{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,29]],"date-time":"2025-09-29T20:26:02Z","timestamp":1759177562039,"version":"3.40.5"},"reference-count":17,"publisher":"Walter de Gruyter GmbH","issue":"2","funder":[{"DOI":"10.13039\/501100002848","name":"Comisi\u00f3n Nacional de Investigaci\u00f3n Cient\u00edfica y Tecnol\u00f3gica","doi-asserted-by":"publisher","award":["Fondecyt 1190009","Fondecyt 11170050","Fondecyt 3190359"],"award-info":[{"award-number":["Fondecyt 1190009","Fondecyt 11170050","Fondecyt 3190359"]}],"id":[{"id":"10.13039\/501100002848","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We develop a discontinuous Petrov\u2013Galerkin scheme with optimal test functions (DPG method)\nfor the Timoshenko beam bending model with various boundary conditions,\ncombining clamped, simply supported, and free ends. Our scheme approximates\nthe transverse deflection and bending moment. It converges quasi-optimally\nin <jats:inline-formula id=\"j_cmam-2020-0048_ineq_9999\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msub>\n                              <m:mi>L<\/m:mi>\n                              <m:mn>2<\/m:mn>\n                           <\/m:msub>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2020-0048_eq_0182.png\"\/>\n                        <jats:tex-math>{L_{2}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> and is locking free. In particular, it behaves well (converges quasi-optimally)\nin the limit case of the Euler\u2013Bernoulli model.\nSeveral numerical results illustrate the performance of our method.<\/jats:p>","DOI":"10.1515\/cmam-2020-0048","type":"journal-article","created":{"date-parts":[[2020,8,18]],"date-time":"2020-08-18T07:03:58Z","timestamp":1597734238000},"page":"373-383","source":"Crossref","is-referenced-by-count":3,"title":["A Locking-Free DPG Scheme for Timoshenko Beams"],"prefix":"10.1515","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5034-6593","authenticated-orcid":false,"given":"Thomas","family":"F\u00fchrer","sequence":"first","affiliation":[{"name":"Facultad de Matem\u00e1ticas , Pontificia Universidad Cat\u00f3lica de Chile , Avenida Vicu\u00f1a Mackenna 4860 , Santiago , Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Carlos","family":"Garc\u00eda Vera","sequence":"additional","affiliation":[{"name":"Facultad de Matem\u00e1ticas , Pontificia Universidad Cat\u00f3lica de Chile , Avenida Vicu\u00f1a Mackenna 4860 , Santiago , Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5171-1457","authenticated-orcid":false,"given":"Norbert","family":"Heuer","sequence":"additional","affiliation":[{"name":"Facultad de Matem\u00e1ticas , Pontificia Universidad Cat\u00f3lica de Chile , Avenida Vicu\u00f1a Mackenna 4860 , Santiago , Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2020,8,5]]},"reference":[{"key":"2023033111200683202_j_cmam-2020-0048_ref_001","doi-asserted-by":"crossref","unstructured":"M.  Baccouch,\nThe local discontinuous Galerkin method for the fourth-order Euler\u2013Bernoulli partial differential equation in one space dimension. Part I: Superconvergence error analysis,\nJ. Sci. Comput. 59 (2014), no. 3, 795\u2013840.","DOI":"10.1007\/s10915-013-9782-0"},{"key":"2023033111200683202_j_cmam-2020-0048_ref_002","doi-asserted-by":"crossref","unstructured":"L.  Beir\u00e3o da Veiga, D.  Mora and R.  Rodr\u00edguez,\nNumerical analysis of a locking-free mixed finite element method for a bending moment formulation of Reissner\u2013Mindlin plate model,\nNumer. Methods Partial Differential Equations 29 (2013), no. 1, 40\u201363.","DOI":"10.1002\/num.21698"},{"key":"2023033111200683202_j_cmam-2020-0048_ref_003","doi-asserted-by":"crossref","unstructured":"V. M.  Calo, N. O.  Collier and A. H.  Niemi,\nAnalysis of the discontinuous Petrov\u2013Galerkin method with optimal test functions for the Reissner\u2013Mindlin plate bending model,\nComput. Math. Appl. 66 (2014), no. 12, 2570\u20132586.","DOI":"10.1016\/j.camwa.2013.07.012"},{"key":"2023033111200683202_j_cmam-2020-0048_ref_004","doi-asserted-by":"crossref","unstructured":"C.  Carstensen, L.  Demkowicz and J.  Gopalakrishnan,\nBreaking spaces and forms for the DPG method and applications including Maxwell equations,\nComput. Math. Appl. 72 (2016), no. 3, 494\u2013522.","DOI":"10.1016\/j.camwa.2016.05.004"},{"key":"2023033111200683202_j_cmam-2020-0048_ref_005","doi-asserted-by":"crossref","unstructured":"F.  Celiker, B.  Cockburn and H. K.  Stolarski,\nLocking-free optimal discontinuous Galerkin methods for Timoshenko beams,\nSIAM J. Numer. Anal. 44 (2006), no. 6, 2297\u20132325.","DOI":"10.1137\/050635821"},{"key":"2023033111200683202_j_cmam-2020-0048_ref_006","doi-asserted-by":"crossref","unstructured":"J.  Chan, N.  Heuer, T.  Bui-Thanh and L.  Demkowicz,\nA robust DPG method for convection-dominated diffusion problems II: Adjoint boundary conditions and mesh-dependent test norms,\nComput. Math. Appl. 67 (2014), no. 4, 771\u2013795.","DOI":"10.1016\/j.camwa.2013.06.010"},{"key":"2023033111200683202_j_cmam-2020-0048_ref_007","unstructured":"L.  Demkowicz, T.  F\u00fchrer, N.  Heuer and X.  Tian,\nThe double adaptivity paradigm (How to circumvent the discrete inf-sup conditions of Babu\u0161ka and Brezzi),\nICES Report 19-07, The University of Texas at Austin, 2019."},{"key":"2023033111200683202_j_cmam-2020-0048_ref_008","doi-asserted-by":"crossref","unstructured":"L.  Demkowicz and J.  Gopalakrishnan,\nAnalysis of the DPG method for the Poisson equation,\nSIAM J. Numer. Anal. 49 (2011), no. 5, 1788\u20131809.","DOI":"10.1137\/100809799"},{"key":"2023033111200683202_j_cmam-2020-0048_ref_009","doi-asserted-by":"crossref","unstructured":"L.  Demkowicz and N.  Heuer,\nRobust DPG method for convection-dominated diffusion problems,\nSIAM J. Numer. Anal. 51 (2013), no. 5, 2514\u20132537.","DOI":"10.1137\/120862065"},{"key":"2023033111200683202_j_cmam-2020-0048_ref_010","doi-asserted-by":"crossref","unstructured":"T.  F\u00fchrer, A.  Haberl and N.  Heuer,\nTrace operators of the bi-Laplacian and applications,\nIMA J. Numer. Anal. (2020), 10.1093\/imanum\/draa012.","DOI":"10.1093\/imanum\/draa012"},{"key":"2023033111200683202_j_cmam-2020-0048_ref_011","doi-asserted-by":"crossref","unstructured":"T.  F\u00fchrer and N.  Heuer,\nFully discrete DPG methods for the Kirchhoff\u2013Love plate bending model,\nComput. Methods Appl. Mech. Engrg. 343 (2019), 550\u2013571.","DOI":"10.1016\/j.cma.2018.08.041"},{"key":"2023033111200683202_j_cmam-2020-0048_ref_012","doi-asserted-by":"crossref","unstructured":"T.  F\u00fchrer, N.  Heuer and A. H.  Niemi,\nAn ultraweak formulation of the Kirchhoff\u2013Love plate bending model and DPG approximation,\nMath. Comp. 88 (2019), no. 318, 1587\u20131619.","DOI":"10.1090\/mcom\/3381"},{"key":"2023033111200683202_j_cmam-2020-0048_ref_013","doi-asserted-by":"crossref","unstructured":"T.  F\u00fchrer, N.  Heuer and F.-J.  Sayas,\nAn ultraweak formulation of the Reissner\u2013Mindlin plate bending model and DPG approximation,\nNumer. Math. 145 (2020), no. 2, 313\u2013344.","DOI":"10.1007\/s00211-020-01116-0"},{"key":"2023033111200683202_j_cmam-2020-0048_ref_014","doi-asserted-by":"crossref","unstructured":"N.  Heuer and M.  Karkulik,\nA robust DPG method for singularly perturbed reaction-diffusion problems,\nSIAM J. Numer. Anal. 55 (2017), no. 3, 1218\u20131242.","DOI":"10.1137\/15M1041304"},{"key":"2023033111200683202_j_cmam-2020-0048_ref_015","doi-asserted-by":"crossref","unstructured":"F.  Lepe, D.  Mora and R.  Rodr\u00edguez,\nLocking-free finite element method for a bending moment formulation of Timoshenko beams,\nComput. Math. Appl. 68 (2014), no. 3, 118\u2013131.","DOI":"10.1016\/j.camwa.2014.05.011"},{"key":"2023033111200683202_j_cmam-2020-0048_ref_016","doi-asserted-by":"crossref","unstructured":"L. K.  Li,\nDiscretization of the Timoshenko beam problem by the p and the h-p versions of the finite element method,\nNumer. Math. 57 (1990), no. 4, 413\u2013420.","DOI":"10.1007\/BF01386420"},{"key":"2023033111200683202_j_cmam-2020-0048_ref_017","doi-asserted-by":"crossref","unstructured":"A. H.  Niemi, J. A.  Bramwell and L. F.  Demkowicz,\nDiscontinuous Petrov-Galerkin method with optimal test functions for thin-body problems in solid mechanics,\nComput. Methods Appl. Mech. Engrg. 200 (2011), no. 9\u201312, 1291\u20131300.","DOI":"10.1016\/j.cma.2010.10.018"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/ahead-of-print\/article-10.1515-cmam-2020-0048\/article-10.1515-cmam-2020-0048.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2020-0048\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2020-0048\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T14:29:56Z","timestamp":1680272996000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2020-0048\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,5]]},"references-count":17,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,11,7]]},"published-print":{"date-parts":[[2021,4,1]]}},"alternative-id":["10.1515\/cmam-2020-0048"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2020-0048","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"type":"print","value":"1609-4840"},{"type":"electronic","value":"1609-9389"}],"subject":[],"published":{"date-parts":[[2020,8,5]]}}}