C.A. Brebbia J.e.F. Telles L.C. Wrobel

Boundary Element Techniques Theory and Applications in Engineering

With 284 Figures

Springer -Verlag Berlin Heidelberg New York Tokyo 1984

C. A. BREBBIA

Dept. of Civil Engineering University of Southampton Southampton S09 5NH United Kingdom

J.C.F. TELLES

L.c. WROBEL

COPPE - Univ. Federal do Rio de Janeiro Programa de Engenharia Civil Caixa Postal 68506 21944-Rio de Janeiro Brazil

ISBN-I3: 978-3-642-48862-7 e-ISBN-13: 978-3-642-48860-3 DOl: 10.1007/978-3-642-48860-3

Library of Congress Cataloging in Publication Data

Brebbia, C.A. Boundary element techniques. Includes index. 1. Boundary value problems. 2. Engineering mathematics. I. Telles, J. C. Faria (Jose Claudio Faria), 1950--. II. Wrobel, L. C. (Luiz Carlos), 1953-. III. Title. TA347.B69B734 1983 620'.0042 83-4827 ISBN-13: 978-3-642-48862-7

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting. re·use of illustrations, broadcasting, reproduc­tion by photocopying machine or similar means, and storage in data banks. Under § 54 of the Gennan Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsge­sellschaft Wort", Munich.

o Springer-Verlag Berlin, Heidelberg 1984

Softcovcr reprint of the hardcover ) st edition 1984

The use of 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.

206113020-543210

The Preface in a Dialogue

The authors have attempted twice to write an appropriate preface to this book but have on both occasions miserably failed to convey in a brief manner what are the main points of the book. This failure is mainly due to the aversion of the authors to prefaces that promise everything but deliver little. Due to the lack of success the best we can offer is a verbatim report of the two meetings.

Act I

(Our authors start to discuss the writing of this preface. The scene is in Rio de Janei­ro beside a swimming pool. The authors are identified by the pseudonyms of Socra­tes, Plato and Aristotle, not for reasons of vanity but because those illustrious philos­ophers somewhat characterize their respective points of view).

SOCRATES: I have been reading some of the literature on Integral equations and Boundary elements recently published and feel most unhappy about the lack of a comprehensive text.

PLATO: Yes. I have just been looking at one that is rather written in a hurry I guess to capitalize on the current interest in the topic. The authors failed to com­prehend the basic principles of the technique.

ARISTOTLE: That is because people write books without having had a first hand experience in the relevant research topic. I always insist you have to look at the problem and build your theory around it!

PLATO: Well, well, Ari. That may well be the case sometimes but you have to re­member that the fundamental mathematical concepts are the essential part of any method.

SOCRATES: I do not think this discussion is leading us anywhere. I propose that we write one book based on our research experience and our fundamental knowl­edge of approximate and basic techniques, trying to blend our past finite element background with the new method and I think we should define the table of contents and preface right now.

PLATO and ARISTOTLE: Hear! Hear!

VI Preface

SOCRATES: I think that we ought to stress that we will write only about things that we have first hand experience in, in a coherent way that will be useful to engineers and other scientists and stressing the formulation without being too mathematical. We should write with integrity and honesty, giving reference to other authors where reference is due, but avoiding mentioning everybody just to be certain that our book is widely advertised. Above all, the book should be clear and useful.

PLATO: I think we should include a good discussion of fundamental ideas, of how integral equations are formed, pointing out that they are like two dimensional shadows of three dimensional objects, ...

SOCRATES: Stop there! Remember you are not 'the' Plato!

PLATO: Sorry, I was carried away.

ARISTOTLE: I think that the book should have many applications so that the reader can learn by looking at them how to use the method.

SOCRATES: I agree. But we should be careful. It is easy to include many illustra­tions and examples in a book in order to disguise its meagre contents. All examples should be relevant.

ARISTOTLE: And we should also include a full computer program to give the reader if so he wishes, a working experience of the technique.

SOCRATES: That is a good idea, provided that the code is well explained and inte­grates with the theory. Any fool can nowadays attach a computer code to a book but requires work and experience to have it properly related to the theory.

PLATO: I wonder if we will write the book. It seems unlikely.

SOCRATES: Yes it does. Does it not? Well, I am going for a swim.

Act II

(The manuscript is finished and the writers are sitting around it. The scene is now in Southampton in April. A timid ray of sun is coming through a window. The writers are spellbound and looking attentively at the manuscript).

PLATO: I cannot believe it! It is really finished!

SOCRATES: Well, not quite. You will see how the publishers will want us to trim it down. They always do as a matter of principle. 20 to 25% I think.*

ARISTOTLE: But that would be a pity! We have been over the manuscript three times. It is perfect!

* Springer-Verlag, to their great credit, accepted the full manuscript. Our apologies -The Authors

Preface VII

PLATO: We can only achieve but pallid reflections of perfection. Still it is a good book.

SOCRATES: You are right. We should fight for it and make them publish the whole work. We have some rights do we not (looks at the contract for a moment and concludes). No, we do not! (meakley) but we can try ...

ARISTOTLE: And once it is published we have to explain to our colleagues that this is a serious book, a work with applications. We should specially stress (i) that the work has a great unity; (ii) the large range of topics covered in depth; (iii) that it is written by those who have used the method; (iv) that it is well written and clear.

PLATO: How are we going to do that?

ARISTOTLE: (downcast) I do not know ...

SOCRATES: I know! We should write a Preface (everybody agrees). Well let us start; "Recent new advances and developments in the field of boundary elements ... etc, etc."

ARISTOTLE and PLATO: Not again!!

The Book

The purpose of this book is to present a comprehensive and up-to-date treatment of the boundary element method (B.E.M.).

The work stresses the non-linear and time-depending applications together with a series of new problems which can now be solved using B.E.M.

The approach followed by the authors is to present the techniques as an out­growth of the finite element method in a way that is simple for engineers to under­stand. The mathematical treatment is always subordinate to the applicability of the technique.

The reader will thus find in this definitive monograph a comprehensive treat­ment of the topic from fundamentals to computer applications, including a fully operational computer program.

The Authors

Contents

Chapter 1 APPROXIMATE METHODS

1.1. Introduction. . . . . 1.2. Basic Definitions. . . . . . . l.3. Approximate Solutions . . . . 1A. Method of Weighted Residuals.

1.4.1. The Collocation Method. 1.4.2. Method of Collocation by Subregions

1.5. Method of Galerkin . . ...... . 1.6. Weak Formulations . . . . . . . . . 1.7. Inverse Problem and Boundary Solutions 1.8. Classification of Approximate Methods

References . Bibliography . . . . . . . . . . . . .

Chapter 2 POTENTIAL PROBLEMS

2.1. Introduction. . . . . . . . 2.2. Elements of Potential Theory 2.3. Indirect Formulation. . . 2A. Direct Formulation 2.5. Boundary Element Method 2.6. Two-Dimensional Problems

2.6.1. Source Formulation 2.7. Poisson Equation . . . . 2.8. Subregions . . . . . . . 2.9. Orthotropy and Anisotropy 2.lO. Infinite Regions . . . . . 2.11. Special Fundamental Solutions 2.l2. Three-Dimensional Problems . 2.13. Axisymmetric Problems. . . . 2.l4. Axisymmetric Problems with Arbitrary Boundary Conditions . 2.15. Nonlinear Materials and Boundary Conditions

2.l5.l. Nonlinear Boundary Conditions References . . . . . . . . . . . . . . .

Chapter 3 INTERPOLA TION FUNCTIONS

3.1. Introduction. . . . . . . . . . . . . . 3.2. Linear Elements for Two-Dimensional Problems

1 2 7

12 13 17 23 25 35 43 44 45

47

47 49 58 61 64 65 70 75 79 82 85 89 92 96 99

102 106 107

109

109 109

x

3.3. Quadratic and Higher-Order Elements . . . . . . 3.4. Boundary Elements for Three-Dimensional Problems

3.4.1. Quadrilateral Elements . . . . . . 3.4.2. Higher-Order Quadrilateral Elements 3.4.3. Lagrangian Quadrilateral Elements 3.4.4. Triangular Elements . . . . . . 3.4.5. Higher-Order Triangular Elements

3.5. Three-Dimensional Cell Elements 3.5.1. Tetrahedron. . . . . . . 3.5.2. Cube. . . . . . . . . .

3.6. Discontinuous Boundary Elements 3.7. Order ofInterpolation Functions

References . . . . . . . . . . . . .

Chapter 4 DIFFUSION PROBLEMS

4.1. Introduction. . . . . . . . . . . 4.2. Laplace Transforms . . . . . . . 4.3. Coupled Boundary Element - Finite Difference Methods 4.4. Time-Dependent Fundamental Solutions 4.5. Two-Dimensional Problems ...

4.5.1. Constant Time Interpolation . 4.5.2. Linear TIme Interpolation . . 4.5.3. Quadratic Time Interpolation 4.5.4. Space Integration. . .

4.6. Time-Marching Schemes . . 4.7. Three-Dimensional Problems 4.8. Axisymmetric Problems . 4.9. Nonlinear Diffusion

References

Chapter 5 ELASTOSTATICS

5.1. Introduction to the Theory of Elasticity 5.1.1. Initial Stresses or Initial Strains.

5.2. Fundamental Integral Statement 5.2.1. Somigliana Identity.

5.3. Fundamental Solutions . . . 5.4. Stresses at Internal Points 5.5. Boundary Integral Equation 5.6. Infinite and Semi-Infinite Regions 5.7. Numerical Implementation 5.8. Boundary Elements . . . . . . 5.9. System of Equations . . . . . . 5.10. Stresses and Displacements Inside the Body 5.11. Stresses on the Boundary . . . 5.12. Surface Traction Discontinuities . . . . .

Contents

118 127 129 131 131 132 134 135 136 136 137 138 140

141

141 142 146 147 150 150 152 153 154 156 164 165 171 174

177

177 183 183 185 187 190 191 195 197 199 201 202 203 204

Contents

5.13. Two-Dimensional Elasticity 5.14. Body Forces ..... .

5.14.1. Gravitational Loads 5.14.2. Centrifugal Load 5.14.3. Thermal Loading .

5.15. Axisymmetric Problems. . 5.15.1. Extension to Nonaxisymmetric Boundary Values.

5.16. Anisotropy References . . . . . . . . . . . . . . . . . . . . . . .

Chapter 6 BOUNDARY INTEGRAL FORMULATION FOR INELASTIC PROBLEMS.

6.1. Introduction. . . . . . . . . 6.2. Inelastic Behavior of Materials . 6.3. Governing Equations. . . . . 6.4. Boundary Integral Formulation 6.5. Internal Stresses . . . . . . . 6.6. Alternative Boundary Element Formulations

6.6.1. Initial Strain. . . . . . . . . . . 6.6.2. Initial Stress. . . . . . . . . . . 6.6.3. Fictitious Tractions and Body Forces

6.7. Half-Plane Formulations 6.8. Spatial Discretization. 6.9. Internal Cells . . . 6.10. Axisymmetric Case.

References . . . . . . .

Chapter 7 ELASTOPLASTICITY . .

7.1. Introduction .......... . 7.2. Some Simple Elastoplastic Relations 7.3. Initial Strain: Numerical Solution Technique.

7.3.1. Examples - Initial Strain Formulation . 7.4. General Elastoplastic Stress-Strain Relations . 7.5. Initial Stress: Outline of Solution Techniques.

7.5.1. Examples: Kelvin Implementation .. 7.5.2. Examples: Half-Plane Implementation

7.6. Comparison with Finite Elements References . . . . . . . . . . . . . . . . . .

Chapter 8 OTHER NONLINEAR MATERIAL PROBLEMS.

8.1. Introduction. . . . . . . . . . . . . 8.2. Rate-Dependent Constitutive Equations. 8.3. Solution Technique: Viscoplasticity . . .

XI

210 217 219 220 222 224 230 230 234

237

237 240 251 253 255 258 258 260 261 262 265 270 274 275

277

277 277 281 282 286 290 292 297 300 304

306

306 306 309

XII

8.4. Examples: Time-Dependent Problems 8.5. No-Tension Materials

References . . . . . . . . . .

Chapter 9 PLA TE BENDING

9.1. Introduction. . . . . 9.2. Governing Equations. . . 9.3. Integral Equations . . . .

9.3.1. Other Fundamental Solutions 9.4. Applications.

References

Chapter 10 WA VE PROPAGA TION PROBLEMS.

10.1. Introduction. . . . . . . . . . . . . . . . . 10.2. Three-Dimensional Water Wave Propagation Problems 10.3. Vertical Axisymmetric Bodies . . . . . 10.4. Horizontal Cylinders of Arbitrary Section 10.5. Vertical Cylinders of Arbitrary Section 10.6. Transient Scalar Wave Equation . . . . 10.7. Three-Dimensional Problems: The Retarded Potential. 10.8. Two-Dimensional Problems References

Chapter 11 VIBRA TIONS .

11.1. Introduction. . . . . . 11.2. Governing Equations. . 11.3. Time-Dependent Integral Formulation 11.4. Laplace Transform Formulation 11.5. Steady-State Elastodynamics 11.6. Free Vibrations References . . . . . . . . . . .

Chapter 12 FURTHER APPLICA TIONS IN FLUID MECHANICS

12.1. Introduction. . . . . . . . 12.2. Transient Groundwater Flow . 12.3. Moving Interface Problems . . 12.4. Axisymmetric Bodies in Cross Flow. 12.5. Slow Viscous Flow (Stokes Flow) . 12.6. General Viscous Flow

12.6.1. Steady Problems 12.6.2. Transient Problems

References . . . . . . . . . .

Contents

312 318 322

324

324 324 326 330 331 336

338

338 339 344 347 350 352 354 356 357

360

360 360 362 363 367 373 375

377

377 377 381 384 386 389 393 395 398

Contents

Chapter 13 COUPLING OF BOUNDARY ELEMENTS WITH OTHER METHODS

13.1. Introduction. . . . . . . . . . . . 13.2. Coupling of Finite Element and Boundary Element Solutions

13.2.1. The Energy Approach 13.3. Alternative Approach. . . . . . . . . 13.4. Internal Fluid Problems. . . . . . . .

13.4.1. Free-Surface Boundary Condition 13.4.2. Extension to Compressible Fluid

13.5. Approximate Boundary Elements 13.6. Approximate Finite Elements References . . . . . . . . . . . . .

Chapter 14 COMPUTER PROGRAM FOR TWO-DIMENSIONAL ELASTOSTA TICS.

14.1. Introduction. . . . . . . . 14.2. Main Program and Data Structure 14.3. Subroutine INPUT. 14.4. Subroutine MATRX 14.5. Subroutine FUNC . 14.6. Subroutine SLNPD 14.7. Subroutine OUTPT 14.8. Subroutine FENC . 14.9. Examples. . . . .

14.9.1. Square Plate 14.9.2. Cylindrical Cavity Problem

References . . . . . . . . . . . . .

Appendix A NUMERICAL INTEGRA TION FORMULAS.

A.1. Introduction . . . . . . . . . . . A.2. Standard Gaussian Quadrature . . .

A.2.1. One-Dimensional Quadrature . A.2.2. Two- and Three-Dimensional Quadrature for Rectangles

and Rectangular Hexahedra. A.2.3. Triangular Domain . . . . . . . . .

A.3. Computation of Singular Integrals . . . . . . A.3.1. One-Dimensional Logarithmic Gaussian

Quadrature Formulas ....... . A.3.2. Numerical Integration over Triangles and Squares

with 1Ir Singularity . . . . . . . . . . . . . A.3.3. Numerical Evaluation of Cauchy Principal Values

References . . . . . . . . . . . . . . . . . . . . . .

XIII

400

400 401 405 409 411 412 414 415 422 424

427

427 428 430 433 435 437 438 439 440 440 446 446

447

447 447 447

447 449 449

449

449 451 454

XIV Contents

Appendix B SEMI-INFINITE FUNDAMENTAL SOLUTIONS 455

B.l. Half-Space 455 B.2. Half-Plane 458 References . 460

Appendix C SOME PARTICULAR EXPRESSIONS FOR TWO-DIMENSIONAL INELASTIC PROBLEMS. 461

SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . . 463