Efficient Rearrangement Algorithms for Shape Optimization on Elliptic Eigenvalue Problems

Authors:
Chiu-Yen Kao;Shu Su
Affiliations:
Department of Mathematics, The Ohio State University Columbus, Columbus, USA 43210 and Department of Mathematics and Computer Science, Claremont McKenna College, Claremont, USA 91711;American Electric Power, Columbus, USA 43215
Venue:
Journal of Scientific Computing
Year:
2013

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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.