Adaptive Finite Element Simulation of Fluid-Structure Interaction with Application to Heart-Valve Dynamics

Abstract

The goal of this work is the development of concepts for the efficient numerical solution of fluid-structure interaction (FSI) problems with applications to heart-valve dynamics. The main motivation for further development in this field is an increasing demand from the medical community for scientifically rigorous investigations of cardiovascular diseases, which are responsible for the major fraction of mortalities in industrialized countries. In this work, the ALE (arbitrary Lagrangian Eulerian) description of fluid equations is utilized for the numerical modeling and simulation of fluid-structure interactions. Using this approach, the fluid equations can easily be coupled with structural deformations. The focal goal is the modeling, numerical analysis, and simulation of prototypical heart-valve dynamics, which requires the investigation of the following issues: the analysis of various fluid-mesh motion techniques, a comparison of different second-order time-stepping schemes, and the prescription of specific boundary conditions on the artificial outflow boundary. To control computational costs, we apply a simplified version of an a posteriori error estimation using the dual weighted residual (DWR) method. This method is used for mesh adaption during the computation. The last, novel aspect comprises a discussion of optimal control problems for wall stress minimization, in which the state is determined by a fluid-structure interaction system. The concepts developed in this work are demonstrated with several numerical tests in two and three dimensions. The programming code is validated by computing several FSI benchmark tests. The focal computation is related to a prototypical two-dimensional aortic heart-valve simulation. The concepts illustrated by this example were developed in cooperation with a cardiologist.