Meshfree Methods for Partial Differential Equations III (eBook)

Preface 5
Contents 6
Local Maximum-Entropy Approximation Schemes 8
1 Introduction 8
2 Formulation 9
3 Properties 17
4 Other Interpretations: Relative Entropy 19
5 Applications in Galerkin Methods 21
6 Conclusions 22
References 22
Genetic Algorithms for Meshfree Numerical Integration 24
1 Introduction 24
2 Problem Formulation and Discretization 26
3 Numerical Integration 37
4 Numerical Examples 41
5 Concluding Remarks 46
References 46
An RBF Meshless Method for Injection Molding Modelling 48
1 Model Equations 48
2 Pressure Field 50
3 Front Displacement 56
4 Conclusions 62
References 62
Strain Smoothing for Stabilization and Regularization of Galerkin Meshfree Methods 64
1 Introduction 64
2 Conforming Strain Smoothing for Rank Instability 65
3 Additional Stabilization for Spurious Nonzero Energy Modes 69
4 Gradient Strain Smoothing Regularization for Material Instability in Strain Localization 72
5 Application to Fragment Penetration Problems 77
6 Conclusions 80
References 81
Fuzzy Grid Method for Lagrangian Gas Dynamics Equations 83
1 Smooth Particle Hydrodynamics 83
2 Fuzzy Grid 85
3 The Unsymmetrical Approach to Modify Weights 86
4 The Symmetrical Approach to Modify Weights 87
5 Numerical Examples 88
6 Conclusion 92
References 92
New Shape Functions for Arbitrary Discontinuities without Additional Unknowns 93
1 Introduction 93
2 Level-Set Method 95
3 Moving Least-Squares Method 95
4 Design of Special Shape Functions 97
5 Coupling 100
6 Numerical Results 102
7 Conclusion 107
References 108
A Meshless BEM for 2-D Stress Analysis in Linear Elastic FGMs 110
1 Introduction 110
2 Formulation of Boundary-Domain Integral Equations 112
3 Transformation of Domain-Integrals into Boundary Integrals 114
4 System of Linear Algebraic Equations 115
5 Computation of Stresses 116
6 Numerical Example 118
7 Conclusions 122
References 123
A Particle-Partition of Unity Method Part VII: Adaptivity 125
1 Introduction 125
2 Partition of Unity Method 126
3 Adaptive Multilevel Solution 133
4 Numerical Results 142
5 Concluding Remarks 150
References 151
Enriched Reproducing Kernel Particle Approximation for Simulating Problems Involving Moving Interfaces 152
1 Introduction 152
2 Enriched Functional Approximations 153
3 Definition of the Continuous Shape Functions with Discontinuous Derivatives 159
4 Enrichment Function Expressed by a Polynomial 161
5 Numerical Results 163
6 Conclusion 166
References 166
Deterministic Particle Methods for High Dimensional Fokker- Planck Equations 168
1 A Polymer Model 168
2 Meshfree Solution Methods 173
3 QMC Product Measures 175
4 QMC Simulations 181
References 185
Bridging Scale Method and Its Applications 187
1 Bridging Scale Fundamentals 187
2 Reduction of the MD Domain 192
3 Cauchy-Born Rule 199
4 Atomistic Model 200
5 Dynamic Crack Propagation in Three Dimensions 200
6 Discussion on the Bridging Scale Method 205
References 206
A New Stabilized Nodal Integration Approach 208
1 Introduction 208
2 Formulation 209
3 Results 214
4 Discussion 217
References 218
Multigrid and M-Matrices in the Finite Pointset Method for Incompressible Flows 219
1 Introduction 219
2 FPM for the Incompressible Navier-Stokes Equation 220
3 Finite Difference Approximation 221
4 Conditions for the Existence of Positive Stencils 224
5 Optimization for Interior Points 228
6 Setting Up the System Matrix 229
7 Algebraic Multigrid 230
8 Numerical Results 231
9 Conclusions and Outlook 233
References 234
Assessment of Generalized Finite Elements in Nonlinear Analysis 235
1 Introduction 235
2 Formulation 236
3 Numerical Examples 241
4 Conclusion 246
References 246
A Meshfree Method for Simulations of Interactions between Fluids and Flexible Structures 248
1 Introduction 248
2 Governing Equations 249
3 Numerical Schemes 251
4 Numerical Examples 257
5 Conclusion 261
References 263
Goal Oriented Error Estimation for the Element Free Galerkin Method 264
1 Introduction 264
2 Basics on Output Oriented Error Estimation in FEM 266
3 Extension to Element Free Galerkin 269
4 Numerical Results 274
5 Concluding Remarks 279
References 280
Bubble and Hermite Natural Element Approximations 282
1 Introduction 283
2 Review of the Natural Element Method 284
3 Hierarchical Bubble Functions in the Natural Element Method 286
4 Hermite-Natural Element Formulation 292
5 Conclusion 295
References 296
Color Plates 298