Document Zbl 1227.78026

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Methods Appl. Mech. Eng.</a> <a href="/?q=in%3A259486" title="Articles in this Issue">199, No. 17-20, 1143-1152 (2010)</a>. </div> <div class="abstract">Summary: We introduce a new discretization scheme for Maxwell equations in two space dimension. Inspired by the new paradigm of Isogeometric analysis introduced in <span class="zbmathjax-textit">T. J. R. Hughes, J. A. Cottrell</span> and <span class="zbmathjax-textit">Y. Bazilevs</span> [Comput. Methods Appl. Mech. Eng. 194, No. 39–41, 4135–4195 (2005; <a href="/1151.74419">Zbl 1151.74419</a>)], we propose an algorithm based on the use of bivariate B-splines spaces suitably adapted to electromagnetics. We construct B-splines spaces of variable interelement regularity on the parametric domain. These spaces (and their push-forwards on the physical domain) form a De Rham diagram and we use them to solve the Maxwell source and eigen problem. Our scheme has the following features: (i) is adapted to treat complex geometries, (ii) is spectral correct, (iii) provides regular (e.g., globally $C^{0}$) discrete solutions of Maxwell equations.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A5982898">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A5982898">Cited in <strong>174</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td class="mono"> 78M25 </td> <td class="space"> Numerical methods in optics (MSC2010) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A78A25" title="MSC2020">78A25</a> </td> <td class="space"> Electromagnetic theory (general) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aisogeometric+analysis">isogeometric analysis</a>; <a href="/?q=ut%3Acompatible+schemes">compatible schemes</a>; <a href="/?q=ut%3AMaxwell+equations">Maxwell equations</a>; <a href="/?q=ut%3Aeigenvalue+problem">eigenvalue problem</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1151.74419">Zbl 1151.74419</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 1227.78026" data-ciurl="/ci/05982898" data-biburl="/bibtex/05982898.bib" data-amsurl="/amsrefs/05982898.bib" data-xmlurl="/xml/05982898.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05982898.pdf" title="Zbl 1227.78026 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/10.1016/j.cma.2009.12.002" aria-label="DOI for “Isogeometric analysis in electromagnetics: B-splines approximation”" title="10.1016/j.cma.2009.12.002">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">Arnold, D. N.; Falk, R. S.; Winther, R., Finite element exterior calculus, homological techniques, and applications, Acta Numer., 15, 1-155 (2006) · <a href="/1185.65204" class="nowrap">Zbl 1185.65204</a></td> </tr><tr> <td>[2]</td> <td class="space">Assous, F.; Ciarlet, P.; Segré, J., Numerical solution to the time-dependent Maxwell equations in two-dimensional singular domains: the singular complement method, J. Comput. Phys., 161, 218-249 (2000) · <a href="/1007.78014" class="nowrap">Zbl 1007.78014</a></td> </tr><tr> <td>[3]</td> <td class="space">Auricchio, F.; Beirão da Veiga, L.; Buffa, A.; Lovadina, C.; Reali, A.; Sangalli, G., A fully “locking-free” isogeometric approach for plane linear elasticity problems: a stream function formulation, Comput. Methods Appl. Mech. Engrg., 197, 160-172 (2007) · <a href="/1169.74643" class="nowrap">Zbl 1169.74643</a></td> </tr><tr> <td>[4]</td> <td class="space">Bazilevs, Y.; Beirão da Veiga, L.; Cottrell, J. A.; Hughes, T. J.R.; Sangalli, G., Isogeometric analysis: approximation, stability and error estimates for $h$-refined meshes, Math. Models Methods Appl. Sci., 16, 1031-1090 (2006) · <a href="/1103.65113" class="nowrap">Zbl 1103.65113</a></td> </tr><tr> <td>[5]</td> <td class="space">Bazilevs, Y.; Calo, V. M.; Hughes, T. J.R.; Zhang, Y., Isogeometric fluid-structure interaction: theory, algorithms, and computations, Comput. Mech., 43, 3-37 (2008) · <a href="/1169.74015" class="nowrap">Zbl 1169.74015</a></td> </tr><tr> <td>[6]</td> <td class="space">L. Beirão da Veiga, A. Buffa, J. Rivas, G. Sangalli, Some estimates for $hpk$; L. Beirão da Veiga, A. Buffa, J. Rivas, G. Sangalli, Some estimates for $hpk$ · <a href="/1222.41010" class="nowrap">Zbl 1222.41010</a></td> </tr><tr> <td>[7]</td> <td class="space">Boffi, D.; Fernandes, P.; Gastaldi, L.; Perugia, I., Computational models of electromagnetic resonators: analysis of edge element approximation, SIAM J. Numer. 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J.R.; Reali, A.; Sangalli, G., Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: comparison of $p$-method finite elements with $k$-method NURBS, Comput. Methods Appl. Mech. Engrg., 197, 4104-4124 (2008) · <a href="/1194.74114" class="nowrap">Zbl 1194.74114</a></td> </tr><tr> <td>[18]</td> <td class="space">Monk, P., Finite Element Methods for Maxwell’s Equations (2003), Oxford University Press: Oxford University Press Oxford · <a href="/1024.78009" class="nowrap">Zbl 1024.78009</a></td> </tr><tr> <td>[19]</td> <td class="space">Nédélec, J.-C., Mixed finite elements in $R^3$, Numer. Math., 35, 315-341 (1980) · <a href="/0419.65069" class="nowrap">Zbl 0419.65069</a></td> </tr><tr> <td>[20]</td> <td class="space">Piegl, L.; Tiller, W., The Nurbs Book (1997), Springer-Verlag: Springer-Verlag New York · <a href="/0868.68106" class="nowrap">Zbl 0868.68106</a></td> </tr><tr> <td>[21]</td> <td class="space">Schumaker, L. L., Spline Functions: Basic Theory (2007), Cambridge Mathematical Library, Cambridge University Press: Cambridge Mathematical Library, Cambridge University Press Cambridge · <a href="/1123.41008" class="nowrap">Zbl 1123.41008</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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