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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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% 2- Multiple authors with multiple affiliations. Complete as follows: author[i] <-> affiliation[i]  
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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.}

% KEYWORDS: replace text with 2-4 keywords, not capitalised, separated by comma, and without a full stop at the end.
\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 CT discretization schemes within a single formulation, published in \cite{arostica2021monolithic}. The main result will be that under appropiate conditions on the rate of domain deformation, a first order time discretization scheme for the CT scheme is unconditionally stable. 

Explicitly, we will show that under certain conditions, our CT scheme is unconditionally energy stable, through the inequality:
\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 case of fluid-solid interactions problem in an idealize geometry, exploiting the splitting nature of the CT scheme with a well-known coupling approach \cite{bertoglio2013sisc}. 

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 one-way (left) to two-ways (right) approach, with $\mathbb{P}_1$ lagrange elements. In arrows the fluid magnitude and direction, while it's surrounds denote the solid effects.}
	\label{fig:comparison_figure}
\end{figure}

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