∇·D = 0.

(9.8)

However, considering any piezoelectric sensor/actuator, which is set up by a piezoelectric material coated at the top and bottom by a metallic layer acting as electrodes, we rewrite (9.8) as ∇ · D = qe

(9.9)

to account for free electric charges qe on the electrodes. By using the relation between electric field intensity E and the scalar electric potential Ve (see (4.60)), (9.7) reads as ) * ρ¨ u − B T [cE ]Bu + [e]T ∇Ve = fV . (9.10) Now, utilizing (9.5) we can transform (9.9) to ) * ∇ · [e]Bu − [εS ]∇Ve = qe .

(9.11)

Therefore, the describing partial differential equations for linear piezoelectricity read as ) * ρ¨ u − B T [cE ]Bu + [e]T ∇Ve = fV (9.12) * ) S ∇ · [e]Bu − [ε ]∇Ve = qe . (9.13)

9.3 Piezoelectric Material Properties Piezoelectric materials can be subdivided into the following three categories [144] 1. Single crystals, like quartz 2. Piezoelectric ceramics like barium titanate (BaTiO3 ) or lead zirconate titanate (PZT) 3. Polymers like PVDF (polyvinylidenfluoride) Since categories 1 and 3 typically show a weak piezoelectric effect, these materials are mainly used in sensor applications (e.g., force, torque or acceleration sensor). For piezoelectric ceramics the electromechanical coupling is large, thus making them attractive for actuator applications. These materials exhibit a polycrystalline structure and the key physical property of these materials is ferroelectricity. In order to provide some physical understanding of the piezoelectric effect, we will perform some discussions about the microscopic structure of piezoceramics, partly following the exposition in [122]. A piezoelectric ceramic material is subdivided into grains consisting of unit cells with

246

9 Piezoelectric Systems

different orientation of the crystal lattice. The unit cells consist of positive and negative charged ions, and their charge center position relative to each other is of main importance for the electromechanical properties. We will call the material polarizable, if an external load, e.g., an electric field can shift these centers with respect to each other. Let us consider BaTiO3 or PZT, which have a polycrystalline structure with grains having different crystal lattice. Above the Curie temperature Tc – for BaTiO3 Tc ≈ 120 oC - 130 oC and for PZT Tc ≈ 250 oC - 350 o C, these materials have the perovskite structure. The cube shape of a unit cell has a side length of a0 and the centers of positive and negative charges coincide (see Fig. 9.3). However, below Tc the unit cell T>Tc

T
a0

c0

a0

a0

a0

a0 2+

4+

Ba

O

Ti

2-

Fig. 9.3. Unit cell of BaTiO3 above and below the Curie temperature Tc

deforms to a tetragonal structure as displayed in Fig. 9.3, e.g., BaTiO3 at room temperature changes its dimension by (c0 − a0 )/a0 ≈ 1 %. In this ferro-

Inital state

E>Ec

E=0

Fig. 9.4. Orientation of the polarization of the unit cells at initial state, due to a strong external electric field and after switching it off

electric phase, the centers of the positive and negative charges differ and the unit cell posses a spontaneous polarization. Since the single dipoles are ran-

9.3 Piezoelectric Material Properties

247

domly oriented, we call this the thermally depoled state or virgin state. This state can be modified by an electric or mechanical loading with significant amplitude. In practice, a strong electric field E ≈ 2 kV/mm will switch the unit cells such that the spontaneous polarization will be more or less oriented towards the direction of the externally applied electric field as displayed in Fig. 9.4. Now, when we switch off the external electric field the ceramic will still exhibit a non-vanishing residual polarization in the macroscopic mean (see Fig. 9.4). We call this the irreversible or remanent polarization and the process as poling. Now let us consider a mechanically unclamped piezoceramic disc at virgin state and load the electrodes by an increasing electric voltage. Initially, the orientation of the polarization within the unit cells is randomly distributed as shown in Fig. 9.5 (state 1). The switching of the domains starts when the

Fig. 9.5. Polarization P as a function of the electric field intensity E

external applied electric field reaches the so-called coercitive intensity Ec 1 . At this state, the increase of the polarization is much faster, until all domains are switched (see state 2 in Fig. 9.5). A further increase of the external electric loading results in an increase of the polarization with a quite smaller slope and the occurring micromechanical process remains reversible. Reducing the external applied voltage to zero will preserve the poled domain structure, and we call the resulting macroscopic polarization the remanent polarization Pr . If the previous excitation has aligned all domains, Pr corresponds to the saturation polarization. Psat . Loading the piezoceramic disc by a negative 1

It has to be noted that in literature Ec often denotes the electric field intensity at zero polarization. According to [122] we define Ec as the electric field intensity at which domain switching occurs.

248

9 Piezoelectric Systems

voltage of an amplitude larger than Ec will initiate the switching process again until we arrive at a random polarization of the domains (see state 4 in Fig. 9.5). A further increase will orient the domain polarization in the new direction of the external applied electric field (see state 5 in Fig. 9.5). Measuring the mechanical strain during such a loading cycle as described above for the electric polarization results in the so-called butterfly curve depicted in Fig. 9.6. Here we also observe that an external applied electric field

Fig. 9.6. Mechanical strain S as a function of the electric field intensity E

intensity E > Ec is required in order to obtain a measurable mechanical strain. The observed strong increase is a superposition of two effects: First, we achieve an increase of the strain due to the domains oriented with the c-axis in the direction of the external electric field. Secondly, the orientation of the domain polarization leads to the macroscopic piezoelectric effect yielding the reversible part of the strain. As soon as all domains are switched (see state 2 in Fig. 9.6), the further increase of the strain just results from the macroscopic piezoelectric effect. A separation of the switching (irreversible) and the piezoelectric (reversible) strain can be best achieved by decreasing the external electric load to zero. Only the strain induced by the alignment of the c-axis remains and we denote this part the saturation strain Ssat . Alternatively, or in addition to, this electric loading, we can perform a mechanical loading, which will also result in switching processes. For a detailed discussion on the occurring effects we refer to [122]. The linear material tensors [cE ], [εS ], and [e], which relate the mechanical and electric quantities, show a certain sparsity pattern according to the crystal structure and polarization of the piezoelectric material. The 6 mm crystal class, which also represents the equivalent class for piezoelectric ceramics, has the following pattern:

9.4 Models for Non-linear Piezoelectricity

⎛

c11 ⎜c ⎜ 12 ⎜ ⎜ c13 E [c ] = ⎜ ⎜ 0 ⎜ ⎜ ⎝ 0 0

c12 c11 c13 0 0 0

⎛

0 0 0 0 e15 ⎜ [e] = ⎝ 0 0 0 e15 0 e31 e31 e33 0 0

c13 c13 c33 0 0 0

⎞ 0 ⎟ 0⎠ 0

0 0 0 c44 0 0

249

⎞ 0 0 ⎟ 0 0 ⎟ ⎟ ⎟ 0 0 ⎟ ⎟ 0 0 ⎟ ⎟ ⎠ c44 0 0 (c11 − c12 )/2

(9.14)

⎞ ε11 0 0 ⎟ ⎜ [εS ] = ⎝ 0 ε11 0 ⎠ . 0 0 ε33

(9.15)

⎛

Properties of some widely used piezoelectric materials are summarized in Table 9.1 (PZT 5A/5H from [144], 3202 (Motorola) from [114]). Table 9.1. Material data for some materials of class 6 mm cE 11 (N/m2 )

cE 12 (N/m2 )

cE 13 (N/m2 )

cE 33 (N/m2 )

cE 44 (N/m2 )

cE 66 (N/m2 )

PZT-5A 12.1 × 1010 75.4 × 109 75.2 × 109 11.1 × 1010 21.1 × 109 22.8 × 109 PZT-5H 12.6 × 1010 79.5 × 109 84.1 × 109 11.7 × 1010 23.0 × 109 23.2 × 109 32032 14.6 × 1010 96.2 × 109 10.2 × 1010 13.8 × 1010 25.5 × 109 24.9 × 109

PZT-5A PZT-5H 3202

PZT-5A PZT-5H 3202

e15 (C/m2 ) 12.3 17.0 15.3

e31 (C/m2 ) –5.4 –6.5 –11.5

e33 (C/m2 ) 15.8 23.3 20.4

ε11 ε33 (Vs/Am) (Vs/Am) 919 ε0 824 ε0 1730 ε0 1437 ε0 1378 ε0 1290 ε0

9.4 Models for Non-linear Piezoelectricity In most actuator applications, piezoceramic materials, e.g., PZT are used, which exhibits a strong non-linear behavior for large signal excitations. This non-linear behavior is characterized by the hysteresis loop of the polarization

250

9 Piezoelectric Systems

(see Fig. 9.5) and the butterfly curve of the mechanical strain (see Fig. 9.6). In general, we can divide the physical/mathematical models into three categories: 1. Thermodynamically consistent models These models are based on a macroscopic view to describe microscopical phenomena, see e.g., [123, 138, 193]. 2. Micromechanical models These models are sometimes also based on thermodynamic fundamentals, however they are constructed by breaking the material down to the size of single grains, see e.g., [11, 69, 195]. 3. Models with hysteresis operator These models are mostly restricted to the actuator working range and consider either the strain or the polarization hysteresis, see e.g., [98, 133, 203]. In a first simplified approach we follow the ideas developed in [198]. Therewith, we just consider the polarization hysteresis and rewrite the linear constitutive law (9.5) by (9.16) D = [e][S] + ε0 E + P . In (9.16) P denotes the polarization modelled by a hysteresis model (see next section) P = H[E] eP with H the hysteresis operator and eP the unit vector of the polarization P being equal to the direction of the applied electric field intensity E. Therewith, the describing PDEs read as follows ) * ρ¨ u − B T [cE ]Bu + [e]T ∇Ve = fV (9.17) ∇ · ([e]Bu − ε0 ∇Ve + P) = qe .

(9.18)

In a second approach, we follow the basic ideas discussed in [123] and decompose the physical quantities in a reversible and an irreversible part. Therefore, we introduce the reversible part Dr and the irreversible part Di of the dielectric displacement according to D = Dr + Di .

(9.19)

In our case, using the general relation between dielectric displacement D, electric field strength E, and polarization P we set Di = Pi . Analogously to (9.19), the mechanical strain S is also broken up into a reversible part Sr and an irreversible part Si S = Sr + Si .

(9.20)

9.4 Models for Non-linear Piezoelectricity

251

The decomposition of the strain S is done in compliance with the theory of elastic-plastic solids with the assumption of very small deformations [15] that is generally true for piezoceramic materials with maximum strains not more than 0.2 %. The reversible parts of mechanical strain Sr and dielectric displacement r D are described by the linear piezoelectric constitutive law (see (9.4) and (9.5)). Now, in contrast to the thermodynamically motivated approaches, we compute the irreversible polarization from the history of the driving electric field E by a hysteresis operator H (see next section) Pi = H[E] eP

(9.21)

with the unit vector of the polarization eP , set equal to the direction of the applied electric field. The butterfly curve for the mechanical strain can also be modelled by an enhanced hysteresis operator. Nevertheless, as it can be seen in Fig. 9.7, the mechanical strain S33 seems to be proportional to the squared dielectric polarization P3 2 (S ∝ P 2 ). The relation S i = β · (H[E])2 , with a model parameter β, seems obvious. However, to keep the model more general, the set-up (9.22) S i = β1 · H[E] + β2 · (H[E])2 + ... + βn · (H[E])n is chosen. Similar to [124] we define the tensor of the irreversibel strains as follows ' β1 · H[E] + β2 · (H[E])2 + · · · [Si ] = 32 (' ( eP eP T − 13 [I] . (9.23) + βn · (H[E])n The parameters β1 . . . βn need to be fitted to measured data. In practice, a fourth-degree polynomial has been sufficient. Furthermore, the entries of the tensor of the piezoelectric modulus are now assumed to be a function of the irreversible dielectric polarization Pi . Here the underlying idea is that the piezoelectric properties of the material only appear once the material is poled. Without any polarization, the domains in the material are not aligned and therefore coupling between the electrical side and the mechanical side does not occur. If the polarization is increased, the coupling also increases. Hence, the following relation is defined [e(Pi )] =

|Pi | Q [e] RT . Psat

(9.24)

Herein, Psat defines the saturation polarization, [e] the tensor of the constant piezoelectric moduli, and [e(Pi )] the tensors of the variable piezoelectric moduli. The two matrices R and Q perform a rotation of [e] in the direction of the

252

9 Piezoelectric Systems

Fig. 9.7. Measured mechanical strain S33 and squared dielectric polarization P32 on a piezoceramic actuator on different axis

polarization eP . Using the solid angles (α1 , α2 , α3 ) of the polarization vector we can compute R by ⎞ ⎛ cos α3 cos α2 cos α1 ⎟ ⎜ ⎟ R=⎜ ⎝ cos α1 cos α3 cos α2 ⎠ cos α2 cos α1 cos α3 and Q according to

 Q=

R 0

 .

0 R Finally, the constitutive relations for the electromechanical coupling can be established (9.25) D = [e(Pi )] Sr + [εS ] E + H[E]eP     n  3 1 (9.26) eP eTP − I [Si ] = βi (H[E])i S = Sr + Si 2 i=0 3 σ = [cE ] Sr − [e(Pi )]T E .

(9.27)

In (9.26) [Si ] denotes the symmetric tensor of the irreversible strain and Si the corresponding six-component vector using Voigt notation. For a further detailed discussion on this model we refer to [89]. Combining the new constitutive relations (9.25) - (9.27) with the governing equations defined by (9.6) and (9.9) we arrive at the following non-linear coupled system of PDEs

9.5 Hysteresis Model

) * * ) ρ¨ u − B T [cE ] Bu − Si + [e(Pi )]T ∇Ve = fV ) ) * * ∇ · [e(Pi )] Bu − Si − [εS ]∇Ve + Pi = qe

253

(9.28) (9.29)

with Pi = H[−∇Ve ]eP     n 3  1 i i T eP eP − I . [S ] = βi (H[−∇Ve ]) 2 i=0 3

9.5 Hysteresis Model One of the most general hysteresis models used is named after F. Preisach, who developed it in 1935. Preisach’s approach was purely intuitive and was based on plausible hypotheses concerning magnetic material behavior [173]. A mathematical-based investigation was performed by M. Krasnoselskii in the 1970s (see e.g., [132]). In order to get some physical as well as mathematical understanding, let us investigate some properties of Preisach’s hysteresis model. Thus, we consider an infinite set of elementary hysteresis operators Rβ,α , where each of them can be represented by a rectangular loop (see Fig. 9.8). Since we want to model

Fig. 9.8. Rectangular hysteresis loop

the hysteresis within dielectric materials, we choose for the input quantity the normalized electric field intensity e and for the output quantity the normalized polarization p according to e=

E Esat

p=

P . Psat

(9.30)

In (9.30) Esat denotes the saturated electric field intensity and Psat the saturated electric polarization. In Fig. 9.8 α and β are the up and down switching

254

9 Piezoelectric Systems

values and according to these switching values, the input will lead to an output value +1 or −1. Restricting the switching values to α ≥ β and |α|, |β| ≤ 1 leads to the following set S (see Fig. 9.9) (α, β) ∈ S with S = {(α, β) ∈ IR2 , |α|, |β| ≤ 1, β ≤ α} .

(9.31)

Therewith, we describe the class of hysteresis loops with closed major loop

Fig. 9.9. Set S for possible switching values α and β

[155]. Now, the Preisach operator for the electric polarization p computes as  p(t) = ℘(α, β)Rβ,α (e(t)) dαdβ . (9.32) S

In (9.32) ℘ denotes the Preisach function, which defines the shape of the hysteresis loops and fulfills the following properties [155] & 6 ≥ 0 for (α, β) ∈ S ℘(α, β) (9.33) = 0 for (α, β) ∈ /S  ℘(α, β) dαdβ = 1 (9.34) S

℘(−β, −α) = ℘(α, β) .

(9.35)

Now let us assume that the input e(t) increases monotonically up to a value of e1 at t = t1 . Thus, all Rβ,α operators with α less than e1 switch up, which means that their outputs take on the value of +1. Within the set S of possible (α, β) values, we will obtain a straight line parallel to the β-axis with α = e1 (see Fig. 9.10a). In the next step, we assume that the input e(t) starts to decrease monotonically to a value of e2 at t2 . Now, all Rβ,α operators with down-switching values β larger than e2 will turn back, so that their output

9.5 Hysteresis Model

255

takes on the value of −1. This leads to a straight line parallel to the α-axis with value β = e2 , which is illustrated in Fig. 9.10a. Therefore, as illustrated in Fig. 9.10a, we can subdivide the region S into S + (p takes on the value of +1) and S − (p takes on the value of −1).

(a) e(t) increases till e1 and decreases till e2

(b) Staircase line L(t)

Fig. 9.10. Decomposing into S + and S −

For the general case, a staircase line L(t) will subdivide S into S + and S − (see Fig. 9.10b) according to the following two rules: •

A monotonically increasing input signal e(t) defines a straight line parallel to the β-axis with value e(t).

•

A monotonically decreasing input signal e(t) defines a straight line parallel to the α-axis with value e(t).

Therefore, the horizontal lines represent relative maxima and the vertical lines relative minima. In addition, by storing the local maxima and minima, the hysteresis can be uniquely constructed. Due to the wiping-out property, not all relative maxima and minima have to be stored. This property states that each local input maximum wipes out the vertical L(t) whose α values are below this maximum, and each local minimum wipes out the vertices whose β values are above the minimum [155]. The wiping out is best illustrated by an input signal e(t) as displayed in Fig. 9.11. Only the relative maxima e1 and e3 as well as relative minima e2 and e4 have to be stored. All other maxima (minima) will be intermediately stored during the process in a list, but will be deleted due to the wiping-out property. Furthermore, the Preisach model fulfills the congruence property [155], which states that all minor hysteresis loops corresponding to back-and-forth variations of inputs between the same two consecutive extrema are congruent (see Fig. 9.12).

256

9 Piezoelectric Systems

Fig. 9.11. Input signal e(t) for illustration of the wiping out property

Fig. 9.12. Congruency property of the hysteresis model

For the numerical computation of the Preisach operator, the following efficient evaluation has been developed. With e1 , ..., en those relative input extrema that have not been wiped out yet at time t, the value of the output at time t computes as p(t) = E(−e1 , e1 ) + 2

n−1 

E(ei , ei+1 ) ,

(9.36)

i=1

with E(ei , ei+1 ) the Everett function (see Fig. 9.13)  E(e1 , e2 ) = ℘(α, β) dαdβ .

(9.37)

T (e1 ,e2 )

For the simplest Preisach function ℘(α, β) = 1/2, the Everett function computes as 1 E(e1 , e2 ) = (e2 − e1 )2 sgn(e2 − e1 ) . (9.38) 4

9.6 Numerical Computation

257

Fig. 9.13. Computation of the Everett function E (e1 , e2 )

Thus, we have an efficient model for taking into account ferroelectric hysteresis within piezoelectric materials. For a detailed discussion concerning hysteresis operators in PDEs, and especially their identification from measured data, we refer to [112].

9.6 Numerical Computation In the following we will first derive the discrete form of the linear piezoelectric partial differential equations applying the FE method. In the second step, we will investigate the numerical modelling of ferroelectric hysteresis within piezoelectric materials and finally, we will present simulation results. 9.6.1 Linear Case Let us consider a simple setup (without loss of generality) as displayed in Fig. 9.14. The strong formulation for the piezoelectric system reads as follows:

Fig. 9.14. Setup for the formulation

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9 Piezoelectric Systems

Given: u0 u˙ 0 V0 ρ, cij , eij , εij Find: u(t), Ve (t)

:Ω :Ω :Ω :Ω

→ IRd → IRd → IR → IR .

¯ × [0, T ] → IRd :Ω

) * ¨ − B T [cE ]Bu − [e]T ∇Ve = 0 ρu ) * ∇ · [e]Bu − [εS ]∇Ve = 0 .

(9.39) (9.40)

Boundary conditions u = 0 on Γeu × (0, T ) Ve = 0 on ΓeV1 × (0, T ) Ve = V0 on ΓeV2 × (0, T ) n · [σ] = 0 on Γnσ × (0, T ) n · D = 0 on ΓnD × (0, T ) . Initial condition u(r, 0) = u0 , r ∈ Ω ˙ 0) = u˙ 0 , r ∈ Ω . u(r, In this strong formulation of the piezoelectric partial differential equations A denotes the surface area covered by the loaded electrode. The variational formulation for this case with u
and ψ defining appropriate test functions is   

T E ˜ e dΩ = 0 ¨ dΩ + (Bu ) [c ]Bu dΩ + (Bu
)T [e]T BV ρu · u Ω

Ω

Ω



 ˜ (Bψ) e(Bu) dΩ −

˜ T [εS ] BV ˜ e dΩ = 0 (9.41) (Bψ)

T

Ω

Ω

with B˜ = (∂/∂x, ∂/∂y, ∂/∂z)T . Now, using standard nodal finite elements for the mechanical displacement u and electric potential Ve (nn denotes the number of nodes with unknown displacement and unknown electric potential)

u ≈ uh =

nd  nn 

Na uia ei =

i=1 a=1

Ve ≈ Veh =

nn  a=1

Na Vea

nn  a=1

⎞ Na 0 0 ; Na = ⎝ 0 Na 0 ⎠ (9.42) 0 0 Na ⎛

Na ua

(9.43)

9.6 Numerical Computation

259

as well as for the test functions u
and ψ, we obtain ( [6, 146]) ⎛   nn  nn  ⎝ ρNTa Nb dΩ u ¨ b + BaT [cE ]Bb dΩ ub a=1 b=1

Ω

Ω

 +

⎞ BaT [e]T B˜b dΩ Veb ⎠ = 0

(9.44)

Ω

nn  nn  a=1

⎛ ⎝

b=1





T B˜a e Bb dΩ ub −

Ω

In (9.44) B˜a computes as

B˜aT [εS ] B˜b dΩ Veb = 0 . Ω

⎛ ∂Na ⎞ ∂x

⎜ ∂N ⎟ a ⎟ B˜a = ⎜ ⎝ ∂y ⎠ .

(9.45)

∂Na ∂z

Introducing damping with a damping matrix Cu (see Sect. 3.7.2) we may write (9.44) and (9.45) in matrix form            u¨ u˙ 0 Cu 0 Ku KuV u Mu 0 + + = , fe 0 0 0 0 KTuV −KV Ve V¨e V˙e (9.46) with the matrices Mu , Cu , Ku as given in Sect. 3.7.1, KV in Sect. 4.6 and f e a right hand side due to the electric Dirichlet boundary conditions. The matrix KuV computes as KuV =

ne 

 BpT [e]T B˜q dΩ

keuV ; keuV = [kpq ] ; kpq =

e=1

Ωe

For the time discretization, a Newmark algorithm as described in Sect. 2.5.2 is used. 9.6.2 Non-linear Case Here, we will only consider the simplified non-linear model for piezoelectricity described by the system of PDEs defined in (9.17), (9.18) and note that the more general model (see (9.28), (9.29)) can be treated in a similar manner. In a straightforward application of the FE method, one would treat the non-linear term ∇ · P = ∇ · H[E] just as a right hand term within a fixed-point iteration. However, this approach will lead to a very poor convergence resulting in a large number of iteration steps or even nonconvergence. Therefore, we modify (9.18)

260

9 Piezoelectric Systems

by introducing a differential permittivity tensor [εd ]. We first decompose the electric displacement vector D at time step n + 1 to the value of the previous time step n and an incremental displacement vector ∆D Dn+1 = Dn + ∆D .

(9.47)

We now define the incremental displacement vector ∆D according to (9.16) by (9.48) ∆D = [e][∆S] + [εd ]∆E and use the differential permittivity tensor [εd ] to represent 0 E + P . For the computation of Dn we use (9.16) in its original form Dn = [e][Sn ] + 0 En + Pn .

(9.49)

We can now rewrite Maxwell’s equation for the electrostatic field in the case of a piezoelectric material as follows ∇ · Dn+1 = ∇ · (Dn + ∆D) = 0 .

(9.50)

By using (9.47) - (9.49) we obtain ) ** ) * ) ∇ · [e][Sn ] + 0 En + Pn + [e] [Sn+1 ] − [Sn ] − [εd ] ∇Ven+1 − ∇Ven = 0 (9.51) which results in ) * ) * ∇ · [e][Sn+1 ] − [εd ]∇Ven+1 = ∇ · (ε0 [I] − [εd ])∇Ven − Pn (9.52) with [I] the identity tensor. The entries of the differential permittivity tensor [εd ] = diag (εd1 , εd2 , εd3 ) are computed according to (9.48) ' ( * ) n+1 n+1 n n S − D − S − e D ijl i i jl jl ∆Di − eijl ∆Sjl εdi = = . (9.53) ∆Ei Ein+1 − Ein Now, the new system of coupled PDEs to be solved at time step n + 1 reads as follows ) * ¨ − B T [cE ]Bu − [e]T ∇Ve = 0 ρu (9.54) * ) * ) n+1 d n+1 d n n = ∇ · (ε0 [I] − [ε ])∇Ve − P . (9.55) − [ε ]∇Ve ∇ · [e]S Applying the FE method to the above equations and using the Newmark scheme for time discretization will lead to the following non-linear algebraic system of equations  ∗   n+1    uk+1 Ku KuV 0 = , (9.56) n KNL2 V k Ve − f k+1 Ve n+1 KT −KNL1 uV

Vk

k+1

9.7 Numerical Examples

261

In (9.56) n denotes the time step counter, k the iteration counter and K∗u computes as Ku + 1/(γH ∆t2 )Mu . The matrices Ku , Mu and KuV compute NL2 as for the linear case, and the non-linear matrices KNL1 V k , KV k as well as the non-linear right-hand side f k+1 can be calculated as follows KNL1 Vk

=

ne 

 e

k

; k = [kpq ] ; kpq =

e=1

KNL2 Vk =

ne 

Ωe

 ke ; ke = [kpq ] ; kpq =

e=1

f k+1 =

ne 

B˜pT [εk d ]B˜q dΩ

e

) * B˜pT ε0 [I] − [εdk ] B˜q dΩ

Ωe

 ˜ e n+1 ] dΩ Np B˜T H[−BV k

f e ; f e = [f p ] ; f p =

e=1

Ωe

[εdk ] = diag (εd1 , εd2 , εd3 ) εdi =

' ( ˜ e n+1 )i − (BV ˜ e n ]i − ε0 (BV ˜ e n )i ˜ e n+1 ]i − H[−BV H[−BV k k ˜ e n+1 )i + (BV ˜ e n )i −(BV k

.

9.7 Numerical Examples In the following, we will discuss in the first example the computation of the electric impedance of a piezoelectric disc. In the second example we will perform computations for large signal loading of a piezoelectric transducer using the nonlinear formulation as described in Sec. 9.6.2. 9.7.1 Computation of Impedance Curve We consider a transducer as shown in Fig. 9.15, i.e., a rotational symmetric disc with electrodes on top and bottom. In order to obtain the whole impedance characteristics in one simulation run, a special technique must be used. The model of the transducer is excited by an electric voltage pulse and its charge response is computed by means of transient analysis. In general, the following equation for the impedance characteristics Z(ω) is valid Z(ω) =

U (ω) U(ω) = , I(ω) jωQ(ω)

(9.57)

where U (ω), I(ω), and Q(ω) are Fourier spectra of the voltage, the current, and the charge time signal, respectively. One can see that the impedance characteristics can be calculated by dividing the spectrum of the voltage signal (excitation) by the jω-multiple of the computed spectrum of the electric

262

9 Piezoelectric Systems

Symmetry axis Upper electrode

u, q

Lower electrode Fig. 9.15. Piezoelectric transducer

charge. Within a postprocessing step we can compute the electric charge as a function of time  '  ( ˜ e · dΓ . [e]Bu − [εS ]BV (9.58) D · dΓ = q(t) = Γe

Γe

In (9.58) Γe denotes the electrode surface, e.g., of the loaded electrode and u and Ve the FE solution quantities. After performing the Fourier transformation of q(t) we obtain Q(ω). The voltage signal must obviously have a frequency spectrum that is able to excite the transducer in the interesting frequency range. It is recommended to choose a signal whose spectrum is nonzero up to a frequency of at least 10fr (one order higher than the resonance frequency fr of the transducer). Due to the rotational symmetry, the transducer is modelled by an axisymmetric formulation. In addition, the planar symmetry of the transducer is utilized in order to reduce the size of the model (see Fig. 9.16). The geometrical dimensions of the electrode are neglected, i.e., it is modelled as an infinitely thin layer of surface nodes forming an equipotential area. The well-known piezoelectric ceramic material PZT-5A is assumed (see Table 9.1). According to Fig. 9.16, the radius of the transducer is R = 10 mm and its thickness is D = 2 mm. Since the estimated resonance frequency can be computed by c , (9.59) fr = 2D we obtain, with c ≈ 4 000 m/s, a value of 1 MHz. Hence, the excitation voltage signal should have a nonzero spectrum up to 10 MHz. The selected signal has a form of a triangular pulse (see Fig. 9.17). Due to the fact that a linear analysis is applied here, the magnitude of the pulse has no influence on the resulting impedance characteristics. Therefore, a unit pulse is taken for simplicity. The boundary conditions applied to the model are illustrated in Fig. 9.18. No displacement in the horizontal direction is allowed along the symmetry

9.7 Numerical Examples

263

Fig. 9.16. Model of the piezoelectric transducer

Fig. 9.17. Electric voltage excitation of the transducer

axis and, correspondingly, no displacement in the vertical direction is allowed at the symmetry plane. Furthermore, the symmetry plane serves as the reference surface for electric potential, i.e., zero electric potential is prescribed there. Finally, the upper surface, which represents the electrode, is an equipotential area, and the electric load is applied there. For the discretization of the

Fig. 9.18. Boundary conditions applied to the model of the transducer

264

9 Piezoelectric Systems

simulation domain, linear finite elements have been used. Due to the simple domain, a mapped meshing with five elements in thickness and 50 elements in the radial direction is performed. The time step size ∆t is set to 20 ns and a total number of 8192 time steps have been computed. Therefore, we achieve a frequency resolution of about 3 kHz and ten time samples for the triangular excitation charge. After the simulation, a Fourier transformation for both the computed

Fig. 9.19. Computed impedance characteristics of the piezoelectric transducer

charge signal and the excitation voltage signal is performed. The computed impedance according to (9.57) is displayed in Fig. 9.19. One can clearly see the resonance and antiresonance points of the various vibration modes of the transducer. Furthermore, the principal thickness mode of the transducer appears at the frequency of about 1 MHz, but it is disturbed by additional modes. When designing such a piezoelectric transducer, this effect can be suppressed by increasing the diameter/thickness ratio, which leads to a better decoupling of the particular vibration modes. 9.7.2 Non-linear Case In this section we will demonstrate measured and simulated data for largesignal loading of a piezoelectric disc (see Fig. 9.20), using for the numerical simulation the scheme described in Sect. 9.6.2. The disc transducer can be described by an axisymmetric model similar to the previous example (see Fig. 9.16). The geometrical dimensions of the elec-

9.7 Numerical Examples

u~ P

265

D

2R Fig. 9.20. Setup for the large-signal loading

trode are neglected. According to Fig. 9.20, the radius of the transducer is R = 5.5 mm and its thickness is D = 0.2 mm. The input voltage is a sine signal with an amplitude of 50 V and a frequency of 1 MHz. According to measurements of the hysteresis curve, the following parameters are used for the Preisach model: Parameter

Value

Saturation of electric field intensity Esat 2.0 MV/m Saturation of electric polarization Psat 0.04 C/m2 The displacement in the r-direction along the axis of symmetry and the displacement in the z-direction along the symmetry plane (z = 0) are set to zero. On the top electrode, the input signal is prescribed (inhomogeneous Dirichlet boundary condition) and along the symmetry plane (z = 0) the electric potential is set to zero. For the discretization of the simulation domain linear finite elements have been used. Due to the simple domain, a mapped meshing with four elements in thickness and 80 elements in the radial direction is performed. The time step size ∆t is set to 25 ns (which means a total of about 50 time samples per fundamental period). Changes in the state of ferroelectric polarization are mainly responsible for the characteristics of the piezoceramic disc. In order to overcome within the experiement strong thermal influences due to the heat generation caused by internal friction in the piezoceramic material, a sine burst signal with a small pulse/pause ratio is used for excitation. In Fig. 9.21(a) the time signal of the input current and voltage can be observed, whereas in Fig. 9.21(b) the spectral rates of the higher-order harmonics related to the fundamental frequency are plotted. These higher-order harmonics and the nonsymmetric response signal are caused by the ferroelectric hysteresis.

9 Piezoelectric Systems 50

50 40

Voltage (measured) Current (measured) Current (simulated)

30 20

10

10

0 -10 -20

0 -10

-30 -40 -50

-30 -40

I (mA)

40 30 20

U (V)

266

-20

-50 0

0.5

1

1.5

2

t (µs)

(a) Input voltage and current

(b) Higher-order frequency spectra Fig. 9.21. Measured and simulated data for the dynamic load case

10 Computational Aeroacoustics

A large amount of the total noise in our daily lives is generated by turbulent flows (e.g., airplanes, cars, air conditioning systems, etc.). The physics behind the generation process is quite complicated and still not fully understood. The use of numerical simulation tools is one important way to analyze the generation of flow-induced sound, e.g., [206]. In the following, we will first discuss the arising requirements for numerical schemes, when computing flow-induced noise. Then, we will focus on Lighthill’s acoustic analogy and derive an FE formulation of the inhomogeneous wave equation. Finally, we will present the numerical computation of a co-rotating vortex pair and the comparison to the analytical solution.

10.1 Requirements for Numerical Schemes Since the beginning of computational aeroacoustics (CAA) several numerical methodologies have been proposed, each of these trying to overcome the challenges that the specific problems under investigation pose for an effective and accurate computation of the radiated sound. The difficulties which have to be considered for the simulation of flow noise problems include [86, 87]: •

Energy disparity and acoustic inefficiency: There is a large disparity between the energy of the flow in the non-linear field and the radiated acoustic energy of an unsteady flow. In general, the total radiated power of a turbulent jet scales with O(v 8 /c5 ) (v is the characteristic flow velocity and c the speed of sound), and for a dipole source arising from pressure fluctuations on surfaces inside the flow scales with O(v 6 /c3 ). This shows that an aeroacoustics process at low Mach number is rather a poor sound emitter.

•

Length scale disparity: Large disparity also occurs between the size of an eddy in the turbulent flow and the wavelength of the generated acoustic

268

10 Computational Aeroacoustics

noise. Low Mach number eddies have a characteristic length scale l, velocity v, a life time l/v and a frequency ω. This eddy will then radiate acoustic waves of the same characteristic frequency, but with a much larger length scale, which scales as follows l l = . (10.1) v Ma In (10.1) Ma denotes the Mach number, which is defined by the ratio of the characteristic velocity v over the speed of sound c λ∝c

Ma =

v . c

•

Preservation of multipole character: The numerical analysis must preserve the multipole structure of the acoustic source in order to resolve the whole structure of the source.

•

Dispersion and dissipation: The discrete form of the acoustic wave equation cannot precisely represent the dispersion relation of the acoustic sound. Numerical discretization in space and time converts the original nondispersive system into a dispersive discretized one, which exhibits wave phenomena of two kinds: 1. Long wavelength components approaching the solution of the original PDE as the grid is refined. 2. Short wavelength components (spurious waves) without counterpart in the original PDE evolving in the numerical scheme disturbing the solution. The wave equation shows a non-dissipative behavior; as such, dissipative errors must be avoided by a numerical implementation, in which both the amplitude and phase of the wave are of crucial importance.

•

Difficulties in non-linear wave phenomena: In turbulent flows having a high speed, non-linear effects will play a role since the wave equation has to be solved over a long range, which induces then dissipation of acoustic energy and refraction.

•

Flows with high Mach and Reynolds number: Aeroacoustic problems often involve both high Mach and Reynolds numbers. Flows at a high Mach number may induce new non-linear sources and convective effects while flows at a high Reynolds number introduce multiple scale difficulties due to the disparity between the acoustic wavelength λ and the size of the energy dissipating eddies.

10.1 Requirements for Numerical Schemes

•

269

Simulation of unbounded domains: As a main issue for the simulation of unbounded domains using interior methods remains the boundary treatment which needs to be applied to avoid the reflection of the outgoing waves on the truncating boundary of the computational domain (see Sect. 5.5).

Currently available aeroacoustic methodologies overcome only some of these broad range of numerical and physical issues, which restricts their applicability, making them, in many cases, problem dependent methodologies. The application of Direct Numerical Simulation (DNS) is becoming more feasible with the permanent advancement in computational resources. However, due to the large disparities of length and time scales between fluid and acoustic fields, DNS remains restricted to low Reynolds number flows. In a DNS, all relevant scales of turbulence are resolved and no turbulence modelling is employed. Therefore, although some promising work has been done in this direction [68], the simulation of practical problems involving high Reynolds numbers requires very high resolutions which are still far beyond the capabilities of current supercomputers [211]. Hence, hybrid methodologies have established as the most practical methods for aeroacoustic computations, due to the separate treatment of the fluid and the acoustic computations. In these schemes, the computational domain is split into a non-linear source region and a wave propagation region, and different numerical schemes are used for the flow and acoustic computations. Herewith, first a turbulence model is used to compute the unsteady flow in the source region. Secondly, from the fluid field, acoustic sources are evaluated which are then used as input for the computation of the acoustic propagation. In these coupled simulations it is generally assumed that no significant physical effects occur from the acoustic to the fluid field. Figure 10.1 shows typical numerical methods which are employed when using any of these hybrid methodologies. Among the group of hybrid approaches, integral methods remain widely used in CAA for solving open problems in large acoustic domains like airframe noise, landing gear noise simulation, fan (turbines) noise, rotor noise, etc. One reason which motivates the use of integral formulations in such applications is that, in general, their acoustic sources can be considered to be compact and only an extension of the acoustic solution at a few points in the far field is desired. Therefore, in such cases, integral methods based on Lighthill’s acoustic analogy, Curle’s formulation, Ffowcs Williams and Hawkings (FW-H) formulation, Kirchhoff method or extension thereof are computationally cheaper than interior methods where a whole discretization of the acoustic domain is required (see e.g., [34, 50, 63, 150]). On the other hand, for interior aeroacoustic problems, where non-compact solid boundaries are present, or if structural/acoustic effects are considered, it is more appropriate to use an acoustic interior method to account for the interactions between the solid surfaces and the flow-induced noise directly in the acoustic simulation. In such cases, integral formulations would require a

270

10 Computational Aeroacoustics

priori knowledge of a hard-wall Green’s function that is not known for complex geometries [165]. Furthermore, integral methods do not allow for a straightforward inclusion of the elastic effects of structures in the flow. An additional advantage of interior methods is that they can also be used to include the effects of wave propagation in non-uniform background flows. Among the interior methods we find those based on Linearized Euler Equations (LEE) [12, 55], Acoustic Perturbation Equation (APE) [62], FE formulations of Lighthill’s acoustic analogy [165], as well as the linearized perturbed compressible equations (LPCE) [196] (for a discussion on these methods see [61]). Fig. 10.1 depicts the general configuration when using these methods. Herewith, ΩF denotes the area where the flow field is firstly computed and where the acoustic sources are interpolated from the fluid mesh to the acoustic mesh. In order to accurately resolve the source terms, unsteady computational fluid dynamics (CFD) schemes are required. Mainly used turbulence models are LES (Large Eddy Simulation), DES (Detached Eddy Simulation) and SAS (Scale Adaptive Simulation). For a detailed discussion on these 7 methods we refer to [35]. The acoustic propagation region is given by ΩF ΩA where the acoustic field is computed in the second step by solving the inhomogeneous wave equation or a corresponding set of equations depending on the CAA methodology followed. Since interior methods require the whole discretization of the propagation domain, usually they are used to compute the radiated sound until an intermediate region in the far field (i.e., until ΓA in Fig. 10.1), before moving to an integral formulation in which the acoustic solution from the interior method at the interface is used as input for computing pressure levels at the far field. Such a combined scheme has been presented in [151] using LEE for the intermediate solution and a Kirchhoff method for the far field noise.

10.2 Lighthill’s Analogy and its Extension Let us start our derivation of Lighthill’s analogy by introducing the two main equations of fluid dynamics: mass conservation and momentum conservation. The differential formulation of the mass conservation (continuity) equation reads as follows ∂ρ ∂ρvi + = 0. (10.2) ∂t ∂xi In (10.2) ρ denotes the density of the fluid and vi the i-th component of the flow velocity vector v. The momentum conservation equation is expressed by ρ

∂vi ∂vi ∂p ∂τij ∂ + ρvj =− − =− (pδij + τij ) , ∂t ∂xj ∂xi ∂xj ∂xj

(10.3)

where p denotes the total pressure within the flow, [τ ] the viscous stress tensor and δ the Kronecker symbol. For Newtonian fluids, we can express [τ ] by

10.2 Lighthill’s Analogy and its Extension

271

Fig. 10.1. Schematic depicting some of the possible strategies when using an aeroacoustic hybrid approach

 τij = −µ

∂vi ∂vj + ∂xj ∂xi



∂vk 2 + µδij 3 ∂xk

(10.4)

with µ the dynamic viscosity. In the first step, we multiply (10.2) by vi (before we substitute the index i by j) and add the so obtained equation to (10.3). In addition, we substitute p in (10.3) by p − p0 with p0 the mean pressure. This operation is allowed, since the relation ∂(p − p0 )/∂xj = ∂p/∂xj holds. Therewith, we obtain ∂vi ∂ ∂ρ ∂vi ∂ρvj ρ =− + vi ((p − p0 )δij − τij ) + vi + ρvj ∂xj  ∂t  ∂t  ∂xj  ∂xj ∂ρvi ∂ (ρvi vj ) ∂t ∂xj

(10.5)

By introducing the momentum flux tensor [τ I ] by τijI = ρvi vj + (p − p0 )δij − τij

(10.6)

∂τijI ∂ρvi =− . ∂t ∂xj

(10.7)

we may write (10.5) as

272

10 Computational Aeroacoustics

For linear acoustics, the momentum flux tensor [τ I ] reduces to τij0 = (p − p0 )δij

(10.8)

∂ ∂ρvi + (p − p0 ) = 0 . ∂t ∂xi

(10.9)

and (10.7) reads as

In addition, we define the acoustic pressure p
, the acoustic density ρ
and its relation (just fulfilled for linear acoustics) p
= p − p0 ;

ρ
= ρ − ρ0 ;

p
= c2 ρ


(10.10)

with c the speed of sound. Substituting p − p0 in (10.9) by c2 ρ
and applying ∂/∂xi to this equation, results in ∂ ∂ ∂2 (ρvi ) + 2 (c2 ρ
) = 0 . ∂t ∂xi ∂xi

(10.11)

According to the mass conservation equation, we may rewrite (10.11) as   1 ∂2 ∂ 2 ρ
∂2 2
∂2 − 2 (c ρ ) = − 2 (c2 ρ
) = 0 . (10.12) ∂t2 ∂xi c2 ∂t2 ∂xi Since we have neglected the flow and there are no other excitations, no acoustic sound field will be generated, since the solution of (10.12) is ρ
= ρ−ρ0 = 0. Lighthill’s analogy is based on the fact that the sound generated by the flow in a real fluid is exactly equivalent to that produced in the ideal, linear acoustic field, forced by the stress distribution Lij = τijI − τij0

) * = ρvi vj + (p − p0 ) − c2 (ρ − ρ0 ) δij − τij ,

(10.13)

where [L] denotes the Lighthill stress tensor. We can now rewrite (10.7) as the momentum equation for the ideal, linear acoustic medium subjected to the externally applied stress according to (10.13) ∂τij0 ∂(τijI − τij0 ) ∂ρvi + =− (10.14) ∂t ∂xj ∂xj Therewith, we obtain * ∂ρvi ∂ ) 2 ∂Lij + c (ρ − ρ0 ) = − . ∂t ∂xi ∂xj

(10.15)

Eliminating the momentum density ρvi similarly as for the linear case, we obtain

10.2 Lighthill’s Analogy and its Extension



1 ∂2 ∂2 − c2 ∂t2 ∂x2i

 (c2 (ρ − ρ0 )) =

∂ 2 Lij . ∂xi ∂xj

The integral formulation of (10.16) is given by  1 ∂Lij (ξ, t − r/c) ∂2 dξ dη dζ c2 (ρ − ρ0 )(x, t) = 4π ∂xi ∂xj r

273

(10.16)

(10.17)

Ω1

with the source coordinates ξ = (ξ, η, ζ)T , the observation coordinates x = (x1 , x2 , x3 )T and the distance vector r = (x1 − ξ, x2 − η, x3 − ζ)T . The integral formulation given in (10.17) is no longer correct, if any solid and/or elastic bodies (in rest or moving) are present within the simulation domain. For such situations, the acoustic field has to fulfill the correct boundary condition and we no longer can use Green’s function for free radiation. Now, the idea is, to derive an extended form of Lighthill’s equation, which is defined in the whole simulation domain, and thus Green’s function for free radiation can be applied to obtain an integral formulation [65]. We will assume a domain with a solid body, which is located within the considered domain Ω. The volume of the body is denoted by ΩS and its surface by ΓS (see Fig. 10.2). We describe the surface of the body with a

Fig. 10.2. Computational domain Ω with a solid body ΩS

function f (x, t)

⎧ f < 0 for x ∈ ΩS ⎪ ⎪ ⎨ f (x, t) = f > 0 for x ∈ Ω \ ΩS ⎪ ⎪ ⎩ f = 0 for x on ΓS

(10.18)

e.g., a breathing sphere: f (x, t) = |x−x0 |2 −R02 (t). Additionally, we introduce the Heaviside function H(f (x, t)) 1 for x ∈ Ω \ ΩS H(f (x, t)) = (10.19) 0 for x ∈ ΩS .

274

10 Computational Aeroacoustics

First of all, we multiply the continuity equation (see (10.2)) with H(f )  
∂ρ ∂ + (ρvi ) H(f ) = 0 (10.20) ∂t ∂xi and rewrite it in the following form ∂ ∂H(f ) ∂
∂H(f ) (ρ H(f )) − ρ
+ ((ρvi )H(f )) − (ρvi ) = 0 . (10.21) ∂t ∂t ∂xi ∂xi The Heaviside function H(f ) fulfills two important properties [96] ∂H(f ) ∂f = δ(f ) ∂xi ∂xi ∂H(f ) ∂f ∂f = δ(f ) = −viB δ(f ) ∂t ∂t ∂xi

(10.22) (10.23)

with viB the velocity of the solid body in Ω. Using (10.22) and (10.23) simplifies (10.21) to * ∂f ∂(ρ
H(f )) ∂(ρvi H(f )) ) + = ρ(vi − viB ) + ρ0 viB δ(f ) . ∂t ∂xi ∂xi

(10.24)

This equation is an extension of the continuity equation, which is fulfilled all over Ω. As can be seen, the right-hand side of (10.24) is just on ΓS different from zero and the left side is zero within ΩS . Applying similar operations to the momentum conservation equation, we obtain ∂ ∂ (ρvi H(f )) + (ρvi vj + p
δij − τij ) H(f ) ∂t ∂xj * ∂f ) δ(f ) . = ρvi (vj − vjB ) + p
δij − τij ∂xj

(10.25)

Using the definition of the Lighthill tensor Lij (see 10.13)) we may write (10.25) as ∂ 2
∂ (ρvi H(f )) + (c ρ H(f )) ∂t ∂xi * ∂f ∂Lij H(f ) ) =− + ρvi (vj − vjB ) + p
δij − τij δ(f ) .(10.26) ∂xj ∂xj Furthermore, we rewrite (10.24) in the following form * ∂f ∂(ρvi H(f )) ) ∂(ρ
H(f )) = ρ(vi − viB ) + ρ0 viB δ(f ) − . ∂t ∂xi ∂xi

(10.27)

10.3 Finite Element Formulation

275

Applying ∂/∂xi to (10.26) and using the relation according to (10.27), we arrive at the extended form of Lighthill’s equation   * 1 ∂2 ∂2 ) 2
− 2 c ρ H(f ) c2 ∂t2 ∂xi   ) * ∂f ∂ ∂2 B
ρvi (vj − vj ) + p δij − τij (Lij H(f )) − δ(f ) = ∂xi ∂xj ∂xi ∂xj   * ∂f ∂ ) ρ(vi − viB ) + ρ0 viB + δ(f ) . (10.28) ∂t ∂xi According to the relations ∂f = nj ∂xj

∂f = ni ∂xi

the terms (vj − vjB ) and (vi − viB ) vanishes in (10.28). Since the extended form of Lighthill’s equation is valid throughout the whole domain Ω, we can use Green’s function for free radiation to obtain a solution. Therefore, the integral form, which is known as the Ffowcs Williams and Hawkings equation [65], reads as  ∂ Lij (p
δij − τij ) ∂2 dΩ − dΓ c2 ρ
H(f ) = ∂xi ∂xj 4π r ∂xi 4π r ΓS

Ω\ΩS

+

∂ ∂t

-

ρ0 viB dΓ . 4π r

(10.29)

ΓS

10.3 Finite Element Formulation We will perform a volume discretization of Lighthill’s equation (see 10.16) by applying the finite element method (FEM). Therewith, any solid–elastic body will be implicitly taken into account, and there is no need to use the extended form of Lighthill’s equation as given by (10.28). Let us start at the strong formulation of (10.16), where we substitute ρ
by p
/c2 Given: Lij : Ω × (0, T ) → IR c : Ω → IR p
0 : Ω → IR p˙
0 : Ω → IR ¯ × [0, T ] → IR Find: p
: Ω

276

10 Computational Aeroacoustics

1 ∂ 2 p
∂ 2 p
∂ 2 Lij − = c2 ∂t2 ∂x2i ∂xi ∂xj

(10.30)

p
(r, 0) = p
, r ∈ Ω 0 0

p˙
(r, 0) = p˙
, r ∈ Ω In ( 10.30) Ω define the total computational domain, which consists of ΩF ∪ΩA as displayed in Fig. 10.1. Therewith, ΩF denotes the computational domain for the flow computation, where we evaluate the acoustic source terms, and ΩA the acoustic propagation domain. At this stage we assume that the Lighthill tensor [L] is a known quantity, e.g., obtained by a flow computation using a large eddy simulation (LES). In the first step, we multiply (10.30) by an appropriate test function w and integrate over the whole domain Ω (corresponding in Fig. 10.1 to ΩF ∪ ΩA )    1 ∂ 2 p
∂ 2 p
∂ 2 Lij w 2 − − dΩ = 0 . (10.31) c ∂t2 ∂x2i ∂xi ∂xj Ω

Now, we apply Green’s integral theorem to the second spatial derivative of p
as well as Lij . This operation will result in the following relations  Ω

 w Ω

∂ 2 p
w 2 dΩ = ∂xi ∂ 2 Lij dΩ = ∂xi ∂xj

 ΓS ∪ΓA

 w

∂p
dΓ − w ∂n

∂Lij ni dΓ − ∂xj

ΓS



∂w ∂p
dΩ ∂xi ∂xi

(10.32)

∂w ∂Lij dΩ . ∂xi ∂xj

(10.33)

Ω

 ΩF

We want to emphasize that the boundary integral (10.33) is just over the surface ΓS of any solid–elastic body, whereas in (10.32) we have to integrate over ΓS as well as over ΓA , which limits the computational domain (see Fig. 10.1). Now we can substitute ∂Lij /∂xj within the first term on the right-hand side of (10.33) by (10.15) and obtain     ∂ 2
∂Lij ∂ρvi − w ni dΓ = w − (c ρ ) ni dΓ . (10.34) ∂xj ∂t ∂xi ΓS ΓS Since on a solid surface vi ni = 0 is fulfilled, we see that the surface integral term over ΓS reduces to    ∂Lij ∂ρ
∂p
dΓ = − dΓ . (10.35) w ni dΓ = − c2 w w ∂xj ∂n ∂n ΓS ΓS ΓS Therewith, we can rewrite (10.31) as

10.3 Finite Element Formulation



1 ∂ 2 p
w + c2 ∂t2

Ω



∂w ∂p
dΩ − ∂xi ∂xi

Ω

 =−

 w ΓS ∪ΓA

∂w ∂Lij dΩ − ∂xi ∂xj

 w ΓS

ΩF

277

∂p
dΓ ∂n

∂p
dΓ . ∂n

(10.36)

Combining the surface integrals results in a single boundary integral just performed over the outer boundary ΓI , on which we, e.g., apply absorbing boundary conditions of first order (see Sect. 5.5.1). Therewith, we utilize the relation ∂p
∂p
=− (10.37) c ∂n ∂t and arrive at the weak form of (10.30): Find p
∈ H 1 such that     1 ∂ 2 p
∂w ∂p
1 ∂p
∂w ∂Lij w + dΩ + dΩ w dΓ = − c2 ∂t2 ∂xi ∂xi c ∂t ∂xi ∂xj Ω

Ω

ΓA

ΩF

for any w ∈ H 1 . Using standard nodal finite elements, we approximate the continuous acoustic pressure p
as well as the test function w by p
≈ p
= h

neq 

Na p
a

(10.38)

Na wa .

(10.39)

a=1 neq

w ≈ wh =



a=1

Thus, (10.38) is transformed to the following semidiscrete Galerkin formulation (10.40) Mp¨
n+1 + Cp˙
n+1 + Kp
n+1 = f n+1 with p¨
= ∂ 2 p
/∂t2 , p˙
= ∂p
/∂t, p
the nodal unknowns of the acoustic pressure and n the time step counter. The matrices as well as right-hand side vector compute as follows: M=

ne 

 me ; me = [mpq ] ; mpq =

e=1

Ωe



nΓI

C=



e=1

1 Np Nq dΩ c2

ce ; ce = [cpq ] ; cpq = ΓIe

1 Np Nq dΓ c

278

10 Computational Aeroacoustics

K=

ne 

 ke ; ke = [kpq ] ; kpq =

e=1

f n+1 =

ne  e=1

Ωe

 f e ; f e = [fp ] ; fp = Ωe

∂Np ∂Nq dΩ ∂xi ∂xi n+1

∂Np ∂Lij ∂xi ∂xj

dΩ .

In the above equations ne is the number of finite elements,  nΓI the number of finite surface elements along the outer boundary ΓI and the finite element assembly operator. The time discretization is performed by applying a standard Newmark algorithm as described in Sec. 2.5.2. By performing a harmonic analysis, it is possible to compute the sound radiation for specific frequency components present in the acoustic sources. In this way, we obtain the complex acoustic pressure at each node in the computational domain. For deriving the harmonic formulation of the implementation, we can simply apply a Fourier-transformation to the semidiscrete Galerkin formulation from (10.40), since the matrices M, C and K are frequency independent. The resulting complex algebraic system of equations is given by * ) (10.41) − ω 2 M + iωC + K pˆn+1 = fˆn+1 , where the source term fˆ represents the complex nodal acoustic sources, which are obtained by applying a Fourier transformation to the dataset of transient nodal sources interpolated from the fluid grid to the acoustic grid. Great care has to be taken by the interpolation of the computed nodal sources from the fine flow grid to the coarser acoustic grid. Within the FE formulation we perform an integration over the volume (corresponds to the computational flow region) and project the results to the nodes of the fine flow grid, which has to be interpolated to the coarser acoustic grid. Therewith, our interpolation has to be conservative in order to preserve the total acoustic energy. As illustrated in Fig. 10.3, we have to find for each nodal source fkF in which finite element of the acoustic grid it is located. Then, we compute from the global position (xk , yk ) its position (ξk , ηk ) in the reference element. This is in the general case a non-linear mapping and is solved by a Newton scheme. Now, with this data we can perform a bilinear interpolation and add the contribution of fkF to the nodes of the acoustic grid by using the standard finite element basis functions Ni fiA = Ni (ξk , ηk )fkF . Therewith, by this procedure the interpolation preserves the overall sum of the acoustic source.

10.4 Validation: Co-Rotating Vortex Pair

279

Fig. 10.3. Standard conservative interpolation on a quadrilateral mesh

10.4 Validation: Co-Rotating Vortex Pair In this section we validate the FE implementation of Lighthill’s acoustic analogy, by computing the far field caused by an incompressible, purely unsteady vortical flow. The weak formulation of Lighthill’s inhomogeneous wave equation is forced with the acoustic sources obtained from the hydrodynamic field induced by a co-rotating vortex pair. The resulting acoustic field represents the basic acoustic field generated by turbulent shear flows, jet flows, edge tones, etc. [139, 171]. The analytical solution for the acoustic far field, used for the validation of the acoustic results, has been obtained employing the method of matched asymptotic expansion (MAE), first presented in [159]. The computation of flow-induced noise from a co-rotating vortex pair has been widely used by other authors in the past as a benchmark for the validation of their numerical methods (see e.g., [54, 58, 62, 139, 148]). A schematic of the corotating vortex pair is presented in Fig. 10.4. It consists of two point vortices separated by a fixed distance of 2r0 with circulation intensity Γ . The vortices rotate around each other with a period T = 8π 2 r02 /Γ . Each vortex induces on the other a velocity vθ = Γ/(4πr0 ). The configuration results in a rotating speed ω = Γ/(4πr02 ), and rotating Mach number M ar = vθ /c = Γ/(4πr0 c) = 2πr0 /T c. The rotating non-circular streamlines are directly associated with the hydrodynamic field of the rotating quadrupole [139]. The incompressible, inviscid flow can be determined numerically by the evaluation of a complex potential function Φ(z, t) [58, 62] Φ(z, t) =

Γ Γ b2 Γ ln(z − b) + ln(z + b) = ln z 2 (1 − 2 ) , 2πi 2πi 2πi z

(10.42)

where z = reiθ and b = r0 eiωt . The hydrodynamic velocity field required for the evaluation of the acoustic source term from (10.41), is obtained by differentiating (10.42) with respect to z as Γ z ∂Φ(z, t) = ux − iuy = . (10.43) ∂z iπ z 2 − b2

280

10 Computational Aeroacoustics

Fig. 10.4. Schematic diagram of corotating vortices

In the acoustic computation a linear propagation is assumed outside the fluid region, governed by the homogeneous acoustic wave equation. The analytical solution of the acoustic far-field pressure fluctuations, used to validate the numerical results, is obtained using a matched asymptotic expansion (MAE) [159], and computes as p
=

ρ0 Γ 4 [J2 (kr) cos(Ψ ) − Y2 (kr) sin(Ψ )] , 64π 3 r04 c2

(10.44)

where k = 2ω/c, J2 (kr), Y2 (kr) are the second-order Bessel functions of the first and second kind and Ψ = 2(ωt − θ). For the validation, the flow field is evaluated in a numerical domain with dimensions 200 m × 200 m. This domain corresponds to the region where the acoustic sources for the inhomogeneous wave equation are computed. The acoustic propagation is computed in a larger domain with dimensions 400 m × 400 m. For evaluating the complex potential function the spinning radius is chosen to be r0 = 1 m, the circulation intensity Γ = 1.00531 m2 /s and the speed of sound c = 1 m/s. This results in a wave length λ ≈ 39 m and a rotating Mach number M ar = 0.08. In the two-step computation, the first step consists of evaluating the velocity field and the acoustic sources on the fine fluid grid. Secondly, after interpolation of the acoustic sources from the fine fluid grid to the coarser acoustic grid, we solve the inhomogeneous wave equation using the resulting sources to compute the acoustic propagation. An additional issue for the computation of the noise generated by the corotating vortex pair is a singularity of the velocity field at the point vortices. A vortex core model of Scully type [62,139,192] has been applied to desingularize the tangential velocity field, in order to allow the computation of the velocity gradient on these regions. In the region around the corotating vortices with

10.4 Validation: Co-Rotating Vortex Pair

281

dimensions 6 m × 6 m, the acoustic element size is chosen to be ha = 0.1 m, which compared to the fluid discretization, corresponds to ha /hf = 5. Outside this area both fluid and acoustic meshes are correspondingly coarsened in the radial directions. A comparison of the numerical acoustic field obtained for the main frequency component of the problem, f = 1/T = 0.026 Hz, with the analytical solution is presented in Fig. 10.6. Good agreement in both the spiral pattern as well as in amplitudes is found except at the center of the computational domain, where the analytical solution has been truncated according to the vortex core model. Figure 10.5 compares the decay of the acoustic pressure along the positive x-axis between the numerical and analytical values.

Fig. 10.5. Decay of the acoustic pressure values along the x-axis

282

10 Computational Aeroacoustics

(a) Numerical computation

(b) Analytical solution Fig. 10.6. Comparison of sound pressure field at frequency f = 0.026 Hz obtained numerically using Lighthill’s acoustic analogy with analytical solution obtained using MAE method. Distance scale in meters

11 Algebraic Solvers

In recent years, many different formulations using Lagrange (nodal) as well as N´ed´elec (edge) finite elements for the numerical computation of Maxwell’s equations have been published, e.g., [23, 126]. The resulting algebraic system of equations is mostly solved by applying the conjugate gradient method with incomplete Cholesky factorization as preconditioner (ICCG). However, the number of necessary iterations of ICCG increases strongly with the number of unknowns. Recently, investigations have been done to adapt multigrid (MG) methods for the fast solution of 3D electromagnetic field problems (e.g., [9, 92, 191]). In this section, we will give a detailed discussion on geometric and algebraic multigrid methods specially adapted for Maxwell’s equations in the quasistatic case. Since a most robust solution strategy is a preconditioned conjugate gradient (PCG) solver with an appropriate multigrid method as preconditioner, we will start with a brief description of the PCG method. For a basic introduction into multigrid methods we refer to [36, 185, 186].

11.1 Preconditioned Conjugate Gradient (PCG) Method Let us consider the algebraic system of equations of the form Kh u h = f h .

(11.1)

Therein Kh ∈ IRnh ×nh denotes the system matrix, f h ∈ IRnh the righthand side and uh ∈ IRnh the solution vector of the unknown nodal (edge) quantity (usually the magnetic vector potential). The entries of Kh are given by kij = (Kh )ij ∈ IRp×p with p defining the number of unknowns per node (edge). The number of unknowns nh is related to the usual discretization parameter h by nh = O(h−d ), with d = 2, 3 the spatial dimension. The system matrix Kh is supposed to be sparse and symmetric positive definite (SPD) as is, in fact, the case for the used discretization of Maxwell’s equations. In

284

11 Algebraic Solvers

general, nh is quite large and due to limited memory resources, iterative solvers have to be used instead of direct ones. However, the convergence of iterative solvers strongly depends on the condition number κ of the system matrix Kh κ(Kh ) =

λmax (Kh ) , λmin (Kh )

(11.2)

with λmax and λmin the largest and the smallest eigenvalue of Kh , respectively. In general, the convergence rate decreases when κ gets large. Since Kh stems from an FE discretization of a second-order partial differential equation (PDE), the condition number κ(Kh ) typically behaves like O(h−2 ). In order to cope with large condition numbers, we apply a symmetric preconditioner Ch to (11.1), i.e., −1 C−1 (11.3) h K h u h = Ch f h , with the properties • •

C−1 h is an approximate inverse of Kh , and C−1 h can be applied very fast

Consequently, the condition number of the preconditioned system is much smaller than the original one. Furthermore, the preconditioned system (11.3) is solved via a Krylov subspace method, i.e., conjugate gradient (CG) or quasiminimal residual (QMR) method, see [187]. The standard method for solving (11.1) is to apply the preconditioned conjugate gadient (PCG) method as given in Algorithm 1. Algorithm 1 Preconditioned conjugate gradient (PCG) method k=0 r 0 = Kh u0h − f h Solve Ch d0 = −r 0 s0 = −d0 r 1 = r0 while r k+1  > r 0  do k )T sk αk = (d(r k ) T K dk h

uk+1 = ukh + αk dk h k+1 r = r k + αk Kh dk Solve Ch sk+1 = rk+1 k+1 T k+1 β k = (r (rk ))T ssk

dk+1 = −sk+1 + β k dk k =k+1 end while

11.2 Multigrid (MG) Method

285

The following result gives a bound on the number of iterations that are sufficient for a prescribed desired error reduction ε: Theorem 11.1. Let Kh ∈ IRnh ×nh and Ch ∈ IRnh ×nh be SPD with the relation γ˜1 Ch v, v ≤ Kh v, v ≤ γ˜2 Ch v, v ∀v ∈ IRnh , and γ˜1 , γ˜2 > 0. The starting solution error ||u − uh0 ||Kh is reduced by a factor ε applying

' ( +

I(ε) = ln ε−1 + (ε−2 + 1) / ln ρ˜−1 ', (8 ', ( γ ˜2 γ ˜2 PCG iterations with ρ˜ = γ ˜1 − 1 γ ˜1 + 1 . In Theorem 11.1, || · ||Kh denotes the energy norm induced by the energy inner product, computed as ||w||2Kh = Kh w, w for w ∈ IRnh . According to γ1 , which is equivalent to κ(C−1 Theorem 11.1 the factor γ˜2 /˜ h Kh ) should be as close as possible to 1 in order to obtain fast convergence. Of course the γ1 = 1, but this theoretically best choice would be Ch = Kh , yielding γ˜2 /˜ would have the consequence to solve Kh sk+1 = r k+1 in the PCG method. Therefore, we have to find a preconditioner Ch with γ˜2 /˜ γ1 ≈ 1 such that Ch sk+1 = rk+1 can be solved in a very fast way. The conventional choice is to use incomplete Cholesky (IC) factorization, i.e., Ch = RT R with R containing the entries of the upper triangular matrix of the factorization of Kh but with the same structure as Kh (possible low fill in is allowed). In [29] it is shown that under special assumptions, the condition number κ(C−1 h Kh ) when using IC as preconditioner behaves like O(h−1 ). However, κ still depends on the discretization parameter h. This fact leads us to MG methods for which it can be shown that the number of necessary iterations does not depend on the mesh parameter h [82]. Furthermore, it was shown in [111] that a most robust solution strategy (concerning the quality of the FE mesh) for (11.1) is PCG with a geometric multigrid preconditioner.

11.2 Multigrid (MG) Method Multigrid methods improve the convergence by using information not only on the computational grid on which the system of equations is supposed to be solved, but also on a (usually hierarchical) sequence of coarser grids. In order to outline the construction of a MG preconditioner we explain this by means of a two-grid method. The indices H and h are related to the coarse and fine grid of an FE discretization, respectively. The linear mappings (with nh > nH and nh , nH the number of unknowns on the fine and coarse grid, respectively) IhH : IRnh → IRnH

and

h IH : IRnH → IRnh

(11.4)

are called restriction and prolongation operators. Therefore, the two-grid algorithm is performed as follows:

286

11 Algebraic Solvers

1. Smooth ν1 times on the fine grid Kh , uh , f h 2. Calculate the defect d h = f h − Kh u h 3. Restrict the defect dh onto the coarse grid dH = IhH dh 4. Solve the coarse grid problem KH v H = dH 5. Prolongate the coarse grid correction v H to the fine grid h v h = IH vH

6. Correct uh by v h , i.e., uh = uh + v h 7. Smooth ν2 times on the fine grid Kh , uh , f h By replacing the exact solution of the coarse grid problem in step 4 itself by a two-grid approximation, we arrive at the recursive definition of a multigrid cycle (see Fig. 11.1). The motivation for this approach comes from examining Grid l Presmoothing Grid l-1

Postsmoothing Direct solver

Grid 2

Grid 1 (coarse)

Prolongation Restriction

Fig. 11.1. MG solution algorithm (V-Cycle)

the error of the numerical solution in the frequency domain. High-frequency errors, which include local variations in the solution, are well eliminated by simple iterative smoothing methods (e.g., Gauss–Seidel smoother). Once this is achieved, further fine-grid iterations would only result in a convergence degradation. Therefore, the solution is transferred to a coarser grid by using an appropriate projection operator IhH . On this grid, the low-frequency errors of the fine grid manifest themselves as relatively high-frequency errors, and are thus eliminated efficiently again using simple iterative smoothing methods. If the coarsest grid has been reached, the equation has to be solved exactly (e.g., direct solver). Consequently, each grid level is responsible for eliminating a particular frequency bandwidth of the error.

11.3 Geometric MG Method

287

The MG iteration operator Mh mapping the k-th MG iteration error ek = uh − ukh (uh being the exact solution of (11.1)) onto the (k + 1)-th MG iteration error ek+1 = uh − uk+1 h ek = Mh ek+1

(11.5)

is in the two-grid case given by *ν2 ) * pre ν1 ) h H Mh = Shpost Ih − IH K−1 , H Ih Kh (Sh )

(11.6)

provided that the coarse grid system (step 4) is solved exactly (see e.g., [82]). In (11.6) Ih ∈ IRnh ×nh denotes the identity matrix, and Shpre , Shpost the smoothing operators, e.g., Gauss–Seidel forwards and backwards, respectively Shpre = Ih − τh (Lh + Dh )−1 Kh Shpost = Ih − τh (Lh + Dh )−T Kh , with Lh the lower triangular part of Kh , Dh = diag(Kh ) and τh a relaxation factor. In the general situation, the MG iteration operator Mh on the finest grid can be defined in the following iterative process (by denoting the coarsest level by 1 and the finest by ) Mh = Ml *ν2 ) ) q Iq − Iq−1 Mq = Sqpost (Iq−1 − Mq−1 ) *) *ν1 q−1 Kq Sqpre , K−1 q−1 Iq with q = 2, 3, ..., , M1 := 0. As already mentioned in the previous section, a most robust solution strategy for (11.1) is PCG with a geometric MG preconditioner. This means that solving Ch sk+1 = r k+1 (see Algorithm 1) corresponds to applying p MG cycles to Kh sk+1 = r k+1 . Using the introduced iteration operator Mh and setting the starting value s0k+1 to zero, we obtain the iteration error according to (11.5) k+1 k+1 K−1 − sk+1 = (Mh )p (K−1 − sk+1 ) p 0 h r h r k+1 = (Ih − (Mh )p ) K−1 . sk+1 p h r

(11.7)

k+1 , the preBy setting the so-obtained solution spk+1 equal to sk+1 = C−1 h r conditioner Ch takes the form

Ch = Kh (Ih − (Mh )p )

−1

.

(11.8)

11.3 Geometric MG Method In contrast to standard FE techniques, geometric MG methods are not based on a fixed FE mesh that describes the unknown field variable accurately

288

11 Algebraic Solvers

enough. Geometric MG techniques start at a very coarse spatial discretization T1 of the computational domain. By dissecting the elements of T1 , a finer discretization T2 is obtained as shown in Fig. 11.2. This refinement process can

Fig. 11.2. Adaptive refinement of an initial mesh

include either all elements (uniform refinement) or only an appropriately selected part of the elements (adaptive refinement). By repeating the refinement, we obtain a hierarchy of FE discretizations T1 , ..., T for which the systems Kq uq = f q q = 1, 2, ..., 

(11.9)

are assembled and a MG cycle as described in Sect. 11.2 can be performed. The generation of an appropriate coarse mesh and the adaptive refinement for the subsequent finer FE discretizations levels is a challenging task, which needs the full data exchange between the geometric modeler, the mesh generator and the error estimator [194]. 11.3.1 Geometric MG for Edge Elements The two essential parts for a successful application of geometric MG to Maxwell’s equations are the choice of the prolongation and of the smoothing operator. In order to determine the prolongation operator Iqq+1 , we consider the refinement of a face Γ q on an edge tetrahedron element at level q. Dissecting the tetrahedron at level q into 8 tetrahedra at level (q + 1), each face Γ q is divided into 4 new faces Γ1q+1 , .., Γ4q+1 (Fig. 11.3). The prolongation operator Iqq+1 must guarantee that the magnetic flux across Γ q is equal to the one across Γ1q+1 + .. + Γ4q+1 [191]  (∇ × u) · dΓ = Γq

 4  k=1 q+1 Γk

(∇ × u) · dΓ .

(11.10)

11.3 Geometric MG Method

289

Fig. 11.3. Prolongation of the edge degrees of freedom from level q to level q + 1

By applying Stoke’s theorem (Appendix B.9), we obtain for each face Γkq+1 the relation 1 u · ds = u · ds . (11.11) 4 Γkq

Γkq+1

By exploiting the degrees of freedom of the FE formulation (see Fig. 11.3), we obtain 1 q (u + uq2 + uq3 ) 4 1 1 = (uq1 + uq2 + uq3 ) 4 1 q = (u1 + uq2 + uq3 ) 4 1 q = (u1 + uq2 + uq3 ) . 4

uq+1 − uq+1 + uq+1 = 1 7 6

(11.12)

uq+1 + uq+1 − uq+1 2 3 8

(11.13)

uq+1 − uq+1 + uq+1 5 9 4 uq+1 + uq+1 + uq+1 7 8 9

(11.14) (11.15)

In addition, since the magnetic vector potential u is constant along the edge, we may write 1 q u 2 1 1 = uq2 2 1 = uq3 . 2

uq+1 = uq+1 = 1 2

(11.16)

uq+1 = uq+1 3 4

(11.17)

uq+1 = uq+1 5 6

(11.18)

Combining the above results allows us to define the transfer operator as follows ⎛

Iqq+1

⎞T 0.5 0.5 0.0 0.0 0.0 0.0 0.25 0.25 −0.25 = ⎝ 0.0 0.0 0.5 0.5 0.0 0.0 −0.25 0.25 0.25 ⎠ , 0.0 0.0 0.0 0.0 0.5 0.5 0.25 −0.25 0.25

which fulfills the requirements of flux conservation. The restriction operator q is chosen to be that transposed to Iqq+1 , i.e., Iq+1

290

11 Algebraic Solvers q Iq+1 = (Iqq+1 )T .

(11.19)

For the construction of the smoothing operator, we have to consider the fact that the Sobolev space H0 (curl) has a Helmholtz decomposition of the form (11.20) H0 (curl ) = N (curl ) + N (curl )⊥ , with N (curl ), N (curl )⊥ the kernel of the curl operator and its orthogonal complement. The smoothing operator has to damp out errors in both spaces efficiently, see [92]. In [9] it has been shown that overlapping block-smoothers, which collect all edges sharing a common vertex in a block, have this property and show a convergence rate independent of the error reduction value . Therefore, a simple Gauss–Seidel method does not behave well for Maxwell problems, but due to [9] and [92] it is known that properly designed blockGauss–Seidel iterations do the job. Each block is assigned to a vertex of the mesh and connects all edges sharing this node. Since each of the edges of the mesh is associated to more than one node, no standard block-Gauss–Seidel smoother but an overlapping technique must be applied. Therefore, all degrees of freedoms belonging to edges with a common node are smoothed together. This can be achieved by first introducing a connectivity matrix Rj , whose entries are zeros and ones that allows the corresponding subblock Kjq to be picked out of our system matrix Kq (q denotes the MG level). The dimension of Rj is (nj × ne ) with nj the number of edges belonging to node j and ne the total number of edges (unknowns) in the mesh. Using this matrix, we can pick out the quadratic sub-blocks Kjq of the matrix Kq as follows Kjq = Rj Kq RTj .

(11.21)

Each of these small matrices has to be inverted in the preparation phase of the multigrid method. One step of the block-Gauss–Seidel iteration with initial approximation ujq,i is defined as ujq,i+1 = ujq,i + RTj (Kjq )−1 Rj (f q − Kq ujq,i )

j = 1, ..., n ,

(11.22)

with i the iteration counter. It has to be mentioned that not the whole residual f q − Kq ujq,i has to be computed at each step, but only the few components picked out with Rj . Therefore, one block-Gauss–Seidel step is not much more expensive than a simple Gauss–Seidel step. 11.3.2 Case Study In order to demonstrate the advantages of the presented scheme, TEAM (Testing Electromagnetic Analysis Methods) Workshop problem 20 is considered. Therefore, the convergence behavior of the multigrid solvers is compared to standard approaches. The TEAM Workshop problem 20 is a 3D, non-linear, and static magnetic field problem [161]. The structure of this problem, consisting of a center pole, a yoke, and a coil, is displayed in Fig. 11.4.

11.3 Geometric MG Method

291

Nested Multigrid First, the structure is discretized with a coarse mesh T1 of linear edge tetrahedron elements, which is shown in Fig. 11.4. Thereby, the symmetries of the

Fig. 11.4. Coarse FE discretization of TEAM problem 20 (without air)

problem in the xz-plane and the yz-plane are exploited. By dissecting each edge tetrahedron element of level 1 into 8 elements of level 2, a new refined mesh T2 is obtained. This procedure is repeated until a mesh T4 is produced that is accurate enough to describe the magnetic field. In Table 11.1 the generated hierarchy of FE meshes is shown. In order to achieve a good initial Table 11.1. FE hierarchy of TEAM problem 20 Grid level Edge elements Edges (dofs) 1 3 050 3 800 2 24 500 30 500 3 196 000 236 000 4 1 570 000 1 900 000

guess for the non-linear iteration procedure on the finer levels, the problem is first solved on the coarser grids and the solution is prolongated to the finer meshes and used as a start approximation for the non-linear iteration process. By this nested-multigrid approach, the number of costly iterations at the finer grids is considerably reduced [90]. Convergence of the MG-PCG solver The most time-consuming part of the computation process is the solution of the matrix equation system at the finest mesh T4 . Thereby, the MG solver

292

11 Algebraic Solvers

is compared to a standard solution technique, based on a CG method with adapted block preconditioning (PCCG). In Fig. 11.5 the convergence behavior of both methods for a matrix equation system at the finest level with 1 900 000 edges is displayed. The MG solver achieves the requested normal-

Fig. 11.5. Number of iterations versus normalized residual

ized residual of 10−6 after 13 iterations and 890 s, whereas the PCCG with block preconditioning needs 180 iterations and 9860 s. Thereby, an SGI ORIGIN with a RS12000 processor (300 MHz) is used. If applying a conventional CG solver with incomplete Cholesky preconditioning (ICCG) to the matrix equation, more than 1000 iterations would be necessary and therefore much higher simulation times arise [70]. Optimal complexity To show the optimal complexity of the MG solver, the number of necessary iterations and the solution time to achieve a normalized residual of 10−6 at each discretization level are displayed in Table 11.2. Since the number of MG iterations is almost independent of the size of the FE meshes and the time for a single MG iteration increases linearly with the number of unknowns, a linear dependency between the degrees of freedom and the solution time can be detected. Accumulated solution times of nested MG and a conventional approach In Table 11.3 the convergence behavior of the non-linear iteration process is compared for different excitations Θ (given in Ampere-turns). For small Θ the

11.4 Algebraic MG Method

293

Table 11.2. Necessary MG-PCCG iterations and simulation times to reduce the normalized residuum of algebraic system of equations to 10−6 Level Iterations Time (s) 1 10 1.5 12 11 2 12 105 3 13 890 4

number of necessary iterations at the coarser levels is low, but if the excitation increases, a higher number of iterations is necessary. On the other hand, due to the nested MG approach, the number of iterations on the finer levels is almost independent of the strength of the non-linearity. Since the iterations on the finer grids are the most time-consuming parts of the computation process, Table 11.3. Necessary non-linear iterations for different excitations Θ (Computer: SGI, RS12000 processor 300 MHz) Θ (Ampere-turns) 1 000 3 000 4 500 5 000

Iterat. 1 2 4 3 11 6 15 6 17 6

at level Accum. time Accum. time 3 4 nested MG (s) conventional (s) 3 2 8 300 45 400 5 3 9 570 114 500 5 3 9 600 154 000 5 3 9 600 173 600

also the accumulated simulation time is almost independent of the strength of the magnetic non-linearity. In the seventh column of Table 11.3 the accumulated simulation time of a conventional approach, which means PCCG solvers for the resultant matrix equation systems, without exploiting the coarse grid information at the finer grids, is displayed. Thereby, a considerable advantage of the nested MG technique can be clearly seen.

11.4 Algebraic MG Method In contrast to geometric MG, algebraic multigrid (AMG) needs no FE discretization with hierarchical grids. The matrices Kq with q = 1, ..,  on the different levels are set up only by knowledge of the matrix Kh = Kl obtained from the FE discretization. In recent years, a lot of different approaches were published, e.g., [27, 131, 185], which mostly concern the scalar case of matrix equation systems arising from a nodal finite element discretization. Geometric MG methods suffer from the inherent need of a hierarchical FE mesh (see [82]), and thus algebraic multigrid (AMG) methods are of special interest, if at least one of the following cases arises:

294

• • •

11 Algebraic Solvers

The discretization provides no hierarchy of FE meshes, which would be essential for the geometric MG method. This is the case for many FE codes, especially commercial ones. The coarsest grid of a geometric multigrid method is too large to be solved efficiently by a direct or classical iterative solver. Classical iterative solvers are not efficient enough.

AMG methods try to mimic their geometric counterpart, but only rely on the information available on a given single grid (for the pioneering work on AMG see [185]). While within a geometric MG solver the construction of a matrix hierarchy is rather simple if a hierarchy of grids is available (see e.g., [82]), this task is not as easy if either the matrix only or the information on the finest grid is available. The classical AMG approach assumes an SPD system matrix that is additionally an M matrix [185]. For such matrix classes a matrix hierarchy can be constructed, imitating the geometric counterpart well. It can be easily shown that the information of an SPD system matrix is not enough in order to construct an efficient and robust AMG method. Therefore, we assume the knowledge of the underlying PDE, the FE discretization scheme and additional information on the given FE mesh. Therefore, such enhanced AMG methods are able to reproduce the behavior of geometric MG methods even for Maxwell’s equation, although the system matrices are not M matrices here. For a detailed discussion we refer to [179]. First, we have to perform the coarsening process to extract from the given system matrix (arising from the FE discretization) matrices with decreasing dimension. The key point of the coarsening process is to construct an auxiliary matrix on which the coarsening is performed. Therefore, we can always guarantee a coarsening that is appropriate and, in addition, very fast. Furthermore, we have to define the smoothing operator and the restriction (prolongation) operator for the transfer of data between the different hierarchy levels based on the auxiliary matrix. 11.4.1 Auxiliary Matrix Let us assume that the system matrix Kh stems from an FE discretization on the FE mesh ωh = (ωhn , ωhe ), with ωhn , |ωhn | = Mh being the set of nodes and ωhe being the set of edges (see Fig. 11.6). An edge is defined as a pair of indices for which the connection of the two corresponding points is a geometric edge. For instance, let i, j ∈ ωhn be the indices of the nodes xi , xj ∈ IRd then the edge is given by eij = (i, j) ∈ ωhe , and the corresponding geometric edge vector can be expressed by aij = xi − xj ∈ IRd .

(11.23)

11.4 Algebraic MG Method

295

Fig. 11.6. Clipping of an FE mesh in 2D

The first task we are concerned with is the construction of an auxiliary matrix Bh ∈ IRMh ×Mh with the following properties & if i = j, bij ≤0 (Bh )ij = (11.24) if i = j . 1 − j =i bij ≥ 0 The entries of Bh should be defined in such a way that the distance and parameter jumps of the variational forms are reflected. The matrix pattern of Bh can be constructed via different objectives: Bh reflects the geometric FE mesh, which is of importance for an edge element discretization, or Bh reflects the matrix pattern of the system matrix Kh , which is useful for nodal FE discretization. 11.4.2 Coarsening Process The auxiliary matrix Bh is a sparse M matrix and therefore the coarsening process for Bh is straightforward and can be done in a robust way. We know that Bh represents a virtual FE mesh ωh = (ωhn , ωhe ). Such a virtual FE mesh can be split into two disjoint sets of nodes, i.e., n n ωhn = ωC ∪ ωFn , ωC ∩ ωFn = ∅ , n and fine grid nodes ωFn . The splitting is with sets of coarse grid nodes ωC usually performed such that no coarse grid nodes are connected directly and that the number of coarse grid nodes is as large as possible (see Fig. 11.7). In order to perform a coarsening algorithm, let us introduce the following sets

Nhi = {j ∈ ωhn : |bij | = 0 , i = j} , Shi = {j ∈ Nhi : |bij | > coarse (Bh , i, j) , i = j} , Shi,T = {j ∈ Nhi : i ∈ Shj } , where Nhi is the set of neighbors for node i, Shi denotes the set of strong connections and Shi,T is related to the set of nodes that have a strong connection to node i, respectively. The cutoff (coarsening) function is chosen as, e.g.,

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Fig. 11.7. Illustration of coarsening

⎧ + ⎨ θ · |bii ||bjj | , coarse (Bh , i, j) = θ · maxl =i |bil | , ⎩ θ,

see [212] , see [185] , see [130] ,

(11.25)

with an appropriate θ ∈ [0, 1]. In addition, we define the local sets i n = ωC ∩ Nhi , ωC

ωFi = ωFn ∩ Nhi

(11.26)

and Ehi = {(i, j) ∈ ωhe : j ∈ Nhi } .

(11.27)

The coarsening algorithm is described in Algorithm 2. Algorithm 2 Coarsening phase n ωC ← ∅,

ωFn ← ∅

n while ωC ∪ ωFn = ωhn do n ∪ ωFn )) i ← Pick(ωhn \ (ωC i,T i,T if |Sh | + |Sh ∩ ωFn | = 0 then n ωFn ← ωhn \ ωC else n n ← ωC ∪ {i} ωC n n n ) ωF ← ωF ∪ (Shi,T \ ωC end if end while

Therein the function n i ← Pick(ωhn \ (ωC ∪ ωFn ))

returns a node i for which the number |Shi,T | + |Shi,T ∩ ωFn | is maximal.

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297

Fig. 11.8. Example with anisotropic mesh and parameter jump (material 1 in elements with number 1 and 2; material 2 in elements with numbers 3 and 4)

Example: Let us consider the FE mesh of Fig. 11.8. The auxiliary matrix is defined on an finite element r by the setting brij =

νr aij 2

i = j ,

with νr the material parameter and aij the geometric edge vector (see (11.23)). Let us assume the following entries for row 5 of the assembled auxiliary matrix b51 = −202 b52 = −2 b53 = −200

b54 = −400 b57 = −100 b55 = 10n09 b58 = −101 b56 = −2 b59 = −1 .

Using the coarsening function of [185] (see (11.25)) with θ = 0.25, we obtain Nh5 = {1, . . . , 4, 6, . . . , 9} Sh5 = {1, 3, 4, 7, 8} Sh5,T = {4, 6, 7} . For the construction of set Sh5,T we assumed Sh1 Sh2 Sh3 Sh4

= {3} = {1} = {1} = {1, 3, 5, 7, 8}

Sh6 Sh7 Sh8 Sh9

= {1, 2, 5, 8} = {4, 5, 8} = {7, 9} = {7, 8} .

A special coarsening algorithm is the agglomeration technique, where θ is set to 0. Consequently, Nhi = Shi = Shi,T for all i = 1, . . . , Mh . Furthermore, n H we call (Ihi )M i=1 (|ωC | = MH < Mh ) a disjoint splitting for the agglomeration method if M H Ihi ∩ Ihj = ∅, Ihi = ωhn , i=1

is valid, see Fig. 11.9.

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Fig. 11.9. Virtual FE mesh with a feasible agglomeration

If an appropriate prolongation Qh for Bh is defined then a coarse auxiliary matrix is computed by BH = (Qh )T Bh Qh , n e n n and BH represents again a virtual FE mesh ωH = (ωH , ωH ), with ωH = ωC . It can be shown that BH is again an M-matrix if the prolongation operator Qh fulfills certain criteria [185]. Thus the coarsening process can be applied recursively. Finally, it is assumed that the degrees of freedom on the coarse grid are numbered first. For instance, the nodes are reordered like n , ωFn ) ωhn = (ωC

(similarly for edges) and as a consequence the system matrix can be written as   KCC KCF Kh = . KTCF KF F 11.4.3 Prolongation Operators n For a given splitting ωhn = ωC ∪ ωFn the optimal prolongation operator is given by the Schur complement, i.e.,

˜ ˜T KH = KCC − KCF K−1 F F KF C = Ph Kh Ph with

T P˜h = (IH , −KCF K−1 FF ) . The prolongation operator P˜h can hardly be realized in practice since the expression −KCF K−1 F F involves the inverse of KF F , which in turn implies a global transport of information. In addition, the coarse grid matrix KH becomes dense. The goal of an AMG method is to approximate P˜h by some prolongation operator Ph , which acts only locally and therefore produces a sparse coarse-grid matrix.

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299

11.4.4 Smoother and Coarse-grid Operator An essential point in MG methods is the smoothing operator Sh ∈ IRNh ×Nh that reduces the high-frequency error components. Typically, a particular smoother works for certain classes of matrices. It is shown in [32] that a point Gauss–Seidel or point Jacobi smoother is appropriate for FE discretizations with Lagrange FE functions for scalar elliptic PDEs of second order. Analogously, the block-Gauss–Seidel and block Jacobi smoother work well for the block counterpart, e.g., discretization of Maxwell’s equation with nodal finite elements. For the edge FE discretization we use a patch smoother. The coarse grid matrix KH is usually constructed by Galerkin’s method, i.e., (11.28) KH = PhT Kh Ph . After a successful setup, an AMG-cycle can be performed as usual (see e.g., [82]). For instance in Algorithm 3 a V (νF , νB )-cycle with variable preand post-smoothing steps is described. The variable CoarseLevel stores the number of levels generated by the coarsening process until the size of the system is smaller than CoarseGrid. Algorithm 3 V(νF , νB )-cycle K ← K,

f ← f,

AMGStep(K, u, f , )

u ← u

if  = CoarseLevel then u ← CoarseGridSolver (L LT , f  ) Return else d ← 0, w+1 ← 0 u ← SνF (u , f  ) d ← f  − K u d+1 ← (P )T d AMGStep(K+1, w+1 , d+1 ,  + 1) w ← P w+1 u ← u + w u ← SνB (u , f  ) end if

In the following subsections we specialize the abstract algorithms, define the components for nodal and edge FE discretization and additionally propose a method for complex symmetric systems. Prior to that we mention that static, transient, and non-linear analysis of a given problem results in the solution of linear systems (11.1). Therefore, we restrict the discussion to the linear analysis. Other applications can be found in [116, 117, 119, 120, 182].

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11.4.5 AMG for Nodal Elements First we consider (4.120) and use nodal elements for discretization. Note that the following approach includes the scalar case (p = 1; e.g., scalar potential equation). Construction of virtual FE meshes: The definition of the auxiliary matrix Bh plays an important role for this problem class. The classical approach uses (Bh )ij = −kij ∞ for i = j , with  ∞ the maximum norm. The diagonal entries of Bh are computed according to (11.24). Now the degrees of freedom per node of the system matrix have to be related to an entry in the auxiliary matrix, which in turn implies that the matrix pattern of Kh and of Bh has to be equal, i.e., kij ∞ = 0 ⇔ |bij | = 0 . Construction of coarse FE spaces: The simplest prolongation operator is given by ⎧ n if i = j ∈ ωC , ⎪ ⎨ Ip 1 n n · I if i ∈ ω , j ∈ Shi,T ∩ ωC , (Ph )ij = |S i,T ∩ωn | p (11.29) F C h ⎪ ⎩0 else , with Ip ∈ IRp×p the p-dimensional identity matrix. The AMG method shows a better convergence behavior as compared to (11.29) with the following discrete harmonic extension ⎧ n ⎨ Ip ) * if i = j n∈ ωC , i −1 if i ∈ ωF , j ∈ ωC , (11.30) (Ph )ij = −kii kij + cij ⎩ 0 else , with cij =

 )  i p∈ωF

kpq

*−1

kip kpj .

i q∈ωC

However, the increasing memory requirement and the slower application compared to the prolongation (11.29) is the major drawback of the discrete harmonic extension. Note that the entries of the prolongation operators are matrix valued, e.g., (Ph )ij ∈ IRp×p , like the entries of the system matrix Kh . Smoothing operator: We use a block-Gauss–Seidel method as smoothing operator, e.g., [32], or a patch-block Gauss–Seidel method, e.g., [131]. The latter should be used for anisotropic problems.

11.4 Algebraic MG Method

301

11.4.6 AMG for Edge Elements The second class originates from an FE discretization with edge FE functions of the variational form (4.112). Construction of virtual FE-meshes: According to [92], the refinement of the FE mesh can be performed on the nodes as is usually done for Lagrange FE functions. We use this fact and base our coarsening on an auxiliary matrix Bh , which is constructed for instance by the finite element wise setting brij = −

νr aij 2

i = j and (i, j) ∈ ωhe ,

with νr the reluctivity of the material. Again the diagonal elements are computed via (11.24). Example: Let us consider the FE mesh of Fig. 11.8 and choose element r = 1. We get the following element matrix ⎛ ⎞ 2.5 −0.5 −1 0 ⎜−0.5 2.5 0 −1 ⎟ ⎟ B1h = 100 · ⎜ ⎝ −1 0 2.5 −0.5⎠ . 0 −1 −0.5 2.5 The entries (Bh1 )14 , (Bh1 )23 , (Bh1 )41 , and (Bh1 )32 are zero, i.e., there is no diagonal edge in the virtual FE mesh. Let us recall that an FE mesh is represented by ωh = (ωhn , ωhe ) , i.e., the set of nodes ωhn and the set of edges ωhe . The coarse grid is defined n n by identifying each coarse grid node j ∈ ωC with an index k ∈ ωH . This is expressed by the index map ind (.) as n n = ind (ωC ). ωH e can be constructed if we invest in a special A useful set of coarse grid edges ωH prolongation operator Qh for the auxiliary matrix Bh . The prolongation operator Qh is constructed such that each fine grid node prolongs exactly from one coarse grid node, so that one arrives at a partition of ωhn into clusters, each of them being represented by a coarse grid variable. We extend the index n n → ωH defined above onto the whole fine set ωhn by assigning map ind : ωC to all fine grid nodes of a cluster the coarse grid index of the representative n . ind : ωhn → ωH

A consequence is that ind (i) = ind (j) iff i, j ∈ ωhn prolongate from the same coarse grid variable. We define an agglomerate (cluster) Ihi of a grid point i ∈ ωhn by (see Fig. 11.10)

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Ihi = {j ∈ ωhn | ind (j) = ind (i)} ⊂ Nhi , and hence the set of coarse grid nodes can be written as n = {ind (i) | i ∈ ωhn } . ωH

The prolongation operator Qh has only 0 and 1 entries by construction, i.e.,

Fig. 11.10. Virtual FE mesh with a feasible agglomeration and coarse-grid edges

(Qh )ij =

1 0

i ∈ ωhn , j = ind (i) otherwise .

(11.31)

Now, a coarse-grid edge only exists if there is at least one fine edge connecting the agglomerates Ihi and Ihk with i = k (see Fig. 11.10), i.e., ∃r ∈ Ihi , ∃s ∈ Ihk such that (r, s) ∈ ωhe . Note that a decrease of the number of edges in the coarsening process is not proved in general, but a decrease is heuristically given, if the average number of nonzero entries of Bh does not grow too fast. Construction of coarse FE spaces: The construction of the prolongation operator Ph : IRNH → IRNh , is delicate because of the kernel of the curl -operator consisting of all gradient fields. Ph is defined for e as i = (i1 , i2 ) ∈ ωhe , j = (j1 , j2 ) ∈ ωH ⎧ if j = (ind (i1 ), ind (i2 )), ⎨1 if j = (ind (i2 ), ind (i1 )), (Ph )ij = −1 (11.32) ⎩ 0 otherwise ,

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303

by assuming a positive orientation of an edge j = (j1 , j2 ) from j1 to j2 if j1 < j2 holds. The constructed prolongation operator Ph has full rank, because the coarse grid edges prolongate to NH distinct fine-grid edges by construction. For a detailed discussion see [180]. Smoothing operator: To complete the components for an AMG method for edge element FE discretizations, we need an appropriate smoother. We consider two different types of smoothers for Kh . The first one was suggested in [9]. This is a block-Gauss–Seidel smoother where all edges that belong to Ehi (see (11.27)) are smoothed simultaneously for all i ∈ ωhn (see Fig. 11.11).

Fig. 11.11. Detail view of a virtual FE mesh

Another kind of smoother was suggested in [92]. A mathematically equivalent formulation is outlined in Algorithm 4. Therein the vector ge,i ∈ IRNh h Algorithm 4 Hybrid smoother uh ← GaussSeidel(Kh , f h , uh ) for all i ∈ ωhn do ((f h −Kh uh ),ge,i h ) uh ← uh + · g e,i e,i h ,g h ) (Kh ge,i h end for

is defined by ge,i h

=

grad h g n,i h

⎧ ⎨1 = −1 ⎩ 0

∈ IRMh , (g n,i ) = δij . with a vector gn,i h h j

if j < i (i, j) ∈ Ehi , if j > i (i, j) ∈ Ehi , otherwise ,

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11.4.7 AMG for Time-harmonic Case In the harmonic case the time derivative of the magnetic vector potential is substituted by ∂A ˆ, → jω A ∂t ˆ the complex with j the complex number, ω the angular frequency and A magnetic vector potential. Therefore, we have to apply the AMG method to a complex valued and symmetric algebraic system of equations with system matrix im (11.33) Kh = Kre h + jKh . im In (11.33) Kre h denotes the real part and Kh the imaginary part of the system matrix. The application to scalar potential equations has been presented in [183], and adaption to the magnetic vector potential formulation is straightforward as shown below.

Construction of virtual FE-meshes: The auxiliary matrix is defined to be real valued. This means that we set up Bh for an edge element discretization as defined in Sect. 11.4.6. For a nodal element discretization we can use the procedure described in Sect. 11.4.5 for Kre h . Construction of coarse FE spaces: For the construction of a coarse-grid operator KH we define the system prolongation to be real valued and computed as defined in Sect. 11.4.5 as well as Sect. 11.4.6. Therefore, we get T im re im KH = PhT Kh Ph = PhT Kre h Ph + jPh Kh Ph = KH + jKH .

The prolongation Qh is also taken from the real-valued realization correspondingly. Smoothing operator: In the case of an algebraic system of equations arising from a nodal FE discretization we apply a block Jacobi or Gauss–Seidel smoother in the complex variant. The complex version of the smoother proposed in [9] is used for an edge FE discretization. 11.4.8 Case Studies In order to gain robustness and efficiency, the proposed AMG methods were used as a preconditioner in the conjugate gradient (CG) method for the static and eddy current case (in the time domain) and in the quasi-minimal residual (QMR) method for the time-harmonic case. The iteration was stopped as soon as the error in the preconditioner energy norm has been reduced by a factor 10−6 for the PCG method. In the time-harmonic case (QMR solver) we use the stopping criterion as follows

11.4 Algebraic MG Method

305

f h − Kh uh 2 ≤ 10−6 f h 2 . For all calculations, a V (2, 2)-cycle has been applied and the coarsest matrix equation is solved by a Cholesky factorization, if the degrees of freedom less than 500. All computations were done on a PC with a Pentium 1.7 GHz chip. A good measure for the speed of coarsening is the so-called grid complexity, which is given by L 

GC(Kh ) =

Mi

i=1

M1

,

(11.34)

with L the number of levels and Mi the number of nodes (edges) for level i. This number is close to 1, if the reduction of unknowns is done very fast. If the number is very large then the coarsening is usually very slow. A second measure that is more related to the memory consumption and arithmetic costs is the operator complexity, i.e., L 

OC(Kh ) =

N M Ei · Ni

i=1

N M E1 · N1

,

(11.35)

where N M Ei denotes the average number of nonzero system matrix entries on level i and Ni the number of unknowns on this level. This number gives an idea of how much memory is used with respect to the finest grid. The same applies for the arithmetic costs. Again this number is close to 1 if only a small amount of memory is required. The abbreviation MB denotes the amount of memory used. The computations with Lagrange and N´ed´elec FE functions were always done on the same FE mesh. We want to emphasize that Nh = p |ωhn | for node (static case: p = 3; eddy current case: p = 4) and Nh = |ωhe | for edge FE discretization. Static Analysis For the computational domain we consider the geometry of TEAM 20 (see Fig. 11.12), which has been discretized by tetrahedron elements. Table 11.4 displays evaluated grid complexity GC and operator complexity OC as well as the required memory for the nodal and edge case. Nh defines the number of unknowns. It can be clearly seen that the required memory scales optimally with the number of unknowns and the values for OC and GC are close to 1. The number of iterations as well as elapsed CPU times are shown in Table 11.5. The short time for performing the setup makes the AMG solvers very attractive for non-linear problems.

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Fig. 11.12. FE mesh of TEAM 20 (without air region)

Table 11.4. TEAM 20: Complexities and memory requirement Nh GC OC MB Nodes Edges Nodes Edges Nodes Edges Nodes Edges 1.263 2.253 8.022 16.217 56.673 122.762

1.4 1.3 1.3

1.2 1.2 1.2

1.07 1.06 1.07

1.02 1.02 1.03

1.5 10 65

3 24 173

Table 11.5. TEAM 20: CPU times and number of iterations Setup (s) Solve (s) Iter Nh Nodes Edges Nodes Edges Nodes Edges Nodes Edges 1.263 2.253 8.022 16.217 56.673 122.762

0.2 0.4 1.7

0.2 0.5 2.9

0.2 2.0 30.7

0.2 2.8 35.3

18 27 48

9 16 24

Transient Analysis In order to show the performance of the proposed enhanced AMG methods for an eddy current problem, we present results of 3D magnetic field computations for a simplified MRI scanner with a z-gradient coil, as shown in Fig. 11.13 [175]. Here, gradient and magnet coils are assumed as smeared cylindrical coils. Furthermore, only the three inner cryostat cylinders are modeled. Table 11.6 displays the values for the grid complexity GC, the operator complexity

11.4 Algebraic MG Method

307

Fig. 11.13. FE mesh of a simplified MRI scanner (not the full air region is displayed)

OC and the required memory. Again a very good performance with optimal memory requirement can be found. Table 11.6. MRI scanner: Complexities and memory requirement Nh GC OC MB Nodes Edges Nodes Edges Nodes Edges Nodes Edges 27.834 61.342 88.053 197.375 162.882 368.131

1.2 1.2 1.2

1.2 1.2 1.2

1.06 1.06 1.05

1.03 1.03 1.03

75 250 510

94 330 639

The performance of the proposed AMG solvers concerning the number of iterations and CPU time is shown in Table 11.7. Since in this example we Table 11.7. MRI scanner: CPU times and number of iterations Setup (s) Solve (s) Iter Nh Nodes Edges Nodes Edges Nodes Edges Nodes Edges 27.834 61.342 2.5 88.053 197.375 7.7 162.882 368.131 14.6

1.5 5.8 10.1

10.3 31.2 64.1

7.6 38.2 75.2

15 15 16

10 15 15

have no parameter jump in the reluctivity, the number of iterations remains quite constant, which results in an optimal convergence rate.

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Time-harmonic Analysis Finally, we show the performance of the AMG method for the harmonic analysis. As a case study, we have chosen a configuration of a coil and a centered iron core surrounded by air. The established mesh can be seen in Fig. 11.14.

Fig. 11.14. FE mesh of an iron core and surrounding air (broken open at yz-plane element boundaries, coil not displayed)

The core diameter was fixed at 2 mm and the coil thickness was set to 1 mm. At an excitation frequency f of 500 Hz, a relative permeability µr = 1000 and a conductivity of γ = 107 S/m one calculates a penetration depth of δ = 0.22 mm, which corresponds to approximately 1/5 of the core radius. For a different conductivity of γ = 105 S/m one gets an eddy current penetration depth of δ = 2.2 mm, which is equal to a full penetration of the core. For the discretization, we have used at least ten finite elements per penetration depth. In the first step we performed calculations on varying grids and have listed the evaluated characteristic solver data in Table 11.8. Therein, the operator and grid complexity are close to 1, which indicates a fast coarsening, and in addition small memory requirements for the AMG preconditioner. Furthermore, it can be noticed that the solution time scales approximately linearly with the number of edges Nh . In the second step we performed computations for different material parameters to investigate on the robustness of our new solver. In Figs. 11.15 and 11.16 the convergence behavior of our QMR-AMG solver is displayed. The typical QMR behavior—a short remaining at a constant relative error over a few iteration steps—can be detected clearly. Summarizing, we can note that the proposed QMR-AMG solver is very robust against parameter jumps.

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Table 11.8. Case study: Performance and complexity for different FE meshes Nh

Iter Setup (s) Solver (s) OC

18.708 30 107.793 30 245.418 32

0.39 2.55 6.04

4.81 31.33 81.35

GC

1.026 1.216 1.031 1.263 1.029 1.211

Fig. 11.15. Convergence behavior with different conductivities γ and fixed relative permeability µr = 1000 for the iron core

As a practical example, the results of a 3D magnetic field computation for an electric transformer are shown. In Fig. 11.17 the model including the finite element discretization is displayed.

Fig. 11.16. Convergence behavior with different conductivities γ and fixed relative permeability µr = 1 for the iron core

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Fig. 11.17. FE mesh of an electric transformer (no air region is displayed)

Due to symmetries, it is only necessary to simulate one quarter of the full configuration by applying proper boundary conditions. The core is made of iron (µr = 1000) and has a conductivity of γ = 106 S/m. The number of coil windings, respectively, the size of the current have been chosen in way to ensure a maximum current density of 4 A/mm2 to avoid an unnecessary heating of the coil. Therefore, with 100 windings one gets a current amplitude of 30 A, considering an inner coil radius of 55 mm, an outer coil radius of 60 mm and a coil length of 150 mm. In Table 11.9 one can see the characteristic solver data, e.g., setup and solution times as well as operator and grid complexity for different FE meshes. Again, the grid and operator complexity values are close to 1. However, it should be mentioned that for an optimal multigrid method, the iteration numbers are independent of the number of unknowns. In our case, a small increase of the iterations can be determined. Table 11.9. Electric transformer: Performance and complexity for different FE meshes Nh 9.115 17.977 32.835 485.451

Iter Setup (s) Solver (s) OC 22 24 30 52

0.24 0.34 0.63 11.35

1.38 3.2 7.47 274.23

1.017 1.020 1.021 1.024

GC 1.196 1.206 1.195 1.181

11.4 Algebraic MG Method

311

Fig. 11.18. Main magnetic induction in the iron core of the electric transformer

The presented AMG solvers are well suited for the efficient solution— both concerning CPU time and memory requirements—of algebraic systems of equations arising from nodal as well as edge FE discretizations of Maxwell’s equations. In particular, the presented algorithms for the coarsening strategy make the solvers very attractive also for non-linear electromagnetic field problems, since the setup time can be kept very small. Further research is concentrated on improvements of the prolongation operators to obtain even better convergence rates. One possible method was proposed in [212] (so-called smoothed aggregation), which could be applied for our problem classes. If even more speedup is required for practical applications, the presented AMG methods can be parallelized on distributed-memory computers. The first promising results can be found in [80, 81].

12 Industrial Applications

12.1 Electrodynamic Loudspeaker The electrodynamic loudspeaker to be investigated is shown in Fig. 12.1. A cylindrical, small, light, voice coil is suspended freely in a strong radial magnetic field, generated by a permanent magnet. The magnet assembly, consisting of pole plate and magnet pot, helps to concentrate most of the magnetic flux within the magnet structure and, therefore, into the narrow radial air gap. When the coil is loaded by an electric voltage, the interaction between the magnetic field of the permanent magnet and the current in the voice coil results in an axial Lorentz force. The voice coil is wound onto a former, which is attached to the rigid, light, cone diaphragm in order to couple the forces more effectively to the air and, hence, to permit acoustic power to be radiated from the assembly. The main function of the spider and the surround is to allow free axial movement of the moving coil driver, while nonaxial movements are suppressed almost completely. Since in the case of a loudspeaker

Fig. 12.1. Schematic of an electrodynamic cone loudspeaker

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the interaction with the ambient fluid must not be neglected, the electrodynamic loudspeaker represents a typical coupled magnetomechanical system immersed in an acoustic fluid. This is, why for the detailed finite element modelling of these moving-coil drivers the magnetic, the mechanical as well as the acoustic fields including their couplings have to be considered as one system, which cannot be separated. Furthermore, electrodynamic loudspeakers in the low-frequency range under large-signal conditions show a strongly non-linear behavior, which is caused mostly by two factors—the inhomogeneity of the magnetic field in the air gap, i.e., magnetic non-linearities, and the non-linearity of the suspension stiffness, i.e., mechanical non-linearities. These non-linearities are caused by the large vibration amplitudes, especially at low frequencies. For large input powers the distortions increase rapidly and reach the same order of magnitude as the fundamental. To reduce the efforts in the development of electrodynamic loudspeakers, precise and efficient computer modelling tools have to be used. With these computer simulations, the costly and lengthy fabrication of a large number of prototypes, required in optimization studies by conventional experimental design, can be reduced tremendously. For many applications an equivalent electromechanical circuit model has been developed (see e.g., [209]). However, the main drawback of these simulation models is that the circuit-element parameters have to be determined empirically by measurements on a prototype. Therefore, we will demonstrate that a model based on the partial differential equations including all coupling terms and solved by an appropriate FE method (see Sects. 7 and 8) can totally fulfill the needs of an engineer. Such a method just requires the geometry of the loudspeaker as well as material data of each part. In the following, we will first discuss finite element models for the smalland large-signal behavior. Later, comparisons between simulation results and accordingly measured data are shown for verification purposes. The main focus will be on the demonstration of the practical usability of this scheme within the industrial computer-aided engineering of electrodynamic loudspeakers. 12.1.1 Finite Element Models Small-Signal Computer Model The finite element discretization of the electrodynamic loudspeaker under small-signal conditions is shown in Fig. 12.2. Here, the voice coil is discretized by so-called magnetomechanical coil elements based on the motional emf-term method (see Sect. 7.3.4), which solves the equations governing the electromagnetic and mechanical field quantities and takes account of the full coupling between these fields. Due to the concentration of the magnetic flux within the magnet assembly, the magnet structure and only a small ambient region have to be discretized by magnetic finite elements. Furthermore, the surround,

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315

spider, diaphragm, and former are modelled by mechanical finite elements. Finally, the surrounding fluid region in front of the loudspeaker is discretized by acoustic finite elements. The fluid region is surrounded by infinite elements for allowing open-domain computation (see Sect. 5.5.1), which have to be located in the far field of the moving-coil driver in order to work correctly. The input level of these simulations is 1 W referred to 4 Ω.

Fig. 12.2. Small-signal finite element model of an electrodynamic loudspeaker

Large-Signal Computer Model The finite element discretization of the electrodynamic loudspeaker under large-signal conditions is shown in Fig. 12.3. The following modifications have been performed in comparison to the above-explained small-signal computer model: 1. To take into account the variation of the force factor for a coil under large excursions (i.e., magnetic non-linearities), the magnetomechanical coil elements discretizing the voice coil of the loudspeaker are based on the moving material method (see Sect. 7.3.4). The force factor is defined by αf = BlN , with B the magnetic induction in the air gap, l the length of one winding and N the number of turns of the voice coil, which are located in the homogeneous magnetic field.

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2. Furthermore, first simulation results showed that the mechanical nonlinearities, i.e., the geometric non-linearity as a result of large displacements and the material non-linearity due to a non-linear stress–strain relationship have to be taken into consideration only for the spider. Therefore, to allow a more efficient computation of the large-signal behavior the diaphragm and the surround are discretized by finite elements solving linear mechanics. 3. Finally, measurements have shown that the distortion factors of the near field and diaphragm acceleration are in excellent agreement. Due to this correlation a modified axisymmetric finite element model has been applied, in which acoustic elements were eliminated completely (see Fig. 12.3). The influence of the surrounding air, which consists of mass-loading effects and damping due to the sound emission, is now realized by so-called spring elements. These elements have been located on the outside boundary of the surround and diaphragm.

Fig. 12.3. Large-signal finite element model of an electrodynamic loudspeaker

12.1.2 Verification of Computer Models The verification of the computer models described above has been performed by comparing simulation results with corresponding measured data. In the first step, the most important small-signal results (frequency dependencies of the electrical input impedance, diaphragm acceleration and axial sound pressure levels as well as Thiele-Small parameters [174]) were considered. As can be seen in Fig. 12.4, good agreement between simulation results and measured data was achieved.

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Next, the force–displacement characteristics were measured and compared with simulations (see Fig. 12.5). After this basic validation of the large-signal computer model, the total harmonic distortion (THD) factors of the voice coil currents and diaphragm accelerations at large-signal conditions have been calculated, which compute as + pˆ2 + pˆ2 + ... THD = + 2 2 2 3 2 , (12.1) pˆ1 + pˆ2 + pˆ3 + ... with pˆi the amplitude of the i-th harmonic. In addition, we define the pˆ2 k2 = + 2 2 pˆ1 + pˆ2 + pˆ23 + ... pˆ3 k3 = + . 2 2 pˆ1 + pˆ2 + pˆ23 + ...

(12.2) (12.3)

The input level of these simulations was 32 W referred to 4 Ω. As can be seen in Fig. 12.5, the good agreement of measured and simulated results over a wide frequency range validates the large-signal model depicted in Fig. 12.3.

Fig. 12.4. Comparison of simulated and measured small-signal results: (a) Frequency dependency of electrical input impedance Z, (b) Axial small-signal sound response level SPL at 1 m distance

12.1.3 Numerical Analysis of the Non-linear Loudspeaker Behavior Measurements as well as simulation results show that at frequencies f < 60 Hz the odd-order harmonics and at higher frequencies the even-order harmonics dominate. The large advantage of computer modelling is the separation of the different non-linearities for the different components of the loudspeaker. In this way, the influence of the different non-linear effects on the loudspeaker

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Fig. 12.5. Comparison of simulated and measured large-signal results: (a) Forcedisplacement characteristic of the loudspeaker, (b) Total harmonic distortion (THD) of diaphragm acceleration at an input power of 32 W

behavior can be very efficiently extracted and researched in the simulation. For example, simulations showed that the magnetic non-linearities cause notable quadratic distortion factors at frequencies f > 60 Hz, whereas the mechanical non-linearities cause the rapid increase of the even-order harmonics in the lower-frequency range (see Fig. 12.6). Furthermore, both mechanical and magnetic non-linearities are responsible for the cubic distortion factor.

Fig. 12.6. Numerical investigation of distortion factors of diaphragm acceleration at an input power of 32 W: (a) Quadratic distortion factor k2 , (b) Cubic distortion factor k3

In the next step of the numerical analysis, the influence of design parameters of the magnet system on the distortion factors has been investigated.

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Simulations considering only magnetic non-linearities showed that large coil flux variations result in notable odd-order harmonics. On the other hand, a nonsymmetric magnetic field in the air gap causes large coil offsets resulting in significant even-order harmonics. Further computations showed that the position of the permanent magnet has a big influence on the symmetry of the magnetic field in the air gap and therefore can be used in the optimization of the system (see Fig. 12.7b). Furthermore, it could be shown that the transient

Fig. 12.7. Finite element models: (a) Original magnet system, (b) Optimized magnet system, (c) Original and optimized spider

magnetic field of the current-carrying voice coil must not be neglected at largesignal conditions. To reduce the influence of the coil field under large-signal conditions on the symmetry of the force factor, the whole magnet pot has to be saturated and the upper air gap above the pole plate has to be increased (see Fig. 12.7b). These design modifications result in a much more symmetric decrease of the force factor (see Fig. 12.8a). Furthermore, to minimize the variation of the force factor, i.e., to raise the so-called jump-out excursion, the thickness of the pole plate has to be reduced (see Fig. 12.7b and Fig. 12.8a). Finally, since this design modification results in a smaller efficiency of the loudspeaker, the width of the permanent magnet has to be increased. After the above-explained numerical analysis of magnetic non-linearities, the influence of design parameters of the spider on the distortion factors caused by the mechanical non-linearities has been investigated. Simulations considering magnetic and mechanical non-linearities showed that a larger spider height results in a more linear force–displacement characteristic and significantly smaller odd-order harmonics Furthermore, a continuous displacement of each midpoint of the spider grooves causes a more symmetric force–displacement characteristic resulting in smaller even-order harmonics. For a more detailed discussion we refer to [174].

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Normalized force - factor Bl (%) 0 -10 -20 -30 -40

Displacement (mm) 8 Original 4

Optimized

Original -50 -60 4 6 -10 -6 -4 0 10 a) Axial coil displacement (mm)

Optimized

0 -4 -8 -20 b)

-10

0 10 Force (N)

20

Fig. 12.8. Comparison of the original and optimized loudspeaker: (a) Simulated coil flux variation (normalized to the original small-signal value), (b) Simulated force-displacement characteristic

12.1.4 Computer Optimization of the Non-linear Loudspeaker Behavior In the course of this computer optimization, the knowledge of the sensitivity studies explained in the previous section was put into a new prototype to reduce the even- and odd-order harmonics under large-signal conditions. As can be seen in Fig. 12.9, significant smaller distortion factors were achieved. In particular, cubic distortion factors could be reduced tremendously. For example, at a frequency of 20 Hz the improvement is 70% with respect to the original loudspeaker. This significant reduction of cubic harmonics is in accordance with studies concerning the subjective perception of low-frequency distortions [72]. According to [72], odd-order harmonics are above all responsible for the deterioration of the sound quality. Furthermore, the important ancillary condition of a similar small-signal behavior with respect to the original loudspeaker must be fulfilled. Small-signal simulations resulted in an acceptable reduction in efficiency of 0.5 dB. Furthermore, the numerically predicted improvements in the large-signal behavior of the loudspeaker could be successfully confirmed by measurements on the new prototype (see Fig. 12.9). Therefore, it can be stated that the presented simulation scheme is well suited to the industrial computer-aided design of electrodynamic loudspeakers, since an optimization with a significantly reduced number of prototypes can be achieved.

12.2 Noise Computation of Power Transformers The sound emission of power transformers conflicts more and more with tightened low emission standards, which must be fulfilled, especially at night.

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Fig. 12.9. Comparison of simulated and measured distortion factors of the optimized loudspeaker (at an input power of 16 W): (a) Total harmonic distortion (THD), (b) Quadratic distortion factor k2 , (c) Cubic distortion factor k3

Therefore, the prediction and reduction of these sound emissions is of increasing interest for the electrical power industry. The transformer noise is mainly caused by the following sources [51, 88]: 1. The no-load noise caused by magnetostrictive strain of core laminations. 2. The noise produced by fans or oil pumps. 3. The load-controlled noise caused by Lorentz forces resulting from the interaction between the magnetic stray field of one current-carrying winding and the total electric currents in the conductors of the other winding. These forces cause vibrations of the winding and result in acoustic radiations with twice the line frequency (100 Hz or 120 Hz). During recent decades the magnetic noise caused by magnetostrictive strain of the core laminations and the noise of fans have been investigated and considerably decreased [101, 125, 178]. Therefore, the coil-emitted noise (see item 3 above) is of increasing interest. At the moment, approximate empirical prediction formulas, which primarily depend only on the rated power of the transformer, represent the state-of-the-art. However, the main disadvantage of these prediction formulas is that accurate parameters on the load-controlled noise are not available. 12.2.1 Finite Element Models The goal of the numerical simulation is to predict very precisely the emitted sound of loaded transformers within a test hall. In the first step of the modelling scheme the outermost winding surface displacements are calculated by an acoustic-magnetomechanical finite element model of one winding of the

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Fig. 12.10. Overview of the developed calculation scheme

oil-filled transformer. Due to rotational symmetry of the winding and symmetric load (as a result of the ideal measurement condition in the factory test field), a 2D finite element model based on axisymmetric elements can be used (see Fig. 12.11). In the finite element model, the voltage-loaded conductors of the winding are discretized using magnetomechanical coil elements. These elements solve the equations governing the electric circuit, the magnetic as well as the mechanical field equations, and take account of the full coupling between these fields (see Sect. 7). All winding clamping and insulation materials between each coil as well as the winding support platforms are modelled using magnetomechanical finite elements, which solve the coupled magnetic and mechanical field equations. Instead of a complete model of the highly permeable core by magnetic finite elements, this computer model was simplified by applying Neumann boundary conditions at the boundary of the winding window (see Fig. 12.11). Furthermore, the surrounding oil within the tank is discretized using purely acoustic finite elements and magnetic-acoustic finite elements (solving the magnetic as well as the acoustic partial differential equation without any coupling). Finally, the tank is modelled using standard mechanical finite elements.

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Fig. 12.11. Axisymmetric acoustic-magnetomechanical finite element model of one winding of the oil-filled power transformer

Additionally, the following aspects have to be considered for the precise computer simulation of the winding vibrations of loaded power transformers: •

To measure the load-controlled noise in a factory test field, the transformer has to be operated at short-circuit and at rated currents. In this case, due to the small voltage during the short-circuit test, the core-emitted noise can be neglected and, therefore, a clear distinction between the no-load noise and the load-controlled noise can be achieved. In the simulations, this effect was taken into account by modelling the innermost low-voltage (LV) winding as a voltage-loaded coil with an external voltage of zero. The high-voltage (HV) winding and both in-series connected, outermost tapping windings, however, are loaded with the measured short-circuit voltage in a star connection.

•

Furthermore, the core clamping supports have been ignored in the finite element model to reduce the effort. Therefore, the influence of these supports, which consists of an additional axial stiffness of the winding, is realized by so-called spring elements. As shown in Fig. 12.11, these spring elements have been located at the outside boundary of the upper and lower winding support platform. In the simulations, a stiffness of 85 MN/m has been used, which is in accordance with the experience of the transformer

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manufacturers. •

Finally, since measurement results revealed a big influence of the tapchanger position on the measured vibrations and sound pressure levels, the following simulations have been performed for three nominal positions: – Tap-changer position 1: The HV winding and both tapping windings are connected in series. – Tap-changer position 2: The HV winding and the coarse tapping winding are connected in series. – Tap-changer position 3: Only the HV winding is connected.

For the computation of the amplitude of the winding surface displacements, a dynamic analysis using a sinusoidal 50 Hz (or 60 Hz) excitation signal for the voltage between the two supply terminals of the high-voltage winding was performed. It should be noted that further input parameters are the geometry of the power transformer, the density, modulus of elasticity, Poisson’s ratio and loss factor for the mechanical materials (tank, conductors, insulation and clamping materials), the electrical conductivity for the conductors as well as the density and bulk modulus for the surrounding oil. After the computation of the response signals (current in the conductors of both windings as well as mechanical displacements of the outermost winding), the Fourier transform of the output signals has to be calculated. Finally, the 100-Hz (or 120-Hz) component of the spectrum must be extracted. In the second step, the previously calculated winding surface displacements are now taken as mechanical excitation in a 3D acoustic-mechanical finite element model of the complete oil-filled tank (see Fig. 12.12). Furthermore, in the 3D finite element model the 120 degree phase shift between the three windings is taken into account. In this model the tank, the core clamping as well as the connections between core clamping and top of the tank, are modelled using mechanical finite elements. Furthermore, the surrounding oil within the tank is discretized using acoustic finite elements. Finally, the additional stiffness of the core clamping supports is again implemented using spring elements. Since the load-controlled noise is primarily a simple 100-Hz (or 120-Hz) tone, a harmonic analysis has been performed. In the last step the previously calculated tank-surface vibrations are now applied as mechanical excitation in the final acoustic simulations to calculate the radiated transformer noise. Here, the radiation within closed rooms such as a high-voltage laboratory are calculated (see Fig. 12.10). For the computation of the sound radiation within a high-voltage laboratory, a 3D acousticmechanical finite element model has been set up (see Fig. 12.13). Here, the tank of the transformer is discretized using mechanical finite elements. Furthermore, the surrounding air within the test hall is modelled using acoustic finite elements. The walls of a typical high-voltage laboratory are not covered

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325

Fig. 12.12. Three-dimensional acoustic-mechanical finite element model of an oilfilled transformer

with any absorbing material. Therefore, in these simulations, the transformer was assumed to be positioned within a hall with ideally reflecting walls. In this computer simulation 640 000 3D finite elements have been used.

Fig. 12.13. Three-dimensional acoustic-mechanical finite element model of the transformer tank and the high-voltage laboratory

12.2.2 Verification of the Computer Models The verification of the computer models described above has been performed by comparing simulation results with corresponding measured data. It should be noted that due to the complexity of the sound emission of the loaded

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power transformer, analytic calculations are unavailable and therefore cannot be used for verification purposes. 12.2.3 Verification of the Calculated Winding and Tank-surface Vibrations In the first step, the axisymmetric finite element model has been verified by comparing measured and calculated short-circuit currents, mechanical eigenfrequencies as well as winding surface accelerations. In Table 12.1, the short-circuit currents obtained by measurements as well as simulations are shown for two tap-changer positions. The good agreement between measured and calculated values (the deviation is within 1.25%) validates again the developed coil-modelling scheme.

Table 12.1. Measured and simulated short-circuit currents (LV Low voltage; HV High voltage) Measurement Simulation (A) (A) Current Current Current Current

in in in in

LV winding, at tap-changer position 1 HV winding, at tap-changer position 1 LV winding, at tap-changer position 2 HV winding, at tap-changer position 2

825 159 825 180

832 161 825 181

Next, the measured and calculated transfer functions and mechanical eigenfrequencies of the complete winding system mounted on the core have been compared (see Fig. 12.14). The deviation at the second eigenfrequency is due to the fact that the actual winding does not show an exact axisymmetric construction. Furthermore, it should be noted that the winding was axially excited at the upper winding support platform and the resulting radial coil acceleration of the outermost winding was measured as well as simulated. In a final verification of the axisymmetric finite element model, the simulation results have been compared with corresponding measured winding surface accelerations. Here, the axial accelerations on the upper winding support platform and the radial vibrations on the outermost fine tapping winding were measured using an oil-resistant piezoelectric accelerometer. The measurements showed that the sensitivity of these sensors against electromagnetic interferences of the high-voltage winding is negligible. Furthermore, the deviations of subsequent measurements were within a range of ±1.5%. In Table 12.2, the winding vibrations of the transformer without a tank, in Table 12.3, the normalized accelerations of the transformer with an oil-filled tank are compared, respectively. In the case of the transformer with an oil-filled tank (see

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327

Fig. 12.14. Comparison of measured and simulated transfer functions and mechanical eigenfrequencies of the winding mounted on the core

Table 12.3), measurements as well as simulations reveal that the surrounding oil does not influence the axial accelerations of the winding support platform. However, due to the mass-loading effect of the surrounding oil, the radial coil acceleration amplitudes are nearly halved when compared to the vibrations ignoring the oil-filled tank. In summary, it can be stated that an axisymmetric finite element model precisely predicts the winding surface accelerations of a loaded power transformer for both configurations, with and without an oil-filled tank.

Table 12.2. Transformer without oil-filled tank: Measured and simulated winding accelerations (m/s2 ) (m/s2 ) Radial coil acceleration at tap-changer position 1 0.047 Axial clamping acceleration at tap-changer position 1 0.036 0.021 Radial coil acceleration at tap-changer position 2 Axial clamping acceleration at tap-changer position 2 0.031

0.046 0.037 0.019 0.032

Finally, the 3D acoustic-mechanical finite element model of the oil-filled tank has been verified by comparing the calculated tank side wall accelerations with values measured at nine different positions (see Table 12.4).

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Table 12.3. Transformer with oil-filled tank: Measured and simulated winding accelerations (normalized to the corresponding result without an oil-filled tank (see Table 12.2)) Measurement Simulation Radial coil acceleration at tap-changer pos. 1 Axial clamping acceleration at tap-changer pos. 1

0.59 0.96

0.62 0.99

Table 12.4. Measured and simulated tank accelerations Position Measurement Simulation Position Measurement Simulation (m/s2 ) (m/s2 ) (m/s2 ) (m/s2 ) 1 2 3 4 5

0.16 0.063 0.14 0.055 0.04

0.13 0.075 0.1 0.07 0.05

6 7 8 9

0.029 0.01 0.02 0.04

0.03 0.015 0.025 0.045

12.2.4 Verification of the Sound-field Calculations After these basic validations of the computational models, the A-weighted sound-power level [145] of the short-circuited transformer was measured in accordance with the European standard EN 60551 [204] and compared with the corresponding acoustic simulations. This standard requires that the Aweighted sound-pressure levels around the transformer have to be measured at a distance of 0.3 m from the tank surface and at half the tank height. Furthermore, these measurements have to be performed in a typical high-voltage laboratory at a transformer manufacturer. Due to this fact, the 3D finite element model for the calculation of the radiated noise within closed rooms, as shown in Fig. 12.13, has been used. An A-weighted sound power level of 66.5 dB(A) was calculated from the simulated sound-pressure levels. Considering the fact that the reproducibility of the sound-pressure measurements lies within a range of ±1 dB, a good agreement between measurement and simulation was achieved (see Table 12.5). Furthermore, as can be seen from Table 12.5, in this case the deviation between measured and calculated sound power level is considerably smaller than those resulting from the current-prediction formulas for the load-controlled noise. Therefore, it can be concluded that this pure finite element scheme is well suited to the computation of the load-controlled noise of oil-filled power transformers, which are operated within a typical high-voltage laboratory. 12.2.5 Influence of Tap-changer Position As a first application, the influence of the tap-changer position on the winding and tank surface vibrations as well as on the A-weighted sound-power level

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Table 12.5. Radiation within closed rooms: Measured, simulated, and predicted sound-power levels Sound-power level (dB(A)) Sound-pressure measurement according to [204] Finite element simulation Empirical prediction formula according to [204] Empirical prediction formula according to [177]

68 66.5 63.5 60.5

has been investigated. Table 12.6 shows the simulation results at tap-changer position 1 (HV winding and both tapping windings are connected in series) and at tap-changer position 3 (only the HV winding is connected). Simulations as well as measurements reveal that the radial vibrations of the outermost, fine tapping winding are greatly decreased at tap-changer position 3 as compared to position 1.

Table 12.6. Influence of the tap-changer position obtained by simulation Tap-changer Tap-changer position 1 position 3 Radial coil acceleration (m/s2 ) Axial winding clamping acceleration(m/s2 ) Tank side wall acceleration (m/s2 ) A-weighted sound power level (dB(A))

0.033 0.044 0.025 59

0.009 0.033 0.009 45

With these simulations it was found that the radial magnetic volume forces acting on the innermost LV and HV winding are almost independent of the tap-changer position. Therefore, this reduction of the radial coil vibrations at tap-changer position 3 is based on the noncurrent-carrying, outermost, fine tapping winding. On the other hand, simulation results revealed that the axial magnetic volume forces acting on the LV and HV winding are larger at tap-changer position 3 when compared to position 1. This is caused by the edge-fringing effect and is responsible for the fact that the axial winding clamping vibrations are almost independent of the tap-changer position (see Table 12.6). Furthermore, due to the decrease of the radial surface accelerations of the outermost winding at tap-changer position 3, the tank-surface vibrations and, therefore, the calculated A-weighted sound-power level are greatly reduced. Therefore, in contrast to [177], where it is assumed that only axial winding vibrations are responsible for the load-controlled noise, these simulations clearly show that the radial coil vibrations also have a significant influence on the coil-emitted noise.

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12.2.6 Influence of Stiffness of Winding Supports In a second application, the influence of the stiffness of the winding and core supports on the load-controlled noise has been investigated. This stiffness has been modelled in the finite element simulations by applying mechanical spring elements, which were located at the outside boundary of the upper and lower winding support platform (see Fig. 12.11). As expected, the simulations reveal that neglecting this axial stiffness causes significantly increased axial clamping accelerations and slightly decreased radial coil accelerations. Therefore, greatly increased sound-pressure levels result (see Table 12.7). These results indicate that this stiffness has a strong influence on the radiated transformer noise effect, which can be used in the optimization of the system.

Table 12.7. Influence of stiffness of winding supports obtained by simulation (SPL Sound-pressure level) Radial coil Axial winding SPL in acceleration acceleration 0.3 m (m/s2 ) (m/s2 ) (dB) With stiffness of winding support 85 MN/m Without stiffness of winding support

0.046 0.044

0.037 0.62

56.8 82.3

12.3 Fast-switching Electromagnetic Valves Modern fast-acting solenoidal valve applications demand further improvements of the operation speed and reproducibility of the opening and closing phase. The development goes towards lightweight construction in combination with sophisticated energizing concepts. This gives rise to structural vibration as well as sound-emission problems whose elimination by means of passive damping is not sufficient. A dynamic analysis of the switching behavior with sufficient precision taking into consideration all significant physical effects can be done only by a numerical analysis. The reproduction of the dynamics of fast-switching solenoidal valves proves to be highly non-linear. Both the magnetization state of the ferromagnetic armature material and the induced eddy current distribution inside the armature during its accelerated motion within an inhomogeneous magnetic field are accounted for. In addition to the numerical calculation scheme presented in Sect. 7.3, we have to take impact dynamics into account to fully model a valve-switching cycle. Impact/contact problems are different from Neumann and Dirichlet boundary problems since the contact constraints are unknown in time and space and have to be determined as a part of the solution.

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Fig. 12.15. Magnetic field region of the actuator including impact region

12.3.1 Modelling and Solution Strategy To precisely compute the dynamic behavior of a solenoidal actuator a numerical calculation scheme has to be able to handle the electromagnetic field, the mechanical field as well as the coupling terms. The strong coupling is necessary because of the interdependency between the position and velocity of moving mechanical parts and the inductivity of the solenoid as well as eddy current induction. Therefore, we are able to reproduce the dominant influence of eddy currents on the dynamics as well as the damping effect during the bouncing period at valve opening and closing. Magnetic Field The governing equation describing the magnetic field can be derived from Maxwell’s equations for the quasistatic case (neglecting the displacement currents) using the magnetic vector potential as state variable (see Sect. 4.2.1). To precisely compute the magnetic field within a solenoidal actuator, the following physical phenomena have to be taken into account: •

B–H curve: The measured B–H curve data are approximated by an enhanced smoothing spline technique to guarantee a smooth approximation of the curve as well as of its derivative. The resulting non-linear magnetic equation is solved by a Newton method with a line search algorithm (see Sect. 4.7.5).

•

Voltage-loaded coil: For a voltage-loaded coil the additional circuit equation is simultaneously solved with the partial differential equation of the magnetic field (see Sect. 7.3.1).

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Mechanical Field and Contact Mechanics For the mechanical field, Navier’s equation is solved with the mechanical displacement as state variable (Sect. 3.7). The impact of moving parts on the static parts of the valves at the switching of the valve state results in a noisy and unreliable bouncing, known as hard landing. A correct physical modelling of the impact as well as the overcoming of the inherent convergence problem of dynamic contact problems can be solved only by using a contact algorithm that satisfies all kinematic and kinetic conservation laws as can be found, e.g., in [8]. Including frictionless contact mechanics into the system starts with a contact-detection algorithm based on the normal distance of the two contacting bodies. In the case of satisfied contact conditions, the contact pressure in the normal direction is applied by using an exponential contact pressure–displacement relation pc (u) = p0 (l − g(u))m .

(12.4)

Here, p is the contact pressure and g the gap length, both in the normal direction and is dependent on the nodal displacements u within the discretized system. The surface hardness p0 can be considered as a penalty parameter to incorporate the contact constraints. The exponent m can be derived analytically using a statistical treatment of Hertzian microcontacts and is verified by measurements within a range of 2.0 to 3.3 [215]. The constant l can be interpreted as the surface roughness and can be used to realize contact at a finite gap length as described later. Applying now the FE method to the continuum mechanics of contact, we have to add the contact force vector f c (u) =

ne 

 f e (u) with

fe =

e=1

pc (u)N dΩ

(12.5)

Ωe

to the right-hand side of the algebraic system of equations. The nodal contact force vector f c is given by the assembly of the element contact force vectors f e over all ne contact elements currently in closed contact. With pc we denote the contact pressure determined in (12.4). To achieve quadratic convergence in solving the non-linear mechanical system with the Newton method we have to add further to the linear stiffness matrix (see Sect. 3.7.1) the tangential contact stiffness matrix

f c (un+1 )

, (12.6) Kc =

∂u n+1 u

which is obtained by an exact and complete linearization of the contact force vector using a directional derivative with respect to the nodal displacements u at time step (n + 1).

12.3 Fast-switching Electromagnetic Valves

333

Fig. 12.16. Eddy-current-induced hysteresis of magnetic flux, magnetic force, and coil inductivity

12.3.2 Actuator Characteristics The direct acting and normally closed solenoid valve investigated can be characterized as a short-stroke valve with a compact design at a volume of 28 cm3 and an air-gap cross-sectional area of 61 mm2 . Valve opening is forced by the electromagnet against the force of the return spring with a spring constant of 6.7 kN/m and a pretensioning of 18 N. The limited stroke of max. 45 µm causes only a small dependency of the static actuator properties on the armature position in the complete operation range of the actuator. With increasing coil current and magnetic saturation, the inductivity of the electromagnet is decreasing in compliance with the decreasing permeability. At the same time, the gain in magnetic force diminishes due to the saturation. Investigating, furthermore, the actuator dynamics, the eddy-current-induced hysteresis can be acquired by using a high-level signal harmonic excitation at different fre-

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quencies (Fig. 12.16). The B–H magnetization curves of ferromagnetic valve components used for the numerical analysis have been adapted to the hysteresis measurement results. This procedure is necessary since some components are manufactured with particular mechanical and heat treatment whereby changing their original magnetic properties and a direct measurement of the saturation curves of small parts is full of uncertainty. Comparing simulation results using the adapted B–H magnetization curves with measurement results at higher frequencies, it can be shown that magnetic hysteresis effects are negligible in comparison to the eddy-current-induced hysteresis (Fig. 12.16).

Fig. 12.17. Effect of eddy current induction on the frequency response

Eddy current induction increases with rising frequency and affects the diffusion rate and extension of the magnetic field inside the armature as well as its temporal expansion, known as the skin effect (see Sect.4.2.2). Eddy currents are therefore a critical factor concerning the dynamic behavior of electromagnetic actuators since they constrain the operation range of the actuator. The magnitude and phase of the actuator frequency response (magnetic force over applied coil voltage, respectively current) is shown in Fig. 12.17. The transfer function shows a cutoff frequency of 860 Hz for the current-fed coil, which is beyond the operation range of the valve. In summary, the dynamic magnetic force is a complex function of the system’s operation point defined by the parameters:

12.3 Fast-switching Electromagnetic Valves

• • • •

335

Armature position and velocity Applied coil current Magnetization state of the ferromagnetic material Induced eddy current distribution inside the armature

12.3.3 Actuator Dynamics Investigating the actuator dynamics, it is instructive to consider first two types of coil energizing: A current-fed coil and a voltage-fed coil. Connecting a current source to the coil results in the shortest pull-in time since, due to the applied current, no voltage feedback effect can become active and the electromagnetic compensation process can be neglected. This forces the fastest movement of the armature and can be approximately realized by a digital controller unit. On the other hand, the use of a voltage source leads to much longer time constants due to the delayed rise of the coil current. As a result, applying a current, respectively voltage, Heaviside function to the coil, the current (voltage)-fed coil has minimal pull-in time of 0.16 ms (0.35 ms) and a minimum response delay time of 0.08 ms (0.26 ms) as shown in Fig. 12.18.

Fig. 12.18. Simulation results: Valve-opening times and strike velocity

No further reduction of the pull-in time can be achieved without using a more sophisticated energizing concept or reduction of eddy current induction through design features. But although eddy currents are undesired, since they increase the switching times, they are desired to damp the bouncing of the armature. A short pull-in time comes with high strike velocity at the end

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of the valve-opening process (Fig. 12.18). Therefore, there is an optimization conflict between obtaining short valve-opening times and reducing bouncing effects caused by high impact velocities to achieve a reproducible valve function without reopening. 12.3.4 Dynamics Optimization I: Electrical Premagnetization The actuator dynamic can be affected basically by •

Forcing the mechanical compensation process by enhancement of the dynamic magnetic force

•

Forcing the electrical compensation process by enhancement of the energizing power during the pull-in time

Fig. 12.19. Stroke and velocity profile for several levels of electrical premagnetization

One method to accelerate the pull-in time of the valve is to premagnetize the actuator by a permanent magnet or electrically using a coil-current control unit. Premagnetization causes a magnetization state inside the armature at increased magnetic field intensity and reduced permeability. The lower

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Fig. 12.20. Effect of electrical premagnetization on the valve dynamics

permeability again leads to an increased magnetic field diffusion velocity and therefore to an accelerated rise of the magnetic force at a following coil excitation spike. The magnetic force caused by premagnetization itself as well as its accelerated rise will exceed the level of the return spring force at an earlier time stage (Fig. 12.19), resulting in a reduced response delay time. In contrast, the actual time of movement (valve needle flight time) itself is almost not influenced (Fig. 12.20). 12.3.5 Dynamics Optimization II: Overexcitation Further improvements in the valve dynamics can be achieved by applying a high current peak to the coil at the start of the valve-opening phase, known as overexcitation. The current spike acts until the end of the opening process. Afterwards, the coil current is reduced and controlled to the nominal current. The profit in higher magnetic force is low, since we get into the region of magnetic saturation, but due to the fact that the permeability decreases in this region, the diffusion velocity of the magnetic field into the armature can be increased. As a consequence thereof we obtain an accelerated rise of the magnetic force, which results in shorter pull-in times (Fig. 12.21). Applying different levels of overexcitation, the accelerated magnetic field diffusion into the armature material can be made visible by simulation (Fig. 12.22). The level of overexcitation is mainly limited by the thermal capacity of the actuator system. 12.3.6 Switching Cycle Modern applications of solenoidal actuators use a digital controller, which executes complex algorithms to provide a wide variation of the switching

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Fig. 12.21. Effect of coil overexcitation on the valve dynamics

Fig. 12.22. Magnetic field diffusion inside the armature for different levels of overexcitations at one point in time

times and precise actuator control. The modulation of the current according to a predefined profile permits control of the electromagnetic force and thus of the valve state. In the dosing valve system under investigation, the objective is to shorten the opening and closing time, since in these periods an undefined fluid dosage takes place. The coil current is controlled in an ON/OFF mode by a pulsewidth modulation generated by the controller using a so-called soft switching technique where the lower coil voltage level is zero (Fig. 12.23). Soft switching gives the coil current lower ripple and achieves a smoother output profile of the magnetic force. The complete control profile for a valve operation cycle can be subdivided as follows:

12.3 Fast-switching Electromagnetic Valves

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Fig. 12.23. Valve-switching cycle: Control profile and system response

1. Electrical premagnetization: Using the concept of electrical premagnetization as presented in the previ-

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ous section with a coil current of 1 A, the valve-opening process can start from a higher magnetization level. 2. Valve opening: The control accomplishes the valve opening by applying a momentary high-voltage overexcitation spike to build a high-density magnetic flux field almost instantaneously, resulting in a steeper magnetic force gradient to shorten the needle flight time. 3. Hold phase: During the hold phase—the actual dosing period—the nominal coil current of 3 A creates the holding force at the end of the stroke. 4. Valve closing: After turning off the driving coil current with a damping negative voltage spike, the spring force returns the armature to its initial position. In a final simulation, the complete switching cycle of the valve was reproduced (Fig. 12.23). Combining the concepts of premagnetization as well as overexcitation to optimize the actuator dynamics, the pure valve needle flight time at valve opening can be reduced to 200 µs.

12.4 Cofired Piezoceramic Multilayer Actuators Piezoceramic actuators are widely used for high-precision positioning systems. Their almost infinite resolution (in the nanometer range) and their very good repeatability predestine these actuators for the usage in linear stages, camera shutters or printer heads [210]. Rapid improvements in the performance of the ceramic materials used, make new smart designs possible and offer new application fields for piezoceramic actuators. The advantage of the piezoelectric actuator lies in the enormous force density and the high dynamics. In order to achieve improved deflections, hundreds of thin piezoceramic layers are stacked. Therefore, the mechanical displacement of the individual layers sum up, while the electric driving levels can be reduced due to a parallel switching of the ceramic sheets. The setup is similar to that used for multifoil-capacitors, as shown in Fig. 12.24. The highly dynamic deflection of these complex structures necessitates a sophisticated design in order to guarantee an effective operation and a large number of duty cycles. Due to the thin ceramic layers, with a thickness of about 100 µm, strong electric fields are established in the ceramic material at typical driving voltages of about 200 V. Therefore, these actuators show a strongly non-linear response mainly caused by the ferroelectric nature of the ceramic materials. These effects, which are responsible for the actuator’s performance, have to be considered during the design process. Therefore, non-linear material relations as well as ferroelectric hysteresis effects have to be considered.

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Fig. 12.24. Piezoceramic multilayer stack actuator

12.4.1 Setup of Multilayer Stack Actuators The thin ceramic layers used for the multilayer actuators are fabricated by tape casting using the so-called doctor blade technique (see [158]) in order to achieve a certain thickness of the layers. On these green sheets the internal electrodes consisting of silver (Ag) and palladium (Pd) are applied by screen printing leading to layers of a few micrometers thickness. After the cutting process the stacks are assembled and sintered together into a monolithic block. Therefore, these structures are called cofired. Due to the absence of adhesive layers the actuators provide a high stiffness and a fast response time. The internal electrodes of the actuator are arranged interdigitally, so that insulation gaps towards the outer side walls can be formed. External electrodes are placed at those sides where only internal electrodes of a single polarity are conducted to each actuator surface. After the manufacturing process the dipole orientations of the grains are statistically distributed, so no piezoelectric effect can be detected. By applying a strong electric field to the external electrodes a poling of the domains occurs, leading to a remanent polarization within the structure, so that the material becomes piezoelectric. Therefore, the piezoelectric effect in these materials is based on the ferroelectric hysteresis effect. The best performance of these actuators can be achieved if the polarization vector is homogeneous throughout the whole transducer, oriented in the principal working direction. However, due to the interdigitally arranged electrodes a homogeneous electric-field distribution cannot be achieved. Regions that are unpolarized, and therefore inactive, as well as regions with polarization vectors deviating strongly from the working axis occur, thus leading to strong mechanical stresses within the structure. While piezoceramic materials are able to handle large compressive stresses up to 300 MPa, they are rather sensitive concerning tensile stresses. This problem can be overcome primarily by simply applying a prestressing to the actuator, thus reducing the

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tensile stresses during the load cycle. On the other hand, locally restricted tensile stress peaks caused by an inhomogeneous electric field distribution may lead to cracks and a breakdown of the whole structure. The fatigue behavior of the ferroelectric material is very complicated, especially under highly dynamic driving conditions. A variety of models have been developed allowing the fatigue behavior of these structures to be analyzed for quasistatic loading conditions from a micro-mechanical point of view [3]. These models are very complex and their numerical simulation would require an extremely high computational effort, thus not allowing the whole complex multilayer structure to be handled. Furthermore, due to the fact that they are restricted to static analysis, dynamic effects that influence the stress distribution and material behavior dramatically, cannot be taken into account. 12.4.2 Finite Element Model The numerical analysis has been performed for a multilayer stack actuator consisting of 18 active ceramic layers, as shown in Fig. 12.25. Inactive zones are placed at the top and bottom of the structure. The symmetry of the setup has been used in our model by applying appropriate boundary conditions, therefore reducing the number of elements. The multilayer stack actuator has been modelled using up to 120 000 hexahedron elements, thus leading to 450 000 unknowns that have to be solved numerically. For this model linear as well as non-linear calculations have been performed to analyze the stress and electric field distribution at the tips of the internal electrodes. The FE formulation used is based on a direct coupling of the electric and mechanical field equations as discussed in Sect. 9. For the numerical analysis the influence of ferroelectric hysteresis has been studied. Therefore, typical values for the saturation field strength and polarization, as well as for the coercitive field intensity have been used. Due to the fact that the problem is highly non-linear a harmonic or modal analysis cannot be employed, even if the input signals are sinusoidal. Instead, the calculations are performed in the time domain using a Newmark time stepping algorithm (see Sect. 9.6). The non-linear iterative calculation scheme has to be executed for each time step using the solution of the previous time step as the initial guess. Therefore, the dynamic analysis of a complete piezoceramic stack actuator is made available. By using a sinusoidal input voltage with a peak value of 200 V and a frequency of 300 Hz we observed the displacement signal of the actuator. For this setup, measurements and numerical simulations have been performed. The excitation of the actuator has been realized using a high-power voltage amplifier allowing large capacitive loads to be driven. The displacements have been obtained offline by integration of the velocity signal, which was measured using a laser vibrometer.

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Fig. 12.25. Finite element model (dimensions in mm)

12.4.3 Measured and Simulated Results In Figs. 12.26 – 12.29 the measured and simulated results for the displacements are plotted showing the time signals as well as the frequency spectra. Due to the ferroelectric hysteresis higher-order harmonics are showing up.

Fig. 12.26. Measured displacement of the actuator

By comparing the measured and simulated spectral rates, higher-harmonic peaks of comparable magnitude can be detected. The results retrieved from numerical analysis show a slightly stronger influence of the ferroelectric effect. This is mainly caused by the used model parameters. The polarization hysteresis loop is normally measured in a quasistatic case. Due to the higher frequency applied, a reduction in hysteresis takes place due to the fact that frictional effects slightly suppress the local depolarization effects.

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Fig. 12.27. Simulated displacement of the actuator

Fig. 12.28. Frequency spectra of measured displacement

However, as can be seen from the results, the exactness of the simulations depends primary on the exactness of the material parameters, which have to be determined by appropriate measurements. Especially in the case of piezoceramic layers used for multilayer actuators that are fabricated by tape casting, process restrictions limit the sheet’s thickness. Thus, the manufacturing of probes with idealized geometries as required for measurements according to the standards [38, 105] becomes almost impossible. Therefore, we have developed a new method using a parameter-optimization technique for determining the linear parameters from measurements on samples with no special geometries [114]. Furthermore, we are going to improve this method to allow the large-signal parameter identification using inverse techniques. Combining both techniques, measurement and simulation, makes further improvements in the performance and duration of life of multilayer actuators possible.

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Fig. 12.29. Frequency spectra of simulated displacement

12.5 Capacitive Micromachined Ultrasound Transducers 3D ultrasound imaging is an area of extensive research and of great interest for many industrial applications ranging from nondestructive testing to medical-imaging systems (see e.g., [75]). To obtain a 3D ultrasound image, a 2D array of transducer elements with electronic focusing and beam steering is needed. Using standard piezoelectric transducer elements will result in difficulties concerning fabrication as well as electronic connections. Capacitive micromachined ultrasound transducers (CMUTs) may overcome many of the drawbacks of piezoelectric transducers, since CMUTs can be fabricated by adding a few technological steps to a standard CMOS process [2,57,128]. Due to small size and the possibility of integrating signal-processing electronics on a chip [129], these transducers may be an attractive alternative to standard ultrasound transducers. Figure 12.30 shows the top view of a CMOS testchip with 4 transducer arrays, each containing 19 capacitive transducers. The transducers are used in the transmitting as well as receiving mode, so that a classical pulse-echo mode can be performed. The principle setup of such an array is shown in Fig. 12.31 and the detail of one cell in Fig. 12.32. By applying a short voltage signal to the electrodes of each transducer cell, the membranes are deflected by the resulting electrostatic force, and an acoustic pulse is generated that propagates into the fluid. Now, the same transducer cells are used for measuring the reflected acoustic pulse, since the membranes will be deflected according to the fluid loading, resulting in a change of their capacitances.

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Fig. 12.30. Top view of a CMOS chip with four arrays, each containing 19 capacitive transducers

Fig. 12.31. Principle setup of cells

Fig. 12.32. Detailed view to a capacitive cell

12.5.1 Requirements to Numerical Simulation Scheme The design of such micromachined transducers strongly depends on the availability of appropriate computer-aided engineering (CAE) tools, since the fabrication of each prototype is quite costly. In addition, the signal-to-noise ratio is still too low compared to piezoelectric transducers, and a lot of research has still to be done [147]. Therefore, precise computer simulations are needed to analyze and, furthermore, optimize the dynamic behavior as well as efficiency of such transducers. However, the precise numerical computation of CMUTs is a quite complicated task, since one has to deal with several challenging problems, which can be summarized as follows: •

Multifield-Problem: We have to deal with a multifield problem consisting of the electrostatic, mechanical and acoustic field including their couplings (see Sect. 6 and Sec. 8).

•

Geometric aspect-ratio: A typical electrostatic cell has the following dimensions:

12.5 Capacitive Micromachined Ultrasound Transducers

347

Thickness of cell-membrane : 400 nm - 1 µm Side-length of cell-membrane : 20 µm - 40 µm Air-gap : 200 nm - 1 µm Assuming an air-gap of 200 nm and the wavelength λ in water at f = 1 MHz ( λ = c/f = 1.5 mm), we achieve at an aspect-ratio of 1.5 10−3 wavelength = = 7500 , air-gap 200 10−9 which clearly results in a big problem concerning the computational grid generation. An enhanced solution approach for this problem will be the use of nonmatching grids as described in Sect. 8.3.2. •

Non-linearities: According to the setup, different non-linearities of the involved physical fields have to be considered. Furthermore, the coupling mechanisms between the physical fields have to be taken into account, which are highly non-linear in practice. – Large deflections: If the deflections are in the range of the thickness of the structure, we have to consider the geometric non-linear case for the mechanical computation (see example in Sect. 3.7.4). – Stress-Stiffening effect The fabrication process of CMUTs leads to a large prestressing within the structure, which results in a stiffer structure behavior. This effect is fully considered within the geometric non-linear case, see Sect. 3.7.3. – Electric field The electrostatic force leads to a deformation of the electrode-membrane structure, and therefore, introduces a kind of geometric non-linearity in the electrostatic field PDE. This means that we always have to compute the electrostatic field on the actual configuration by performing an updated Lagrangian formulation (see Sect. 6). – Electrostatic force In a first case, let us consider a parallel plate capacitor, for which the electrostatic force computes as follows

FelC =

ε0 A U 2 . 2 d2

(12.7)

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In (12.7) U denotes the applied electric voltage, A the electrode surface, d the distance between the electrodes and ε0 the permittivity of the air. In a second case, we consider a parallel plate capacitor with insulation layer having a relative permittivity of εr . The electrostatic force at the interface between the insulation layer and air calculates by

FelI =

ε0 A U2 (εr −1) . 2 (d2 + d1 /εr )2 (12.8)

Therewith, the ratio of the two forces is FelI (d1 + d2 )2 ≈ εr ≈ εr C d22 Fel and we clearly see the importance of considering both parts of the electrostatic force, the force due to the electric charges on the electrodes as well as the interface force at changing permittivity. As described in Sect. 6.1, we will apply the method of virtual work, which considers the overall electrostatic force by evaluating (6.24) within an FE formulation. In the following, we will analyze the dynamic behavior of such transducers, especially their problems concerning crosstalk. 12.5.2 Single CMUT Cell The first numerical case study concentrates on a single CMUT cell. The computational domain consisting of the cell, the silicon wafer and the fluid domain as displayed in Fig. 12.33. The focus of the investigation in this section is the snap-in curve of an individual cell and its sensitivity to the variation of the geometric dimensions. Within a static analysis, we obtain a snap-in voltage of 194 V (see Fig. 12.34) compared to a measured snap-in voltage of 192 V [37]. However, the measurements of the geometric dimensions are not very accurate and in addition due to the fabrication process the dimensions will vary from cell to cell. In order to obtain the above mentioned agreement between measured and simulated snap-in voltage, we had to adapt the air-gap size as well as the membrane radius in the range of about 8 %. The change of the snap-in voltage as a function of the air-gap size as well as membrane radius is displayed in Fig. 12.35 and Fig. 12.36. Later we will emphasize that the material parameters will also have a strong influence on the computed results. Applying a transient analysis to our FE model (now also taking into account the fluid region) by exciting the CMUT cell by a short pulse u(t), we can compute the electric input impedance as follows

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Fig. 12.33. Setup of the CMUT cell 0

Center displacement ( nm)

Ŧ20 Ŧ40 Ŧ60 Ŧ80 Ŧ100 Ŧ120 Ŧ140 Ŧ160 Ŧ180 0

20

40

60 80 100 Bias voltage (V)

120

140

Fig. 12.34. Center displacement of membrane

Z(ω) =

FFT(u(t)) . j ω FFT(Q(t))

(12.9)

In (12.9) ω denotes the angular frequency, j the imaginary unit, FFT() the fast Fourier transformation and Q(t) the obtained total charge at the top electrode. Figure 12.37 shows the amplitude of the electric input impedance as a function of frequency. It can be clearly seen that the immersion of the CMUT cell into water strongly damps the resonance and antiresonance peaks and furthermore shifts them to lower frequencies due to the large mass loading of the water as compared to air.

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Fig. 12.35. Center displacement of membrane: variation of air-gap

Fig. 12.36. Center displacement of membrane: variation of membrane radius

12.5.3 CMUT Array In a second numerical case study we consider a CMUT array consisting of 9 cells. Due to symmetry, we just have modelled 4.5 cells (see Fig. 12.38). Each individual cell has the geometric setup as described in the previous section (see Fig. 12.33). The main focus of this investigation is the analysis of the cross-talk between the individual cells. Therefore, we just excite the center cell (cell 1) with a short sine-burst and compute the mechanical vibration at the center of cells 2 to cell 5. With this result, we compute the cross-talk as follows

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Fig. 12.37. Input impedance in air and water

Fig. 12.38. Setup of the CMUT with nine cells

cross-talk = 20 log10

uirms u1rms

i ∈ [2, 3, 4, 5] .

(12.10)

In (12.10) urms denotes the root mean square of the mechanical displacement u(t). By neglecting the fluid domain, the occurring cross-talk must be purely mechanical, and as demonstrated in Table 12.8, the resulting values are quite small. However, if we additionally consider the fluid domain (CMUT array immersed in water) the cross-talk increases tremendously (see Table 12.9).

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Table 12.9. Total cross-talk Cell 2 Cell 3 Cell 4 Cell 5 Level (dB) -20.2 -22.1 -23.3 -23.8

Therefore, we can conclude that the dominant cross-talk between the individual cells is due to the acoustic coupling. 12.5.4 Controlled CMUT Array For the 3D simulation one of the four arrays (each consisting of 19 capacitive transducer cells), as shown in Fig.12.30, was considered. The used finite ele-

Fig. 12.39. Detail of the finite element model; the membranes are marked by the numbers 1–7

ment model consists of a quarter of one array (see Fig. 12.39). The thickness of the membranes is 1 µm and the gap between the electrodes has a size of 500 nm. We apply a dc voltage of 10 V to the electrodes and for excitation a single period of a sine burst with a frequency of 5 MHz and amplitude of 10 V. To study the acoustic crosstalk between the individual membranes, only the center membrane is excited and the deformations of the neighboring membranes are computed. As an example of the simulation results the center displacements of membranes 4 and 7 are shown in Fig. 12.40 together with the dynamic response of the driving membrane 1. The observed significant crosstalk between the individual elements leads to a reduction of the effective membrane deflections when all membranes are driven in parallel. This can be

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Fig. 12.40. Center displacements of driving membrane 1 (solid line), membrane 4 (dashed line) and membrane 7 (dotted line)

seen from Fig. 12.41, where the corresponding center displacements of membrane 1 are shown, and is in agreement with the results obtained for the 2D simulations.

Fig. 12.41. Center displacements of membrane 1: all seven membranes (solid line) and just membrane 1 excited (dashed line)

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In the next step, the radiated pressure signal of the array is investigated. Of course, the long ring-down time of the membranes also shows up in the pressure signal. In the case that all membranes are driven in parallel, this results in a main, high-amplitude signal, which is followed by a secondary signal of lower amplitude (Fig. 12.42).

Fig. 12.42. Pressure signal of the array: all membranes (solid line) and just membrane 1 excited (dashed line)

When only the center membrane is excited, however, this secondary signal is of the same order in amplitude as the primary signal (Fig. 12.42). As a consequence, the bandwidth is significantly reduced for the single driven membrane. In order to decrease the ring-down time of the membranes, we have designed and applied controllers to each capacitive transducer of the array. Due to the quadratic dependency of the electrostatic force on the deflection of the membranes, no standard PID controller can be used. Instead, a non-linear controller has been designed. In each outer iteration step k (see Sect. 6.2.1) the change of the capacitance of each transducer is computed from the mechaniand used as the input of the controller. The controller cal displacements un+1 k algorithm then calculates the voltage for each transducer, which is a direct input value for the electric source vector f u . In Fig. 12.43 the dynamic response of the center membrane, when all membranes are driven in parallel, is depicted. As can be seen, the controller strongly decreases the ring-down time of the membrane. Furthermore, the use of the controller strongly decreases the acoustic crosstalk between the membrane elements.

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Fig. 12.43. Center displacements of uncontrolled (solid line) and controlled (dashed line) membrane 1, when all membranes are excited

This can be seen in Fig. 12.44, where the center deflection of membrane 2 is shown, in the case that only membrane 1 is excited. Therefore, the secondary

Fig. 12.44. Center displacements of uncontrolled (solid line) and controlled (dashed line) membrane 2, when just membrane 1 is excited

signal in the acoustic pressure, as observed for the uncontrolled case, is no longer present for the controlled membrane array. This is shown in Fig. 12.45 for the case that all membranes are driven in parallel. As a consequence a

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Fig. 12.45. Pressure signal of uncontrolled (solid line) and controlled (dashed line) array, when all membranes are excited

smoothing effect of the controller is also observed in the frequency spectrum (Fig. 12.46).

Fig. 12.46. Frequency spectrum of uncontrolled (solid line) and controlled (dashed line) array, when all membranes are excited

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12.6 High-Intensity Focused Ultrasound High-power ultrasound sources have found their way into a wide variety of applications, ranging from medical ultrasound, like lithotripsy or HIFU-therapy (High-Intensity Focused Ultrasound), to ultrasonic cleaning or welding and sonochemistry [115]. In contrast to ultrasonic applications with sources which radiate low amplitude pressure waves, the appearance of non-linear effects like sawtooth and shock formation is observed at high-power ultrasonic sound generators. In order to speed up the design process of these high-power sources appropriate numerical simulation tools are required. These have to take into account not only the electro-mechanical and the fluid–solid coupling, but also must be able to simulate the propagation of finite amplitude waves through lossy and compressible media. Previous investigations on the numerical simulation of non-linear wave propagation problems are mainly based on the KZK (Khokhlov-ZabolotskayaKuznetsov), NPE (non-linear progressive wave equation), or the Burgers equation, to mention only the most popular ones. Various methods for the solution of the above equations can be found in the literature, ranging from spectral [1] to time domain algorithms [141], finite difference schemes (FD) and finite element approaches (FEM) [219] as well as operator splitting methods [41]. We will apply the computational scheme as described in Sect. 5.4.2, which numerically solves Kuznetsov’s equation by an enhanced FE formulation. 12.6.1 Piezoelectric Transducer and Input Impedance The principle setup of the acoustic power transducer is shown in Fig. 12.47. Due to the geometric focusing of the lens, high acoustic intensity can be achieved in the focus region. The piezoelectric transducer has a diameter of 60 mm and the radius of curvature of the lense is 55 mm, which results in a focal distance of 70 mm. The operating frequency of the transducer is at 1.7 MHz. As the starting point in our investigations impedance calculations have been performed for the piezoelectric disc transducer. This was mainly performed to establish all necessary and unknown material parameters. The piezoelectric disc has a thickness of 1.2 mm and a diameter of 60 mm. The simulated and the measured electric impedance of the piezoelectric transducer is displayed in Fig. 12.48. Due to good agreement over a wide frequency range, we can trust our used material parameters. Next, the impedance of the whole HIFU source was simulated with water load. The finite element model used in the impedance simulation of the complete HIFU source is shown in Fig. 12.49. The computed electric impedance in water is displayed in Fig. 12.50. 12.6.2 Pressure Pulse Computation In the numerical simulation of the HIFU antenna, the piezoelectric and the fluid–solid coupling as well as the non-linear wave propagation in the fluid

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Fig. 12.47. Principle setup of high-power ultrasound source

Fig. 12.48. Impedance of the M453 piezo transducer

must be taken into account. Therefore, a complete finite element model has been setup in which the piezoelectric transducer, the lens, the matching layer and the water has been discretized (see Fig. 12.51). The FE model consisted of about one million elements for both linear and non-linear acoustic simulations. Near the sound source 20 elements per wavelength have been used. To account for the generation of higher harmonics, the spatial discretization was increased in propagation direction to 40 elements per wavelength in the focus region. Therewith, at least 8 elements per wavelength λ are guaranteed for the first

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Fig. 12.49. FE model of the HIFU source used for the impedance calculations

Fig. 12.50. Electrical input impedance of the HIFU source loaded with water

Fig. 12.51. Finite element mesh (for display reasons, just a coarse mesh is shown)

4 harmonics. For the excitation of the piezoelectric transducer a sine burst at 1.7 MHz and varying amplitude, as shown in Fig. 12.52, has been used. The simulation results were observed at several points on the rotational axis between the source and the focus region. Transient analyses were performed with 13500 time steps with a time step size of 4 ns.

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U (V)

20 0 Ŧ20 Ŧ40 Ŧ60 0

0.5

1

1.5 t (Ps)

2

2.5

3

Fig. 12.52. Excitation voltage for pressure measurements

p (MPa)

0.5

lin. simulation measurement

0

Ŧ0.5 46

47

48

49

t (P s)

50

51

52

53

Fig. 12.53. Pressure pulse signal for low intensity measurement and linear simulation at the focal distance

First, we considered pressure pulses due to a low excitation voltage of U = 9Vpp . As can be seen from Figs. 12.53 and 12.54, linear simulation results and measurements compare very well. For the high-intensity measurement, an excitation voltage of U = 133Vpp has been used. It should be noted that also for the higher excitation voltage the piezoelectric transducer still behaved totally linear. Therefore, any distortions in the pressure signal stem from non-linear effects in the fluid only. The comparison of measurements with

12.6 High-Intensity Focused Ultrasound 0

361

lin. simulation measurement

Ŧ10

A (dB)

Ŧ20 Ŧ30 Ŧ40 Ŧ50 Ŧ60 Ŧ70

1

2

3 f (MHz)

4

5

6

Fig. 12.54. Frequency spectra of pressure pulse for low intensity measurement and linear simulation at the focal distance

non-linear simulation results is shown in Figs. 12.55 and 12.56. The nonlinear distortion of the sine wave due to the generation of higher harmonics is clearly visible. In the simulations a damping value of 0.22 dB/MHz2m has been used. 12.6.3 High-Power Pulse Sources for Lithotripsy In this section we will discuss the numerical computation of two high-power pulse sources used for lithotripsy application: a piezoelectric driven pulse source and an electromagnetic pulse source. In such applications we have up to 80 MPa in the focus, and we need a quite fine grid within the acoustic domain in order to resolve the higher harmonics forming the high-pressure pulse. We will start with the piezoelectric high power pulse source as shown in Fig. 12.57 (for a discussion on the physics we refer to [53]). In parallel connected piezoceramic discs are placed on a spherical surface. The driving voltage is provided by decharging a capacitor. Due to geometrical focusing, high-power ultrasound is achieved in the focal region of the source. A simulation of the non-linear sound field was performed for the whole fluid domain. To account for the higher harmonics, we have chosen about 200 finite elements per fundamental wavelength, resulting in about 2.9 million bilinear quadrilateral elements for the axisymmetric setup. The sound pressure signal at the surface of the source is found by measuring the radiated pressure

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nlin. simulation, B/A=5 measurement

6

p (MPa)

4 2 0 Ŧ2 Ŧ4 46

47

48

49

t (P s)

50

51

52

53

Fig. 12.55. Pressure pulse signal for high-intensity measurement and non-linear simulation at the focal distance

signal near the surface of the source, and is used for the simulation (see Fig.

0

nlin. simulation measurement

Ŧ10

A (dB)

Ŧ20 Ŧ30 Ŧ40 Ŧ50 Ŧ60 Ŧ70

1

2

3 f (MHz)

4

5

6

Fig. 12.56. Frequency spectra of pressure pulse for high-intensity measurement and non-linear simulation at the focal distance

12.6 High-Intensity Focused Ultrasound

363

Fig. 12.57. Piezoelectric high-power pulse source DL100.

12.58). Measured, linear and non-linear simulated pressure signals in the focus

Fig. 12.58. Sound pressure signal at the surface of the source used in the simulation.

region of the source are compared in Fig. 12.59. The maximum sound pressure in the measurement is 75.6 MPa, the maximum sound pressure in the non-linear simulation is 69.7 MPa. In comparison to the non-linear result, the linear simulation has been performed using the same simulation parameters. The maximum sound pressure here is 46.2 MPa. The sound pressure is therefore raised by a factor of 33.7% due to the non-linear behavior of the sound wave in the fluid domain.

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measurement nonlinear simulation linear simulation

p (MPa)

60

40

20

0

Ŧ20 38

39

40

41

42 43 t (P s)

44

45

46

47

Fig. 12.59. Comparison between measured, non-linear- and linear-simulated sound pressure level in the focal region of the source

The second power source for lithotripsy is based on an electromagnetic principle, and its schematic setup is displayed in Fig. 12.60. When the slab

Fig. 12.60. Schematic of an electromagnetic driven acoustic power source

coil is loaded by a capacitor discharge, eddy currents are induced in the metallic membrane. The interaction between these eddy currents and the overall magnetic field results in a magnetic volume force (Lorentz force) acting on

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365

the membrane. Therewith, the membrane-rubber structure is deformed and an acoustic pulse is radiated into the fluid and focused by the lens. For the numerical simulation a finite element grid width of 90µm (corresponds to about 70 finite elements per fundamental wavelength) was used for the acoustic domain. Since in this case, we have to consider the non-linearities within the electromagnetic transducer, we perform the numerical simulation in two steps: 1. Transducer Computation: Since the non-linearities of the acoustic field near the transducer can be neglected, we compute the acoustic pressure using the linear acoustic wave equation. Therewith, we fully take into account the fluid loading of the transducer. For modelling the electromagnetic transducer we consider all relevant non-linearities ( updated Lagrangian formulation for the magnetic field, geometric non-linearity for the aluminum membrane and the non-linear electromagnetic force term, see Sect. 7). 2. Non-linear Wave Propagation Computation: In a second run, we fully solve Kuznetsov’s non-linear wave equation using the computed pressure near the transducer obtained from the first simulation step. The measured and simulated pressure signals in the focus region of the source are shown in Fig. 12.61. The dispersion at the beginning and the de-

Fig. 12.61. Comparison between measured and simulated sound pressure level in the focal region of the electromagnetic pulse source

creasing slope of the simulated pressure pulse as compared to the measured one indicates that the mesh size and the time step have to be further reduced for a more precise computer simulation.

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12.7 Noise Generated from a Flow around a Square Cylinder In the following, we will demonstrate the applicability of the developed calculation scheme for computational aeroacoustics as described in Sect. 10. Therewith, we will investigate in the noise generated by a flow around a square cylinder as displayed in Fig. 12.62. We denote with ΩF the fluid domain, where we will evaluate the acoustic sources on the fine flow grid and transform them to the coarser acoustic grid. In order to analyze our scheme especially concerning the transfer of the acoustic sources on the much finer flow grid to the coarser acoustic grid, we will start with 2D transient as well as harmonic acoustic computations. In the second part we will also perform full 3D computation of the acoustic field. In both cases, the acoustic computations are based on 3D large eddy simulations (LES) of the flow.

Fig. 12.62. Principle setup for the computation of the flow around a square cylinder

12.7.1 3D Flow and 2D Acoustic Computations As explained, in the first step we will perform a 2D acoustic computation at a cross section and investigate in the different simulation parameters. Figure 12.63 represents the configuration chosen for the simulations, where ΩF denotes the 3D domain for the flow computation and ΩAc defines the cross section at z = 2D (D denotes the side length of the cylinder) for the 2D acoustic simulation. The square cylinder has a side-length D of 20 mm and a height of 4D. This cylinder is positioned in a box with Lx = 40D, Ly = 11D and Lz = 4D as displayed in Fig. 12.63 for the flow computation (corresponds to ΩF ).

12.7 Noise Generated from a Flow around a Square Cylinder

367

Fig. 12.63. Principle setup for the aeroacoustic computation

Flow Field computation In the present case study, a large eddy simulation based on Smagorinsky model was performed with the code FASTEST-3D [56] to resolve the flow field. Since in this case the height of the square cylinder is equal to the height of the overall flow domain, we achieve a turbulent flow, which do not vary strongly in zdirection. The numerical domain for the problem is displayed in Fig. 12.63 and the boundary conditions are listed in Table 12.10. Spatial discretization of the numerical grid is Lx × Ly × Lz = 192 × 96 × 128, with a stretching factor of 1.05 downstream from the cylinder, 1.35 upstream from the cylinder and 1.18 in span wise direction from the cylinder. This results in a total number of 2.3 million control volumes. For the case, where inlet velocity ux = 10 m/s, D = 20 mm and fluid air at 25 oC we achieve a Reynolds number around 13000. Second-order spatial discretization was used with Crank-Nicolson time stepping scheme and a step size of tf = 10µs. For the Smagorinsky constant Cs we choose a value of 0.065. Table 12.10. Boundary condition for square cylinder simulation Position

Boundary Condition

x=0 x = 40D y = 0, y = 11D z = 0, z = 4D cylinder

inlet, ux = 10 m/s convective exit symmetry periodic no slip

Investigation of the interpolated acoustic nodal sources Initially, before computing the radiated sound field, several characteristic points within the turbulent flow region have been chosen to investigate the properties of the interpolated acoustic sources. Herewith, we have analyzed the interpolated acoustic nodal sources at locations depicted in Fig. 12.64.

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Fig. 12.64. Schematic representation of source region depicting the square cylinder with selected points (distance scale in millimeters)

Initially, analyzes are carried out at two different characteristic points, p5 = (x, y) = (0.1 m , 0.01 m) and p6 = (x, y) = (0.1 m , 0.0 m), located at 5D in the downstream direction along the fringe of the cylinder and along the x-axis, respectively. Figure 12.65 presents the results for these two points. Their characteristics confirm the measurements from a turbulent flow around a square cylinder [5]. For the point located along the fringe, among other frequency modes, we find the 65 Hz component, which is the actual main component for this tonal problem, observed in the vortical structures in the fluid computation. Therewith, we have a fundamental wavelength of 343/65 ≈ 5.3 m. This aspect has also been mentioned in the 2D investigation presented in [149], concerning the sound generation in a laminar flow past an elongated square cylinder. However, for the point located exactly along the x-axis we find, as main frequency component, a value which is twice higher than the main frequency of 65 Hz. This fact can be associated with the combination at the center line in the downstream flow of the upper and lower main vortices, each having a frequency of 65 Hz. Significant differences in the source values are found at a point p7 = (x, y) = (0.2 m , 0.06 m) located outside the central region of the wake, as presented in Fig. 12.66. In this case several other frequency components are found, but still the first two dominant components are present. 2D Acoustic Field Computations In the following, transient and harmonic investigations concerning the flowinduced noise propagation are presented. The acoustic field is computed in the cross section ΩAc located at z = 2D (see Fig. 12.63). In the first step, we will perform a transient analysis of the flow induced acoustic field. Figure 12.67, depicts two different spatial discretizations of the source region (corresponds to the total computational flow region) of the acoustic domain from which results are evaluated. The mapped mesh has an

12.7 Noise Generated from a Flow around a Square Cylinder 2

369

along fringes of cylinder in the center of wake

1 0

RHS (kg / s2)

Ŧ1 Ŧ2 Ŧ3 Ŧ4 Ŧ5 Ŧ6 Ŧ7 Ŧ8 0.06

0.07

0.08

0.09

0.1

0.11

0.12

t (s)

(a) Time domain 1

along the fringes of cylinder In center of wake

0.9

normalized source values

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

50

100

150

200

250

300

350

400

450

500

f (Hz)

(b) Frequency spectrum Fig. 12.65. Acoustic source values as a function of time as well as frequency at points p5 and p6 located at 5 × D in the downstream direction along the fringe of the cylinder and along the x-axis, respectively

element size h = 10 mm. For the second mesh, we set the element size near the cylinder equal to the ones used in the flow computation (h = 1 mm) and coarse it until reaching an element size of h = 10 mm (see Fig. 12.67 (b)). For the transient case, the radius of the complete acoustic domain has been chosen to be r = 40 m. Hereby, it is possible to investigate the acoustic

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(a) Time domain

(b) Frequency spectrum Fig. 12.66. Acoustic source values in time and frequency domain at point p7 = (x, y) = (0.2 m , 0.06 m) located outside the central region of the wake

solution during several periods, without any influence from spurious reflections of waves impinging not orthogonally on the acoustic boundary (we apply first order absorbing boundary conditions). The discretization of the farthermost elements in the acoustic region ΩAc , located near the absorbing boundary, corresponds to about 7 elements per fundamental acoustic wave length, which is for the main frequency component of 65 Hz about 5.3 m. The total number

12.7 Noise Generated from a Flow around a Square Cylinder

371

(a) Mapped mesh

(b) Progressively coarsened mesh Fig. 12.67. Two different acoustic coupled regions used for the evaluation in transient computations

of second-order quadrilateral elements in the cross sectional domain ΩAc is 41.584. The time step size for the acoustic computation is chosen to be ta = 100 µs, which corresponds to about 12 time samples for the largest frequency occurring in the source region (f ≈ 400 Hz). Figure 12.68 presents the results at a point located 5 m away from the cylinder using the two different discretizations in the source region (see Fig. 12.67). Simulations performed with the mapped mesh in the source region, produce a slight better quality in the solution when compared with the results obtained using the progressively coarsened mesh. Contour plots of the acoustic pressure in the near and the far field are presented in Fig. 12.69. In the near field a strong radiation in the upstream and in the downstream is observed. Here it is important to mention that although this does not seem to correspond to the expected dipole radiation in the cross-flow direction, it is from the reciprocal oscillation of these two radiation patterns that the far field dipole characteristic originates. The contour far field pressure from Fig. 12.69 b) at time t = 140 ms helps to understand this sound generation mechanism. Near the square cylinder the acoustic field shows the strong radiation in the upstream and downstream direction whereas in the far field the expected dipole radiation for this problem dominates. In the second step, we perform a harmonic analysis using a computational domain as depicted in Fig. 12.70. The dimensions of the complete acoustic domain without considering the PML region are Lx × Ly = 4.4 m×3.3 m with the cylinder located at (x, y) = (2 m, 0 m). The PML region and its discretization is also shown in Fig. 12.70, to emphasize that a small and coarse mesh with 612 elements suffices for this region to obtain accurate results. For direct comparison with the transient results, an initial harmonic computation has been performed using a numerical domain with the same discretization in the

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(a) Using progressively coarsened mesh

(b) Using mapped mesh Fig. 12.68. Acoustic pressure values in time for a point (x, y) = (0 m, 5 m) located 5 m away along the y-axis

source region as in the transient case (see Fig. 12.67 (a)). The computational mesh exhibits a total number of 8.488 second-order quadrilateral finite elements, including the PML region. Figure 12.71 presents the amplitude and phase values of the acoustic pressure field in the whole acoustic domain, computed for the main frequency component, f = 65 Hz. Similar to the transient results, we find the expected dipole sound radiation in the acoustic field. Di-

12.7 Noise Generated from a Flow around a Square Cylinder

373

(a) Acoustic near-field around the square cylinder (normalized pressure).

(b) Acoustic pressure distribution at time t = 140 ms (normalized). Distance scale in m. Fig. 12.69. Acoustic pressure field obtained from transient analysis

rectivity patterns obtained from the transient and harmonic analyses at a radius r = 1 m away from the cylinder are directly compared in Fig. 12.72. In this comparison, only small numerical differences are noticeable, which can be due to the evaluation of the amplitudes from the transient pressure signals. Moreover, these results have demonstrated that the excellent performance of the PML allows to use a small numerical domain, which in all directions en-

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compasses just a fraction of the acoustic wave length for the main frequency component (about λ/3, since for f = 65 Hz, we obtain for the wavelength λ ≈ 5.3 m).

Fig. 12.70. Schematic of numerical domain used in harmonic computations. Distance scale in meters

Fig. 12.71. Contour visualization of harmonic results for frequency f = 65 Hz

12.7 Noise Generated from a Flow around a Square Cylinder

375

Fig. 12.72. Comparison of directivity patterns at radius r = 1.0 m between harmonic and transient computations

In the following, additional harmonic acoustic computations are performed using several spatial discretizations to evaluate the robustness of the conservative interpolation scheme and its influence on the results. Hereby, we interpolate the computed acoustic nodal sources from the fine flow grid containing 91.492 cell points, to four different acoustic meshes with a total number of nodes in the source region ranging from 10.307 to 590. As previously mentioned, all computations are performed using second-order quadrilateral elements ( about 8 finite elements per element). Figure 12.73 presents the four different discretization of the acoustic source region used for the investigation, where Fig. 12.73 (b) corresponds to the mesh previously used for comparison with the transient results. The finest numerical grid corresponds to a ratio between the acoustic and fluid discretization sizes, ha /hf ≈ 5. Figure 12.73 (d) is a extremely coarse mesh containing 590 nodes, which corresponds to a ratio ha /hf ≈ 20 in the region directly around the cylinder and ha /hf ≈ 60 at the outermost regions. The aim for using this latter numerical grid is mainly to estimate the limits of the interpolation for achieving acceptable results. Figure 12.73 also presents the discretization ratios for the different meshes with respect to the wave length λmin obtained for the 400 Hz component found in the acoustic nodal sources (see Fig. 12.66). Figure 12.74 presents the results from the flow-induced noise computations for the four numerical grids. Except for the coarsest computation, where amplitudes are significantly smaller, all other results are in very good agreement with each other. Even for the coarse discretization from Fig. 12.73 (c), where ha /hf is around ten near the cylinder and 50 at outer regions, results still show very similar amplitudes in comparison to the results from the finest numeri-

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(a) 10307 nodes, λmin /h ≈ 187

(b) 5480 nodes, λmin /h ≈ 84

(c) 1616 nodes, 22 < λmin /h < 84

(d) 590 nodes, 20 < λmin /h < 40

Fig. 12.73. Different spatial discretizations of acoustic coupled region as used for sensitivity analysis

cal grid. For this flow-induced noise computation, this fact demonstrates that the conservative interpolation scheme makes the implementation a robust and efficient approach, producing good results even for very coarse acoustic resolutions in comparison to the CFD discretization, which significantly reduces the computational cost for the acoustic simulation.

Fig. 12.74. Directivity patterns from flow-induced noise harmonic computation using different grids. Amplitudes 1 m away from cylinder

12.7 Noise Generated from a Flow around a Square Cylinder

377

12.7.2 3D Flow and 3D Acoustic Computations In this section, we will present the noise generated from the flow around a wall mounted square Cylinder. Flow Field Computation The flow field computations are performed with the commercial CFD code ANSYS-CFX. In this case, the height of the square cylinder is as in the previous computation equal to 4D, but now the height of the overall flow computational domain is extended to 11D. The same inlet velocity profile with ux = 10 m/s is applied, which results in a Reynolds number Re of about 13000. We have applied for the modelling of the turbulence the SAS (Scale Adaptive Simulation) model [156]. This method allows coarser numerical grids than those used in LES computations, causing a shorter computation time. The boundary conditions used in the simulation are presented in Table 12.11. The spatial discretization of the computational flow domain results in about 1.1 million cells, and the time step size has been set to 10 µs. Figure 12.75 displays results for the velocity, pressure and eddy viscosity fields for the flow around the wall mounted cylinder. Table 12.11. Boundary conditions used for fluid computation Position

Boundary Condition

x=0 x = 40D y = 0, y = 11D z = 11D wall

block inlet profile, 10 m/s convective exit boundary symmetry boundary condition symmetry boundary condition no slip boundary condition

Acoustic Field Computation The acoustic field has just been computed for the main frequency component within a time-harmonic analysis. In the source region, the acoustic mesh consists of about 10000 3D finite elements and the overall acoustic mesh including the PML layer has 200000 3D finite elements. Figure 12.76 presents the configuration of the simulation domain showing the monitoring points and 12.77 the iso-surfaces of the acoustic pressure for f = 58 Hz, which was the main frequency component found in the simulation. The outermost iso-surface corresponds to 44 dB. This result shows the typical dipole characteristic with opposite phases expected for this tonal noise problem. Further directivity analysis has been done on the principal planes at several radii from the square cylinder. Figure 12.77 a) depicts the position of the

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Fig. 12.75. Transient flow field around wall-mounted cylinder

Fig. 12.76. Schematic drawing of the acoustic domain used for the harmonic computation showing points used for directivity analysis. Distance scale in meters

monitoring points at radius r = 1.0 m. Directivity plots from these planes at two different distances are shown in Fig. 12.78. The directivity patterns at the bottom XZ-plane as well as those on the cross-flow YZ-plane show the dipole characteristic of the problem. From the cross-flow plots it can be

12.7 Noise Generated from a Flow around a Square Cylinder

379

Fig. 12.77. Iso-surface of acoustic pressure at frequency f = 58 Hz normalized with phase and clipped through YZ-plane. Outer iso-surface corresponds to p
= 3 mPa. PML region is shown dotted

Fig. 12.78. Directivity patterns at radii r = 1.0 m and r = 0.75 m on the three principal planes. Left: XZ-plane (bottom plane). Right: YZ and XY planes (crossflow and stream-wise planes)

observed that the higher amplitudes (48 dB at r = 0.75 m) are located at the opposite points near the bottom plane and the lower one right above the cylinder (11 dB). On the other hand, the directivity plot on the stream-wise plane presents much lower sound pressure levels in comparison to the other

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two plots. For the bottom XZ-plane plot at r = 0.75 m a strong influence of the turbulent field is observed in the downstream direction, where the acoustic field is not yet homogeneous.

A Norms

A.1 Vector Norms Definition A.1. Vector norm: A vector norm on IRn is a function || || : IRn → IR, which fulfills the following properties: (i) (ii) (iii) (iv)

||x|| ≥ 0 for all x ∈ IRn ||x|| = 0 iff x = 0 ||x + y|| ≤ ||x|| + ||y|| ||αx|| = |α| ||x||

We use the H¨ older or p-norms, which are defined by  ||x||p =

n 

1/p |xi |p

with p ≥ 1 .

(A.1)

i=1

Therefore, we compute, e.g., the 1-norm, 2-norm, and ∞-norm as follows ||x||1 =

n 

|xi |

(A.2)

i=1

 n 

1/2 2

|xi |

(A.3)

||x||∞ = maxi |xi | .

(A.4)

||x||2 =

i=1

The p-norms have the following useful properties • • •

|xT y| ≤ ||x||p ||y||q , p1 + 1q = 1 (H¨ older inequality) T |x y| ≤ ||x||2 ||y|| √2 (Cauchy–Schwarz inequality) ||x||2 ≤ ||x||1 ≤ n||x||2

382

• •

A Norms

√ ||x||∞ ≤ ||x||2 ≤ n||x||∞ ||x||∞ ≤ ||x||1 ≤ n||x||∞

With the help of norms we can define a distance on a vector space, and furthermore, we call a vector space with a norm a normed space.

A.2 Matrix Norms Definition A.2. Matrix norm: A matrix norm on IRn×m is a function || || : IRn×m → IR, which fulfills the following properties: (i) (ii) (iii) (iv)

||A|| ≥ 0 for all A ∈ IRn×m ||A|| = 0 iff A = 0 ||A + B|| ≤ ||A|| + ||B|| for all A , B ∈ IRn×m ||αA|| = |α| ||A|| for all α ∈ IR and A ∈ IRn×m

The matrix norm associated to the vector p-norm is defined by the operator norm ||Ax||p . (A.5) ||A||p = supx =0 ||x||p Other matrix norms are ⎞1/2 ⎛ n  m  ||A||F = ⎝ |aij |2 ⎠

Frobenius or F-norm

(A.6)

|aij |

column sum norm

(A.7)

|aij |

row sum norm .

(A.8)

i=1 j=1

||A||1 = maxj ||A||∞ = maxi

n  i=1 m  j=1

B Scalar and Vector Fields

Definition B.1. (Scalar field) If we assign to each point in IR3 defined by the vector r a scalar quantity V (r) (e.g., electric potential, temperature, acoustic velocity potential), then V is called a scalar field. For the illustration of scalar fields we use iso-lines in the 2D case and isosurfaces in the 3D case, where the scalar quantity V (r) is constant (Fig. B.1).

Fig. B.1. Illustration of a scalar field V with the help of equipotential surfaces

Definition B.2. (Vector field) If we assign to each point IR3 defined by the vector r a vector quantity F(r) (e.g., electric field, magnetic field, mechanical deformation), then F is called a vector field.

384

B Scalar and Vector Fields

Vector fields are divided into irrotational vector fields (e.g., electrostatic field) and solenoidal vector fields (e.g., magnetic field) as shown in Fig. B.2 (see also Sects. B.12 and B.13).

Fig. B.2. (a) Solenoidal vector field; (b) Irrotational vector field

Fig. B.3. Lines of force for the vector field F(r)

The lines of force (see Fig. B.3) are defined by F(r) × dr = 0 ,

(B.1)

which means that in each point of the lines the field vector F is parallel to the tangential vector. In the following, we try to compute the lines of force for the vector field F(r) =

r r3

(B.2)

with the help of (B.1). Since we are just interested in the direction, we have to solve r × dr = 0 . (B.3) By using a Cartesian coordinate system, we obtain

B Scalar and Vector Fields

r = xex + yey + zez dr = dxex + dyey + dzez ,

and

⎛ ⎞

ex ey ez y dz − z dy

r × dr =

x y z

= ⎝ z dx − x dz ⎠ .

dx dy dz x dy − y dx

385

(B.4) (B.5) (B.6)

(B.7)

Therefore, we can formulate the following three relations y dz = z dy

(B.8)

z dx = x dz x dy = y dx .

(B.9) (B.10)

We now search for the line of force including point P0 (x0 , y0 , z0 ). Integration of (B.8) results in y z ln = ln (B.11) y0 z0 and z0 y = y0 z .

(B.12)

Analogously, we can compute the solutions of the other two differential equations z0 x = x0 z y0 x = x0 y .

(B.13) (B.14)

From (B.13) a plane through point P0 containing the y-axis, and from (B.14) a plane through point P0 containing the z-axis is defined. The intersection of the two planes leads to a straight line through the origin, and therefore we obtain the vector field drawn in Fig. B.4, which corresponds, e.g., to the vector field of an electric charge.

Fig. B.4. Lines of force of the vector field F(r) = r/r 3

386

B Scalar and Vector Fields

B.1 The Nabla (∇) Operator First, we recall that a scalar function may depend on one or more variables, e.g., using Cartesian coordinates, a function can be denoted by f = f (x, y, z) . The partial derivatives read as ∂f ∂f ∂f , , . ∂x ∂y ∂z The nabla operator ∇ is defined in Cartesian coordinates by ⎛ ∂ ⎞ ∂x

⎜ ∂ ⎟ ∂ ∂ ∂ ⎟ ex + ey + ez = ⎜ ∇= ⎝ ∂y ⎠ , ∂x ∂y ∂z

(B.15)

∂ ∂z

where ex , ey and ez are the unit vectors in x-, y-, and z-directions. The interaction between the nabla operator and a scalar or a vector field yields its geometric significance.

B.2 Definition of Gradient, Divergence, and Curl We introduce a scalar function V with nonzero first-order partial derivatives with respect to the coordinates x, y, and z, and a vector field F with components Fx , Fy , and Fz . Then, the following operations are defined: 1. Gradient of a scalar: ⎛ ∂V ⎞ ∂x

⎜ ∂V ⎟ ⎟ grad V = ∇V = ⎜ ⎝ ∂y ⎠ . ∂V ∂z

As can be seen, the result of this operation is a vector. 2. Divergence of a vector: div F = ∇ · F =

∂Fy ∂Fz ∂Fx + + . ∂x ∂y ∂z

Therefore, the result of this operation is a scalar value.

B.3 The Gradient

387

3. Curl of a vector:

⎛ ∂Fz

ex ey ez

∂y −

⎜ ∂Fx curl F = ∇ × F =

∂/∂x ∂/∂y ∂/∂z

= ⎜ ⎝ ∂z −

∂Fy

Fx Fy Fz ∂x −

∂Fy ∂z ∂Fz ∂x

⎞ ⎟ ⎟. ⎠

∂Fx ∂y

The result of taking the curl of a vector is again a vector.

B.3 The Gradient We will consider the scalar function V (x, y, z) with its partial derivatives ∂V /∂x, ∂V /∂y, ∂V /∂z and dependent on a point P = (x, y, z). In the first step we calculate the total differential of V dV =

∂V ∂V ∂V dx + dy + dz . ∂x ∂y ∂z

(B.16)

Now, we define a point P
infinitely close to P by P
= (x+ dx, y + dy, z +dz). By calculating the vector dP = P
− P, which has the components dP = ( dx, dy, dz)T , we can write (B.16) as   ∂V ∂V ∂V dV = ex + ey + ez · ( dxex + dyey + dzez ) (B.17) ∂x ∂y ∂z = ∇V · dP . (B.18) For the geometrical illustration of the gradient, consider an equipotential surface, i.e., a surface with V = const. (see Fig. B.5). Hence, for all differential

Fig. B.5. The gradient is orthogonal to a constant potential surface

displacements from P to P
on this surface dV = 0 holds, and therefore, ∇V · dP = 0 .

(B.19)

388

B Scalar and Vector Fields

From the definition of the scalar product it is clear that ∇V and dP are orthogonal. In this situation the displacement from P to P
points into the direction of increasing V , as shown in Fig. B.6, and the scalar product ∇V · dP is positive.

Fig. B.6. Geometrical representation of the gradient

From the foregoing arguments, we conclude that ∇V is a vector, perpendicular to the surface on which V is constant and that it points in the direction of increasing V . As an example we consider a function r(x, y, z), which defines the distance of a point P from the origin (0, 0, 0). The surface r = const. is a sphere of radius r with center (0, 0, 0), whose equation is given by + r = x2 + y 2 + z 2 . Therefore, the gradient calculates as ∂r x x = + = ∂x r x2 + y 2 + z 2 y ∂r = ∂y r ∂r z = ∂z r xex + yey + zez r ∇r = = . r r Geometrically speaking, ∇r points in the direction of increasing r, or towards spheres with radii larger than r.

B.4 The Flux

389

B.4 The Flux Definition B.3. (Flux) The vector field F(r) and a corresponding surface Γ as shown in Fig. B.7 are given. The vector n denotes the normal unit vector of the differential surface dΓ. Therefore, the differential flux dψ through dΓ is defined by dψ = F · dΓ = F · n dΓ . (B.20) The total flux ψ computes as  F · dΓ .

ψ=

(B.21)

Γ

In the following, we want to compute the flux ψ of the vector field F(r) = r

Fig. B.7. Flux through the surface Γ

through the square Γ with side length h according to Fig. B.8. With the normal unit vector n = ex and dΓ = dy dzex we obtain h h (−hex + yey + zez ) · ex dy dz

ψ= 0

0

h h = −h

dydz 0 3

0

= −h . The total flux ψ through a closed surface S is given by

(B.22)

390

B Scalar and Vector Fields

Fig. B.8. Flux ψ through the square with area h2

F · dΓ

ψ=

(B.23)

Γ

and defines whether we have sources (ψ > 0) or sinks (ψ < 0) within Γ . A very important property of the flux ψ defined by a closed surface is given by (see Fig. B.9) F · dΓ + F · dΓ = F · dΓ . (B.24) Γ1 ∪Γ0

Γ2 ∪Γ0

Γ1 ∪Γ2

Fig. B.9. Flux through the closed surface Γ1 ∪ Γ2

B.5 Divergence Definition B.4. (Divergence) The vector field F(r) is given. If we divide the flux ψ, defined by a closed surface Γ , by the corresponding volume Ω and let the volume Ω tend to zero, then the obtained value is called the divergence (source density)

B.5 Divergence

div F = lim

Ω→0

1 Ω

Γ

dψ

F · dΓ = . dΩ r

391

(B.25)

Let us now consider the closed surface of a differential cube (see Fig. B.10) and the general vector field F(r) = Fx ex + Fy ey + Fz ez . In the first step, let

Fig. B.10. Flux through a cube

us compute the differential flux through the hatched surfaces F · dΓ = [F(x + dx/2, y, z) − F(x − dx/2, y, z)] · ex dy dz 

 ∂Fx dx ∂Fx dx − Fx (x, y, z) − ≈ Fx (x, y, z) + dy dz ∂x 2 ∂x 2 ∂Fx dx dy dz . (B.26) = ∂x Analogously, we obtain the contribution of the other two directions, and thus, the differential flux   ∂Fx ∂Fy ∂Fz dψ = + + dx dy dz . (B.27) ∂x ∂y ∂z Since the differential volume dΩ is equal to dx dy dz, we end up with the following expression for the divergence of a vector field in Cartesian coordinates ∂Fy ∂Fz ∂Fx + + , (B.28) div F = ∂x ∂y ∂z or, by using the nabla operator, div F = ∇ · F .

(B.29)

392

B Scalar and Vector Fields

B.6 Divergence Theorem (Gauss Theorem) By the definition of the divergence (see B.25) we get dψ = ∇ · F dΩ  ψ= ∇ · F dΩ .

(B.30) (B.31)

Ω

On the other hand, we have the relation for the flux ψ according to (B.23). Combining these two expressions for the flux results in  ψ= ∇ · F dΩ = F · dΓ . (B.32) Γ (Ω)

Ω

This equality between the two integrals tells us that the flux of the vector F through the closed surface Γ is equal to the volume integral of the divergence of F over the volume Ω enclosed by the surface Γ .

Fig. B.11. Radial vector field

Consider a radial vector field F as shown in Fig. B.11, and assume that the magnitude of F is constant in all points on a sphere centered at P . To compute the flux of the vector field F through a spherical shell of radius R, we note that ds and F are colinear and in the same direction F · dΓ = F dΓ = 4πR2 F . ψ= S

Γ

From the divergence theorem, (the flux is nonzero) we conclude ∇ · F = 0 .

B.7 The Circulation The circulation of a vector field F(r) along a closed contour C is given by the closed-line integral

B.8 The Curl

393

F · ds .

Z=

(B.33)

C

Therefore, the important property follows (see Fig. B.12) F · dr + F · dr = F · dr . C1 ∪C0

C2 ∪C0

(B.34)

C1 ∪C2

If the circulation along a closed curve C is not equal to zero, then we say the

Fig. B.12. Circulation along the closed line C1 ∪ C2

closed line contains eddies.

B.8 The Curl Definition B.5. (Curl) We consider a point defined by r (Fig. B.13), in which the curl of the vector field F has to be computed. Furthermore, we define a closed line C enclosing the area Γ and consider the circulation along C. If the area Γ tends to zero, we obtain the definition of the curl by  F · ds dZ = . (B.35) n · curl F = lim C Γ →0 Γ dΓ The vector curl F is obtained by a separation in the three directions of the unit vectors ex , ey , and ez curl F = (ex · curl F)ex + (ey · curl F)ey + (ez · curl F)ez .

(B.36)

The circulation for the differential square in Fig. B.14 is given by dZx = ( F(x, y, z − dz/2) − F(x, y, z + dz/2) ) · ey dy + ( F(x, y + dy/2, z) − F(x, y − dy/2, z) ) · ez dz   ∂Fz ∂Fy ≈ − dydz . ∂y ∂z

(B.37)

394

B Scalar and Vector Fields

Fig. B.13. Curl in a point defined by r

Therefore, we obtain the x-component of curl F with dΓ = dy dz ex · curl F =

∂Fy ∂Fz − . ∂y ∂z

(B.38)

Analogously, the y- and z-component of curl F can be computed, and the full

Fig. B.14. x-component of curl F

vector in Cartesian coordinates reads as ⎛ ∂Fz ∂y

−

⎜ ∂Fx curl F = ⎜ ⎝ ∂z − ∂Fy ∂x

−

∂Fy ∂z ∂Fz ∂x ∂Fx ∂y

⎞ ⎟ ⎟, ⎠

(B.39)

B.9 Stoke’s Theorem

395

or with the help of the nabla operator

ex ey ez

∂ ∂ ∂

curl F = ∇ × F = ∂x ∂y ∂z

.

Fx Fy Fz

(B.40)

B.9 Stoke’s Theorem We consider the vector field F on the surface Γ with fixed oriented contour C as shown in Fig. B.15. For a differential surface dΓν , we obtain according

Fig. B.15. Vector field F on the surface Γ with fixed oriented contour C

to (B.35) dZν = n(rν ) · curl F(r) dΓν  Zν = curl F(r) · dΓν ,

(B.41)

Γν



and

curl F · dΓ .

Z=

(B.42)

Γ

Furthermore, according to the definition of the circulation Z (see B.33), we get the following relation  F · dr = curl F · dΓ . (B.43) Z= C

Γ

396

B Scalar and Vector Fields

For a radial vector field F as shown in Fig. B.11, the closed-line integral along a circle C of constant radius F · ds C

is zero, and therefore, the curl of this vector field ∇ × F is zero, too.

B.10 Green’s Integral Theorems The integral theorems of Green can be derived from the divergence theorem. For this purpose, we first introduce the Laplace operator by ∆ = ∇·∇ =

∂2 ∂2 ∂2 + + . ∂x2 ∂y 2 ∂z 2

(B.44)

This differential operator can be applied to scalar as well as vector quantities ∆V = div (grad V )

(B.45)

∆F = (∆Fx )ex + (∆Fy )ey + (∆Fz )ez .

(B.46)

Setting a vector F equal to V1 ∇V2 and using the divergence theorem, we obtain according to (B.32)  ∇ · (V1 ∇V2 ) dΩ = (V1 ∇V2 ) · dΓ . (B.47) Ω

Γ

Since the term ∇ · (V1 ∇V2 ) can be expressed by (see B.54 below) ∇ · (V1 ∇V2 ) = V1 ∆V2 + ∇V1 · ∇V2 ,

(B.48)

we get the following integral theorem, called Green’s first integral theorem   ∂V2 dΓ . (B.49) V1 ∆V2 dΩ + ∇V1 · ∇V2 dΩ = V1 ∂n Ω Ω Γ By substituting V1 with V2 and vice versa in (B.47) and subtracting the resulting equation from (B.47), we achieve Green’s second integral theorem   -   ∂V2 ∂V1 − V2 V1 ∆V2 dΩ − V2 ∆V1 dΩ = V1 dΓ . (B.50) ∂n ∂n Ω V Γ In addition, Green’s first integral theorem in vector form is  (∇ × u · ∇ × v − u · ∇ × ∇ × v) dΩ Ω

 (u × ∇ × v) · n dΓ ,

= Γ

(B.51)

B.12 Irrotational Vector Fields

397

and Green’s second integral theorem in vector form reads as  (u · ∇ × ∇ × v − v · ∇ × ∇ × u) dΩ Ω

 (v × ∇ × u − u × ∇ × v) · n dΓ .

=

(B.52)

Γ

B.11 Application of the Operators By using the definitions of gradient, divergence, and curl in Cartesian coordinates, the following relations hold: ∇(V1 V2 ) = V1 ∇V2 + V2 ∇V1 ∇ · (V F) = V ∇ · F + F · ∇V

(B.53) (B.54)

∇ · (F1 × F2 ) = F2 · ∇ × F1 − F1 · ∇ × F2

(B.55)

∇ × (V F) = V ∇ × F − F × ∇V ∆F = ∇(∇ · F) − ∇ × (∇ × F) .

(B.56) (B.57)

These relations combine the essential differential operators and build up a basis for the description of physical fields.

B.12 Irrotational Vector Fields We consider a vector field F, which is given as the gradient of a scalar potential F = ∇V . The computation of a line integral from point A to point B yields B (∇V ) · dr = V (B) − V (A) .

(B.58)

A

Therefore, for any closed contour within this vector field, the following relation holds (∇V ) · dr = 0 . (B.59) This result proves that any vector field that can be expressed by the gradient of a scalar potential is irrotational. Furthermore, the local quantity, given by the curl of the vector field, is zero ∇ × ∇V = 0 .

(B.60)

398

B Scalar and Vector Fields

Fig. B.16. Domain for solenoidal vector field

B.13 Solenoidal Vector Fields We will consider the solenoidal vector field ∇ × F for a domain as displayed in Fig. B.16. This domain shall consist of two subdomains defined by their surfaces Γ1 and Γ2 with their related contours C1 and C2 . By using Stoke’s theorem, we obtain the following relation (∇ × F) · dΓ = (∇ × F) · n1 dΓ + (∇ × F) · n2 dΓ Γ Γ1 Γ2 F · dr + F · dr = C1

= 0.

C2

(B.61)

Thus, the total flux (global quantity) is zero, and, furthermore, the local solenoidality, too ∇ · (∇ × F) = 0 . (B.62)

C Appropriate Function Spaces

Let us define the derivative of order α with respect to the multi-index α, with |α| = i αi and αi ∈ IN, as follows Dα v :=

∂ |α| v . n · · · xα n

1 ∂xα 1

(C.1)

For example, the partial derivatives of order 2 in IR2 can be written as Dα v with α = (2, 0), α = (1, 1) or α = (0, 2), since |α| = α1 + α2 = 2 is fulfilled for all three cases α = (2, 0) α = (1, 1) α = (0, 2)

∂2v ∂x21 ∂2v Dα v = ∂x1 ∂x2 ∂2v Dα v = . ∂x22 Dα v =

Definition C.1. Continuously differentiable functions: Let Ω be a closed domain in IRn and let C(Ω) denote the space of continuous functions on Ω. Now, the space of up to order m continuously differentiable functions is given by (C.2) C m (Ω) = {v : Ω → IR | Dα v ∈ C(Ω), |α| ≤ m} . If the function v is infinitely often continuously differentiable on Ω, we write v ∈ C ∞ (Ω). For the function u(x) shown Fig. C.1 the following inclusions hold (with v(x) = u
(x)) v ∈ C0 u ∈ C1 .

400

C Appropriate Function Spaces

Fig. C.1. Example of a C 1 function

Definition C.2. Square integrable functions: Let Ω be a closed domain in IRn . Then, the function u is called square integrable, if it fulfills the following relation  |u(x)|2 dx < ∞ . (C.3) Ω



We denote L2 (Ω) = {u : Ω → IR |

| u(x) |2 dx < ∞} .

(C.4)

Ω

For example, the function f (t) with the definition ⎧ ⎨ 1 for 0 < x < 2 f (t) = 0 for x = 0 ⎩ −1 for −2 ≤ x < 0

(C.5)

belongs to the space L2 (−2, 2) (see Fig. C.2). u(x)

1

-2

0

2

x

-1

Fig. C.2. Function u(x) = sgn(x) in the interval (–2,2)

Analogously to the above definition, we obtain the definition for Lp (Ω)spaces for p ∈ [1, ∞).

C Appropriate Function Spaces

401

Definition C.3. Lp (Ω)-spaces: Let Ω be a closed domain in IRn . Then, the space of p-integrable functions is given by  Lp (Ω) = {u : Ω → IR| |u(x)|p dx < ∞} . (C.6) Ω

Let us assume that the function u has a continuous derivative u
. According to the formula for partial integration, we have for each continuously differentiable function ϕ with ϕ(a) = ϕ(b) = 0 the following relation b

b




u(x)ϕ (x) dx = − a

u
(x)ϕ(x) dx .

(C.7)

a

With the help of (C.7), we can define the derivative of functions, which have no finite derivative in the classical sense. If u and w denote integrable functions that fulfill the following relation b

b




u(x)ϕ (x) dx = − a

w(x)ϕ(x) dx

(C.8)

a

for all differentiable functions ϕ with ϕ(a) = ϕ(b) = 0, then the function w is called the derivative of u in the weak sense (with respect to x). The function

Fig. C.3. Example of a function in H 1 (a, b)

u defined by (see Fig. C.3) & u(x) =

x + 1 for −1 ≤ x ≤ 0 1 − x for 0 < x ≤ 1

will have no derivative in the classical sense at x = 0. Applying partial integration for differentiable functions ϕ(x) with ϕ(−1) = ϕ(1) = 0, we obtain

402

C Appropriate Function Spaces

1

0




u(x)ϕ (x) dx = −1

1




(x + 1)ϕ (x) dx + −1

(1 − x)ϕ
(x) dx

0

0 =−

ϕ(x) dx + (x + 1)ϕ(x)|0−1

−1

1 −

(−1)ϕ(x) dx + (1 − x)ϕ(x) |10 0

⎡ = −⎣

0

1 ϕ(x) dx +

−1

0

⎤ (−1)ϕ(x) dx⎦ + ϕ(0) − ϕ(0) .    =0

Therefore, in the weak sense of differentiation we obtain & 1 for −1 ≤ x < 0
u (x) = −1 for 0 < x ≤ 1 with an arbitrary value for u
(0). Definition C.4. Sobolev space: Let Ω be a domain in IRn . The functional space (C.9) Wpm (Ω) = {u ∈ Lp (Ω)|Dα u ∈ Lp , |α| ≤ m} is called Sobolev space Wpm (Ω). The partial derivatives of u are defined in the weak sense. The appropriate norms on Sobolev spaces are defined by ⎛ ||u||Wpm (Ω) = ⎝



⎞1/p



|Dα u|p dx⎠

(C.10)

Ω |α|≤m

and its semi-norm by ⎛ |u|Wpm (Ω) = ⎝





⎞1/p |Dα u|p dx⎠

.

(C.11)

Ω |α|=m

If we restrict p to two, then we obtain a Hilbert space (W2m (Ω) = H m (Ω)) with the scalar product ⎛ ⎞   Dα uDα v ⎠ dx . (C.12) (u, v) = ⎝ Ω

|α|≤m

C Appropriate Function Spaces

403

For example, the function u(x) is in the space H 1 (a, b), if u
(x) exists and is within the space L2 (a, b). The norm is computed via 3 4 b b 4 4 (C.13) ||u||H 1 (a,b) = 5 (u(x))2 dx + (u
(x))2 dx , a

its semi-norm by |u|H 1 (a,b)

a

3 4 b 4 4 = 5 (u
(x))2 dx ,

(C.14)

a

and the scalar product as follows b (u, v)H 1 (a,b) =

b u(x)v(x) dx +

a

u
(x)v
(x) dx .

(C.15)

a

Definition C.5. Let Ω be a domain in IRn and denote by C0∞ (Ω) the space of infinitely often differentiable functions with zero boundary values. Then we write for the closure of C0∞ (Ω) with respect to the H 1 norm H01 (Ω) = C0∞ (Ω)

H 1 (Ω)

⊂ H 1 (Ω) .

(C.16)

Definition C.6. Partial Integration: Let Ω ⊂ IRn , n = 2, 3 be a domain with smooth boundary Γ . Then, for any u, v ∈ H 1 (Ω) the following relation holds    ∂v ∂u v dx = uv n · ei ds − u dx . (C.17) ∂xi ∂xi Γ Ω

Ω

¯ the considered domain Ω with In (C.17) n denotes the outer normal and Ω boundary Γ . By a multiple application of (C.17), we arrive at Green’s formula    ∂u v ds − (∇u)T ∇v dx ∆u v dx = ∂n Ω

Γ

for all u ∈ H 2 (Ω) and v ∈ H 1 (Ω).

Ω

(C.18)

D Solution of Nonlinear Equations

In this section we are concerned with the solution of systems of nonlinear equations. As an example, we will consider the nonlinear Poisson equation, given as follows −∇ · ε(|∇u|)∇u − f = 0 u=0

on Γ .

(D.1) (D.2)

This defines a nonlinear operator F that allows us to rewrite (D.1) and (D.2) as F (u) = 0 . (D.3) The weak formulation of (D.1) and (D.2) for all test functions v ∈ H01 reads as   ε(|∇u|)∇v · ∇u dΩ − vf dΩ = 0 . (D.4) Ω

Ω

By applying the finite element method, we arrive at the following algebraic system K(u)u = f , (D.5) with the matrix K ∈ IRn×n , f ∈ IRn , u ∈ IRn and n the number of unknowns. Since we cannot solve (D.5) explicitly, we have to establish an approximate solution by setting up a series uk (k = 0, 1, 2, 3, ..) that is supposed to converge to the correct solution. Concerning the rate of convergence, we will restrict the discussion to the following types: Definition D.1. Convergence: Let u∗ ∈ IRn be the exact solution. Then •

uk converges towards u∗ q-quadratically (q stands for quotient), if there exists a C such that ||uk+1 − u∗ || ≤ C||uk − u∗ ||2 .

(D.6)

406

•

D Solution of Nonlinear Equations

uk converges towards u∗ q-linearly with the q-factor σ ∈ (0, 1), if ||uk+1 − u∗ || ≤ σ||uk − u∗ || .

(D.7)

In general, a q-quadratically convergent algorithm is preferable to a qlinearly convergent one. However, we always have to take into account the numerical cost for one iteration. Therefore, in some cases the method with the slower convergence rate can even be faster. Since we solve (D.5) numerically by computing a series of approximating solutions uk , the question of the stopping criterion is of great importance. In general, we distinguish between the following two types of stopping criteria: 1) Error criterion: We take the solutions of two successive iteration steps and define an absolute accuracy εabs by ||uk+1 − uk ||2 < εabs ,

(D.8)

and a relative accuracy εrel by ||uk+1 − uk ||2 < εrel ||uk+1 ||2 ,

(D.9)

which has to be achieved. However, in some analysis the true solution may still be far away, although the above-defined stopping criteria are fulfilled. This may particularly occur in the solution methods that have to use a line search (see Sect. D.1) to avoid possible divergence during early steps of the iteration process or due to nonmonotonic material relations. Then, it can happen that the control parameter becomes very small, which results in almost no difference between uk+1 and uk . 2) Residual criterion: By computing the residual of the obtained solution, we can define an absolute accuracy εabs res by ||K(uk+1 )uk+1 − f ||2 < εabs res ,

(D.10)

as well as a relative accuracy εrel res by ||K(uk+1 )uk+1 − f ||2 < εrel res ||f ||2 .

(D.11)

As shown in Fig. D.1, according to the problem type, this stopping criterion may also be reached too early. As a consequence of the above discussion, it is preferable to check both stopping criteria.

D.1 Fixed-point Iteration

407

||K(u )u - f ||2 h

h

h

eres

abs

u

*

uk+1

Fig. D.1. Obtained solution uk+1 is still far away from the true solution u∗

D.1 Fixed-point Iteration The simplest method of solving (D.5) is to rewrite it as a fixed-point equation u = K−1 (u)f .

(D.12)

This will result in the following sequence uk+1 = K−1 (uk )f K(uk )uk+1 = f .

(D.13) (D.14)

Thus, we can write the damped fixed-point iteration method as follows K(uk )∆u = f − K(uk )uk = r(uk ) uk+1 = uk + η∆u .

(D.15) (D.16)

The nodal vector r(u) is known as the residual of the problem and a solution is given by the set of nodal values u, for which the residual is zero. The scalar parameter η ∈ [0, 1] is introduced to control the possible divergence during the early steps of the iteration process or due to nonmonotonic material relations. A common algorithm to compute η is a line search (see [227]) defined by |G(η)| → min ,

(D.17)

G(η) = ∆u · r(uk + η∆u) .

(D.18)

with One simple method of approximating the optimal η is as follows

408

D Solution of Nonlinear Equations

1. Evaluate g1 = G(0.1) and g2 = G(1.0) 2. Calculate the straight line l(g1 , g2 ) between g1 and g2 3. Calculate the value η =

10g1 −g2 10·(g2 −g1 )

for which l(g1 , g2 ) = 0 holds

A graphical interpretation of the fixed-point method is given in Fig. D.2.

Fig. D.2. Graphical interpretation for solving a nonlinear equation using the fixedpoint method

D.2 Newton’s Method Let us introduce the following linearization of the nonlinear operator F (u) at uk F (u) ≈ F(uk ) + F
(uk )[s] (D.19) with uk+1 = uk + s. The term F
(uk )[s] denotes the Frech´et - derivative of the nonlinear operator F at uk in the direction of s and is defined as follows Definition D.2. Frech´ et - derivative: Let X and Y be two normed vector spaces and D ⊂ X an open domain. The operator F : D → Y is differentiable in the sense of Frech´et at x, iff there exists an operator A : X → Y, so that for all y ∈ D F (y) = F (x) + A(y − x) + R(x, y) ,

D.2 Newton’s Method

with lim

y→x

409

||R(x, y)|| =0 ||y − x||

is fulfilled. A is the Frech´et derivative F
(x). Therefore, Newton’s method reads as F
(uk )[s] = −F(uk )

(D.20)

uk+1 = uk + s .

(D.21)

Analogously to the fixed-point method, a line-search parameter may accelerate the convergence, and in addition may guarantee a global convergence of the Newton method. A graphical interpretation of Newton’s method is displayed in Fig. D.3.

Fig. D.3. Graphical interpretation for solving a nonlinear equation using Newton’s method

To derive the Frech´et derivative F
and Newton’s method for the nonlinear Poisson equation given in (D.1), we first compute the difference between F (u+ s) and F (u) in the weak formulation for arbitrary test functions v ∈ H01   ε(|∇(u + s)|)∇v · ∇(u + s) dΩ − ε(|∇u|)∇v · ∇u dΩ . (D.22) Ω

Ω

Now, we will add to and at the same time subtract from (D.22) the term
ε(|∇(u)|)∇v · ∇(u + s) dΩ, and obtain Ω

410

D Solution of Nonlinear Equations



 (ε(|∇(u + s)|) − ε(|∇u|)) ∇v · ∇(u + s) dΩ + Ω

ε(|∇u|)∇v · ∇s dΩ . Ω

The term ε(|∇(u + s)|) − ε(|∇u|) can be approximated as follows ε(|∇(u + s)|) − ε(|∇u|) ≈ ε
(|∇u|) (|∇(u + s)| − |∇u|) .

(D.23)

Now, let us investigate the term (|∇(u + s)| − |∇u|) |∇(u + s)|2 − |∇u|2 (D.24) |∇(u + s)| + |∇u| ∇u · ∇u + ∇s · ∇s + 2∇u · ∇s − ∇u · ∇u (D.25) = |∇(u + s)| + |∇u| ∇u · ∇s . (D.26) ≈ |∇u|

|∇(u + s)| − |∇u| =

With this result, we can write  (ε(|∇(u + s)|) − ε(|∇u|)) ∇v · ∇(u + s) dΩ Ω

 ≈

ε
(|∇u|)

Ω

∇u · ∇s ∇v · ∇u dΩ . |∇u|

(D.27)

Summarizing the above results, we conclude that the Frech´et derivative F
(uk )[s] in the weak formulation of the PDE for a test function v is given by   ∇uk · ∇s ∇v · ∇uk dΩ . ε(|∇uk |)∇v · ∇s dΩ + ε
(|∇uk |) (D.28) |∇uk | Ω

Ω

Therefore, by using (D.20) as well as (D.21), we obtain Newton’s method for the nonlinear Poisson equation   ∇uk · ∇s ε(|∇uk |)∇v · ∇s dΩ + ε
(|∇uk |) ∇v · ∇uk dΩ |∇uk | Ω Ω  = vf dΩ  ∀v ∈ H01 (Ω) − ε(|∇uk |)∇v · ∇uk dΩ Ω

uk+1 = uk + s .

(D.29)

By apply the finite element method to the above equation, we will arrive at the appropriate algebraic system of equations.

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Index

absorbing boundary condition 175 acoustic averaged energy density 147 averaged intensity 147 averaged power 147 density 140 energy density 146 energy flux 146 field 139 impedance 148 intensity 146 linear wave equation 145 non-linear wave equation 156 overall sound pressure level (OSPL) 155 particle velocity 140 pressure 140 quantities 146 sound-field impedance 148 sound-intensity level 154 sound-power level 154 sound-pressure level (SPL) 154 spherical spreading law 151 velocity potential 145 actuator see mechatronic adiabatic 143 bulk modulus 144 compressibility 144 aeroacoustics 267 co-rotating vortex pair 279 Ffowcs Williams and Hawkings equation 275 Lighthill’s analogy 270

agglomeration technique see coarsening algebraic multigrid see multigrid Amp` ere 95 approximation B–H curve 129 auxiliary matrix 294 B–H curve see approximation balanced reduced and selective integration 85 Biot–Savart’s law 108 boundary condition 9 Dirichlet 9 essential 10 natural 10 Neumann 9 bulk viscosity 156 Burger’s equation 159 butterfly curve 248 circulation 392 CMUT see micromachined co-rotating vortex pair 279 coarse grid operator see grid coarsening agglomeration technique 297 function 295 process 295 coil current-loaded 131 voltage-loaded 211 computational aeroacoustics 267 condition number 284

424

Index

configuration deformed 55 initial 55 congruency 255 conjugate gradient (PCG) method see preconditioned conservation of mass 140 of momentum 141 contact mechanics condition 332 pressure-displacement relation 332 tangent stiffness matrix 332 continuity equation 140 convergence 405 Coulomb-gauge see gauge coupling aeroacoustics 267 electromagnetics-mechanics 207 electrostatics-mechanics 195 mechanics-acoustics 229 piezoelectrics 243 coupling mechanisms 4 Courant-Friedrich-Levi condition 40 Crank-Nicolson scheme 39 curl 387, 393 damping 69 modal 69 Rayleigh model 69 density 54 derivative Frech´et 408 global/local 29 weak sense 401 design process 3 CAE-based 3 experimental-based 3 diamagnetic 105 dielectric remanent 109 differential operator 54 diffusion equation 103 diffusivity of sound 159 displacement current density divergence 386, 390 theorem 392 elasticity modulus electric

60

96

charge density 94 conductivity 94, 108 current 95 current density 94 field intensity 94 flux density 94 permittivity 94 polarization 94, 245 scalar potential 104 specific resistivity 108 electrodynamic loudspeaker see loudspeaker electromagnetic energy 210 field 93 force 209 interface conditions 109 quasistatic field 101 electromagnetic-mechanical system 207 calculation scheme 216 electromotive force 98 electrostatic energy 196 field 104 force 196 electrostatic-mechanical system 195 calculation scheme 202 Enhanced assumed strain method 83 error a posteriori 46 a priori 46 discretization 45 dispersion 170 interpolation 170 pollution 170 Euler equation 141 Eulerian coordinate 55 Everett function 256 Faraday 96 Fay solution 192 ferroelectricity 109, 245 ferromagnetic 105 Ffowcs Williams and Hawkings equation 275 filter 1/3 octave 152 octave 152

Index finite element 7 assembling procedure 32 compatible 23 conforming 23 edge 43 formulation 9 hexahedral 27 infinite element 174 isoparametric 21 Lagrangian 20 method 7 mortar 166 N´ ed´ elec 43 nodal 20 quadrilateral 23 tetrahedral 26 triangular 26 finite element/boundary element method 219 fixed-point iteration 407 flux 389 force electromagnetic 209 electrostatic 196 formulation strong 9 variational 10 weak 10 Fubini solution 191 functional spaces 399 Lp 400 continuously differentiable 399 Hilbert 402 Sobolev 402 square integrable 399 weighted Sobolev 120 Galerkin 10 method 10 semidiscrete formulation 12 gauge 102 Gauss 99 Gauss theorem see divergence geometric multigrid see muiltigrid gradient 387 deformation 55 displacement 56 of a scalar 386 Green’s integral theorem 396

scalar form 396 vector form 396 grid coarse 295 coarse-grid operator complexity 305 fine 295

299

harmonic distortion 317 Helmholtz decomposition 63, 290 Hook’s law 60 Hu-Washizu principle 83 hysteresis 253 Preisach model 253 incompatible modes method 81 induced electric voltage 134 inductance see magnetic infinite finite elements 174 interpolation conservativ 278 function 22 irrotational 99, 384 vector field 397 isothermal process 143 Jacobi 29 matrix 29 Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation 160 Kuznetsov’s equation 156 Lagrange multiplier 220 Lagrangian coordinate 55 updated formulation 208 Lam´e parameters 60 Lighthill’s analogy 270 line search 407 litotripsy 361 local support 21 locking 78 effect 77 membrane 79 Poisson 78 shear 78 Lorentz force 4, 95 loss factor 69

425

426

Index

loudspeaker

4, 218, 313

magnetic field intensity 94 flux 96, 132 hard material 105 hysteresis 106 inductance 132 induction 94 permeability 94, 104 reluctivity 105 remanent field 105 scalar potential 107 soft material 105 vector potential 101 magnetic valve 207, 330 overexcitation 337 premagnetization 336 switching cycle 337 magnetization 94 magnetomechanical system see electromagnetic-mechanical system Maxwell’s equations 93 mechanical acceleration 54 axisymmetric stress–strain 63 contact 332 damping see damping field 51 plane strain 61 plane stress 62 strain 56 stress 51 stress-stiffening effect 347 yield stress 66 mechanical-acoustic system 229 calculation scheme 233 mechatronic 1 actuator 1 sensor 1 micromachined capacitive ultrasound array (CMUT) 345 motional electromotive force 99, 207 method 219, 222, 314 moving body electric field 203 magnetic field 207

moving coil current-loaded 218 voltage-loaded 218 moving-material method 221, 223, 315 moving-mesh method 203, 221 multigrid 283 algebraic 293 geometric 287 method 285 nested 291 multilayer actuator see piezoelectric nabla operator 386 Navier’s equations 54 Newmark scheme 41 Newton method 408 electromagnetics 125 mechanics 72 non-matching grids acoustics 166 mechanics-acoustics 234 norms 381 H¨ older 381 matrix 382 p-norms 381 vector 381 numerical computation electromagnetics 114 electromagnetics-mechanics 216 electrostatics 113 electrostatics-mechanics 202 geometric non-linear case 70 linear acoustics 161 linear elasticity 66 mechanics-acoustics 233 non-linear acoustics 163 non-linear electromagnetics 125 piezoelectrics 257 numerical integration 30 Gaussian quadrature 30 operator complexity 305 nonlinear 405 paramagnetic 105 parameter of non-linearity 157 partial differential equation 9 hyperbolic 40

Index parabolic 36 patch test 82 penalty formulation 119 penetration depth see skin depth perfectly matched layer (PML) 176 permeability see magnetic permeability piezoelectric 243 ceramics 245 cofired multilayer 340 direct effect 243 inverse effect 243 systems 243 Poisson ratio 60 polarization irreversibel 247 permanent 247 saturation 247 poling 247 polymers 245 power transformer 320 preconditioned conjugate gradient (PCG) method 283 predictor-corrector algorithm 38, 42 Preisach function 254 model 253 operator 254 prestressing 205 principle of virtual work 196, 199, 209, 213 process adiabatic 143 isothermal 143 prolongation 285 operator 285, 298 Rayleigh damping model see damping remanent magnetic field see magnetic restriction 285 operator 285 saturation strain 248 scalar acoustic velocity potential electric potential 104 field 383 magnetic potential 107 sensor see mechatronic

145

427

shape function see interpolation shear modulus 60 shear viscosity 156 shock-formation distance 192 single crystals 245 skin depth 103 effect 102 smoothing overlapping block-smoothers 290 block-Gauss-Seidel 290 Gauss-Seidel backforward 287 Gauss-Seidel forward 287 hybrid 303 operator 299 post 286 pre 286 Sobolev space see functional spaces solenoidal 100, 384 vector field 398 solid/fluid interface 230 sound velocity 139 spherical spreading law see acoustic SPL see acoustic stack actuator see piezoelectric state equation 143 Stoke’s theorem 395 stopping criterion 406 error 406 residual 406 strain see mechanical strain tensor Green–Lagrangian 59 linear 59 stress tensor 1st Piola–Kirchhoff 71 2nd Piola–Kirchhoff 71 Cauchy 53 stress-stiffening effect 347 surface integration 43 TEAM (Testing Electromagnetic Analysis Methods) 290 tensor of dielectric constants 244 of elasticity moduli 60, 244 of piezoelectric moduli 244 test function 9 thermal strain 66

428

Index

time discretization 35 effective mass formulation (hyperbolic) 42 effective mass formulation (parabolic) 38 effective stiffness formulation (hyperbolic) 42 effective stiffness formulation (parabolic) 38 explicit (hyperbolic) 41 explicit (parabolic) 39 implicit (hyperbolic) 41 implicit (parabolic) 39 transducing mechanisms 3 transformer see power transformer trapezoidal difference scheme 37 ultrasound high intensity focused (HIFU)

357

litotripsy

361

vector field 383 irrotational 384 solenoidal 384 virtual work see principle Voigt notation 54, 59 wave longitudinal 65, 139 number 148 plane 148 shear 65 spherical 150 weighted regularization 120 Westervelt equation 160 wiping-out 255 yield stress

66