Nodal force error and its reduction for material point methods

Abstract

This work shows that with the presence of a prestress, the discretized momentum equation using the material point method (MPM) or the dual domain material point (DDMP) method is inconsistent with the continuum version of the equation. This inconsistency leads to incorrect wave speed predictions in prestressed materials and numerical instabilities. The source of this inconsistency is the nodal force calculation using the material points.To address this issue, we introduce an auxiliary stress field, which can be regarded as a local background stress. We then reformulate the nodal force calculation, such that it only depends on the local stress variation. The effect of the prestress is then subtracted away in the nodal force calculation, and the consistencies of MPM and DDMP are then restored.This reformulation only requires simple modifications to the original nodal force in both MPM and DDMP. After implementing this reformulated nodal force in a numerical code and comparing the results with manufactured solutions, we found significant improvements not only in the accuracy of the solutions, but also in the numerical stability for both MPM and DDMP. With the reformulated nodal force, the DDMP results converge rapidly as the number of material points increases.