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% TITLE: replace text with your abstract title WITHOUT full stop
\title{On monolithic and Chorin-Temam schemes for incompressible flows in moving domains}
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%\author[1]{Jerem\'ias Garay}
%\author[2]{Second B. Author}
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% \author[1]{Jerem\'ias Garay}
\author[1]{Reidmen Ar\'ostica}
% \author[1]{David Nolte}
\author[1]{Crist\'obal Bertoglio}

\affil[1]{{Bernoulli Institute}, {University of Groningen},
{Groningen}, The Netherlands}
%\affil[cmm]{{Center for Mathematical Modeling}, {Universidad de Chile}, {Santiago}, Chile}
%\affil[tub]{{Department of Fluid Dynamics}, {Technische Universit\"at Berlin}, {Berlin}, Germany}

%\affil[2]{{Bernoulli Institute}, {University of Groningen},
%{Groningen}, The Netherlands}

%\affil[2]{{Biomedical Imaging Center}, {Pontificia Universidad Cat\'olica de Chile},
%{Santiago}, Chile}


%\affil[3]{{School of Biomedical Engineering}, {Universidad  de Valparaiso},
%{Valparaiso}, Chile}
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%\affil[4]{{Department of Mathematical Engineering}, {Universidad de Concepci\'on},
%{Concepci\'on}, Chile}
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%\affil[5]{{Department of Mechanical Engineering}, {Universidad T\'ecnica Federico Santa Mar\'ia},
%{Santiago}, Chile}
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\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.}

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\keywords{numerical schemes, stability analysis, incompressible flows, fluid-structure interaction}


\begin{document}

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. 

We will show the extension to a CT scheme. Namely, in that case the following inequality can be shown:
\begin{equation}
\int_{\Omega^0} \frac{\rho J^{n+1}}{2\tau} \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X} -\int_{\Omega^0} \frac{\rho J^{n}}{2\tau} \vert \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} \leq - \int_{\Omega^0} J^{\star} 2 \mu \vert \epsilon^{\star} (\tilde{\mathbf{u}}) \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} \frac{\tau J^n}{2 \rho} \vert Grad(p^n) H^n \vert^2 \, \text{d} \mathbf{X}
\end{equation}
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}.
%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.
\begin{figure}[!hbtp]
	\centering
	\includegraphics[width=0.8\textwidth]{figs/comparison_two_ways_to_one_way.png}
	\caption{Comparison two-ways (left) to one-way (right) approach, with $\mathbb{P}_1$ lagrange elements. In arrows the fluid magnitude and direction, driven by an hyperelastic solid actively contracting (grey, with decreased opacity).}
	\label{fig:comparison_figure}
\end{figure}

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