Toolbox is available at FSI Toolbox.

## 1. Fluid Structure Interaction

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

### 1.1. 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.

Figure 1 : FSI case example

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

### 1.2. 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 $\Omega^*$.

Figure 2 : ALE map

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

$\begin{eqnarray*} \mathcal{A}^t : \Omega^* &\longrightarrow & \Omega^{t} \\ \mathbf{x}^* &\longmapsto & \mathbf{x} \left(\mathbf{x}^*,t \right) = \mathcal{A}^t \left(\mathbf{x}^*\right) \end{eqnarray*}$

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

$\begin{eqnarray*} \mathbf{x} : \left[t_0,t_f \right] &\longrightarrow & \Omega^t \\ t &\longmapsto & \mathbf{x} \left(\mathbf{x}^*,t \right) \end{eqnarray*}$

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

## 2. Lid-driven Cavity Flow benchmark

This test case has originally been proposed by [MokWall].

Computer codes, used for the acquisition of results, are from Vincent [Chabannes].

This benchmark has aso been realized by [Gerbeau], [Vàzquez], [Kuttler] and [Kassiotis].

### 2.1. Problem Description

We study here an incompressible fluid flowing into a cavity, where its walls are elastic. We use the following geometry to represent it.

Figure 1 : Geometry of lid-driven cavity flow.

The domain $\Omega_f^*$ is define by a square $[0,1]^2$, $\Gamma^{i,*}_f$ and $\Gamma^{o,*}_f$ are respectively the flow entrance and the flow outlet. A constant flow velocity, following the $x$ axis, will be imposed on $\Gamma_f^{h,*}$ border, while a null flow velocity will be imposed on $\Gamma_f^{f,*}$. This last point represent also a non-slip condition for the fluid.

Furthermore, we add a structure domain, at the bottom of the fluid one, named $\Omega_s^*$. It is fixed by his two vertical sides $\Gamma_s^{f,*}$, and we denote by $\Gamma_f^{w,*}$ the border which will interact with $\Omega_f^*$.

During this test, we will observe the displacement of a point $A$, positioned at $(0;0.5)$, into the $y$ direction, and compare our results to ones found in other references.

#### 2.1.1. Boundary conditions

Before enunciate the boundary conditions, we need to describe a oscillatory velocity, following the x axis and dependent of time.

$v_{in} = 1-cos\left( \frac{2\pi t}{5} \right)$

Then we can set

• on $\Gamma^{h,*}_f$, an inflow Dirichlet condition : $\boldsymbol{u}_{f} = ( v_{in}, 0 )$

• on $\Gamma^{i,*}_f$, an inflow Dirichlet condition : $\boldsymbol{u}_{f} = ( v_{in}(8 y -7) ,0)$

• on $\Gamma^{f,*}_f$, a homogeneous Dirichlet condition : $\boldsymbol{u}_{f} = \boldsymbol{0}$

• on $\Gamma^{o,*}_f$, a Neumann condition : $\boldsymbol{\sigma}_{f} \boldsymbol{n}_f = \boldsymbol{0}$

• on $\Gamma^{f,*}_s$, a condition that imposes this boundary to be fixed : $\boldsymbol{\eta}_{s} = \boldsymbol{0}$

• on $\Gamma^{e,*}_s$, a condition that lets these boundaries be free from constraints : $\boldsymbol{F}_{s} \boldsymbol{\Sigma}_s \boldsymbol{n}_s = \boldsymbol{0}$

To them we also add the fluid-structure coupling conditions on $\Gamma_{fsi}^*$ :

$\frac{\partial \boldsymbol{\eta_{s}} }{\partial t} - \boldsymbol{u}_f \circ \mathcal{A}^t_f = \boldsymbol{0} \quad \left( \text{Velocities continuity}\right)$
$\boldsymbol{F}_{s} \boldsymbol{\Sigma}_{s} \boldsymbol{n}^*_s + J_{\mathcal{A}^t_f} \boldsymbol{F}_{\mathcal{A}^t_f}^{-T} \hat{\boldsymbol{\sigma}}_f \boldsymbol{n}^*_f = \boldsymbol{0} \quad \left( \text{ Constraint continuity}\right)$
$\boldsymbol{\varphi}_s^t - \mathcal{A}^t_f = \boldsymbol{0} \quad \left( \text{Geometric continuity}\right)$

#### 2.1.2. Initial conditions

To realize the simulations, we used a time step $\Delta t$ equals to $0.01$ s.

### 2.2. Inputs

The following table displays the various fixed and variables parameters of this test-case.

 Name Description Nominal Value Units $E_s$ Young’s modulus $250$ $N.m^{-2}$ $\nu_s$ Poisson’s ratio $0$ dimensionless $\rho_s$ structure density $500$ $kg.m^{-3}$ $\mu_f$ viscosity $1\times 10^{-3}$ $m^2.s^{-1}$ $\rho_f$ density $1$ $kg.m^{-3}$

### 2.3. Outputs

$(\boldsymbol{u}_f, p_f, \boldsymbol{\eta}_s, \mathcal{A}_f^t)$, checking the fluid-structure system, are the output of this problem.

### 2.4. Discretization

To discretize space, we used $P_N~-~P_{N-1}$ Taylor-Hood finite elements.

For the structure time discretization, Newmark-beta method is the one we used. And for the fluid time discretization, we used BDF, at order $q$.

### 2.5. Implementation

All the codes files are into FSI

### 2.6. Results

We begin with a $P_2~-~P_1$ approximation for the fluid with a geometry order equals at $1$, and a fluid-structure stable interface.

Then we retry with a $P_3~-~P_2$ approximation for the fluid with a geometry order equals at $2$, and a fluid-structure stable interface.

Finally we launch it with the same conditions as before, but with a deformed interface.

#### 2.6.1. Conclusion

First at all, we can see that the first two tests offer us similar results, despite different orders uses. At contrary, the third result set are better than the others.

The elastic wall thinness, in the stable case, should give an important refinement on the fluid domain, and so a better fluid-structure coupling control. However, the deformed case result are closer to the stable case made measure.

### 2.7. Bibliography

References for this benchmark
• [MokWall] DP Mok and WA Wall, Partitioned analysis schemes for the transient interaction of incompressible flows and nonlinear flexible structures, Trends in computational structural mechanics, Barcelona, 2001.

• [Chabannes] Vincent Chabannes, Vers la simulation numérique des écoulements sanguins, Équations aux dérivées partielles [math.AP], Université de Grenoble, 2013.

• [Gerbeau] J.F. Gerbeau, M. Vidrascu, et al, A quasi-newton algorithm based on a reduced model for fluid-structure interaction problems in blood flows, 2003.

• [Vazquez] J.G. Valdés Vazquèz et al, Nonlinear analysis of orthotropic membrane and shell structures including fluid-structure interaction, 2007.

• [KuttlerWall] U. Kuttler and W.A. Wall, Fixed-point fluid–structure interaction solvers with dynamic relaxation, Computational Mechanics, 2008.

• [Kassiotis] C. Kassiotis, A. Ibrahimbegovic, R. Niekamp, and H.G. Matthies, Nonlinear fluid–structure interaction problem ,part i : implicit partitioned algorithm, nonlinear stability proof and validation examples, Computational Mechanics, 2011.