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cote interne IRCAM: Bensoam09a; None / None; National audience; This paper deals with the theoretical and numerical treatment of the unilateral dynamic contact problem between two arbitrary elastic bodies without friction. In addition to the classical variational statement that arises from static problems, a dynamic contact condition is needed and found by adjusting the balance laws of physical quantities to the impenetrability condition. In the context of infinitesimal deformation, a reciprocal formulation is then used to reduce this well-posed problem to one involving Green functions defined only on contact surfaces. It is then often possible to approximate the system using considerably fewer unknowns than with finite difference algorithms. The ability of the method to predict the contact interaction between two elastic bodies, irrespective of the material constitution and geometry, is highlighted by analytical and numerical simulations.

This paper investigates the reliability of an algorithm that implements the Green function method in free vibration analysis of Euler-Bernoulli beams. The investigation is concerned with the robustness of the algorithm with respect to the occurrence of numerical singularities in the calculation procedure of mode shapes. The problem is studied for beams supported with an arbitrary number of intermediate translational springs, which can be understood as a generalization of the cases when the beam is without elastic supports and when the beam rests on intermediate rigid supports. The problem of numerical singularities arises from the fact that the elements of the modal vector have to be expressed in terms of an "arbitrarily" chosen referential element of that vector, whose value can vanish if it coincides closely enough with a node of the sought mode shape. The problem is generally tackled here with the introduction of a fictitious spring of a vanishingly small stiffness, and the robustness of the algorithm depends crucially on the appropriate placement of that spring. This paper presents several useful guidelines for the implementation of computer code based on these principles and its reliability is demonstrated through examples.

We aim here at determining the Green function for general Schrödinger operators on product networks. The first step consists in expressing Schrödinger operators on a product network as sum of appropriate Schrödinger operators on each factor network. Hence, we apply the philosophy of the separation of variables method in PDE, to express the Green function for the Schrödinger operator on a product network using Green functions on one of the factors and the eigenvalues and eigenfunctions of some Schrödinger operator on the other factor network. We emphasize that our method only needs the knowledge of eigenvalues and eigenfunctions of one of the factors, whereas other previous works need the spectral information of both factors. We apply our results to compute the Green function of Pm×Sh , where Pm is a Path with m vertices and Sh is a Star network with h+1 vertices.