Document Zbl 1067.74573

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II: Multiscale modeling and quality control in multidimensional case.</strong> <i>(English)</i> <a class="label nowrap" href="/1067.74573">Zbl 1067.74573</a> </h2> <div class="source"> <a href="/serials/64" title="Journal Profile">Comput. Methods Appl. Mech. Eng.</a> <a href="/?q=in%3A33873" title="Articles in this Issue">126, No. 1-2, 17-38 (1995)</a>. </div> <div class="abstract">Summary: We present a multigrid method for multidimensional periodic heterogeneous media. Based on homogenization theory, special intergrid transfer operators have been developed to simulate low-frequency response of the boundary value problem with oscillatory coefficients. An adaptive strategy is developed to form a nearly optimal two-scale computational model consisting of the finite element mesh entirely constructed on the microscale in the regions identified by the idealization error indicators, while elsewhere the modeling level is only sufficient to capture the response of the homogenized medium. Numerical experiments show the usefulness of the proposed adaptive multilevel procedure for predicting a detailed response of composite specimens.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A1500337">Cited in <strong>47</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><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%3A74Q05" title="MSC2020">74Q05</a> </td> <td class="space"> Homogenization in equilibrium problems of solid mechanics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M55" title="MSC2020">65M55</a> </td> <td class="space"> Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Ahomogenization+theory">homogenization theory</a>; <a href="/?q=ut%3Aintergrid+transfer+operators">intergrid transfer operators</a>; <a href="/?q=ut%3Aadaptive+multilevel+procedure">adaptive multilevel procedure</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 1067.74573" data-ciurl="/ci/01500337" data-biburl="/bibtex/01500337.bib" data-amsurl="/amsrefs/01500337.bib" data-xmlurl="/xml/01500337.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/01500337.pdf" title="Zbl 1067.74573 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/10.1016/0045-7825(95)00812-F" aria-label="DOI for “Multigrid method for periodic heterogeneous media. II: Multiscale modeling and quality control in multidimensional case.”" title="10.1016/0045-7825(95)00812-F">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">Benssousan, A.; Lions, J. L.; Papanicoulau, G., Asymptotic Analysis for Periodic Structures (1978), North Holland: North Holland Amsterdam · <a href="/0404.35001" class="nowrap">Zbl 0404.35001</a></td> </tr><tr> <td>[2]</td> <td class="space">Sanchez-Palencia, E.; Zaoui, A., Homogenization techniques for composites, (Homogenization Techniques for Composite Media (1985), Springer-Verlag: Springer-Verlag Berlin) · <a href="/0619.00027" class="nowrap">Zbl 0619.00027</a></td> </tr><tr> <td>[3]</td> <td class="space">Bakhvalov, N. S.; Panasenko, G. P., Homogenization of Periodic Medium Processes (1984), Nauka: Nauka Moscow · <a href="/0607.73009" class="nowrap">Zbl 0607.73009</a></td> </tr><tr> <td>[4]</td> <td class="space">Hashin, Z., Theory of composite materials, (Wend, F. W.; Liebowitz, H.; Perrone, N., Mechanics of Composite Materials (1991), John Wiley: John Wiley New York) · <a href="/0542.73092" class="nowrap">Zbl 0542.73092</a></td> </tr><tr> <td>[5]</td> <td class="space">Kröner, E., Bounds of effective elastic moduli of diordered materials, J. Mech. Phys. Solids, 25, 137 (1977) · <a href="/0359.73020" class="nowrap">Zbl 0359.73020</a></td> </tr><tr> <td>[6]</td> <td class="space">Fish, J.; Belsky, V., Multigrid method for periodic heterogeneous media. Part 1: Convergence studies for one-dimensional case, Comput. Methods Appl. Mech. Engrg., 126, 1 (1995) · <a href="/1067.74574" class="nowrap">Zbl 1067.74574</a></td> </tr><tr> <td>[7]</td> <td class="space">Brandt, Achi, Multi-level adaptive solutions to boundary-value problems, Math. Comput., 31, 333-390 (1978) · <a href="/0373.65054" class="nowrap">Zbl 0373.65054</a></td> </tr><tr> <td>[8]</td> <td class="space">McCormick, S. F.; Thomas, J. W., The fast adaptive composite grid (FAC) method for elliptic equations, Math. Comput., 46, 439-456 (1986) · <a href="/0594.65078" class="nowrap">Zbl 0594.65078</a></td> </tr><tr> <td>[9]</td> <td class="space">Heroux, Michael A.; Thomas, J. W., A comparison of FAC and PCG methods for solving composite grid problems, Commun. Appl. Numer. Methods, 8, 573-583 (1992) · <a href="/0761.65019" class="nowrap">Zbl 0761.65019</a></td> </tr><tr> <td>[10]</td> <td class="space">Zienkiewicz, O. C.; Zhu, J. Z., Superconvergent derivative recovery techniques and a posteriori error estimation in the finite element method. Part I: A general superconvergent recovery technique (1991), Institute of Numerical Methods in Engineering, University College of Swansea: Institute of Numerical Methods in Engineering, University College of Swansea UK, CR/671/91 · <a href="/0755.65119" class="nowrap">Zbl 0755.65119</a></td> </tr><tr> <td>[11]</td> <td class="space">Fish, J.; Markolefas, S., Adaptive s-method for linear elastostatics, Comput. Methods Appl. Mech. Engrg., 104, 363-396 (1993) · <a href="/0775.73248" class="nowrap">Zbl 0775.73248</a></td> </tr><tr> <td>[12]</td> <td class="space">Guedes, J. M.; Kikuchi, N., Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods, Comput. Methods Appl. Mech. Engrg., 83, 143-198 (1990) · <a href="/0737.73008" class="nowrap">Zbl 0737.73008</a></td> </tr><tr> <td>[13]</td> <td class="space">Fish, J.; Wagiman, A., Multiscale finite element method for a locally nonperiodic heterogeneous medium, Computat. Mech.: Int. J., 21, 1-17 (1993) · <a href="/0779.73058" class="nowrap">Zbl 0779.73058</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. 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