Fast and stable Gauss-Newton optimization of IPC barrier energy

Barrier terms for Incremental Potential Contact (IPC) energy are crucial for maintaining an intersection and inversion free simulation trajectory. However, existing formulations which directly use distance for measuring the state of contact can restrict implementation design and performance. This is because numerical eigendecompositions are required for guaranteeing positive semi-definiteness of the Hessian of the barrier energy during optimization with Projected-Newton solvers, and alternative Gauss-Newton methods suffer from significantly reduced convergence rates. We rewrite the barrier function of IPC to derive an efficient approximation its Hessian, which can then be used for simultaneous construction and projection to positive semi-definite state. The key idea is to formulate a simplicial geometric measure of contact using mesh boundary elements, where analytic eigensystems are obtained for minimising the proposed barrier energy with superlinear convergence to retain much of the advantage of Newton-type methods. Our approach is suitable for standard second order unconstrained optimization strategies for IPC based collision processing, minimizing nonlinear nonconvex functions where the Hessian may be indefinite. The result is a 3-times speedup over the standard full-space IPC barrier formulation based on direct Euclidean distance measures. We further apply our analytic proxy eigensystems to produce an entirely GPU-based implementation of IPC with significant further acceleration.