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

Abstract

SUMMARYThe 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 to be extremely useful in solving solid mechanics problems involving large deformations and/or fragmentation of structures, problem domains that 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 that 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.

Evaluation of material point method for use in geotechnics

“…However, the paper of Sulsky et al [2] was one of the first to describe an algorithm for the procedure and associated all the relevant properties, especially those related to the constitutive model, with each material point rather than the grid nodes. The method has been subsequently developed and applied by a number of researchers, including Sulsky et al [3], Bardenhagen et al [4], Chen et al [5], Love and Sulsky [6], Steffen et al [7,8] and Burghardt et al [9].…”

A convected particle domain interpolation technique to extend applicability of the material point method for problems involving massive deformations

“…These functions are analytically calculated as (26) in which (r n 1x ,r n 1y ) and (r n 2x ,r n 2y ) are, respectively, the components of vectors r n 1 and r n 2 which define the particle domain at the current time step (Figure 2). …”

Mitigating kinematic locking in the material point method

“…Significant research has focused on applications of this type (e.g., [11,12,17,34,59,61,40]), but much work has also been done in additional areas-pushing the capabilities of the MPM and exploring new applications. These include applications in geotechnical engineering [3,60,66], fracture and material failure [15,19,20,41,44,45,53,55], contact, material interaction, and penetration [6,7,23,24,27,33,38,46,42,64,65], as well as general implementation considerations [5,25,28,29,36,48,50].…”