Stability Analysis of Explicit MPM

Abstract

In this paper we analyze the stability of the explicit material point method (MPM). We focus on PIC, APIC, and CPIC transfers using quadratic and cubic splines in two and three dimensions. We perform a fully three-dimensional Von Neumann stability analysis to study the behavior within the bulk of a material. This reveals the relationship between the sound speed, CFL number, and actual time step restriction and its dependence on discretization options. We note that boundaries are generally less stable than the interior, with stable time steps generally decreasing until the limit when particles become isolated. We then analyze the stability of a single particle to derive a novel time step restriction that stabilizes simulations at their boundaries. Finally, we show that for explicit MPM with APIC or CPIC transfers, there are pathological cases where growth is observed at arbitrarily small time steps sizes. While these cases do not necessarily pose a problem for practical usage, they do suggest that a guarantee of stability may be theoretically impossible and that necessary but not sufficient time step restrictions may be a necessary and practical compromise.