A reduced unified continuum formulation for vascular fluid–structure interaction

Abstract

We recently derived the unified continuum and variational multiscale formulation for fluid–structure interaction (FSI) using the Gibbs free energy as the thermodynamic potential. Restricting our attention to vascular FSI, we now reduce this formulation in arbitrary Lagrangian–Eulerian (ALE) coordinates by adopting three common modeling assumptions for the vascular wall. The resulting semi-discrete formulation, referred to as the reduced unified continuum formulation, achieves monolithic coupling of the FSI system in the Eulerian frame through a simple modification of the fluid boundary integral. While ostensibly similar to the semi-discrete formulation of the coupled momentum method introduced by Figueroa et al., its underlying derivation does not rely on an assumption of a fictitious body force in the elastodynamics sub-problem and therefore represents a direct simplification of the ALE method. Furthermore, uniform temporal discretization of the entire FSI system is performed via the generalized-α scheme. In contrast to the predominant approach yielding only first-order accuracy for pressure, we collocate both pressure and velocity at the intermediate time step to achieve uniform second-order temporal accuracy. In conjunction with quadratic tetrahedral elements, our methodology offers higher-order temporal and spatial accuracy for quantities of clinical interest, including pressure and wall shear stress. Furthermore, without loss of consistency, a segregated predictor multi-corrector algorithm is developed to preserve the same block structure as for the incompressible Navier–Stokes equations in the implicit solver’s associated linear system. Block preconditioning of a monolithically coupled FSI system is therefore made possible for the first time. Compared to alternative preconditioners, our three-level nested block preconditioner, which achieves improved representation of the Schur complement, demonstrates robust performance over a wide range of physical parameters. We present verification of our methodology against Womersley’s deformable wall theory and additionally develop practical modeling techniques for clinical applications, including tissue prestressing. We conclude with an assessment of our combined FSI technology in two patient-specific cases.