A unified stability analysis of meshless particle methods

A unified stability analysis of meshless methods with Eulerian and Lagrangian kernels is presented. Three types of instabilities were identified in one dimension: an instability due to rank deficiency, a tensile instability and a material instability which is also found in continua. The stability of particle methods with Eulerian and Lagrangian kernels is markedly different: Lagrangian kernels do not exhibit the tensile instability. In both kernels, the instability due to rank deficiency can be suppressed by stress points. In two dimensions the stabilizing effect of stress points is dependent on their locations. It was found that the best approach to stable particle discretizations is to use Lagrangian kernels with stress points. The stability of the least-squares stabilization was also studied.