Advertisement

Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Shape design sensitivity of a membrane

  • 128 Accesses

  • 36 Citations

Abstract

The dependence of the static response and the eigenvalues of a membrane on its shape is characterized. A transformation function is defined to determine the shape of the membrane. Differential operator properties and transformation techniques of integral calculus are employed to show that the static response and the eigenvalues of the system depend in a continuous and differentiable way on the shape of the membrane. Explicit and computable formulas are presented for the derivative (first variation) of the structural response and the eigenvalues with respect to the shape. A rigorous proof is provided, and the shape design sensitivity of a typical integral functional is determined.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Haug, E. J., andRousselet, B.,Design Sensitivity Analysis in Structural Mechanics, I: Static Response Variations, Journal of Structural Mechanics, Vol. 8, pp. 17–41, 1980.

  2. 2.

    Haug, E. J., andRousselet, B.,Design Sensitivity Analysis in Structural Mechanics, II: Eigenvalue Variations, Journal of Structural Mechanics, Vol. 8, pp. 161–186, 1980.

  3. 3.

    Rousselet, B., andHaug, E. J.,Design Sensitivity Analysis of Shape Variation, Optimization of Distributed Parameter Structures, Vol. 2, Edited by E. J. Haug and J. Céa, Sijthoff and Noordhoff, Alphen aan den Rijn, Holland, 1981.

  4. 4.

    Rousselet, B.,Shape Design Sensitivity Methods for Structural Mechanics, Optimization of Distributed Parameter Structural Systems, Edited by E. J. Haug and J. Céa, Sijthoff and Noordhoff, Alphen aan den Rijn, Holland, 1981.

  5. 5.

    Rousselet, B., andHaug, E. J.,Design Sensitivity Analysis in Structural Mechanics, III: Effects of Shape Variations for Plate and Plane Elasticity, Journal of Structural Mechanics (to appear).

  6. 6.

    Rousselet, B.,Static and Dynamic Loads, Pointwise Constraints in Structural Optimization, Proceedings of the Colloquium on Optimization: Theory and Algorithms, Confolant, France, 1981 (to appear).

  7. 7.

    Rousselet, B.,Dynamic Response in Shape Optimization, Communication at 3rd IFAC Symposium on the Control of Distributed Parameter System, Toulouse, France, 1982.

  8. 8.

    Céa, J.,Problems of Shape Optimal Design, Optimization of Distributed Parameter Structures, Vol. 2, Edited by E. J. Haug and J. Céa, Sijthoff and Noordhoff, Alphen aan den Rijn, Holland, 1981.

  9. 9.

    Chenais, D.,Sur une Famille de Variétés à Bord Lipschitziennes: Application à un Problème d'Identification de Domaines, Annales de l'Institut Fourier, Vol. 27, pp. 201–231, 1977.

  10. 10.

    Necas, J.,Les Méthodes Directes dans la Théorie des Equations Elliptiques, Académie Tchécoslovaque des Sciences, Prague, Czechoslovakia, 1967.

  11. 11.

    Rousselet, B.,Optimal Design and Eigenvalue Problems, Proceedings of the 8th IFIP Conference, Würzburg, Germany, 1977; Springer-Verlag, Berlin, Germany, 1978.

  12. 12.

    Courant, R. S., andHilbert, D.,Methods of Mathematical Physics, Wiley-Interscience, New York, New York, 1953.

  13. 13.

    Micheletti, A. M.,Perturbazione dello Spettro di un Operatore Ellitico di Tipo Variazionale in Relazione ad una Variazione del Campo, Annali di Mathematica Pura e Applicata, Vol. 47, pp. 267–282, 1973.

  14. 14.

    Rousselet, B.,Identification de Domaines et Problèmes de Valeurs, Thèse de Spécialité, Université de Nice, Nice, France, 1977.

  15. 15.

    Murat, F., andSimon, J.,Sur le Contrôle par un Domaine Géométrique, Publication du Laboratoire d'Analyse Numérique, Université de Paris 6, Paris, France, 1976.

  16. 16.

    Bendali, A.,Existence et Régularisation dans un Problème d'Identification de Domaine: Application à l'Analyse Numérique d'un Cas Modèle, Thèse de Spécialité, Université d'Alger, Alger, Algérie, 1975.

  17. 17.

    Céa, J.,Une Méthode Numérique pour la Recherche d'un Domaine Optimal, IMAN Publication, Université de Nice, Nice, France, 1977.

  18. 18.

    Céa, J., Gioan, A., andMichel, J.,Adaptation de la Méthode du Gradient à un Problème d'Identification de Domaine, Computing Methods in Applied Science and Engineering, Vol. 2, Springer-Verlag, Berlin, Germany, 1974.

  19. 19.

    Dervieux, A., andPalmerio, B.,Une Formule de Hadamard dans des Problèmes d'Identification de Domaines, Comptes Rendus de l'Académie des Sciences, Vol. 280, pp. 1697–1700, 1975, and Vol. 280, pp. 1761–1764, 1975.

  20. 20.

    Djadane, A.,Régularité d'une Fonctionnelle de Domaines et Application à l'Analyse Numérique d'un Problème Modèle, Thèse de Spécialité, Université d'Alger, Alger, Algérie, 1975.

  21. 21.

    Joseph, D. D.,Parameter and Domain Dependence of Eigenvalues of Elliptic Partial Differential Equations, Archive of Rational Mechanics and Analysis, Vol. 35, pp. 169–177, 1977.

  22. 22.

    Licari, J. P., andWarner, W. H.,Domain Dependence of Eigenvalues of Vibrating Plates, SIAM Journal on Applied Mathematics, Vol. 24, pp. 383–395, 1973.

  23. 23.

    Palmerio, B., andDervieux, A.,Hadamard's Variational Formula for a Mixed Problem and an Application to a Problem Related to a Signorini-Like Variational Inequality, INRIA, Rocquencourt, Le Chesnay, France, Report No. 339, 1979.

  24. 24.

    Hadamard, J.,Mémoire sur le Problème d'Analyse Relatif à l'Equilibre des Plaques Elastiques Encastrées, Mémoire des Savants Etrangers, Vol. 33, pp. 515–629, 1908.

  25. 25.

    Kato, T.,Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, Germany, 1966.

  26. 26.

    Céa, J.,Lectures in Optimization, Theory and Algorithms, Springer-Verlag, Berlin, Germany, 1978.

  27. 27.

    Aubin, J. F.,Approximation of Elliptic Boundary Problems, Wiley-Interscience, New York, New York, 1972.

  28. 28.

    Dieudonne, J. A.,Eléments d'Analyse, I, Fondements de l'Analyse Moderne, Gauthiers-Villars, Paris, France, 1969.

  29. 29.

    Lions, J. L., andMagenes, E.,Problèmes aux Limites Nonhomogènes et Applications, Vol. 1, Dunod, Paris, France, 1968.

  30. 30.

    Zolesio, J. P.,The Material Derivative (or Speed) Method for Shape Optimization, Optimization of Distributed Parameter Structures, Vol. 2, Edited by E. J. Haug and J. Céa, Sijthoff and Noordhoff, Alphen aan den Rijn, Holland, 1981.

  31. 31.

    Germain, P.,Cours de Mécanique des Milieux Continus, Masson, Paris, France, 1973.

  32. 32.

    Simon, J.,Variations par Rapport au Domaine dans des Problèmes aux Limites, Publication du Laboratoire d'Analyse Numérique, Université de Paris, 6, Paris, France, 1980.

  33. 33.

    Pironneau, O.,Optimum Design with Lagrangian Finite Elements: Design of an Electromagnet, Computer Methods in Applied Mechanics and Engineering, Vol. 15, pp. 207–308, 1978.

  34. 34.

    Rousselet, B.,Dependence of Eigenvalues with Respect to Shape, Optimization of Distributed Parameter Structural Systems, Vol. 2, Edited by E. J. Haug and J. Céa, Sijthoff and Noordhoff, Alphen aan den Rijn, Holland, 1981.

  35. 35.

    Haug, E. J., andArora, J. S.,Applied Optimal Design, John Wiley and Sons, New York, New York, 1979.

Download references

Author information

Additional information

The author is indebted to Professor E. J. Haug for his comments and stimulating interest in the paper.

Communicated by E. J. Haug

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Rousselet, B. Shape design sensitivity of a membrane. J Optim Theory Appl 40, 595–623 (1983). https://doi.org/10.1007/BF00933973

Download citation

Key Words

  • Optimization
  • partial differential equations
  • elasticity
  • control
  • design sensitivity