A stable poro-mechanical formulation for Material Point Methods leveraging overlapping meshes and multi-field ghost penalisation

TL;DR

This paper introduces a stable poro-mechanical formulation for the Material Point Method that overcomes instability issues related to inf-sup condition violations and system conditioning by using overlapping meshes and ghost penalisation.

Contribution

It proposes a novel MPM formulation employing overlapping meshes and ghost penalisation to ensure stability and improve solvability in coupled solid-water problems.

Findings

Stable pressure and displacement fields achieved.

Improved system condition number demonstrated.

Validated with elastic and elasto-plastic numerical examples.

Abstract

The Material Point Method (MPM) is widely used to analyse coupled (solid-water) problems under large deformations/displacements. However, if not addressed carefully, MPM u-p formulations for poro-mechanics can be affected by two major sources of instability. Firstly, inf-sup condition violation can arise when the spaces for the displacement and pressure fields are not chosen correctly, resulting in an unstable pressure field. Secondly, the intrinsic nature of particle-based discretisation makes the MPM an unfitted mesh-based method, which can affect the system's condition number and solvability, particularly when background mesh elements are poorly populated. This work proposes a solution to both problems. The inf-sup condition is avoided using two overlapping meshes, a coarser one for the pressure and a finer one for the displacement. This approach does not require stabilisation of the primary equations since it is stable by design and is particularly valuable for low-order shape functions. As for the system's poor condition number, a face ghost penalisation method is added to both the primary equations, which constitutes a novelty in the context of MPM mixed formulations. This study frequently makes use of the theories of functional analysis or the unfitted Finite Element Method (FEM). Although these theories may not directly apply to the MPM, they provide a robust and logical basis for the research. These rationales are further supported by three numerical examples, which encompass both elastic and elasto-plastic cases and drained and undrained conditions.