An Efficient Multilevel Preconditioned Nonlinear Conjugate Gradient Method for Incremental Potential Contact

2026-04-21

AI Summary

While incremental potential contact (IPC) guarantees intersection-free simulations, its Newton-based solvers incur high computational costs due to Hessian assembly and linear solves. Existing preconditioned nonlinear conjugate gradient (PNCG) methods struggle with convergence in scenarios involving high stiffness and dense contacts. This work proposes MAS-PNCG, which integrates a multilevel additive Schwarz (MAS) preconditioner into nonlinear optimization. The method employs sparse-input Woodbury updates to dynamically adapt to changing contact sets without reconstructing the preconditioner at every step, introduces a Hessian-aware two-dimensional subspace minimization to replace heuristic search directions, and incorporates fast subdomain-level conservative continuous collision detection (CCD) to alleviate global step-size restrictions while preserving non-penetration guarantees. Compared to state-of-the-art Newton-PCG solvers (GIPC/StiffGIPC), MAS-PNCG achieves speedups of up to 5.66× and 2.07×, significantly enhancing contact simulation efficiency.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning
• Multiagent Systems: Distributed Problem Solving

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Incremental Potential Contact (IPC) guarantees intersection-free simulation but suffers from high computational costs due to the expensive Hessian assembly and linear solves required by Newton's method. While Preconditioned Nonlinear Conjugate Gradient (PNCG) avoids Hessian assembly, it has historically struggled with poor convergence in stiff, contact-rich scenarios due to the lack of effective preconditioners; simple Jacobi preconditioners fail to capture the global coupling, while advanced hierarchy-based preconditioners like Multilevel Additive Schwarz (MAS) are computationally prohibitive to rebuild at every nonlinear iteration. We present MAS-PNCG, a method that unlocks the power of hierarchical preconditioning for nonlinear optimization. Our key technical innovation is a Sparse-Input Woodbury update algorithm that incrementally adapts the fine-level MAS components to rapidly evolving contact sets. This bypasses the need for full preconditioner rebuilds, reducing maintenance cost to near-zero while capturing the complex spectral properties of the contact system. Furthermore, we replace heuristic PNCG search directions with a Hessian-aware 2D subspace minimization that optimally combines the preconditioned gradient and previous direction. We also apply a fast per-subdomain conservative CCD method that ensures penetration-free trajectories while avoiding overly restrictive global step sizes. Experiments demonstrate that our MAS-PNCG outperforms state-of-the-art Newton-PCG solvers, GIPC and StiffGIPC, both preconditioned with MAS up to 5.66$\times$ and 2.07$\times$ respectively.

Problem

• Incremental Potential Contact
• Nonlinear Conjugate Gradient
• Preconditioning
• Contact Simulation
• Computational Efficiency

Innovation

• Multilevel Preconditioning
• Incremental Potential Contact
• Nonlinear Conjugate Gradient
• Woodbury Update
• Conservative CCD