On the optimization of higher eigenvalues

Youri V. Egorov

doi:10.1016/j.crme.2004.03.008

On the optimization of higher eigenvalues

Presented by: Évariste Sanchez-Palencia

Youri V. Egorov ¹

¹ Université Paul Sabatier, UFR MIG, labo MIP, UMR 5640, 118, route de Narbonne, 31062 Toulouse cedex 4, France

Comptes Rendus. Mécanique, Volume 332 (2004) no. 8, pp. 673-678.

Abstract
Résumé

A new approach is proposed for the study of the classical Lagrange problem on the optimal form of a column with clamped ends and a fixed volume. It is proved that there exists a column with the maximal possible value of the kth eigenvalue and that such a column is unique. A method for the numerical solution is proposed. The method is based on the analysis of the critical points of a non-linear functional related to the Lagrange problem.

On propose une nouvelle approche au problème classique de Lagrange de la forme optimale de la colonne encastrée aux volume et hauteur fixés. On prouve qu'il existe une colonne avec la valeur de la k-ième valeur propre maximale et que telle colonne est unique. La méthode est basée sur l'étude des points critiques d'une fonctionnelle nonlinéaire. Une méthode pour la solution numerique du problème est proposée.

Received: 2004-02-24
Accepted: 2004-03-09
Published online: 2004-05-28

DOI: 10.1016/j.crme.2004.03.008

Keywords: Computational solid mechanics, Lagrange problem, Optimization of eigenvalues, Optimal column
Mots-clés : Mécanique des solides numérique, Problème de Lagrange, Optimisation de valeurs propres, Colonne optimale

Author's affiliations:

Youri V. Egorov ¹

¹ Université Paul Sabatier, UFR MIG, labo MIP, UMR 5640, 118, route de Narbonne, 31062 Toulouse cedex 4, France

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     title = {On the optimization of higher eigenvalues},
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Youri V. Egorov. On the optimization of higher eigenvalues. Comptes Rendus. Mécanique, Volume 332 (2004) no. 8, pp. 673-678. doi : 10.1016/j.crme.2004.03.008. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2004.03.008/

References
Cited by

[1] A.P. Seyranian The Lagrange problem on optimal design of a column, Uspekhi Mekh., Volume 2 (2003), pp. 45-96

[2] N. Olhoff Optimization of vibrating beams with respect to higher order natural frequencies, J. Struct. Mech., Volume 4 (1976) no. 1, pp. 87-122

[3] N. Olhoff Maximizing higher order eigen-frequencies of beams with constraints on the design geometry, J. Struct. Mech., Volume 5 (1977) no. 2, pp. 107-134

[4] Yu.V. Egorov On the Lagrange problem about the strongest circular column, C. R. Acad. Sci. Paris, Ser. I, Volume 335 (2002), pp. 997-1002

[5] Yu.V. Egorov On the Lagrange problem about the optimal form of columns, Dokl. RAN, Volume 392 (2003) no. 5, pp. 1-5

[6] Yu.V. Egorov On the Lagrange problem about hollow circular columns, C. R. Mecanique, Volume 331 (2003), pp. 699-704

[7] Yu.V. Egorov On the Lagrange problem about the optimal form for circular hollow columns, Russian J. Math. Phys., Volume 10 (2003) no. 4, pp. 411-421

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