| Preface | 5 |
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| Contents | 6 |
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| A Particle Strategy for Solving the Fokker- Planck Equation Modelling the Fiber Orientation Distribution in Steady Recirculating Flows Involving Short Fiber Suspensions | 8 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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