Improving estimates of critical time-steps for discrete element simulations

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Granular materials are almost ubiquitous in our daily lives and include soil particles, pharmaceuticals in solid dosage forms, tea, coffee and powdered food ingredients, e.g., flour, bran, salt, sugar or condensed milk. Researchers investigating granular materials often use computer simulations to study their behaviour in detail. One such software tool, discrete element modelling (DEM), has become extremely popular in the last 20 years due to its power and flexibility and its popularity continues to grow year-on-year. DEM is based on a time-stepping algorithm: some calculations are performed, then time is incremented by a tiny time-step before the calculations are repeated. The size of this time-step determines how quickly the simulation may be run; it is therefore advantageous to choose the largest possible time-step. However, there is a limiting value - the 'critical' time-step - beyond which the simulation becomes unstable and the results become invalid. Unfortunately, the methods used to estimate the critical time-step at present are crude and different approaches can lead to greatly differing estimates. The lack of an accurate method to estimate critical time-steps for non-trivial simulations means that large factors of safety are required. This is why small and unnecessarily conservative time-steps are often adopted which causes simulations to run slowly.The overall aim of this project is to improve upon existing approaches for estimating critical time-steps for DEM simulations. This overarching aim can be divided into four objectives. Firstly, bounds will be calculated on the critical time-step for the simplest possible DEM simulation with only two idealised particles. Once this objective has been fully met, objectives two and three involve extending this analysis to systems of many particles and including complications in the basic discrete element model. These objectives will be achieved using a well-established approach for analysing the stability of nonlinear dynamical systems. The final objective is to critically evaluate the current methods for estimating critical time-steps by comparison with the findings of this study.This study has many potential benefits. Being able to estimate critical time-steps more accurately will allow the factors of safety applied to simulation time-step to be reduced. This has potentially huge implications for efficiency: simulation durations could be reduced from days to several hours. It will also become feasible to run larger, more ambitious simulations than was formerly the case. For example, a researcher who is barely able to run a simulation containing 100,000 particles might be able to increase the number of particles five-fold, without a commensurate increase in the duration of their simulation, by simply choosing a less conservative time-step. As the results of this study will be published openly and disseminated widely, this research will also be useful for increasing the efficiency of other related multi-body simulation codes. Furthermore, there are obvious environmental benefits as DEM simulations at all scales may be run in less time if the time-step can be increased without compromising the stability of the simulation.