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 second-order Crank–Nicholson, respectively, 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 Krylovmultigrid type.
For validation and evaluation of the accuracy of the proposed methodology, we present corresponding results for a newset 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 deformations. 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 future aneurysm rupture shall be suppressed by an optimal setting for the implanted stent geometry.
