When  written in an Eulerian frame, the conservation laws of continuum mechanic are  similar for fluids and solids leading to a single set of variables for a monolithic formulation; the only difference is  in the expression of the stress tensors.   Such monolithic formulations are well adapted to large displacement fluid-structure configurations, but stability is a challenging problem because of moving geometries.

   In this talk the method and its discretization are presented, stability is discussed for an implicit in time finite element method in space by showing that energy decreases with time.  The key numerical ingredient is the Characterics-Galerkin method coupled with a powerful mesh generator.

   A numerical section discusses implementation issues and presents a few simple tests.

We will also present an numerical implementation of the same with contact where the variational inequality is solved by the semi-smooth Newton method.

   The case of a ball drop in a fluid with rebound at the bottom will be simulated in 2D.

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