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Q.</a></div> <h2 class="title"> <strong>Discontinuous finite-element phase-field modeling of polycrystalline grain growth with convection.</strong> <i>(English)</i> <a class="label nowrap" href="/1173.80308">Zbl 1173.80308</a> </h2> <div class="source"> <a href="/serials/62" title="Journal Profile">Comput. Math. Appl.</a> <a href="/?q=in%3A188761" title="Articles in this Issue">52, No. 5, 721-734 (2006)</a>. </div> <div class="abstract">Summary: This paper presents a discontinuous finite-element computational framework for the numerical modeling of polycrystalline microstructure formation during solidification processing, taking into account the effect of convection. An important aspect of the discontinuous formulation is that the interelement continuity is relaxed for all the field variables or more precisely the continuity across the elements is weakly enforced, rather than strongly enforced as done in conventional finite-element methods. Because of this local nature, the discontinuous finite-element method enjoys certain advantages over the traditional finite difference and finite-element methods, in particular, for large scale numerical simulations of physical phenomena in thermal and fluids systems. The solution procedure involves an element-by-element iterative sweep. The discontinuous finite-element formulation is presented for a fully coupled set of nonlinear equations including the Navier-Stokes equations, the phase-field equation, the thermal equation, and the crystal orientation equation, which describe the physical phenomena associated with solidification processing. Both single grain and polygrain models are considered. Numerical results are presented that illustrate the effect of convection on the microstructure formation during polycrystalline solidification.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5207604">Cited in <strong>2</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A80M10" title="MSC2020">80M10</a> </td> <td class="space"> Finite element, Galerkin and related methods applied to problems in thermodynamics and heat transfer </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></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3ASolidification">Solidification</a>; <a href="/?q=ut%3Adendrite">dendrite</a>; <a href="/?q=ut%3Apolycrystalline+grain+growth">polycrystalline grain growth</a>; <a href="/?q=ut%3Aconvection">convection</a>; <a href="/?q=ut%3Adiscontinuous+finite+element">discontinuous finite element</a></div> <!-- Modal used to show zbmath metadata in different output formats--> <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">&times;</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;"> <!-- alert for connection errors etc --> </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"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1173.80308" data-ciurl="/ci/05207604" data-biburl="/bibtex/05207604.bib" data-amsurl="/amsrefs/05207604.bib" data-xmlurl="/xml/05207604.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05207604.pdf" title="Zbl 1173.80308 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.camwa.2006.10.010" aria-label="DOI for “Discontinuous finite-element phase-field modeling of polycrystalline grain growth with convection”" title="10.1016/j.camwa.2006.10.010">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">Caginalp, G.; Fife, P., Phase-field methods for interfacial boundaries, Physical Review B, 33, 7792-7794 (1986)</td> </tr><tr> <td>[2]</td> <td class="space">Caginalp, G.; Xie, W., Phase-field and sharp-interface alloy models, Physical Review E, 48, 1897-1909 (1993)</td> </tr><tr> <td>[3]</td> <td class="space">Kobayashi, R., Modeling and numerical simulations of dendritic crystal growth, Physica D, 63, 410-423 (1993) &middot; <a href="/0797.35175" class="nowrap">Zbl 0797.35175</a></td> </tr><tr> <td>[4]</td> <td class="space">Karma, A.; Rappel, W.-J., Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics, Physical Review E, 53, 3017-3020 (1996)</td> </tr><tr> <td>[5]</td> <td class="space">Warren, J. A.; Kobayashi, R.; Lobkovsky, A. E.; Carter, W. C., Extending phase field models of solidification to polycrystalline materials, Acta Materialia, 51, 6035-6058 (2003)</td> </tr><tr> <td>[6]</td> <td class="space">Kobayashi, R.; Warren, J. A.; Carter, W. C., A continuum model of grain boundaries, Physica D, 140, 141-150 (2000) &middot; <a href="/0956.35123" class="nowrap">Zbl 0956.35123</a></td> </tr><tr> <td>[7]</td> <td class="space">B.Q. Li, Discontinuous Finite Element Methods in Fluid Dynamics and Heat Transfer, Springer-Verlag (to appear).; B.Q. Li, Discontinuous Finite Element Methods in Fluid Dynamics and Heat Transfer, Springer-Verlag (to appear).</td> </tr><tr> <td>[8]</td> <td class="space">Cockburn, B.; Karniadakis, G. E.; Shu, C. W., The development of discontinuous Galerkin methods, (Cockburn, B.; Karniadakis, G.; Shu, C. W., Discontinuous Galerkin Methods: Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, 11 (2000 February), Springer- Verlag: Springer- Verlag New York), 3-50 &middot; <a href="/0989.76045" class="nowrap">Zbl 0989.76045</a></td> </tr><tr> <td>[9]</td> <td class="space">Beckermann, C.; Diepers, H.-J.; Steinbach, I.; Karma, A.; Tong, X., Modeling melt convection in phase-field simulations of solidification, Journal of Computational Physics, 15, 468-496 (1999) &middot; <a href="/0960.82015" class="nowrap">Zbl 0960.82015</a></td> </tr><tr> <td>[10]</td> <td class="space">Ai, X.; Li, B. Q., Solution of hyperbolic heat conduction problems by discontinuous Galerkin finite element method, Engineering Computations, 21, 6, 577-597 (2004) &middot; <a href="/1074.80004" class="nowrap">Zbl 1074.80004</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&nbsp;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"> &copy; 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 &amp; 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">&nbsp;<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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