Controlling the thermal and electric fields in isotropic and anisotropic media

In this study, we theoretically propose cloaking and concentration devices allowing simultaneous control of electric and thermal fields in spherically inhomogeneous layered medium with isotropic and anisotropic layers. The above combination of layers (isotropic and anisotropic ones) is obtained by the transformation coordinate approach applied to a spherically inhomogeneous layered medium which contains isotropic and anisotropic layers. It is shown that in steady-state conditions, both thermal and electric fields can pass smoothly around the targeted area while preventing any disturbance in the surrounding medium. The constitutive parameters of both fields have been determined analytically. In this work, we have combined two different methodologies to achieve cloaking in ideal state and in homogenized structure for cylindrical and spherical cases. Numerical validation of the obtained solutions using COMSOL software is performed in this study.


Introduction
The use of transformation media has attracted significant attention from the scientific community as a tool to achieve invisibility [1,2]. Mathematical methods describing transformations of electromagnetic waves in such media preserves the form invariance of Maxwell's equations under coordinate transformation and enables several manipulation strategies such as concentration, rotation, and absorption [3][4][5][6][7][8][9][10][11][12]. Invisibility cloaking and concentration have been extended to various fields, including electromagnetic, thermal, acoustic, etc [13][14][15][16][17][18][19][20][21][22][23][24]. While earlier proposed cloaks had complex constitutive properties, such as inhomogeneous, anisotropic, and singular, which presented experimental challenges, Han's demonstration showed that thermal cloaking and concentration phenomena can be achieved using natural materials [25]. Researchers have also performed experimental studies on invisibility cloaking for electromagnetic, thermal, plasmonic, and acoustic fields [26][27][28][29][30][31]. Later, the idea of simultaneously modeling two different physical fields to make a hidden object invisible emerged [32]. Effective properties of considered fields were calculated using several techniques such as curve linear transformation, optimization methods, some numerical approaches, etc [33][34][35][36][37][38]. Jiang proposed an intelligent bifunctional device that can switch the cloaking and concentration characteristics by varying temperature [39], and further developments have been made in bifunctional cloaks [40][41][42][43]. Note that in the field of cloaking devices, previous research has primarily focused on either isotropic mediums [44] or anisotropic mediums [37]. In this study, we present a novel approach that combines both types of mediums into a single device, thereby demonstrating the effectiveness of an efficient bifunctional cloak capable of simultaneously propagating electrical and thermal fields around a cloaked region without any distortion.
As electric and thermal fields have wide-ranging applications in our daily lives, such as novel wave concentrators [45], thermal-electric sensors [46], solar systems [47], metamaterial synthesis [48] as well as in medical fields [49], we propose a bilayer structure model that includes both electric and thermal fields. In our study, we investigate the invisibility cloaking and concentration of both electric and thermal fields. To achieve this, we employ a model consisting of two layers, with one layer utilizing an isotropic medium and the other one utilizing an anisotropic medium. Specifically, in our proposed model consisting of two shells, the first shell is composed of an anisotropic material, and the second shell is composed of an isotropic material. The design and analysis of the model has been achieved for the first time using the transformation coordinate approach, applied to a comparable two-layered medium with two isotropic layers. Indeed, combinations of anisotropic and isotropic layers have not been considered previously.
To facilitate prospective implementation, we achieve the homogenization process of the first shell by employing acetone and a glass wool board. The validation of the proposed analytical models is conducted through suitable simulations using COMSOL software.

Methodology
Metamaterials have been used for developing unusual electromagnetic devices performing exotic properties at will [1][2][3][4]. Since we recognize the unique electromagnetic characteristics associated with various physical states such of matter as solids, fluids, and gases, it's important to note that alterations in environmental conditions can lead to changes in the electromagnetic material's physical properties. In this context, the inherent traits of these elements will also play a significant role in the practical implementation of our proposed cloaking model. Additionally, it makes sense to pursue cloaking under steady conditions as a means of addressing this concern more effectively.
In the steady state, the term ¶L ¶t becomes zero, where Λ stands for potential and temperature. Consequently, we are pursuing to simultaneously achieve the steady-state cloaking for both electric and heat fields within a single structure. In other words, our objective is to realize bifunctional cloaking under steady-state conditions. To achieve this, we have employed transformational modeling for both orthogonal coordinate systems (cylindrical and spherical) to enable cloaking and concentration phenomena. The geometrical configuration of the proposed model has been partitioned into four distinct regions.
Consider three circles with radii r , 1 r 2 and r 3 symmetrically enclosed in a rectangle of dimensions  is the electric conductivity of appropriate mediums. The main focus of the cloak is shell 1 and shell 2 being of their anisotropic and isotropic characteristics. Notice that in the previous findings, only a handful of works may have presented a cloak with diverse characteristics across different regions from an anisotropic perspective. Indeed, researchers have designed cloaks in both isotropic and anisotropic media within a single domain [37,44], introducing novel effects, distinct techniques, and more. The duel coupling of isotropic and anisotropic mediums in the bifunctional cloak extend the material range for the scientific community to perform the experimental realization. In the finite element simulations, we have described the electric and thermal effects of the proposed bifunctional cloaking. Here, the proposed cloak is investigated both in cylindrical and spherical coordinate systems. First, we discuss the mathematical modeling of cylindrical cloak.

Mathematical modeling of cylindrical cloak
With only one physical domain, a significant amount of research on cloaking has been undertaken. The phenomenon of bifunctionality has the potential to catalyze a breakthrough in the realm of invisibility cloaking. We are investigating a two-dimensional (2D) framework that is exclusively influenced by electric and thermal fields. The mathematical model governing the dynamics of electric and thermal fields can be formulated in terms of Laplace's equation [16]: where λ represents the electric or thermal conductivity and Λ signifies the absolute temperature or voltage. Within a cylindrical coordinate system, the general solution of equation (1) can be expressed as: and g z i are the constants to be determined by boundary conditions where i stands for different regions. Further, q represents the angle between a given physical quantity's orientation and the direction of the applied field and L i cyl shows the conduction of heat (electricity) in different regions, in such a way that = i 1 shows vacuum r r Applying the variation of index i in equation (2), gives: for the region , 5 for the region . 5  As the inner region is empty (vacuum), therefore the value of l cyl 1 will be zero. The thermal conductivity of shell 2 will be assumed equal to the background one = k k cyl b cyl 2 due to the uniform flow while the electrical conductivity of shell 1 approaches to zero to avoid the electric effects (a dielectric). Solving the above system of equation for thermal conductivity i.e. in above system put l = k . i cyl i cyl As k cyl 3 is the thermal conductivity of background, therefore, we will use 'b' as subscript instead of '3'. Thus, by solving the above system simultaneously, we get r r Equations (9) and (10) depicts that both shells have isotropic properties. Our objective is to study the bifunctional cloak along the two adjacent isotropic and anisotropic shells. As a consequence, we suppose that shell 1 is an anisotropic medium. In this situation, we apply the coordinate transformation to calculate the properties of shell 1 [4]: This transformation leads the conductivity as a diagonal tensor, that is: During the transformation process only the radial component is modified which gives the thermal conductivity of shell 1 as: As a result, the conductivities for both shells are as follows respectively:

Mathematical modeling of spherical cloak
In this section, we will perform the analytical derivation for spherical coordinates. Consider a 2D framework that influenced by thermal and electrical fields. The general spherical solution of the Laplace equation is given as [43]: Where L i sph represents the electric(heat) conduction in the region i, h z i and g z i are the coefficients of the z th order Legendre polynomial G z and can be determined by using the boundary conditions and θ is the angle due to the applied field direction.
Then by varying i in equation (16), we have: L h r f r r = < < ( ) ( ) a cos , for the region 0 , 17  consider the influence of uniform thermal wave propagation within shell 2 and background that is Thus, we obtain from equation (18): Similar to the previous case, it can be demonstrated that shell 1 can exhibit an anisotropic behavior in an alternate coordinate system. For instance, by employing the coordinate transformation described in [1], we can illustrate this phenomenon as: we arrive at the tensor form of conductivity of shell 1: We consider the radial component of the above tensor for thermal and electrical conductivities: As a result, the conductivities for each shell are as follows, respectively:

Results and discussion
We employed the COMSOL MULTIPHYSICS software to simulate our model, focusing on a steady-state situation that encompasses both thermal and electrical fields. To investigate the cloaking and concentration phenomena within the proposed bifunctional cloak, we analyzed the model under three distinct cases: the ideal scenario, the case involving homogenization of anisotropic medium, and the point source excitation.

Cylindrical ideal cloak
Note that all the geometrical and analytical parameters in this study are assumed to be constant unless explicitly stated otherwise. The radii of the three circles are denoted as are r = m 0.008 , 1 r = m 0.01 2 and r = m 0.012 , 3 and the properties of shell 1 and shell 2 are analytically determined as outlined in equations (14) and (15), respectively.
Regarding the source configuration, the left boundary is set as source at 1 V and 373.15 K for the electric and thermal fields, respectively. Conversely, the right boundary is maintained at 0 V and 273. 15  Furthermore, to redirect thermal waves within shell 1 and achieve uniform behavior in shell 2 and the background, k cyl 2 must be equal to k . b cyl Effective source and sink conditions must be established to govern the propagation of electric and heat field waves.
The simulation results for the electric and thermal fields are depicted in figures 2(a) and (b), respectively. The streamlines (horizontal lines) describe the pathways of electric current (heat flux), while the contour lines (vertical lines) portray the distribution of electric potential (absolute temperature).
In figure 2(a), it is evident that the field waves flow uniformly both in the background and shell 2. Both field flows are directed through shell 1 and shell 2, circumventing the vacuum region without any significant agitation or mixing, as elucidated. As a result, we observe a cloaking phenomenon where both current and heat flux are redirected. Additionally, the contour lines amplify the potential and temperature within the vacuum for the electric and thermal fields, respectively.

Homogenization of anisotropic medium for cylindrical cloak
As discussed earlier, the properties of shell 1 exhibit anisotropy due to equation (12). The pertinent parameters are analytically computed in this study. To realize such properties, we have employed a composite material for the homogenization of the thermal material, illustrated in  The ideal case model proposed earlier has been modified, as detailed in the subsection. Following these updates, the simulation results are presented in figure 4. These results vividly illustrate the attainment of the envisaged cloaking and concentration phenomena. Notably, the contour lines demonstrate concentration without dispersion, while the streamlines'wave pattern exemplifies invisibility cloaking.

Point source excitations for cylindrical cloak
To substantiate our theoretical findings, we employ a validation test known as 'point source excitation' within this study. In this test, a point-like source voltage (or temperature) is assignedto a specific location, thereby generating waves propagating from that point. Copper material is employed as the source point in this experiment. As depicted in figure 5, the phenomena of cloaking and concentration are strikingly evident. Notably, a highly efficient cloaking effect is observed without introducing any discernible disturbance or distortion to the wave pattern.
The waves generated by the point source material propagate uniformly throughout the surrounding regions, excluding the cloaking areas. Specifically, the electric flow waves encompassed by the circle of radius R 3 gradually divert through shell 2, circumventing the shell 1. Meanwhile, the thermal waves entering the circle  with radius R 2 are seamlessly guided through shell 1 cloaking the vacuum region. Figures 5(a) and (c) correspond to point source excitations within the electric field, while figures 5(b) and (d) depict the thermal effects. Our observations clearly highlight the effective cloaking and concentration attributes inherent in the proposed bifunctional cloak having anisotropic and isotropic regions.

Ideal state and homogenization of anisotropic medium for spherical cloak
Considering the 2D scenario, the cross-section of our model closely resembles that in both cylindrical and spherical coordinate systems. Consequently, the parameters and boundary conditions remain consistent, encompassing insulation, source, sink, and the identical inner vacuum region. The constitutive properties of shell 1 and shell 2 are defined in equations (23) and (24) respectively, and closely match the characteristics of polyurethane. Following these adjustments, the obtained result is depicted in figure 6.
Firstly, the ideal state of invisibility cloaking and concentration phenomena within the spherical coordinate system is elucidated in figures 6(a) and (c). The stream lines visually convey the cloaking effect, while the behavior of contour lines signifies an enhancement in the potential and temperature within the vacuum region. As intended, the contour lines within the confined area converge towards the vacuum, devoid of any resulting distortion in the surrounding flow.
The properties of shell 1 exhibit anisotropy in the transformed coordinates, as we have previously indicated in the cylindrical case. To homogenize shell 1 and achieve these distinctive attributes, we have utilized composite materials. Specifically, glass wool boards and liquid acetone are incorporated within shell 1. The surface of shell 1 comprises glass wool boards, possessing a thermal conductivity of 0.023 W/(m·K) and an electrical conductivity of 0.01 mS/m. Additionally, acetone-filled ellipses are seamlessly integrated into the surface. These ellipses are filled with liquid acetone, which exhibits a thermal conductivity of 0.16 W/(m·K) and an electrical conductivity of 0.4 mS m −1 [51]. Furthermore, the ellipse is rotated by an angle of 6 degrees. By applying Milton's effective medium theory [52], we deduce the effective thermal conductivity of shell 1 to be:  The ideal scenario model has been refined, and the resulting simulated outcomes are depicted in figures 6(b) and (d). Within these graphs, horizontal black and blue lines illustrate the trajectories of current and heat flow respectively. Notably, these lines traverse linear paths across the background. Meanwhile, the redirection of electric current and heat flux occurs around the inner region without inducing any distortion. The behavior of the vertical lines signifies the distribution of temperature and voltage.

Point source excitations of spherical cloak
In this phase of our research, we also conduct a point source test to validate the numerical simulation results for the case of a spherical cloak. In this test, a source voltage (or temperature) is applied to a point within the material, resulting in the generation of waves. A copper material is utilized as the point source for this test. The effectiveness of cloaking and concentration can be clearly observed using the point source test.We have observed efficient and perfect cloaking with no disruptions or distortions in the wave pattern. The contour lines of the electric and thermal fields are tightly focused within the vacuum region, where the electric and thermal waves originate, and propagate in all directions, as depicted in figure 7. Specifically, figures 7(a) and (c) showcase the point source test in a homogenized anisotropic medium, while figures 7(b) and (d) illustrate the point source test in an ideal case.
The acquired cloaking results from the point source tests are displayed in figure 7. It is evident that the proposed cloaks effectively achieve cloaking and concentration, showcasing satisfactory performance.

Conclusion
We have presented a bifunctional cloak that can simultaneously perform cloaking and concentration for the electric and thermal fields in a single structure having anisotropic and isotropic media.We have performed four kinds of simulations for both ideal state and homogenized structure in cylindrical and spherical coordinate systems. In all the cases, efficient cloaking and concentration phenomenon are observed. We have observed that analytically determined isotropic properties of electric field and anisotropic properties for thermal field are perfectly support to manipulate the waves of both fields under our proposed boundary conditions.
To validate the efficiency of the proposed structure, we have conducted a point source test which showed that the current and heat flux are manipulated at will around the inner region while the electric potential and temperature of the vacuum are concentrated in the inner shell. Overall, the results of our study indicate that the proposed bifunctional cloak is a promising approach for simultaneous manipulation of two physical fields in a single structure having anisotropic and isotropic layers. This study may be extended with multiple layers of anisotropic and isotropic media.Thermal and electrical cloaking offer promising medical applications. In hyperthermia treatment, heat can be precisely concentrated for targeted therapy. Medical imaging quality could improve with reduced interference through electrical cloaking. Neuromodulation, like deep brain stimulation, might benefit from better-targeted electric fields.