A Three-Dimensional Arbitrary Grid Material Point Method for Large Deformation Problems with Geometrically Complex Boundaries

The Material Point Method (MPM) is a hybrid “mesh-particle” approach for large deformation simulations that avoids the high computational costs and boundary challenges of traditional meshfree particle methods. However, bottlenecks arise when modeling geometrically complex boundaries, especially in three dimensions. In this article, a three-dimensional arbitrary grid material point method (3D-AGMPM) is proposed to construct complex boundaries using polyhedral grid cells. This novel method establishes a unified framework for various cell types. In the particle-to-grid mapping procedure, the 3D-Wachspress basis functions are introduced as shape functions for arbitrary grid cells, while an improved hash-cell based particle localization algorithm is proposed to enhance computational efficiency. Nonlinear frictional boundary conditions are proposed in a trial-correction framework to model the frictional interaction between the body and complex geometric surfaces. In addition, a contact algorithm for polyhedral cells is proposed to handle contacts among multiple bodies within the arbitrary grid. Several benchmarks demonstrate the effectiveness of the proposed 3D-AGMPM for handling problems involving complex geometric boundaries. Furthermore, practical case studies, such as the Wangjiayan landslide, highlight its robustness and potential in addressing large-scale, complex engineering problems. Due to the simple implementation, flexibility, and high efficiency, the proposed 3D-AGMPM shows promise as a powerful tool for solving large deformation problems with geometrically complex boundaries.