The discrete element method (DEM) typically uses an explicit numerical integration scheme to solve the equations of motion. However, like all explicit schemes, the scheme is only conditionally stable, with the stability determined by the size of the time step. Currently, there are no comprehensive techniques for estimating appropriate DEM time steps when a nonlinear contact interaction is used. It is common practice to apply a large factor of safety to these estimates to ensure stability, which unnecessarily increases the computational cost of these simulations. This work introduces an alternative framework for selecting a stable time step for nonlinear contact laws, specifically for the Hertz-Mindlin contact law. This approach uses the fact that the discretised equations of motion take the form of a nonlinear map and can be analysed as such. Using this framework, we analyse the effects of both system damping and the initial relative velocity of collision on the critical time step for a Hertz-Mindlin contact event between spherical particles.

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SUMMARYThe discrete element method, developed by Cundall and Strack, typically uses some variations 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 discrete element method 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 q 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.

In this article, we will introduce the phenomenon known as the Painlevé paradox and further discuss the associated coupled phenomena, jam and lift-off. We analyze under what conditions the Painlevé paradox can occur for a general two-body collision using a framework that can be easily used with a variety of impact laws, however, in order to visualize jam and lift-off in a numerical simulation, we choose to use a recently developed energetic impact law as it is capable of achieving a unique forward solution in time. Further, we will use this framework to derive the criteria under which the Painlevé paradox can occur in a forced double-pendulum mechanical system. First, using a graphical technique, we will show that it is possible to achieve the Painlevé paradox for relatively low coefficient of friction values, and second we will use the energetic impact law to numerically show the occurrence of the Painlevé paradox in the double-pendulum system.

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