Natural Element Method for the Simulation of Structures and Processes

Foreword ix

Acknowledgements xi

Chapter 1. Introduction 1

1.1. SPH method 3

1.2. RKPM method 5

1.3. MLS based approximations 10

1.4. Final note 11

Chapter 2. Basics of the Natural Element Method 13

2.1. Introduction 13

2.2. Natural neighbor Galerkin methods 14

2.3. Exact imposition of the essential boundary conditions 22

2.4. Mixed approximations of natural neighbor type 27

2.5. High order natural neighbor interpolants 40

Chapter 3. Numerical Aspects 49

3.1. Searching for natural neighbors 49

3.2. Calculation of NEM shape functions of the Sibson type 52

3.3. Numerical integration 71

3.4. NEM on an octree structure 80

Chapter 4. Applications in the Mechanics of Structures and Processes 93

4.1. Two- and three-dimensional elasticity 93

4.2. Indicators and estimators of error: adaptivity 96

4.3. Metal extrusion 107

4.4. Friction stir welding 113

4.5. Models and numerical treatment of the phase transition: foundry and treatment of surfaces 123

4.6. Adiabatic shearing, cutting, and high speed blanking 136

Chapter 5. A Mixed Approach to the Natural Elements 159

5.1. Introduction 159

5.2. The Fraeijs de Veubeke variational principle for linear elastic problems 161

5.3. Field decomposition 164

5.4. Discretization 166

5.5. Discretized equations 170

5.6. Matrix solution for linear elastic problems 172

5.7. Numerical integration 176

5.8. Linear elastic patch tests 178

5.9. Application 1: pure bending of a linear elastic beam 182

5.10. Application 2: square domain with circular hole 185

5.11. Mixed approach to nonlinear problems 187

5.12. Step-by-step solution of the discretized nonlinear equations 192

5.13. Example of an elastoplastic material 195

5.14. Application: pure bending of an elastoplastic beam 196

5.15. Conclusion 199

Chapter 6. Flow Models 201

6.1. Natural element method in fluid mechanics: updated Lagrangian approach 201

6.2. Free and moving surfaces 202

6.3. Short-fiber suspensions flow 206

6.4. Breaking dam problem 208

6.5. Multi-scale approaches 209

Chapter 7. Conclusion 225

Bibliography 227

Index 239