# Extension of B-spline Material Point Method for unstructured triangular grids using Powell-Sabin splines

## TL;DR

This paper extends the B-spline Material Point Method to unstructured triangular grids by integrating Powell-Sabin splines, reducing grid-crossing errors and achieving higher-order convergence in simulations.

## Contribution

It introduces a novel extension of BSMPM to unstructured meshes using Powell-Sabin splines, enhancing applicability and accuracy.

## Key findings

- Significant reduction in grid-crossing errors.
- Higher-order convergence demonstrated in numerical tests.
- Effective application on unstructured triangular grids.

## Abstract

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 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 (PS) 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.