A time-stepping scheme for inelastic collisions - Numerical handling of the nonoverlapping constraint

We propose here a numerical scheme to compute the motion of rigid bodies with a non-elastic impact law. The method is based on a global computation of the reaction forces between bodies. Those forces, whose direction is known since we neglect friction effects, are identified at the discrete level with a scalar which plays the role of a Kuhn-Tucker multiplier associated to a first-order approximation of the non-overlapping constraint, expressed in terms of velocities. Since our original motivation is the handling of the non-overlapping constraint in fluid-particle direct simulations, we paid a special attention to stability and robustness. The scheme is proved to be stable and robust. As regards its asymptotic behaviour, a convergence result is established in the case of a single contact. Some numerical tests are presented to illustrate the properties of the algorithm. Firstly, we investigate its asymptotic behaviour in a situation of non-uniqueness, for a single particle. The two other sets of results show the good behaviour of the scheme for large time steps.