Mechelen Robin
M2 SMI
2022/2023

# Fluid-Structure Interaction Project

## Introduction

The problem here is a fluid-structure interaction problem. We have a linear pressure on one side of the fluid, which will apply a pressure on a solid wall embedded in $y=0$.

To solve this problem, we will first use the finite volume method to get the value of the pressure on the wall. Then, in a second step, we will couple the measured pressure with the wall and apply the finite element method to obtain the displacement of the solid element.

$$
\left\{
\begin{array}{ll}
-\frac{\partial^2 u}{\partial x^2} = - \frac{P}{T} \\
u(0)=0 \\
-(\frac{\partial^2 P}{\partial x^2}+\frac{\partial^2 P}{\partial y^2})=0 \\
P(x=0)=P(y)
\end{array}
\right.
$$

Here is a diagram representing the problem :


## Finite Volume Method

By applying the finite volume metohde knowing all these parameters. We can determine the pressure induced on the cells H1, H2 and H3. These are directly on the solid surface which will allow us to continue the resolution of the problem with the finite elements.

In the resolution in finite volume, to make it "simpler", I will rename the cells

|Old Cells |New Cells |
| -------- | -------- |
| H1 | H7 |
| H2 | H4 |
| H3 | H1 |
| H4 | H8 |
| H5 | H5 |
| H6 | H2 |
| H7 | H9 |
| H8 | H6 |
| H9 | H3 |

As seen on the previous diagram, the pressure exerted on cells H6, H7 and H8 is linear : $P(y)=y$.

It varies linearly on all this side.

Thanks to the system of equations in the introduction we can determine the "neighbors" of each cell.

Starting from this equation :

$-(\frac{\partial^2 P}{\partial x^2}+\frac{\partial^2 P}{\partial y^2})=0$

Then using the Green-Ostrogradski theorem we get :

$\displaystyle \int_{\partial k} \frac{\partial P}{\partial n} \, \mathrm{d}S =-P_{k-ny}-P_{k-1}+4P_{k}-P_{k+1}-P_{k+ny}$

This formula will allow us to program the assembly of the matrix thanks to the neighbors.

By doing a numerical resolution by the Gauss method of a system of the form $AX=B$. We can then deduce a vector with the values of the pressure in each cell.

## Finite Element Method

Let us now apply the finite element method to the problem.
First we compute the local stiffness matrix associated to this problem. Then we assemble it to obtain the global stiffness matrix $K$. We know the condition $u(0)=0$. It is enough to solve in the same way as before the system $KU=F$ by taking as value in $F$ the values of the elements of $P$ which act on H1, H2, H3.

By solving this system for 3 elements we can plot the displacement of the solid as a function of the length of the bar.


By increasing the number of cells, thus also increasing the number of elements, we obtain a graph with more points, thus more precise.


These results seem consistent. The finite volume method coupled with finite elements is a good approach to simulate this kind of problem.