Boundary Elements: Theory and Applications: Theory and Applications
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Table of contents
- Table of Contentsxi
- Prefacevii
- Chapter 1. Introduction1
- 1.1 Scope of the book1
- 1.2 Boundary Elements and Finite Elements2
- 1.3 Historical development of the BEM5
- 1.4 Structure of the book7
- 1.5 CD-ROM contents9
- 1.6 References9
- Chapter 2. Preliminary Mathematical Concepts13
- 2.1 Introduction13
- 2.2 The Gauss-Green theorem13
- 2.3 The divergence theorem of Gauss15
- 2.4 Green's second identity16
- 2.5 The adjoint operator17
- 2.6 The Dirac delta function18
- 2.7 References23
- Problems23
- Chapter 3. The BEM for Potential Problems in Two Dimensions25
- 3.1 Introduction25
- 3.2 Fundamental solution26
- 3.3 The direct BEM for the Laplace equation28
- 3.4 The direct BEM for the Poisson equation34
- 3.5 Transformation of the domain integrals to boundary integrals36
- 3.6 The BEM for potential problems in anisotropic bodies39
- 3.7 References44
- Problems46
- Chapter 4. Numerical Implementation of the BEM47
- 4.1 Introduction47
- 4.2 The BEM with constant boundary elements49
- 4.3 Evaluation of line integrals53
- 4.4 Evaluation of domain integrals57
- 4.5 The Dual Reciprocity Method for Poisson's equation58
- 4.6 Program LABECON for solving the Laplace equation with constant boundary elements61
- 4.7 Domains with multiple boundaries85
- 4.8 Program LABECONMU for domains with multiple boundaries86
- 4.9 The method of subdomains96
- 4.10 References102
- Problems104
- Chapter 5. Boundary Element Technology105
- 5.1 Introduction105
- 5.2 Linear elements107
- 5.3 The BEM with linear boundary elements111
- 5.4 Evaluation of line integrals on linear elements115
- 5.5 Higher order elements129
- 5.6 Near-singular integrals136
- 5.7 References140
- Problems142
- Chapter 6. Applications143
- 6.1 Introduction143
- 6.2 Torsion of non-circular bars143
- 6.3 Deflection o f elastic membranes174
- 6.4 Bending o f simply supported plates178
- 6.5 Heat transfer problems181
- 6.6 Fluid flow problems187
- 6.7 Conclusions193
- 6.8 References197
- Problems198
- Chapter 7. The BEM for Two-Dimensional Elastostatic Problems201
- 7.1 Introduction201
- 7.2 Equations o f plane elasticity201
- 7.3 Betti ' s reciprocal identity211
- 7.4 Fundamental solution213
- 7.5 Stresses due to a unit concentrated force219
- 7.6 Boundary tractions due to a unit concentrated force220
- 7.7 Integral representation o f the solution221
- 7.8 Boundary integral equations224
- 7.9 Integral representation o f the stresses228
- 7.10 Numerical soiution of the boundary integral equations230
- 7.11 Body forces234
- 7.12 Program ELBECON for solving the plane elastostatic problem with constant boundary elements241
- 7.13 References279
- Problems281
- Appendix A: Derivatives of r285
- Appendix B: Gauss Integration289
- B.1 Gauss integration of a regular function289
- B.2 Integrals with a logarithmic singularity298
- B.3 Double integrals of a regular function298
- B.4 Double singular integrals305
- B.5 References307
- Appendix C: Answers to selected problems309
- Author Index321
- Subject Index325
task was to organize and present the material in such a way so that the book becomes "user-friendly" and easy to comprehend, taking into account only the mathematics and mechanics to which students have been exposed during their undergraduate studies. This effort led to an innovative, in many aspects, way of presenting
BEM, including the derivation of fundamental solutions, the integral representation of the solutions and the boundary integral equations for various governing differential
equations in a simple way minimizing a recourse to mathematics with which the student is not familiar. The indicial and tensorial notations, though they facilitate the author's work and allow to borrow ready to use expressions from the literature, have been avoided in the present book. Nevertheless, all the necessary preliminary mathematical concepts have been included in order to make the book complete and self-sufficient.
Throughout the book, every concept is followed by example problems, which have been worked out in detail and with all the necessary clarifications. Furthermore, each chapter of the book is enriched with problems-to-solve. These problems serve a threefold purpose. Some of them are simple and aim at applying and better understanding the presented theory, some others are more difficult and aim at extending the theory to special cases requiring a deeper understanding of the concepts, and others are small projects which serve the purpose of familiarizing the student with BEM programming and the programs contained in the CD-ROM.
The latter class of problems is very important as it helps students to comprehend the usefulness and effectiveness of the method by solving real-life engineering problems. Through these problems students realize that the BEM is a powerful computational tool and not an alternative theoretical approach for dealing with physical problems. My experience in teaching BEM shows that this is the students' most favorite type of problems. They are delighted to solve them, since they integrate their knowledge and make them feel confident in mastering BEM.
The CD-ROM which accompanies the book contains the source codes of all the computer programs developed in the book, so that the student or the engineer can use them for the solution of a broad class of problems. Among them are general potential problems, problems of torsion, thermal conductivity,
deflection of membranes and plates, flow of incompressible fluids, flow through porous media, in isotropic or anisotropic, homogeneous or composite bodies, as well as plane elastostatic problems in simply or multiply connected domains. As one can readily find out from the variety of the applications, the book is useful for engineers of all disciplines. The author is hopeful that the present book will introduce the reader to BEM in an easy, smooth and pleasant way and also contribute to its
dissemination as a modern robust computational tool for solving engineering problems.