Review of Computational Approaches to Optimization Problems in Inhomogeneous Rods and Plates

Abstract

In this paper, we review computational approaches to optimization problems of inhomogeneous rods and plates. We consider both the optimization of eigenvalues and the localization of eigenfunctions. These problems are motivated by physical problems including the determination of the extremum of the fundamental vibration frequency and the localization of the vibration displacement. We demonstrate how an iterative rearrangement approach and a gradient descent approach with projection can successfully solve these optimization problems under different boundary conditions with different densities given.

Keywords

Inhomogeneous rods and plates / Bi-Laplacian / Optimization of eigenvalues / Localization of eigenfunctions / Rearrangement

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Funding

Division of Mathematical Sciences(MS-1853701)