An Introduction to Boundary Element Methods

Prem K. Kythe

• Publisher: CRC Press
• Year: 2020
• Print ISBN: 9780367449148
• eBook ISBN: 9781000142082
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

The finite element and the boundary element methods are the two most important developments in numerical mathematics to occur in this century. Many engineering and mathematics graduate curricula now include a course in boundary element methods. Such a course must cover numerical methods, basic methodology to real problems, and interactive computer usage. Both theory and applications, necessary for applied courses, are available in this new textbook. An Introduction to Boundary Element Methods is logically organized and easy to read. The topics are carefully selected and meticulously presented. Applications are described for use in identifying potential problems and for heat transfer, diffusion equations, linear elasticity, water waves, ocean acoustics, acoustic scattering, aerodynamics, porous media, and simple laminar flows. More than 20 computer subroutines help develop and explain the computational aspect of the subject. Hundreds of figures, exercises, and solved examples supplement text and help clarify important information. The computer programs have been tested on some benchmark problems. Even in single precision the results are more accurate and better than those obtained from available Fortran programs.

1

Mathematical Preliminaries

We will first cite some useful formulas and results, and discuss the notation used in this book. Proofs for these results can be found in textbooks on calculus, linear algebra, differential equations, and analysis. This chapter should, therefore, be either glanced through quickly or studied carefully depending on the reader’s background.

1.1. Results from Calculus

Integration. The formula

${\displaystyle {\int }_0^1{x}^{\mu -1}{(1-x^{\lambda })}^{\nu -1}dx}=\frac{\text{1}}{\text{λ}}B(\mu /\text{λ},\nu ),$

(1.1a)

where ℛμ > 0, ℛv > 0, λ > 0, and B(x, y) is the beta function, yields, e.g., for positive integers m and n

${\displaystyle {\int }_0^1{x}^m{(1-x)}^{n}dx}=\frac{m!n!}{(m+n+1)!}$

(1.1b)

Integration by Parts. Let f, g ∈ C²(a, b), and h(x) ∈ C(a, b). Then using integration by parts we have

${\displaystyle {\int }_a^{b}g\frac{df}{dx}dx}=-{\displaystyle {\int }_a^{b}f\frac{dg}{dx}dx}+fg{{\displaystyle |}}_a^b,$

(1.2a)

${\displaystyle {\int }_{\mathrm{a...}}^{}}$

Table of contents

• Cover
• Title Page
• Copyright Page
• Errata
• Dedication
• The Author
• Table of Contents
• Preface
• Notation
• Introduction
• Chapter 1: Mathematical Preliminaries
• Chapter 2: Variation and Weighted Residual Methods
• Chapter 3: One–Dimensional Problems
• Chapter 4: Fundamental Solutions
• Chapter 5: Potential Problems
• Chapter 6: Linear Elasticity
• Chapter 7: Ocean Acoustics and Helmholtz Equation
• Chapter 8: Some Fluid Flows and D’Arcy Equation
• Chapter 9: Domain Integrals
• Chapter 10: Transient Problems
• Appendix A: Variational Calculus
• Appendix B: Potential Theory
• Appendix C: Gauss Quadrature
• Appendix D: Software Development
• Selected Answers
• Index