JGS2: Near Second-order Converging Jacobi/Gauss-Seidel for GPU Elastodynamics

TL;DR

This paper introduces a GPU algorithm for elastodynamics that achieves near second-order convergence, combining the rapid convergence of Newton's method with the parallel efficiency of Jacobi, significantly improving simulation quality and speed.

Contribution

A novel GPU algorithm that mitigates overshoot to attain near second-order convergence while maintaining high parallelizability, outperforming existing methods in elastodynamics simulations.

Findings

Achieves convergence rates comparable to Newton's method.

Outperforms state-of-the-art GPU methods by 50-100 times.

Maintains high parallelism with marginal runtime overhead.

Abstract

In parallel simulation, convergence and parallelism are often seen as inherently conflicting objectives. Improved parallelism typically entails lighter local computation and weaker coupling, which unavoidably slow the global convergence. This paper presents a novel GPU algorithm that achieves convergence rates comparable to fullspace Newton's method while maintaining good parallelizability just like the Jacobi method. Our approach is built on a key insight into the phenomenon of overshoot. Overshoot occurs when a local solver aggressively minimizes its local energy without accounting for the global context, resulting in a local update that undermines global convergence. To address this, we derive a theoretically second-order optimal solution to mitigate overshoot. Furthermore, we adapt this solution into a pre-computable form. Leveraging Cubature sampling, our runtime cost is only marginally higher than the Jacobi method, yet our algorithm converges nearly quadratically as Newton's method. We also introduce a novel full-coordinate formulation for more efficient pre-computation. Our method integrates seamlessly with the incremental potential contact method and achieves second-order convergence for both stiff and soft materials. Experimental results demonstrate that our approach delivers high-quality simulations and outperforms state-of-the-art GPU methods with 50 to 100 times better convergence.