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

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.