Francisco Chinesta, born in 1966 in Valencia (Spain), obtained his M.S. in Mechanical Engineering (1990) and later his Ph.D. degree supervised at the ENS Cachan in France. He is currently Professor of Computational Mechanics at the Ecole Centrale of Nantes (France) and titular of the EADS Corporate Foundation International Chair on Advanced Modeling of Composites Manufacturing Processes.

• Foreword ixAcknowledgements xiChapter 1. Introduction 11.1. SPH method 31.2. RKPM method 51.3. MLS based approximations 101.4. Final note 11Chapter 2. Basics of the Natural Element Method 132.1. Introduction 132.2. Natural neighbor Galerkin methods 142.3. Exact imposition of the essential boundary conditions 222.4. Mixed approximations of natural neighbor type 272.5. High order natural neighbor interpolants 40Chapter 3. Numerical Aspects 493.1. Searching for natural neighbors 493.2. Calculation of NEM shape functions of the Sibson type 523.3. Numerical integration 713.4. NEM on an octree structure 80Chapter 4. Applications in the Mechanics of Structures and Processes 934.1. Two- and three-dimensional elasticity 934.2. Indicators and estimators of error: adaptivity 964.3. Metal extrusion 1074.4. Friction stir welding 1134.5. Models and numerical treatment of the phase transition: foundry and treatment of surfaces 1234.6. Adiabatic shearing, cutting, and high speed blanking 136Chapter 5. A Mixed Approach to the Natural Elements 1595.1. Introduction 1595.2. The Fraeijs de Veubeke variational principle for linear elastic problems 1615.3. Field decomposition 1645.4. Discretization 1665.5. Discretized equations 1705.6. Matrix solution for linear elastic problems 1725.7. Numerical integration 1765.8. Linear elastic patch tests 1785.9. Application 1: pure bending of a linear elastic beam 1825.10. Application 2: square domain with circular hole 1855.11. Mixed approach to nonlinear problems 1875.12. Step-by-step solution of the discretized nonlinear equations 1925.13. Example of an elastoplastic material 1955.14. Application: pure bending of an elastoplastic beam 1965.15. Conclusion 199Chapter 6. Flow Models 2016.1. Natural element method in fluid mechanics: updated Lagrangian approach 2016.2. Free and moving surfaces 2026.3. Short-fiber suspensions flow 2066.4. Breaking dam problem 2086.5. Multi-scale approaches 209Chapter 7. Conclusion 225Bibliography 227Index 239

Francisco Chinesta, born in 1966 in Valencia (Spain), obtained his M.S. in Mechanical Engineering (1990) and later his Ph.D. degree supervised at the ENS Cachan in France. He is currently Professor of Computational Mechanics at the Ecole Centrale of Nantes (France) and titular of the EADS Corporate Foundation International Chair on Advanced Modeling of Composites Manufacturing Processes.

• Foreword ixAcknowledgements xiChapter 1. Introduction 11.1. SPH method 31.2. RKPM method 51.3. MLS based approximations 101.4. Final note 11Chapter 2. Basics of the Natural Element Method 132.1. Introduction 132.2. Natural neighbor Galerkin methods 142.3. Exact imposition of the essential boundary conditions 222.4. Mixed approximations of natural neighbor type 272.5. High order natural neighbor interpolants 40Chapter 3. Numerical Aspects 493.1. Searching for natural neighbors 493.2. Calculation of NEM shape functions of the Sibson type 523.3. Numerical integration 713.4. NEM on an octree structure 80Chapter 4. Applications in the Mechanics of Structures and Processes 934.1. Two- and three-dimensional elasticity 934.2. Indicators and estimators of error: adaptivity 964.3. Metal extrusion 1074.4. Friction stir welding 1134.5. Models and numerical treatment of the phase transition: foundry and treatment of surfaces 1234.6. Adiabatic shearing, cutting, and high speed blanking 136Chapter 5. A Mixed Approach to the Natural Elements 1595.1. Introduction 1595.2. The Fraeijs de Veubeke variational principle for linear elastic problems 1615.3. Field decomposition 1645.4. Discretization 1665.5. Discretized equations 1705.6. Matrix solution for linear elastic problems 1725.7. Numerical integration 1765.8. Linear elastic patch tests 1785.9. Application 1: pure bending of a linear elastic beam 1825.10. Application 2: square domain with circular hole 1855.11. Mixed approach to nonlinear problems 1875.12. Step-by-step solution of the discretized nonlinear equations 1925.13. Example of an elastoplastic material 1955.14. Application: pure bending of an elastoplastic beam 1965.15. Conclusion 199Chapter 6. Flow Models 2016.1. Natural element method in fluid mechanics: updated Lagrangian approach 2016.2. Free and moving surfaces 2026.3. Short-fiber suspensions flow 2066.4. Breaking dam problem 2086.5. Multi-scale approaches 209Chapter 7. Conclusion 225Bibliography 227Index 239

Francisco Chinesta, born in 1966 in Valencia (Spain), obtained his M.S. in Mechanical Engineering (1990) and later his Ph.D. degree supervised at the ENS Cachan in France. He is currently Professor of Computational Mechanics at the Ecole Centrale of Nantes (France) and titular of the EADS Corporate Foundation International Chair on Advanced Modeling of Composites Manufacturing Processes.

• Foreword ixAcknowledgements xiChapter 1. Introduction 11.1. SPH method 31.2. RKPM method 51.3. MLS based approximations 101.4. Final note 11Chapter 2. Basics of the Natural Element Method 132.1. Introduction 132.2. Natural neighbor Galerkin methods 142.3. Exact imposition of the essential boundary conditions 222.4. Mixed approximations of natural neighbor type 272.5. High order natural neighbor interpolants 40Chapter 3. Numerical Aspects 493.1. Searching for natural neighbors 493.2. Calculation of NEM shape functions of the Sibson type 523.3. Numerical integration 713.4. NEM on an octree structure 80Chapter 4. Applications in the Mechanics of Structures and Processes 934.1. Two- and three-dimensional elasticity 934.2. Indicators and estimators of error: adaptivity 964.3. Metal extrusion 1074.4. Friction stir welding 1134.5. Models and numerical treatment of the phase transition: foundry and treatment of surfaces 1234.6. Adiabatic shearing, cutting, and high speed blanking 136Chapter 5. A Mixed Approach to the Natural Elements 1595.1. Introduction 1595.2. The Fraeijs de Veubeke variational principle for linear elastic problems 1615.3. Field decomposition 1645.4. Discretization 1665.5. Discretized equations 1705.6. Matrix solution for linear elastic problems 1725.7. Numerical integration 1765.8. Linear elastic patch tests 1785.9. Application 1: pure bending of a linear elastic beam 1825.10. Application 2: square domain with circular hole 1855.11. Mixed approach to nonlinear problems 1875.12. Step-by-step solution of the discretized nonlinear equations 1925.13. Example of an elastoplastic material 1955.14. Application: pure bending of an elastoplastic beam 1965.15. Conclusion 199Chapter 6. Flow Models 2016.1. Natural element method in fluid mechanics: updated Lagrangian approach 2016.2. Free and moving surfaces 2026.3. Short-fiber suspensions flow 2066.4. Breaking dam problem 2086.5. Multi-scale approaches 209Chapter 7. Conclusion 225Bibliography 227Index 239