The Material Point Method (MPM) is a numerical technique that combines a fixed Eulerian background grid and Lagrangian point masses to simulate materials which undergo large deformations. Within the original MPM, discontinuous gradients of the piecewise-linear basis functions lead to the so-called grid-crossing errors when particles cross element boundaries. Previous research has shown that B-spline MPM (BSMPM) is a viable alternative not only to MPM, but also to more advanced versions of the method that are designed to reduce the grid-crossing errors. In contrast to many other MPM-related methods, BSMPM has been used exclusively on structured rectangular domains, considerably limiting its range of applicability. In this paper, we present an extension of BSMPM to unstructured triangulations. The proposed approach combines MPM with \(C^1\)-continuous high-order Powell–Sabin spline basis functions. Numerical results demonstrate the potential of these basis functions within MPM in terms of grid-crossing-error elimination and higher-order convergence.

材质点方法（MPM）是一种将固定的欧拉背景网格和拉格朗日点质量结合起来的数值技术，以模拟经受大变形的材料。在原始MPM内，分段线性基函数的不连续梯度会在粒子穿过元素边界时导致所谓的网格交叉误差。先前的研究表明，B样条MPM（BSMPM）不仅是MPM的可行替代方案，而且是该方法的更高级版本的一种可行的替代方法，旨在减少网格交叉误差。与许多其他与MPM相关的方法相比，BSPMM仅用于结构化矩形域，极大地限制了其适用范围。在本文中，我们提出了BSMPM到非结构化三角剖分的扩展。提议的方法将MPM与\（C ^ 1 \） -连续的高阶Powell-Sabin样条基函数。数值结果证明了在消除网格交叉误差和高阶收敛方面，这些基本函数在MPM中的潜力。