Establishing stable time-steps for DEM simulations of non-collinear planar collisions with linear contact laws

The discrete element method, developed by Cundall and Strack, typically uses some variation of the central difference numerical integration scheme. However, like all explicit schemes, the scheme is only conditionally stable, with the stability determined by the size of the time-step. The current methods for estimating appropriate DEM time-steps are based on many assumptions; therefore large factors of safety are usually applied to the time-step to ensure stability which substantially increases the computational cost of a simulation. This work introduces a general framework for estimating critical time-steps for any planar rigid body subject to linear damping and forcing. A numerical investigation of how system damping, coupled with non-collinear impact, affects the critical time-step is also presented. It is shown that the critical time-step is proportional to $\sqrt{\frac{m}{k}}$ if a linear contact model is adopted, where $m$ and $k$ represent mass and stiffness, respectively. The term which multiplies this factor is a function of known physical parameters of the system. The stability of a system is independent of the initial conditions.