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

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 (FEMs). However, similar to most met...

Full description

Saved in:
Bibliographic Details
Published in:International journal for numerical methods in engineering 2008-11, Vol.76 (6), p.922-948
Main Authors: Steffen, Michael, Kirby, Robert M., Berzins, Martin
Format: Article
Language:English
Subjects:
Exact sciences and technology
Fracture mechanics (crack, fatigue, damage...)
Fundamental areas of phenomenology (including applications)
material point method
mesh-free methods
meshless methods
particle methods
Physics
quadrature
smoothed particle hydrodynamics
Solid mechanics
Structural and continuum mechanics
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 (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. Copyright © 2008 John Wiley & Sons, Ltd.
ISSN:0029-5981
1097-0207
DOI:10.1002/nme.2360