An optimal control strategy to design passive thermal cloaks of arbitrary shape

TL;DR

This paper presents a numerical framework for designing passive thermal cloaks of arbitrary shapes using optimal control of the heat equation, applicable in static and transient regimes, with demonstrated effectiveness through numerical tests.

Contribution

It introduces a novel optimal control approach for passive thermal cloaking of arbitrary shapes, including theoretical analysis and numerical implementation.

Findings

Existence of optimal diffusivity coefficients for static and transient cases

Derivation of first-order optimality conditions

Successful numerical demonstrations on complex 2D objects

Abstract

In this paper we describe a numerical framework for achieving passive thermal cloaking of arbitrary shapes in both static and transient regimes. The design strategy is cast as the solution of an optimal control problem (OCP) for the heat equation where the coefficients of the thermal diffusivity matrix take the role of control functions and the distance between the uncloaked and the cloaked field is minimized in a suitable observation domain. The control actions enter bilinearly in the heat equation, thus making the resulting OCP nonlinear, and its analysis nontrivial. We show that optimal diffusivity coefficients exist both for the static and the transient case; we derive a system of first-order necessary optimality conditions; finally, we carry out their numerical approximation using the Finite Element Method. A series of numerical test cases assess the capability of our strategy to tackle passive thermal cloaking of arbitrarily complex two-dimensional objects.