Preconditioning a Fluid--Structure Interaction Problem Using Monolithic and Block Domain Decomposition Methods for the Fluid

TL;DR

The paper investigates how to efficiently precondition a monolithically coupled fluid–structure interaction system by comparing monolithic and SIMPLE-based block preconditioners for the fluid subproblem within the FaCSI framework. It employs two-level overlapping Schwarz methods to approximate inverses and evaluates performance on a patient-specific artery, finding that monolithic fluid preconditioning yields fewer Newton iterations and better robustness at higher flow rates, with strong parallel scaling. While SIMPLE variants reduce setup time, they incur longer solve times, making monolithic preconditioning generally more effective for this application. The study demonstrates the practical impact of fluid-subproblem preconditioning choices on the overall efficiency of cardiovascular FSI simulations.

Abstract

A fluid-structure interaction (FSI) problem is solved via a monolithic coupling of the fluid, structure, and geometry subproblems. The iterative GMRES solver is accelerated with the FaCSI block preconditioner. In the FaCSI factorization, the fluid subproblem is approximated using either a monolithic preconditioner or the block preconditioner SIMPLE. Two-level overlapping Schwarz methods are then used to approximate the arising inverses. The robustness and scalability of the monolithic and SIMPLE preconditioners are compared for a realistic patient-specific artery. The results indicate that the monolithic preconditioning of the fluid subproblem performs better than the SIMPLE approach. Different flow rates are tested and parallel strong scaling has been evaluated.

Figures (3)

• Figure 1: Boundary conditions of fluid and solid domain. For the fluid subproblem, a parabolic-like inflow profile is prescribed on the inlet (in pink). On the outflow, a pressure boundary condition is prescribed. For the solid subproblem, the $z=0$ and $z=L$ planes are held in $z$ direction. Additional points on the respective inflow and outflow planes are held in $x$-$z$ or $y$-$z$ direction to prevent rigid body motions and to ensure that the system is statically determinate.
• Figure 2: Left: Prescribed flowrate on inlet of fluid domain. We consider a range from 2 to 6 cm$^{3}$/s for Test (a). For the steady phase up to 0.2 s the pressure is not influenced by the flow rate due to the absorbing boundary condition. Right: The prescribed reference pressure $p_{\text{ref}}$ leads to the pressure measured at the outlet. It follows the fluid flow rate ramp prescribed (left). For Test (b) the flow rate prescribed by the heart beat also influences the pressure value after the steady phase.
• Figure 3: Comparison of monlithic and block preconditioner. Left: Resulting average iteration count per Newton step per time step for varying inflow flow rate using Test (a) in \ref{['fig: results pressure and area']} for 200 time steps. On average 6 Newton iterations per time step are needed. Right: Strong scaling using Test (b) in \ref{['fig: results pressure and area']} for 600 time steps. The full FSI system has 2 230 418 degrees of freedom. Total time consists of setup and solve time. On average 8 Newton iterations per time step are needed.