An Arbitrary Lagrangian-Eulerian (ALE) formulation is applied in a fully
coupled monolithic way, considering the fluid-structure interaction (FSI) problem
as one continuum. The mathematical description and the numerical schemes are
designed in such a way that general constitutive relations (which are realistic for
biomechanics applications) for the fluid as well as for the structural part can be easily
incorporated. We utilize the LBB-stable finite element pairs Q2P1 and P+
2 P1 for
discretization in space to gain high accuracy and perform as time-stepping the 2nd
order Crank-Nicholson, respectively, a new modified Fractional-Step-q-scheme for
both solid and fluid parts. The resulting discretized nonlinear 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 direct or iterative
solvers, preferably of Krylov-multigrid type.
For validation and evaluation of the accuracy and performance of the proposed
methodology, we present corresponding results for a new set of FSI benchmark 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 deformations. Then, as an example of FSI in biomedical problems, the
influence of endovascular stent implantation on 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 future aneurysm
rupture shall be suppressed by an optimal setting of the implanted stent geometry.