Committee Chair

Belinskiy, Boris P.

Committee Member

Kong, Lingju; Cox, Christopher; Wang, Jin; Nichols, Roger

Department

Dept. of Computational Science

College

College of Engineering and Computer Science

Publisher

University of Tennessee at Chattanooga

Place of Publication

Chattanooga (Tenn.)

Abstract

We find an optimal mass of a structure described by a Sturm-Liouville (S-L) problem with a spectral parameter in the boundary conditions. While previous work on the subject focused on a somewhat simplified model, we consider a more general S-L problem. We use the calculus of variations approach to determine a set of critical points of a corresponding mass functional, yet these critical points - which we call \textit{predesigns} - do not necessarily themselves represent meaningful solutions. It is natural to expect a mass to be real and positive. To this end, we additionally introduce a set of solvability conditions on the S-L problem data, confirming that these critical points represent meaningful solutions we refer to as \textit{designs}. We further present the analytic continuation of these predesigns in regards to the spectral parameter as well as a discussion of the stability of these (pre)designs. We present a code that allows us to for the given data of the S-L problem check conditions of solvability, plot the design, and calculate the value of the functional that represents the optimal mass.

Acknowledgments

UTC Graduate School and Department of Mathematics

Degree

Ph. D.; A dissertation submitted to the faculty of the University of Tennessee at Chattanooga in partial fulfillment of the requirements of the degree of Doctor of Philosophy.

Date

12-2022

Subject

Calculus of variations; Differential equations

Keyword

spectral analysis; calculus of variations; ordinary differential equations; partial differential equations

Document Type

Doctoral dissertations

DCMI Type

Text

Extent

xi, 71 leaves.

Language

English

Rights

http://rightsstatements.org/vocab/InC/1.0/

License

http://creativecommons.org/licenses/by/4.0/

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