On convex numerical schemes for inelastic contacts with friction

Abstract

This paper reviews the different existing Contact Dynamics schemes for the simulation of granular media, for which the discrete incremental problem is based on the resolution of convex problems. This type of discretization has the great advantage of allowing the use of standard convex optimization algorithms. In the case of frictional contacts, we consider schemes based on a convex relaxation of the constraint as well as a fixed point scheme. The model and the computations leading to the discrete problems are detailed in the case of convex, regular but not necessarily spherical particles. We prove, using basic tools of convex analysis, that the discrete optimization problem can be seen as a minimization problem of a global discrete energy for the system, in which the velocity to be considered is an average of the pre- and post-impact velocities. A numerical study on an academic test case is conducted, illustrating for the first time the convergence with order 1 in the time step of the different schemes. We also discuss the influence of the convex relaxation of the constraint on the behavior of the system. We show in particular that, although it induces numerical dilatation, it does not significantly modify the macrosopic behavior of a column collapse en 2d. The numerical tests are performed using the code SCoPI.