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

Giuliano Pretti, Robert E. Bird, Nathan D. Gavin, William M. Coombs, Charles E. Augarde

TL;DR

The paper tackles two major instabilities in MPM poro-mechanics: inf-sup violations between displacement and pressure spaces and ill-conditioning from the unfitted, small-cut nature of particle-based discretisation. It resolves these by using overlapping meshes (pressure on a coarser grid and displacement on a finer grid, i.e., $Qk_{SD}$-$Qk$) to achieve stability by design and by introducing a face ghost penalty to improve conditioning on boundary-cut elements. The approach is analyzed through references to unfitted-FEM theory and validated via three numerical experiments involving elastic and elasto-plastic behaviour under drained and undrained conditions. Results show oscillation-free pressure fields and well-conditioned Jacobians, demonstrating robustness across large deformations and complex boundary interactions. This method offers a stable, implicit MPM framework for coupled solid-fluid problems without injecting stabilization into the primary equations themselves, enhancing reliability for geotechnical and biomechanical simulations.

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.

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

TL;DR

The paper tackles two major instabilities in MPM poro-mechanics: inf-sup violations between displacement and pressure spaces and ill-conditioning from the unfitted, small-cut nature of particle-based discretisation. It resolves these by using overlapping meshes (pressure on a coarser grid and displacement on a finer grid, i.e., -) to achieve stability by design and by introducing a face ghost penalty to improve conditioning on boundary-cut elements. The approach is analyzed through references to unfitted-FEM theory and validated via three numerical experiments involving elastic and elasto-plastic behaviour under drained and undrained conditions. Results show oscillation-free pressure fields and well-conditioned Jacobians, demonstrating robustness across large deformations and complex boundary interactions. This method offers a stable, implicit MPM framework for coupled solid-fluid problems without injecting stabilization into the primary equations themselves, enhancing reliability for geotechnical and biomechanical simulations.

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.
Paper Structure (32 sections, 50 equations, 11 figures, 3 tables, 2 algorithms)

This paper contains 32 sections, 50 equations, 11 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Fundamentals of mixture theory
  3. Kinematics of the phases
  4. Mass conservation
  5. Balance of rate of linear momentum
  6. Constitutive relationships
  7. Continuum weak form and arising requirements
  8. From strong to weak form of equations
  9. Linearisation
  10. Stable MPM discretisation
  11. MPM algorithm
  12. The small-cut issue in the MPM
  13. Face ghost stabilisation
  14. Qk$_{SD}$-Qk spatial discretisation
  15. sMPM shape functions
  16. ...and 17 more sections

Figures (11)

  • Figure 1: MPM step phases.
  • Figure 2: Algorithmic selection of boundary elements and facets (top row) for finer (left) and coarser (right) meshes, and stabilised facets (bottom row).
  • Figure 3: Combination of GIMPM functions on overlapping meshes. Thick grey lines illustrate the finer mesh, while white dashed lines represent the coarser one.
  • Figure 4: Illustration of the Terzaghi mono-dimensional problem. Permeable surfaces are designed by the dashed line.
  • Figure 5: Graphical comparison between analytical formula and numerical solutions obtained with the PPP stabilisation and Q1$_{SD}$-Q1 meshes.
  • ...and 6 more figures