Analysis and reduction of quadrature errors in the material point method (MPM)

Abstract

SUMMARYThe material point method (MPM) has demonstrated itself as a computationally effective particle method for solving solid mechanics problems involving large deformations and/or fragmentation of structures, which are sometimes problematic for finite element methods (FEMs). However, similar to most methods that employ mixed Lagrangian (particle) and Eulerian strategies, analysis of the method is not straightforward. The lack of an analysis framework for MPM, as is found in FEMs, makes it challenging to explain anomalies found in its employment and makes it difficult to propose methodology improvements with predictable outcomes.In this paper we present an analysis of the quadrature errors found in the computation of (material) internal force in MPM and use this analysis to direct proposed improvements. In particular, we demonstrate that lack of regularity in the grid functions used for representing the solution to the equations of motion can hamper spatial convergence of the method. We propose the use of a quadratic B-spline basis for representing solutions on the grid, and we demonstrate computationally and explain theoretically why such a small change can have a significant impact on the reduction in the internal force quadrature error (and corresponding 'grid crossing error') often experienced when using MPM.

Solving time‐dependent PDEs using the material point method, a case study from gas dynamics

“…while this is a reasonable approximation for compressible flow, and was first used by Kim [11], it represents a departure from the standard MPM approach for solid mechanics, in which the volumes associated with particles are tracked, see [16], for an analysis of this case. The particle's mass is calculated from the density and the volume of the particle as…”

“…This method of force calculation has been developed here as being more appropriate for compressible gas dynamics as it assumes that the particles within a cell have the same volume. Analysis and investigation of alternative methods for solid mechanics and different approaches to gas dynamics are provided by [6,13,16].…”

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

“…For example, B-splines have been shown to decrease quadrature errors and improve the spatial convergence rates for many MPM problems [11]. A typical one-dimensional quadratic B-spline can be constructed by convolving piecewise-constant basis functions with themselves:…”

“…While CD was shown to have the lowest error of the time-stepping methods in their tests, the method showed no temporal error convergence in the regions of time-step selection where their simulations were stable. This paper seeks to use and extend the perspective on the spatial errors gained in [11,12] to understand the lack of temporal convergence demonstrated in [7]. The inspiration for connecting the spatial error characteristics with the temporal error characteristics lies outside the MPM literature.…”

Hysteretic Model for the Explicit Material Point Method

“…In this work the shape functions used are cubic BSplines [4], [5] . They have been shown to reduce the quadrature errors and the grid crossing errors associated with discontinuous shape functions.…”