Advanced Numerical Simulation Methods from CAD data directly to simulation results

【实例截图】

【核心代码】

Table of contents
Preface xi
About the author xv
1 Introduction 1
1 A brief history of simulation 1
1.1 The world’s first simulation 1
1.2 Emergence of mathematics and mechanics 3
1.3 Computer age 4
2 Basic steps in simulation 9
2.1 Geometry description 9
2.2 Approximation of the unknown 11
2.3 Solution 12
2.4 Recovery of the results 12
3 A change of paradigm: towards a more efficient and
accurate simulation 12
4 Organization of the text 13
2 Stage 1: Basis functions 17
1 One-dimensional basis functions 17
1.1 Lagrange and Serendipity functions 17
1.2 From B-splines to NURBS 22
2 Two-dimensional basis functions 33
2.1 Lagrange and Serendipity functions 33
2.2 B-splines 38
2.3 NURBS 38
2.4 T-splines 40
3 Programming 44
4 The NURBS toolkit 45
5 Summary and conclusions 51
3 Stage 2: Geometry 55
1 Coordinate systems 55
1.1 Coordinate transformation 55
2 Curves 56
2.1 Mapping with Serendipity/Lagrange basis functions 56
2.2 Mapping with NURBS 57
vi Table of contents
3 Programming 58
3.1 NURBS toolkit 58
3.2 Geometry functions 59
3.3 Examples 63
3.4 Example 1: Circular arc 63
3.5 Example 2: Horseshoe tunnel 65
3.6 Example 3: Plate with hole 67
4 Surfaces 68
4.1 Mapping with Serendipity/Lagrange basis functions 69
4.2 Mapping with NURBS basis functions 70
4.3 Programming 71
5 Surface of revolution 73
5.1 Example 1: Cylindrical surface 74
5.2 Example 2: Spherical surface 75
5.3 Example 3: Bell shaped surface 77
6 Lofted surfaces 79
7 NURBS surfaces with cutouts 81
7.1 Analysis suitable trimmed NURBS surfaces 82
8 Infinite NURBS patch 87
8.1 Example 92
9 Summary and conclusions 93
4 Stage 3: Computer Aided Design 95
1 Introduction 95
2 IGES data structure 98
3 How CAD programs describe geometry – entity types 100
3.1 Line entity (110) 100
3.2 Surface of revolution entity (120) 100
3.3 Rational B-spline entity (126) 101
3.4 Rational B-spline surface entity (128) 101
3.5 Boundary entity (141) 101
4 NURBS surfaces 102
5 Trimmed NURBS surfaces 104
6 Summary and conclusions 113
5 Stage 4: Introduction to numerical simulation 117
1 One-dimensional simulation 117
1.1 Ritz method 119
1.2 Approximation 121
2 Steps in the simulation 126
2.1 Description of the geometry 126
2.2 Description of known values 126
2.3 Convergence tests 127
2.4 Approximation of unknown 128
2.5 P-refinement or order elevation 128
2.6 H-refinement, the Finite Element Method 128
2.7 Knot insertion, isogeometric method 132
Table of contents vii
2.8 K-refinement 132
2.9 Summary and conclusions 133
3 2-D simulation, plane stress and plane strain 135
3.1 Boundary Conditions (BC) 138
3.2 Using one NURBS patch 140
3.3 Comparison with classical FEM 141
3.4 Example 142
3.5 Multiple NURBS patches 143
3.6 Bezièr elements 147
3.7 Trimmed NURBS patches 148
3.8 Convergence test 150
4 Summary and conclusions 150
6 Stage 5: Plates and shells 153
1 Kirchhoff plate 153
1.1 Plates 154
1.2 Examples 158
2 Kirchhoff shells 161
2.1 Example 1: Scordelis roof 162
2.2 Example 2: Trimmed Scordelis roof 164
2.3 Example 3: Arched Scordelis roof 166
3 Multiple patches 168
3.1 Assembly 168
3.2 Example 169
4 Summary and conclusions 172
7 Stage 6: Integral equations 175
1 Introduction 175
2 Trefftz method 176
2.1 Example 180
2.2 Conclusions 183
3 Integral equations 183
3.1 Theorem of Betti 187
3.2 Rigid body trick 191
3.3 Conclusions 193
4 Numerical solution of integral equations 193
4.1 Nyström method 193
4.2 Galerkin method 196
4.3 Collocation 196
4.4 Discretisation 197
5 Summary and conclusions 199
8 Stage 7: The boundary element method for plane problems 201
1 Introduction 201
2 Classical isoparametric approach 201
2.1 Numerical evaluation of integrals 203
viii Table of contents
3 NURBS based approach 206
3.1 Boundary conditions 210
4 Assembly of multiple patches 211
4.1 Pure Neumann problem 211
4.2 Mixed Neumann/Dirichlet problem 211
4.3 Symmetry 212
5 Postprocessing 213
5.1 Results on the boundary 213
5.2 Results inside the domain 214
6 Programming 216
7 Examples 227
7.1 Potential problem: Flow past isolator 227
7.2 Elasticity problem: Circular excavation in infinite domain 229
7.3 Practical example: Horseshoe tunnel 231
8 Conclusions 233
9 Stage 8: The boundary element method
for three-dimensional problems 235
1 Introduction 235
2 Numerical integration 236
2.1 Regular integration 236
2.2 Determination of the optimal number of Gauss points 237
2.3 Regular integration 238
2.4 Nearly singular integration 239
2.5 Weakly singular integration 243
2.6 Infinite patches 246
3 Symmetry 247
4 Multiple patches 249
5 Postprocessing 249
5.1 Stress recovery 249
5.2 Internal stress computation 251
6 Test examples 252
6.1 Infinite tunnel 252
6.2 Loading on infinite half-space 253
7 Examples 253
7.1 Infinite tunnel in infinite domain near tunnel face 253
7.2 Finite tunnel in a semi-infinite domain 254
7.3 Branched tunnel 255
8 Conclusions 260
10 Stage 9: The boundary element method with volume effects 263
1 Introduction 263
2 Effect of body forces and initial strain 263
2.1 Body forces 264
2.2 Effect of initial strain 265
2.3 Solution 266
Table of contents ix
3 Implementation for plane problems 266
3.1 Geometry definition 266
3.2 Computation of the volume integral 268
4 Implementation for 3-D problems 268
4.1 Geometry definition 269
4.2 Computation of the volume integral 270
5 Iterative solution algorithm 270
6 Inclusions 271
6.1 Computation of body force 273
6.2 Steps in the analysis 275
7 Inelastic behavior 276
7.1 Yield conditions 277
7.2 Determination of plastic strain increment 278
8 Implementation of plasticity for plane problems 280
8.1 Determination of plastic zone 280
8.2 Computation of the volume term 284
8.3 Numerical integration 286
8.4 Internal stress computation 287
8.5 Extension of plastic zone during iteration 287
9 Implementation for 3-D problems 288
9.1 Determination of the plastic zone 288
9.2 Computation of the volume term 289
9.3 Numerical integration 289
10 Programming 289
11 Example 293
12 Summary and conclusions 295
11 Stage 10: The time domain 297
1 Introduction 297
1.1 Bernoulli beam with mass 297
2 Solutions in the frequency domain 298
2.1 Numerical solution 299
3 Solutions in the time domain 301
3.1 Finite difference method 301
3.2 Newmark method 302
4 Programming 306
5 Summary and conclusions 309
Appendix: Fundamental solutions 311
1 Stress solution 
(x,y) 312
2 Derived solution for displacement S(x,y) 313
3 Derived solution for traction R(x,y) 314
4 Derived solution for displacement S(x,y) 316
5 Derived solution for traction R(x,y) 317
6 Derivatives of kernel S(x,y) 319
7 Derivatives of kernel R(x,y) 320
Subject index 325