Stability of the kinematically coupled \beta-scheme for fluid-structure interaction problems in hemodynamics

TL;DR

This paper demonstrates that the kinematically coupled eta-scheme for fluid-structure interaction in hemodynamics is unconditionally stable across all relevant parameters, overcoming the instability issues of classical schemes due to the added-mass effect.

Contribution

It introduces and proves the unconditional stability of the eta-scheme, a novel partitioned approach for FSI problems in hemodynamics, regardless of physical and geometric parameters.

Findings

The eta-scheme is unconditionally stable for all parameters.

Numerical results confirm stability on complex benchmark problems.

The scheme avoids the added-mass instability present in classical methods.

Abstract

It is well-known that classical Dirichlet-Neumann loosely coupled partitioned schemes for fluid-structure interaction (FSI) problems are unconditionally unstable for certain combinations of physical and geometric parameters that are relevant in hemodynamics. It was shown in \cite{causin2005added} on a simple test problem, that these instabilities are associated with the so called ``added-mass effect''. By considering the same test problem as in \cite{causin2005added}, the present work shows that a novel, partitioned, loosely coupled scheme, recently introduced in \cite{MarSun}, called the kinematically coupled $\beta$-scheme, does not suffer from the added mass effect for any $\beta \in [0,1]$, and is unconditionally stable for all the parameters in the problem. Numerical results showing unconditional stability are presented for a full, nonlinearly coupled benchmark FSI problem, first considered in \cite{formaggia2001coupling}.