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BOUNDARY ELEMENTS: THEORY AND APPLICATIONS
The author's ambition for this publication was to make BEM accessible to the student as well as to the professional engineer. For this reason, his main
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.
Introduction . Scope of the book. Boundary elements and finite elements. Historical development of the BEM. Structure of the book. CD-ROM contents. Preliminary Mathematical Concepts . The Gauss-Green theorem. The divergence theorem of Gauss. Green's second identity. The adjoint operator. The Dirac delta function. The BEM for Potential Problems in Two Dimensions . Fundamental solution. The direct BEM for the Laplace equation. The direct BEM for the Poisson equation. Transformation of the domain intergrals to boundary intergrals. The BEM for potential problems in anisotropic bodies. Numerical Implementation of the BEM . The BEM with constant boundary elements. Evaluation of line integrals. Evaluation of domain integrals. The dual Reciprocity Method for Poisson's equation. Program LABECON for solving the Laplace equation with constant boundary elements. Domains with multiple boundaries. Program LABECONMU for domains with multiple boundaries. The method of subdomains. Boundary Element Technology . Linear elements. The BEM with linear boundary elements. Evaluation of line integrals on linear elements. Higher order elements. Near-singular intergrals. Applications . Torsion of non-circular bars. Deflection of elastic membranes. Bending of simply supported plates. Heat transfer problems. Fluid flow problems. The BEM for Two-Dimensional Elastostatic Problems . Equations of plane elasticity. Betti's reciprocal identity. Fundamental solution. Stresses due to a unit concentrated force. Boundary tractions due to a unit concentrated force. Intergral representation of the solution. Boundary intergral equations. Intergral representation of the stresses. Numerical solution of the boundary intergral equations. Body forces. Program ELBECON for solving the plane elastostatic problem with constant boundary elements.