FEM multigrid techniques for fluid–structure interaction with application to hemodynamics

Abstract

We present special finite element and multigrid techniques for solving prototypical cerebral aneurysm hemodynamics problems numerically. An arbitrary Lagrangian–Eulerian (ALE) formulation is employed for this fluid–structure interaction (FSI) application. We utilize the well-known high order finite element pair Q2P1 for discretization in space to gain high accuracy and robustness and perform as time-stepping a fully implicit second order accurate time integrator. The resulting nonlinear discretized algebraic system is solved by an iterative Newton solver which approximates the Jacobian matrix by the divided difference approach, and the resulting linear system is solved by means of Krylov type and geometrical multigrid solvers with a Vanka-like smoother. The aim of this paper is to study the interaction of the elastic walls of an aneurysm with the geometrical shape of an implanted stent structure for prototypical 2D configurations. Preliminary results for the stent-assisted occlusion of a cerebral aneurysm and a qualitative analysis of the behavior of the elasticity of the walls vs. the geometrical details of the stent for prototypical flow situation are presented. Additionally, our approach is designed in such a way that complicated realistic constitutive relations for biomechanics applications for blood vessel simulations can be easily integrated.