Meshfree Methods for Partial Differential Equations II (eBook)

Griebel (Eds), Meshfree Methods for Partial Differential Equations II (LNCSE 43)

Preface 5
Contents 6
A Particle Strategy for Solving the Fokker- Planck Equation Modelling the Fiber Orientation Distribution in Steady Recirculating Flows Involving Short Fiber Suspensions 8
1 Introduction 8
2 A Particle Discretisation 11
3 Numerical Examples 17
4 Conclusions 21
References 22
Extended Meshfree Method for Elastic and Inelastic Media 24
1 Introduction 24
2 Review of Extended Meshfree Method 25
3 Extended Meshfree Method for Elastic Boundary Value Problems 30
4 Extended Meshfree Method for Inelastic Boundary Value Problems 35
5 Conclusions 42
6 Appendix 43
References 43
Meshfree Petrov-Galerkin Methods for the Incompressible Navier- Stokes Equations 46
1 Introduction 46
2 Stabilisation Schemes 47
3 The Stabilisation Parameter 49
4 Numerical Results 56
5 Conclusion 59
References 60
The a-shape Based Natural Element Method in Solid and Fluid Mechanics 62
1 Introduction 62
2 Natural Neighbour Galerkin Methods 63
3 Natural Element Methods for Incompressible Media 65
4 The a-shape based Natural Element Method 69
5 Numerical Examples 70
6 Closing Remarks 74
References 75
A Particle-Partition of Unity Method Part VI: A p- robust Multilevel Solver 78
1 Introduction 78
2 Partition of Unity Method 80
3 Multilevel Solution of Resulting Linear System 84
4 Numerical Results 94
5 Concluding Remarks 97
References 98
Enriched Reproducing Kernel Approximation: Reproducing Functions with Discontinuous Derivatives 100
1 Introduction 100
2 Enriched Reproducing Kernel Particle Approximation 102
3 The Case of a Function with Discontinuous Derivatives 104
4 Properties of the Moment Matrix 105
5 About the Resulting Shape Function and its Derivatives 107
6 Numerical Example 111
7 Conclusion 113
References 114
Reproducing Kernel Element Interpolation: Globally Conforming Im/ Cn/ Pk Hierarchies 116
1 Introduction 116
2 Reproducing Kernel Element Method 117
3 Globally Conforming Im/Cn/Pk Hierarchy I 122
4 Globally Conforming Im/Cn/Pk Hierarchy II 127
5 Numerical Examples 133
6 Closure 137
References 138
Multi-scale Analysis Using Two Influence Radii in EFGM 140
1 Introduction 140
2 P-like Adaptivity Analysis 142
3 Multi-Scale Analysis Using Two Nodal Points Layouts 144
4 Conclusion 154
References 154
Solution of a Dynamic Main Crack Interaction with a System of Micro- Cracks by the Element Free Galerkin Method 156
1 Introduction 156
2 The EFG Method for Dynamic Linear Elastic Fracture Mechanics 157
3 Solution of Multi-Crack Problems by the EFG Method 160
4 Individual Crack Subjected to Pulse Loading 164
5 Main Crack Propagation in a Field of Interacting Flaws under Dynamic Pulse Loading 168
6 Conclusions 173
References 174
Finite Cover Method for Physically and Geometrically Nonlinear Problems 176
1 Introduction 176
2 Finite Cover Method 177
3 Application to Evolution Problems of Failure Surfaces 182
4 Application to Finite Deformation Problems 189
5 Conclusion 194
References 196
A Numerical Scheme for Solving Incompressible and Low Mach Number Flows by the Finite Pointset Method 198
1 Introduction 198
2 Governing Equations 200
3 Numerical Scheme 201
4 Numerical Tests 207
References 212
SPH Renormalized Hybrid Methods for Conservation Laws: Applications to Free Surface Flows 214
1 Introduction 214
2 Classical Recipes 215
3 Weak Formulation 219
4 Application to Euler Equations 226
5 Applications 227
References 235
Discontinuous Radial Basis Function Approximations for Meshfree Methods 238
1 Introduction 238
2 Radial Basis Function Approximation 240
3 Approximation for Discontinuous Functions by Radial Basis Function Interpolation 241
4 Discrete Equations for Discontinuous Radial Basis Function Approximation 245
5 Example 247
6 Conclusions 256
References 258
Treating Moving Interfaces in Thermal Models with the C- NEM 262
1 Introduction 262
2 Problem Formulation 264
3 The Constrained Natural Element Method (C-NEM) 265
4 C-NEM Discretization 269
5 Numerical Example 270
6 Conclusion 272
References 274
Bridging Scale Particle and Finite Element Methods 278
1 Introduction 278
2 Review of Multiscale Simulation Methods 280
3 Concurrent Bridging Scale Method 281
4 Bridging Scale Computations for Localized Failure 291
5 Conclusions 294
References 295
Color Plates 299