Fast Multipole Boundary Element Method: Theory and Applications in Engineering

✍ Scribed by Yijun Liu

• Publisher: Cambridge University Press
• Year: 2009
• Tongue: English
• Leaves: 255
• Edition: 1
• Category: Library

✦ Synopsis

The fast multipole method is one of the most important algorithms in computing developed in the 20th century. Along with the fast multipole method, the boundary element method (BEM) has also emerged, as a powerful method for modeling large-scale problems. BEM models with millions of unknowns on the boundary can now be solved on desktop computers using the fast multipole BEM. This is the first book on the fast multipole BEM, which brings together the classical theories in BEM formulations and the recent development of the fast multipole method. Two- and three-dimensional potential, elastostatic, Stokes flow, and acoustic wave problems are covered, supplemented with exercise problems and computer source codes. Applications in modeling nanocomposite materials, bio-materials, fuel cells, acoustic waves, and image-based simulations are demonstrated to show the potential of the fast multipole BEM. This book will help students, researchers, and engineers to learn the BEM and fast multipole method from a single source.

✦ Table of Contents

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 13
Acknowledgments......Page 17
Acronyms Used in This Book......Page 19
1.2 Why the Boundary Element Method?......Page 21
1.3 A Comparison of the Finite Element Method and the Boundary Element Method......Page 22
1.5 Fast Multipole Method......Page 23
1.6 Applications of the Boundary Element Method in Engineering......Page 24
1.7 An Example – Bending of a Beam......Page 25
1.8.1 Integral Equations......Page 29
1.8.2 Indicial Notation......Page 30
1.8.3 Gauss Theorem......Page 31
1.8.6 Fundamental Solutions......Page 32
1.8.7 Singular Integrals......Page 33
Problems......Page 35
2.1 The Boundary-Value Problem......Page 37
2.2 Fundamental Solution for Potential Problems......Page 38
2.3 Boundary Integral Equation Formulations......Page 39
2.4 Weakly Singular Forms of the Boundary Integral Equations......Page 43
2.5 Discretization of the Boundary Integral Equations for 2D Problems Using Constant Elements......Page 44
2.6.1 Linear Elements......Page 46
2.6.2 Quadratic Elements......Page 49
2.7 Discretization of the Boundary Integral Equations for 3D Problems......Page 50
2.8 Multidomain Problems......Page 54
2.9.2 Transformation to Boundary Integrals......Page 55
2.10 Indirect Boundary Integral Equation Formulations......Page 56
2.11 Programming for the Conventional Boundary Element Method......Page 58
2.12.1 An Annular Region......Page 59
2.12.2 Electrostatic Fields Outside Two Conducting Beams......Page 60
2.12.4 Electrostatic Field Outside a Conducting Sphere......Page 63
Problems......Page 65
3 Fast Multipole Boundary Element Method for Potential Problems......Page 67
3.1 Basic Ideas in the Fast Multipole Method......Page 68
3.2 Fast Multipole Boundary Element Method for 2D Potential Problems......Page 70
3.2.1 Multipole Expansion (Moments)......Page 71
3.2.2 Error Estimate for the Multipole Expansion......Page 73
3.2.4 Local Expansion and Moment-to-Local Translation......Page 74
3.2.6 Expansions for the Integral with the F Kernel......Page 76
3.2.7 Multipole Expansions for the Hypersingular Boundary Integral Equation......Page 77
3.2.8 Fast Multipole Boundary Element Method Algorithms and Procedures......Page 78
3.2.9 Preconditioning......Page 84
3.3 Programming for the Fast Multipole Boundary Element Method......Page 85
3.3.2 Subroutine tree......Page 87
3.3.3 Subroutine fmmbvector......Page 89
3.3.6 Subroutine dwnwrd......Page 90
3.4 Fast Multipole Formulation for 3D Potential Problems......Page 91
3.5.1 An Annular Region......Page 94
3.5.2 Electrostatic Fields Outside Conducting Beams......Page 95
3.5.4 Electrostatic Field Outside Multiple Conducting Spheres......Page 98
3.5.5 A Fuel Cell Model......Page 99
3.5.6 Image-Based Boundary Element Method Models and Analysis......Page 100
Problems......Page 103
4 Elastostatic Problems......Page 105
4.1 The Boundary-Value Problem......Page 106
4.2 Fundamental Solution for Elastostatic Problems......Page 107
4.3 Boundary Integral Equation Formulations......Page 108
4.4 Weakly Singular Forms of the Boundary Integral Equations......Page 111
4.5 Discretization of the Boundary Integral Equations......Page 112
4.6 Recovery of the Full Stress Field on the Boundary......Page 113
4.7 Fast Multipole Boundary Element Method for 2D Elastostatic Problems......Page 115
4.7.1 Multipole Expansion for the U Kernel Integral......Page 117
4.7.3 Local Expansion and Moment-to-Local Translation......Page 118
4.7.5 Expansions for the T Kernel Integral......Page 119
4.7.6 Expansions for the Hypersingular Boundary Integral Equation......Page 120
4.8 Fast Multipole Boundary Element Method for 3D Elastostatic Problems......Page 121
4.9 Fast Multipole Boundary Element Method for Multidomain Elasticity Problems......Page 124
4.10.1 A Cylinder with Pressure Loads......Page 128
4.10.2 A Square Plate with a Circular Hole......Page 130
4.10.3 Multiple Inclusion Problems......Page 131
4.10.4 Modeling of Functionally Graded Materials......Page 133
4.10.5 Large-Scale Modeling of Fiber-Reinforced Composites......Page 135
4.11 Summary......Page 137
Problems......Page 138
5 Stokes Flow Problems......Page 139
5.2 Fundamental Solution for Stokes Flow Problems......Page 140
5.3 Boundary Integral Equation Formulations......Page 141
5.4 Fast Multipole Boundary Element Method for 2D Stokes Flow Problems......Page 144
5.4.1 Multipole Expansion (Moments) for the U Kernel Integral......Page 146
5.4.3 Local Expansion and Moment-to-Local Translation......Page 147
5.4.5 Expansions for the T Kernel Integral......Page 148
5.4.6 Expansions for the Hypersingular Boundary Integral Equation......Page 149
5.5 Fast Multipole Boundary Element Method for 3D Stokes Flow Problems......Page 150
5.6.1 Flow That Is Due to a Rotating Cylinder......Page 153
5.6.2 Shear Flow Between Two Parallel Plates......Page 155
5.6.3 Flow Through a Channel with Many Cylinders......Page 158
5.6.4 A Translating Sphere......Page 161
5.6.5 Large-Scale Modeling of Multiple Particles......Page 162
5.7 Summary......Page 164
Problems......Page 165
6 Acoustic Wave Problems......Page 166
6.1 Basic Equations in Acoustics......Page 167
6.2 Fundamental Solution for Acoustic Wave Problems......Page 170
6.3 Boundary Integral Equation Formulations......Page 172
6.4 Weakly Singular Forms of the Boundary Integral Equations......Page 174
6.5 Discretization of the Boundary Integral Equations......Page 176
6.6 Fast Multipole Boundary Element Method for 2D Acoustic Wave Problems......Page 177
6.7 Fast Multipole Boundary Element Method for 3D Acoustic Wave Problems......Page 179
6.8.1 Scattering from Cylinders in a 2D Medium......Page 183
6.8.2 Radiation from a Pulsating Sphere......Page 184
6.8.3 Scattering from Multiple Scatterers......Page 185
6.8.4 Performance Study of the 3D Fast Multipole Boundary Element Method Code......Page 186
6.8.5 An Engine-Block Model......Page 187
6.8.6 A Submarine Model......Page 189
6.8.8 A Human-Head Model......Page 190
6.8.9 Analysis of Sound Barriers – A Half-Space Acoustic Wave Problem......Page 192
Problems......Page 194
A.1 2D Potential Boundary Integral Equations......Page 197
A.2 2D Elastostatic Boundary Integral Equations......Page 198
A.3 2D Stokes Flow Boundary Integral Equations......Page 201
B.1 A Fortran Code of the Conventional Boundary Element Method for 2D Potential Problems......Page 204
B.2 A Fortran Code of the Fast Multipole Boundary Element Method for 2D Potential Problems......Page 212
B.3 Sample Input File and Parameter File......Page 240
References......Page 243
Index......Page 253