Analysis and Reduction of Quadrature Errors in the Material Point Method (MPM)

Summary

The theoretical details of one-dimentional cubic B-splines for representing grid basis functions

Abstract

The 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. However, like most methods which 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 finite element methods, 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 of 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 significant impact on the reduction of the internal force quadrature error (and corresponding “grid crossing error”) often experienced when using the material point method. Copyright c 2008 John Wiley & Sons, Ltd.

Author

Michael Steffen
Robert M. Kirby
Martin Berzins

School of Computing and Scientific Computing and Imaging Institute, University of Utah

Journal/Conference

International Journal for Numerical Methods in Engineering 2008.

j20232 / survey