Fluid-structure interaction problem

General
1

Hello,
I would like to simulate the Navier Stokes flow of a fluid in a channel, where the channel has an obstacle in it, and the obstacle is an elastic body. I wrote a code inspired by D. Kamensky’s ‘Lecture notes for MAE 207: Finite element analysis for coupled problems,’ but the code is unstable as I iterate in time.

Here are some details:

  • The problem is defined on a 2d manifold \Omega, given by a rectangle with an ellipse in its center, and the ellipse has a circular hole, see screenshot. The green region is the one where the fluid velocity \bf v and pressure p are defined. The orange region is the one where the deformation field \bf u_{\rm el} of the elastic body is defined. In the green region, the mesh deformation field of the ALE formulation \bf u_{\rm msh}, is also defined.

    Screenshot 2025-07-27 at 18.31.00
    Screenshot 2025-07-27 at 18.31.001394×1208 37.8 KB

  • I iterate in time as follows:

  1. Given the velocity \bf v^{n-1}, p^{n-3/2} calculated with the Crank Nicholson (CN) scheme at the preceding time steps, I solve for the displacement {\bf u}_{\rm el}^n of the elastic problem, \partial_i P_{ji}({\bf u}_{\rm el}^n) = 0, with the BC that {\bf u}_{\rm el}^n at the circle vanishes (the elastic body is pinned at the circle), and that the force exerted on the ellipse is given by the force exerted by the fluid: \nu_j P_{ij} d \lambda = \sigma_{ij} ({\bf v}^{n-1}, p^{n-3/2})n_j dl, where \nu d\lambda and n dl are the normals and length elements in the reference and current configurations, see Slaughter ‘The linearized theory of elasticity’. Here P is the stress tensor of the Kirkhoff-St Venant elastic model.
  2. Given {\bf u}_{\rm el}^n, I solve for the mesh displacement field {\bf u}_{\rm msh}^n, by solving for a ficticious elastic problem, similar to the Kirchoff St Venant, see Slaughter ‘The linearized theory of elasticity’.
  3. I solve for \bf v^n, p^{n-1/2} according to the CN scheme.

I did not put additional details here because I don’t want to make the post too long. If you have any questions, I will give you more details.

If I iterate in time steps (1-3), the elastic displacement field blows up, and so do the other ones. This may be due that in the first steps of the CN scheme for the Navier Stokes equations, the pressure field blows up and strongly oscillates before settling to a finite value, so this compresses too much the elastic body.

Do you have any suggestions or ideas?
Thanks