Selection of an appropriate time integration scheme for the discrete element method (DEM)

With increasing computer power simulation methods addressing discrete problems in a broad range of scientific fields become more and more available. The discrete element method is one of these discontinuous approaches used for modeling granular assemblies. Within this method the dynamics of a system of particles is modeled by tracking the motion of individual particles and their interaction with their adjacencies over time. For the interaction of particles, force models need to be specified. The resulting equations of motion are of coupled ordinary differential configuration, which are usually solved by explicit numerical schemes. In large-scale systems like avalanches, planetary rings, hoppers or chemical reactors vast numbers of particles need to be addressed. Therefore, integration schemes need to be accurate on the one hand, but also numerically efficient on the other hand. This numerical efficiency is characterized by the method's demand for memory and CPU-time. In this paper a number of mostly explicit numerical integration schemes are reviewed and applied to the benchmark problem of a particle impacting a fixed wall as investigated experimentally by Gorham and Kharaz [Gorham, D. A., & Kharaz, A. H. (2000). The measurement of particle rebound characteristics. Powder Technology, 112(3), 193–202]. The accurate modeling which includes the correct integration of the equations of motion is essential. In discrete simulation methods the accuracy of properties on the single particle level directly influence the global properties of the granular assembly like velocity distributions, porosities or flow rates, whereas their correct knowledge is often of key interest in engineering applications. The impact experiment is modeled with simple force displacement approaches which allow an analytical solution of the problem. Aspects discussed are the dependency of the step size on the accuracy of certain collision properties and the related computing time. The effect of a fixed time step is analyzed. Guidelines for the efficient selection of an integration scheme considering the additional computational cost by contact detection and force calculation are presented.