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The Material Point Method (MPM) has become a cornerstone of physics-based simulation, widely used in geomechanics and computer graphics for modeling phenomena such as granular flows, viscoelasticity, fracture mechanics, etc. Despite its versatility, the original MPM suffers from cell-crossing instabilities caused by discontinuities in particle-grid transfer kernels. Existing solutions mostly mitigate these issues by adopting smoother shape functions, but at the cost of increased numerical diffusion and computational overhead due to larger kernel support. In this paper, we propose a novel C 2 -continuous compact kernel for MPM that achieves a unique balance in terms of stability, accuracy, and computational efficiency. Our method integrates seamlessly with Affine Particle-In-Cell (APIC) and Moving Least Squares (MLS) MPM, while only doubling the number of grid nodes associated with each particle compared to linear kernels. At its core is an innovative dual-grid framework, which associates particles with grid nodes exclusively within the cells they occupy on two staggered grids, ensuring consistent and stable force computations. We demonstrate that our method can be conveniently implemented using a domain-specific language, Taichi, or based on open-source GPU MPM frameworks, achieving faster runtime and less numerical diffusion compared to quadratic B-spline MPM. Comprehensive validation through unit tests, comparative studies, and stress tests demonstrates the efficacy of our approach in conserving both linear and angular momentum, handling stiff materials, and scaling efficiently for large-scale simulations. Our results highlight the transformative potential of compact, high-order kernels in advancing MPM's capabilities for stable, accurate, and high-performance simulations.
Liu et al. (Sun,) studied this question.