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

2026-04-20

AI Summary

This work proposes an implicit compact-kernel material point method (CK-MPM) that integrates implicit time integration with compactly supported kernel functions to simultaneously achieve smoothness, locality, low numerical dissipation, and high contact accuracy in large-deformation solid mechanics. Traditional material point methods struggle to balance these competing requirements, and the suitability of compact kernels within implicit frameworks has remained unclear. The proposed CK-MPM preserves locality while fulfilling the smoothness necessary for robust large-deformation simulations. This study presents the first validation of compact kernels in an implicit setting, demonstrating significantly reduced stress noise and numerical dissipation. Compared to quadratic B-spline MPM, CK-MPM enhances contact locality, eliminates artificial gaps and premature contact artifacts, and maintains comparable accuracy and computational efficiency.

Technology Category

• Machine Learning: Kernel Methods
• Data Mining & Knowledge Management: Data Compression
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

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.

Problem

• Material Point Method
• Compact Kernel
• Implicit Formulation
• Computational Solid Mechanics
• Numerical Stability

Innovation

• Implicit MPM
• Compact-kernel
• Cell-crossing instability
• Contact accuracy
• Numerical diffusion