Numerical minimization of eigenmodes of a membrane with respect to the domain

Oudet, Édouard

ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 3, pp. 315-330.

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

[1] G. Allaire, Shape optimization by the homogenization method. Springer-Verlag, New York (2001). | MR | Zbl

[2] G. Allaire, F. Jouve and A.M. Toader, A level-set method for shape optimization. C. R. Acad. Sci. Paris 334 (2002) 1125-1130. | MR | Zbl

[3] M. Bendsoe, Optimization of structural Topology, Shape and Material. Springer (1995). | MR | Zbl

[4] M. Bendsoe and C. Mota Soares, Topology optimization of structures. Kluwer Academic Press, Dordrechts (1993). | MR

[5] G. Buttazzo and G. Dal Maso, An Existence Result for a Class of Shape Optimization Problems. Arch. Ration. Mech. Anal. 122 (1993) 183-195. | MR | Zbl

[6] M.G. Crandall and P.L. Lions, Viscosity Solutions of Hamilton-Jacobi Equations. Trans. Amer. Math. Soc. 277 (1983) 1-43. | MR | Zbl

[7] 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. | JFM

[8] S. Finzi Vita, Constrained shape optimization for Dirichlets problems: discretization via relaxation. Adv. Math. Sci. Appl. 9 (1999) 581-596. | MR | Zbl

[9] H. Hamda, F. Jouve, E. Lutton, M. Schoenauer and M. Sebag, Représentations non structurées en optimisation topologique de formes par algorithmes évolutionnaires 8 (2000).

[10] A. Henrot, Minimization problems for eigenvalues of the Laplacian. J. Evol. Eq. 3 (2003) 443-461. | MR | Zbl

[11] A. Henrot and E. Oudet, 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. | MR | Zbl

[12] A. Henrot and E. Oudet, Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditions. Arch. Ration. Mech. Anal. 169 (2003) 73-87. | MR | Zbl

[13] A. Henrot and M. Pierre, Optimisation de forme (in preparation).

[14] E. Krahn, Über eine von Rayleigh formulierte Minimaleigenshaft des Kreises. Math. Ann. 94 (1925) 97-100. | EuDML | JFM | MR

[15] E. Krahn, Über Minimaleigenshaften der Kugel in drei und mehr Dimensionen. Acta Comm. Univ. Dorpat. A9 (1926) 1-44. | JFM

[16] S. Osher and F. Santosa, Level set methods for optimization problems involving geometry and constraints: frequencies of a two-density inhomogeneous drum. J. Comput. Phys. 171 (2001) 272-288. | MR | Zbl

[17] 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. | MR | Zbl

[18] E. Oudet, $Quelques$ résultats en optimisation de forme et stabilisation. Prépublication de l'Institut de recherche mathématique avancée, Strasbourg (2002).

[19] M. Pierre and J.M. Roche, Numerical simulation of tridimensional electromagnetic shaping of liquid metals. Numer. Math. 65 (1993) 203-217. | EuDML | MR | Zbl

[20] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics. Ann. Math. Stud. 27 (1952). | MR | Zbl

[21] J.A. Sethian, Level Set Methods and Fast Marching Methods. Cambridge University Press (1999). | MR | Zbl

[22] J. Sokolowski and J.P. Zolesio, Introduction to shape optimization: shape sensitivity analysis. Springer, Berlin, Springer Ser. Comput. Math. 10 (1992). | MR | Zbl

[23] B.A. Troesch, Elliptical membranes with smallest second eigenvalue. Math. Comp. 27 (1973) 767-772. | MR | Zbl

[24] S.A. Wolf and J.B. Keller, Range of the first two eigenvalues of the laplacian. Proc. Roy. Soc. Lond. A 447 (1994) 397-412. | MR | Zbl