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\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 fluidsolid coupling, e.g. \cite{astorinochoulyfernandez09, 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, letallecmouro01, Burtschell2017} while no analysis has been reported for ChorinTemam (CT) schemes, being an alternative option when requirements such a low time computations are needed, specially for future industrial applications.


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 fluidsolid coupling, e.g. \cite{astorinochoulyfernandez09, bertoglio2013sisc}.




The goal of this talk is to present the finding of wellposedness and unconditional energy balance of the iNSEALE for several reported CT discretization schemes within a single formulation, published in \cite{arostica2021monolithic}. The main result to show 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.


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, letallecmouro01, Burtschell2017} while no analysis has been reported for ChorinTemam (CT) schemes, being a feasible alternative when requirements such a low time computations are needed.




Our finding will be supplemented with an application case of fluidsolid interactions problem in an idealize geometry, exploiting the splitting nature of the CT scheme with a wellknown coupling approach \cite{bertoglio2013sisc}.


The goal of this talk is to present the finding of wellposedness and unconditional energy balance of the iNSEALE 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 fluidsolid interactions problem in an idealize geometry, exploiting the splitting nature of the CT scheme with a wellknown 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 oneway (left) to twoways (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}




\newpage






\bibliography{biblio_merged.bib}




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