132 lines
6.8 KiB
TeX
132 lines
6.8 KiB
TeX
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%% CMBE TEMPLATE FOR ABSTRACT SUBMISSION
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%% USE CLASS cmbe17.cls ** DO NOT MODIFY CLASS FILE **
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% The extended abstract should consist of a short abstract (SUMMARY)
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% and 2-4 keywords, followed by a short article. The length of the
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% extended abstract must not be longer than four pages incl. references.
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%
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% Guidelines to complete each section are provided below.
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% References are managed using thebibliography with cite_key and
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% the \bibitem{cite_key} command. Make sure they are arranged by
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% Consult the conference website for submission information.
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\documentclass{cmbe21}
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% additional packages
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\usepackage{amsmath,amsfonts,amssymb}
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\usepackage{graphicx,wrapfig}
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% comment if not needed
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\usepackage[hyphens]{url}
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\usepackage{hyperref}
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\usepackage{color}
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% TITLE: replace text with your abstract title WITHOUT full stop
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\title{On monolithic and Chorin-Temam schemes for incompressible flows in moving domains}
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% AUTHOR/AFFILIATION: handled by authblk.
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% Use only one of the two following methods for author listing. Delete or comment out the other.
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% Add/remove authors/affiliations as necessary, complete following the template without adding additional superscript/footnotes
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% 1- Authors have the same affiliation:
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% ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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%\author{First A. Author}
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%\author{Second B. Author}
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%\author{Third C. Author}
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%\affil{Affiliation, Postal Address, \texttt{\{First,Second,Third\}@affil}}
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% 2- Multiple authors with multiple affiliations. Complete as follows: author[i] <-> affiliation[i]
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% ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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%\author[1]{Jerem\'ias Garay}
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%\author[2]{Second B. Author}
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%\author[3]{Third C. Author}
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%\author[2]{Fourth D. Author}
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% \author[1]{Jerem\'ias Garay}
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\author[1]{Reidmen Ar\'ostica}
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% \author[1]{David Nolte}
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\author[1]{Crist\'obal Bertoglio}
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\affil[1]{{Bernoulli Institute}, {University of Groningen},
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{Groningen}, The Netherlands}
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%\affil[cmm]{{Center for Mathematical Modeling}, {Universidad de Chile}, {Santiago}, Chile}
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%\affil[tub]{{Department of Fluid Dynamics}, {Technische Universit\"at Berlin}, {Berlin}, Germany}
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%\affil[2]{{Bernoulli Institute}, {University of Groningen},
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%{Groningen}, The Netherlands}
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%\affil[2]{{Biomedical Imaging Center}, {Pontificia Universidad Cat\'olica de Chile},
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%{Santiago}, Chile}
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%\affil[3]{{School of Biomedical Engineering}, {Universidad de Valparaiso},
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%{Valparaiso}, Chile}
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%\affil[4]{{Department of Mathematical Engineering}, {Universidad de Concepci\'on},
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%{Concepci\'on}, Chile}
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%\affil[5]{{Department of Mechanical Engineering}, {Universidad T\'ecnica Federico Santa Mar\'ia},
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%{Santiago}, Chile}
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%\affil[6]{{Joint last authors}, {in alphabetical order}}
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%\affil[1]{Affiliation 1, Postal Address, \texttt{First@affil1}}
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%\affil[2]{Affiliation 2, Postal Address, \texttt{\{Second,Fourth\}@affil2}}
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%\affil[3]{Affiliation 3, Postal Address, \texttt{Third@affil3}}
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% SUMMARY: replace text with a short summary
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\summary{Several time discretized domain for the incompressible Navier-Stokes equations (iNSE) in moving domains have been proposed in literature. Here, we introduce a unified formulation that combines different approaches found in literature, allowing a common well posedness and time stability analysis. It can be therefore shown that only a particular choice of numerical schemes ensure such properties under some restrictions. The analysis will be shown for Chorin-Temam schemes using the insight found in the monolithic case. Results are supported from numerical simulations and its usage in fluid-solid interaction problems in cardiac geometries will be presented.}
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% KEYWORDS: replace text with 2-4 keywords, not capitalised, separated by comma, and without a full stop at the end.
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\keywords{numerical schemes, stability analysis, incompressible flows, fluid-structure interaction}
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\begin{document}
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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}.
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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.
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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.
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We will show the extension to a CT scheme. Namely, in that case the following inequality can be shown:
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\begin{equation}
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\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}
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\end{equation}
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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.
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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, fernandez-gerbeau-grandmont-06}, see example in Figure \ref{fig:comparison_figure}.
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%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.
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\begin{figure}[!hbtp]
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\centering
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\includegraphics[width=0.8\textwidth]{figs/comparison_two_ways_to_one_way.png}
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\caption{Two-ways FSI (left) and one-way FSI (right), on an ellipsoid. In arrows the fluid velocity magnitude and direction, driven by an hyperelastic active solid (in grey, with decreased opacity).}
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\label{fig:comparison_figure}
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\end{figure}
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\newpage
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\bibliography{biblio_merged.bib}
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\bibliographystyle{unsrt}
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\end{document}
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