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

Nicaise, S.; Merabet, I.

doi:10.1090/mcom/3094

A mixed discontinuous finite element method for folded Naghdi’s shell in Cartesian coordinates
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by S. Nicaise and I. Merabet PDF

Math. Comp. 86 (2017), 1-47 Request permission

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

Douglas N. Arnold and Franco Brezzi, Locking-free finite element methods for shells, Math. Comp. 66 (1997), no. 217, 1–14. MR 1370847, DOI 10.1090/S0025-5718-97-00785-0
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.
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, DOI 10.1007/s10659-010-9269-2
M. Bernadou, Méthode d’élements finis pour les problèmes de coques minces, Dunod, 1994.
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, DOI 10.1016/S0045-7825(97)00327-7
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, DOI 10.1016/0045-7825(89)90054-6
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 (English, with English and French summaries). MR 2922471, DOI 10.1007/s10659-011-9354-1
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, DOI 10.1093/imanum/drs009
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 (French, with English and French summaries). MR 1446590, DOI 10.1016/S0764-4442(97)86955-8
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, DOI 10.1023/A:1015270504666
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. MR 2218963, DOI 10.1137/050624339
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, DOI 10.1016/S0377-0427(97)00234-3
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, DOI 10.1007/978-0-387-75934-0
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. MR 1654586, DOI 10.1137/S0036142996302918
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
Philippe G. Ciarlet, Mathematical elasticity. Vol. III, Studies in Mathematics and its Applications, vol. 29, North-Holland Publishing Co., Amsterdam, 2000. Theory of shells. MR 1757535
A. Cubier, Numerical analysis and simulation of thin shells junction, Thèse de Doctorat. Université Pierre et Marie Curie, 1994.
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, DOI 10.1007/978-3-642-16408-8
Hanen Ferchichi and Saloua Mani Aouadi, Numerical stability for a dynamic Naghdi shell, Appl. Math. Comput. 216 (2010), no. 9, 2483–2506. MR 2653062, DOI 10.1016/j.amc.2010.03.098
Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
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, DOI 10.1090/S0025-5718-10-02360-4
F. Hecht and O. Pironneau, Freefem++, www.freefem.org 1 (1990), 1–230.
Oana Iosifescu, Regularity for Naghdi’s shell equations, Math. Mech. Solids 5 (2000), no. 4, 453–465. MR 1793505, DOI 10.1177/108128650000500404
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, DOI 10.1016/0045-7825(95)00870-X
E. Sanchez-Palencia J. Sanchez-Hubert, Coques élastiques minces, Masson, 1997.
P. M. Naghdi, Foundations of elastic shell theory, Progress in Solid Mechanics, Vol. IV, North-Holland, Amsterdam, 1963, pp. 1–90. MR 0163488
I. Paris, On the reliability of triangular finite elements for shells, Thèse de Doctorat. Université Pierre et Marie Curie, 2006.
Juhani Pitkäranta, The problem of membrane locking in finite element analysis of cylindrical shells, Numer. Math. 61 (1992), no. 4, 523–542. MR 1155337, DOI 10.1007/BF01385524
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.