Spectral element method approximation of fluid-structure interaction in hemodynamics

The mathematical modeling and numerical simulation of the cardiovascular system, or in a more specific case, blood flow in arteries, can be of help in understanding blood flow pathologies. One of the main causes of death in the spectrum of cardiovascular diseases is atherosclerosis, also known as "Arteriosclerotic Vascular Disease". It is the condition in which an artery wall thickens as the result of a build-up of fatty materials such as cholesterol. This disease can produce the narrowing of the arteries, therefore diminishing the blood supply of the organs it feeds. The numerical simulation of the effect of obstructions and stiffening in the blood flow in arteries can be of great assistance to medical doctors in devising better ways of treatment. Also, the a priori knowledge of the local hemodynamics in the presence of a stent or a bypass can help minimizing post-surgical complications or, for instance, contribute to the better design of bypasses.

The mathematical description of blood flow in an artery is done by coupling a model describing the blood flow with another model describing the artery's wall. This setting fits a bigger class of problems called fluid-structure interaction (FSI) problems. In the case of large arteries, the blood flow can be modeled by the incompressible Navier-Stokes equations and the artery's wall by a viscoelastic equation (such as the generalized string model, independent ring model or the StVenant Kirchoff model). The numerical methods found in the literature to treat this kind of problem are usually the finite element method for the space discretization combined with a time discretization technique and a modular or non-modular approach for the coupled FSI problem. The main difficulties of simulating this problem are:

• the discretization of the system of equations in a domain that evolves in time,
• the solution algorithm for the coupled problem and
• the techniques to solve the linear system that appear in the process of calculating the solution.

The method we propose to solve the FSI problem follows the Arbitrary Lagrangian Eulerian approach combined with the spectral element method for the space discretization and stiffly stable time integrators. The motivation for the use of spectral elements is in their good properties of convergence (as well as low dissipation and dispersion) and the assumption that the blood flow is smooth enough.

This project has two main goals. The first is to develop efficient algorithms to tackle the fluid-structure problem in hemodynamics using spectral elements. This involves, among other sub-problems, the definition of a high order ALE transformation, the choice of robust methods to solve the fluid-structure interaction problem, the parallelization of the code to solve this problem. The second is to apply the developed code to real life artery geometries and to develop a reliable tool to, in the future, collaborate with medical doctors in simulating real patient pathologies.

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Gonçalo Pena, LCM-CMUC
Christophe Prud'homme, LJK-UJF
Vincent Chabannes, LJK-UJF