On Discretization of Second-order Derivatives in Smoothed Particle Hydrodynamics

<!DOCTYPE html> <html lang="en" dir="ltr"> <head>  <script async src="https://www.googletagmanager.com/gtag/js?id=G-P63WKM1TM1"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-P63WKM1TM1'); </script>  <script type="text/javascript" > (function(m,e,t,r,i,k,a){m[i]=m[i]||function(){(m[i].a=m[i].a||[]).push(arguments)}; m[i].l=1*new Date(); for (var j = 0; j < document.scripts.length; j++) {if (document.scripts[j].src === r) { return; }} k=e.createElement(t),a=e.getElementsByTagName(t)[0],k.async=1,k.src=r,a.parentNode.insertBefore(k,a)}) (window, document, "script", "https://mc.yandex.ru/metrika/tag.js", "ym"); ym(55165297, "init", { clickmap:false, trackLinks:true, accurateTrackBounce:true, webvisor:false }); </script> <noscript><div><img src="https://mc.yandex.ru/watch/55165297" style="position:absolute; left:-9999px;" alt="" /></div></noscript>    <title>On Discretization of Second-order Derivatives in Smoothed Particle Hydrodynamics</title> <meta name="description" content="On Discretization of Second-order Derivatives in Smoothed Particle Hydrodynamics"> <meta name="keywords" content="Heat conduction, Meshfree methods, Smoothed ParticleHydrodynamics (SPH), Second-order derivatives."> <meta name="viewport" content="width=device-width, initial-scale=1, minimum-scale=1, maximum-scale=1, user-scalable=no"> <meta charset="utf-8"> <meta name="citation_title" content="On Discretization of Second-order Derivatives in Smoothed Particle Hydrodynamics"> <meta name="citation_author" content="R. Fatehi"> <meta name="citation_author" content="M.A. Fayazbakhsh"> <meta name="citation_author" content="M.T. Manzari"> <meta name="citation_publication_date" content="2008/04/23"> <meta name="citation_journal_title" content="International Journal of Mechanical and Mechatronics Engineering"> <meta name="citation_volume" content="2"> <meta name="citation_issue" content="4"> <meta name="citation_firstpage" content="428"> <meta name="citation_lastpage" content="431"> <meta name="citation_pdf_url" content="https://publications.waset.org/14882/pdf"> <link href="https://cdn.waset.org/favicon.ico" type="image/x-icon" rel="shortcut icon"> <link href="https://cdn.waset.org/static/plugins/bootstrap-4.2.1/css/bootstrap.min.css" rel="stylesheet"> <link href="https://cdn.waset.org/static/plugins/fontawesome/css/all.min.css" rel="stylesheet"> <link href="https://cdn.waset.org/static/css/site.css?v=150220211555" rel="stylesheet"> </head> <body> <header> <div class="container"> <nav class="navbar navbar-expand-lg navbar-light"> <a class="navbar-brand" href="https://waset.org"> <img src="https://cdn.waset.org/static/images/wasetc.png" alt="Open Science Research Excellence" title="Open Science Research Excellence" /> </a> <button class="d-block d-lg-none navbar-toggler ml-auto" type="button" data-toggle="collapse" data-target="#navbarMenu" aria-controls="navbarMenu" aria-expanded="false" aria-label="Toggle navigation"> <span class="navbar-toggler-icon"></span> </button> <div class="w-100"> <div class="d-none d-lg-flex flex-row-reverse"> <form method="get" action="https://waset.org/search" class="form-inline my-2 my-lg-0"> <input class="form-control mr-sm-2" type="search" placeholder="Search Conferences" value="" name="q" aria-label="Search"> <button class="btn btn-light my-2 my-sm-0" type="submit"><i class="fas fa-search"></i></button> </form> </div> <div class="collapse navbar-collapse mt-1" id="navbarMenu"> <ul class="navbar-nav ml-auto align-items-center" id="mainNavMenu"> <li class="nav-item"> <a class="nav-link" href="https://waset.org/conferences" title="Conferences in 2024/2025/2026">Conferences</a> </li> <li class="nav-item"> <a class="nav-link" href="https://waset.org/disciplines" title="Disciplines">Disciplines</a> </li> <li class="nav-item"> <a class="nav-link" href="https://waset.org/committees" rel="nofollow">Committees</a> </li> <li class="nav-item dropdown"> <a class="nav-link dropdown-toggle" href="#" id="navbarDropdownPublications" role="button" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false"> Publications </a> <div class="dropdown-menu" aria-labelledby="navbarDropdownPublications"> <a class="dropdown-item" href="https://publications.waset.org/abstracts">Abstracts</a> <a class="dropdown-item" href="https://publications.waset.org">Periodicals</a> <a class="dropdown-item" href="https://publications.waset.org/archive">Archive</a> </div> </li> <li class="nav-item"> <a class="nav-link" href="https://waset.org/page/support" title="Support">Support</a> </li> </ul> </div> </div> </nav> </div> </header> <main> <div class="container mt-4"> <div class="row"> <div class="col-md-9 mx-auto"> <form method="get" action="https://publications.waset.org/search"> <div id="custom-search-input"> <div class="input-group"> <i class="fas fa-search"></i> <input type="text" class="search-query" name="q" placeholder="Author, Title, Abstract, Keywords" value=""> <input type="submit" class="btn_search" value="Search"> </div> </div> </form> </div> </div> <div class="row mt-3"> <div class="col-sm-3"> <div class="card"> <div class="card-body"><strong>Commenced</strong> in January 2007</div> </div> </div> <div class="col-sm-3"> <div class="card"> <div class="card-body"><strong>Frequency:</strong> Monthly</div> </div> </div> <div class="col-sm-3"> <div class="card"> <div class="card-body"><strong>Edition:</strong> International</div> </div> </div> <div class="col-sm-3"> <div class="card"> <div class="card-body"><strong>Paper Count:</strong> 33093</div> </div> </div> </div> <div class="card publication-listing mt-3 mb-3"> <h5 class="card-header" style="font-size:.9rem">On Discretization of Second-order Derivatives in Smoothed Particle Hydrodynamics</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/search?q=R.%20Fatehi">R. Fatehi</a>, <a href="https://publications.waset.org/search?q=M.A.%20Fayazbakhsh"> M.A. Fayazbakhsh</a>, <a href="https://publications.waset.org/search?q=M.T.%20Manzari"> M.T. Manzari</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Discretization of spatial derivatives is an important issue in meshfree methods especially when the derivative terms contain non-linear coefficients. In this paper, various methods used for discretization of second-order spatial derivatives are investigated in the context of Smoothed Particle Hydrodynamics. Three popular forms (i.e. "double summation", "second-order kernel derivation", and "difference scheme") are studied using one-dimensional unsteady heat conduction equation. To assess these schemes, transient response to a step function initial condition is considered. Due to parabolic nature of the heat equation, one can expect smooth and monotone solutions. It is shown, however in this paper, that regardless of the type of kernel function used and the size of smoothing radius, the double summation discretization form leads to non-physical oscillations which persist in the solution. Also, results show that when a second-order kernel derivative is used, a high-order kernel function shall be employed in such a way that the distance of inflection point from origin in the kernel function be less than the nearest particle distance. Otherwise, solutions may exhibit oscillations near discontinuities unlike the "difference scheme" which unconditionally produces monotone results. <iframe src="https://publications.waset.org/14882.pdf" style="width:100%; height:400px;" frameborder="0"></iframe> <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/search?q=Heat%20conduction" title="Heat conduction">Heat conduction</a>, <a href="https://publications.waset.org/search?q=Meshfree%20methods" title=" Meshfree methods"> Meshfree methods</a>, <a href="https://publications.waset.org/search?q=Smoothed%20ParticleHydrodynamics%20%28SPH%29" title=" Smoothed ParticleHydrodynamics (SPH)"> Smoothed ParticleHydrodynamics (SPH)</a>, <a href="https://publications.waset.org/search?q=Second-order%20derivatives." title=" Second-order derivatives."> Second-order derivatives.</a> </p> <p class="card-text"><strong>Digital Object Identifier (DOI):</strong> <a href="https://doi.org/10.5281/zenodo.1083799" target="_blank">doi.org/10.5281/zenodo.1083799</a> </p> <a href="https://publications.waset.org/14882/on-discretization-of-second-order-derivatives-in-smoothed-particle-hydrodynamics" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/14882/apa" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">APA</a> <a href="https://publications.waset.org/14882/bibtex" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">BibTeX</a> <a href="https://publications.waset.org/14882/chicago" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">Chicago</a> <a href="https://publications.waset.org/14882/endnote" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">EndNote</a> <a href="https://publications.waset.org/14882/harvard" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">Harvard</a> <a href="https://publications.waset.org/14882/json" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">JSON</a> <a href="https://publications.waset.org/14882/mla" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">MLA</a> <a href="https://publications.waset.org/14882/ris" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">RIS</a> <a href="https://publications.waset.org/14882/xml" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">XML</a> <a href="https://publications.waset.org/14882/iso690" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">ISO 690</a> <a href="https://publications.waset.org/14882.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">3088</span> </span> <p class="card-text"><strong>References:</strong></p> <br>[1] L. Lucy, "A numerical approach to the testing of fission hypothesis," Astrophysical Journal, vol. 82, pp. 1013-1020, 1977. <br>[2] R. Gingold and J. Monaghan, "Smoothed particle hydrodynamics: Theory and application to nonspherical stars," Astrophysical Journal, vol. 181, pp. 275-389, 1977. <br>[3] 脭脟枚脭脟枚, "Kernel estimates as a basis for general particle methods in hydrodynamics," Journal of Computational Physics, vol. 46, pp. 429- 453, 1982. <br>[4] J. Monaghan, "Simulating free surface flows with sph," Journal of Computational Physics, vol. 110, pp. 399-406, 1994. <br>[5] H. Takeda, S. Miyama, and M. Sekiya, "Numerical simulation of viscous flow by smoothed particle hydrodynamics," Progress of Theoretical Physics, vol. 92, no. 5, pp. 939-960, 1994. <br>[6] J. Morris, P. Fox, and Y. Zhu, "Modeling low reynolds number incompressible flows using sph," Journal of Computational Physics, vol. 136, no. 1, pp. 214-226, 1997. <br>[7] J. Monaghan, "Smoothed particle hydrodynamics," Reports on Progress in Physics, vol. 68, pp. 1703-1759, 2005. <br>[8] O. Flebbe, S. Munzel, H. Herold, H. Riffert, and H. Ruder, "Smoothed particle hydrodynamics-physical viscosity and the simulation of accretion disks," The Astrophysical Journal, vol. 431, pp. 214-226, 1994. <br>[9] S. Watkins, A. Bhattal, N. Francis, T. J.A., and Whitworth, "A new prescription for viscosity in smoothed particle hydrodynamics," Astronomy and Astrophysics, vol. 119, pp. 177-187, 1996. <br>[10] J. Jeong, M. Jhon, J. Halow, and J. van Osdol, "Smoothed particle hydrodynamics: Applications to heat conduction," Computer Physics Communications, vol. 153, no. 1, pp. 71-84, 2003. <br>[11] J. Bonet and T. Lok, "Variational and momentum preservation aspects of Smooth Particle Hydrodynamic formulations," Computer Methods in Applied Mechanics and Engineering, vol. 180, no. 1-2, pp. 97-115, 1999. <br>[12] S. Nugent and H. Posch, "Liquid drops and surface tension with smoothed particle applied mechanics," Physical Review E, vol. 62, no. 4, pp. 4968-4975, 2000. <br>[13] J. Bonet, S. Kulasegaram, M. Rodriguez-Paz, and M. Profit, "Variational formulation for the smooth particle hydrodynamics (SPH) simulation of fluid and solid problems," Computer Methods in Applied Mechanics and Engineering, vol. 193, no. 12-14, pp. 1245-1256, 2004. <br>[14] Y. Mele'an, L. Sigalotti, and A. Hasmy, "On the SPH tensile instability in forming viscous liquid drops," Computer Physics Communications, vol. 157, no. 3, pp. 191-200, 2004. <br>[15] H. L'opez and L. Sigalotti, "Oscillation of viscous drops with smoothed particle hydrodynamics," Physical Review E, vol. 73, no. 5, p. 51201, 2006. <br>[16] A. Chaniotis, D. Poulikakos, and P. Kououtsakos, "Remeshed smoothed particle hydrodynamics for the simulations of viscous and heat conducting flows," Journal of Computational Physics, vol. 182, pp. 67-90, 2002. <br>[17] P. Cleary, "Modelling confined multi-material heat and mass flows using sph," Applied Mathematical Modelling, vol. 22, pp. 981-993, 1998. <br>[18] J. Monaghan and J. Lattanzio, "A refined particle method for astrophysical problems," Astronomy and Astrophysics, vol. 149, no. 1, pp. 135-143, 1985. <br>[19] I. Schoenberg, "Contributions to the problem of approximation of equidistant data by analytic functions. Part A - On the problem of smoothing or graduation. A first class of analytic approximation formulas," Quarterly of Applied Mathematics, vol. 4, pp. 45-99, 1946. </div> </div> </div> </main> <footer> <div id="infolinks" class="pt-3 pb-2"> <div class="container"> <div style="background-color:#f5f5f5;" class="p-3"> <div class="row"> <div class="col-md-2"> <ul class="list-unstyled"> About <li><a href="https://waset.org/page/support">About Us</a></li> <li><a href="https://waset.org/page/support#legal-information">Legal</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/WASET-16th-foundational-anniversary.pdf">WASET celebrates its 16th foundational anniversary</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Account <li><a href="https://waset.org/profile">My Account</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Explore <li><a href="https://waset.org/disciplines">Disciplines</a></li> <li><a href="https://waset.org/conferences">Conferences</a></li> <li><a href="https://waset.org/conference-programs">Conference Program</a></li> <li><a href="https://waset.org/committees">Committees</a></li> <li><a href="https://publications.waset.org">Publications</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Research <li><a href="https://publications.waset.org/abstracts">Abstracts</a></li> <li><a href="https://publications.waset.org">Periodicals</a></li> <li><a href="https://publications.waset.org/archive">Archive</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Open Science <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Open-Science-Philosophy.pdf">Open Science Philosophy</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Open-Science-Award.pdf">Open Science Award</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Open-Society-Open-Science-and-Open-Innovation.pdf">Open Innovation</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Postdoctoral-Fellowship-Award.pdf">Postdoctoral Fellowship Award</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Scholarly-Research-Review.pdf">Scholarly Research Review</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Support <li><a href="https://waset.org/page/support">Support</a></li> <li><a href="https://waset.org/profile/messages/create">Contact Us</a></li> <li><a href="https://waset.org/profile/messages/create">Report Abuse</a></li> </ul> </div> </div> </div> </div> </div> <div class="container text-center"> <hr style="margin-top:0;margin-bottom:.3rem;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" class="text-muted small">Creative Commons Attribution 4.0 International License</a> <div id="copy" class="mt-2">© 2024 World Academy of Science, Engineering and Technology</div> </div> </footer> <a href="javascript:" id="return-to-top"><i class="fas fa-arrow-up"></i></a> <div class="modal" id="modal-template"> <div class="modal-dialog"> <div class="modal-content"> <div class="row m-0 mt-1"> <div class="col-md-12"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> </div> </div> <div class="modal-body"></div> </div> </div> </div> <script src="https://cdn.waset.org/static/plugins/jquery-3.3.1.min.js"></script> <script src="https://cdn.waset.org/static/plugins/bootstrap-4.2.1/js/bootstrap.bundle.min.js"></script> <script src="https://cdn.waset.org/static/js/site.js?v=150220211556"></script> <script> jQuery(document).ready(function() { /*jQuery.get("https://publications.waset.org/xhr/user-menu", function (response) { jQuery('#mainNavMenu').append(response); });*/ jQuery.get({ url: "https://publications.waset.org/xhr/user-menu", cache: false }).then(function(response){ jQuery('#mainNavMenu').append(response); }); }); </script> </body> </html>

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On Discretization of Second-order Derivatives in Smoothed Particle Hydrodynamics