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 mitigate these issues by adopting smoother shape functions, but at the cost of increased 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 between stability 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. To further accelerate performance, we present a GPU-optimized implementation inspired by state-of-the-art massively parallel MPM techniques, achieving an additional $2\times$ speedup in G2P2G transfers over 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, high-performance simulations, paving the way for more computationally efficient applications in computer graphics and beyond.

翻译：物质点法已成为基于物理仿真的基石，广泛应用于岩土力学和计算机图形学中，用于模拟颗粒流、粘弹性、断裂力学等现象。尽管其通用性强，原始MPM存在由粒子-网格传递核函数不连续性引起的单元穿越不稳定性。现有解决方案通过采用更平滑的形函数来缓解这些问题，但代价是由于更大的核函数支撑域导致计算开销增加。本文提出一种新颖的$C^2$连续紧支撑核函数用于MPM，在稳定性和计算效率之间实现了独特的平衡。该方法与仿射粒子网格法和移动最小二乘MPM无缝集成，同时与线性核函数相比，每个粒子关联的网格节点数仅增加一倍。其核心是一个创新的双网格框架，该框架将粒子与两个交错网格上其占据单元内的网格节点进行关联，确保了一致且稳定的力计算。为进一步提升性能，我们提出了一种受先进大规模并行MPM技术启发的GPU优化实现，在G2P2G传输上相比二次B样条MPM实现了额外的$2\times$加速。通过单元测试、对比研究和压力测试的综合验证，证明了我们的方法在保持线性和角动量、处理刚性材料以及大规模仿真高效扩展方面的有效性。我们的结果凸显了紧支撑高阶核函数在提升MPM稳定高性能仿真能力方面的变革潜力，为计算机图形学及其他领域更高效计算的应用铺平了道路。