_Book_Fast多极边界元法理论及其在工程中的应用

【实例简介】_Book_Fast多极边界元法理论及其在工程中的应用

快速边界元

【实例截图】

【核心代码】

Contents
Preface page xi
Acknowledgments xv
Acronyms Used in This Book xvii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 What Is the Boundary Element Method? 1
1.2 Why the Boundary Element Method? 1
1.3 A Comparison of the Finite Element Method and the
Boundary Element Method 2
1.4 A Brief History of the Boundary Element Method and
Other References 3
1.5 Fast Multipole Method 3
1.6 Applications of the Boundary Element Method in
Engineering 4
1.7 An Example – Bending of a Beam 5
1.8 Some Mathematical Preliminaries 9
1.8.1 Integral Equations 9
1.8.2 Indicial Notation 10
1.8.3 Gauss Theorem 11
1.8.4 The Green’s Identities 12
1.8.5 Dirac δ Function 12
1.8.6 Fundamental Solutions 12
1.8.7 Singular Integrals 13
1.9 Summary 15
Problems 15
2 Conventional Boundary Element Method for Potential
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 The Boundary-Value Problem 17
v
vi Contents
2.2 Fundamental Solution for Potential Problems 18
2.3 Boundary Integral Equation Formulations 19
2.4 Weakly Singular Forms of the Boundary Integral Equations 23
2.5 Discretization of the Boundary Integral Equations for 2D
Problems Using Constant Elements 24
2.6 Using Higher-Order Elements 26
2.6.1 Linear Elements 26
2.6.2 Quadratic Elements 29
2.7 Discretization of the Boundary Integral Equations for 3D
Problems 30
2.8 Multidomain Problems 34
2.9 Treatment of the Domain Integrals 35
2.9.1 Numerical Integration Using Internal Cells 35
2.9.2 Transformation to Boundary Integrals 35
2.9.3 Use of Particular Solutions 36
2.10 Indirect Boundary Integral Equation Formulations 36
2.11 Programming for the Conventional Boundary Element
Method 38
2.12 Numerical Examples 39
2.12.1 An Annular Region 39
2.12.2 Electrostatic Fields Outside Two Conducting Beams 40
2.12.3 Potential Field in a Cube 43
2.12.4 Electrostatic Field Outside a Conducting Sphere 43
2.13 Summary 45
Problems 45
3 Fast Multipole Boundary Element Method for Potential
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1 Basic Ideas in the Fast Multipole Method 48
3.2 Fast Multipole Boundary Element Method for 2D Potential
Problems 50
3.2.1 Multipole Expansion (Moments) 51
3.2.2 Error Estimate for the Multipole Expansion 53
3.2.3 Moment-to-Moment Translation 54
3.2.4 Local Expansion and Moment-to-Local Translation 54
3.2.5 Local-to-Local Translation 56
3.2.6 Expansions for the Integral with the F Kernel 56
3.2.7 Multipole Expansions for the Hypersingular
Boundary Integral Equation 57
3.2.8 Fast Multipole Boundary Element Method
Algorithms and Procedures 58
3.2.9 Preconditioning 64
3.2.10 Estimate of the Computational Complexity 65
Contents vii
3.3 Programming for the Fast Multipole Boundary Element
Method 65
3.3.1 Subroutine fmmmain 67
3.3.2 Subroutine tree 67
3.3.3 Subroutine fmmbvector 69
3.3.4 Subroutine dgmres 70
3.3.5 Subroutine upward 70
3.3.6 Subroutine dwnwrd 70
3.4 Fast Multipole Formulation for 3D Potential Problems 71
3.5 Numerical Examples 74
3.5.1 An Annular Region 74
3.5.2 Electrostatic Fields Outside Conducting Beams 75
3.5.3 Potential Field in a Cube 78
3.5.4 Electrostatic Field Outside Multiple Conducting
Spheres 78
3.5.5 A Fuel Cell Model 79
3.5.6 Image-Based Boundary Element Method Models and
Analysis 80
3.6 Summary 83
Problems 83
4 Elastostatic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.1 The Boundary-Value Problem 86
4.2 Fundamental Solution for Elastostatic Problems 87
4.3 Boundary Integral Equation Formulations 88
4.4 Weakly Singular Forms of the Boundary Integral
Equations 91
4.5 Discretization of the Boundary Integral Equations 92
4.6 Recovery of the Full Stress Field on the Boundary 93
4.7 Fast Multipole Boundary Element Method for 2D
Elastostatic Problems 95
4.7.1 Multipole Expansion for the U Kernel Integral 97
4.7.2 Moment-to-Moment Translation 98
4.7.3 Local Expansion and Moment-to-Local
Translation 98
4.7.4 Local-to-Local Translation 99
4.7.5 Expansions for the T Kernel Integral 99
4.7.6 Expansions for the Hypersingular Boundary Integral
Equation 100
4.8 Fast Multipole Boundary Element Method for 3D
Elastostatic Problems 101
4.9 Fast Multipole Boundary Element Method for Multidomain
Elasticity Problems 104
viii Contents
4.10 Numerical Examples 108
4.10.1 A Cylinder with Pressure Loads 108
4.10.2 A Square Plate with a Circular Hole 110
4.10.3 Multiple Inclusion Problems 111
4.10.4 Modeling of Functionally Graded Materials 113
4.10.5 Large-Scale Modeling of Fiber-Reinforced
Composites 115
4.11 Summary 117
Problems 118
5 Stokes Flow Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.1 The Boundary-Value Problem 120
5.2 Fundamental Solution for Stokes Flow Problems 120
5.3 Boundary Integral Equation Formulations 121
5.4 Fast Multipole Boundary Element Method for 2D Stokes
Flow Problems 124
5.4.1 Multipole Expansion (Moments) for the U Kernel
Integral 126
5.4.2 Moment-to-Moment Translation 127
5.4.3 Local Expansion and Moment-to-Local Translation 127
5.4.4 Local-to-Local Translation 128
5.4.5 Expansions for the T Kernel Integral 128
5.4.6 Expansions for the Hypersingular Boundary Integral
Equation 129
5.5 Fast Multipole Boundary Element Method for 3D Stokes
Flow Problems 130
5.6 Numerical Examples 133
5.6.1 Flow That Is Due to a Rotating Cylinder 133
5.6.2 Shear Flow Between Two Parallel Plates 135
5.6.3 Flow Through a Channel with Many Cylinders 138
5.6.4 A Translating Sphere 141
5.6.5 Large-Scale Modeling of Multiple Particles 142
5.7 Summary 144
Problems 145
6 Acoustic Wave Problems . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.1 Basic Equations in Acoustics 147
6.2 Fundamental Solution for Acoustic Wave Problems 150
6.3 Boundary Integral Equation Formulations 152
6.4 Weakly Singular Forms of the Boundary Integral
Equations 154
6.5 Discretization of the Boundary Integral Equations 156
Contents ix
6.6 Fast Multipole Boundary Element Method for 2D Acoustic
Wave Problems 157
6.7 Fast Multipole Boundary Element Method for 3D Acoustic
Wave Problems 159
6.8 Numerical Examples 163
6.8.1 Scattering from Cylinders in a 2D Medium 163
6.8.2 Radiation from a Pulsating Sphere 164
6.8.3 Scattering from Multiple Scatterers 165
6.8.4 Performance Study of the 3D Fast Multipole
Boundary Element Method Code 166
6.8.5 An Engine-Block Model 167
6.8.6 A Submarine Model 169
6.8.7 An Airbus A320 Model 170
6.8.8 A Human-Head Model 170
6.8.9 Analysis of Sound Barriers – A Half-Space Acoustic
Wave Problem 172
6.9 Summary 174
Problems 174
APPENDIX A: Analytical Integration of the Kernels . . . . . . . . . . . . 177
A.1 2D Potential Boundary Integral Equations 177
A.2 2D Elastostatic Boundary Integral Equations 178
A.3 2D Stokes Flow Boundary Integral Equations 181
APPENDIX B: Sample Computer Programs . . . . . . . . . . . . . . . . . . 184
B.1 A Fortran Code of the Conventional Boundary Element
Method for 2D Potential Problems 184
B.2 A Fortran Code of the Fast Multipole Boundary Element
Method for 2D Potential Problems 192
B.3 Sample Input File and Parameter File 220
References 223
Index 233