A High Order Material Point Method

Abstract

The classical material point method (MPM) developed in the 90s is known for drawbacks which affect the quality of results. The movement of material points from one element to another leads to non-physical oscillations known as ‘grid crossing errors’. Furthermore, the use of material points as integration points renders a numerical quadrature rule of limited quality. Different solutions have been proposed in recent years to overcome these drawbacks. In this paper the approach of combining quadratic B-spline basis functions with a reconstruction based quadrature rule is pursued to solve these numerical problems. High-order B-spline basis functions solve the problem of grid crossing completely, whereas the considered reconstruction based quadrature rule reduces the quadrature error observed with MPM. In addition, the use of quadratic B-splines leads to a more accurate piecewise linear approximation of the stress field compared to the piecewise constant one obtained with linear Lagrangian basis functions commonly used with MPM. Two 1D benchmarks are considered involving large deformations, a vibrating bar and a column under self-weight. They render excellent results when adopting this high-order MPM.