Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-28T03:14:28.021Z Has data issue: false hasContentIssue false
  • English
  • Français

Un problème Spectral Issu d'un Couplage Elasto-Acoustique

Published online by Cambridge University Press:  15 April 2002

Mario Durán  and
Jean-Claude Nédélec
Mario Durán
Affiliation:
Centre de Mathématiques Appliquées, École Polytechnique, 91128 Palaiseau, France. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile. ([email protected])
Jean-Claude Nédélec
Affiliation:
Centre de Mathématiques Appliquées, École Polytechnique, 91128 Palaiseau, France. ([email protected])
Get access

Abstract

We are interested in the theoretical study of a spectral problem arising in a physical situation, namely interactions of fluid-solid type structure. More precisely, we study the existence of solutions for a quadratic eigenvalue problem, which describes the vibrations of a system made up of two elastic bodies, where a slip is allowed on their interface and which surround a cavity full of an inviscid and slightly compressible fluid. The problem shall be treated like a generalized eigenvalue problem. Thus by using some functional analysis results, we deduce the existence of solutions and prove a spectral asymptotic behavior property, which allows us to compare the spectrum of this coupled model and the spectrum associated to the problem without transmission between the fluid-solid media.

Keywords

Couplage élasto-acoustiqueproblème quadratique de valeurs propres.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 4 , July 2000 , pp. 835 - 857
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

H. Brezis, Analyse Fonctionnelle. Masson, Paris (1983).
C. Conca and M. Durán, On some class of elliptic spectral problems. Rapport interne, Facultad de Matemáticas, Universidad Católica de Chile, PUC-FM/99-07 (1999).
de Figueiredo, D., Positive solutions of semilinear elliptic problems. Differential Equations Proceedings, Lect. Notes Math. 957 (1982) 34-84. CrossRef
N. Dunfort and J.T. Schwartz, Linear Operators. Part II: Spectral Theory. Wiley-Interscience, New-York (1964).
M. Durán, Étude théorique et numérique de quelques problèmes de type fluide-solide, Partie I. Thèse de doctorat à l'École Polytechnique, Paris (1996).
M. Durán, Mathematical and numerical analysis of an elastic-acoustic coupling, in Proceeding of Second ECCOMAS Conference on Numerical Methods in Engineering, J. Wiley & Sons Ltd. (1996) 888-893.
V. Hutson and J.S. Pym, Applications of Functional Analysis and Operator Theory. Academic Press, London (1980).
T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag, Berlin (1976).
Lene, F. and Leguillon, D., Étude de l'influence d'un glissement entre les constituants d'un matériau composite sur ses coefficients de comportements effectifs. J. Mécanique 20 (1981) 509-536.
J. Necas, Les Méthodes Directes en Théorie des Equations Elliptiques. Masson, Paris (1967).
P.-A. Raviart and J.-M. Thomas, Introduction à l'Analyse Numérique des Equations aux Dérivées Partielles. Masson, Paris (1983).