Efficient and Robust ADMM Methods for Dynamics and Geometry Optimization

Abstract

We present novel ADMM-based methods for efficiently solving problems in a variety of applications in computer graphics. First, in the domain of physics-based animation we propose new techniques for simulating elastic bodies subject to dissipative forces. Second, in the field of geometry optimization we introduce a new algorithm for quasi-static deformation and surface parameterization. In each of these applications our proposed methods robustly converge to accurate solutions, and do so faster than existing algorithms. We achieve this by using key insights to address and overcome many limitations of standard solvers. Here we highlight the features of our two new algorithms for the aforementioned problems. Then we summarize our preliminary investigations into new techniques we designed in pursuit of faster and more reliable convergence in parameterization problems. Our first method is one for incorporating dissipative forces into optimization-based time integration schemes, which hitherto have been applied almost exclusively to systems with only conservative forces. We represent such forces using dissipation functions that may be nonlinear in both positions and velocities, enabling us to model a range of dissipative effects including Coulomb friction, Rayleigh damping, and power-law dissipation. To improve accuracy and minimize artificial damping, we provide an optimization-based version of the second-order accurate TR-BDF2 integrator. Finally, we present a method for modifying arbitrary dissipation functions to conserve linear and angular momentum, allowing us to eliminate the artificial angular momentum loss caused by Rayleigh damping. Our second method is designed to efficiently solve geometry optimization problems. We observe that in this domain existing local-global solvers such as ADMM struggle to resolve large rotations such as bending and twisting modes, and large distortions in the presence of barrier energies. We propose two improvements to address these challenges. First, we introduce a novel local-global splitting based on the polar decomposition that separates the geometric nonlinearity of rotations from the material nonlinearity of the deformation energy. The resulting ADMM-based algorithm is a combination of an L-BFGS solve in the global step and proximal updates of element stretches in the local step. We also introduce a novel method for dynamic reweighting that is used to adjust element weights at runtime for improved convergence. With both improved rotation handling and element weighting, our WRAPD algorithm is considerably faster than state-of-the-art approaches for quasi-static simulations. WRAPD is also much faster at making early progress in parameterization problems, making it valuable as an initializer to jump-start second-order algorithms. Finally, we investigate two possible extensions to WRAPD for accelerated convergence in parameterization problems. The first extension, P-WRAPD, leverages progressive reference shape updates similar to Liu et al. [2018] to bound distortion. We show that this yields minor improvements in a subset of examples. Our second extension, N-WRAPD, uses a non-scalar weighting scheme that independently assigns unique weights for every mode of deformation. This method shows promising preliminary results. Although slightly less stable, N-WRAPD generally converges significantly faster at a rate comparable to second-order algorithms.