Abstract
This presentation focuses on the numerical simulation of granular materials and suspensions. Contacts between grains lead to singular interactions for which adapted numerical schemes must be developed. We take as our starting point models of the type Dynamique des contacts developed by J. J. Moreau, using non-smooth convex analysis. Friction models between grains then lead to complex non-convex optimization problems. The aim of the presentation is to show how algorithms based, at each instant, on convex optimization problems can be used to obtain numerical simulations of large numbers of particles in long time, taking friction and lubrication into account. In these schemes, contact forces are obtained implicitly, as Lagrange multipliers associated with constraints. The fundamental principle of dynamics is then obtained (in a discretized version) from the Euler (optimality) equations of the minimization problem. In the case of frictionless grains, with or without a lubrication model, we return to affine constraint minimization problems. To introduce friction (Coulomb model) between grains, we have to consider minimization problems under conical stress, for which the stress is therefore non-derivable. We will illustrate this work with numerical simulations, showing that the resulting schemes can be used to study, for example, the macroscopic behavior of granular materials.
