{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T18:38:55Z","timestamp":1790879935158,"version":"4.1.0"},"reference-count":16,"publisher":"American Mathematical Society (AMS)","issue":"275","license":[{"start":{"date-parts":[[2012,2,7]],"date-time":"2012-02-07T00:00:00Z","timestamp":1328572800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    This paper deals with the finite element approximation of the vibration modes of a laminated plate modeled by the Reissner-Mindlin equations; DL3 elements are used for the bending terms and standard piecewise linear continuous elements for the in-plane displacements. An\n                    <italic>a priori<\/italic>\n                    estimate of the regularity of the solution, independent of the plate thickness, is proved for the corresponding load problem. This allows using the abstract approximation theory for spectral problems to study the convergence of the proposed finite element method. Thus, optimal order error estimates including a double order for the vibration frequencies are obtained under appropriate assumptions. These estimates are independent of the plate thickness, which leads to the conclusion that the method is locking-free. Numerical tests are reported to assess the performance of the method.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2011-02456-7","type":"journal-article","created":{"date-parts":[[2011,2,7]],"date-time":"2011-02-07T11:04:28Z","timestamp":1297076668000},"page":"1239-1264","source":"Crossref","is-referenced-by-count":3,"title":["Numerical analysis of a finite element method to compute the vibration modes of a Reissner-Mindlin laminated plate"],"prefix":"10.1090","volume":"80","author":[{"given":"Ricardo","family":"Dur\u00e1n","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rodolfo","family":"Rodr\u00edguez","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Frank","family":"Sanhueza","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2011,2,7]]},"reference":[{"issue":"6","key":"1","doi-asserted-by":"publisher","first-page":"1276","DOI":"10.1137\/0726074","article-title":"A uniformly accurate finite element method for the Reissner-Mindlin plate","volume":"26","author":"Arnold, Douglas N.","year":"1989","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"35-36","key":"2","doi-asserted-by":"publisher","first-page":"4767","DOI":"10.1016\/S0045-7825(00)00335-2","article-title":"Analysis of mixed finite elements for laminated composite plates","volume":"190","author":"Auricchio, F.","year":"2001","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","unstructured":"F. Auricchio and E. Sacco, A mixed-enhanced finite element for the analysis of laminated composite plates, Internat. J. Numer. Methods Engrg., 44 (1999) 1481\u20131504.","DOI":"10.1002\/(SICI)1097-0207(19990410)44:10<1481::AID-NME554>3.0.CO;2-Q"},{"key":"4","isbn-type":"print","first-page":"641","article-title":"Eigenvalue problems","author":"Babu\u0161ka, I.","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0444703659"},{"key":"5","doi-asserted-by":"crossref","unstructured":"K.J. Bathe and E.N. Dvorkin, A four-node plate bending element based on Mindlin\/Reissner plate theory and a mixed interpolation, Internat. J. Numer. Methods Engrg., 21 (1985) 367\u2013383.","DOI":"10.1002\/nme.1620210213"},{"key":"6","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3172-1","volume-title":"Mixed and hybrid finite element methods","volume":"15","author":"Brezzi, Franco","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0387975829"},{"issue":"4","key":"7","doi-asserted-by":"publisher","first-page":"427","DOI":"10.1007\/s002110050036","article-title":"On the locking phenomenon for a class of elliptic problems","volume":"67","author":"Chenais, D.","year":"1994","journal-title":"Numer. 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Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"198","key":"10","doi-asserted-by":"publisher","first-page":"561","DOI":"10.2307\/2153202","article-title":"On mixed finite element methods for the Reissner-Mindlin plate model","volume":"58","author":"Dur\u00e1n, Ricardo","year":"1992","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"11","series-title":"Classics in Mathematics","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-66282-9","volume-title":"Perturbation theory for linear operators","author":"Kato, Tosio","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/354058661X"},{"key":"12","series-title":"Mathematical Surveys and Monographs","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/085","volume-title":"Spectral problems associated with corner singularities of solutions to elliptic equations","volume":"85","author":"Kozlov, V. A.","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/0821827278"},{"key":"13","doi-asserted-by":"crossref","unstructured":"O.O. Ochoa and J.N. Reddy, Finite element analysis of composite laminates, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992.","DOI":"10.1007\/978-94-015-7995-7"},{"key":"14","first-page":"292","article-title":"A mixed finite element method for 2nd order elliptic problems","author":"Raviart, P.-A.","year":"1977"},{"key":"15","unstructured":"J.N. Reddy, Energy and Variational Methods in Applied Mechanics, Wiley, New York, 1984."},{"key":"16","unstructured":"J.N. Reddy, Mechanics of Laminated Composite Plates \u2013 Theory and Analysis, CRC Press, Boca Raton, 1997."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2011-80-275\/S0025-5718-2011-02456-7\/S0025-5718-2011-02456-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2011-80-275\/S0025-5718-2011-02456-7\/S0025-5718-2011-02456-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:51:09Z","timestamp":1776790269000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2011-80-275\/S0025-5718-2011-02456-7\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,2,7]]},"references-count":16,"journal-issue":{"issue":"275","published-print":{"date-parts":[[2011,7]]}},"alternative-id":["S0025-5718-2011-02456-7"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2011-02456-7","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2011,2,7]]}}}