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Direct and Indirect Boundary Integral Equation Methods




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ISBN 9780849306396
Published September 28, 1999 by Chapman and Hall/CRC
216 Pages

 
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Book Description

The computational power currently available means that practitioners can find extremely accurate approximations to the solutions of more and more sophisticated mathematical models-providing they know the right analytical techniques. In relatively simple terms, this book describes a class of techniques that fulfill this need by providing closed-form solutions to many boundary value problems that arise in science and engineering.
Boundary integral equation methods (BIEM's) have certain advantages over other procedures for solving such problems: BIEM's are powerful, applicable to a wide variety of situations, elegant, and ideal for numerical treatment. Certain fundamental constructs in BIEM's are also essential ingredients in boundary element methods, often used by scientists and engineers.
However, BIEM's are also sometimes more difficult to use in plane cases than in their three-dimensional counterparts. Consequently, the full, detailed BIEM treatment of two-dimensional problems has been largely neglected in the literature-even when it is more than marginally different from that applied to the corresponding three-dimensional versions.
This volume discusses three typical cases where such differences are clear: the Laplace equation (one unknown function), plane strain (two unknown functions), and the bending of plates with transverse shear deformation (three unknown functions). The author considers each of these with Dirichlet, Neumann, and Robin boundary conditions. He subjects each to a thorough investigation-with respect to the existence and uniqueness of regular solutions-through several BIEM's. He proposes suitable generalizations of the concept of logarithmic capacity for plane strain and bending of plates, then uses these to identify contours where non-uniqueness may occur. In the final section, the author compares and contrasts the various solution representations, links them by means of boundary operators, and evaluates them for their suitability for numeric computation.

Table of Contents

Introduction
THE LAPLACE EQUATION
Notation and Prerequisites
The Fundamental Boundary Value Problems
Green's Formulae
Uniqueness Theorems
The Harmonic Potentials
A Classification of Boundary Integral Equation Methods
The Classical Indirect Method
The Alternative Indirect Method
The Modified Indirect Method
The Refined Indirect Method
The Direct Method
The Substitute Direct Method
PLANE STRAIN
Notation and Prerequisites
The Fundamental Boundary Value Problems
The Betti and Somigliana Formulae
Uniqueness Theorems
The Elastic Potentials
Properties of the Boundary Operators
The Classical Indirect Method
The Alternative Indirect Method
The Modified Indirect Method
The Refined Indirect Method
The Direct Method
The Substitute Direct Method
BENDING OF ELASTIC PLATES
Notation and Prerequisites
The Fundamental Boundary Value Problems
The Betti and Somigliana Formulae
Uniqueness Theorems
The Plate Potentials
Properties of the Boundary Operators
Boundary Integral Equation Methods
WHICH METHOD?
Notation and Prerequisites
Connections between the Indirect Methods
Connections between the Direct and Indirect Methods
Overall View and Conclusions
APPENDIX

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Author(s)

Biography

Constanda\, Christian

Reviews

"The text is written clearly and the proofs are given in detail."
M. Aron, Proceedings of the Edinburgh Mathematical Society, Vol. 44, 445-448, 2001

"…the book offers a comprehensive treatment of the subject matter and constitutes a very useful source of information for mathematicians and other scientists interested in boundary integral equation methods.
M. Aron, Proceedings of the Edinburgh Mathematical Society, Vol. 44, 445-448, 2001