An Efficient Arbitrary Grid Material Point Method for Problems With Nonconforming Boundary Conditions

In the standard material point method (MPM), a Cartesian background grid is typically used to solve equations of motion. This can make imposing boundary conditions a challenging task when the boundary of the material domain does not align with the grid edge, as these nonconforming boundary conditions are difficult to apply directly to the nodes of the Cartesian grid. In this paper, we propose the Arbitrary Grid Material Point Method (AGMPM) to efficiently solve problems involving a nonconforming boundary conditions by converting them into conforming boundary conditions. In the AGMPM, boundaries with arbitrary geometries are constructed by using arbitrary convex polygonal grid cells. The Wachspress coordinates are introduced as the shape functions for these grid cells. To impose boundary conditions on the arbitrary grid, two specific types of boundary conditions are proposed as examples: roller boundary conditions, which are two-sided constraints, and rigid-wall-boundary conditions, which are single-sided constraints. These boundary conditions are extensions of those used in the standard MPM. To improve computational efficiency during the particle-to-mesh mapping, an efficient search algorithm based on the bucket search method is presented. Several numerical examples are studied to verify the proposed AGMPM, demonstrate its potential and flexibility in solving engineering problems, and showcase its improved accuracy compared to the standard MPM when dealing with nonconforming boundary conditions.