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

doi:10.1007/s42967-022-00242-w

Communications on Applied Mathematics and Computation ›› 2024, Vol. 6 ›› Issue (1): 236-256.doi: 10.1007/s42967-022-00242-w

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

Weitao Chen¹, Chiu-Yen Kao²

1. Department of Mathematics, University of California at Riverside, 900 University Ave, Riverside, CA, 92521, USA;
2. Department of Mathematical Sciences, Claremont McKenna College, 850 Columbia Ave, Claremont, CA, 91711, USA

Received:2022-08-17 Revised:2022-12-05 Published:2024-04-16
Contact: Chiu-Yen Kao,E-mail:ckao@cmc.edu;Weitao Chen,E-mail:weitaoc@ucr.edu E-mail:ckao@cmc.edu;weitaoc@ucr.edu
Supported by:
Weitao Chen is partially supported by the DMS-1853701. Chiu-Yen Kao’s work is supported in part by the DMS-2208373.

Abstract

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.

Key words: Inhomogeneous rods and plates, Bi-Laplacian, Optimization of eigenvalues, Localization of eigenfunctions, Rearrangement

Weitao Chen, Chiu-Yen Kao. Review of Computational Approaches to Optimization Problems in Inhomogeneous Rods and Plates[J]. Communications on Applied Mathematics and Computation, 2024, 6(1): 236-256.

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

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