From 7496ea4254e4f7678b2e5bdd3636a2e522731f82 Mon Sep 17 00:00:00 2001 From: reidmen Date: Mon, 11 Apr 2022 10:42:11 +0200 Subject: [PATCH] Updated tex for cbme2022 --- cbme2022/CMBE21_ale.tex | 26 ++++++++++++++++-- .../figs/comparison_two_ways_to_one_way.png | Bin 0 -> 1099696 bytes 2 files changed, 23 insertions(+), 3 deletions(-) create mode 100644 cbme2022/figs/comparison_two_ways_to_one_way.png diff --git a/cbme2022/CMBE21_ale.tex b/cbme2022/CMBE21_ale.tex index 0a80ffe..82a439d 100644 --- a/cbme2022/CMBE21_ale.tex +++ b/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 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 \bibliography{biblio_merged.bib} diff --git a/cbme2022/figs/comparison_two_ways_to_one_way.png b/cbme2022/figs/comparison_two_ways_to_one_way.png new file mode 100644 index 0000000000000000000000000000000000000000..56fa6954eac09c7e91755621d8b8995b1749a698 GIT binary patch literal 1099696 zcmZ6z1yGh<)Gka&gLHSNw3Kv*ba#hzm!xz_cO%`6bO=(?APs_aN;jO%KXbnSd(S=N zm`~l$UTa;oj8IaLLO~=zgn)oRk&zZxfq;PXfPjGBhlc`x60ybl5Bvn_q9P>CqSl{2swkTH6Hz0*Ctb2c(P&6?nu;k&L*Any0~0Hk^^#z{OzA z_O#eu{tb<+W(Fd>NhnF z)b?j747h*KvPO%dgXfJPa*}2g$RLAGMTx4x!U#T^jr)PmG-C3^>+bhHc_^45OfCWW zzt?jnYc(V^G;|OqDcovD3gIme+6jRm``5ffvcp>>cGmw~x3NDw(^0?^DDVV}<{M3V z9~lU_R=yyE?+XuDVbyN4E9v zVtvE8bwwd;rl@m5;;p3k$YDMDElU}HjulMeMUC)?2TiQEuq1e1{AIUem(Zl919LIBm z_8>JLl^)4yXW1@^pa*jg73qYC29oyrUOr~OJGe!g4agI+Tk83fD|I{hw5ME}_v$xGF-StU)CvbZ)KzU_GPOCcW 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