corrections

cbme2022/CMBE21_ale.tex

@@ -23,6 +23,7 @@
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 % TITLE: replace text with your abstract title WITHOUT full stop
@@ -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, 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. 
+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. 
 
-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 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.
-
+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}. \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 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.}
+	\caption{Comparison one-way (left) to two-ways (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}
 
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+%\newpage
 
 
 \bibliography{biblio_merged.bib}