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The Material Point Method (MPM) is a hybrid Eulerian-Lagrangian simulationtechnique for solid mechanics with significant deformation. Structuredbackground grids are commonly employed in the standard MPM, but they may giverise 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 numericalerrors develop due to the inherent C0 continuity property of the interpolationfunction, which causes discontinuous gradients across element boundaries. Priorefforts in constructing C1 continuous interpolation functions have either notbeen adapted for unstructured grids or have only been applied to 2D triangularmeshes. In this study, an Unstructured Moving Least Squares MPM (UMLS-MPM) isintroduced to accommodate 2D and 3D simplex tessellation. The central idea isto incorporate a diminishing function into the sample weights of the MLSkernel, ensuring an analytically continuous velocity gradient estimation.Numerical analyses confirm the method's capability in mitigating cell crossinginaccuracies and realizing expected convergence.

Material Point Methods on Unstructured Tessellations: A Stable Kernel Approach With Continuous Gradient Reconstruction