Determining the Optimal Geometry for Producing Isotherms on a Boundary Surface

Abstract

A shape-design problem is examined in this study that consists of determining the optimal boundary shape of a conductive body with a heating object or heating substrate that will yield a uniform boundary temperature. The Levenberg–Marquardt method and commercial software are used in this shape-design algorithm. The validity of the design analysis is verified using numerical experiments. Without considering the constraint of the domain area, different test cases examined previously by Mayeli et al. (“Inverse Shape Design for Heat Conduction Problems via the Ball Spine Algorithm,” Numerical Heat Transfer, Part B: Fundamentals, Vol. 69, No. 3, 2016, pp. 249–269) are reconsidered in this work to justify the validity and superiority of the present algorithm. The estimated results in the present work are then compared with the results given by Mayeli et al. It is found that the difficulty in choosing the best value for the overall underrelaxation factor reported by Mayeli et al. can be avoided by using the present algorithm, and it needs fewer iterations than reported by Mayeli et al. Next, when the constraint of the domain area is considered, the numerical experiments reveal that the optimal boundary shape with uniform temperature requirement can always be obtained.