Symmetric Galerkin Boundary Element Method

Contents......Page 6
Foreword......Page 12
Preface......Page 14
1.1 Boundary Element Method......Page 18
1.1.1 Approximations and Solution......Page 19
1.1.2 The Green\'s function G(P,Q)......Page 21
1.1.3 Singular and Hypersingular Integrals......Page 22
1.1.4 Numerical Solution: Collocation and Galerkin......Page 23
1.1.5 Symmetric Galerkin BEM......Page 24
1.2 An Application Example: Automotive Electrocoating......Page 25
1.2.1 Engineering Optimization......Page 26
1.2.2 Electrocoating Simulation......Page 27
1.3 Visualization......Page 28
1.3.1 Virtual Reality......Page 29
1.3.3 The MechVR......Page 31
1.4 Other Boundary Techniques......Page 32
1.4.2 Meshless and Mesh-Reduction Methods......Page 33
1.5 A Brief History of Galerkin BEM......Page 36
2.1 Boundary Potential Equation......Page 39
2.2 Boundary Flux Equation......Page 44
2.3 Elasticity......Page 46
2.4.1 Approximations......Page 49
2.4.2 Collocation......Page 51
2.4.3 Galerkin Approximation......Page 53
2.5 Hypersingular Integration: an example......Page 55
2.5.1 Collocation: C[sup(1)] Condition......Page 56
2.5.2 Galerkin: C[sup(0)]......Page 58
3.1 Introduction......Page 61
3.2.1 Coincident Integration......Page 64
3.2.2 Coincident: Symbolic Computation......Page 67
3.2.3 Adjacent Integration......Page 69
3.2.4 Cancellation of log(ε[sup(2)])......Page 72
3.2.5 Adjacent: shape function expansion......Page 73
3.2.6 Numerical Tests......Page 74
3.3 Higher Order Interpolation......Page 76
3.3.1 Integral of G......Page 77
3.3.2 Integral of ∂G/∂n and ∂G/∂N......Page 78
3.4 Other Green\'s functions......Page 79
3.5 Corners......Page 80
3.6 Nonlinear Boundary Conditions......Page 81
3.7 Concluding Remarks......Page 83
4.1 Preliminaries......Page 84
4.2 Linear Element Analysis......Page 88
4.2.1 Nonsingular Integration......Page 89
4.2.2 Coincident Integration......Page 90
4.2.3 Coincident CPV integral......Page 97
4.2.4 Edge Adjacent Integration......Page 98
4.2.5 Vertex Adjacent Integration......Page 103
4.2.6 Proof of Cancellation......Page 105
4.3 Higher Order Interpolation......Page 107
4.4 Hypersingular Boundary Integral: Quadratic Element......Page 108
4.4.2 r Expansion......Page 109
4.4.3 First Integration......Page 110
4.4.4 Edge Integration......Page 111
4.6 Anisotropic Elasticity......Page 112
4.6.1 Anisotropic Elasticity Boundary Integral Formulation......Page 114
4.6.2 T Kernel: Coincident Integration......Page 115
4.6.3 Spherical Coordinates......Page 120
4.6.5 Edge Adjacent Integration......Page 121
5.1 Introduction......Page 123
5.2 Gradient Equations......Page 126
5.2.1 Limit Evaluation in two dimensions......Page 128
5.2.2 Example: Surface Stress......Page 132
5.2.3 Limit Evaluation in three dimensions......Page 135
5.3.1 Introduction......Page 137
5.3.2 Hermite Interpolation......Page 138
5.3.3 Iterative Solution......Page 139
6.1 Introduction......Page 143
6.2 Axisymmetric Formulation......Page 145
6.3 Singular Integration......Page 148
6.3.2 Coincident Integration......Page 149
6.3.4 Log Integral Transformation......Page 150
6.3.5 Analytic integration formulas......Page 152
6.4.1 Gradient Equations......Page 153
6.4.2 Coincident Integration......Page 154
6.5 Numerical Results......Page 156
7.1 Introduction......Page 158
7.2 Symmetric Galerkin Formulation......Page 160
7.3 Interface and Symmetry......Page 162
7.3.1 Multiple Interfaces......Page 164
7.3.4 Computational Aspects......Page 165
7.4 Numerical Examples......Page 166
7.5 Remarks......Page 168
8.1 Introduction......Page 170
8.2 Boundary Integral Equations......Page 171
8.3 Galerkin Residuals and Error Estimates......Page 173
8.4 Self Adaptive Strategy......Page 174
8.4.2 Element Refinement Criterion......Page 175
8.4.3 Global Error Estimation......Page 176
8.5 Numerical Example......Page 177
8.6 BEAN Code......Page 180
9.1 Introduction......Page 184
9.2 Fracture parameters: Stress intensity factors (SIFs) and T-stress......Page 185
9.3.1 Basic SGBEM formulation for 2D elasticity......Page 186
9.3.2 Fracture analysis with the SGBEM......Page 188
9.4 On Computational Methods for Evaluating Fracture Parameters......Page 191
9.5.1 Basic Formulation......Page 192
9.5.2 Auxiliary Fields for T-stress......Page 193
9.5.3 Determination of T-stress......Page 195
9.5.4 Auxiliary Fields for SIFs......Page 196
9.5.5 Determination of SIFs......Page 197
9.5.6 Crack-tip elements......Page 198
9.5.7 Numerical implementation of the M-integral......Page 199
9.6.1 Infinite plate with an interior inclined crack......Page 200
9.6.2 Slanted edge crack in a finite plate......Page 204
9.6.3 Multiple interacting cracks......Page 205
9.6.4 Various fracture specimen configurations......Page 206
10.1 Introduction......Page 210
10.2 Steady State Heat Conduction......Page 211
10.2.2 Symmetric Galerkin Formulation......Page 212
10.2.3 Treatment of Singular and Hypersingular Integrals......Page 215
10.3 Evaluation of singular double integrals......Page 216
10.3.1 Coincident Integration......Page 217
10.3.2 Edge Adjacent Integration......Page 223
10.3.3 Vertex Adjacent Integration......Page 225
10.3.4 Numerical Example......Page 227
10.4 Transient heat conduction in FGMs......Page 229
10.4.1 Basic Equations......Page 230
10.4.2 Green\'s Function......Page 231
10.4.3 Laplace Transform BEM (LTBEM) Formulation......Page 232
10.4.4 Numerical Implementation of the 3D Galerkin BEM......Page 234
10.4.5 Numerical Inversion of the Laplace Transform......Page 235
10.4.6 Numerical Examples......Page 236
10.5 Concluding Remarks......Page 237
11.1 Introduction......Page 239
11.2.3 Geometry......Page 240
11.2.4 Boundary Conditions (BCs)......Page 241
11.2.6 Results......Page 242
11.3.1 Menus......Page 243
11.4.1 Menus......Page 244
11.5 General Instructions......Page 246
11.6 Troubleshooting......Page 247
11.6.1 Error Message Meanings......Page 248
11.7 Sample Problems......Page 249
A.1 Dirac Delta function......Page 252
A.3 Derivative, Gradient, Divergence and Laplacian......Page 253
A.6 Green\'s Identities......Page 254
A.8 Free Space Green\'s function......Page 255
B.2 Gaussian rule for One-dimensional non-singular integration......Page 257
C.1 Maple Script: Coincident......Page 261
C.2 Maple Script: Edge Adjacent......Page 263
C.3 Maple Script: Vertex Adjacent......Page 264
References......Page 266
E......Page 284
W......Page 285