A Multi-Resolution Material Point Method Based on Penalty Formulation

The computational cost of the material point method (MPM) primarily arises from the information transfer between material points and the grid. In some high-precision simulations, using a globally high-resolution background grid and sufficient material particles per cell (PPC) can severely reduce computational efficiency. Many problems in geomechanics focus on local large deformations or failures, making global refinement an unsuitable choice. To address these issues, this study introduces a novel multi-resolution material point method (MR-MPM). This approach constructs MPM models at different resolution levels using bounded material points that connect these levels. By enforcing positional deviations of bounded material points through the penalty function, the high-resolution and low-resolution models are linked together, achieving local refinement. During implementation, it is only necessary to map the penalty forces of bounded material points to the corresponding level background grids. No other modifications are required, and no local equations need to be solved. Furthermore, this study derives a penalty factor value that eliminates the influence of material elastic modulus and background grid spacing, effectively simplifying parameter adjustment. Finally, a series of classical numerical examples, including elastic and elasto-plastic cases, are used to verify the algorithm's accuracy, convergence, and efficiency. The results predicted by MR-MPM closely match finite element reference solutions and demonstrate good convergence. Compared to globally refined MPM, MR-MPM significantly improves computational efficiency. These features give MR-MPM great potential in large-scale simulation analyses involving large local deformation, such as tunnel excavation, submarine landslides, and similar events.