Distillation of the material point method cell crossing error leading to a novel quadrature-based C0 remedy

The material point method (MPM) is a hybrid particle-mesh scheme conceptually placed between mesh-based and mesh-free methods, combining aspects of both and well suited to solving large deformation engineering problems involving history-dependent material models. A unique drawback of the MPM is the cell crossing error whereby material points crossing grid cells suddenly produce spurious stress oscillations leading to significant errors. After distilling the necessary conditions required for the cell crossing error, a novel quadrature-based C0 enhancement scheme is proposed. The partitioned quadrature material point method (PQMPM) effectively mitigates the cell crossing error while retaining the linear Lagrangian basis and Dirac density function of the original MPM, rendering it easily integrable into existing MPM codebases. One and two-dimensional examples illustrate the elimination of the cell crossing error, improvement of solution accuracy (particularly reliable stress prediction), local application of the scheme across dynamically varying arbitrary spatial and temporal regions and the reasonable additional computational costs against increased accuracy. Contrasting existing enhanced MPM schemes, the PQMPM only incurs additional computational cost exactly when and where cell crossing errors would otherwise occur and minimally diffuses local effects.