Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

A mixed discontinuous finite element method for folded Naghdi's shell in Cartesian coordinates


Authors: S. Nicaise and I. Merabet
Journal: Math. Comp. 86 (2017), 1-47
MSC (2010): Primary 74K25; Secondary 65N30, 74S05
DOI: https://doi.org/10.1090/mcom/3094
Published electronically: March 3, 2016
MathSciNet review: 3557792
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this work a mixed method and its DG formulation are proposed to solve Naghdi's equations for a thin linearly elastic shell. The unknowns of the problem are the displacement of the points of the middle surface, the rotation field of the normal vector to the middle surface of the shell and a Lagrange multiplier which is introduced in order to enforce the tangency requirement on the rotation. In addition to existence and uniqueness results of solutions of the continuous and the discrete problems we derive an a priori error estimate. We further propose an a posteriori estimator that yields an upper bound and a lower bound of the error. Numerical tests that validate and illustrate our approach are given.


References [Enhancements On Off] (What's this?)

  • [1] Douglas N. Arnold and Franco Brezzi, Locking-free finite element methods for shells, Math. Comp. 66 (1997), no. 217, 1-14. MR 1370847 (97c:73061), https://doi.org/10.1090/S0025-5718-97-00785-0
  • [2] K.-J. Bathe and L. W. Ho, Some results in the analysis of thin shell structures, Nonlinear finite element analysis in structural mechanics (W. Wunderlich, E. Stein, and K.-J. Bathe, eds.), Springer-Verlag, Berlin-New York, 1981, pp. xiii+777.
  • [3] Faker Ben Belgacem, Christine Bernardi, Adel Blouza, and Frekh Taallah, On the obstacle problem for a Naghdi shell, J. Elasticity 103 (2011), no. 1, 1-13 (English, with English and French summaries). MR 2771397 (2012b:49011), https://doi.org/10.1007/s10659-010-9269-2
  • [4] M. Bernadou, Méthode d'élements finis pour les problèmes de coques minces, Dunod, 1994.
  • [5] Michel Bernadou and Annie Cubier, Numerical analysis of junctions between thin shells. II. Approximation by finite element methods, Comput. Methods Appl. Mech. Engrg. 161 (1998), no. 3-4, 365-387. MR 1642790 (99k:73111b), https://doi.org/10.1016/S0045-7825(97)00327-7
  • [6] Michel Bernadou, Séverine Fayolle, and Françoise Léné, Numerical analysis of junctions between plates, Comput. Methods Appl. Mech. Engrg. 74 (1989), no. 3, 307-326. MR 1020628 (90j:73062), https://doi.org/10.1016/0045-7825(89)90054-6
  • [7] Christine Bernardi, Adel Blouza, and Frédéric Hecht, An incompressible fluid in a weakly deformable shell. Part I: Analysis of the model, J. Elasticity 108 (2012), no. 1, 29-47. MR 2922471, https://doi.org/10.1007/s10659-011-9354-1
  • [8] Christine Bernardi, Frédéric Hecht, Hervé Le Dret, and Adel Blouza, A posteriori analysis of a finite element discretization of a Naghdi shell model, IMA J. Numer. Anal. 33 (2013), no. 1, 190-211. MR 3020955, https://doi.org/10.1093/imanum/drs009
  • [9] Adel Blouza, Existence et unicité pour le modèle de Nagdhi pour une coque peu régulière, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 7, 839-844. MR 1446590 (97m:73038), https://doi.org/10.1016/S0764-4442(97)86955-8
  • [10] A. Blouza and H. Le Dret, Nagdhi's shell model: existence, uniqueness and continuous dependence on the midsurface, J. Elasticity 64 (2001), no. 2-3, 199-216 (2002). MR 1907796 (2003a:74039), https://doi.org/10.1023/A:1015270504666
  • [11] Adel Blouza, Frédéric Hecht, and Hervé Le Dret, Two finite element approximations of Naghdi's shell model in Cartesian coordinates, SIAM J. Numer. Anal. 44 (2006), no. 2, 636-654 (electronic). MR 2218963 (2007b:74083), https://doi.org/10.1137/050624339
  • [12] James H. Bramble and Tong Sun, A locking-free finite element method for Naghdi shells, J. Comput. Appl. Math. 89 (1998), no. 1, 119-133. MR 1625959 (99h:65186), https://doi.org/10.1016/S0377-0427(97)00234-3
  • [13] Susanne C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954 (2008m:65001)
  • [14] Dominique Chapelle and Rolf Stenberg, Stabilized finite element formulations for shells in a bending dominated state, SIAM J. Numer. Anal. 36 (1999), no. 1, 32-73 (electronic). MR 1654586 (99j:65199), https://doi.org/10.1137/S0036142996302918
  • [15] Philippe G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174 (58 #25001)
  • [16] Philippe G. Ciarlet, Mathematical elasticity. Vol. III: Theory of shells, Studies in Mathematics and its Applications, vol. 29, North-Holland Publishing Co., Amsterdam, 2000. MR 1757535 (2001j:74060)
  • [17] A. Cubier, Numerical analysis and simulation of thin shells junction, Thèse de Doctorat. Université Pierre et Marie Curie, 1994.
  • [18] Dominique Chapelle and Klaus-Jürgen Bathe, The Finite Element Analysis of Shells--Fundamentals, 2nd ed., Computational Fluid and Solid Mechanics, Springer, Heidelberg, 2011. MR 3234588
  • [19] Hanen Ferchichi and Saloua Mani Aouadi, Numerical stability for a dynamic Naghdi shell, Appl. Math. Comput. 216 (2010), no. 9, 2483-2506. MR 2653062, https://doi.org/10.1016/j.amc.2010.03.098
  • [20] Vivette Girault and Pierre-Arnaud Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and algorithms, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. MR 851383 (88b:65129)
  • [21] Thirupathi Gudi, A new error analysis for discontinuous finite element methods for linear elliptic problems, Math. Comp. 79 (2010), no. 272, 2169-2189. MR 2684360 (2011h:65223), https://doi.org/10.1090/S0025-5718-10-02360-4
  • [22] F. Hecht and O. Pironneau, Freefem++, www.freefem.org 1 (1990), 1-230.
  • [23] Oana Iosifescu, Regularity for Naghdi's shell equations, Math. Mech. Solids 5 (2000), no. 4, 453-465. MR 1793505 (2001j:74062), https://doi.org/10.1177/108128650000500404
  • [24] J. Pitkäranta, Y. Leino, O. Ovaskainen, and J. Piila, Shell deformation states and the finite element method: a benchmark study of cylindrical shells, Comput. Methods Appl. Mech. Engrg. 128 (1995), no. 1-2, 81-121. MR 1376906 (96j:73085), https://doi.org/10.1016/0045-7825(95)00870-X
  • [25] E. Sanchez-Palencia J. Sanchez-Hubert, Coques élastiques minces, Masson, 1997.
  • [26] P. M. Naghdi, Foundations of elastic shell theory, Progress in Solid Mechanics, Vol. IV, North-Holland, Amsterdam, 1963, pp. 1-90. MR 0163488 (29 #790)
  • [27] I. Paris, On the reliability of triangular finite elements for shells, Thèse de Doctorat. Université Pierre et Marie Curie, 2006.
  • [28] Juhani Pitkäranta, The problem of membrane locking in finite element analysis of cylindrical shells, Numer. Math. 61 (1992), no. 4, 523-542. MR 1155337 (93b:65178), https://doi.org/10.1007/BF01385524
  • [29] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-refinement Techniques, Wiley-Teubner Series Advances in Numerical Mathematics, Wiley-Teubner, Chichester, Stuttgart, 1996.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 74K25, 65N30, 74S05

Retrieve articles in all journals with MSC (2010): 74K25, 65N30, 74S05


Additional Information

S. Nicaise
Affiliation: LAMAV, Université de Valenciennes et du Hainaut-Cambrésis, Valenciennes Cedex 9, France
Email: serge.nicaise@univ-valenciennes.fr

I. Merabet
Affiliation: Lab. LMA, Université de Ouargla, Ouargla 30000, Algérie
Email: merabet.ismail@univ-ouargla.dz

DOI: https://doi.org/10.1090/mcom/3094
Keywords: DG method, a posteriori analysis, Naghdi's shell
Received by editor(s): January 23, 2015
Received by editor(s) in revised form: June 19, 2015
Published electronically: March 3, 2016
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society