Improving estimates of critical time-steps for discrete element simulations

颗粒材料在我们的日常生活中几乎无处不在，包括土壤颗粒、固体剂型的药物、茶、咖啡和粉状食品配料，如面粉、麸皮、盐、糖或炼乳。研究颗粒状材料的研究人员经常使用计算机模拟来详细研究它们的行为。其中一个这样的软件工具，离散元素建模（DEM），由于其强大的功能和灵活性，在过去的20年里变得非常流行，并且其受欢迎程度逐年增长。DEM基于时间步进算法：先进行一些计算，然后在重复计算之前将时间增加一个微小的时间步长。这个时间步长的大小决定了模拟运行的速度；因此，选择最大可能的时间步长是有利的。然而，有一个极限值——“临界”时间步长——超过这个值，模拟就会变得不稳定，结果就会失效。不幸的是，目前用于估计临界时间步长的方法是粗糙的，不同的方法可能导致巨大差异的估计。缺乏一种精确的方法来估计临界时间步，这意味着需要很大的安全系数。这就是为什么经常采用小而不必要的保守时间步，这会导致模拟运行缓慢。该项目的总体目标是改进现有的方法来估计DEM模拟的关键时间步长。这个总体目标可以分为四个目标。首先，将在只有两个理想粒子的最简单的DEM模拟的临界时间步上计算边界。一旦这一目标得到充分满足，目标二和目标三涉及将这一分析扩展到多粒子系统，并包括基本离散元模型中的复杂性。这些目标将采用一种行之有效的方法来分析非线性动力系统的稳定性。最后的目标是通过与本研究的结果进行比较，批判性地评估当前估计关键时间步长的方法。这项研究有很多潜在的好处。能够更准确地估计临界时间步长，将使应用于模拟时间步长的安全系数降低。这对效率有潜在的巨大影响：模拟持续时间可以从几天减少到几个小时。与以前的情况相比，运行更大、更雄心勃勃的模拟也将变得可行。例如，一个几乎无法运行包含100,000个粒子的模拟的研究人员可以通过简单地选择一个不那么保守的时间步，将粒子的数量增加五倍，而不会相应地增加模拟的持续时间。由于本研究结果将公开发表并广泛传播，本研究也将有助于提高其他相关多体仿真代码的效率。此外，如果可以在不影响模拟稳定性的情况下增加时间步长，所有尺度的DEM模拟都可以在更短的时间内运行，这对环境有明显的好处。

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.