{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T14:32:14Z","timestamp":1787236334526,"version":"build-2736575974"},"reference-count":64,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Multiscale Model. Simul."],"published-print":{"date-parts":[[2013,1]]},"abstract":"<jats:p>We present a $\\Gamma$-convergence analysis of the quasicontinuum method focused on the behavior of the approximate energy functionals in the continuum limit of a harmonic and defect-free crystal. The analysis shows that, under general conditions of stability and boundedness of the energy, the continuum limit is attained provided that the continuum---e.g., finite-element---approximation spaces are strongly dense in an appropriate topology and provided that the lattice size converges to zero more rapidly than the mesh size. The equicoercivity of the quasicontinuum energy functionals is likewise established with broad generality, which, in conjunction with $\\Gamma$-convergence, ensures the convergence of the minimizers. We also show under rather general conditions that, for interatomic energies having a clusterwise additive structure, summation or quadrature rules that suitably approximate the local element energies do not affect the continuum limit. Finally, we propose a discrete patch test that provides a practical means of assessing the convergence of quasicontinuum approximations. We demonstrate the utility of the discrete patch test by means of selected examples of application.<\/jats:p>","DOI":"10.1137\/120895354","type":"journal-article","created":{"date-parts":[[2013,8,1]],"date-time":"2013-08-01T10:08:59Z","timestamp":1375351739000},"page":"766-794","source":"Crossref","is-referenced-by-count":14,"title":["A $\\Gamma$-Convergence Analysis of the Quasicontinuum Method"],"prefix":"10.1137","volume":"11","author":[{"given":"Malena I.","family":"Espan\u0342ol","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dennis M.","family":"Kochmann","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sergio","family":"Conti","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michael","family":"Ortiz","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2013,8,1]]},"reference":[{"key":"atypb1","unstructured":"R. A. Adams and J. J. F. Fournier,\n                      Sobolev Spaces\n                      , 2nd ed., Pure Appl. Math. (Amst.) 140, Elsevier\/Academic Press, Amsterdam, 2003."},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1137\/S0036141003426471"},{"key":"atypb3","first-page":"709","author":"Arevalo C.","year":"2010","journal-title":"Heidelberg"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-005-0391-4"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmps.2010.02.008"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1007\/s10704-011-9660-4"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1534"},{"key":"atypb8","doi-asserted-by":"crossref","unstructured":"A. Braides and M. S. Gelli,\n                      The passage from discrete to continuous variational problems: A nonlinear homogenization process\n                      , in Nonlinear Homogenization and Its Applications to Composites, Polycrystals and Smart Materials, NATO Sci. Ser. II Math. Phys. Chem. 170, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2004, pp. 45-63.","DOI":"10.1007\/1-4020-2623-4_3"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/0707006"},{"key":"atypb10","doi-asserted-by":"crossref","unstructured":"S. C. Brenner and L. R. Scott,\n                      The Mathematical Theory of Finite Element Methods\n                      , Texts Appl. Math. 15, Springer-Verlag, New York, 1994.","DOI":"10.1007\/978-1-4757-4338-8"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/s00526-011-0441-8"},{"key":"atypb12","unstructured":"P. G. Ciarlet,\n                      The Finite Element Method for Elliptic Problems\n                      , Studies in Mathematics and Its Applications, North-Holland, Amsterdam, New York, 1978."},{"key":"atypb13","unstructured":"D. Cioranescu and P. Donato,\n                      An Introduction to Homogenization\n                      , Oxford Lecture Ser. Math. Appl. 17, The Clarendon Press, Oxford University Press, New York, 1999."},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1088\/0953-8984\/18\/19\/008"},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"G. Dal Maso,\n                      An Introduction to $\\Gamma$-Convergence\n                      , Progr. Nonlinear Differential Equations Appl. 8, Birkha\u0308user Boston, Boston, MA, 1993.","DOI":"10.1007\/978-1-4612-0327-8"},{"key":"atypb16","first-page":"49","author":"Daw M. S.","year":"1990","journal-title":"Berlin"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.29.6443"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmps.2010.06.011"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1137\/090767005"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-009-0276-z"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2010.07.008"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.95.060202"},{"key":"atypb23","first-page":"1040","volume":"221","author":"Lu W. E","year":"2013","journal-title":"Mem. Amer. Math. Soc."},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmps.2008.09.017"},{"key":"atypb25","unstructured":"L. C. Evans,\n                      Partial Differential Equations\n                      , 2nd ed., Grad. Stud. Math. 19, American Mathematical Society, Providence, RI, 2010."},{"key":"atypb26","doi-asserted-by":"crossref","unstructured":"M. Finnis,\n                      Interatomic Forces in Condensed Matter\n                      , Oxf. Ser. Mater. Model., Oxford University Press, Oxford, New York, 2003.","DOI":"10.1093\/acprof:oso\/9780198509776.001.0001"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1088\/0965-0393\/1\/4\/006"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1137\/080722151"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2011014"},{"key":"atypb30","unstructured":"T. J. R. Hughes,\n                      The Finite Element Method: Linear Static and Dynamic Finite Element Analysis\n                      , Dover Publications, Mineola, NY, 2000."},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-5096(01)00034-5"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.90.226102"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmps.2007.09.005"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-02-01456-4"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1137\/050636772"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.21429"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1137\/080743391"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-012-0582-8"},{"key":"atypb39","unstructured":"S. Mallat,\n                      A Wavelet Tour of Signal Processing. The Sparse Way\n                      , 3rd ed., With contributions from G. Peyre\u0301, Elsevier\/Academic Press, Amsterdam, 2009."},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.93.165503"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1016\/j.actamat.2005.02.046"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1088\/0965-0393\/18\/1\/015003"},{"key":"atypb43","first-page":"2817","author":"Miller R.","year":"1997","journal-title":"Berlin"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1016\/S0013-7944(98)00047-2"},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1088\/0965-0393\/6\/5\/008"},{"key":"atypb46","doi-asserted-by":"publisher","DOI":"10.1016\/S1631-073X(02)02494-9"},{"key":"atypb47","unstructured":"M. J. P. Musgrave,\n                      Crystal Acoustics: Introduction to the Study of Elastic Waves and Vibrations in Crystals\n                      , Acoustical Society of America, Melville, NY, 2003."},{"key":"atypb49","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-012-0592-6"},{"key":"atypb50","doi-asserted-by":"publisher","DOI":"10.1088\/0965-0393\/7\/5\/309"},{"key":"atypb51","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmps.2006.08.005"},{"key":"atypb52","first-page":"1535","author":"Scha\u0308ffner M.","year":"2013","journal-title":"MRS Proceedings"},{"key":"atypb53","doi-asserted-by":"publisher","DOI":"10.1137\/100792421"},{"key":"atypb54","doi-asserted-by":"publisher","DOI":"10.1137\/110844544"},{"key":"atypb55","doi-asserted-by":"publisher","DOI":"10.1557\/PROC-538-465"},{"key":"atypb56","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.80.742"},{"key":"atypb57","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.31.5262"},{"key":"atypb58","doi-asserted-by":"publisher","DOI":"10.1557\/JMR.1999.0300"},{"key":"atypb59","doi-asserted-by":"publisher","DOI":"10.1080\/01418619608243000"},{"key":"atypb60","doi-asserted-by":"publisher","DOI":"10.1021\/la9508912"},{"key":"atypb61","doi-asserted-by":"publisher","DOI":"10.1016\/S0020\u20107683(99)00095\u20105"},{"key":"atypb62","first-page":"1","volume":"10","author":"Venturini G.","year":"2012","journal-title":"International Journal for Multiscale Computational Engineering"},{"key":"atypb63","unstructured":"J. H. Weiner,\n                      Statistical Mechanics of Elasticity\n                      , 2nd ed., Dover Publications, Mineola, NY, 2002."},{"key":"atypb64","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2009.10.015"},{"key":"atypb65","unstructured":"O. C. Zienkiewicz and R. L. Taylor,\n                      The Finite Element Method for Solid and Structural Mechanics\n                      , 6th ed., Elsevier Butterworth-Heinemann, Amsterdam, Boston, 2005."}],"container-title":["Multiscale Modeling &amp; Simulation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/120895354","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:23:28Z","timestamp":1787232208000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/120895354"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,1]]},"references-count":64,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2013,1]]}},"alternative-id":["10.1137\/120895354"],"URL":"https:\/\/doi.org\/10.1137\/120895354","relation":{},"ISSN":["1540-3459","1540-3467"],"issn-type":[{"value":"1540-3459","type":"print"},{"value":"1540-3467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,1]]}}}