Mathematical Physics - Volume II - Numerical Methods

III

Numerical method - SPH

159

Chapter 5 Review of Development of the Smooth Particle Hydrody namics (SPH) Method 161 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16.1 5.2 Basic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 6.5 5.3 Conservation Equations . . . . . . . . . . . . . . . . . . . . . . . .16.7 5.4 Kernel Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 6.9 5.5 Variable Smoothing Length . . . . . . . . . . . . . . . . . . . . . 1. 7.0 5.6 Neighbour Search . . . . . . . . . . . . . . . . . . . . . . . . . . .17.1 5.7 SPH Shortcomings . . . . . . . . . . . . . . . . . . . . . . . . . 1. 7.2 5.7.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . 1. 7.2 5.8 Derivation of Normalised Corrected Gradient SPH formula . . . . 1. 7.2 5.9 Tensile Instability . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 7.6 5.9.1 Stability analysis of conventional (Eulerian) SPH . . . . . 1. 7.8 5.10 Zero-Energy Modes . . . . . . . . . . . . . . . . . . . . . . . . . .18.1 5.11 Non-Local properties of SPH . . . . . . . . . . . . . . . . . . . 1. 8.2 5.11.1 Theoretical Background of Strain-Softening . . . . . . . . 1. 8.3 5.11.2 Main Aspects of the Smoothed Particle Hydrodynamics (SPH) Method as Nonlocal Regularisation Method . . . . .18.7 5.11.3 Numerical Experiments for the Evaluation of the SPH Metho1d89 5.11.4 Numerical Results of the Strain-Softening in FE . . . . . 1. 9.3 5.11.5 Strain-Softening in SPH Numerical Results of the Strain Softening in SPH . . . . . . . . . . . . . . . . . . . . . . 1. 9.4 5.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20.1

Bibliography

203

IV

Numerical method - CMD

213

Chapter 6 Introduction to

Computational Mechanics of Discontinua 215 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. 1.5 6.2 Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . .22.1 6.2.1 Basic Idea of MD . . . . . . . . . . . . . . . . . . . . . . 2. 2.2 6.2.2 Empirical Interatomic Potentials . . . . . . . . . . . . . . 2. 2.3