The Aggregated Material Point Method (AgMPM)

Abstract

The Material Point Method (MPM) has been shown to be an effective approach for analysing large deformation processes across a range of physical problems. However, the method suffers from a number of spurious artefacts, such as a widely documented cell crossing instability, which can be mitigated by adopting basis functions with higher order continuity. The larger stencil of these basis functions exacerbate a less widely discussed issue - small cuts. The small cut issue is linked to the arbitrary interaction between the physical body and the background mesh that is used to assemble and solve the governing equations in the MPM. There is the potential for degrees of freedom near the boundary of the body to have very small contributions from material points, which causes two problems: (i) artificially large accelerations/displacements at the boundary and (ii) ill conditioning of the global linear system. This paper provides a new mesh Aggregated MPM, or AgMPM, that mitigates the small cut issue by forming aggregated elements, tying the ill-behaved degrees of freedom to well posed interior elements. Implicit quasi-static and explicit dynamic formulations are provided and demonstrated through a series of numerical examples. The approach does not introduce any new numerical parameters and can be applied to implementations that adopt a lumped mass matrix. Aggregation is shown to significantly improve the stability of implicit implementations of the MPM, often at a lower computational cost compared to standard, non-aggregated, implementations. The technique improves the energy conservation and the stress field of explicit dynamic MPMs.