Comptes Rendus
no. 4
Mathematical Problems in Mechanics
Equivalence estimates for a class of singular perturbation problems
[Estimations d'équivalence pour une classe de problèmes de perturbations singulières]

Présenté par : Philippe G. Ciarlet

Sheng Zhang1
1 Department of Mathematics, Wayne State University, Detroit, MI 48202, USA
Comptes Rendus. Mathématique, Volume 342 (2006) no. 4, pp. 285-288.

Nous donnons des estimations d'équivalence de la solution d'un problème de perturbations singulières pour des modèles de coques qui englobent les modèles de Koiter et de Naghdi. Deux de ces estimations sont valables pour les problèmes de coques dits intermédiaires, la troisième s'applique à des coques de type membrane/cisaillement. Quelques unes de ces équivalences sont connues, mais d'autres équivalences donnent des résultats précis pour des solutions de problèmes de perturbations singulières.

We give some equivalence estimates on the solution of a singular perturbation problem that represents, among other models, the Koiter and Naghdi shell models. Two of the estimates apply to intermediate shell problems and the third is for membrane/shear dominated shells. From these equivalences, many known and some new sharp estimates on the solutions of the singular perturbation problems easily follow.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2005.10.004

Sheng Zhang 1

1 Department of Mathematics, Wayne State University, Detroit, MI 48202, USA 
@article{CRMATH_2006__342_4_285_0,
     author = {Sheng Zhang},
     title = {Equivalence estimates for a class of singular perturbation problems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {285--288},
     publisher = {Elsevier},
     volume = {342},
     number = {4},
     year = {2006},
     doi = {10.1016/j.crma.2005.10.004},
     language = {en},
}
TY  - JOUR
AU  - Sheng Zhang
TI  - Equivalence estimates for a class of singular perturbation problems
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 285
EP  - 288
VL  - 342
IS  - 4
PB  - Elsevier
DO  - 10.1016/j.crma.2005.10.004
LA  - en
ID  - CRMATH_2006__342_4_285_0
ER  - 
%0 Journal Article
%A Sheng Zhang
%T Equivalence estimates for a class of singular perturbation problems
%J Comptes Rendus. Mathématique
%D 2006
%P 285-288
%V 342
%N 4
%I Elsevier
%R 10.1016/j.crma.2005.10.004
%G en
%F CRMATH_2006__342_4_285_0
Sheng Zhang. Equivalence estimates for a class of singular perturbation problems. Comptes Rendus. Mathématique, Volume 342 (2006) no. 4, pp. 285-288. doi : 10.1016/j.crma.2005.10.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.10.004/

[1] F. Auricchio; L. Beirão da Veiga; C. Lovadina Remarks on the asymptotic behavior of Koiter shells, Comput. & Structures, Volume 80 (2002), pp. 735-745

[2] C. Baiocchi; C. Lovadina A shell classification by interpolation, Math. Models Methods Appl. Sci., Volume 12 (2002), pp. 1359-1380

[3] J. Bergh; J. Löfström Interpolation Space: An Introduction, Springer-Verlag, 1976

[4] A. Blouza; F. Brezzi; C. Lovadina Sur la classification des coques linéairement élastiques, C. R. Acad. Sci. Paris, Ser. I, Volume 328 (1999), pp. 831-836

[5] D. Caillerie Étude générale d'un type de problèmes raides et de perturbation singulière, C. R. Acad. Sci. Paris, Ser. I, Volume 323 (1996), pp. 835-840

[6] D. Chapelle; K.J. Bathe The Finite Element Analysis of Shells – Fundamentals, Springer, 2003

[7] P.G. Ciarlet Mathematical Elasticity, vol. III: Theory of Shells, North-Holland, 2000

[8] C.A. DeSouza; E. Sanchez-Palencia Complexification phenomena in an example of sensitive singular perturbation, C. R. Mecanique, Volume 332 (2004), pp. 605-612

[9] D. Leguillon; J. Sanchez-Hubert; E. Sanchez-Palencia Model problem of singular perturbation without limit in the space of finite energy and its computation, C. R. Acad. Sci. Paris, Ser. IIb, Volume 327 (1999), pp. 485-492

[10] J.L. Lions Perturbations singulières dans les problèmes aux limites et en contrôle optimal, Lecture Notes in Math., vol. 323, Springer-Verlag, 1973

[11] E. Sanchez-Palencia On a singular perturbation going out of the energy space, J. Math. Pures Appl., Volume 79 (2000), pp. 591-602

[12] Z. Schuss Singular perturbations and the transition from thin plate to membrane, Proc. Amer. Math. Soc., Volume 58 (1976), pp. 139-147

  • Sheng Zhang Analysis of a discontinuous Galerkin method for the bending problem of Koiter shell, Numerische Mathematik, Volume 133 (2016) no. 2, p. 333 | DOI:10.1007/s00211-015-0747-0
  • M. Chipot; S. Guesmia; A. Sengouga Singular perturbations of some nonlinear problems, Journal of Mathematical Sciences, Volume 176 (2011) no. 6, p. 828 | DOI:10.1007/s10958-011-0439-y
  • SHENG ZHANG AN ASYMPTOTIC ANALYSIS ON THE FORM OF NAGHDI TYPE ARCH MODEL, Mathematical Models and Methods in Applied Sciences, Volume 18 (2008) no. 03, p. 417 | DOI:10.1142/s0218202508002747
  • Sheng Zhang Analysis of Finite Element Domain Embedding Methods for Curved Domains using Uniform Grids, SIAM Journal on Numerical Analysis, Volume 46 (2008) no. 6, p. 2843 | DOI:10.1137/060671681
  • Sheng Zhang On the accuracy of Reissner–Mindlin plate model for stress boundary conditions, ESAIM: Mathematical Modelling and Numerical Analysis, Volume 40 (2006) no. 2, p. 269 | DOI:10.1051/m2an:2006014

Cité par 5 documents. Sources : Crossref

⁎ This work was partially supported by NSF grant DMS-0513559.

Commentaires - Politique