Section: New Results

Mathematical and numerical analysis of fluid-structure interaction problems

Participants : Matteo Aletti, Ludovic Boilevin-Kayl, Chen-Yu Chiang, Miguel Ángel Fernández Varela, Jean-Frédéric Gerbeau, Céline Grandmont, Damiano Lombardi, Marc Thiriet, Marina Vidrascu.

In [15] a reduced order modeling method is developed to simulate multi-domain multi-physics problems. In particular we considered the case in which one problem of interest, described by a generic non-linear partial differential equation is coupled to one or several problems described by a set of linear partial differential equations. In order to speed up the resolution of the coupled system, a low-rank representation of the Poincaré-Steklov operator is built by a reduced-basis approach. A database for the secondary problems is built when the interface condition is set to be equal to a subset of the Laplace-Beltrami eigenfunctions on the surface. The convergence of the method is analysed and several 3D fluid-fluid and fluid-structure couplings are presented as numerical experiments.

In [43] we study an unsteady nonlinear fluid–structure interaction problem. We consider a Newtonian incompressible two-dimensional flow described by the Navier-Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear wave equation or a linear beam equation. We prove existence of a unique local-in-time strong solution. In the case of the wave equation or a beam equation with inertia of rotation, this is, to our knowledge the first result of existence of strong solutions for which no viscosity is added. One key point, is to use the fluid dissipation to control, in appropriate function spaces, the structure velocity.

In [26] a fluid-structure interaction solver based on 3D Eulerian monolithic formulation for an incompressible Newtonian fluid coupled with a hyperelastic incompressible solid has been implemented, verified, and validated. It is based on a Eulerian formulation of the full system. After a fully implicit discretization in time, displacement is eliminated and the variational equation is solved for the velocity and pressure. Its main application in medicine is venous flow in inferior limbs.