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 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.
Highlights
The Material Point Method (MPM) has proven to be successful in solving complex engineering problems that involve large deformations, multi-phase interactions and historydependent material behaviour
We propose an extension of B-spline MPM (BSMPM) to unstructured triangulations to combine the benefits of Bsplines with the geometric flexibility of triangular grids
We would like to remark that this paper focuses on PS-splines, other options such as refinable C1 splines [21] can be used to extend MPM to unstructured triangular grids
Summary
The Material Point Method (MPM) has proven to be successful in solving complex engineering problems that involve large deformations, multi-phase interactions and historydependent material behaviour. The DDMP method replaces the gradients of the piecewise-linear Lagrange basis functions in standard MPM by smoother ones. The B-spline Material Point Method (BSMPM) [28, 29] solves the problem of grid-crossing errors completely by replacing piecewise-linear Lagrange basis functions with higher-order B-spline basis functions. The main advantage of higher-order B-spline basis functions over piecewise-linear Lagrange basis function is, that they have at least C0-continuous gradients which preclude grid-crossing errors from the outset. On structured rectangular grids, adopting B-spline basis functions within MPM eliminates grid-crossing errors and yields higher-order spatial convergence [3,11,28,29, 37].
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