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International audience; We consider a discrete mechanical system subjected to perfect time-dependent unilateral constraints, which dynamics is described by a second order measure differential inclusion. The transmission of the velocity at impacts is given by a minimization property of the kinetic energy with respect to the set of kinematically admissible post-impact velocities. We construct a sequence of feasible approximate positions by using a time-stepping algorithm inspired by a kind of Euler discretization of the differential inclusion. We prove the convergence of the approximate trajectories to a solution of the Cauchy problem and we obtain as a by-product a global existence result.

International audience; In computational structural dynamics, particularly in the presence of nonsmooth behavior, the choice of the time-step and the time integrator has a critical impact on the feasibility of the simulation. Furthermore, in some cases, as in the case of a bridge crane under seismic loading, multiple time-scales coexist in the same problem. In that case, the use of multi-time scale methods is suitable. Here, we propose a new explicit–implicit heterogeneous asynchronous time integrator (HATI) for nonsmooth transient dynamics with frictionless unilateral contacts and impacts. Furthermore, we present a new explicit time integrator for contact/impact problems where the contact constraints are enforced using a Lagrange multiplier method. In other words, the aim of this paper consists in using an explicit time integrator with a fine time scale in the contact area for reproducing high frequency phenomena, while an implicit time integrator is adopted in the other parts in order to reproduce much low frequency phenomena and to optimize the CPU time. In a first step, the explicit time integrator is tested on a one-dimensional example and compared to Moreau-Jean’s event-capturing schemes. The explicit algorithm is found to be very accurate and the scheme has generally a higher order of convergence than Moreau-Jean’s schemes and provides also an excellent energy behavior. Then, the two time scales explicit–implicit HATI is applied to the numerical example of a bridge crane under seismic loading. The results are validated in comparison to a fine scale full explicit computation. The energy dissipated in the implicit–explicit interface is well controlled and the computational time is lower than a full-explicit simulation.

International audience; We consider a system of rigid bodies subjected to unilateral constraints with soft contact and dry friction. When the constraints are saturated, velocity jumps may occur and the dynamics is described in generalized coordinates by a second-order measure differential inclusion for the unknown configurations. Observing that the right velocity obeys a minimization principle, a time-stepping algorithm is proposed. It allows to construct a sequence of approximate solutions satisfying at each time-step a discrete contact law which mimics the behaviour of the system in case of collision. In case of tangential contact, dry friction may lead to indeterminacies such as the famous Painlevé’s paradoxes. By a precise study of the asymptotic properties of the scheme, it is shown that the limit of the approximate trajectories exhibits the same kind of indeterminacies.

We consider a discrete mechanical system subjected to perfect unilateral contraints characterized by some geometrical inequalities $f_\alpha(q) >=0$, $\alpha \in {1,...,v}$, with $v >=1$. We assume that the transmission of the velocities at impacts is governed by a Newton’s impact law with a restitution coefficient $e \in [0, 1]$, allowing for conservation of kinetic energy if $e=1$, or loss of kinetic energy if $e \in [0, 1)$, when the constraints are saturated. Starting from a formulation of the dynamics as a first order measure-differential inclusion for the unknown velocities, time-stepping schemes inspired by the proximal methods can be proposed. Convergence results in the single-constraint case $( v = 1)$ are recalled and extended to the multi-constraint case $( v > 1)$, leading to new existence results for this kind of problems.

International audience; We calculate numerically the motion of a slender bar dropped on a rigid foundation. For the computation the bar is discretized by a system of rigid bodies linked by spiral springs or by a pair of linear springs. We assume that the impact is frictionless and we model it by Newton's law. We compute the motion by using either an event-driven method based on the detection of impacts or a time-stepping scheme avoiding the detection of impacts. We calculate also the apparent restitution coefficient and we compare our results with the experimental and numerical results of Stoianovici and Hurmuzlu.