Decoupling and balancing of space and time errors in the material point method (MPM)

SUMMARY The Material Point Method (MPM) is a computationally effective particle method with mathematical roots in both particle-in-cell and finite element type methods. The method has proven itself extremely useful in solving solid mechanics problems involving large deformations and/or fragmentation of structures, problem domains which are sometimes problematic for finite element type methods. Recently, the MPM community has focused significant attention on understanding the basic mathematical error properties of the method. Complementary to this thrust, in this paper we show how spatial and temporal errors are typically coupled within the MPM framework. In an attempt to overcome the challenge to analysis that this coupling poses, we take advantage of MPM’s connection to finite element methods by developing a “moving-mesh” variant of MPM which allows us to use finite element type error analysis to demonstrate and understand the spatial and temporal error behaviors of MPM. We then provide an analysis and demonstration of various spatial and temporal errors in MPM and in simplified MPM-type simulations. Our analysis allows us to anticipate the global error behavior in MPM type methods and allows us to estimate the time-step where spatial and temporal errors are balanced. Larger time-steps result in solutions dominated by temporal errors and show second-order temporal error convergence. Smaller time-steps result in solutions dominated by spatial errors, and hence temporal refinement produces no appreciative change in the solution. Based upon our understanding of MPM from both analysis and numerical experimentation, we are able to provide to MPM practitioners a collection of guidelines to be used in the selection of simulation parameters that respect the interplay between spatial (grid) resolution, number of particles and time step. Copyright c ° 2009 John Wiley & Sons, Ltd.