Inverse Design and Experimental Verification of Metamaterials for Thermal Illusion Using Genetic Algorithms

Zonggang He(何宗堽)$^{1}$, Kun Yuan(袁坤)$^{1}$, Guohuan Xiong(熊国欢)$^{1,2\hyperlink{s}{*}}$, and Jian Wang(王健)$^{1\hyperlink{s}{*}}$

doi:10.1088/0256-307X/40/10/104402

⇦Chinese Physics Letters, 2023, Vol. 40, No. 10, Article code 104402⇨ Inverse Design and Experimental Verification of Metamaterials for Thermal Illusion Using Genetic Algorithms Zonggang He (何宗堽)1, Kun Yuan (袁坤)1, Guohuan Xiong (熊国欢)1,2*, and Jian Wang (王健)1* Affiliations 1College of Physical Science and Technology, Yangzhou University, Yangzhou 225002, China 2 School of Physical Science and Technology, Nanjing Normal University, Nanjing 210046, China Received 29 July 2023; accepted manuscript online 18 September 2023; published online 1 October 2023 ^*Corresponding authors. Email: phyghx@yzu.edu.cn; wangjian@yzu.edu.cn Citation Text: He Z G, Yuan K, Xiong G H et al. 2023 Chin. Phys. Lett. 40 104402 Abstract Thermal metamaterials offer a promising avenue for creating artificial materials with unconventional physical properties, such as thermal cloak, concentrator, rotator, and illusion. However, designs and fabrication of thermal metamaterials are of challenge due to the limitations of existing methods on anisotropic material properties. We propose an evolutionary framework for designing thermal metamaterials using genetic algorithm optimization. Our approach encodes unit cells with different thermal conductivities and performs global optimization using the evolution-inspired operators. We further fabricate the thermal functional cells using 3D printing and verify their thermal illusion functionality experimentally. Our study introduces a new design paradigm for advanced thermal metamaterials that can manipulate heat flows robustly and realize functional thermal metadevices without anisotropic thermal conductivity. Our approach can be easily applied to fabrications in various fields such as thermal management and thermal sensing.

DOI:10.1088/0256-307X/40/10/104402 © 2023 Chinese Physics Society Article Text Metamaterials[1] provide extra degrees of freedom in designing miscellaneous artificial materials with unachievable physical properties that do not exist ordinarily. Compared to their counterpart in the wave-nature acoustic[2] and optical systems,[3] thermal metamaterials[4,5] based on diffusive heat transfer can also exhibit[6-11] great potential on flexible manipulation of heat flow and precise control of temperature. Thermal functionalities such as thermal cloak,[4,12] concentrator,[13,14] rotator,[15] and illusion[16-28] can be realized using thermal metamaterials. Among them, thermal illusion aims to mimic the exterior thermal behavior[24] of an equivalent reference of heat sources and transform the internal actual heat sources to different illusion signatures. Several approaches have been proposed in designs of thermal metamaterials for thermal illusion. The illusion signatures in thermal scattering from an arbitrary object to multiple virtual objects have been demonstrated[26] using the scattering cancelation method. The illusion reshaping an arbitrary thermal object into another different one has been demonstrated[27,29] by the method of transformation thermotics. In addition, a topology optimization scheme[30-36] is proposed to design thermotics for camouflaging the external temperature field. However, practical and flexible designs of thermal illusion structures remain a challenge[20] despite the above progress. Designs of metamaterials with certain thermal functionalities are a reverse or inverse problem[37-39] mathematically. It is usually difficult to solve because of the highly nonlinear relationships between design parameters and desired thermal functionalities. Direct inverse solution methods such as the scattering cancellation[26] and the transformation thermotics[27,29] lack general adaptability and require anisotropic tensor parameters that are hard to realize. The topology optimization scheme[30-36] is heavily dependent on parameter initialization[40] and may fall into local minima. In this Letter, we present an evolution-inspired roadmap to design thermal illusion metamaterials using genetic algorithm (GA)[41-45] within the approach of block cells encoded with different thermal conductivities. Unlike the conventional optimization methods,[20] which may suffer from local optima and parameter initialization,[40] the evolutionary scheme employs a population-based approach that mimics the biological evolution process. It randomly generates an initial population of metamaterial entities and applies operators of mutation, crossover and natural selection to produce offspring that inherits and improves their genetic material. Therefore, the evolutionary scheme can effectively search large solution space in parallel and find optimal or near-optimal solutions. The metamaterial configurations with the desired functions can emerge during this evolutionary process. We further fabricate and experimentally validate thermal functional cells of our designed metamaterials using 3D printing technology. Our study pioneers a new design paradigm for advanced thermal metamaterials, enabling robust manipulation of heat flows and ease of realization of functional thermal metadevices free of anisotropic thermal conductivity. Our approach may have wide applications in various fields, including thermal management and thermal sensing with ease of fabrication. Model and Methods. Figure 1 shows the evolutionary design of thermal illusion metamaterials using GA. The structural design layout is depicted in Fig. 1(a). The objective is to camouflage two real heat sources $Q_{1}$ and $Q_{2}$ as five virtual heat sources $Q_{1}',\ldots, Q_{5}'$ in the design domain $\varOmega_{\rm D}$. Ideally, the thermal illusion[20] should make the temperature distribution $T$ in the external area $\varOmega_{\rm out}$, induced by the two actual heat sources $Q_{1}$ and $Q_{2}$, identical to the target illusion temperature field $T_{\rm ref}$ in a homogenous material, generated by the five virtual heat sources $Q_{1}',\ldots, Q_{5}'$. To implement the design approach using GA, the square design region $\varOmega_{\rm D}$ is divided into a grid of $n\times n$ blocks. To examine the influence of size, we consider two different grid sizes for the design area $20\times 20$ and $40\times 40$, respectively. Each cell is composed of either polydimethylsiloxane (PDMS) or aluminum (Al), encoded by 0 or 1, respectively. The external region $\varOmega_{\rm out}$ is evenly filled with PDMS, forming a continuous layer.

Fig. 1. The evolutionary design process for thermal illusion metamaterials. (a) Structural design layout with real heat sources $Q_{1}$ and $Q_{1}$ denoted by the red circles and illusion heat sources $Q_{1}',\ldots, Q_{5}'$ by the blue triangles. The binary grids in the design square $\varOmega_{\rm D}$ with $n\times n$ dimensions. (b) Major steps involved in generating new structures using GA: (i) initial population, (ii) ranking and selection of parent structures based on the fitness function, (iii) new offspring using the breeding template $T_{\rm e}$.

We solve the steady heat conduction equation with five virtual heat sources to obtain the target temperature fields $T_{\rm ref}$ in a homogenous material with a single thermal conductivity, \begin{align} \nabla \cdot (-\kappa_{0} \cdot \nabla T_{\rm ref})=\sum\limits_{i=1}^5 {Q_{i}' }. \tag {1} \end{align} Here $\kappa_{0}$ is the isotropic thermal conductivity of PDMS and $Q_{i}'$ denotes the virtual heat sources. We use the steady temperature field as the target of optimization. To extend the applicability of the designed metamaterial[46] to the transient process of heat conduction, it is necessary to use the sampled transient temperature field distribution as the target temperature field and employ the multi-objective optimization[47] by genetic algorithm. The temperature fields $T$ of the designed metamaterials with two real heat sources are obtained by solving the heat conduction equation \begin{align} \nabla \cdot (-\kappa \cdot \nabla T)=\sum\limits_{i=1}^2 {Q_{i}}, \notag\\ \kappa=\begin{cases} \kappa_{0} & {r}\in {\rm PDMS} \\ \kappa_{1} & {r}\in {\rm Al,} \end{cases} \tag {2} \end{align} where the value of thermal conductivity $\kappa$ depends on its position $r$. During our simulations, the temperature on the boundary of the external region $\varOmega_{\rm out}$ is set to 293 K. The deviation between the temperature fields of the candidate metamaterials designed and the illusionary target can be evaluated by means of the fitness function \begin{align} f=\int_{\varOmega_{\rm out}}{|{T-T_{\rm ref}}|}^{2}dS, \tag {3} \end{align} where the integration is performed over the external region $\varOmega_{\rm out}$. Figure 1(b) illustrates the schematic representation of the primary steps involved in generating evolutionary metamaterial configuration through breeding using GA. Initial populations of $N_{\rm pop} =100$ and $N_{\rm pop} =400$ are selected with the random distribution of 0 and 1 for the grid size of $20\times 20$ and $40\times 40$, respectively. The Laplace heat conduction Eq. (2) is utilized to compute the temperature fields for each entity. The fitness function (3) is evaluated to rank the members of the population based on their corresponding values. Following this ranking, the top half of entities exhibiting high fitness are selected and subsequently preserved as parents for the succeeding generation. Thirdly, the crossover offspring metamaterials are generated as ${\rm offspring}=C\cdot T_{\rm e}+D({1-T_{\rm e}})$ through utilization of the breeding template $T_{\rm e}$ as exemplified in Fig. 1(b). Here $C$ and $D$ represent the preserved parent meta-configurations. A mutation is introduced to the grid cell of the new offspring of configuration with a probability of 0.1%. The flowchart of design process using GA is presented in Table 1. Compared to transformation thermodynamics,[9-11] the above genetic evolution algorithm circumvents the limitations of materials with non-uniform and anisotropic thermal conductivity, thereby enabling the realization of the designed thermal function in ordinary natural materials. Unlike traditional gradient topology optimization,[30-36] the evolutionary algorithm significantly expands the configuration search space through evolutionary operators as shown in Fig. 1(b) and reduces dependence on initial parameters, thus facilitating to find satisfying structures in the entire configuration space. Conversely, a challenge of the genetic evolutionary algorithm is its strong dependence on the target temperature field, which is based on the solution of the heat conduction equation (1) with predefined boundary conditions. The results obtained by transformation thermodynamics show a lower sensitivity to the initial conditions of the target temperature field. We suggest that this challenge can be overcome by applying multi-objective genetic optimization algorithms.[47]

Table 1. Design of thermal illusion metamaterials using the GA framework.
Input: $M$: the shape ($n\times n$) of model,
$P$: the size of population,
$t$: the number of iterations
Output: Solution $X$
1: generate initial population randomly
2: for iteration $t = 0$, 1, 2, ..., $t$ do
3: for population $p = 0$, 1, 2, ..., $P$ do
4: compute the fitness value f for each entity in population
5: select half of entities according to the highest fitness $f$
6: for candidate $i = 0$, 1, 2, ..., $p/2$ do
7: crossover on the selected candidates and generate new candidates
8: mutation on new candidates
9: return the best solution $X$

Fig. 2. Dynamic evolution of generations in GA for designing metamaterials of thermal illusion: (a) mean fitness of the objective function over time, (b) average distance between individuals evolving with the optimization.

Results. Figure 2 shows the dynamic evolution of generations in GA for the design of metamaterials for thermal illusion. We compare the thermal illusion effects of the two block sizes, $20\times 20$ and $40\times 40$, respectively. Figure 2(a) displays the evolution of mean fitness defined by Eq. (3) over generations, which indicates the convergence of the GA to the final solution. It is revealed that the mean values of the objective function $f$ decrease rapidly as the number of iterations for GA increases. This observation suggests that the population converges towards the optimal object function $f=0$, as genetic evolutionary operations continue. The mean fitness value for a grid size of $20\times 20$ stabilizes after the 150th generation. Similarly, a larger grid size of $40\times 40$ reaches a stable state after the 170th generation. Due to the high computational complexities involved, we choose a total of 200 iterations for the evolution process in our calculation. In addition, Fig. 2(b) shows the average distance[45] between individuals within each generation, which reflects the diversity in each iteration. The average distance is a metric for the exploration extent within the parameter space, where a smaller value implies a narrow search and a risk of the evolutionary algorithm being stuck in a local minimum. Figure 2(b) reveals significant fluctuations in the average distance, indicating a broad exploration[45] of the configuration space within each population. A possible explanation for the large fluctuations of average distance for the grid size $20\times 20$ is that the search space is relatively smaller, so that the structural configurations of each individual entity vary greatly across generations due to the evolutionary operators. In our simulations, we select the best structural configuration with the lowest mean fitness value as the final solution of the configuration. Figure 3 shows the designed metamaterials and the simulated temperature fields that camouflage two real heat sources as five virtual ones using the GA approach. Figure 3(a) (left) depicts the temperature field $T_{0}$ for a single PDMS material with two heat sources. Figure 3(a) (right) presents the target reference temperature field $T_{\rm ref}$ for a single PDMS material with five virtual heat sources. Figures 3(b) and 3(c) (left) demonstrate the optimal structural configurations of the square design area $\varOmega_{D}$ after 200 evolutionary steps. The grid sizes considered in the investigation were $20\times 20$ and $40\times 40$, respectively. Figures 3(b) and 3(c) (Center) illustrate the simulated temperature fields of two real heat sources, utilizing the designed metamaterials. For comparison, Figs. 3(b) and 3(c) (right) show the temperature field difference $|{T-T_{\rm ref}}|$ between the simulated temperature field $T$ and the reference temperature field $T_{\rm ref}$. The temperature difference in the external region between the camouflaging field generated by the metamaterials for the five virtual heat sources and the reference field for the five real heat sources is less than 0.2 K. Therefore, it is difficult to distinguish the five real heat sources from the virtual heat source created by the metamaterials at the center, marked by white squares. Thus, we achieve the effective realization of thermal illusion from two real heat sources to five virtual heat sources through the proficient design of metamaterials composed of PDMS and aluminum.

Fig. 3. Metamaterials and simulated temperature fields using the GA approach. The white lines in (b) and (c) delineate the designed square area $\varOmega_{\rm D}$. Panel (a) shows the temperature field $T_{0}$ (left) and $T_{\rm ref}$ (right) for the two and five real heat sources, respectively. Panels (b) and (c) display the optimized structures for the design area $\varOmega_{\rm D}$ with $20\times 20$ and $40\times 40$, respectively. The center of each panel shows the corresponding temperature field. The right side shows the difference between the simulated temperature field $T$ and the reference temperature field $T_{\rm ref}$.

We simulate two different grid sizes $20\times 20$ and $40\times 40$, respectively, to investigate the size effect on the evolutionary algorithm, as shown in Figs. 3(b) and 3(c). The figures show that we implement thermal illusion from two real heat sources to five virtual heat sources in both grid sizes using the GA method. However, a smaller grid block size and a higher grid density can improve the desired thermal performance at the expense of an exponential increase in the search space. For example, the search space expands exponentially using GA because the state space is given by $2^{N}$, where $N=n\times n$ is the total number of grids and $n$ is the grid size of the design area $\varOmega_{D}$. Moreover, we must consider the minimum feature size that is related to the precision of the 3D printing process. To corroborate the simulations presented in Fig. 3, we have undertaken further experimental validations. The three-dimensional configuration of the experimental metamaterials, generated using COMSOL, is illustrated in Fig. 4(a). The aluminum thermal metamaterials were fabricated via the 3D printing, utilizing the generated DXF file. The resulting experimental sample is shown in Fig. 4(b). Considering the accuracy of 3D printing, we select the metamaterials with the grid size of $20\times 20$ shown in Fig. 3(b) as our experimental model, even though larger grid size can achieve better performance of thermal illusion. The experimental sample has a size of 4 mm and a thickness of 4 mm for each grid, resulting in a square block size of 80 mm for the $20\times 20$ grid configuration. The remaining section of the thermal metamaterials has been filled with a thermal compound composed of PDMS (Dowsil TC-5021), possessing a thermal conductivity value of $\kappa =3.3\,{\rm W/m}\cdot{\rm K}$. The outer disc area $\varOmega_{\rm out}$ with a diameter of $d=150$ mm is uniformly filled with PDMS and subsequently enveloped by an aluminum strip, which serves as thermal connector for the external heat sinks. The assembled metamaterials are then positioned atop a polystyrene foam disc of 150 mm in diameter and 20 mm in thickness. Two $7\,{\rm mm}\times 5\,{\rm mm}$ ceramic heating plates with thickness of 1.2 mm are placed at the designated locations of heat sources on the metamaterials, and are powered by a DC power supply. The entire set is placed in a water tank, which serves as a steady external heat sink at a constant temperature of 295 K.

Fig. 4. Experimental validation of the designed metamaterials for thermal illusion: (a) the designed 3D thermal metamaterials, (b) experimental thermal metamaterials composed of aluminum fabricated by 3D printing, (c) the measured temperature fields of thermal metadevices captured using an infrared camera. The pentagonal pattern of the temperature field of panel (c) matches with the target temperature field in Fig. 3(a), thereby demonstrating the successful thermal illusion function from two real heat sources to five virtual heat sources.

The steady-state temperature fields of the thermal metadevices were captured utilizing an infrared camera (FLIR E75). The temperature fields measured in Fig. 4(c) were highly consistent with the simulated temperature field distributions shown in Fig. 3(b). We can find that the outside pentagonal pattern of the temperature field matches well with the target temperature field in Fig. 3(a), demonstrating the successful thermal illusion function from two real heat sources to five virtual heat sources. Therefore, these experimental results have confirmed the successful thermal illusion from two actual heat sources to five virtual sources, achieved by using the metamaterials designed through the GA approach. However, some deviation between the simulation and experimental results may exist, due to the difficulty of precisely controlling the temperature during the experiments. In the following, we provide some possible explanations of the mechanisms behind the inverse design and optimization of thermal metamaterials with thermal illusion function using the genetic algorithm. As shown in Fig. 1(a), the algorithm combines unit cells with spatially discretized thermal conductivity. The algorithm numerically solves the heat conduction equation (2) to find the optimal structural configuration with temperature fields close to the target ones given by Eq. (1), thus obtaining the optimized configuration that achieves the thermal illusion function. From the structures of thermal metamaterials obtained in Fig. 4(a), we can observe that the genetic algorithm produces spatially coarse-grained anisotropic materials with thermal conductivity, which implement the thermal function. In summary, we have presented an evolutionary method for designing functional metamaterials that can achieve thermal illusion by using genetic algorithms and topology optimization. The block-based design of thermal metamaterials offers a high level of efficiency and effectiveness, which avoids the need for anisotropic thermal conductivity. Our study introduces a new design paradigm for advanced thermal metamaterials, enabling robust manipulation of heat flows and easy realization of functional thermal metadevices. Our method can be generalized to the complex problem of designing metamaterials, which involves multi-objective optimization for a flexible number of virtual sources and three-dimensional thermal functional metamaterial designs. Practically, the block-based metamaterials have great potential for wide applications in diverse fields such as energy management, thermal regulation, and cloaking devices with other fabrication techniques, such as electrospinning. Acknowledgment. This work was supported by the National Natural Science Foundation of China (Grant No. 11875047). References Metamaterials and Negative Refractive IndexBroadband Acoustic Cloak for Ultrasound WavesRay-optics cloaking devices for large objects in incoherent natural lightShaped graded materials with an apparent negative thermal conductivityThermal Metamaterial: Fundamental, Application, and OutlookDiffusion metamaterialsTransforming heat transfer with thermal metamaterials and devicesA Brief Review on Thermal Metamaterials for Cloaking and Heat Flux ManipulationExperimental Demonstration of a Bilayer Thermal CloakTransformation thermodynamics: cloaking and concentrating heat fluxDesign of square-shaped heat flux cloaks and concentrators using method of coordinate transformationHeat Flux Manipulation with Engineered Thermal MaterialsTopology-optimized thermal metamaterials traversing full-parameter anisotropic spaceRobustly printable freeform thermal metamaterialsFlexible Janus Functional Film for Adaptive Thermal CamouflageThermal camouflaging metamaterialsIllusion thermotics with topology optimization3D Printed Meta‐Helmet for Wide‐Angle Thermal CamouflagesAn Adaptive and Wearable Thermal Camouflage DeviceThermal illusion with twinborn-like heat signaturesIllusion ThermoticsIllusion thermal device based on material with constant anisotropic thermal conductivity for location camouflageFull Control and Manipulation of Heat Signatures: Cloaking, Camouflage and Thermal MetamaterialsConverting the patterns of local heat flux via thermal illusion deviceCamouflage thermotics: A cavity without disturbing heat signatures outsideIllusion thermodynamics: A camouflage technique changing an object into another one with arbitrary cross sectionComputational design of metadevices for heat flux manipulation considering the transient regimeOptimizing the structural topology of bifunctional invisible cloak manipulating heat flux and direct currentTopology-optimized thermal carpet cloak expressed by an immersed-boundary level-set method via a covariance matrix adaptation evolution strategyTopology optimization of thermal cloak using the adjoint lattice Boltzmann method and the level-set methodTopology optimization of bilayer thermal scattering cloak based on CMA-ESTopology optimization of thermal cloaks in euclidean spaces and manifolds using an extended level set methodDeep learning based design of thermal metadevicesInverse design in nanophotonicsThermal transparency with periodic particle distribution: A machine learning approachReinforcement learning approach to thermal transparency with particles in periodic latticesBio‐Inspired Morphological Evolution of Metastructures with New Operation ModalitiesGenetic Algorithms and Machine LearningA review on genetic algorithm: past, present, and futureDesigning DNA-grafted particles that self-assemble into desired crystalline structures using the genetic algorithmA transformation theory for camouflaging arbitrary heat sourcesMulti-objective optimization using genetic algorithms: A tutorial

[1]	Smith D R, Pendry J B, and Wiltshire M C 2004 Science 305 788
[2]	Zhang S, Xia C, and Fang N 2011 Phys. Rev. Lett. 106 024301
[3]	Chen H S, Zheng B, Shen L, Wang H P, Zhang X M, Zheludev N I, and Zhang B L 2013 Nat. Commun. 4 2652
[4]	Fan C Z, Gao Y, and Huang J P 2008 Appl. Phys. Lett. 92 251907
[5]	Wang J, Dai G, and Huang J 2020 iScience 23 101637
[6]	Zhang Z R, Xu L J, Qu T, Lei M, Lin Z K, Ouyang X P, Jiang J H, and Huang J P 2023 Nat. Rev. Phys. 5 218
[7]	Li Y, Li W, Han T C, Zheng X, Li J X, Li B W, Fan S H, and Qiu C W 2021 Nat. Rev. Mater. 6 488
[8]	Peralta I, Fachinotti V D, and Álvarez H J C 2020 Adv. Eng. Mater. 22 1901034
[9]	Xu L J and Huang J P 2023 Transformation Thermotics and Extended Theories, 1st edn (Singapore: Springer)
[10]	Yeung W S and Yang R J 2022 Introduction to Thermal Cloaking Theory and Analysis in Conduction and Convection 1st edn (Singapore: Springer)
[11]	Huang J P 2020 Theoretical Thermotics Transformation Thermotics and Extended Theories for Thermal Metamaterials 1st edn (Singapore: Springer)
[12]	Han T C, Bai X, Gao D L, Thong J T L, Li B W, and Qiu C W 2014 Phys. Rev. Lett. 112 054302
[13]	Guenneau S, Amra C, and Veynante D 2012 Opt. Express 20 8207
[14]	Yu G X, Lin Y F, Zhang G Q, Yu Z, Yu L L, and Su J 2011 Front. Phys. 6 70
[15]	Narayana S and Sato Y 2012 Phys. Rev. Lett. 108 214303
[16]	Sha W, Hu R, Xiao M, Chu S, Zhu Z, Qiu C W, and Gao L 2022 npj Comput. Mater. 8 179
[17]	Sha W, Xiao M, Zhang J, Ren X, Zhu Z, Zhang Y, Xu G, Li H, Liu X, Chen X, Gao L, Qiu C W, and Hu R 2021 Nat. Commun. 12 7228
[18]	Liu Y D, Zuo H Y, Xi W, Hu R, and Luo X B 2022 Adv. Mater. Technol. 7 2100821
[19]	Hu R, Xi W, Liu Y D, Tang K C, Song J L, Luo X B, Wu J Q, and Qiu C W 2021 Mater. Today 45 120
[20]	Sha W, Zhao Y, Gao L, Xiao M, and Hu R 2020 J. Appl. Phys. 128 045106
[21]	Peng Y G, Li Y, Cao P C, Zhu X F, and Qiu C W 2020 Adv. Funct. Mater. 30 2002061
[22]	Hong S, Shin S M, and Chen R K 2020 Adv. Funct. Mater. 30 1909788
[23]	Zhou S L, Hu R, and Luo X B 2018 Int. J. Heat Mass Transfer 127 607
[24]	Hu R, Zhou S L, Li Y, Lei D Y, Luo X B, and Qiu C W 2018 Adv. Mater. 30 1707237
[25]	Hou Q W, Zhao X P, Meng T, and Liu C L 2016 Appl. Phys. Lett. 109 103506
[26]	Han T C, Bai X, Thong J T L, Li B W, and Qiu C W 2014 Adv. Mater. 26 1731
[27]	Zhu N Q, Shen X Y, and Huang J P 2015 AIP Adv. 5 053401
[28]	Xu L J, Wang R Z, and Huang J P 2018 J. Appl. Phys. 123 245111
[29]	He X and Wu L Z 2014 Appl. Phys. Lett. 105 221904
[30]	Álvarez H J C, Fachinotti V D, Peralta I, and Tourn B A 2019 Numer. Heat Transfer Part A 76 648
[31]	Fujii G and Akimoto Y 2019 Appl. Phys. Lett. 115 174101
[32]	Fujii G and Akimoto Y 2019 Int. J. Heat Mass Transfer 137 1312
[33]	Luo J W, Chen L, Wang Z H, and Tao W Q 2022 Appl. Therm. Eng. 216 119103
[34]	Wang W, Ai Q, Shuai Y, and Tan H 2023 Int. J. Heat Mass Transfer 206 123959
[35]	Xu X Q, Gu X D, and Chen S K 2023 Int. J. Heat Mass Transfer 202 123720
[36]	Ji Q X, Chen X Y, Liang J, Fang G D, Laude V, Arepolage T, Euphrasie S, Iglesias M J A, Guenneau S, and Kadic M 2022 Int. J. Heat Mass Transfer 196 123149
[37]	Molesky S, Lin Z, Piggott A Y, Jin W, Vucković J, and Rodriguez A W 2018 Nat. Photon. 12 659
[38]	Liu B, Xu L, and Huang J 2021 J. Appl. Phys. 129 065101
[39]	Liu B, Xu L, and Huang J 2021 J. Appl. Phys. 130 045103
[40]	Zhang Q Y, Barri K, Yu H, Wan Z, Lu W Y, Luo J Z, and Alavi A H 2023 Adv. Intell. Syst. 5 2300019
[41]	Haupt R L and Haupt S E 2004 Practical Genetic Algorithms 2nd edn (Hoboken, NJ: Wiley) pp xvii, 253
[42]	Coley D A 1999 An Introduction to Genetic Algorithms for Scientists and Engineers (Singapore: World Scientific) pp xvi, 227
[43]	Goldberg D E and Holland J H 1988 Mach. Learn. 3 95
[44]	Katoch S, Chauhan S S, and Kumar V 2021 Multimed Tools Appl. 80 8091
[45]	Srinivasan B, Vo T, Zhang Y, Gang O, Kumar S, and Venkatasubramanian V 2013 Proc. Natl. Acad. Sci. USA 110 18431
[46]	Xu L J and Huang J P 2018 Phys. Lett. A 382 3313
[47]	Konak A, Coit D W, and Smith A E 2006 Reliab. Eng. Syst. Saf. 91 992