An Implicit Compact-Kernel Material Point Method for Computational Solid Mechanics

Abstract: The numerical performance of the material point method (MPM) is strongly governed by the particle-grid kernel, which controls the trade-off among smoothness, locality, numerical diffusion, contact accuracy, and computational cost. Although wide-support smooth kernels can effectively suppress cell-crossing instability, they often introduce increased numerical diffusion, artificial contact gaps, and higher transfer cost. In contrast, the suitability of compact-kernel designs for implicit computational solid mechanics remains unclear. In this work, we develop an implicit formulation of the Compact-Kernel Material Point Method (CK-MPM) and assess its performance through benchmark problems in linear and nonlinear solid mechanics, including cantilever bending, Hertzian contact, narrow-clearance free fall, and colliding hyperelastic rings. The results show that implicit CK-MPM retains the advantages of compact support while preserving the smoothness required for robust large-deformation simulation. Compared with linear MPM, it reduces cell-crossing-induced stress noise and excessive numerical dissipation; compared with quadratic B-spline MPM, it improves contact locality and reduces artificial contact gaps and early-contact artifacts while maintaining comparable overall smoothness and accuracy. These results indicate that CK-MPM provides a viable implicit MPM framework for computational mechanics.