MS4.5 - Computational Fluid-structure Interaction

Today, cardiovascular diseases are among the leading causes of death worldwide. With a special focus on the treatment of hypertension and the clinical consequences, thereof, the computational modeling of fluid-structure interaction with pharmaco-mechanical effects becomes vastly relevant. Therefore, the state of the art of fluid-structure interaction is extended to reflect the influence of drugs on the structural properties of arterial walls leading to a fully coupled fluid-structure-chemical interaction, denoted as FSCI. Highly-scalable parallel GDSW (Generalized Dryja–Smith–Widlund) coarse spaces have been implemented in the solver framework FROSch (Fast and Robust Overlapping Schwarz), which is part of the software library Trilinos and can easily be applied to the geometry and structure blocks in an FSI simulation framework. Furthermore, these methods have also been extended to monolithic GDSW-type preconditioners for fluid flow problems; the parallel implementation is also available in FROSch. We plan to solve the resulting FSCI linearized system with a Krylov method preconditioned by the FaCSI preconditioner which was introduced by Deparis, Forti, Grandperrin, and Quarteroni in 2016. The inverses appearing in FaCSI will be approximated by GDSW-type overlapping Schwarz preconditioners. In this talk, first results of FSCI are investigated and compared using a finite element implementation based on our software FEDDLib (Finite Element and Domain Decomposition Library) and Schwarz preconditioners from the Trilinos package FROSch.

The numerical simulation of atherosclerotic plaque growth is computationally prohibitive since it involves a complex cardiovascular fluid-structure interaction (FSI) problem with a characteristic time scale of milliseconds to seconds as well as a plaque growth process governed by reaction-diffusion equations, which takes place over several months. A resolution of the fast (micro) scale over this period can easily require more than a billion time steps, each corresponding to the solution of a computationally expensive FSI problem. To tackle this problem, we combine a temporal homogenization approach with parallel time-stepping. First, a temporal homogenization approach is developed, which separates the problem in an FSI problem on the micro scale and a reaction-diffusion problem on the macro scale. The approach is analyzed in detail for a simplified flow problem and estimates for the homogenization error and the discretization errors on both time scales are given. Second, a parallel time-stepping approach based on the parareal algorithm is applied on the macro scale of the homogenized system. We investigate modifications in the coarse propagator of the parareal algorithm to further reduce the number of expensive micro problems to be solved and test the numerical algorithms in detailed numerical studies.