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Numerical minimization of eigenmodes of a membranewith respect to the domain

Published online by Cambridge University Press:  15 June 2004

Édouard Oudet
Édouard Oudet*
Affiliation:
Institut de Recherche Mathématique Avancée, Université Louis Pasteur, 67084 Strasbourg Cedex, France; [email protected].
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Abstract

In this paper we introduce a numerical approach adapted to the minimizationof the eigenmodes of a membrane with respect to the domain. This method isbased on the combination of the Level Set method of S. Osher and J.A.Sethian with the relaxed approach. This algorithm enables both changing thetopology and working on a fixed regular grid.

Keywords

Shape optimizationeigenvaluelevelsetrelaxation.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 10 , Issue 3 , July 2004 , pp. 315 - 330
Copyright
© EDP Sciences, SMAI, 2004

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References

G. Allaire, Shape optimization by the homogenization method. Springer-Verlag, New York (2001).
Allaire, G., Jouve, F. and Toader, A.M., A level-set method for shape optimization. C. R. Acad. Sci. Paris 334 (2002) 1125-1130. CrossRef
M. Bendsoe, Optimization of structural Topology, Shape and Material. Springer (1995).
M. Bendsoe and C. Mota Soares, Topology optimization of structures. Kluwer Academic Press, Dordrechts (1993).
G. Buttazzo and G. Dal Maso, An Existence Result for a Class of Shape Optimization Problems. Arch. Ration. Mech. Anal. 122 (1993) 183-195.
Crandall, M.G. and Lions, P.L., Viscosity Solutions of Hamilton-Jacobi Equations. Trans. Amer. Math. Soc. 277 (1983) 1-43. CrossRef
G. Faber, Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt. Sitz. Ber. Bayer. Akad. Wiss. (1923) 169-172.
S. Finzi Vita, Constrained shape optimization for Dirichlets problems: discretization via relaxation. Adv. Math. Sci. Appl. 9 (1999) 581-596.
H. Hamda, F. Jouve, E. Lutton, M. Schoenauer and M. Sebag, Représentations non structurées en optimisation topologique de formes par algorithmes évolutionnaires. Actes du 32e Congrès d'Analyse Numérique, Canum. ESAIM Proc. 8 (2000).
Henrot, A., Minimization problems for eigenvalues of the Laplacian. J. Evol. Eq. 3 (2003) 443-461. CrossRef
Henrot, A. and Oudet, E., Le stade ne minimise pas $\lambda _{2}$ parmi les ouverts convexes du plan. C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 417-422. CrossRef
Henrot, A. and Oudet, E., Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditions. Arch. Ration. Mech. Anal. 169 (2003) 73-87. CrossRef
A. Henrot and M. Pierre, Optimisation de forme (in preparation).
E. Krahn, Über eine von Rayleigh formulierte Minimaleigenshaft des Kreises. Math. Ann. 94 (1925) 97-100.
E. Krahn, Über Minimaleigenshaften der Kugel in drei und mehr Dimensionen. Acta Comm. Univ. Dorpat. A9 (1926) 1-44.
Osher, S. and Santosa, F., Level set methods for optimization problems involving geometry and constraints: frequencies of a two-density inhomogeneous drum. J. Comput. Phys. 171 (2001) 272-288. CrossRef
S. Osher and J.A. Sethian, Front propagation with curvature-dependant speed: Algorithms based on Hamilton-Jacobi formulations J. Comput. Phys. 79 (1988) 12-49.
E. Oudet, Quelques résultats en optimisation de forme et stabilisation. Prépublication de l'Institut de recherche mathématique avancée, Strasbourg (2002).
Pierre, M. and Roche, J.M., Numerical simulation of tridimensional electromagnetic shaping of liquid metals. Numer. Math. 65 (1993) 203-217. CrossRef
G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics. Ann. Math. Stud. 27 (1952).
J.A. Sethian, Level Set Methods and Fast Marching Methods. Cambridge University Press (1999).
J. Sokolowski and J.P. Zolesio, Introduction to shape optimization: shape sensitivity analysis. Springer, Berlin, Springer Ser. Comput. Math. 10 (1992).
B.A. Troesch, Elliptical membranes with smallest second eigenvalue. Math. Comp. 27 (1973) 767-772.
Wolf, S.A. and Keller, J.B., Range of the first two eigenvalues of the laplacian. Proc. Roy. Soc. Lond. A 447 (1994) 397-412. CrossRef