Journal of Scientific Computing

, Volume 54, Issue 2, pp 492-512

First online:

Efficient Rearrangement Algorithms for Shape Optimization on Elliptic Eigenvalue Problems

  • Chiu-Yen KaoAffiliated withDepartment of Mathematics, The Ohio State University ColumbusDepartment of Mathematics and Computer Science, Claremont McKenna College Email author 
  • , Shu SuAffiliated withAmerican Electric Power

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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.


Minimal eigenvalue Maximal eigenvalue Maximal ratio of eigenvalues Elliptic operator Shape optimization Rearrangement algorithm Rayleigh quotient