Comparison of two approaches for the discretization of elastodynamic contact problems

The purpose of this Note is to compare two approaches for the discretization of elastodynamic contact problems. First, we introduce an energy conserving method based on a standard midpoint scheme and a contact condition expressed in terms of velocity. The second approach consists in considering an equivalent distribution of the body mass so that the nodes on the contact boundary have no inertia. We prove that this method leads to an energy conservation for the space semi-discretized elastodynamic contact problem. Finally, some numerical results are presented in the two dimensional case. To cite this article: H.B. Khenous et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Exponential stabilization of a flexible structure with a local interior control and under the presence of a boundary infinite memory

In this paper, we deal with the interior stabilization problem of a flexible structure governed by a hyperbolic partial differential equation coupled to two ordinary differential equations. Contrary to the previous works on the system, the boundary control is subject to the presence of an infinite memory term. In order to deal with such a nonlocal term, the minimal state approach is invoked. Specifically, a localized interior control is proposed in order to compensate the infinite memory effect. Thereafter, reasonable assumptions on the memory kernel are evoked so that the closed-loop system is shown to be well-posed thanks to semigroups theory of linear operators. Furthermore, the resolvent method is used to establish the exponential stability of the system.