Critical time step for discrete element method simulations of convex particles with central symmetry

A time step must be selected for any explicit discrete element method (DEM) simulation. This time step must be small enough to ensure a stable simulation but should not be overly conservative for computational efficiency. There are established methods to estimate critical time steps for simulations of spherical particles. However, there is a comparable lack of guidance on choosing time steps for DEM simulations involving nonspherical particles: a fact which is increasingly problematic as simulations of nonspherical particles become more commonplace. In this paper, the eigenvalues of the amplification matrix are used to develop an explicit formula for the critical time step for a range of shapes including ellipsoids, convex superquadrics and convex, central symmetric polyhedra. This derivation is based on a linear analysis and applies to both underdamped and overdamped systems. The dependence on the particle mass and contact stiffness expected for a system of spheres is recovered. For a fixed particle mass, as particle shape becomes increasingly nonspherical, the critical time step decreases nonlinearly. Thus estimating a critical time step by assuming a sphere of equivalent volume may not always be conservative.