On the detection of several obstacles in 2D Stokes flow: Topological sensitivity and combination with shape derivatives

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html lang="en" > <head> <meta name=renderer content=webkit> <meta http-equiv="X-UA-Compatible" content="IE=edge" /> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <meta name="viewport" content="width=device-width,initial-scale=1.0,maximum-scale=1.0,minimum-scale=1.0,user-scalable=no" /> <title>On the detection of several obstacles in 2D Stokes flow: Topological sensitivity and combination with shape derivatives</title> <meta name="keywords" content="Geometric inverse problem, topological sensitivity analysis, topological gradient, shape gradient, Stokes equations, Kohn-Vogelius functional" /> <meta name="hw.ad-path" content="https://www.aimsciences.org/en/article/doi/10.3934/ipi.2016003" /> <meta name="hw.identifier" content="https://www.aimsciences.org/en/article/doi/10.3934/ipi.2016003" /> <meta name="dc.format" content="text/html" /> <meta name="dc.title" content="On the detection of several obstacles in 2D Stokes flow: Topological sensitivity and combination with shape derivatives" /> <meta name="dc.identifier" content="10.3934/ipi.2016003" /> <meta name="dc.contributor" content="Fabien Caubet" /> <meta name="dc.contributor" content="Carlos Conca" /> <meta name="dc.contributor" content="Matías Godoy" /> <meta name="dc.creator" content="Fabien Caubet" /> <meta name="dc.creator" content="Carlos Conca" /> <meta name="dc.creator" content="Matías Godoy" /> <meta name="dc.date" content="Sun May 01 00:00:00 UTC 2016" /> <meta name="dc.keywords" content="Geometric inverse problem, topological sensitivity analysis, topological gradient, shape gradient, Stokes equations, Kohn-Vogelius functional" /> <meta name="citation_journal_title" content="Inverse Problems and Imaging" /> <meta name="citation_publisher" content="Inverse Problems and Imaging" /> <meta name="citation_journal_abbrev" content="IPI" /> <meta name="citation_issn" content="1930-8337" /> <meta name="citation_authors" content="Fabien Caubet, Carlos Conca, Matías Godoy" /> <meta name="citation_date" content="Sun May 01 00:00:00 UTC 2016" /> <meta name="citation_online_date" content=" "/> <meta name="citation_abstract_html_url" content="https://www.aimsciences.org/en/article/doi/10.3934/ipi.2016003?viewType=HTML" /> <meta name="citation_pdf_url" content="https://www.aimsciences.org/data/article/export-pdf?id=5cfa3af2-5c66-4e6e-a088-10bbff937bfb"/> <meta name="citation_public_url" content="https://www.aimsciences.org/en/article/doi/10.3934/ipi.2016003" /> <meta property="og:url" content="https://www.aimsciences.org/article/doi/10.3934/ipi.2016003" > <meta property="og:type" content="article"> <meta property="og:site_name" content="Inverse Problems and Imaging"> <meta property="og:title" content="On the detection of several obstacles in 2D Stokes flow: Topological sensitivity and combination with shape derivatives"> <meta name="twitter:card" content="summary_large_image"> <meta name="twitter:description" content="We consider the inverse problem of detecting the location and the shape of several obstacles immersed in a fluid flowing in a larger bounded domain $\Omega$ from partial boundary measurements in the two dimensional case. The fluid flow is governed by the steady-state Stokes equations. We use a topological sensitivity analysis for the Kohn-Vogelius functional in order to find the number and the qualitative location of the objects. Then we explore the numerical possibilities of this approach and also present a numerical method which combines the topological gradient algorithm with the classical geometric shape gradient algorithm; this blending method allows to find the number of objects, their relative location and their approximate shape." > <meta name="twitter:site" content="Inverse Problems and Imaging"> <meta name="twitter:title" content="On the detection of several obstacles in 2D Stokes flow: Topological sensitivity and combination with shape derivatives"> <meta name="twitter:image" content=""> <meta name="twitter:url" content="https://www.aimsciences.org/article/doi/10.3934/ipi.2016003" > <meta property="og:image" content=""> <meta name="robots" content="noarchive" /> <meta name="dc.publisher" content="Inverse Problems and Imaging"/> <meta name="dc.rights" content="http://creativecommons.org/licenses/by/3.0/"/> <meta name="dc.format" content="application/pdf" /> <meta name="dc.language" content="en" /> <meta name="dc.description" content="We consider the inverse problem of detecting the location and the shape of several obstacles immersed in a fluid flowing in a larger bounded domain $\Omega$ from partial boundary measurements in the two dimensional case. The fluid flow is governed by the steady-state Stokes equations. We use a topological sensitivity analysis for the Kohn-Vogelius functional in order to find the number and the qualitative location of the objects. Then we explore the numerical possibilities of this approach and also present a numerical method which combines the topological gradient algorithm with the classical geometric shape gradient algorithm; this blending method allows to find the number of objects, their relative location and their approximate shape." /> <meta property="og:description" content="We consider the inverse problem of detecting the location and the shape of several obstacles immersed in a fluid flowing in a larger bounded domain $\Omega$ from partial boundary measurements in the two dimensional case. The fluid flow is governed by the steady-state Stokes equations. We use a topological sensitivity analysis for the Kohn-Vogelius functional in order to find the number and the qualitative location of the objects. Then we explore the numerical possibilities of this approach and also present a numerical method which combines the topological gradient algorithm with the classical geometric shape gradient algorithm; this blending method allows to find the number of objects, their relative location and their approximate shape." /> <meta name="dc.subject" content="Geometric inverse problem" /> <meta name="dc.subject" content="topological sensitivity analysis" /> <meta name="dc.subject" content="topological gradient" /> <meta name="dc.subject" content="shape gradient" /> <meta name="dc.subject" content="Stokes equations" /> <meta name="dc.subject" content="Kohn-Vogelius functional" />  <meta name="citation_title" content="On the detection of several obstacles in 2D Stokes flow: Topological sensitivity and combination with shape derivatives" /> <meta name="citation_author" content="Fabien Caubet" /> <meta name="citation_author" content="Carlos Conca" /> <meta name="citation_author" content="Matías Godoy" /> <meta name="citation_keywords" content="Geometric inverse problem, topological sensitivity analysis, topological gradient, shape gradient, Stokes equations, Kohn-Vogelius functional" /> <meta name="citation_year" content="2016"/> <meta name="citation_volume" content="10"/> <meta name="citation_issue" content="2"/> <meta name="citation_firstpage" content="327"/> <meta name="citation_lastpage" content="367"/> <meta name="citation_doi" content="10.3934/ipi.2016003"/> <meta name="citation_id" content="5cfa3af2-5c66-4e6e-a088-10bbff937bfb"/> <meta name="citation_state" content="" /> <link type="text/css" href="/style/web/images/custom/favicon.ico" rel="shortcut icon" /> <link rel="stylesheet" href="/style/web/css/common/bootstrap.min.css"> <link rel="stylesheet" href="/style/web/css/common/swiper.min.css"> <link rel="stylesheet" href="/style/web/css/common/font-awesome.min.css"> <link rel="stylesheet" href="/style/web/css/defercss/article_public.min.css?t=8" type="text/css"/> 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<a href="/ipi/archive" class="mainColor">2016, Volume 10</a>, <a href="/ipi/article/2016/10/2" class="mainColor">Issue 2</a>: 327-367. Doi: <a class="mainColor" href="https://doi.org/10.3934/ipi.2016003" target="_blank">10.3934/ipi.2016003 </a> </div> <div class="fr" id="title-show"> <a href="/ipi/article/2016/10/2" class="this-issue">This issue</a> <a href="/article/doi/10.3934/ipi.2016002" class="page-btn prev-page "><span class="fa fa-angle-left"></span>Previous Article</a> <span class="title-show">Forward and backward filtering based on backward stochastic differential equations</span> <a href="/article/doi/10.3934/ipi.2016004" class="page-btn next-page ">Next Article<span class="fa fa-angle-right"></span></a> <span class="title-show">On a transmission eigenvalue problem for a spherically stratified coated dielectric</span> </div> </div>  <div class="article-left"> <div class="article-app" style="padding:0;"> <footer class="download-pdf " data="5cfa3af2-5c66-4e6e-a088-10bbff937bfb" style="width:100%;margin:20px 0;padding:0;">  <a href="javascript:void(0);"> <div class="pdfView"> <div class="in-bl"> <i class="fa fa-file-pdf-o"></i> <span>PDF  view</span> </div> </div> </a> </footer> </div> <div class="articleEn prerender" id="articleEn"> <div class="article-title left-title clearfix"> <h1>On the detection of several obstacles in 2D Stokes flow: Topological sensitivity and combination with shape derivatives </h1> </div> <ul class="article-author clear"> <li> <a href="javascript:void(0);" class="mainColor" data-relate="fabien.caubet@math.univ-toulouse.fr" data="{{author.authorNameEn}}" type="authors.authorNameEn">Fabien Caubet</a><sup class="authorTag"><a href="javascript:void(0);" class="com-num" data-tagVal="1">1</a>, <a href="mailto:fabien.caubet@math.univ-toulouse.fr" class="com-mail" title="mailto: fabien.caubet@math.univ-toulouse.fr"></a></sup>,  </li> <li> <a href="javascript:void(0);" class="mainColor" data-relate="cconca@dim.uchile.cl" data="{{author.authorNameEn}}" type="authors.authorNameEn">Carlos Conca</a><sup class="authorTag"><a href="javascript:void(0);" class="com-num" data-tagVal="2">2</a>, <a href="mailto:cconca@dim.uchile.cl" class="com-mail" title="mailto: cconca@dim.uchile.cl"></a></sup> <span class="mainColor">and</span>  </li> <li> <a href="javascript:void(0);" class="mainColor" data-relate="mgodoy@dim.uchile.cl" data="{{author.authorNameEn}}" type="authors.authorNameEn">Matías Godoy</a><sup class="authorTag"><a href="javascript:void(0);" class="com-num" data-tagVal="1">1</a>, <a href="mailto:mgodoy@dim.uchile.cl" class="com-mail" title="mailto: mgodoy@dim.uchile.cl"></a></sup> </li> <li><span class="com-drop" onclick="showAuthor(this);"></span></li> </ul> <ul class=" marginT addresswrap about-author">  <li class="article-author-address"> <span class="aff-nmuber">1.</span> <p class="lostOf" > Institut de Mathématiques de Toulouse, Université de Toulouse, F-31062 Toulouse Cedex 9 </p> </li> <li class="article-author-address"> <span class="aff-nmuber">2.</span> <p class="lostOf" > Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 170-3, Correo 3, Santiago </p> </li> </ul>  <div class="com-author-wrap"> <ul class="about-article about-author about-author1"> </ul> <div class="author-note-box"> </div>  <div class="com-author-info gjrq">  <div class="article-date clearfix"> <div><b>Received: </b> <span id="" class="mainColor"> February 2015 </span> </div> <div><b>Revised: </b> <span id="" class="mainColor"> May 2015 </span> </div> <div><b>Published: </b> <span class="mainColor"> May 2016 </span> </div> </div> </div>  </div> </div> </div>  <div class="article-detail-btns clearfix"> <a href="javascript:void(0);" rel="Abstract" class="desc ac">Abstract <span class="isIntroduction">/ Introduction</span></a> <a href="javascript:void(0);" rel="relative-article" class="relative">Related Papers</a> <a href="javascript:void(0);" rel="citedby-info" class="citeby">Cited by</a> </div> <div class="clearfix" > <ul class="article-tab-box tab-content" id="myTabContent">  <div class="phone-title phone-mark" val="Abstract"> <a class="mainColor " href="javascript:void(0);">Abstract</a> </div> <li id="Abstract" class="articleListBox current prerender" > <h3 class="navTitle hidden" id="Abstract-list">Abstract</h3> <div class="article-abstract article-abstract-2017"> We consider the inverse problem of detecting the location and the shape of several obstacles immersed in a fluid flowing in a larger bounded domain $\Omega$ from partial boundary measurements in the two dimensional case. The fluid flow is governed by the steady-state Stokes equations. We use a topological sensitivity analysis for the Kohn-Vogelius functional in order to find the number and the qualitative location of the objects. Then we explore the numerical possibilities of this approach and also present a numerical method which combines the topological gradient algorithm with the classical geometric shape gradient algorithm; this blending method allows to find the number of objects, their relative location and their approximate shape. </div> <div class="hasIntroduction"> </div> <ul class="article-keyword article-info-en"> <b>Keywords:</b> <li><a class="underHigh mainColor" href="javascript:void(0);" data="Geometric inverse problem" type="keywords.keywordEn">Geometric inverse problem</a>, </li> <li><a class="underHigh mainColor" href="javascript:void(0);" data="topological sensitivity analysis" type="keywords.keywordEn">topological sensitivity analysis</a>, </li> <li><a class="underHigh mainColor" href="javascript:void(0);" data="topological gradient" type="keywords.keywordEn">topological gradient</a>, </li> <li><a class="underHigh mainColor" href="javascript:void(0);" data="shape gradient" type="keywords.keywordEn">shape gradient</a>, </li> <li><a class="underHigh mainColor" href="javascript:void(0);" data="Stokes equations" type="keywords.keywordEn">Stokes equations</a>, </li> <li><a class="underHigh mainColor" href="javascript:void(0);" data="Kohn-Vogelius functional" type="keywords.keywordEn">Kohn-Vogelius functional</a>.</li> </ul> <div class="cssNo"> <b>Mathematics Subject Classification: </b>Primary: 49Q10, 35R30, 49Q12; Secondary: 76D55. </div>  <div class="article-header"> <p class="quot"> <div class="copyCitationInfo" style="width: 0px;height: 0px;opacity:0;overflow: hidden;"></div> <table class="article-tab"> <tr>  <td> <b>Citation:</b><span class="info" id="infoEn"></span> <div class="modal-footer"> <span class="copy-citation" onclick="copyArticle2(this)" title="copy to clipboard"> <img class="shu" src="/style/web/images/article/shu.png" alt="shu" /> </span> </div> </td> </tr> </table> </p> </div> <div style="display:none;">\begin{equation} \\ \end{equation}</div> </li>   <div class="phone-title" val="Relative_Articles"> <a class="mainColor " data-navid="relative-article" href="javascript:void(0);"> Related Papers </a> </div> <li id="relative-article" class="articleListBox article-related-article" ></li> <div class="phone-title" val="Cited_by"> <a class="mainColor " data-navid="citedby-article" href="javascript:void(0);"> Cited by </a> </div> <li id="citedby-info" class="articleListBox article-knowledge-map"></li>   <div class="phone-title" rel="ref" val="References"> <a class="mainColor " data-navid="References" href="javascript:void(0);"> References </a> </div> <li id="References" class="articleListBox"> <h3 class="navTitle" id="References-list">References</h3> <div class="References-wrap"> <table class="reference-tab"> <tr class="document-box" id="b1"> <td valign="top" class="td1"> [1] </td> <td class="td2 105"> <p>G. 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On the detection of several obstacles in 2D Stokes flow: Topological sensitivity and combination with shape derivatives