{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:36:33Z","timestamp":1787330193985,"version":"build-2736575974"},"reference-count":28,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2006,1]]},"abstract":"<jats:p>In this paper, we consider the so\u2010called $hp$\u2010version of discontinuous Galerkin methods for Timoshenko beams. We prove that, when the numerical traces are properly chosen, the methods display optimal convergence uniformly with respect to the thickness of the beam. These methods are thus free from shear locking. We also prove that, when polynomials of degree p are used, all the numerical traces superconverge with a rate of order $h^{2p+1}\/p^{2p+1}$. Numerical experiments verifying the above\u2010mentioned theoretical results are shown.<\/jats:p>","DOI":"10.1137\/050635821","type":"journal-article","created":{"date-parts":[[2006,11,29]],"date-time":"2006-11-29T15:53:41Z","timestamp":1164815621000},"page":"2297-2325","source":"Crossref","is-referenced-by-count":26,"title":["Locking\u2010Free Optimal Discontinuous Galerkin Methods for Timoshenko Beams"],"prefix":"10.1137","volume":"44","author":[{"given":"Fatih","family":"Celiker","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bernardo","family":"Cockburn","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Henryk K.","family":"Stolarski","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,11,24]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01400318"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-97-00785-0"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-004-4134-8"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/0020-7403(71)90017-8"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/0729075"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-01-01317-5"},{"key":"R7","unstructured":"F. Celiker,\n                      Discontinuous Galerkin Methods for Structural Mechanics\n                      , Ph.D. thesis, University of Minnesota, Minneapolis, MN, 2005."},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-005-9057-5"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-06-01895-3"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"F. Celiker, B. Cockburn, S. G\u00fczey, R. Kanapady, S.\u2010C. Soon, H. K. Stolarski, and K. K. Tamma,\n                      Discontinuous Galerkin methods for Timoshenko beams\n                      , in Numerical Mathematics and Advanced Applications, ENUMATH 2003, Springer, 2004, pp. 221\u2013231.","DOI":"10.1007\/978-3-642-18775-9_19"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142997316712"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142900374111"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1620111005"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.2514\/3.60853"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/BF01386420"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(78)90005-1"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(79)90007-0"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/s100920070002"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142999352394"},{"key":"R20","unstructured":"C. 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