السلام عليكم

كتاب

Boundary Elements

BOUNDARY ELEMENTS:

Theory and Applications

BOUNDARY ELEMENTS: Theory and Applications

John T. Katsikadelis

Department of Civil Engineering, National Technical University of Athens,

Athens, Greece

Preface

The last three decades have been marked by the evolution of electronic computers and an enormous and wide-spread availability of computational power. This has boosted the development of computational methods and their application in engi- neering and in the analysis and design of structures, which extend from bridges to aircrafts and from machine elements to tunnels and the human body. New scientific

subfields were generated in all engineering disciplines being described as "Compu- tational", e.g. Computational Mechanics, Computational Fluid Mechanics, Com- putational Structural Analysis, Computational Structural Dynamics etc. The Finite Element Method (FEM) and the Boundary Element Method (BEM) are the most popular of the computational methods. While the FEM has been long established and is most well known in the engineering community, the BEM appeared later offering new computational capabilities with its effectiveness, accuracy and low computational cost.

Although the BEM is taught as a regular course at an ever increasing number of universities, there is a noticeable lack of a textbook which could help students as well as professional engineers to understand the method, the underlying theory and its application to engineering problems. An essential reason is that BEM courses are taught mainly as advanced graduate courses, and therefore much of the under- lying fundamental knowledge of mathematics and mechanics is not covered in the respective undergraduate courses. Thus, the existing books on BEM are addressed rather to academia and researchers who, somehow, have already been exposed to the BEM than to students following a BEM course for the first time and engineers who are using boundary element software in industry.

This observation stimulated the author to write the book at hand. His research in the development of BEM during the last 25 years as well as the experience he ac- quired by teaching for many years the course of Boundary Elements at the Civil Engineering Department of the National Technical University of Athens, Greece,

justify this endeavor. The author's ambition was to make BEM accessible to the student as well 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, includ- ing the derivation of fundamental solutions, the integral representation of the solu- tions and the boundary integral equations for various governing differential equa- tions 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 authors' 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.

In writing the book, topics requiring a detailed study for a deep.and thorough un- derstanding of the BEM, have been emphasized. These are:

(i) The formulation of the physical problem. (ii) The formulation of the mathematical problem, which is expressed by the

governing differential equations and the boundary conditions (boundary value problem).

(iii) The conversion of the differential equations to boundary integral equations. This topic familiarizes the reader with special particular solutions, the so- called fundamental solutions, shows how they are utilized and helps to com- prehend their singular behavior.

(iv) The transformation of domain integrals to boundary line integrals or their elimination, in order to obtain pure boundary integral equations.

(v) The numerical solution of the boundary integral equations. This topic, which covers a significant part of the book, deals with the numerical implementa- tion of BEM rendering a powerful computational tool for solving realistic engineering problems. It contains the discretization of the boundary into elements, the modeling of its geometry, the approximation of the boundary quantities, as well as the techniques for the evaluation of regular and singular line integrals and in general the procedure for approximating the actual problem by a system of linear algebraic equations.

(vi) A detailed description of the FORTRAN programs, which implement the nu- merical procedure for the various problems. The reader is provided with all the necessary information and the know-how so that he can write his own BEM-based computer programs for problems other than those included in the book.

(vii) The use of the aforementioned computer programs for the solution of repre- sentative problems and the study of the behavior of the corresponding physi- cal system.

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 un- derstanding the presented theory, some others are more difficult and aim at extend- ing 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 inte- grate 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 mem- branes and plates, flow of incompressible fluids, flow through porous media, in isotropic or anisotropic, homogeneous or composite bodies, as well as plane elas- tostatic 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 dis- ciplines. 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 modem robust computational tool for solving engineering problems.

In closing, the author would like to express his sincere thanks to his former student and Visiting Assistant Professor at Texas A&M University Dr. Filis Kokkinos for his carefully reading the manuscript and his suggestions for constructive changes. His critic and comments are greatly appreciated. Thanks also belong to my doctoral student Mr. G.C. Tsiatas, M.Sc., for checking the numerical results and the deriva- tion of several expressions.

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