Document Zbl 0778.65076

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K.</a>; <a href="/authors/shephard.mark-s" title="Author Profile">Shephard, M. S.</a></div> <h2 class="title"> <strong>Automated adaptive two-dimensional system for the $hp$-version of the finite element method.</strong> <i>(English)</i> <a class="label nowrap" href="/0778.65076">Zbl 0778.65076</a> </h2> <div class="source"> <a href="/serials/462" title="Journal Profile">Int. J. Numer. Methods Eng.</a> <a href="/?q=in%3A160011" title="Articles in this Issue">32, No. 4, 867-893 (1991)</a>. </div> <div class="abstract">The authors present a strategy for coupling their earlier work in mesh generation [“Automatic mesh generator for use in two-dimensional $h-p$ analysis”, ASCE J. Comp. Civil Eng., 4, 199-220 (1990)] with a commercial product called PROBE [NOETIC Technology Corporation, St. Louis, MO]. This strategy employs both local and global error and convergence rate estimates to determine when elemental mesh size $h$ and polynomial order $p$ must be improved in order to achieve prescribed accuracy with minimal effort.<br class="zbmathjax-paragraph">The authors consider the problem of two-dimensional linear elasticity with prescribed displacements on part of the boundary and prescribed tractions on the rest. The region can be complex, with corners and cutouts inducing singularities into the solution. The error estimation procedures depend on individual treatment of the singularities.<br class="zbmathjax-paragraph">Three types of error estimation are performed. The first is a global error estimate based on convergence rate as the polynomial order $p$ is increased. (A uniform polynomial order $p$ over all mesh elements is used.) The second is an estimate of the error contribution of each singularity. The third is a local error estimate based on comparison of elemental differences between the solution achieved with order $p$ and that with $p+1$. The behavior of these three estimates must be consistent with theory and their values small enough before the solution is accepted. Three numerical examples of increasing complexity are presented in order to illustrate the effectiveness of the overall strategy.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A4380">Myron Sussman (Bethel Park)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A95156">Cited in <strong>8</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65N15" title="MSC2020">65N15</a> </td> <td class="space"> Error bounds for boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65N30" title="MSC2020">65N30</a> </td> <td class="space"> Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65N50" title="MSC2020">65N50</a> </td> <td class="space"> Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A74S05" title="MSC2020">74S05</a> </td> <td class="space"> Finite element methods applied to problems in solid mechanics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A74B05" title="MSC2020">74B05</a> </td> <td class="space"> Classical linear elasticity </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Alocal+error+estimate">local error estimate</a>; <a href="/?q=ut%3Aglobal+error+estimate">global error estimate</a>; <a href="/?q=ut%3Amesh+generation">mesh generation</a>; <a href="/?q=ut%3Aconvergence+rate+estimates">convergence rate estimates</a>; <a href="/?q=ut%3Atwo-dimensional+linear+elasticity">two-dimensional linear elasticity</a>; <a href="/?q=ut%3Asingularities">singularities</a>; <a href="/?q=ut%3Anumerical+examples">numerical examples</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 0778.65076" data-ciurl="/ci/00095156" data-biburl="/bibtex/00095156.bib" data-amsurl="/amsrefs/00095156.bib" data-xmlurl="/xml/00095156.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/00095156.pdf" title="Zbl 0778.65076 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/10.1002/nme.1620320412" aria-label="DOI for “Automated adaptive two-dimensional system for the $hp$-version of the finite element method”" title="10.1002/nme.1620320412">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">’Estimation and control of error based on p-convergence’, in Proc. Int. Conf. on Accuracy Estimates and Adaptive Refinements in Finite Element Computations (ARFEC), Lisbon, Portugal, 1984.</td> </tr><tr> <td>[2]</td> <td class="space">’Estimation and control of error based on p convergence’, in et al. (eds.), Accuracy Estimates and Adaptive Refinements in Finite Element Computations, Wiley, Chichester, U.K., 1986, pp. 61-78.</td> </tr><tr> <td>[3]</td> <td class="space">Szabo, Comp. Methods Appl. Mech. Eng. 55 pp 181– (1986)</td> </tr><tr> <td>[4]</td> <td class="space">Rank, Int. j. numer. methods eng. 24 pp 2087– (1987)</td> </tr><tr> <td>[5]</td> <td class="space">’Automated finite element modelling and simulation’, Ph.D. Thesis, Rensselaer Design Research Center, Rensselaer Polytechnic Institute, Troy, NY, May 1989.</td> </tr><tr> <td>[6]</td> <td class="space">Shephard, Appl. Mech. Rev. 41 pp 169– (1988)</td> </tr><tr> <td>[7]</td> <td class="space">Zienkiewicz, Int. j. numer. methods eng. 28 pp 879– (1988)</td> </tr><tr> <td>[8]</td> <td class="space">and , ’Toward automatic model generation’, in and (eds.), State-of-the-Art Surveys on Computational Mechanics, ASME, New York, 1989, pp. 335-366.</td> </tr><tr> <td>[9]</td> <td class="space">, and (eds.), Accuracy Estimates and Adaptive Refinements in Finite Element Computations. Wiley, Chichester, U.K., 1986.</td> </tr><tr> <td>[10]</td> <td class="space">, and (eds.), Adaptive Methods for Partial Differential Equations, SIAM, Philadelphia, PA, 1989.</td> </tr><tr> <td>[11]</td> <td class="space">PROBE: Theoretical Manual, NOETIC Technology Corporation, 7980 Clayton Road, Suite 205, St. Louis, MO 63117, 1985.</td> </tr><tr> <td>[12]</td> <td class="space">Babuska, SIAM J. Numer. Anal. 18 pp 515– (1981)</td> </tr><tr> <td>[13]</td> <td class="space">’A two-dimensional automatic modeling system for the h-p version of the finite element method’, Ph.D. Thesis, Rensselaer Design Research Center, Rensselaer Polytechnic Institute, Troy, NY, Dec. 1989.</td> </tr><tr> <td>[14]</td> <td class="space">Georges, ASCE J. Comp. Civil Eng. 4 pp 199– (1990)</td> </tr><tr> <td>[15]</td> <td class="space">and , ’The h, p and h-p version of the finite element method of one-dimensional problems, Part 1: The analysis of the p-version’, Technical Report, Institute for Physical Science and Technology, Laboratory for Numerical Analysis, 1978, Note BN-1036.</td> </tr><tr> <td>[16]</td> <td class="space">and , ’The h, p and h-p version of the finite element method of one-dimensional problems, Part 2: The error analysis of the h- and h-p version’, Technical Report, Institute for Physical Science and Technology, Laboratory for Numerical Analysis, 1978, Note BN-1037.</td> </tr><tr> <td>[17]</td> <td class="space">and , ’The h, p and h-p version of the finite element method of one-dimensional problems, Part 3: The adaptive h-p version’, Technical Report, Institute for Physical Science and Technology, Laboratory for Numerical Analysis, 1978. Note BN-1038.</td> </tr><tr> <td>[18]</td> <td class="space">Guo, J. Comp. Mech. 1 pp 21– (1986)</td> </tr><tr> <td>[19]</td> <td class="space">’The p and h-p versions of the finite element method. The state of the art’, in (ed.), Proc. Finite Element Method Workshop, Springer-Verlag, New York, 1987.</td> </tr><tr> <td>[20]</td> <td class="space">Zienkiewicz, Int. j. numer. methods eng. 24 pp 337– (1987)</td> </tr><tr> <td>[21]</td> <td class="space">Guo, J. Comp. Mech. 1 pp 21– (1986)</td> </tr><tr> <td>[22]</td> <td class="space">Williams, J. Appl. Mech. ASME 19 pp 526– (1952)</td> </tr><tr> <td>[23]</td> <td class="space">’Behavior of the solution of an elliptic boundary value problem in a polygonal or polyhedral domain’, in (ed.), Numerical Solution of Partial Differential Equations, 3rd edn, Academic Press, New York, 1976, pp. 207-274. · <a href="https://doi.org/10.1016/B978-0-12-358503-5.50013-0" class="nowrap">doi:10.1016/B978-0-12-358503-5.50013-0</a></td> </tr><tr> <td>[24]</td> <td class="space">Zienkiewicz, Int. j. numer. methods eng. 24 pp 337– (1988)</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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