Design and analysis of arbitrary shaped bifunctional cloaks for multifunctional material composites

The conventional cloaks that execute two physical fields in a single structure are often limited in their applications due to their regular geometry. This work proposes a solution to this problem by introducing an irregular shape bifunctional cloak that shields the cloaked region from both electric and thermal fields. Unlike previous cloaks, the proposed cloak can be designed with arbitrary shapes, which greatly expands its potential applications. The cloak is designed using Transformation Optics (TO) theory to derive the constitutive parameters required to execute the cloaking phenomenon. The numerical simulation setup is then devised based on the calculated parameters, and the functionality of the cloak is thoroughly validated. The results show that the proposed cloak can efficiently cloak a conductive or non-conductive material under both plan wave and point source excitation conditions. Moreover, the ability to cloak conductive materials make it an ideal candidate for engineering and medical applications where regular geometries are restricted.

Researchers in the field of cloaking have been working on developing bifunctional cloaks, which are devices that can simultaneously cloak two different physical fields in a single structure [26][27][28][29][30][31][32][33].While there has been progress in developing cloaks for single incident fields, such as arbitrary shape cloaks, there is still a need for a cloak that can combine different functions, including shielding of conductive or non-conductive material, any type of incident source, and cloaking for a number of sources.Li et al showed in 2008 that invisibility cloaking of electromagnetic waves can be possible with irregular geometries [17], and Li's group designed a cloak based on transformation theory that can exhibit invisibility cloaking of the electric and thermal fields simultaneously [27].Further, petal cloaks and twin cloaks with folding geometry are also a valuable addition in this field [22][23][24].Additionally, Zhang et al proposed an arbitrary shape cloak based on the optimization technique to guide the electric and thermal fields as the first attempt to present the arbitrary-shaped bifunctional cloak [31].An electromagnetic cloak has its applications to reduce the unwanted interaction between two biotelemetry applications: An electromagnetic (EM) cloak is used to reduce the magnetic field inhomogeneities produced during the process of magnetic resonance imaging (MRI) due to the presence of metallic hardware in nearby vicinity [34].Electromagnetic cloak is also regarded as one of the potential technique to suppress the induced current by shielding the conducive material from rest of the radio frequency (RF) circuitry in implantable electronic devices and techniques [35].
The proposed work aims to address this gap in the literature by introducing an arbitrary-shaped bifunctional cloak with versatile cloaking properties.Specifically, we use transformation optics to derive the constitutive parameters for guiding the electric and thermal fields around the cloaked region.We employed an analytical method to calculate the constitutive parameters, ensuring a seamless cloaking behavior.In contrast to existing cloaking devices that are either limited to regular shapes, irregular shape relying on approximation theories, or exhibit a single physical functionality, our approach offers a more comprehensive solution 32].The proposed cloak is capable of shielding both conductive and non-conductive materials from any type of electric and thermal fields in steady-state situations.In addition, the proposed cloak can operate under a variety of excitation sources, making it a potential candidate for practical applications in defense and biomedical fields.
The remainder of the paper is organized as follows: section 2 describes the required parameters for the bifunctional cloak, while section 3 presents the numerical simulations based on the derived parameters.Section 3.1 demonstrates the functionality of the arbitrary-shaped cloak, while section 3.2 shows its capability for shielding conductive or non-conductive materials.Section 4 presents the operation of the cloak for different types of excitation sources.Finally, the paper concludes in section 5 with a summary of the main contributions and future research directions.The proposed cloak having capability to route the EM and thermal waves has potential to alleviate the heating effects in the cloaked region required in most of the biotelemetry applications like MRI or implantable devices etc [36].

Coordinate transformation
The design of the bifunctional cloak's structure is the first step, which consists of four regions with arbitrary shapes, divided by three contours with a period of 2π.These regions are named as inner region, shell 1, shell 2, and background, with shell 1 and shell 2 used to manipulate the thermal and electrical fields, respectively.
To develop the mathematical model for arbitrary shaped bifunctional cloak, the conduction equation governing the time-independent system is considered [28], where f represents the thermal (electric) conductivity and F denote the temperature (voltage).However, it is challenging to formulate the mathematical model in a rectangular coordinate system due to the complexity of the cloak's shape.Therefore, the cylindrical coordinate transformation in the cartesian plane is used to simplify the process.The coordinate transformations for shell 1 and shell 2 are given for the contours ( ) q R , 1 ( ) q R 2 and ( ) q R 3 [32]   ( ) ( ) ( ) ( ) ( ) respectively.Guenneau used equations (2) and (3) to attain the cloaking and concentration states of thermal field, respectively.Here, the idea of arbitrary shape cloak for single functionality is extended to bifunctionality, in the bilayer arbitrary structure.In the context of transformation, the thermal and electrical conductivities of both shells are determined as: where A is the Jacobian matrix, and A T is the transpose of the Jacobian matrix and det(A) represents the determinant of the Jacobian matrix A. The symbol f represents the conductivity of the virtual space and ψ represents the conductivity of physical space.The transformed conductivities ψ will keep the transformed system invariant.First, we find the thermal conductivity of shell 1 by using equations (2) and (4).Suppose the Jacobian matrix for the transformation given in equation ( 2) is denoted by A 1 as: Similarly, to find the electric conductivity of shell 2, we use equations ( 3) and (4), then its Jacobian matrix A 2 is given as: Now, by equation (4) we get the electrical conductivity of shell 2 as: The necessary physical conditions are employed to achieve the desired outcomes.It is limited so that only shell 2 can guide electrical waves, meaning shell 1's electrical conductivity must be close to zero to prevent electrical waves from entering shell 1.Similarly, the manipulation of thermal waves is constrained to shell 1, and to ensure a uniform flow in shell 2 and the background, the thermal conductivity of both regions must be equal with the thermal conductivity of shell 2 denoted by the symbol

Numerical simulation
The extracted thermal and electric parameters in the last section are deployed to design the region 2 and 3, as shown in figure 1, to guide the excited thermal and electric fields, respectively.Overall, the proposed model consists of four regions containing arbitrary shapes as shown in figure 1(b).COMSOL Multiphysics software is used to design and demonstrate the functionality of the proposed cloak.To design an arbitrary cloak three contour equations 9(a)-9c(c) are considered [33] with independent variable theta having step size pi/50.These contours divided the considered rectangle of area 1.05 × 0.7 m 2 into four regions.The inner region consists of the vacuum, the electrical conductivity s 0 and thermal conductivity k 0 are zero i.e. s = = k 0. 0 0 The analytical conductivities of shell 1 and shell 2 are given in equations (7) and (8) respectively.Whereas, for the background, the numerical value of thermal conductivity is considered as The left side of figure 1 is assigned as the source of the respective excited field and the right side is assigned as the sink.The source and sink temperatures are 373.15K and 273.15 K, respectively.Similarly, the source voltage is 1 V and sink voltage is 0 V.The upper and lower boundaries are thermally and electrically insulated.A material loamy sand having thermal conductivity range 0.19 W/(mK)-1.13W/(mK) and electrical conductivity range 0.01 S m −1 -0.03 S m −1 , is suitable according to the required conductivities of the backgrounds [37,38].It is observed that the orientation of the thermal field lines is same before entering and after emerging from the, overall, cloaked geometry (all the four regions).Which demonstrates the functionality of the proposed cloak for the thermal fields.Moreover, within the cloaked region (shell 1), no reflections of thermal fields were observed.All the thermal field entered the shell 1 surround the arbitrary inner region smoothly and propagate through the shell 1 towards the sink without entering or interacting the inner region, the cloaked region.The thermal field lines entered the shell 1are guided around the inner region by the matched thermal conductivity of shell 1, which was calculated and assigned the shell1 to guide the thermal field.So, excited thermal field guided around the inner region in the shell 1 and emerge from the other side towards the sink.Thus, resultant field at the sink side did not have any effects, distortion etc of the cloaked region.
Similarly, the behavior of arbitrary cloak for the electric field is shown in figure 2(b).The shell 2 is assigned to guide the electric field instead of shell 1, so cloaked region for the electric field is larger than the thermal field as it includes shell 1 and inner region.It is noticed that the excited electric field lines, entering for the left side (the source side), are guided smoothly around the cloaked region and transmits towards the sink without any distortion and reflection.The orientation of the electric field lines in the background region around the cloaked region is constant.So, an electric receiver on the right side (the sink side) cannot detect any distortion of the electric field, which again, reinforces the functionality of the proposed arbitrary shape cloak for the electric field as well.
When the electric field lines enter the shell 2, the assigned calculated and matched electric conductivity guides the field lines around the cloaked region and zero reflections of the electric field lines from the cloaked region are observed.The calculated electric conductivity bounds and guides the electric field lines within the assigned region and allows negligible interaction with the other geometry.In this way, it guides the travelling electric fields around the arbitrary shape and helps to propagate out of the cloaked region towards the sink with minimum distortion and in same orientation as it entered the cloaked region.Which demonstrates the accuracy of the calculated analytical electrical conductivities for the arbitrary region.
However, negligible distortions of the electric field lines at fewer locations of the boundary of shell 2 are observed, as highlighted in figure 3. It is noticed that these are the regions where roughness of the arbitrary structure is relatively high.The constituted contours are the functions of the same variable, and step size during the numerical simulation is also same in all the contours.So, due to the difference in circumference of arbitrary regions the smoothness of the geometry varies, circumference is directly proportional to the smoothness.Since circumference of ( ) q R 3 consists of shell 1 and inner region so it has maximum robustness at certain locations where the electric field is reported distorted.It is further analyzed and verified that for the smooth geometries like circle, rectangle, ellipse etc, the same electric field propagating in same calculated conductivities has zero distortion as shown in figures 4 (a)-(c).
In figure 4 (a), a circular shape cloak is designed by applying equations (7) and (8).The excited electric field and value of electrical conductivity are remained same as of figure 2 (b).The behavior of the cloaked region towards the excited electric field is reported ideal.No distorted electric field is reported around the cloaked region and orientation of electric field around the cloaked region is same.Similarly, elliptical and rectangular geometries are considered in figures 4(b) and (c), respectively.Here, again, like figure 4 (a), no distortion of electric field lines is observed around the cloaked region in shell 2. Which demonstrates that smoother the surface, less will be distortion.For the brevity, in the simulation model, the reduction in step size (pi/50) of the contour parameter will result in smooth path and consequently the less distortion.It is analyzed that the proposed arbitrary bifunctional cloak, even with step size of pi/50, has the capability to cloak the inner region with negligible distortion of electric field.As the calculated matched conductivities guide the received fields in the respective assigned bounded regions, i.e, shell 1 for the thermal field and shell 2 for the electric field, the excited field has minimal chances of interacting with the conductive material.Consequently, the excited fields pass through the cloaked region, where matched conductivities guide the received fields, smoothly around the conductive material, maintaining their orientation as they emerge towards the sink, just as they were upon entering the cloaked region.Therefore, it is confirmed that the proposed bifunctional arbitrary cloak efficiently conceals both conductive and nonconductive materials.Additionally, the proposed cloak's effectiveness is independent of the cloaked material.
Realizing this work in an experimental setting is relatively straightforward due to the availability of the background material, which exhibits a wide range of electrical and thermal conductivities as mentioned earlier.Moreover, identifying suitable materials for Shell 1 and Shell 2 is facilitated by the precise nature of the derived parameters.Additionally, it is explicitly explained that modifying the structure automatically yields the corresponding constitutive parameters.Consequently, this feature simplifies the process of selecting appropriate materials for both Shell 1 and Shell 2 during experimentation, regardless of the geometry under consideration.

Point source excitations
It is also analyzed that the proposed arbitrary bifunctional cloak is equally functional for the other excitation sources besides the plane waves.Point source excitation is deployed to verify the numerical and simulated results presented in above sections.All the parameters and geometries are remained same as in previous sections, except the source.The simulated results of the proposed model in the presence of point source are shown in figure 6.The figures 6(a) and (b) report the functionality of the proposed cloak for the thermal and electric point source excitation conditions.It is evident that due to the calculated matched conductivities, the field lines passed around the cloaked region smoothly, showing zero distortion within the respective cloaked region and having constant orientation around the cloaked region in the background.It is reported that the response of the bifunctional arbitrary shape cloak is independent of the excited field.

Conclusion
In conclusion, this research work presents a bifunctional cloak capable of cloaking electric and thermal fields for irregular structures.The analytical calculations of effective parameters for the physical regions involved in cloaking the excited fields are achieved using transformation theory.The numerical results obtained demonstrate the effectiveness of the proposed cloak for arbitrary geometries, with no disturbance or distortion of wave patterns observed even in the presence of conductive or non-conductive materials.The cloak is also source independent, making it suitable for a wide range of applications, including engineering, biomedical, wearable technology, and shielding, where cloaking of electric and thermal fields in arbitrary geometries is required.The proposed bi-functional cloak has applications in biotelemetry applications like MRI, implant, wearable etc to shield the required (cloaked) region from unwanted EM waves and excessive heating effects.

3. 1 .
Results and discussionsThe simulated results of the proposed bifunctional arbitrary cloak are shown in figure 2. The figure 2(a) shows the behavior of cloak for the thermal fields.The thermal field is represented by the symbolic horizontal lines flowing from source (left) to sink (right).

Figure 1 .
Figure 1.(a) This diagram represents the design of the device.In which three contours R 1 (ϴ), R 2 (ϴ) and R 3 (ϴ) are considered, the upper and lower boundaries of the device are insulated while the left boundary has potential 1[V] and temperature is 373.15[K].The right boundary behaves like a sink boundary i.e. it has 0[V] and 273.15[K].(b) The arrows demonstrate the direction of heat flux and electric current.Each region is labelled by a name with respect to its specification.

Figure 2 .
Figure 2. (a) This figure is the result of the thermal field in which horizontal lines show the behaviour of the heat flux.Since no line enters the inner region, these lines are manipulating around the vacuum and come out at their original path.This phenomenon shows that the object becomes cloaked by the thermal field.While effects of the electric field are given in (b), the horizontal lines represent the behaviour of electric current.

3. 2 .
Shielding of conductive materialNext, the functionality and effectiveness of the proposed bifunctional arbitrary cloak to shield the conductive material placed inside the inner region are verified.A highly conductive material, 'Silicon Carbide,' is chosen and placed inside the inner region of the proposed cloak to observe the cloaking effects.The electric conductivity of the material is / s = e S m 1 3 and the thermal conductivity is( ) / = k W mK 450.An elliptical-shaped silicon dioxide material is simulated inside the inner region with a semi-major axis of 25 mm and a semi-minor axis of 18 mm.The results of the simulated structure containing the highly conductive material are shown in figure5.Figures5(a) and (b) display the simulation results of thermal and electrical cloaking of the conductive material, respectively.It is observed that the respective field lines pass around the cloaked region without distortion, and the orientation of the field around the cloak (before entering and after emerging from the cloaked region) remains the same.

Figure 3 .
Figure 3.The distorted area of the electric field is zoomed.

Figure 4 .
Figure 4.The simulation of our proposed model with some geometrical structures of well-defined shapes.

Figure 5 .
Figure 5. Simulations of the arbitrary cloak in the presence of highly conductive material (silicon carbide).The horizontal lines represent that the existence of silicon carbide doesn't influence the invisibility cloaking behaviour of (a) thermal and (b) electric fields.This figure presents that the object is perfectly protected from electric and thermal rays.

Figure 6 .
Figure 6.Schematic graphs of normalised electric and thermal fields.Point source test is used to verify the performance of the proposed cloaking device.Graphs (a) and (b) represent the performance of the electric and thermal fields respectively, in the point source test.The good perfection of cloaking performance in the point source test is verifying the proposed bifunctional cloaking device.