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arXiv · 2307.09303

Heat Transfer Shape Optimization: Stability and Non-Optimality of the Ball

Qinfeng Li·Hang Yang
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Abstract

This paper investigates shape optimization problems in the context of heat transfer, with a focus on the stability and non-optimality of round domains under Robin boundary conditions. Using the flow approach and Steklov eigenvalue estimates, we derive the necessary and sufficient stability conditions for a ball to maximize the averaged heat when the heat source is radially decreasing. Our results show that, counterintuitively, a ball may not be optimal for maximizing the averaged heat under heat convection, even with radially decreasing heat sources located on the center of the ball. Moreover, we identify stability-breaking phenomena by giving precise values of thresholds, which depend on the Robin coefficient, dimension, and volume constraints. Additionally, we demonstrate that a ball can maximize the averaged temperature under certain conditions and we also explore optimal shapes in thin insulation problems.

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Qinfeng Li, Hang Yang. 2023-07-18. Heat Transfer Shape Optimization: Stability and Non-Optimality of the Ball. https://arxiv.org/abs/2307.09303

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