Improvement and Application of Smoothed Particle Hydrodynamics in Elastodynamics

Abstract

This thesis explores the mesh-free numerical method, Smooth Particle Hydrodynamics (SPH), presents improvements to the algorithm and studies its application in solid mechanics problems. The basic concept of the SPH method is introduced and the governing equations are discretised using the SPH method to simulate the elastic solid problems. Special treatments are discussed to improve the stability of the method, such as the treatment for boundary problems, artificial viscosity and tensile instability. In order to improve the stability and efficiency, (i) the classical SPH method has been combined with the Runge-Kutta Chebyshev scheme and (ii) a new time-space Adaptive Smooth Particle Hydrodynamics (ASPH) algorithm has been developed in this thesis.

The SPH method employs a purely meshless Lagrangian numerical technique for spatial discretisation of the domain and it avoids many numerical difficulties related to re-meshing in mesh-based methods such as the finite element method. The explicit Runge-Kutta Chebyshev (RKC) scheme is developed to accurately capture the dynamics in elastic materials for the SPH method in the study. Numerical results are presented for several test examples applied by the RKC-SPH method compared with other different time stepping scheme. It is found that the proposed RKC scheme offers a robust and accurate approach for solving elastodynamics using SPH techniques. The new time-space ASPH algorithm which is combining the previous ASPH method and the RKC schemes can achieve not only the adaptivity of the particle distribution during the simulation, but also the adaptivity of the number of stage in one fixed time step. Numerical results are presented for a shock wave propagation problem using the time-space ASPH method compared with the analytical solution and the results of standard SPH. It is found that using the dynamic adaptive particle refinement procedure with adequate refinement criterion, instead of adopting a fine discretisation for the whole domain, can achieve a substantial reduction in memory and computational time, and similar accuracy is achieved.