Spectral Element Approximation of Fluid-Structure Interaction in Hemodynamics
Résumé
A fluid-structure interaction problem is obtained by coupling the incompressible Navier-Stokes equations with the generalized string model to simulate the blood flow and arterial displacement. The Navier-Stokes equations are expressed in the Arbitrary Lagrangian-Eulerian framework and the structure equations are described using a Lagrangian approach. This talk is divided in two parts. First, we describe the high-order space-time approximation of the incompressible Navier-Stokes equations evolving in a moving domain. We present the high-order Arbitrary Lagrangian Eulerian (ALE) framework in which we solve the fluid-structure interaction problem. We propose an ALE map using a polynomial description of the boundary of the computational domain and the Laplace operator. The construction of the map relies on a straight edge mesh in the reference domain and it conserves the shape of its interior elements. In the second part, we apply the ALE framework to solve a fluid-structure interaction problem for a simple 2D hemodynamics example. Two algorithms are considered: an implicit fully coupled and a semi-implicit. Both approaches are combined with the fix point method. To accelerate its convergence, we consider an Aitken relaxation procedure. Finally, we discuss the ingredients and details of the necessary tools to generalize the presented framework to a three-dimensional setting.