Direct and Indirect Boundary Integral Equation Methods

by Constanda; Christian

• ISBN13: 9780849306396
• ISBN10: 0849306396
• Format: Hardcover
• Copyright: 1999-09-28
• Publisher: Chapman & Hall/

Summary

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

Preface
The Laplace Equation	1	(53)
Notation and prerequisites	1	(6)
The fundamental boundary value problems	7	(2)
Green's formulae	9	(3)
Uniqueness theorems	12	(2)
The harmonic potentials	14	(3)
Properties of the boundary operators	17	(7)
The classical indirect method	24	(5)
The alternative indirect method	29	(2)
The modified indirect method	31	(6)
The refined indirect method	37	(3)
The direct method	40	(8)
The substitute direct method	48	(6)
Plane Strain	54	(67)
Notation and prerequisites	54	(4)
The fundamental boundary value problems	58	(3)
The Betti and Somigliana formulae	61	(5)
Uniqueness theorems	66	(1)
The elastic potentials	67	(4)
Properties of the boundary operators	71	(14)
The classical indirect method	85	(7)
The alternative indirect method	92	(3)
The modified indirect method	95	(7)
The refined indirect method	102	(3)
The direct method	105	(9)
The substitute direct method	114	(7)
Bending of Elastic Plates	121	(24)
Notation and Prerequisites	121	(5)
The fundamental boundary value problems	126	(3)
The Betti and Somigliana formulae	129	(6)
Uniqueness theorems	135	(1)
The plate potentials	136	(3)
Properties of the boundary operators	139	(4)
Boundary integral equation methods	143	(2)
Which Method?	145	(9)
Notation and prerequisites	145	(1)
Connections between the indirect methods	146	(4)
Connections between the direct and indirect methods	150	(2)
Overall view and conclusions	152	(2)
Appendix	154	(46)
A1. Geometry of the boundary curve	154	(3)
A2. Properties of the boundary layer	157	(7)
A3. Integrals with singular kernels	164	(12)
A4. Potential-type functions	176	(6)
A5. Other potential-type functions	182	(8)
A6. Complex singular kernels	190	(5)
A7. Singular integral equations	195	(5)
References	200