We study the asymptotic behavior of the solution of a diffusion problem posed in the union of a cylinder of small diameter and fixed length with another cylinder with much smaller diameter and length. The Dirichlet condition is assumed to hold at both extremities of this domain. Depending on the relative size of the parameters, we show that the boundary condition of the one-dimensional limit problem is a Dirichlet, Fourier or Neumann condition. We also prove a corrector result for every case.
Nous étudions le comportement asymptotique de la solution d'un problème de diffusion posé sur l'union d'un cylindre de petit diamètre et de longueur fixe et d'un autre cylindre de longueur et de diamètre beaucoup plus petits. La condition de Dirichlet est imposée aux deux extrémités. Nous démontrons que selon les valeurs relatives des paramètres, la condition au bord du problème unidimensionnel limite est une condition de Dirichlet, de Fourier ou de Neumann. Nous démontrons aussi dans chaque cas un résultat de correcteur.
Accepted:
Published online:
Juan Casado-Díaz 1; Manuel Luna-Laynez 1; François Murat 2
@article{CRMATH_2004__338_2_133_0,
author = {Juan Casado-D{\'\i}az and Manuel Luna-Laynez and Fran\c{c}ois Murat},
title = {Asymptotic behavior of diffusion problems in a domain made of two cylinders of different diameters and lengths},
journal = {Comptes Rendus. Math\'ematique},
pages = {133--138},
publisher = {Elsevier},
volume = {338},
number = {2},
year = {2004},
doi = {10.1016/j.crma.2003.10.033},
language = {en},
}
TY - JOUR AU - Juan Casado-Díaz AU - Manuel Luna-Laynez AU - François Murat TI - Asymptotic behavior of diffusion problems in a domain made of two cylinders of different diameters and lengths JO - Comptes Rendus. Mathématique PY - 2004 SP - 133 EP - 138 VL - 338 IS - 2 PB - Elsevier DO - 10.1016/j.crma.2003.10.033 LA - en ID - CRMATH_2004__338_2_133_0 ER -
%0 Journal Article %A Juan Casado-Díaz %A Manuel Luna-Laynez %A François Murat %T Asymptotic behavior of diffusion problems in a domain made of two cylinders of different diameters and lengths %J Comptes Rendus. Mathématique %D 2004 %P 133-138 %V 338 %N 2 %I Elsevier %R 10.1016/j.crma.2003.10.033 %G en %F CRMATH_2004__338_2_133_0
Juan Casado-Díaz; Manuel Luna-Laynez; François Murat. Asymptotic behavior of diffusion problems in a domain made of two cylinders of different diameters and lengths. Comptes Rendus. Mathématique, Volume 338 (2004) no. 2, pp. 133-138. doi : 10.1016/j.crma.2003.10.033. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.10.033/
[1] J. Casado-Dı́az, F. Murat, The diffusion equation in a notched beam, in press
[2] J. Casado-Dı́az, M. Luna-Laynez, F. Murat, Diffusion problems in a domain made of two thin cylinders of different diameters and lengths, in preparation
[3] J. Casado-Dı́az, M. Luna-Laynez, F. Murat, Elasticity problems in a domain made of two thin cylinders of different diameters and lengths, in preparation
[4] Asymptotic theory and analysis for displacements and stress distribution in nonlinear straight slender rods, J. Elasticity, Volume 19 (1988), pp. 111-161
[5] Problèmes monotones dans des cylindres de faible diamètre formés de matériaux hétérogènes, C. R. Acad. Sci. Paris, Ser. I, Volume 320 (1995), pp. 1199-1204
[6] Mathematical modelling of rods, Handbook of Numerical Analysis, vol. IV, North-Holland, Amsterdam, 1996
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