Transformation thermodynamics: cloaking and concentrating heat flux

We adapt tools of transformation optics, governed by a (elliptic) wave equation, to thermodynamics, governed by the (parabolic) heat equation. We apply this new concept to an invibility cloak in order to thermally protect a region (a dead core) and to a concentrator to focus heat flux in a small region. We finally propose a multilayered cloak consisting of 20 homogeneous concentric layers with a piecewise constant isotropic diffusivity working over a finite time interval (homogenization approach).