\summary{Several time discretized domain for the incompressible Navier-Stokes equations (iNSE) in moving domains have been proposed in literature. Here, we introduce a unified formulation that combines different approaches found in literature, allowing a common well posedness and time stability analysis. It can be therefore shown that only a particular choice of numerical schemes ensure such properties under some restrictions. The analysis will be shown for Chorin-Temam schemes using the insight found in the monolithic case. Results are supported from numerical simulations and its usage in fluid-solid interaction problems in cardiac geometries will be presented.}
Several works have been reported dealing with numerical solutions of the iNSE in moving domains within the Arbitrary Lagrangian Eulerian formulation (ALE), primarily in the context of fluid-solid coupling, e.g. \cite{astorino-chouly-fernandez-09, bertoglio2013sisc}.
Different choices of time discretization schemes have been reported e.g. \cite{Basting2017, Hessenthaler2017}, nevertheless to the best of the authors knowledge, only a few monolithic schemes have been throughly analyzed, e.g. \cite{Lozovskiy2018, smaldone2014, le-tallec-mouro-01, Burtschell2017} while no analysis has been reported for Chorin-Temam (CT) schemes, being a feasible alternative when requirements such a low time computations are needed.
The goal of this talk is to present the finding of well-posedness and unconditional energy balance of the iNSE-ALE for several reported monolithic discretization schemes within a single formulation, recently published in \cite{arostica2021monolithic}. The main result is that under appropriate conditions on the rate of domain deformation, only some of first order time discretization schemes are unconditionally stable.
for $(\mathbf{u}^n, p^n)$ the velocity/pressure pair solution at time $t^n$, in the reference domain $\Omega^0$, for operators to be specified in the talk.
Our finding will be supplemented with an application to fluid-solid interaction in an idealized cardiac geometry, exploiting the splitting nature of the CT scheme with a well-known coupling approach \cite{bertoglio2013sisc, fernandez-gerbeau-grandmont-06}, see example in Figure \ref{fig:comparison_figure}.
%In such a case, we will exploit the fluid pressure projection step, coupling it with the solid problem in an efficient fashion. Simulations of such a case will be provided, as well as the current research done in more realistic geometries.
\caption{Two-ways FSI (left) and one-way FSI (right), on an ellipsoid. In arrows the fluid velocity magnitude and direction, driven by an hyperelastic active solid (in grey, with decreased opacity).}