Algorithms for fluid-structure interaction problems arising in hemodynamics

We consider the numerical approximation of fluid-structure interaction
(FSI) problems with a particular concern (albeit not exclusive) on hemodynamics applications.

Firstly, we model the blood as an incompressible fluid and the artery wall as an elastic structure. To solve the coupled problem, we propose new semi-implicit algorithms based on inexact block-LU factorization of the linear system obtained after the space-time discretization and linearization of the FSI problem. As a result, the fluid velocity is computed separately from the coupled pressure-structure velocity system at each iteration, hence reducing the computational cost. This approach leads to two different families of methods which extend to FSI problems schemes that were previously adopted for pure fluid problems.

The algorithms derived from inexact factorization methods are compared with other schemes based on two preconditioners for the FSI system.
The first one is the classical Dirichlet-Neumann preconditioner, which has the advantage of modularity (i.e. it allows to reuse existing fluid and structure codes with minimum effort). Unfortunately, its performance is very poor in case of large added-mass effect, as it happens in hemodynamics. Alternatively, we consider a non-modular approach which consists in preconditioning the coupled system with a suitable diagonal scaling combined with an ILUT preconditioner. The system is then solved by a Krylov method. The drawback of this procedure is the loss of modularity. Independently of the preconditioner, the efficiency of semi-implicit algorithms is highlighted. All the methods are tested on two and three-dimensional blood-vessel systems. The algorithm combining the non-modular ILUT preconditioner with Krylov methods proved to be the fastest.

Finally, we improve the structure model by representing the vessel wall as a linear poroelastic medium. Our non-modular approach and the partitioned procedures arising from a domain decomposition viewpoint are extended to fluid-poroelastic structure interactions. Their numerical performance are analyzed and compared on simplified blood-vessel systems.