Section: New Results

Mathematical and numerical analysis of fluid-structure interaction problems

Participants : Matteo Aletti, Faisal Amlani, Miguel Ángel Fernández Varela, Jean-Frédéric Gerbeau, Mikel Landajuela Larma, Damiano Lombardi, Marina Vidrascu.

In [15] a simplified fluid-structure interaction method is proposed in order to deal with the simulation of fluids in elastic pipes. The motivation of this work is the modeling of the blood flow in arterioles. The structure is modeled by a non-linear Koiter shell, without bending. In addition, the presence of active elastic fibers is considered. The structure is lumped into the boundary condition of the fluid problem leading to a generalized Robin boundary condition. A finite elements discretization is proposed and several numerical test cases are presented to assess the properties of the method.

In [45] a reduced order modeling method is investigated to deal with 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. An online update is also introduced in order to guarantee stability and robustness. Several 3D fluid-fluid and fluid-structure couplings are presented as numerical experiments.

In [44] two new numerical methods for incompressible fluid/thin-walled structure interaction problems using unfitted meshes are proposed. The spatial discretization is based on different variants of Nitsche's method with cut elements. The degree of fluid-solid splitting (semi-implicit or explicit) is given by the order in which the space and time discretizations are performed. For the semi-implicit schemes, energy-based stability and a priori error estimates are derived and which guarantee the unconditional stability and optimal accuracy in the energy-norm of one the methods. Stability and a priori error estimates are also derived for one of the explicit schemes. Numerical experiments in a benchmark illustrate the performance of the different methods proposed.