Fig. 2
Three steps to design the camouflaging device. Starting from a pure background (blue coloured) with conductivity κ0, a Cartesian system (x,y) is constructed with its origin O on the background surface. Each step is represented with an arrow of a particular colour. The lines with the same colour are the deformed boundaries of that step. a, b Step 1: space creation. Stretch a small region of height δ (from x0 to x1) at the surface of the background to form a wedge (shaded with grey lines). The resulting conductivity κ′ is dependent on κ0. b, c Step 2: perform the transformation of a unidirectional thermal cloak (shaded with pink and cerulean lines) on the created space to place the object (yellow coloured). The transformation keeps Region A unchanged, and compresses Region A1 and A4 to Region B, Region A2 and A3 to Region C. The resulting conductivity κ′′ is dependent on κ′, thus also dependent on κ0. c, d Step 3: space elimination. Take the limit of δ→0 to eliminate the region used in Step 1 and avoid modification to the background. The conductivity of the device κ′ and κ′′ are now κ0-independent under this operation. The limit taking is indicated with a colour change for the lines in Region A, B and C from light to dark. e We also truncate the bottom to allow contact between the object and the background. The hatched lines represent the orientations of the anisotropic thermal conductivity
