Fast active thermal cloaking through PDE-constrained optimization and reduced-order modelling

Abstract

In this paper, we show how to efficiently achieve thermal cloaking from a computational standpoint in several virtual scenarios by controlling a distribution of active heat sources. We frame this problem in the setting of PDE-constrained optimization, where the reference field is the solution of the time-dependent heat equation in the absence of the object to cloak. The optimal control problem then aims at actuating the space–time control field so that the thermal field outside the obstacle is indistinguishable from the reference field. In particular, we consider multiple scenarios where material’s thermal diffusivity, source intensity and obstacle’s temperature are allowed to vary within a user-defined range. To tackle the thermal cloaking problem in a rapid and reliable way, we rely on a parametrized reduced order model built through the reduced basis method, thus entailing huge computational speedups compared with a high-fidelity, full-order model exploiting the finite-element method while dealing both with complex target shapes and disconnected control domains.

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

“…[21,25,26]). While the analysis carried out in [17] involved a linear OCP in which the distributed control played the role of an additive source, now we turn to the problem of optimizing the material properties-that is, acting on the state operator-to get the desired cloaking.…”

“…Achieving thermal cloaking through PDE-constrained optimization allows to ensure a remarkable flexibility regarding the shape of the obstacle to hide as well as how the cloaking sources are placed. An optimization formulation is combined with a reduced order model to improve performance in [17]. In particular, an active thermal cloak can be modelled by inserting a set of additional control variables into the state heat equation: these act as external sources or forces and do not change the intrinsic properties of the material.…”

Active exterior cloaking for the two-dimensional Helmholtz equation with complex wavenumbers and application to thermal cloaking

“…There should be a trade-off between using a single dipole source and numerous monopole and dipole sources to achieve efficient thermal cloaking in the transient regime, which is what motivated the present work. Numerical optimization techniques can also be used to achieve active exterior cloaking for the heat equation [57]. Our analysis is performed in the frequency regime, where we can extend results of [6] to the Helmholtz equation with complex wavenumbers.…”

Active exterior cloaking and mimicking for the heat equation

“…Our approach, when performed in the frequency domain, relies upon a Helmholtz equation with a complex wavenumber, also valid for wave problems in dispersive media. Our proposal is a source of inspiration for other groups [9].…”