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

TL;DR

This paper explores the stability of spherical shapes in heat transfer optimization, revealing that balls are not always optimal for maximizing heat, with results depending on boundary conditions and problem parameters.

Contribution

It provides new stability criteria and identifies conditions under which balls are non-optimal in heat transfer shape optimization problems.

Findings

Balls may not be optimal for maximizing average heat under Robin boundary conditions.

Stability thresholds depend on Robin coefficient, dimension, and volume.

Conditions are identified where balls maximize the average temperature.

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.