Variational Elastodynamic Simulation

Abstract

Numerical schemes for time integration are the cornerstone of dynamical simulations for deformable solids. The most popular time integrators for isotropic distortion energies rely on nonlinear root-finding solvers, most prominently, Newton’s method. These solvers are computationally expensive and sensitive to ill-conditioned Hessians and poor initial guesses; these difficulties can particularly hamper the effectiveness of variational integrators, whose momentum conservation properties require reliable root-finding. To tackle these difficulties, this paper shows how to express variational time integration for a large class of elastic energies as an optimization problem with a “hidden” convex substructure. This hidden convexity suggests uses of optimization techniques with rigorous convergence analysis, guaranteed inversion-free elements, and conservation of physical invariants up to tolerance/numerical precision. In particular, we propose an Alternating Direction Method of Multipliers (ADMM) algorithm combined with a proximal operator step to solve our formulation. Empirically, our integrator improves the performance of elastic simulation tasks, as we demonstrate in a number of examples.