{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T11:03:01Z","timestamp":1776769381706,"version":"3.51.2"},"reference-count":11,"publisher":"American Mathematical Society (AMS)","issue":"239","license":[{"start":{"date-parts":[[2002,11,20]],"date-time":"2002-11-20T00:00:00Z","timestamp":1037750400000},"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>We consider a bilinear reduced-strain finite element formulation for a shallow shell model of Reissner\u2013Naghdi type. The formulation is closely related to the facet models used in engineering practice. We estimate the error of this scheme when approximating an inextensional displacement field. We make the strong assumptions that the domain and the finite element mesh are rectangular and that the boundary conditions are periodic and the mesh uniform in one of the coordinate directions. We prove then that for sufficiently smooth fields, the convergence rate in the energy norm is of optimal order uniformly with respect to the shell thickness. In case of elliptic shell geometry the error bound is furthermore quasioptimal, whereas in parabolic and hyperbolic geometries slightly enhanced smoothness is required, except for the degenerate cases where the characteristic lines are parallel with the mesh lines. The error bound is shown to be sharp.<\/p>","DOI":"10.1090\/s0025-5718-01-01376-x","type":"journal-article","created":{"date-parts":[[2002,9,20]],"date-time":"2002-09-20T15:46:54Z","timestamp":1032536814000},"page":"923-943","source":"Crossref","is-referenced-by-count":13,"title":["Analysis of a bilinear finite element for shallow shells I: Approximation of inextensional deformations"],"prefix":"10.1090","volume":"71","author":[{"given":"Ville","family":"Havu","sequence":"first","affiliation":[]},{"given":"Juhani","family":"Pitk\u00e4ranta","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,11,20]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"K.J. Bathe, E.N. Dvorkin, A formulation of general shell elements \u2013 the use of mixed interpolation of tensorial components, Int. J. Numer. Methods Engrg. 22 (1986) 697-722.","DOI":"10.1002\/nme.1620220312"},{"key":"2","doi-asserted-by":"crossref","unstructured":"D. Chapelle, K.J. Bathe, Fundamental considerations for the finite element analysis of shell structures, Computers & Structures, 66 (1998) 19-36.","DOI":"10.1016\/S0045-7949(97)00078-3"},{"issue":"11","key":"3","doi-asserted-by":"publisher","first-page":"1637","DOI":"10.1002\/1097-0207(20000820)48:11<1637::aid-nme954>3.0.co;2-h","article-title":"A benchmark study of reduced-strain shell finite elements: quadratic schemes","volume":"48","author":"Malinen, Mika","year":"2000","journal-title":"Internat. J. Numer. Methods Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-5981","issn-type":"print"},{"key":"4","unstructured":"R.H. MacNeal, Finite Elements: Their Design and Performance (Marcel Dekker, 1994)."},{"issue":"11","key":"5","doi-asserted-by":"publisher","first-page":"1637","DOI":"10.1002\/1097-0207(20000820)48:11<1637::aid-nme954>3.0.co;2-h","article-title":"A benchmark study of reduced-strain shell finite elements: quadratic schemes","volume":"48","author":"Malinen, Mika","year":"2000","journal-title":"Internat. J. Numer. Methods Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-5981","issn-type":"print"},{"key":"6","unstructured":"M. Malinen, On geometrically incompatible bilinear shell elements and classical shell models, to appear."},{"key":"#cr-split#-7.1","unstructured":"J. Pitk\u00e4ranta, The first locking-free plane-elastic finite element: historia mathematica, Helsinki University of Technology Institute of Mathematics Research Reports A411 (1999)"},{"key":"#cr-split#-7.2","doi-asserted-by":"crossref","unstructured":"and Comput. Methods Appl. Mech. Engrg. 190 (2000), 1323-1366.","DOI":"10.1016\/S0045-7825(00)00163-8"},{"key":"8","unstructured":"J. Pitk\u00e4ranta, A.-M. Matache, C. Schwab, Fourier mode analysis of layers in shallow shell deformations, Research Report No. 99-18 (ETH Z\u00fcrich, 1999)."},{"issue":"4","key":"9","doi-asserted-by":"publisher","first-page":"523","DOI":"10.1007\/BF01385524","article-title":"The problem of membrane locking in finite element analysis of cylindrical shells","volume":"61","author":"Pitk\u00e4ranta, Juhani","year":"1992","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"1-2","key":"10","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1016\/0045-7825(95)00870-X","article-title":"Shell deformation states and the finite element method: a benchmark study of cylindrical shells","volume":"128","author":"Pitk\u00e4ranta, J.","year":"1995","journal-title":"Comput. Methods Appl. Mech. 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