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.

翻译：物质点法（MPM）是一种适用于大变形固体力学的混合欧拉-拉格朗日模拟技术。标准MPM通常采用结构化背景网格，但在处理复杂几何时可能引发若干精度问题。然而，当使用（二维）非结构化三角形或（三维）四面体背景单元时，会出现显著挑战（例如单元穿越误差）。由于插值函数固有的$\mathcal{C}^0$连续性特性导致单元边界处的梯度不连续，从而产生严重的数值误差。先前构建$\mathcal{C}^1$连续插值函数的尝试要么未适配非结构化网格，要么仅应用于二维三角形网格。本研究提出一种适用于二维和三维单纯形网格划分的非结构化移动最小二乘MPM（UMLS-MPM）。其核心思想是在MLS核函数的样本权重中引入衰减函数，从而确保速度梯度估计在解析意义上的连续性。数值分析证实了该方法在缓解单元穿越误差和实现预期收敛性方面的有效性。