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cbme2022/CMBE21_ale.tex

@@ -96,11 +96,31 @@
 
 \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 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 fluid-solid coupling, e.g. \cite{astorino-chouly-fernandez-09, bertoglio2013sisc}. 
 
-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 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, 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.
 
-Our finding will be supplemented with an application case of fluid-solid interactions problem in an idealize cardiac geometry, exploiting the splitting nature of the CT scheme with a well-known coupling approach \cite{bertoglio2013sisc}.
+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}
+
+\newpage
 
 
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