The fast solution of boundary integral equations / Sergej Rjasanow, Olaf Steinbach.

Format:

Book

Author/Creator:

Rjasanow, Sergej.

Contributor:

Steinbach, Olaf, 1967-

Alumni and Friends Memorial Book Fund.

Series:

Mathematical and analytical techniques with applications to engineering

Language:

English

Subjects (All):

Boundary element methods.

Physical Description:

xi, 279 pages : illustrations ; 25 cm.

Place of Publication:

New York ; London : Springer, [2007]

Summary:

The use of surface potentials to describe solutions of partial differential equations goes back to the middle of the 19th century. Numerical approximation procedures, known today as Boundary Element Methods (BEM), have been developed in the physics and engineering community since the 1950s. These methods turn out to be powerful tools for numerical studies of various physical phenomena which can be described mathematically by partial differential equations.

The Fast Solution of Boundary Integral Equations provides a detailed description of fast boundary element methods which are based on rigorous mathematical analysis. In particular, a symmetric formulation of boundary integral equations is used, Galerkin discretisation is discussed, and the necessary related stability and error estimates are derived. For the practical use of boundary integral methods, efficient algorithms together with their implementation are needed. The authors therefore describe the Adaptive Cross Approximation Algorithm, starting from the basic ideas and proceeding to their practical realization. Numerous examples representing standard problems are given which underline both theoretical results and the practical relevance of boundary element methods in typical computations.

The most prominent example is the potential equation (Laplace equation), which is used to model physical phenomena in electromagnetism, gravitation theory, and in perfect fluids. A further application leading to the Laplace equation is the model of steady state heat flow. One of the most popular applications of the BEM is the system of linear elastostatics, which can be considered in both bounded and unbounded domains. A simple model for a fluid flow, the Stokes system, can also be solved by the use of the BEM. The most important examples for the Helmholtz equation are the acoustic scattering and the sound radiation.

Contents:

1 Boundary Integral Equations 1

1.1 Laplace Equation 2

1.1.1 Interior Dirichlet Boundary Value Problem 10

1.1.2 Interior Neumann Boundary Value Problem 13

1.1.3 Mixed Boundary Value Problem 17

1.1.4 Robin Boundary Value Problem 19

1.1.5 Exterior Dirichlet Boundary Value Problem 21

1.1.6 Exterior Neumann Boundary Value Problem 22

1.1.7 Poisson Problem 24

1.1.8 Interface Problem 26

1.2 Lame Equations 27

1.2.1 Dirichlet Boundary Value Problem 35

1.2.2 Neumann Boundary Value Problem 36

1.2.3 Mixed Boundary Value Problem 37

1.3 Stokes System 40

1.4 Helmholtz Equation 44

1.4.1 Interior Dirichlet Boundary Value Problem 49

1.4.2 Interior Neumann Boundary Value Problem 50

1.4.3 Exterior Dirichlet Boundary Value Problem 52

1.4.4 Exterior Neumann Boundary Value Problem 54

2 Boundary Element Methods 59

2.1 Boundary Elements 59

2.2 Basis Functions 61

2.3 Laplace Equation 65

2.3.1 Interior Dirichlet Boundary Value Problem 65

2.3.2 Interior Neumann Boundary Value Problem 72

2.3.3 Mixed Boundary Value Problem 77

2.3.4 Interface Problem 84

2.4 Lame Equations 87

2.5 Helmholtz Equation 91

2.5.1 Interior Dirichlet Problem 91

2.5.2 Interior Neumann Problem 93

2.5.3 Exterior Dirichlet Problem 97

2.5.4 Exterior Neumann Problem 98

3 Approximation of Boundary Element Matrices 101

3.1 Hierarchical Matrices 101

3.1.1 Motivation 101

3.1.2 Hierarchical clustering 108

3.2 Block Approximation Methods 112

3.2.1 Analytic Form of Adaptive Cross Approximation 112

3.2.2 Algebraic Form of Adaptive Cross Approximation 119

4 Implementation and Numerical Examples 131

4.1 Geometry Description 131

4.1.1 Unit Sphere 131

4.1.2 TEAM Problem 10 131

4.1.3 TEAM Problem 24 134

4.1.4 Relay 134

4.1.5 Exhaust manifold 135

4.2 Laplace Equation 135

4.2.1 Analytical solutions 135

4.2.2 Discretisation, Approximation and Iterative Solution 137

4.2.3 Generation of Matrices 139

4.2.4 Interior Dirichlet Problem 143

4.2.5 Interior Neumann Problem 149

4.2.6 Interior Mixed Problem 155

4.2.7 Inhomogeneous Interface Problem 160

4.3 Linear Elastostatics 162

4.3.1 Generation of Matrices 162

4.3.2 Relay 163

4.3.3 Foam 166

4.4 Helmholtz Equation 168

4.4.1 Analytical Solutions 169

4.4.2 Discretisation, Approximation and Iterative Solution 169

4.4.3 Generation of Matrices 170

4.4.4 Interior Dirichlet Problem 171

4.4.5 Interior Neumann Problem 185

4.4.6 Exterior Dirichlet Problem 191

4.4.7 Exterior Neumann Problem 196

A Mathematical Foundations 199

A.1 Function Spaces 199

A.2 Fundamental Solutions 208

A.2.1 Laplace Equation 208

A.2.2 Lame System 209

A.2.3 Stokes System 210

A.2.4 Helmholtz Equation 212

A.3 Mapping Properties 213

B Numerical Analysis 225

B.1 Variational Methods 225

B.2 Approximation Properties 231

C Numerical Algorithms 239

C.1 Numerical Integration 239

C.2 Analytic Integration 244

C.2.1 Single Layer Potential for the Laplace operator 246

C.2.2 Double Layer Potential for the Laplace operator 249

C.2.3 Linear Elasticity Single Layer Potential 252

C.3 Iterative Solution Methods 256

C.3.1 Conjugate Gradient Method (CG) 256

C.3.2 Generalised Minimal Residual Method (GMRES) 263.

Notes:

Includes bibliographical references (pages [269]-275) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

9780387340418

0387340416

OCLC:

76935732

Online:

Publisher description