1．Introduction 1
1.1 Lagrangian Methods 1
1.2 Eulerian Methods 3
1.3 Hybrid Methods 4
1.3.1 Arbitrary Eulerian-Lagrangian Method and Its Variations 4
1.3.2 Particle-In-Cell Method and Its Variations 5
1.3.3 Material Point Method 6
1.4 Meshfree Methods 7
2．Governing Equations 11
2.1 Description of Motion 11
2.2 Deformation Gradient 14
2.3 Rate of Deformation 16
2.4 Cauchy Stress 17
2.5 Jaumann Stress Rate 19
2.6 Updated Lagrangian Formulation 20
2.6.1 Reynolds'Transport Theorem 20
2.6.2 Conservation of Mass 21
2.6.3 Conservation of Linear Momentum 22
2.6.4 Conservation of Energy 23
2.6.5 Governing Equations 24
2.7 Weak Form of the Updated Lagrangian Formulation 25
2.8 Shock Wave 27
2.8.1 Rankine-Hugoniot Equations 27
2.8.2 Artificial Bulk Viscosity 29
2.9 Detonation Wave 32
2.9.1 CJ Detonation Model 32
2.9.2 ZND Detonation Model 35
3．The Material Point Method 37
3.1 Material Point Discretization 37
3.1.1 Lagrangian Phase 38
3.1.2 Convective Phase 41
3.2 Explicit Material Point Method 42
3.2.1 Explicit Time Integration 43
3.2.2 Explicit MPM Scheme 47
3.2.3 Qualitative Demonstration 53
3.2.4 Comparison Between MPM and FEM 56
3.3 Contact Method 59
3.3.1 Boundary Conditions at Contact Surface 60
3.3.2 Contact Detection 62
3.3.3 Contact Force 64
3.3.4 Numerical Algorithm for Contact Method 67
3.4 Generalized Interpolation MPM and Other Improvements 68
3.4.1 Contiguous Particle GIMP 71
3.4.2 Uniform GIMP 72
3.4.3 Convected Particle Domain Interpolation 74
3.4.4 Dual Domain Material Point Method 75
3.4.5 Spline Grid Shape Function 76
3.5 Adaptive Material Point Method 77
3.5.1 Particle Adaptive Split 77
3.5.2 Adaptive Computational Grid 79
3.6 Non-reflecting Boundary 87
3.7 Incompressible Material Point Method 88
3.7.1 Momentum Equation of Fluid 89
3.7.2 Operator Splitting 89
3.7.3 Pressure Poisson Equations 90
3.7.4 Pressure Boundary Conditions 91
3.7.5 Velocity Update 92
3.8 Implicit Material Point Method 93
3.8.1 Implicit Time Integration 95
3.8.2 Solution of a System of Nonlinear Equations 95
3.8.3 The Jacobian of Grid NodalInternal Force 97
3.8.4 Solution of a Linearized System of Equations 99
4．Computer Implementation of the MPM 104
4.1 Execution of the MPM3D-F90 104
4.2 Input Data File Format of the MPM3D-F90 105
4.2.1 Unit 105
4.2.2 Keywords 105
4.2.3 Global Information 106
4.2.4 Material Model 106
4.2.5 Background Grid 108
4.2.6 Solution Scheme 109
4.2.7 Results Output 109
4.2.8 Bodies 110
4.2.9 Load 111
4.2.10 An Example of Input Data File 111
4.3 Source Files of the MPM3D-F90 113
4.4 Free Format Input 113
4.5 MPM Data Encapsulation 115
4.5.1 Particle Data 115
4.5.2 Grid Data 116
4.5.3 Data Input 119
4.5.4 Data Output 120
4.6 Main Subroutines 121
4.7 Numerical Examples 137
4.7.1 TNT Slab Detonation 137
4.7.2 Taylor Bar Impact 138
4.7.3 Perforation of a Thick Plate 139
4.7.4 Failure of Soil Slope 141
5．Coupling of the MPM with FEM 143
5.1 Explicit Finite Element Method 143
5.1.1 Finite Element Discretization 143
5.1.2 The FEM Formulation in Matrix Form 145
5.1.3 Hexahedron Element 147
5.1.4 Numerical Algorithm for an Explicit FEM 155
5.2 Hybrid FEM and MPM 156
5.3 Coupled FEM and MPM 162
5.3.1 Global Search 164
5.3.2 Local Search 165
5.3.3 Contact Force 166
5.4 Adaptive FEMP Method 168
5.4.1 Discretization Scheme 168
5.4.2 Convers ion Algorithm 169
5.4.3 Coupling Between Remaining Elements and Particles 170
6．Constitutive Models 175
6.1 Stress Update 175
6.2 Strength Models 178
6.2.1 Elastic Model 178
6.2.2 Elastoplastic Models 179
6.2.3 Return Mapping Algorithm 182
6.2.4 J2 Flow Theory 188
6.2.5 Pressure-Dependent Elastoplasticity 196
6.2.6 Newtonian Fluid 205
6.2.7 High Explosive 205
6.3 Equation of State 207
6.3.1 Polytropic Process 207
6.3.2 Nearly Incompressible Fluid 208
6.3.3 Linear Polynomial 209
6.3.4 JWL 210
6.3.5 Mie-Grüneisen 211
6.4 Failure Models 213
6.4.1 Effective Plastic Strain Failure Model 214
6.4.2 Hydrostatic Tensile Failure Model 214
6.4.3 Maximum Principal/Shear Stress Failure Model 214
6.4.4 Maximum Principal/Shear Strain Failure Model 214
6.4.5 Effective Strain Failure Model 215
6.5 Computer Implementation of Material Models 215
6.5.1 Module MaterialData 215
6.5.2 Module MaterialModel 217
7．Multiscale MPM 222
7.1 Governing Equations at Different Scales 222
7.2 Solution Scheme for Concurrent Simulations 224
7.2.1 Preprocessor 224
7.2.2 Central Processing Unit 224
7.3 I nterfacial Treatment 227
7.4 Demonstration 228
8．Applications of the MPM 231
8.1 Fracture Evolution 231
8.2 Impact 236
8.3 Explosion 242
8.4 Fluid-Structure/Solid Interaction 247
8.5 Multiscale Simulation 251
8.6 Biomechanics Problems 259
8.7 Other Problems with Extreme Deformations 261
Bibliography 265
Index 277