Optics Transformation and Homogenization Techniques Applied to the Heat Equation

The metamaterials community has been heavily excited since the publication of two articles by Pendry and Leonhardt in 2006, in which exotic devices, such as invisibility cloaks were proposed to be implemented by space transformation. Indeed, the form invariance of the Maxwell equations allow for an equivalence between a deformed geometry and a material with specific properties. Since then, several experimental studies have shown the feasibility of such transformed devices. The form invariance was also found in other physical domains and the space transformations were applied to many physcial phenomena such acoustic wave propagation, elasto-dynamic wave and surface wave propagation. We present in this work the space transformation applied to the heat equation. Throughout our study, we focus on the transformations leading to thermal invisibility cloaks and thermal concentrators. Those transformed devices are made of anisotropic heterogeneous materials which make them difficult to practically design. Therefore, we make use of the two-scale homogenization theory , allowing to approach the behavior of those devices with an alternate set of isotropic materials. We systematically try to evaluate quantitatively the performance of our approximate devices by defining an effectiveness criterion to achieve high level of mthermal metamaterials engineering. Finally, we present a model of a 50-layer carpet cloak, whose first results are to be expected.