Abstract
This contribution considers the critical time increment (Δtcrit) to achieve stable simulations using particulate discrete element method (DEM) codes that adopt a Verlet-type time integration scheme. The Δtcrit is determined by considering the maximum vibration frequency of the system. Based on a series of parametric studies, Δtcrit is shown to depend on the particle mass (m), the maximum contact stiffness (Kmax), and the maximum particle coordination number (CN,max). Empirical expressions relating Δtcrit to m, Kmax, and CN,max are presented; while strictly only valid within the range of simulation scenarios considered here, these can inform DEM analysts selecting appropriate Δtcrit values.
Highlights
This contribution considers the issue of numerical stability, and applies eigenmode analyses to a database of discrete element modelling (DEM) simulations to show how the particle characteristics, packing and stress level influence the critical time increment calculated from consideration of the maximum eigenfrequency
The simulations examined the effect of particle size, stress level and particle size distribution on the maximum eigenvalue and the critical time increment in DEM simulations
This contribution has extracted mass and contact stiffness information from DEM simulations to construct system mass and stiffness matrices. Eigenvalue decomposition of these matrices was used to determine the maximum frequency in the system from which the critical time step could be directly determined
Summary
Particulate discrete element modelling (DEM) is well established as a research tool in science in general, and in geomechanics in particular; there has been a consistent increase in the number of DEM-related publications published each year over the past 20–25 years [1,2]. Two approaches are used in the literature to determine Dtcrit for DEM simulations; the first is based on the oscillation period of a single degree of freedom system, while the second uses the Rayleigh wave speed In their initial description of the discrete element method Cundall and Strack [13] estimated Dtcrit by considering a single degree of freedom system of a mass m connected to the ground by a spring. The element contact stiffness matrix, kij, connecting particles (nodes) i and j is created in the global coordinate system and the degrees of freedom associated with node i are given by [ui;x, ui;y, ui;z, hi;x, hi;y, hi;z] where ui;s is the translational displacement in direction s and hi;s is the rotation about axis s. As before Eq (7) relates xmax and kmax and x Dtcrit;eig 1⁄4 2 : max ð13Þ
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