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 C0 continuity property of the interpolation function, which causes discontinuous gradients across element boundaries. Prior efforts in constructing C1 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）是一种用于处理大变形固体力学的混合欧拉-拉格朗日仿真技术。标准MPM通常采用结构化背景网格，但在处理复杂几何构型时可能引发若干精度问题。当使用（二维）非结构化三角形或（三维）四面体背景单元时，会面临显著挑战（如单元穿越误差）。由于插值函数固有的C0连续性特性，导致单元边界上的梯度不连续，从而产生大量数值误差。此前构建C1连续插值函数的尝试要么未适用于非结构化网格，要么仅应用于二维三角形网格。本研究提出一种非结构化移动最小二乘物质点法（UMLS-MPM），可适用于二维和三维单纯形剖分。其核心思想是将衰减函数引入MLS核的样本权重中，确保速度梯度估计的解析连续性。数值分析证实该方法能够缓解单元穿越误差并实现预期的收敛性。