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Analysis and reduction of quadrature errors in the material point method (MPM)
Author(s) -
Steffen Michael,
Kirby Robert M.,
Berzins Martin
Publication year - 2008
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.2360
Subject(s) - quadrature (astronomy) , material point method , grid , mathematics , quadratic equation , basis function , finite element method , eulerian path , computation , computer science , spline (mechanical) , reduction (mathematics) , lagrangian , mathematical optimization , algorithm , mathematical analysis , geometry , structural engineering , engineering , electrical engineering
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.