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. Restricting our attention to vascular FSI, we now reduce this arbitrary Lagrangian-Eulerian (ALE) formulation by adopting three assumptions for the vascular wall. The resulting reduced unified continuum formulation achieves monolithic FSI coupling 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, 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. Uniform temporal discretization is performed via the generalized-$\alpha$ 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. 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 improves representation of the Schur complement, demonstrates robust performance over a wide range of physical parameters. We present verification against Womersley's deformable wall theory and additionally develop practical modeling techniques for clinical applications.

References in corpus (12)

• Outflow boundary conditions for 3D simulations of non-periodic blood flow and pressure fields in deformable arteries
• A unified continuum and variational multiscale formulation for fluids, solids, and fluid-structure interaction
• Fluid-structure interaction modeling of blood flow in the pulmonary arteries using the unified continuum and variational multiscale formulation
• Modeling blood flow in viscoelastic vessels: the 1D augmented fluid-structure interaction system
• Simplex Space-Time Meshes in Compressible Flow Simulations
• A continuum and computational framework for viscoelastodynamics: finite deformation linear models
• The nested block preconditioning technique for the incompressible Navier-Stokes equations with emphasis on hemodynamic simulations
• A note on the accuracy of the generalized-$\alpha$ scheme for the incompressible Navier-Stokes equations
• A hybrid variational Allen-Cahn/ALE scheme for the coupled analysis of two-phase fluid-structure interaction
• An energy-stable mixed formulation for isogeometric analysis of incompressible hyper-elastodynamics
• A robust and efficient iterative method for hyper-elastodynamics with nested block preconditioning
• Insights into the performance of loosely-coupled FSI schemes based on Robin boundary conditions