Electron. J. Diff. Equ., Vol. 2015 (2015), No. 206, pp. 1-17.

Piecewise uniform optimal design of a bar with an attached mass

Boris P. Belinskiy, James W. Hiestand, John V. Matthews

Abstract:
We minimize, with respect to the cross sectional area, the mass of a bar given the rate of heat transfer. The bar enhances the heat transfer surface of a larger known mass to which the bar is attached. This article is an extension of a previous publication by two coauthors, where heat transfer from the sides of the bar was neglected and only conduction through its length was considered. The rate of cooling is defined by the first eigenvalue of the corresponding Sturm-Liouville problem. We compare the mass of the computed variable cross-section bar with the mass of a bar with constant cross-sectional area and the same rate of heat transfer, and conclude that a fin design with constant, or near constant, cross-sectional area is best.

Submitted December 14, 2014. Published August 10,2015.
Math Subject Classifications: 62K05, 80A20, 49R05, 35K05, 34B24, 65H10.
Key Words: Optimal design; heat transfer; heat equation; least eigenvalue; Sturm-Liouville problem; calculus of variations; transcendental equation; computer algebra.

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Boris P. Belinskiy
Department of Mathematics
University of Tennessee at Chattanooga
615 Mccallie Avenue
Chattanooga, TN 37403-2598, USA
email: Boris-Belinskiy@utc.edu
James W. Hiestand
College of Engineering
University of Tennessee at Chattanooga
615 Mccallie Avenue
Chattanooga, TN 37403-2598, USA
email: James-Hiestand@utc.edu
John V. Matthews
Department of Mathematics
University of Tennessee at Chattanooga
615 Mccallie Avenue
Chattanooga, TN 37403-2598, USA
email: Matt-Matthews@utc.edu

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