Document Zbl 1530.35214

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We reformulate the inverse problem into an optimization problem using a least squares functional. We prove the existence of an optimal solution to the optimization problem. Then, we perform the asymptotic expansion of the cost function in a simple way using a penalty method. An important advantage of this method is that it avoids the truncation method used in the literature. To reconstruct the obstacle, we propose a fast algorithm based on the topological derivative. Finally, we present some numerical experiments in two- and three-dimensional cases showing the efficiency of the proposed method.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A35Q35" title="MSC2020">35Q35</a> </td> <td class="space"> PDEs in connection with fluid mechanics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35R30" title="MSC2020">35R30</a> </td> <td class="space"> Inverse problems for PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A76S05" title="MSC2020">76S05</a> </td> <td class="space"> Flows in porous media; filtration; seepage </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A49Q10" title="MSC2020">49Q10</a> </td> <td class="space"> Optimization of shapes other than minimal surfaces </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A49K20" title="MSC2020">49K20</a> </td> <td class="space"> Optimality conditions for problems involving partial differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A49K40" title="MSC2020">49K40</a> </td> <td class="space"> Sensitivity, stability, well-posedness </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A49M41" title="MSC2020">49M41</a> </td> <td class="space"> PDE constrained optimization (numerical aspects) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35B40" title="MSC2020">35B40</a> </td> <td class="space"> Asymptotic behavior of solutions to PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35A01" title="MSC2020">35A01</a> </td> <td class="space"> Existence problems for PDEs: global existence, local existence, non-existence </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M32" title="MSC2020">65M32</a> </td> <td class="space"> Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M60" title="MSC2020">65M60</a> </td> <td class="space"> Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A76M21" title="MSC2020">76M21</a> </td> <td class="space"> Inverse problems in fluid mechanics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A76M10" title="MSC2020">76M10</a> </td> <td class="space"> Finite element methods applied to problems in fluid mechanics </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aoptimal+shape+design">optimal shape design</a>; <a href="/?q=ut%3Ainverse+problem">inverse problem</a>; <a href="/?q=ut%3Atime-dependent+Brinkman+equation">time-dependent Brinkman equation</a>; <a href="/?q=ut%3Atopological+sensitivity+analysis">topological sensitivity analysis</a></div> <div class="software"> <h3>Software:</h3><a href="/software/4314">FEniCS</a>; <a href="/software/11183">UFL</a>; <a href="/software/5701">SyFi</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1530.35214" data-ciurl="/ci/07702837" data-biburl="/bibtex/07702837.bib" data-amsurl="/amsrefs/07702837.bib" data-xmlurl="/xml/07702837.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07702837.pdf" title="Zbl 1530.35214 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.3233/ASY-221792" aria-label="DOI for “Optimal shape design for a time-dependent Brinkman flow using asymptotic analysis techniques”" title="10.3233/ASY-221792">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">A.B. 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