# 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

## Access Full Article

top## Abstract

top## How to cite

topKaramian, 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

top- 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).
- 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.
- 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 Structures66 (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).
- 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.
- P. Karamian, Réflexion des singularités dans les coques hyperboliques inhibées. C.R. Acad. Sci. Paris Sér. IIb326 (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).
- 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.
- 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).
- 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.
- 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).
- J. Sanchez-Hubert and É. Sanchez Palencia, 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).

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.