# Fluid Structure Interaction

We will interest now to the different interactions a fluid and a structure can have together with specific conditions.

## Fluid structure model

To describe and solve our fluid-structure interaction problem, we need to define a model, which regroup structure model and fluid model parts.

We have then in one hand the fluid equations, and in the other hand the structure equations.

The solution of this model are \((\mathcal{A}^t, \boldsymbol{u}_f, p_f, \boldsymbol{\eta}_s)\).

## ALE

Generally, the solid mechanic equations are expressed in a Lagrangian frame, and the fluid part in Eulerian frame. To define and take in account the fluid domain displacement, we use a technique name ALE ( Arbitrary Lagrangian Eulerian ). This allow the flow to follow the fluid-structure interface movements and also permit us to have a different deformation velocity than the fluid one.

Let denote \(\Omega^{t_0}\) the calculation domain, and \(\Omega^t\) the deformed domain at time \(t\). As explain before, we want to conserve the Lagrangian and Eulerian characteristics of each part, and to do this, we introduce \(\mathcal{A}^t\) the ALE map.

This map give us the position of \(x\), a point in the deformed domain at time \(t\) from the position of \(x^*\) in the initial configuration \(\Omega^*\).

\(\mathcal{A}^t\) is a homeomorphism, i.e. a continuous and bijective application we can define as

We denote also \(\forall \mathbf{x}^* \in \Omega^*\), the application :

This ALE map can then be retrieve into the fluid-structure model.