Tailoring specific heat and density in the design of thermal transformation media

The transformation optics or acoustics in the design of metamaterials has been successfully applied to a number of physical phenomena. This novel method provides simple routes to attain certain significant effects such as invisibility cloaks, concentrators, and others. Given a mapping between two geometric configurations, the material properties of the transformed domain can be accordingly determined. They are in general anisotropic, position varying and sometimes of extreme values at certain points. This in fact restricts the practicality of real fabrication. Here we are concerned with the transient behavior of heat conduction. An inverse algorithm is presented here in the design of transformed medium based on the invariance of governing equation between physical and virtual configurations. We will demonstrate that the thermal capacity (thermal conductivity and heat capacity) can be tailored in specific forms, spatially uniform, linear or quadratic functions of positions, so that the design can be materialized more easily. Numerical illustrations are also presented for 2D and 3D thermal cloak for transient thermal conduction.