A model problem for boundary layers of thin elastic shells

Philippe Karamian; Jacqueline Sanchez-Hubert; Évarisite Sanchez Palencia

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 34, Issue: 1, page 1-30
  • ISSN: 0764-583X

Abstract

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We consider a model problem (with constant coefficients and simplified geometry) for the boundary layer phenomena which appear in thin shell theory as the relative thickness ε of the shell tends to zero. For ε = 0 our problem is parabolic, then it is a model of developpable surfaces. Boundary layers along and across the characteristic have very different structure. It also appears internal layers associated with propagations of singularities along the characteristics. The special structure of the limit problem often implies solutions which exhibit distributional singularities along the characteristics. The corresponding layers for small ε have a very large intensity. Layers along the characteristics have a special structure involving subspaces; the corresponding Lagrange multipliers are exhibited. Numerical experiments show the advantage of adaptive meshes in these problems.

How to cite

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Karamian, Philippe, Sanchez-Hubert, Jacqueline, and Palencia, Évarisite Sanchez. "A model problem for boundary layers of thin elastic shells." ESAIM: Mathematical Modelling and Numerical Analysis 34.1 (2010): 1-30. <http://eudml.org/doc/197443>.

@article{Karamian2010,
abstract = { We consider a model problem (with constant coefficients and simplified geometry) for the boundary layer phenomena which appear in thin shell theory as the relative thickness ε of the shell tends to zero. For ε = 0 our problem is parabolic, then it is a model of developpable surfaces. Boundary layers along and across the characteristic have very different structure. It also appears internal layers associated with propagations of singularities along the characteristics. The special structure of the limit problem often implies solutions which exhibit distributional singularities along the characteristics. The corresponding layers for small ε have a very large intensity. Layers along the characteristics have a special structure involving subspaces; the corresponding Lagrange multipliers are exhibited. Numerical experiments show the advantage of adaptive meshes in these problems. },
author = {Karamian, Philippe, Sanchez-Hubert, Jacqueline, Palencia, Évarisite Sanchez},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Shells; boundary layers; singular perturbation.; thin shell theory; boundary layer; developable surfaces; propagation of singularities; characteristics; Lagrange multipliers},
language = {eng},
month = {3},
number = {1},
pages = {1-30},
publisher = {EDP Sciences},
title = {A model problem for boundary layers of thin elastic shells},
url = {http://eudml.org/doc/197443},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Karamian, Philippe
AU - Sanchez-Hubert, Jacqueline
AU - Palencia, Évarisite Sanchez
TI - A model problem for boundary layers of thin elastic shells
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 1
SP - 1
EP - 30
AB - We consider a model problem (with constant coefficients and simplified geometry) for the boundary layer phenomena which appear in thin shell theory as the relative thickness ε of the shell tends to zero. For ε = 0 our problem is parabolic, then it is a model of developpable surfaces. Boundary layers along and across the characteristic have very different structure. It also appears internal layers associated with propagations of singularities along the characteristics. The special structure of the limit problem often implies solutions which exhibit distributional singularities along the characteristics. The corresponding layers for small ε have a very large intensity. Layers along the characteristics have a special structure involving subspaces; the corresponding Lagrange multipliers are exhibited. Numerical experiments show the advantage of adaptive meshes in these problems.
LA - eng
KW - Shells; boundary layers; singular perturbation.; thin shell theory; boundary layer; developable surfaces; propagation of singularities; characteristics; Lagrange multipliers
UR - http://eudml.org/doc/197443
ER -

References

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  1. M. Bernadou, Méthodes d'éléments finis pour les problèmes de coques minces. Masson, Paris (1994).  
  2. F. Brezzi and F. M. Fortin, Mixed and hybrid finite elements methods. Springer (1991).  
  3. D. Choï, F.J. Palma, É. Sanchez Palencia and M.A. Vilari no, Remarks on membrane locking in the finite element computation of very thin elastic shells. Math. Modell. Num. Anal.32 (1998) 131-152.  
  4. P.G. Ciarlet, Mathematical elasticity, Vol. III, Theory of shells. North Holland, Amsterdam (to appear).  
  5. D. Chapelle and K.J. Bathe, Fundamental considerations for the finite element analysis of shell structures,Computers and Structures66 (1998) 19-36.  
  6. P. Gérard and É. Sanchez Palencia, Sensitivity phenomena for certain thin elastic shells with edges. Math. Meth. Appl. Sci. (to appear).  
  7. A.L. Goldenveizer, Theory of elastic thin shells. Pergamon, New York (1962).  
  8. P. Karamian, Nouveaux résultats numériques concernant les coques minces hyperboliques inhibées: cas du paraboloïde hyperbolique. C. R. Acad. Sci. Paris Sér. IIb326 (1998) 755-760.  
  9. P. Karamian, Réflexion des singularités dans les coques hyperboliques inhibées. C.R. Acad. Sci. Paris Sér. IIb326 (1998) 609-614.  
  10. 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).  
  11. D. Leguillon, J. Sanchez-Hubert and É. Sanchez Palencia, Model problem of singular perturbation without limit in the space of finite energy and its computation. C.R. Acad. Sci. Paris Sér. IIb327 (1999) 485-492.  
  12. 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.  
  13. 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.  
  14. A.E.H Love, A treatrise on the mathematical theory of elasticity, Reprinted by Dover, New-York (1944).  
  15. J. Pitkaranta and É. Sanchez Palencia, On the asymptotic behavior of sensitive shells with small thickness. C.R. Acad. Sci. Paris Sér. IIb325 (1997) 127-134.  
  16. H.S. Rutten, Theory and design of shells on the basis of asymptotic analysis. Rutten and Kruisman, Voorburg (1973).  
  17. J. Sanchez-Hubert and É. Sanchez Palencia, Introduction aux méthodes asymptotiques et à l'homogénéisation, Masson, Paris (1992).  
  18. J. Sanchez-Hubert and É. Sanchez Palencia, Coques élastiques minces. Propriétés asymptotiques. Masson, Paris (1997).  
  19. J. Sanchez-Hubert and É. Sanchez Palencia, Pathological phenomena in computation of thin elastic shells. Transactions Can. Soc. Mech. Engin.22 (1998) 435-446.  
  20. M. Van Dyke, Perturbation methods in fluid mechanics. Academic Press, New-York (1964).  

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