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A model problem for boundary layers of thin elastic shells

Published online by Cambridge University Press:  15 April 2002

Philippe Karamian ,
Jacqueline Sanchez-Hubert  and
Évarisite Sanchez Palencia
Philippe Karamian
Affiliation:
Laboratoire de Mécanique, Université de Caen, Boulevard Maréchal Juin, 14032 Caen Cedex, France.
Jacqueline Sanchez-Hubert
Affiliation:
Laboratoire de Modélisation en Mécanique, Université Paris VI, 4 place Jussieu, 75252 Paris Cedex 05, France Laboratoire de Mécanique, Université de Caen, Boulevard Maréchal Juin, 14032 Caen Cedex, France.Laboratoire de Modélisation en Mécanique, Université Paris VI, 4 place Jussieu, 75252 Paris Cedex 05, FranceLaboratoire de Modélisation en Mécanique, Université Paris VI, 4 place Jussieu, 75252 Paris Cedex 05, France. (sanchez@lmm.jussieu.fr)
Évarisite Sanchez Palencia
Affiliation:
Laboratoire de Mécanique, Université de Caen, Boulevard Maréchal Juin, 14032 Caen Cedex, France.Laboratoire de Modélisation en Mécanique, Université Paris VI, 4 place Jussieu, 75252 Paris Cedex 05, FranceLaboratoire de Modélisation en Mécanique, Université Paris VI, 4 place Jussieu, 75252 Paris Cedex 05, France. (sanchez@lmm.jussieu.fr)
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Abstract

We consider a model problem (with constant coefficients and simplified geometry) for the boundary layer phenomena which appear in thin shell theoryas the relative thickness ε of the shell tends tozero. For ε = 0 our problem is parabolic, then it is amodel of developpable surfaces. Boundary layers along and across the characteristichave very different structure. It also appears internal layers associatedwith propagations of singularities along the characteristics. The specialstructure of the limit problem often implies solutions which exhibitdistributional singularities along the characteristics. The correspondinglayers for small ε have a very large intensity. Layers alongthe characteristics have a special structure involving subspaces; thecorresponding Lagrange multipliers are exhibited. Numerical experimentsshow the advantage of adaptive meshes in these problems.

Keywords

Shellsboundary layerssingular perturbation.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 1 , January 2000 , pp. 1 - 30
Copyright
© EDP Sciences, SMAI, 2000

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References

M. Bernadou, Méthodes d'éléments finis pour les problèmes de coques minces. Masson, Paris (1994).
F. Brezzi and F. M. Fortin, Mixed and hybrid finite elements methods. Springer (1991).
Choï, D., Palma, F.J., Sanchez Palencia, É. and Vilari, M.A. no, Remarks on membrane locking in the finite element computation of very thin elastic shells. Math. Modell. Num. Anal. 32 (1998) 131-152.
P.G. Ciarlet, Mathematical elasticity, Vol. III, Theory of shells. North Holland, Amsterdam (to appear).
D. Chapelle and K.J. Bathe, Fundamental considerations for the finite element analysis of shell structures, Computers and Structures 66 (1998) 19-36.
P. Gérard and É. Sanchez Palencia, Sensitivity phenomena for certain thin elastic shells with edges. Math. Meth. Appl. Sci. (to appear).
A.L. Goldenveizer, Theory of elastic thin shells. Pergamon, New York (1962).
Karamian, P., Nouveaux résultats numériques concernant les coques minces hyperboliques inhibées: cas du paraboloïde hyperbolique. C. R. Acad. Sci. Paris Sér. IIb 326 (1998) 755-760.
Karamian, P., Réflexion des singularités dans les coques hyperboliques inhibées. C.R. Acad. Sci. Paris Sér. IIb 326 (1998) 609-614.
P. Karamian, Coques élastiques minces hyperboliques inhibées : calcul du problème limite par éléments finis et non reflexion des singularités. Thèse de l'Universté de Caen (1999).
Leguillon, D., Sanchez-Hubert, J. and Sanchez Palencia, É., Model problem of singular perturbation without limit in the space of finite energy and its computation. C.R. Acad. Sci. Paris Sér. IIb 327 (1999) 485-492.
J.L. Lions and É. Sanchez Palencia, Problèmes sensitifs et coques élastiques minces. in Partial Differential Equations and Functional Analysis, in memory of P. Grisvard (J. Céa, D. Chesnais, G. Geymonat, J.L. Lions Eds.), Birkhauser, Boston (1996) 207-220.
J.L. Lions and É. Sanchez Palencia, Sur quelques espaces de la théorie des coques et la sensitivité, in Homogenization and applications to material sciences, Cioranescu, Damlamian, Doneto Eds., Gakkotosho, Tokyo (1995) 271-278.
A.E.H Love, A treatrise on the mathematical theory of elasticity, Reprinted by Dover, New-York (1944).
Pitkaranta, J. and Sanchez Palencia, É., On the asymptotic behavior of sensitive shells with small thickness. C.R. Acad. Sci. Paris Sér. IIb 325 (1997) 127-134.
H.S. Rutten, Theory and design of shells on the basis of asymptotic analysis. Rutten and Kruisman, Voorburg (1973).
J. Sanchez-Hubert and É. Sanchez Palencia, Introduction aux méthodes asymptotiques et à l'homogénéisation, Masson, Paris (1992).
J. Sanchez-Hubert and É. Sanchez Palencia, Coques élastiques minces. Propriétés asymptotiques. Masson, Paris (1997).
Sanchez-Hubert, J. and Sanchez Palencia, É., Pathological phenomena in computation of thin elastic shells. Transactions Can. Soc. Mech. Engin. 22 (1998) 435-446.
M. Van Dyke, Perturbation methods in fluid mechanics. Academic Press, New-York (1964).