JGS2-GQ: Training-free 2nd Jacobi with Gaussian Quadrature

JGS2 is a Jacobi-like GPU simulation algorithm. It avoids the overshooting issue by augmenting each subproblem with a perturbation subspace that predicts the global influence of the local solve. The efficiency of JGS2 is due to Cubature-based subspace integration at each subproblem. Being a data-driven method, Cubature requires a set of representative deformed poses that cover deformations likely to occur in the simulation. This requirement is unlikely for high-resolution deformation with rich local details. Therefore, simulation performance and convergence degenerate when Cubature extrapolates. This paper proposes a training-free subspace integration algorithm based on classic Gaussian quadrature (GQ). We leverage the fact that the subproblem's subspace bases can be well-approximated by a low-degree multivariable polynomial, which suggests GQ an excellent candidate for Cubature substitute. To this end, we introduce a novel algorithm that adaptively generates the integration region for each subproblem. As a result, GQ integration can be analytically retrieved without cumbersome data generation and training. We also show how to handle frictional contact by modifying the pre-computed perturbation subspace. The resulting JGS2-GQ framework is more versatile than the vanilla JGS2 method. It is more stable for large and novel deformations, and is free of data generation and expensive training, while maintaining a near second-order convergence that is comparable to Newton. Performance-wise, JGS2-GQ is as efficient as JGS2, which is three orders faster than classic CPU methods and up to two orders faster than classic GPU algorithms. When novel deformation occurs, JGS2-GQ outperforms JGS2 over 50%.