A comparison of numerical methods for solving multibody dynamics problems with frictional contact modeled via differential variational inequalities

Abstract

We investigate two classes of solvers used to determine the time evolution of large systems of rigid bodies that mutually interact through contact with friction. The contact is modeled through a complementarity condition; the friction is posed as a variational problem. The system dynamics is described by a set of differential algebraic equations coupled with differential variational inequalities (DVI). Upon discretization in time, the complementarity conditions enforced at the velocity level are relaxed to obtain a cone complementarity problem (CCP). The solution of the CCP, which becomes the simulation bottleneck, is found by minimizing an equivalent quadratic optimization problem with conic constraints. Herein, we investigate two classes of solvers for this constrained optimization problem. The projected Gauss-Jacobi (PGJ), projected Gauss-Seidel (PGS), and accelerated projected gradient descent (APGD) methods are exponents of the first class of solvers. They are first order, using only costfunction value and gradient information. The second class of solvers is represented by a symmetric cone interior point (SCIP) method and a primal-dual interior point (PDIP) method. These second order methods rely on a Newton step to identify the descent direction and a line search to compute the step size. All five methods draw on parallel computing on Graphics Processing Unit (GPU) cards; the Newton step employs a sparse parallel GPU solver. Two types of numerical experiments, filling and drafting, are carried out to evaluate the performance of the five solution strategies in terms of convergence rate, accuracy, and computational cost. For consistency, all numerical experiments were performed in the same open source code modified to host the five methods of interest.

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