Résumé
We analyse basic thermal cloaks designed via different geometric transforms applied to thermal cloaking. We evaluate quantitatively the effectiveness of these heterogeneous anisotropic thermal cloaks through the calculation of the standard deviation of the isotherms. The study addresses the frequency regime and we point out the cloak's spectral effectiveness. We find that all these cloaks have comparable effectiveness irrespective of whether or not they have singular conductivity at their inner boundary. However, approximate cloaking with multi-layered cloak critically depends upon the homogenization algorithm and it is shown that the standard deviation varies linearly with the inverse of the number of layers. I n 2006, Pendry, Schurig and Smith 1 showed that upon a transformation applied to a certain region of space, one could obtain a phenomenon of optical invisibility via highly heterogeneous and anisotropic meta-materials whose permittivity and permeability directly result from the space transformation. Leonhardt proposed a similar route to cloaking based upon a conformal map preserving right angles, which makes the required meta-materials isotropic 2 , but this only works for a certain frequency range. Following these works, similar techniques have been used to generalize these concepts to acoustics and mechanics. In 2012, some of us proposed to extend concepts of cloaking to diffusion processes, such as heat fluxes 3 , which led to first experimental demonstrations 4–7 in the transient regime. Interestingly, cloaking can be also applied to mass diffusion in a very similar manner 8 (this may open a new range of biomedical and chemical engineering 9 applications as well as light diffusion 10). This paper is focused on the analysis of the cloaking effectiveness in the case of heat diffusion. Indeed numerous studies point out the solution of ideal cloaking but fewer evaluate in a quantitative way the resulting effectiveness after the homogenization procedure. To reach this goal the effectiveness is here characterized by a root-mean square calculated on the line fluxes, a value which would be zero in the case of a homogeneous medium. Such characteristics allow us to compare different homogenized cloaks and emphasize their advantages and weaknesses. Moreover, rather than analyzing the effectiveness at each time step, we prefer to perform a spectral analysis, on the basis of the classical time-frequency analogy. Hence we work in the frequency or time harmonic regime and investigate the spectral effectiveness of various homogenized cloaks. Such harmonic study can be implemented if one consider a heat source generated by a modulated laser beam for instance. We first investigate both ideal (singular and non singular) cloaks and approximate cloaks. Then, we improve earlier attempts to design multi-layered thermal cloaks 3,11 using the two-scale convergence method introduced in ref. 12 and further developed by ref. 13. We show that our improved design reduces by at least one order of magnitude the standard deviation of thermal isovalues compared to earlier designs. Potential applications are in thermal metamaterials and the management of heat flux. Results Quantitative analysis of different ideal heat cloak effectiveness Space transformation and singularities. Changes from polar coordinates (r, h) to radially stretched polar coordinates (r9, h9) 5 (f(r), h) applied to the heat equation (19) gives us the transformed conductivity and transformed product of heat capacity and density