Numerical techniques for solving the problem of fluid-structure interaction with an
elastic material in a laminar incompressible viscous flow are described. An Arbitrary
Lagrangian-Eulerian (ALE) formulation is employed in a fully coupled monolithic
way, considering the problem as one continuum. The mathematical description and
the numerical schemes are designed in such a way that more complicated constitutive
relations (and more realistic for biomechanics applications) for the fluid as well as
the structural part can be easily incorporated. We utilize the well-known Q2P1
finite element pair for discretization in space to gain high accuracy and perform as
time-stepping the 2nd order Crank-Nicholson, resp., Fractional-Step-θ-scheme for
both solid and fluid parts. The resulting nonlinear discretized algebraic system is
solved by a Newton method which approximates the Jacobian matrices by a divided
differences approach, and the resulting linear systems are solved by iterative solvers,
preferably of Krylov-multigrid type.
For validation and evaluation of the accuracy of the proposed methodology, we
present corresponding results for a new set of FSI benchmarking configurations
which describe the self-induced elastic deformation of a beam attached to a cylinder
in laminar channel flow, allowing stationary as well as periodically oscillating de-
formations. Then, as an example for fluid-structure interaction (FSI) in biomedical
problems, the influence of endovascular stent implantation onto cerebral aneurysm
hemodynamics is numerically investigated. The aim is to study the interaction of
the elastic walls of the aneurysm with the geometrical shape of the implanted stent
structure for prototypical 2D configurations. This study can be seen as a basic step
towards the understanding of the resulting complex flow phenomena so that in fu-
ture aneurysm rupture shall be suppressed by an optimal setting for the implanted stent geometry.