Unstructured Moving Least Squares Material Point Methods: A Stable Kernel Approach With Continuous Gradient Reconstruction on General Unstructured Tessellations

TL;DR

This work addresses the cell-crossing error plaguing MPM on unstructured meshes by introducing Unstructured MLS-MPM (UMLS-MPM), which enforces a continuous velocity gradient via a diminishing function applied to MLS kernel weights. It develops a new MLS-based transfer kernel tailored for 2D and 3D simplex tessellations, extends MLS-MPM to unstructured meshes, and provides analytical and numerical verification across 1D, 2D, and 3D problems showing improved convergence and reduced cross-cell artifacts. The method demonstrates compatibility with APIC for angular momentum conservation and discusses practical limitations related to mesh quality and weight design, while outlining future directions such as dynamic sizing fields and hybrid schemes. Overall, UMLS-MPM enables stable, high-fidelity MPM simulations on general unstructured geometries with potential for broad application in complex boundary problems.

Abstract

The Material Point Method (MPM) is a hybrid Eulerian Lagrangian simulation technique for solid mechanics with significant deformation. Structured background grids are commonly employed in the standard MPM, but they may give rise to several accuracy problems in handling complex geometries. When using (2D) unstructured triangular or (3D) tetrahedral background elements, however, significant challenges arise (\eg, cell-crossing error). Substantial numerical errors develop due to the inherent $\mathcal{C}^0$ continuity property of the interpolation function, which causes discontinuous gradients across element boundaries. Prior efforts in constructing $\mathcal{C}^1$ continuous interpolation functions have either not been adapted for unstructured grids or have only been applied to 2D triangular meshes. In this study, an Unstructured Moving Least Squares MPM (UMLS-MPM) is introduced to accommodate 2D and 3D simplex tessellation. The central idea is to incorporate a diminishing function into the sample weights of the MLS kernel, ensuring an analytically continuous velocity gradient estimation. Numerical analyses confirm the method's capability in mitigating cell crossing inaccuracies and realizing expected convergence.

Figures (27)

• Figure 1: Schematic plot of the zeroth and first ring of neighbors.
• Figure 2: (a) When ${\mathcal{N}}_p^0$ alone is selected as the active nearby nodes, as a particle crosses the cell edge, the nodes indicated by the blue and red boxes are added or removed, respectively. Consequently, the weights there must approach zero to ensure ${\mathcal{C}}^0$ continuity, resulting in kernel degeneration along the edge. (b) Advancing to ${\mathcal{C}}^1$ addresses this issue by incorporating a sufficient number of surrounding nodes to fully encompass the particle.
• Figure 3: 1D meshes: (a) Uniform. (b) Uniform but truncated. (c) Periodically shrinking/expanding.
• Figure 4: Comparison of kernel values and gradient estimations on a uniform 1D mesh (a) with and (b) without applying the diminishing function.
• Figure 5: The negative weight for Node 3 (yellow) when the particle is in the boundary cell and there is no extra layer.

Theorems & Definitions (6)

• Proof
• Proof
• Definition 1
• Definition 2
• Definition 3
• Definition 4