Abstract
In this paper, several efficient rearrangement algorithms are proposed to find the optimal shape and topology for elliptic eigenvalue problems with inhomogeneous structures. The goal is to solve minimization and maximization of the k-th eigenvalue and maximization of spectrum ratios of the second order elliptic differential operator. Physically, these problems are motivated by the frequency control based on density distribution of vibrating membranes. The methods proposed are based on Rayleigh quotient formulation of eigenvalues and rearrangement algorithms which can handle topology changes automatically. Due to the efficient rearrangement strategy, the new proposed methods are more efficient than classical level set approaches based on shape and/or topological derivatives. Numerous numerical examples are provided to demonstrate the robustness and efficiency of new approach.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allaire, G.: Shape Optimization by the Homogenization Method. Springer, New York (2001)
Amstutz, S., Andrä, H.: A new algorithm for topology optimization using a level-set method. J. Comput. Phys. 216, 573–588 (2006)
Bajalinov, E.B.: Linear-Fractional Programming: Theory, Methods, Applications and Software. Kluwer Academic, Boston (2003)
Bendsoe, M., Sigmund, O.: Topology Optimization. Theory. Methods and Applications. Springer, New York (2003)
Burger, M.: A framework for the construction of level set methods for shape optimization and reconstruction. Inverse Probl. 17, 1327–1356 (2001)
Burger, M., Hackl, B., Ring, W.: Incorporating topological derivatives into level set methods. J. Comput. Phys. 194, 344–362 (2004)
Cox, S.: The two phase drum with the deepest bass note. Jpn. J. Ind. Appl. Math. 8, 345–355 (1991)
Cox, S., McLaughlin, J.: Extremal eigenvalue problems for composite membranes I and II. Appl. Math. Optim. 22, 153–167 (1990)
Cox, S.J.: The generalized gradient at a multiple eigenvalue. J. Funct. Anal. 133, 30–40 (1995)
Haber, E.: A multilevel, level-set method for optimizing eigenvalues in shape design problems. J. Comput. Phys. 198, 518–534 (2004)
He, L., Kao, C.-Y., Osher, S.: Incorporating topological derivatives into shape derivatives based level set methods. J. Comput. Phys. 225, 891–909 (2007)
Henrot, A.: Extremum Problems for Eigenvalues of Elliptic Operators. Birkhäuser, Basel (2006)
Ito, K., Kunischm, K., Li, Z.: Level-set function approach to an inverse interface problem. Inverse Probl. 17, 1225–1242 (2001)
Kao, C.-Y., Lou, Y., Yanagida, E.: Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains. Math. Biosci. Eng. 5, 315–335 (2008)
Kao, C.Y., Osher, S., Yablonovitch, E.: Maximizing band gaps in two dimensional photonic crystals by using level set methods. Appl. Phys. B, Lasers Opt. 81, 235–244 (2005)
Kao, C.-Y., Santosa, F.: Maximization of the quality factor of an optical resonator. Wave Motion 45(4), 412–427 (2008)
Krein, M.G.: On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability. In: American Mathematical Society Translations, pp. 163–187 (1955)
Murqat, F., Simon, S.: Etudes de problems d’optimal design. Lect. Notes Comput. Sci. 41, 52–62 (1976)
Osher, J., Santosa, F.: Level set methods for optimization problems involving geometry and constraints. I. Frequencies of a two-density inhomogeneous drum. J. Comput. Phys. 171, 272–288 (2001)
Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)
Rayleigh, J.W.S.: The Theory of Sound, vols. 1, 2. Dover, New York (1945)
Sethian, J., Wiegmann, A.: Structural boundary design via level set and immersed interface methods. J. Comput. Phys. 163, 489–528 (2000)
Sokolowski, J., Zolesio, J.-P.: Introduction to Shape Optimization: Shape Sensitivity Analysis, vol. 10. Springer, Heidelberg (1992)
Su, S.: Numerical approaches on shape optimization of elliptic eigenvalue problems and shape study of human brains. PhD thesis, The Ohio State University (2010)
Wayne, A.: Inequalities and inversion of order. Scr. Math. 12, 164–169 (1946)
Zhu, S., Wu, Q., Liu, C.: Variational piecewise constant level set methods for shape optimization of a two-density drum. J. Comput. Phys. 229, 5062–5089 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kao, CY., Su, S. Efficient Rearrangement Algorithms for Shape Optimization on Elliptic Eigenvalue Problems. J Sci Comput 54, 492–512 (2013). https://doi.org/10.1007/s10915-012-9629-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-012-9629-0