Natural Element Method for the Simulation of Structures and Processes

Computational mechanics is the discipline concerned with the use of computational methods to study phenomena governed by the principles of mechanics. Before the emergence of computational science (also called scientific computing) as a "e;third way"e; besides theoretical and experimental sciences, computational mechanics was widely considered to be a sub-discipline of applied mechanics. It is now considered to be a sub-discipline within computational science. This book presents a recent state of the art on the foundations and applications of the meshless natural element method in computational mechanics, including structural mechanics and material forming processes involving solids and Newtonian and non-Newtonian fluids.

Computational mechanics is the discipline concerned with the use ofcomputational methods to study phenomena governed by the principlesof mechanics. Before the emergence of computational science (alsocalled scientific computing) as a "third way" besides theoreticaland experimental sciences, computational mechanics was widelyconsidered to be a sub-discipline of applied mechanics. It is nowconsidered to be a sub-discipline within computationalscience.This book presents a recent state of the art on the foundationsand applications of the meshless natural element method incomputational mechanics, including structural mechanics andmaterial forming processes involving solids and Newtonian andnon-Newtonian fluids.

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