Stability analysis of explicit MPM
Stability analysis of explicit MPM
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DOI：
10.1111/cgf.14620
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发表时间：
2022-12
影响因子：
2.5
通讯作者：
Song Bai;Craig A. Schroeder
中科院分区：
文献类型：
--
作者：
Song Bai;Craig A. Schroeder
本文分析了显式物质点法（MPM）的稳定性。我们聚焦于在二维和三维空间中使用二次和三次样条的粒子在单元格法（PIC）、任意拉格朗日 - 欧拉粒子在单元格法（APIC）以及耦合粒子在单元格法（CPIC）的输运过程。我们进行了完整的三维冯·诺依曼稳定性分析，以研究材料内部的特性。这揭示了声速、库朗数（CFL数）与实际时间步长限制之间的关系，以及该关系对离散化选项的依赖。我们注意到，边界通常比内部稳定性差，稳定的时间步长一般会逐渐减小，直至粒子孤立时达到极限。随后，我们分析单个粒子的稳定性，推导出一种新的时间步长限制，以确保模拟在边界处的稳定性。最后，我们表明，对于采用APIC或CPIC输运的显式MPM，存在一些特殊情况，即即便时间步长任意小，仍会出现增长现象。虽然这些情况在实际应用中不一定构成问题，但它们确实表明，从理论上保证稳定性或许是不可能的，而必要但不充分的时间步长限制可能是一种必要且实际的折衷办法。
In this paper we analyze the stability of the explicit material point method (MPM). We focus on PIC, APIC, and CPIC transfers using quadratic and cubic splines in two and three dimensions. We perform a fully three‐dimensional Von Neumann stability analysis to study the behavior within the bulk of a material. This reveals the relationship between the sound speed, CFL number, and actual time step restriction and its dependence on discretization options. We note that boundaries are generally less stable than the interior, with stable time steps generally decreasing until the limit when particles become isolated. We then analyze the stability of a single particle to derive a novel time step restriction that stabilizes simulations at their boundaries. Finally, we show that for explicit MPM with APIC or CPIC transfers, there are pathological cases where growth is observed at arbitrarily small time steps sizes. While these cases do not necessarily pose a problem for practical usage, they do suggest that a guarantee of stability may be theoretically impossible and that necessary but not sufficient time step restrictions may be a necessary and practical compromise.