Solving variational inequalities and cone complementarity problems in nonsmooth dynamics using the alternating direction method of multipliers

Abstract

This work presents a numerical method for the solution of variational inequalities arising in nonsmooth flexible multibody problems that involve set-valued forces. For the special case of hard frictional contacts, the method solves a second order cone complementarity problem. We ground our algorithm on the Alternating Direction Method of Multipliers (ADMM), an efficient and robust optimization method that draws on few computational primitives. In order to improve computational performance, we reformulated the original ADMM scheme in order to exploit the sparsity of constraint jacobians and we added optimizations such as warm starting and adaptive step scaling. The proposed method can be used in scenarios that pose major difficulties to other methods available in literature for complementarity in contact dynamics, namely when using very stiff finite elements and when simulating articulated mechanisms with odd mass ratios. The method can have applications in the fields of robotics, vehicle dynamics, virtual reality, and multiphysics simulation in general.

Modular and Parallelizable Multibody Physics Simulation via Subsystem-Based ADMM

“…3) Full ADMM (FADMM): State-of-the-art implementations of ADMM algorithms [39] can be used to solve physics simulation, which is specified in [29]. The main difference with our algorithm is that they require solving of the fullsystem size matrix for each iteration.…”

“…In contrast, by utilizing the structural peculiarity of ADMM, our proposed framework can handle all the constraints in a decoupled manner for each iteration phase, thereby not only substantially improving the algorithmic efficiency but also allowing for its extension to a wide range of multibody systems. We also note that [21], [29], [30] employ ADMM structure in simulation. However, their full system level approaches still require large-sized matrix operations.…”