L. Gaul, M. Kogl, M. Wagner

Boundary Element Methods for Engineers and Scientists

Springer-Verlag Berlin Heidelberg GmbH

Engineering ONLINE LIBRARY

http://www.springer.de/engine/

Lothar Gaul · Martin Kogl • Marcus Wagner

Boundary Element Methods for Engineers and Scientists An Introductory Course with Advanced Topics

With 135 Figures

Springer

Prof. Dr.-Ing. habil. Lothar Gaul Full Professor and Head Institute A of Mechanics, University of Stuttgart. Former Dean Process Engineering and Engineering Cybernetics, University of Stuttgart. Former Dean Mechanical Engineering, University of the Federal Armed Forces University Hamburg. e-mail: gaul@mecha.uni-stuttgart.de

Dr.-Ing. Martin Kogl Currently working as post-doctoral researcher at the Department of Structures and Foundation Engineering, University of Sao Paulo, Brazil. Former research coworker at the Institute A of Mechanics, University of Stuttgart. e-mail: research@martin-koegl.de

Dr.-Ing. Marcus Wagner Currently research engineer at BMW Group, Munich. Former post-doctoral fellow, Division of Mechanics and Computation, Stanford University, Stanford, CA, USA. Former research assistant at the Institute A of Mechanics, University of Stuttgart. e-mail: marcus@m-m-wagner.de

Universitat Stuttgart Institut A fur Mechanik Pfaffenwaldring 9 70550 Stuttgart Germany

ISBN 978-3-642-05589-8 ISBN 978-3-662-05136-8 (eBook) DOI 10.1007/978-3-662-05136-8

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Preface

Over the past decades, the Boundary Element Method has emerged as a ver­satile and powerful tool for the solution of engineering problems, presenting in many cases an alternative to the more widely used Finite Element Method. As with any numerical method, the engineer or scientist who applies it to a practical problem needs to be acquainted with, and understand, its basic principles to be able to apply it correctly and be aware of its limitations. It is with this intention that we have endeavoured to write this book: to give the student or practitioner an easy-to-understand introductory course to the method so as to enable him or her to apply it judiciously. As the title suggests, this book not only serves as an introductory course, but also cov­ers some advanced topics that we consider important for the researcher who needs to be up-to-date with new developments.

This book is the result of our teaching experiences with the Boundary Element Method, along with research and consulting activities carried out in the field. Its roots lie in a graduate course on the Boundary Element Method given by the authors at the university of Stuttgart. The experiences gained from teaching and the remarks and questions of the students have contributed to shaping the 'Introductory course' (Chapters 1-8) to the needs of the stu­dents without assuming a background in numerical methods in general or the Boundary Element Method in particular. This part can be used both for self-study or as a basis for a graduate university course on the method. The remaining chapters cover more advanced topics and are aimed at researchers who need to learn morc about extensions and alternative approaches to the 'classical' method. We sincerely hope that we have succeeded in the task to find an approach that is suitable for a first introduction to the method, since we believe that the first contact is of great importance in awaking the students' interest and to motivate them to delve deeper into the subject.

Before embarking on the Boundary Element Method, we have included two chapters on the mathematical basis and continuum physics; this way the reader will not have to consult other textbooks to be able to follow ours. We assume that the reader is familiar with the basic concepts of vector algebra and calculus as taught in any undergraduate course on engineering mathe­matics - the chapter on mathematics is intended only to introduce some con­cepts that might not be covered in those courses. The chapter on continuum

VI Preface

physics attempts to describe in a concise way the basic physical principles that lead to the differential equations on which we base our numerical solu­tion procedures. While standard textbooks on numerical methods commonly omit this subject, we feel that an understanding of the physical principles is of great importance for successfully solving engineering problems, not only for the correct choice of the model and the pre-processing, but also for the interpretation and critical judgement of the results.

While the classical Boundary Element Method, described in the first part of this book, is the most widely known and universally employed approach, the last two decades have seen the emergence of a number of complementary and concurrent developments in Boundary Element research, dealing with problems where the classical method presents difficulties or fails completely. We have therefore included two of these additional methods in the book; we believe their study is of great importance for the practitioner or researcher who needs or wants to extend his or her knowledge about the method: the Dual Reciprocity Method and the Hybrid Boundary Element Methods.

In the first two chapters of Part II, we give an introductory course on the Dual Reciprocity Method and describe in detail the solution of the equations of motion. Later, we will apply the method to more advanced problems. The Dual Reciprocity Method is an extremely versatile tool for the transformation of general source terms to the boundary, a problem that occurs in many Boundary Element applications (coupled field problems, dynamic analysis, inhomogeneities, non-linearities, to name but a few). It is not a separate Boundary Element Method in itself but rather an extension to the classical method which can be used in the many cases in which we have to deal with sources that cannot be transformed to the boundary by other means.

In Part III, we present a new development in the field of symmetric for­mulations, the Hybrid Boundary Element Methods. These methods are based upon variational principles rather than upon weighted residual formulations. The name 'hybrid' describes the fact that we use a Boundary Element-type approximation with fundamental solutions for the domain field, and a Finite Element-type approximation on the boundary. We will describe two formu­lations of the method for elastodynamics and acoustics, and also give an introduction to fluid-structure interaction.

The insight and experience that have contributed to this book could only be gained by an intensive cooperation in the framework of a research group on the Boundary Element Method. The authors are indebted and would like to thank their colleagues and ex-colleagues Dr. Christian Fiedler, Dr. Wolfgang Wenzel, Dr. Friedrich Moser, and Dipl.-Ing. Matthias Fischer at the Institute A of Mechanics at the University of Stuttgart for their cooperation and the fruitful discussions on the subject. We would also like to thank our Brazil­ian colleagues Prof. Ney Dumont of PUC Rio de Janeiro and Prof. Marcos Noronha of the University of Sao Paulo (USP), with whom an intensive ex­change of researchers and ideas has taken place. The authors appreciate the

Preface VII

continuous advice of Professor Patrick Selvadurai, McGill University Mon­treal, during his stay as recipient of the Alexander von Humboldt Senior Scientist Award at the Institute A of Mechanics. We are as well indebted to his kind wife Mrs. Sally Selvadurai for her thorough proofreading and editing of the manuscript. We shall not forget our students who have made valuable contributions through a number of excellent theses.

This research would not have been possible without the excellent pro­motion by the Deutsche Forschungsgemeinschaft (DFG -- German Research Foundation) and the Deutscher Akademischer Austausch Dienst (DAAD -German Academic Exchange Service), who have financed a number of our research projects on the Boundary Element Method.

Stuttgart, November 2002

Lothar Gaul Martin K ogl

Marcus Wagner

Contents

Part I. The Direct Boundary Element Method

1. Introduction.............................................. 3 1.1 Numerical Solution of Engineering Problems. . . . . . . . . . . . . . . 3 1.2 Historic Development of Boundary Element Method ... . . . . . 6 1.3 The Method of Weighted Residuals . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.1 Collocation Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.2 Method of Moments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.3 Galerkin's :\lethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.4 Collocation by Subregions. . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.5 Least Squares. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10 1.3.6 Summary........................................ 11

1.4 Boundary Elements vs. Finite Elements .. . . . . . . . . . . . . . . . .. 11 1.4.1 General Features. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12 1.4.2 Comparison of FE and BE Formulations ............ 13

1.5 Boundary Integral Method for 1-D Differential Equation. . . .. 16 1.6 General Boundary Element Approach. . . . . . . . . . . . . . . . . . . .. 20

2. Mathematical Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23 2.1 Notation.............................................. 23

2.1.1 Vector Notation and Matrix Representation of Vectors 23 2.1.2 Index Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 25

2.2 Theory of Distributions ................................. 26 2.2.1 Definition and Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 27 2.2.2 Dirac Distribution and Heaviside Function. . . . . . . . . .. 27

2.3 Integral Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 29 2.4 Time-Harmonic ;vIotion ................................. 30 2.5 Introduction to the Calculus of Variations. . . . . . . . . . . . . . . .. 31

3. Continuum Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35 3.1 Introduction........................................... 35

3.1.1 Continuous l\ledia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35 3.1.2 Variables and Equations of State. . . . . . . . . . . . . . . . . .. 36 3.1.3 Crystal Structure of a Continuous Medium. . . . . . . . .. 37

3.2 Elastodynamics........................................ 38

X Contents

3.2.1 Kinematics...................................... 39 3.2.2 Kinetics......................................... 43 3.2.3 Conservation of Mass - The Continuity Equation. . . .. 44 3.2.4 Conservation of Linear Momentum ................. 46 3.2.5 Conservation of Angular Momentum. . . . . . . . . . . . . . .. 47 3.2.6 Conservation of Energy ........................... 48 3.2.7 Constitutive Equation ............................ 50 3.2.8 Field Equations and Boundary Conditions. . . . . . . . . .. 53 3.2.9 Elastodynamic Wave Equations. . . . . . . . . . . . . . . . . . .. 55 3.2.10 Waves in Anisotropic Materials .................... 56

3.3 Heat Conduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 60 3.3.1 First Law of Thermodynamics. . . . . . . . . . . . . . . . . . . .. 60 3.3.2 Second Law of Thermodynamics ................... 61 3.3.3 Field Equations of Heat Conduction . . . . . . . . . . . . . . .. 63 3.3.4 Boundary and Initial Conditions ................... 64

3.4 Electrodynamics........................................ 65 3.4.1 Maxwell's Equations in Vacuum. . . . . . . . . . . . . . . . . . .. 65 3.4.2 Electromagnetic Wave Equations. . . . . . . . . . . . . . . . . .. 67 3.4.3 Electrostatic Field in Macroscopic Media. . . . . . . . . . .. 68 3.4.4 Field Equation for Dielectrics . . . . . . . . . . . . . . . . . . . . .. 68 3.4.5 Electric Jump Conditions at Interfaces. . . . . . . . . . . . .. 70 3.4.6 Electric Boundary Conditions for an Ideal Conductor. 70

3.5 Thermoelasticity....................................... 71 3.5.1 First Law of Thermodynamics. . . . . . . . . . . . . . . . . . . .. 71 3.5.2 Second Law of Thermodynamics ................... 72 3.5.3 Constitutive Equations. . . . . . . . . . . . . . . . . . . . . . . . . . .. 73 3.5.4 Field Equations of Thermoelasticity ................ 76 3.5.5 Simplified Thermoelastic Theories. . . . . . . . . . . . . . . . .. 77

3.6 Acoustics.............................................. 79 3.6.1 The Acoustic Wave Equation. . . . . . . . . . . . . . . . . . . . .. 80 3.6.2 Constitutive Equations. . . . . . . . . . . . . . . . . . . . . . . . . . .. 81 3.6.3 The Velocity Potential . . . . . . . . . . . . . . . . . . . . . . . . . . .. 86 3.6.4 Boundary Conditions in Acoustics. . . . . . . . . . . . . . . . .. 86 3.6.5 Radiation Condition in Infinite Domains ............ 87

3.7 Piezoelectricity......................................... 89 3.7.1 Polarisation in Piezoelectrics. . . . . . . . . . . . . . . . . . . . . .. 90 3.7.2 Energy Balance and Thermodynamic Potentials. . . . .. 91 3.7.3 Constitutive Equations. . . . . . . . . . . . . . . . . . . . . . . . . . .. 92 3.7.4 Field Equations of Piezoelectricity. . . . . . . . . . . . . . . . .. 93

4. Boundary Element Method for Potential Problems ....... 95 4.1 Introduction........................................... 95 4.2 BE Formulation of Laplace's Equation . . . . . . . . . . . . . . . . . . .. 96

4.2.1 Inverse Formulation of Differential Equation. . . . . . . .. 96 4.2.2 Green's Representation Formula. . . . . . . . . . . . . . . . . . .. 98

Contents XI

4.2.3 Fundamental Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . .. 99 4.2.4 Boundary Integral Equation of the 2-D Problem. . . . .. 99 4.2.5 Discretisation of the Boundary ..................... 104 4.2.6 The Collocation Method .......................... 107 4.2.7 Modelling of Discontinuous Fluxes .................. 110

4.3 Example: Steady-State Heat Conduction .................. 114 4.3.1 Calculation of System Matrices .................... 114 4.3.2 Assembly and Solution of Equations ................ 120 4.3.3 The Analytical Solution ........................... 121

4.4 Calculation of Solution in the Domain ..................... 122 4.4.1 Potential in the Domain ........................... 122 4.4.2 Flux in the Domain ............................... 123

4.5 Poisson's Equation Treatment of Source Terms ........... 123 4.5.1 Calculation of Domain Integral by Cell Integration ... 124 4.5.2 Calculation of Domain Integral by Transformation to

a Boundary Integral .............................. 127 4.5.3 Calculation of the Unknown Boundary Variables ..... 130

4.6 Indirect Calculation of Diagonal Entries of H .............. 131 4.7 Concentrated Sources ................................... 131 4.8 Sub domains ........................................... 132 4.9 Orthotropic Heat Conduction ............................ 134 4.10 Example: Coupling of Orthotropic and Isotropic Subdomains 136

4.10.1 Calculation of Matrix Elements H78 and G78 ........ 138 4.10.2 Analytical Solution ............................... 140

5. Boundary Element Method for Elastic Continua .......... 141 5.1 Integral Formulation of Equation of Motion ................ 141

5.1.1 Method of Weighted Residuals ..................... 141 5.1.2 Inverse Statement ................................ 142 5.1.3 Reciprocal Work Principle ......................... 145 5.1.4 Somigliana's Identity ............................. 146

5.2 Derivation of Boundary Integral Equation ................. 149 5.2.1 Boundary Extension Around Load Point ............ 149 5.2.2 Integration over Boundary Extension FE ............. 151 5.2.3 Boundary Integral Equation ....................... 152

5.3 Kumerical Implementation ............................... 152 5.3.1 Discretisation with Boundary Elements ............. 153 5.3.2 Approximation of Boundary Variables ............... 156 5.3.3 Matrix Assembly ................................. 157

5.4 Implementation of Anisotropic Fundamental Solutions ...... 160 5.4.1 Fundamental Solutions in Elastostatics .............. 160 5.4.2 Interpolation Scheme for Anisotropic Fundamental So-

lution ........................................... 161 5.4.3 Evaluation of Anisotropic Fundamental Solution ..... 165

5.5 Static Piezoelectricity ................................... 168

XII Contents

5.5.1 Contracted Notation .............................. 168 5.5.2 Boundary Element Formulation .................... 170 5.5.3 Numerical Example ............................... 171

6. Numerical Integration .................................... 175 6.1 Regular Integration in I-D ............................... 176

6.1.1 Simpson's Rule ................................... 176 6.1.2 Gaussian Quadrature ............................. 179

6.2 Regular Integration in 2-D and 3-D ....................... 183 6.2.1 Multidimensional Gaussian Quadrature ............. 183 6.2.2 Numerical Integration over Boundary Elements ...... 184

6.3 Weakly Singular Integration ............................. 187 6.3.1 Weak Singularity in 2-D BEM ..................... 187 6.3.2 Weak Singularity in 3-D BEM ..................... 188

6.4 Strongly Singular Integration ............................ 193 6.4.1 Strong Singularity in 2-D BEM .................... 193 6.4.2 Strong Singularity in 3-D BEM .................... 199

Part II. The Dual Reciprocity Method

7. DRM for Potential Problems and Elastodynamics ........ 207 7.1 Introduction ........................................... 207 7.2 Dual Reciprocity Formulation for Poisson's Equation ........ 209

7.2.1 Dual Representation Formula ...................... 209 7.2.2 Boundary Integral Equation and Discretisation ....... 211 7.2.3 Coefficients a and System of Equations ............. 213 7.2.4 Calculation of Interior Values ...................... 214

7.3 The Wave Equation ..................................... 214 7.4 The Helmholtz Equation ................................ 217 7.5 Transient Heat Conduction .............................. 218 7.6 Anisotropic Elastodynamics .............................. 219

7.6.1 Representation Formula ........................... 220 7.6.2 Dual Reciprocity Formulation ...................... 220 7.6.3 Boundary Integral Equation and System of Equations. 222

7.7 Particular Solutions ..................................... 223 7.7.1 Types of Interpolation Functions ................... 223 7.7.2 Interpolation Functions in Anisotropic Analysis ...... 225

8. Solution of the Equations of Motion ...................... 227 8.1 System of Equations .................................... 227 8.2 The Elliptic Problem: Static and Time-Harmonic Analysis ... 228

8.2.1 Solution of the Equations .......................... 228 8.2.2 Numerical Example ............................... 229

8.3 The Eigenproblem: Free Vibration Analysis ................ 231

Contents XIII

8.3.1 Solution of the Eigenvalue Problem ................. 231 8.3.2 Numerical Example ............................... 232

8.4 The Hyperbolic Problem: Transient Analysis ............... 237 8.4.1 Direct Analysis .................................. 238 8.4.2 Modal Superposition .............................. 240 8.4.3 Instabilities in Time Integration .................... 243 8.4.4 Numerical Example ............................... 250

8.5 The Parabolic Problem: Transient Heat Conduction ......... 253

9. Dynamic Piezoelectricity ................................. 255 9.1 Dual Reciprocity Formulation ............................ 255 9.2 Coefficients a and System of Equations ................... 257 9.3 Solution of Piezoelectric Equations ....................... 258 9.4 ~umerical Example ..................................... 260

10. Coupled Thermoelasticity ................................ 263 10.1 Representation Formulae ................................ 263

10.1.1 Elastic Representation Formula .................... 264 10.1.2 Thermal Representation Formula ................... 265 10.1.3 Thermoelastic Representation Formula .............. 265

10.2 Dual Reciprocity Formulation ............................ 267 10.3 The Coefficients a ..................................... 269 10.4 System of Equations .................................... 271 10.5 Solution of Thermoelastic Equations ...................... 272

10.5.1 Stationary Thermoelasticity ....................... 273 10.5.2 "Cncoupled Quasi-Static Thermoelasticity ............ 273 10.3.3 Theory of Thermal Stresses ........................ 273 10.5.4 Coupled Quasi-Static Thermoelasticity .............. 274 10.5.5 Fully Coupled Thermoelasticity .................... 275

10.6 Numerical Examples .................................... 275 10.6.1 Stationary Thermoelasticity ....................... 275 10.6.2 Quasi-Static Therrnoelasticity ...................... 278

Part III. Hybrid Boundary Element Methods

11. Variational Principles of Continuum Mechanics ........... 283 11.1 Virtual Quantities in Continuum :\!Iechanics ................ 283

11.1.1 Virtual Work .................................... 284 11.1.2 Strain Energy .................................... 285 11.1.3 Complementary Strain Energy ..................... 286

11.2 Single-Field Principles .................................. 287 11.2.1 Elastostatics ..................................... 287 11.2.2 Elastodynamics .................................. 289

11.3 Generalised Principles ................................... 292

XIV Contents

11.3.1 Elastostatics ..................................... 292 11.3.2 Elastodynamics .................................. 293

12. The Hybrid Displacement Method ........................ 297 12.1 Introduction with Laplace's Equation ..................... 297 12.2 Discretisation of the First Variation ....................... 299

12.2.1 Boundary Approximation ......................... 300 12.2.2 Domain Approximation ........................... 300 12.2.3 Domain Modification ............................. 302

12.3 Matrix Formulation for Potential Problems ................ 303 12.3.1 Example for Laplace's Equation ................... 304

12.4 The HDBEM for Elastodynamics ......................... 310 12.4.1 Transformation to the Frequency Domain ........... 313 12.4.2 Discretisation of the First Variation ................. 315 12.4.3 Matrix Formulation in Elastodynamics .............. 317 12.4.4 Computation of Field Points ....................... 319

12.5 Numerical Implementation ............................... 319 12.5.1 The L Matrix ................................... 319 12.5.2 The Vector of Equivalent Nodal Loads .............. 320 12.5.3 The F Matrix ................................... 321 12.5.4 Efficient Calculation of the F Matrix ............... 332

12.6 The Concept of Rigid Body Motion in Elastostatics ......... 336 12.7 Harmonic Bending Waves of Beams ....................... 338

12.7.1 Hybrid Integral Formulation for the Beam ........... 341 12.7.2 Example: Viscoelastic Beam Element ............... 348

13. The Hybrid Stress Method for Acoustics ................. 351 13.1 Hellinger-Reissner Principle for Acoustics .................. 351 13.2 Matrix Formulation ..................................... 354

13.2.1 Efficient Field Point Computation .................. 356 13.3 Numerical Implementation for Acoustics ................... 360 13.4 Orthogonality Properties of the HSBEM .................. 362

13.4.1 Coordinate Sets and Transformation Relations ....... 362 13.4.2 Finite Domains .................................. 364

- * 13.4.3 Determination of the Fundamental Solution Matrix CPo 367 13.4.4 Infinite Domains ................................. 373

13.5 Applications in Acoustics ................................ 375 13.5.1 Interior Problems ................................. 376 13.5.2 An Exterior Problem ............................. 378

13.6 Fluid-Structure Interaction in the Frequency Domain ....... 381 13.6.1 Model of Acoustic Fluid-Structure Interaction ........ 381 13.6.2 Coupled Multi-Field Problem ...................... 385 13.6.3 Sound Radiation by Bending Waves ................ 390

Contents XV

14. The Hybrid Boundary Element Method in Time Domain. 399 14.1 Time-domain Formulation for Elastodynamics .............. 400

14.1.1 Non-Singular Formulation of the HDBEM ........... 401 14.1.2 Positioning of the Load Points ..................... 404 14.1.3 Transformation of the Mass Matrix to the Boundary .. 405 14.1.4 Numerical Examples .............................. 406

14.2 Time-domain Formulation for Acoustics ................... 413 14.2.1 Hybrid Boundary Element Formulation for the Fluid .. 414 14.2.2 Approximation of Domain and Boundary Variables ... 415 14.2.3 Boundary Integral Formulation ..................... 416 14.2.4 System of Discretised Equations .................... 417 14.2.5 Numerical Examples .............................. 419

14.3 Fluid-Structure Interaction in the Time Domain ............ 421

Part IV. Appendices

A. Properties of Elastic Materials ............................ 431

B. Fundamental Solutions ................................... 433 B.1 Potential Problems ..................................... 433

B.1.1 Laplace's Equation ............................... 433 B.1.2 The Helmholtz Equation .......................... 437 B.1.3 Anisotropic Form of Laplace's equation ............. 440

B.2 Elastomechanics ........................................ 442 B. 2.1 Isotropic Elastostatics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 B.2.2 Anisotropic Elastostatics .......................... 445 B.2.3 Isotropic Time-Harmonic Elastodynamics ........... 448 B.2.4 Anisotropic Elastodynamics ....................... 451

B.3 Piezoelectricity ......................................... 451 B.3.1 Static Piezoelectricity ............................. 452 B.3.2 Dynamic Piezoelectricity .......................... 453

c. Particular Solutions ...................................... 455 C.1 Poisson's Equation ..................................... 455 C.2 Anisotropic Heat Conduction ............................ 457 C.3 Isotropic Elastostatics ................................... 457 C.4 Anisotropic Elastostatics ................................ 459 C.5 Piezoelectricity ......................................... 459

D. The Bott-Duffin Inverse .................................. 461

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

Index ......................................................... 477