Simultaneous Thermal–Electrical Cloak and Camouflage Via Level-Set-Based Topology Optimization Free

Devices capable of concealing objects from various detection techniques have attracted significant interest since the pioneering work of Pendry et al. [1] and Leonhardt [2]. The original concept of optical cloaking has been extended into multiple fields, including thermal [3,4], electrical [5], acoustic [6], and mechanical cloaks [7]. These engineered cloaks have a wide range of practical applications, such as guiding heat flux away from heat-sensitive devices to extend their lifespan [8] and cloaking sensors without affecting their ability to measure incoming signals [9].

The coordinate transformation [10,11] and scattering cancelation [4,12] are two commonly employed methods to design such metadevices. The essence of coordinate transformation lies in the fact that to maintain the invariance of the domain equation, the relevant physical properties, e.g., thermal or electrical conductivity tensor, have to be modified properly. As a consequence, the metadevices obtained from the above methods often exhibit high material anisotropy and inhomogeneity, making the physical realization quite difficult. One remedy to this issue is to employ layered structures [13,14] or composites with the aid of an effective medium theory [15,16]. Nevertheless, it is, if not impossible, quite challenging to not lose any accuracy. We briefly mention some notable research endeavors using the above methods. Imran et al. [14] and Dede et al. [17] investigated the heat flow control devices taking into account the convection effect. Vemuri et al. [18] succeeded in guiding the conductive heat flux by properly positioning and orienting the nominally isotropic material. An ultrathin but nearly perfect direct current electric cloak was derived in Ref. [19]. An intelligent thermal metadevice was developed in Ref. [20], which can function as a cloak or concentrator depending on external stimulus. Han et al. [5] devised a bilayer cloak using bulk natural materials to manipulate DC currents. The research studies in Refs. [4,12,21–23] placed their emphasis on experimental validation of thermal and electrical metadevices with various functionalities. Interested readers are referred to Ref. [24] for a comprehensive review of metamaterials carrying various cloaking functionalities.

The majority of existing research on manipulating diffusive or wave systems concerns only one single physics, be it optical or thermal phenomena. Recently, there has been an emerging interest in going one step further to achieve multiphysics cloaking functionalities. Moccia et al. [25] realized the independent manipulation of heat and electrical current, pushing one step further toward the “transformation multiphysics.” By combining coordinate transformation together with effective medium theory, a bunch of thermal-electric bifunctional metadevices was developed in Refs. [15,16,26,27]. Zhang and Shi [28] focused on the cloak with arbitrary shape for thermal and electric manipulations using scattering cancelation. Based on transformation optics, a cloaking of thermoelectric transport was presented in Ref. [29] coupling the thermal and electric flow using the Seebeck effect. A metasurface carpet was given in Ref. [30] to simultaneously cloak electromagnetic, acoustic, and water waves, which are all governed by the same Helmholtz equations. Interested readers are referred to Ref. [31] for a complete review of recent advances in manipulating thermal and DC fields.

As a powerful tool to find the optimal material layout to maximize a certain performance, topology optimization stands as an effective alternative method to design various thermal and electrical manipulation devices. Some other quantative methods can also be applied towards various optimization problems [32–37]. Peralta et al. [38,39] employed the density-based optimization method to design a thermal concentrator without disturbing the external reference temperature field. A number of papers [3,40,41] dealt with the topology optimization of thermal and electrical metadevices with a level-set representation using the covariance matrix adaptation evolution strategy to search for optimal designs. Xu et al. [42,43] systematically investigated the topology optimization of thermal cloaks in Euclidean spaces and manifolds using an extended level-set method, marking the first conformal thermal cloak on a freeform surface. Seo et al. [44] proposed a multiscale topology optimization method using only a single variable for heat flux control. Sha et al. [45] developed an illusion device using the density-based topology optimization, where the heat source is camouflaged while keeping the exterior temperature field unaltered. Recent work from Sha et al. [46] combined the transformation thermotics and topology optimization to successfully design thermal metadevices with omnidirectional thermal functionalities. Zhu et al. [47] applied the multiphysics transformation and resorted to density-based topology optimization to realize the local highly anisotropic thermal and electrical conductivity. Recently, data-driven approaches have emerged as an effective alternative to design various thermal metamaterials [48,49]. They also have been employed for other application scenarios, e.g., thermoelectric device design [50,51], and human–machine interaction applications [52–55].

This study aims to design a bifunctional thermal and electrical cloak using level-set-based topology optimization [56]. With the full capability to manipulate thermal and electrical flows simultaneously, several potential applications can be envisioned. For instance, the figure of merit of the thermoelectric materials could be improved greatly to enhance the conversion efficiency [25,29,57]. A thermal-electric “Janus” device can be expected that cloaks heat flow in one direction while concentrating electrical flow in an orthogonal direction [23,58]. The objective function will be the least-square error of the temperature and electrical potential fields from the current design with predefined reference fields in a specific evaluation domain. Two bulk materials with distinct thermal and electrical conductivity, i.e., iron and aluminum, will be properly distributed in the design domain to minimize the temperature perturbation in the evaluation domain. The clear boundaries between different material phases make it easy to generate 3D models and conduct physical experiments. The optimized iron/aluminum configuration exhibits thermal and electrical cloaking functionality simultaneously. The robustness of the proposed method is demonstrated against different inflow direction combinations of thermal and electrical conduction (parallel and orthogonal). The dependency of the final optimized configuration on the initial designs is also briefly investigated. Another contribution is that, building on a similar formulation, we successfully developed a thermal–electrical camouflage device, where a sensor becomes invisible yet still functions to receive undisturbed temperature or electrical potential signals. This example demonstrates the flexibility and strength of the proposed method in effectively manipulating multiphysics fields.

This article is organized as follows: Sec. 2 will provide the problem formulation and shape sensitivity analysis. Numerical examples containing various cloaking and camouflaging designs will be presented in Sec. 3. Section 4 will contain some discussions with closing remarks.

The normal velocity $Vn$ can be derived from shape sensitivity analysis using a typical adjoint method. Once $Vn$ is obtained, it will be plugged into Eq. (5) to solve for the new $Φ$⁠, thus driving the boundary evolution. The detailed shape sensitivity analysis is provided in Sec. 2.2.

As a gradient-based optimization method, the shape sensitivity analysis [60,61] needs to be conducted to provide the necessary information to guide the design update iteratively. Under the level-set framework, the ultimate goal of shape sensitivity analysis is to obtain the normal velocity field $Vn$⁠, which will be plugged into Eq. (5) to solve for a new level-set function $Φ$ and thus a better design. In this study, we employ the material derivative method [63] the and adjoint method [64] to derive the shape derivative.

In this section, several numerical examples are provided to demonstrate the feasibility of the proposed method. Specifically, the first two examples deal with bifunctional thermal–electrical cloaks with parallel or orthogonal inflow directions. The dependency of optimized cloaks on initial configurations is also briefly investigated. The third example is concerned with a thermal–electrical camouflage, where a sensor is made invisible yet still sensing. To better visualize the temperature and electrical field comparisons, the temperature and electrical potential fields are normalized in the following manner: $Tin=(Ti−Tl)/(Th−Tl)$⁠, where $Ti=T,T0$⁠, or $Tref$⁠. $Vin=(Vi−Vl)/(Vh−Vl)$⁠, where $Vi=V,V0$⁠, or $Vref$⁠. The hot and cold ends are 10 K and 0 K, respectively. The high and low voltages are 10 V and 0 V, respectively.

This section and the next section are concerned with bifunctional thermal–electrical cloaks. For these examples, $Ωout$ is fixed with iron. $Ωins$ is occupied with an insulator. An optimal iron/aluminum layout is sought later in the design domain $ΩD$⁠. The material properties of iron and aluminum can be seen in Table 1.

The first numerical example deals with a bifunctional thermal and electrical cloak sharing the same inflow directions as exactly depicted in Fig. 1. As can be seen from Fig. 3, left column, the reference temperature and electrical fields with uniform gradient are generated by filling both $ΩD$ and $Ωins$ with iron. The white lines represent the temperature and potential contours. When an insulator is introduced into $Ωins$ shown in Fig. 3, right column, the temperature and potential fields in $Ωout$ are disturbed with nonparallel contour lines. If a thermal camera and a voltmeter are placed inside $Ωout$ monitoring the temperature and voltage changes, respectively, the presence of the insulator would be detected, meaning it is not cloaked. To mitigate these perturbations in $Ωout$ and cloak the insulator in $Ωins$⁠, an optimal iron and aluminum layout in the design domain $ΩD$ is sought later via the level-set-based topology optimization scheme as detailed in Sec. 2.

The optimized iron–aluminum layout in the design domain $ΩD$ is shown in Fig. 4. A clear boundary between iron and aluminum is maintained. The optimized cloak only employs bulk natural materials and does not exhibit material inhomogeneity and anisotropy as commonly observed from the transformation thermotics method [10,11]. It is worth mentioning that the optimized thermal–electrical cloak structure is close to the pure thermal cloak structure obtained in Refs. [42,43]. This can be attributed to the following two facts. First, pure heat conduction and electrical conduction are both governed by Laplace-type equations with similar boundary conditions as detailed in Sec. 2.1. Second, the thermal conductivity ratio of aluminum to iron is 3.54, which is close to the electrical conductivity ratio of 3.81. Starting with the same initial guess, the optimal bifunctional thermal–electrical cloak result should not go far from that of the pure thermal cloak result. The ersatz material model is adopted in the optimization formulation. To be specific, the central insulator region is also included in the finite element analysis but with a very tiny thermal conductivity $kins=1×10−5W/(mK)$ and electrical conductivity $σins=1×10−5S/m.$ The normalized temperature and potential fields are displayed in Fig. 5. The temperature and electrical voltage contours have been restored to parallel as they are when the whole domain $Ω$ is filled with iron. The objective function $J=4×10−3$⁠, which is tiny enough to indicate that the insulator is effectively cloaked from being detected by measuring the temperature and potential field perturbation in $Ωout$⁠. Compared with bare insulators, the optimal iron/aluminum in $ΩD$ can reduce 99.6% of the total temperature and potential distortions, as can be seen in Table 2.

The convergence history plot for the simultaneous thermal and electrical cloak with parallel inflow directions is displayed in Fig. 6. The objective function value decreases monotonically as the iteration number goes up. There is no volume constraint enforced on either the iron or the aluminum in the optimization formulation. Aluminum accounts for 24.74% of the whole design domain area in $ΩD$ for the final optimal configuration.

Upon analysis of Fig. 6, it is evident that the optimized structure displays some dependence on the initial design. To examine this behavior more closely, two distinct initial designs are explored. Figure 7 displays the optimized result with a four-hole initial design. The objective function $J=2.9×10−3$⁠, reducing 99.71% compared with the bare reference design as shown in Table 3. The temperature and voltage disturbance are eliminated and indicated by the restored parallel contour lines shown in Fig. 7, bottom row. Another optimization result starts with a bilayer initial guess as shown in Fig. 8. The objective function $J=4.7×10−3$⁠, achieving a 99.53% reduction in objective function compared with the bare reference case as observed in Table 4. Again, the reproduced parallel contour lines seen in Fig. 8, bottom row, serve as an implication that the central circular insulator is indeed cloaked thermally and electrically. To sum up, the proposed method does show relatively heavy dependency on the initial designs. Nevertheless, since this is a least-square error optimization problem, all the final optimized designs would make sense as long as the final objective function $J$ is small enough. One added advantage is that you can select an optimized design that is more manufacturing-friendly.

To show the robustness of the proposed method, the second numerical example focuses on a similar problem-setting but with orthogonal inflow directions. Specifically, the boundary conditions stay the same for thermal conduction phenomena. For the electrical conduction simulation, the current flows from the bottom edge toward the top edge. The left and right boundary edges are set to be electrically insulated. As shown in Fig. 9, right column, the insulator causes both temperature and electric potential field disturbance in the outer domain $Ωout$⁠, indicated by the nonparallel contour lines. The optimized structure is exhibited in Fig. 10. Again, the temperature and potential fields in $Ωout$ with uniform gradient are reproduced as shown in Fig. 11. The objective function $J=3.77×10−4$⁠, which is small enough to safely claim that the insulator is cloaked thermally and electrically. As shown in Table 5, the objective function can drop 99.96%, thanks to the optimal iron/aluminum layout in $ΩD$ compared with the bare insulator case.

In the optimization formulation, we did not enforce a volume percentage equality constraint on either of the two material phases. The rationales behind this treatment are two-fold. On the one hand, the two material phases in the design domain $ΩD$⁠, specifically, iron and aluminum for thermal–electrical cloaks, magnesium alloy and copper for thermal–electrical camouflage (see Sec. 3.3), respectively, are relatively common and inexpensive from the standpoint of material cost. Thus, there is no strong desire to save one material phase usage over the other. On the other hand, since the optimization formulation is a least-square error minimization problem, it will be deemed as a successful thermal–electrical cloak as long as the objective function $J$ is small enough. The volume fraction for each material phase will be determined by the optimization algorithm, driven by correct shape sensitivity analysis as detailed in Sec. 2.2. The correctness of shape sensitivity analysis is further backed up by the fact that the objective function decreases monotonically as the iteration number adds up, as shown in Figs. 6 and 12. In the presented numerical examples for thermal–electrical cloaks, it seems that more iron is desirable to eliminate the temperature and voltage disturbance. Essentially, the optimal materials distribution (iron/aluminum/air) in the union of domain $ΩD$ and $Ωins$ should render an effective thermal and electrical conductivity, which are, if not identical, quite close to those of pure iron material. It then makes sense that iron is more desirable, and a small portion of aluminum is introduced to compensate for the insulator with extremely low thermal and electrical conductivity. According to the authors’ experience, enforcing a volume percentage constraint on, say, the iron material phase would result in a different objective function value and final topology. Adding an extra constraint will likely lead to another (local) optimum and make the optimization slightly more complicated.

In this example, we push forward one step further from a thermal–electrical cloak to a thermal–electrical camouflage. For the bifunctional cloak, the disturbance in temperature and electrical potential field on the evaluation domain $Ωout$ is eliminated. However, there are still some perturbations on $Ωins$ and $ΩD$⁠. In other words, the insulator is not merged into the background and still can be detected if there is a thermal camera and multimeter capturing temperature and electrical field distribution of the whole field $Ω$⁠. For the bifunctional camouflage device, we aim to eliminate the disturbance on the whole computational domain $Ω$⁠. In addition, we replace the insulator with a sensor and make it invisible to an external detector. Yet, the sensor can still measure the undisturbed signals, e.g., temperature and electrical potential.

For the camouflage design, the geometry and boundary conditions are the same as depicted in Fig. 1. The background material is magnesium alloy. The insulator domain is filled with a sensor made of stainless steel. Some copper material is introduced into the design domain $ΩD$ to regulate thermal conductivity and electrical conductivity distribution to mitigate the disturbance. The material properties are displayed in Table 6. Note that the outer domain $Ωout$ is always fixed with magnesium alloy for this numerical example. The window function $g(x)$ from Sec. 2.2 takes a value of 1 everywhere inside $Ω$ since we are aiming to mitigate the disturbance all over $Ω$⁠.

As shown in Fig. 13, left column, parallel temperature and potential contour lines are generated by filling $ΩD$ and $Ωins$ with background material magnesium alloy. The arrow refers to the heat flux vector and electrical field. When a sensor made of stainless steel is introduced into $Ωins$ shown in Fig. 13, right column, it causes scattering in the whole computational domain $Ω$⁠, making the sensor detectable via an out-of-plane thermal camera or voltmeter.

The topology optimization result for thermal–electrical camouflage is exhibited in Fig. 14 with a large copper ring structure serving as the initial design. The final optimized structure ends up with a much thinner near-ring for copper. The objective function $J=4×10−3$⁠. Apparent distortions on normalized temperature and potential fields for initial design can be observed in Fig. 14, left column. These disturbances are greatly eliminated over the whole computational domain $Ω$⁠, indicated by parallel temperature and potential contour lines in Fig. 14, right column. The sensor is now camouflaged in the background, and it becomes difficult to detect thermally and electrically. Yet, it can still sense the undisturbed temperature and potential signals. To the best of the authors’ knowledge, this is among the first works to achieve simultaneous sensing and camouflaging in multiphysics fields from the standpoint of topology optimization.

Upon carefully checking Fig. 14, right column, we can notice that the contour lines are not fully parallel, particularly near the inner circle. Some explanation will be given here. First, as a result-driven method, the proposed level-set-based topology optimization aims to minimize the least-square error for temperature and electrical potential fields over the whole computational domain $Ω$⁠. Such formulation does not require a strict point-to-point field match as long as the integral value is small enough. Second, it is quite challenging to match a field with another. For the reference temperature and electrical potential fields, there is only one single homogeneous material magnesium alloy occupying the whole domain $Ω$⁠. The optimized camouflage device comprises three material phases: magnesium alloy, stainless steel, and copper. Along the material interfaces, there will be a jump in material properties, i.e., thermal conductivity and electrical conductivity, which will inevitably have an impact on heat and electrical flow. This phenomenon is also commonly seen in results obtained using other approaches [65]. Although there are still some deviations at certain locations, it would be difficult for the infrared camera or multimeter to detect such tiny perturbations. The optimized structure performs tremendously better than the initial guess, as illustrated in Fig. 14. Note that the contour lines or flux arrow lines are only for better visualization purposes. When the detection happens, there will be no such marks for the camera or multimeter to look at as a reference.

In this study, a level-set-based topology optimization is proposed to design simultaneous thermal–electrical cloaks and camouflage. Formulated as a least-square error minimization problem, the optimization algorithm successfully found the optimal material layout in the design domain $ΩD$ to eliminate the temperature and potential field disturbances in domain $Ωout$ (cloaking) or whole domain $Ω$ (camouflaging). The proposed method is advantageous in that it only employs naturally occurring bulk materials and does not introduce material anisotropy, which would greatly facilitate the physical realization compared with popular transformation thermotics methods [10,11].

A number of numerical examples are provided in this article. Several thermal–electrical cloaks are given in Secs. 3.1 and 3.2. The proposed method is robust in generating effective cloak designs no matter whether the thermal and electrical inflows are parallel or orthogonal. The objective functions are small enough (in the magnitude of $10−3)$⁠. The restored parallel temperature and potential contour lines indicate that the central insulator is successfully cloaked. Two additional optimization cases are given to investigate the dependency on the initial guesses. Even though the proposed method shows a relatively strong dependency on the initial designs, all the optimized cloak designs would make sense as long as the objective function is small enough. One added benefit is that you can select an optimal cloak design with superior manufacturability for subsequent physical validation. Section 3.3 details a thermal–electrical camouflage device in which a sensor is camouflaged into the background but still detects undisturbed temperature and electrical potential signals.

The thermal–electrical cloaks and camouflage designs presented in this study are essentially background-dependent and unidirectional. Although omnidirectional cloaking and camouflage devices can find broader practical applications as they are not sensitive to where the heat or electrical flow comes from, the research on directional cloaking and camouflage still holds great scientific significance. First, omnidirectional metadevices used to manipulate various diffusive systems often involve coordinate transformation to a certain degree. There are obvious shortcomings associated with this approach. It typically results in high material non-homogeneity and anisotropy, making the physical realization difficult [10,11]. Researchers often resort to multilayer structure or effective medium theory to approximate the thermal and electrical conductivity profile at the cost of losing some accuracy. Also, it is difficult for coordinate transformation-based approaches to cloak arbitrary freeform regions, different from topology optimization counterparts as reported in Ref. [43]. All these characteristics would somehow limit the applications of omnidirectional metadevices. Second, the presented unidirectional cloaking device is well suited for application scenarios where apriori knowledge of background field distribution is known, and the heat/electrical sources do not change with time and location. For example, in consumer electronics heat transfer, there will be heat-generating and heat-sensitive components. Under steady operating conditions, the power source location will not change, giving us a fixed heat flow direction. Moreover, the heat-sensitive component that is thermally cloaked can take a complex shape as it will have to accommodate other components. The directional designs from the proposed level-set-based topology optimization would then be advantageous to be more flexible in handling arbitrary clocking regions [43].

However, there are still some aspects that could be improved in the future. First, experimental validation needs to be conducted to make the results more convincing. Second, as the sensors to be cloaked or camouflaged can take arbitrary shapes, it is desired to design conformal multiphysics cloaks or camouflaging devices on manifolds. Conformal mapping has been introduced into the level-set-based topology optimization community [67,68]. It shows great prospects in solving various optimization problems on freeform surfaces.

This work was partially supported by the National Science Foundation under grants CMMI-1762287 and PFI-RP-2213852; the Ford University Research Program (URP) under award 2017-9198R; and the Office of the Vice President for Research (OVPR) at Stony Brook University through the Summer 2022 and Fall 2023 Seed Grant programs.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.