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Engineering the accurate distortion of an object's temperature-distribution signature

It is up to now a challenge to control the conduction of heat. Here we develop a method to distort the temperature distribution signature of an object at will. As a result, the object accurately exhibits the same temperature distribution signature as another object that is predetermined, but actually does not exist in the system. Our finite element simulations confirm the desired effect for different objects with various geometries and compositions. The underlying mechanism lies in the effects of thermal metamaterials designed by using this method. Our work is of value for applications in thermal engineering.


I. INTRODUCTION
One can see an object because of its specific optical scattering pattern. Then, if one adjusts Object A's optical scattering pattern by using some peculiar methods, the adjusted scattering pattern causes him/her to "see" Object B. Since Object B actually does not exist, this phenomenon produces an optical illusion [1][2][3]. So far, the concept of optical illusion has been extended from optics [1][2][3], to electromagnetics [4][5][6][7], and to acoustics [8] where wave equations dominate. To empower unprecedent applications in thermal engineering, here we attempt to reveal the counterpart of optical illusions in the field of heat conduction (Fig. 1) where diffusion equations play a key role instead. We call this counterpart as "thermal illusions" (Fig. 1). However, which method can be used for achieving thermal illusions? One of the candidate methods is the coordinate transformation approach.
In 2008, one of us [9] extended the coordinate transformation approach to the field of heat conduction, and proposed a class of thermal metamaterials with novel thermal properties that cannot be found in nature or chemical compound. This approach provides a way to steer heat flow and change temperature distribution patterns almost at will. As a result, a lot of thermal metamaterials with novel thermal properties have come to appear, such as cloaks (which are used to let heat flow around an object as if the object does not exist) [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23], concentrators (which are used to concentrate heat into a specific region) [11,12], inverters (which are used to apparently let heat flow from the region of low temperature to the region of high temperature) [9,24], and rotators (which are used to rotate the flow of heat as if it comes from a different angle) [12,24].
In this work, we shall develop the coordinate transformation approach for heat conduction [9,10], and propose a kind of illusion device made of thermal metamaterials, which causes an object to apparently possess the same temperature distribution pattern as another object, thus producing the so-called thermal illusion (Fig. 1). Our finite element simulations in two dimensions confirm the desired effect for different objects with various geometries and compositions. This kind of illusion device paves a different way for controlling heat conduction.  Fig. 1(c) is same as Fig. 1(a), but we add an illusion device.
As a result, with the help of the illusion device, the pencil outside the illusion device in Fig. 1(c) apparently has the same temperature distribution pattern as the key in Fig. 1(b), thus yielding the thermal illusion. Clearly, the illusion device is the key to obtain this kind of thermal illusion. But, how to design the illusion device? We offer relevant details in Fig. 2. Fig. 2 just shows the illusion device occupying Regions 2-3 embedded in a background of thermal conductivity κ 0 , which can be used to produce the desired thermal illusion: the medium of κ 1 occupying Region 1 apparently produces the same temperature distribution pattern as the medium of κ 4 occupying Region 4. Here Regions 1-2 both have a shape of trapezoid, Region 3 contains one rectangle and two triangles, and Region 4 is in the shape of hexagon, which has the same shape, size and location as the total area of Regions 1-3. For achieving this kind of thermal illusion, the medium of κ 2 (complementary medium) occupying Region 2 thermally cancels the medium occupying Region 1, and the medium of κ 3 (restoring medium) occupying Region 3 thermally takes place of the medium occupying Region 4. Then, we are in a position to obtain κ 2 and κ 3 , in order to obtain the illusion device.
To proceed, without loss of generality, we start by considering the heat conduction equation for the steady state where κ is the thermal conductivity and T is the temperature. Since Eq. (1) remains form invariant under coordinate transformations [25], we can then directly apply the coordinate transformation approach to Eq. (1) [9]. As a result, the new thermal conductivity κ ′ in the transformed coordinates satisfies the following relation [9,10], where J is the Jacobian transformation matrix between the original and distorted coordinates, J t is the transposed matrix of J, and det(J) is the determinant of J.
As shown in Fig. 2(a), Region 2 should be used to thermally cancel Region 1, which can be achieved by folding the geometry of Region 1 into Region 2. So, according to Eq. (2), the thermal conductivity of Region 2, κ 2 , is given by Here, J 12 is the Jacobian transformation matrix which is determined by the coordinates transfor-mation between Region 1 [with coordinates (x 1 , y 1 )] and Region 2 [with coordinates (x 2 , y 2 )], On the other hand, since Region 3 of Fig. 2(a) is used to replace the whole area corresponding to Region 4 of Fig. 2(b), according to Eq. (2), the thermal conductivity of Region 3, κ 3 , is given by where J 43 is the Jacobian transformation matrix determined by the coordinates transformation between Region 3 [with coordinates (x 3 , y 3 )] and Region 4 [with coordinates (x 4 , y 4 )],

III. SIMULATION RESULTS
In order to show the desired thermal illusion convincingly, we are in a position to perform finite element simulations (based on commercial software COMSOL Multiphysics).
For simulations, we set the following coordinates transformation between Region 1 (x 1 , y 1 ) and Region 2 (x 2 , y 2 ), Then, the substitution of Eqs.
for the upper obtuse triangle of Region 3, for the lower obtuse triangle of Region 3, for the rectangle of Region 3.
So, plugging Eqs. (9)-(12) into Eqs. (5)-(6) gives κ 3 . Fig. 3 shows our simulation results; in each panel of Fig. 3, the temperature at the left and right boundary is respectively set to be 400 and 300 K. In Fig. 3(a), Region 1 within the background of κ 0 = 40 W/ (m·K) is fully occupied by Object A of κ A = 1 W/(m· K) in the shape of trapezoid; In Fig. 3(b), Region 4 within the same background is fully occupied by Object B of κ B = 400 W/(m· K) in the shape of hexagon; Fig. 3(c) is same as Fig. 3(a), but involves an illusion device that occupies Regions 2-3. Clearly, Fig. 3(b) and Fig. 3(c) show the same temperature distribution pattern outside either Region 4 of Fig. 3(b) or Regions 1-3 of Fig. 3(c), thus yielding the desired thermal illusion.
Then, we discuss the case of different shapes and compositions. Fig. 3(d-f) is same as Fig. 3 Fig. 3(k). In other words, the illusion device proposed in this work can be used to get thermal illusions not only for the whole area of Object A [ Fig. 3(c,f,i)], but also for a part of Object A [ Fig. 3(l)].
According to Fig. 3, we may turn to the conclusion that this illusion device is capable of realizing various kinds of thermal illusion, which are independent of the shape and composition of Object A and Object B. But, what is the underlying mechanism? As mentioned in Fig. 2, this mechanism lies in the complementary and restoring effects of thermal metamaterials. In particular, thermal conductivities of these thermal metamaterials should be not only anisotropic, but also negative, as implied by Eqs. (3) and (5). Since a negative thermal conductivity of a material means that heat flows from the region of low temperature to the region of high temperature, in reality one must apply an external work to get the negative thermal conductivity as required by the second law of thermodynamics. In fact, an electric refrigerator is just a sort of "material" whose effective thermal conductivity is negative because heat is brought from the interior of the refrigerator (namely, the region of low temperature) to the exterior of the refrigerator (i.e., the region of high temperature) due to the input of electric power. On the same footing, for experimental demonstration of our proposal about thermal illusions, Peltier effects [26,27] may help to get negative values of thermal conductivities.

IV. CONCLUSIONS AND DISCUSSION
By using the coordinate transformation approach for heat flow [9,10]    In (j), the thermal wall just serves as Object A, which contains Region 1. (k) shows Region 4,