Table Of ContentC.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, reproduc
tion 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 "Verwertungsge
sellschaft 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 Janei
ro beside a swimming pool. The authors are identified by the pseudonyms of Socra
tes, Plato and Aristotle, not for reasons of vanity but because those illustrious philos
ophers 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 com
prehend 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 re
member 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 knowl
edge 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 illustra
tions 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 inte
grates 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 out
growth of the finite element method in a way that is simple for engineers to under
stand. The mathematical treatment is always subordinate to the applicability of the
technique.
The reader will thus find in this definitive monograph a comprehensive treat
ment 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
1.2. Basic Definitions. . . . . . . 2
l.3. Approximate Solutions . . . . 7
1A. Method of Weighted Residuals. 12
1.4.1. The Collocation Method. 13
1.4.2. Method of Collocation by Subregions 17
1.5. Method of Galerkin . . ...... . 23
1.6. Weak Formulations . . . . . . . . . 25
1.7. Inverse Problem and Boundary Solutions 35
1.8. Classification of Approximate Methods 43
References . 44
Bibliography . . . . . . . . . . . . . 45
Chapter 2 POTENTIAL PROBLEMS 47
2.1. Introduction. . . . . . . . 47
2.2. Elements of Potential Theory 49
2.3. Indirect Formulation. . . 58
2A. Direct Formulation 61
2.5. Boundary Element Method 64
2.6. Two-Dimensional Problems 65
2.6.1. Source Formulation 70
2.7. Poisson Equation . . . . 75
2.8. Subregions . . . . . . . 79
2.9. Orthotropy and Anisotropy 82
2.lO. Infinite Regions . . . . . 85
2.11. Special Fundamental Solutions 89
2.l2. Three-Dimensional Problems . 92
2.13. Axisymmetric Problems. . . . 96
2.l4. Axisymmetric Problems with Arbitrary Boundary Conditions . 99
2.15. Nonlinear Materials and Boundary Conditions 102
2.l5.l. Nonlinear Boundary Conditions 106
References . . . . . . . . . . . . . . . 107
Chapter 3 INTERPOLA TION FUNCTIONS 109
3.1. Introduction. . . . . . . . . . . . . . 109
3.2. Linear Elements for Two-Dimensional Problems 109
x Contents
3.3. Quadratic and Higher-Order Elements . . . . . . 118
3.4. Boundary Elements for Three-Dimensional Problems 127
3.4.1. Quadrilateral Elements . . . . . . 129
3.4.2. Higher-Order Quadrilateral Elements 131
3.4.3. Lagrangian Quadrilateral Elements 131
3.4.4. Triangular Elements . . . . . . 132
3.4.5. Higher-Order Triangular Elements 134
3.5. Three-Dimensional Cell Elements 135
3.5.1. Tetrahedron. . . . . . . 136
3.5.2. Cube. . . . . . . . . . 136
3.6. Discontinuous Boundary Elements 137
3.7. Order ofInterpolation Functions 138
References . . . . . . . . . . . . . 140
Chapter 4 DIFFUSION PROBLEMS 141
4.1. Introduction. . . . . . . . . . . 141
4.2. Laplace Transforms . . . . . . . 142
4.3. Coupled Boundary Element - Finite Difference Methods 146
4.4. Time-Dependent Fundamental Solutions 147
4.5. Two-Dimensional Problems ... 150
4.5.1. Constant Time Interpolation . 150
4.5.2. Linear TIme Interpolation . . 152
4.5.3. Quadratic Time Interpolation 153
4.5.4. Space Integration. . . 154
4.6. Time-Marching Schemes . . 156
4.7. Three-Dimensional Problems 164
4.8. Axisymmetric Problems . 165
4.9. Nonlinear Diffusion 171
References 174
Chapter 5 ELASTOSTATICS 177
5.1. Introduction to the Theory of Elasticity 177
5.1.1. Initial Stresses or Initial Strains. 183
5.2. Fundamental Integral Statement 183
5.2.1. Somigliana Identity. 185
5.3. Fundamental Solutions . . . 187
5.4. Stresses at Internal Points 190
5.5. Boundary Integral Equation 191
5.6. Infinite and Semi-Infinite Regions 195
5.7. Numerical Implementation 197
5.8. Boundary Elements . . . . . . 199
5.9. System of Equations . . . . . . 201
5.10. Stresses and Displacements Inside the Body 202
5.11. Stresses on the Boundary . . . 203
5.12. Surface Traction Discontinuities . . . . . 204
Contents XI
5.13. Two-Dimensional Elasticity 210
5.14. Body Forces ..... . 217
5.14.1. Gravitational Loads 219
5.14.2. Centrifugal Load 220
5.14.3. Thermal Loading . 222
5.15. Axisymmetric Problems. . 224
5.15.1. Extension to Nonaxisymmetric Boundary Values. 230
5.16. Anisotropy 230
References . . . . . . . . . . . . . . . . . . . . . . . 234
Chapter 6 BOUNDARY INTEGRAL FORMULATION FOR
INELASTIC PROBLEMS. 237
6.1. Introduction. . . . . . . . . 237
6.2. Inelastic Behavior of Materials . 240
6.3. Governing Equations. . . . . 251
6.4. Boundary Integral Formulation 253
6.5. Internal Stresses . . . . . . . 255
6.6. Alternative Boundary Element Formulations 258
6.6.1. Initial Strain. . . . . . . . . . . 258
6.6.2. Initial Stress. . . . . . . . . . . 260
6.6.3. Fictitious Tractions and Body Forces 261
6.7. Half-Plane Formulations 262
6.8. Spatial Discretization. 265
6.9. Internal Cells . . . 270
6.10. Axisymmetric Case. 274
References . . . . . . . 275
Chapter 7 ELASTOPLASTICITY . . 277
7.1. Introduction .......... . 277
7.2. Some Simple Elastoplastic Relations 277
7.3. Initial Strain: Numerical Solution Technique. 281
7.3.1. Examples - Initial Strain Formulation . 282
7.4. General Elastoplastic Stress-Strain Relations . 286
7.5. Initial Stress: Outline of Solution Techniques. 290
7.5.1. Examples: Kelvin Implementation .. 292
7.5.2. Examples: Half-Plane Implementation 297
7.6. Comparison with Finite Elements 300
References . . . . . . . . . . . . . . . . . . 304
Chapter 8 OTHER NONLINEAR MATERIAL PROBLEMS. 306
8.1. Introduction. . . . . . . . . . . . . 306
8.2. Rate-Dependent Constitutive Equations. 306
8.3. Solution Technique: Viscoplasticity . . . 309
XII Contents
8.4. Examples: Time-Dependent Problems 312
8.5. No-Tension Materials 318
References . . . . . . . . . . 322
Chapter 9 PLA TE BENDING 324
9.1. Introduction. . . . . 324
9.2. Governing Equations. . . 324
9.3. Integral Equations . . . . 326
9.3.1. Other Fundamental Solutions 330
9.4. Applications. 331
References 336
Chapter 10 WA VE PROPAGA TION PROBLEMS. 338
10.1. Introduction. . . . . . . . . . . . . . . . . 338
10.2. Three-Dimensional Water Wave Propagation Problems 339
10.3. Vertical Axisymmetric Bodies . . . . . 344
10.4. Horizontal Cylinders of Arbitrary Section 347
10.5. Vertical Cylinders of Arbitrary Section 350
10.6. Transient Scalar Wave Equation . . . . 352
10.7. Three-Dimensional Problems: The Retarded Potential. 354
10.8. Two-Dimensional Problems 356
References 357
Chapter 11 VIBRA TIONS . 360
11.1. Introduction. . . . . . 360
11.2. Governing Equations. . 360
11.3. Time-Dependent Integral Formulation 362
11.4. Laplace Transform Formulation 363
11.5. Steady-State Elastodynamics 367
11.6. Free Vibrations 373
References . . . . . . . . . . . 375
Chapter 12 FURTHER APPLICA TIONS IN
FLUID MECHANICS 377
12.1. Introduction. . . . . . . . 377
12.2. Transient Groundwater Flow . 377
12.3. Moving Interface Problems . . 381
12.4. Axisymmetric Bodies in Cross Flow. 384
12.5. Slow Viscous Flow (Stokes Flow) . 386
12.6. General Viscous Flow 389
12.6.1. Steady Problems 393
12.6.2. Transient Problems 395
References . . . . . . . . . . 398
