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@ 100,27 +101,24 @@ Several works have been reported dealing with numerical solutions of the iNSE in




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.




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.


The goal of this talk is to present the finding of wellposedness and unconditional energy balance of the iNSEALE 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.




Explicitly, we will show that under certain conditions, our CT scheme is unconditionally energy stable, through the inequality:


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 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.




Our finding will be supplemented with an application to fluidsolid interaction in an idealized cardiac geometry, exploiting the splitting nature of the CT scheme with a wellknown coupling approach \cite{bertoglio2013sisc}. \textcolor{red}{cita el paper original de fernandez, gerbau y grandmont porfa}


%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.}


\caption{Comparison oneway (left) to twoways (right) approach \textcolor{red}{no deberia ser al reves?}, 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}




\newpage


%\newpage






\bibliography{biblio_merged.bib}




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