Efficient multigrid based solvers for B-spline MPM

Abstract

The Material Point Method (MPM) has been applied successfully to problems in engineering which involve large deformations and history-dependent material behavior. However, the classical method suffers from some shortcomings which influence the quality of the numerical solution significantly. High-order B-spline basis functions solve the problem of so-called ‘grid crossing errors’ completely due to their higher continuity at inter-element boundaries. Adopting a consistent mass matrix instead of its lumped counterpart, which is common practice in standard MPM, further improves the convergence properties of the MPM. However, solving a linear system of equations resulting from a B-spline discretization is considered a challenging task. In this paper, we present a solution technique using p-multigrid methods to efficiently solve linear systems arising in B-spline MPM.