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The thermal cloak has special thermal insulation performance because of its unique anisotropic thermal conductivity tensor. Constrained by simplistic geometric designs, traditional thermal cloaks face limitations in achieving precise thermal regulation for complex structures with different shapes. The hexagonal and dodecahedral thermal cloaks are quite rare, largely due to their complex transformation equations. In this study, the transformation equations for the hexagonal and dodecahedral cloaks are derived by the rotation matrix, and their thermal conductivity tensors are further obtained according to the theory of transformation thermotics. The derivation method is applicable to any two and three-dimensional thermal invisibility cloaks with geometric symmetry. Furthermore, the numerical verification shows that both hexagonal and dodecahedral thermal cloaks could avoid heat flow in their thermal invisibility region. This proves the correctness of the derivation method and the thermal conductivity tensor calculated. The hexagonal and dodecahedral thermal cloaks obtained in this paper could provide uniform temperature field for different applications preventing the field of non-uniform stress and deformation in actual.
Thermal metamaterials make it possible to actively manipulate heat phenomena of artificial systems and govern heat diffusion processes [1]. Various thermal metamaterials with different functionalities have already been demonstrated due to the development of transformation thermotics [2]. Thermal cloak, as one of the thermal metamaterials, provides a zero-temperature gradient inside a central region without disturbing the temperature distribution in the surrounding region. So, such cloak is of an application in hiding or protecting objects from being detected or destroyed [3]. The thermal cloaks have been extensively studied since Pendry et al. [4] designed an artificial metamaterial based on the formal invariance of Maxwell’s equation and achieved electromagnetic invisibility. Fan et al. [5] introduced the concept of electromagnetic cloak into the thermal field for the first time and predicted the thermal invisibility cloak theoretically. Narayan et al. [6] synthesized two materials with very different thermal conductivity into a spiral multilayer structure and made a prototype of a thermal cloak. Various cloak has been studied, such as the bifunctional cloak [7], switchable thermal cloak [8] and the microstructured thermal cloak [9]. Realizing anisotropic thermal conductivities as expected is essential for thermal cloak. Jian Z et al. [10] found that periodic dimpling of silicon film could reduce thermal conductivity and constructed a rectangular cloak in this way. Yang S et al. [11] proposed a single-particle structure to effectively manipulate the thermal conductivity by appropriately adjusting the form factor or area fraction of the single particle. Sha et al. [12] took the local thermal conductivity tensor as input, carried out the free form design of topological functional units by topology optimization, and then directly 3D printed and assembled them to design three free-form thermal devices. Ji et al. [13] propose a machine learning based thermal cloak consisting of a finite number of layers with isotropic materials. Han et al. [14] demonstrate the design of full-parameter omnidirectional thermal metadevices with anisotropic geometry, which is experimentally confirmed via three proof-of-concept experiments. Li et al. [15] suggest a de-homogenization approach that uses optimal multi-rank laminates to provide closed-form solutions for any imaginable thermal manipulation device and create thermal cloaks, rotators, and concentrators. A forward conformality-assisted tracing method is proposed, which uses a conformal mesh composed of orthogonal streamlines and isotherms to produces free-form metamaterials using only isotropic media [16]. At the microscopic level, how to adjust the thermal conductivity is also a research hotspot [17–20], which provides a new idea for the actual preparation of thermal invisibility cloak.
Hexagon has been used in elastic metamaterials [21] dual-band terahertz metamaterial absorber [22] and mechanical metamaterials [23]. Dodecahedron is one of the densest known packings of the non-tiling Platonic in three-dimensional Euclidean space [24]. The mechanical metamaterials [25] of dodecahedron is widely researched such as their topology-property relationships [26] and mechanical properties [27–32]. However, rare researchers have deduced the thermal insulation performance of hexagonal and dodecahedral thermal cloaks. In this study, the thermal conductivity tensors of hex-agonal and dodecahedral thermal cloaks are derived to provide uniform temperature field for different applications to prevent non-uniform stress and deformation from nonuniform heat flow in actual situation. The method in this paper could be applied to any two-dimensional or three-dimensional cloak with geometric symmetry.
Hexagonal Thermal CloakDerivation of the Hexagonal Cloak’s Thermal Conductivity Tensor
If there is no heat resource and at steady state, the heat conduction equation is:∇·λ0∇T=0where λ0 represents the thermal conductivity of the medium, T is the temperature. Since the heat conduction equation has formal invariance [33], Equation 1 could be transformed in other space as Equation 2:∇′·λ′∇′T′=0where λ′,T′ represent the thermal conductivity coefficients and temperatures in the transformed spaces. According to the theory of transformation thermotics [34], the relationship between transformation space and original space is:λ′=A·λ0·ATdetAwhere A is the Jacobian transformation matrix, reflecting the geometric changes from the original space to the transformation space, and its components are:Aij=∂xi′∂xjwhere xi′ denotes the three coordinate components x’, y’, z’ in the transformation space, xj represents the three coordinate components x, y, z in the original space.
Three steps would be carried out to obtain the thermal conductivity of the thermal cloak. Firstly, the geometric transformation equations of the cloak with complex shape should be established based on the rotation matrix which is obtained by the specific rotation method. Secondly, the Jacobian transformation matrix A could be formed according to the geometric transformation equations. Finally, the thermal conductivity coefficients λ′ of the cloak could be obtained by Equation 3.
The thermal conductivity tensor of hexagonal thermal invisibility cloak designed as Figure 1 is derived as follows. The cloak with anisotropic thermal conductivity tensors is set in the pink region so that the gray region in the middle is heat stealthy. The width of the hexagonal cloak and the middle invisibility region are 2s2 and 2s1, respectively. Furthermore, the cloak is divided into six regions Ω1−6 because of the geometric symmetry.
The sketch map of the hexagon thermal invisibility cloak (region Ω1−6).
Hexagonal diagram with a center and axes labeled X' and Y'. Sections are numbered one to six around the center. A red line labeled S₁ extends towards section four, while a green line labeled S₂ also points to section four. The sections are colored pink with the central hexagon in gray.
The transformation formula of region Ω1 is easily established as Equation 5:x′y′=s2−s1s200s2−s1s2+s1xxy+s10
The corresponding Jacobian matrix of region Ω1 could be calculated as Equation 6:A1=s2−s1s20−s1yx2s2−s1s2+s1x
Then the thermal conductivity relationship of region Ω1 between transformation space and original space is established as Equation 7 according to Equations 3, 6:λ1′=1−s1x′−s1y′x′2−s1y′x′2x′4+s12y′2x′4−s1x′3λ0
The region Ω2 could be obtained by the rotating the region Ω1 with 60° anticlockwise. The transformation equation of region Ω2 could be set up by multiplying the transformation equation of region Ω1 by the rotation matrix P=12‐323212.x′y′=s2−s1s2+32s1x+3y−32s1x+3y−32s1x+3ys2−s1s2+12s1x+3yxy+12s132s1
The corresponding Jacobian matrix and the thermal conductivity of region Ω2 could be calculated as Equation 9 and Equations 10–13 according to Equations 3, 4, 8 as region Ω1:A2=s2−s1s2+23s1yx+3y2−23s1xx+3y2−2s1yx+3y2s2−s1s2+2s1xx+3y2λ2′=λ211′λ212′λ212′λ222′λ0λ211′=x′+3y′−2s1x′+3y′+43s1y′x′+3y′2+12s12x′2+y′2x′+3y′3x′+3y′−2s1λ212′=λ221′=−2s1y′+3x′x′+3y′2−43s12x′2+y′2x′+3y′−2s1x′+3y′3λ222′=x′+3y′−2s1x′+3y′+4s1x′x′+3y′2+4s12x′2+y′2x′+3y′3x′+3y′−2s1
In the same way, the left regions’ transformation equations could be established by multiplying the rotation matrix P in turn. Their Jacobian matrixes could be calculated. Finally, the thermal conductivities could be deduced as Equations 14–26.λ3′=λ311′λ312′λ321′λ322′λ0λ311′=3y′−x′−2s13y′−x′+43s1y′3y′−x′2+12s12x′2+y′23y′−x′33y′−x′−2s1λ312′=λ321′=2s1y′−3x′3y′−x′2+43s12x′2+y′23y′−x′−2s13y′−x′3λ322′=3y′−x′−2s13y′−x′+−4s1x′3y′−x′2+4s12x′2+y′23y′−x′33y′−x′−2s1λ4′=1+s1x′s1y′x′2s1y′x′2x′4+s12y′2x′4+s1x′3λ0λ5′=λ511′λ512′λ521′λ522′λ0λ511′=x′+3y′+2s1x′+3y′−43s1y′x′+3y′2+12s12x′2+y′2x′+3y′3x′+3y′+2s1λ512′=λ521′=2s1y′+3x′x′+3y′2−43s12x′2+y′2x′+3y′+2s1x′+3y′3λ522′=x′+3y′+2s1x′+3y′−4s1x′x′+3y′2+4s12x′2+y′2x′+3y′3x′+3y′+2s1λ6′=λ611′λ612′λ621′λ622′λ0λ611′=x′−3y′−2s1x′−3y′−43s1y′x′−3y′2+12s12x′2+y′2x′−3y′3x′−3y′−2s1λ612′=λ621′=2s1x′−3y′x′−3y′2+43s12x′2+y′2x′−3y′−2s1x′−3y′3λ622′=x′−3y′−2s1x′−3y′+4s1x′x′−3y′2+4s12x′2+y′2x′−3y′3x′−3y′−2s1
Thermal Flux Insulation Numerical Verification of the Hexagonal Cloak
The correctness of the above thermal conductivity tensors of hexagonal thermal cloak will be verified directly in the numerical simulation. We take one example of the hexagonal thermal cloak model with s1=12m,s2=16m, embedding in the square background filed with length of 60 m, which are established in the heat transfer module of the COMSOL Multiphysics software. For the hexagon with thermal cloak effects, the calculated thermal conductivity tensors λ′ of the cloak shell are input in the material properties setting module. The thermal conductivity coefficient of the original space λ0 , the background and the region inside the cloak are set to 1 W·m‐1K‐1 to highlight the role of the derived anisotropic thermal conductivity tensor. Besides, the hexagon without thermal cloak effects, whose thermal conductivity coefficient of the whole region is 1 W·m‐1K‐1, is set as a comparison. The boundary conditions are set as 400 K for the left boundary and 300 K for the right boundary of the background. Other boundaries of the background are set to thermal insulation.
The distribution of temperature with heat flow lines and isotherms are shown in Figure 2. At the case of the hexagon with cloak effect, it could be seen in Figures 2A,C that the heat flows from the left boundary to the right boundary, parallel to each other in the background filed and concentrates in the cloak shell so that both the heat flow and the temperature gradient inside the cloak are zero because of the anisotropic thermal conductivity of the cloak shell. However, the heat flux passes through the whole hexagon without the cloak effect as Figure 2B and a gradient temperature is shown in Figure 2D. The temperature of the center line along the X-axis of the hexagon with and without the cloak is displayed in Figure 3, which further illustrated that the uniform temperature field inside the cloak with more obvious temperature gradient in the cloak shell region because of the hexagonal cloak effect. Both Figures 2, 3 demonstrate that the background temperature field remain consistent with the cloak effect or not. This is attributable to the fact that after the heat flux bypasses the cloak, it reverts to its original path in the external region of the cloak. This indicates that the hexagonal thermal cloak derived in this paper could effectively redirect heat flux without exerting influence on the background temperature field, which proves the correctness of the calculation method of the hexagonal thermal invisibility cloak and its thermal conductivity tensors.
(A,B) The distribution of temperature and heat flow lines with and without the cloak effect. (C,D) The isotherms of the hexagon thermal cloak with and without the cloak effect.
Temperature distribution diagrams with a hexagonal shape in the center, shown in four panels (A, B, C, D). Arrows indicate heat flow direction, with a gradient from red (400K) to blue (300K). Panels A and B show distinct flow patterns, while C and D depict varied temperature gradients across the hexagon.
The temperature distribution on the center line along the X-axis of the hexagon with and without the thermal cloak effect.
Graph showing temperature in Kelvin versus the X-axis in meters. It compares plots with and without cloak effects. Purple circles indicate data with the cloak effect, and orange diamonds indicate data without. The graph includes regions labeled as background, cloak shell, and inside cloak, with varying background colors.Dodecahedral Thermal CloakDerivation of the Dodecahedral Cloak’s Thermal Conductivity Tensor
Different from the derivation of two-dimensional hexagonal thermal cloak, dodecahedron is a three-dimensional thermal cloak. Figure 4 illustrates the sketch map of rhombic dodecahedron thermal cloak. The rhombic dodecahedron thermal invisibility cloak is divided into 12 regions called as Ω1−12 according to the symmetry. In Figure 4A, the regions marked in blue are facing us and the regions marked in orange are the faces facing away. Figures 4B,C are two facets of the cloak. The yellow shell indicates the cloak setting region and the purple region in the middle is the heat hiding region. The location of region Ω1 is special which is parallel to the Y-Z plane and the X-axis passes through its midline point as Figure 5 shown.
(A) The sketch map of the rhombic dodecahedron thermal invisibility cloak (region Ω1−12). (B,C) are two facets of the cloak.
Diagram showing a polyhedral structure divided into sections, labeled with numbers one to twelve. Section A depicts a three-dimensional view with each section numbered. Section B shows a purple hexagonal face with three axes labeled X’, Y’, and Z’. Section C provides a side view with the same axes. The structure is primarily yellow with internal divisions outlined.
The schematic diagram of region Ω1 of the rhombic dodecahedron cloak.
Technical diagram showing two polygonal shapes labeled A and B on a three-dimensional coordinate system with axes Z and X. The shapes are connected by vectors O to O1 and O2, with marked angles φ and θ, and dimensions r1, r2, l, h, a, and b.
The green and pink diamonds represent the inner boundary R1θ,φ and outer boundary R2θ,φ of the rhombic dodecahedron thermal cloak in region Ω1, respectively. In triangles OO2Q and OAQ, Equation 27 is obtained by trigonometric functions:cosφ=al,sinθ=lr2
Therefore, the inner and outer boundaries of region Ω1 could be expressed as Equation 28:R1=bsinθcosφ,R2=asinθcosφ
For a three-dimensional heat cloak, the transformation equation of compressing region 0<r<R2θ,φ into region R1′θ′,φ′<r<R2′θ′,φ′ is Equation 29:r′=R2θ,φ−R1θ,φR2θ,φr+R1θ,φθ′=θφ′=φ
The region Ω1 is chosen as the minimum rotation element because the transformation equation of it is easily to set up in Cartesian coordinates:x′y′z′=a−ba000a−ba+bx000a−ba+bxxyz+b00
The transformation equations of region Ω3, Ω2, Ω4 could be set up by rotating the region Ω1 clockwise about the Y-axis with 90°,180°,270° as Figure 6 noted. Using region Ω2 as an example, the transformation equation of region Ω2 could be calculated by multiplying Equation 30 by the relevant rotation matrix P2=‐10001000‐1 as Equation 31:x′y′z′=a−ba000a−ba+b−x000a−ba+b−xxyz+−b00
The schematic diagram of region Ω1−4.
A geometric 3D shape with four numbered faces, labeled one to four in blue. The shape appears twisted, and is translucent yellow. An XYZ coordinate axis is displayed at the bottom left.
The rotation of the remaining regions is more special, no longer around the axis. They are divided into two categories, one by region Ω1 rotation as Figure 7A shown, and the other by region Ω2 rotation as Figure 7B. Use region Ω5 as an example. The axis of the rotation of region Ω5 is an edge of region Ω1, as Figure 8 shown. The region Ω5 could be got by rotating region Ω1 with 120° clockwise about the specified axis u5=63,33,0. The corresponding rotation matrix is P5=1222−122202212−22−12. The transformation equations of region Ω5 is Equation 32:x5′=b2+x3bσ1+a−ba−y2a−b2a+bx+z+2y2−2a−b4a−bzσ1y5′=2b2−xσ2−zσ2+ya−ba+bx+z+2yz5′=b2+z3bσ1+a−ba−y2a−b2a+bx+z+2y2−2a−b4a−bxσ1where σ1=2x+2z+22y, σ2=2bx2+z2+2y2+a‐ba4‐2a‐b4a.
(A) Regions Ω5, Ω8, Ω9,Ω11 are obtained by rotating region Ω1. (B) Regions Ω6, Ω7, Ω10,Ω12 are obtained by rotating region Ω2.
Two 3D geometric models labeled (A) and (B) with colored faces and black numbers. Model (A) has blue sections numbered 1, 5, 8, 9, 11 with green borders. Model (B) has pink sections numbered 2, 6, 7, 10, 12 with orange borders. Each has a coordinate axis labeled X’, Y’, Z’.
The rotation of region Ω5 (the axis of rotation u5 is marked in red).
Diagram of an angled purple structure labeled with numbers 1 and 5. A close-up inset shows a red line along the joint. A three-axis coordinate system in the bottom left is marked X', Y', and Z'.
In the similar way, the transformation equations of remain regions could be obtained by rotating the region Ω1 and region Ω2 with 120° counterclockwise or clockwise about their edges in the X-Y plane. Tables 1, 2 are the corresponding rotation axis and matrix of the region obtained by rotating region Ω1 and region Ω2, respectively. Similarly, their Jacobian matrixes could be obtained by the transform equations. Finally, their thermal conductivity tensors could be calculated. The calculating progress is achieved by MATLAB.
The rotation axis and rotation matrix of the region obtained by rotating region Ω1.
Regions obtained by rotating region Ω1
Rotation axis
Rotation matrix
Region Ω5
u5=63,33,0
P5=1222−122202212−22−12
Region Ω8
u8=63,−33,0
P8=12−22−12−220−221222−12
Region Ω9
u9=63,33,0
P9=122212220−22−1222−12
Region Ω11
u11=63,−33,0
P11=122212−22022−12−22−12
The rotation axis and rotation matrix of the region obtained by rotating region Ω2.
Region obtained by rotating region Ω2
Rotation axis
Rotation matrix
Region Ω6
u6=−63,−33,0
P6=1222−122202212−22−12
Region Ω7
u7=−63,33,0
P7=12−22−12−220−221222−12
Region Ω10
u10=−63,−33,0
P10=122212220−22−1222−12
Region Ω12
u12=−63,33,0
P12=12−2212−22022−12−22−12
For the thermal cloak with less regular geometries or non-symmetric shapes, the present method is applicable too. For an any unregular region, the cloak could be divided into n sub-regions Ωi=1,2,3..n., which are no longer obtained by rotating the same “region Ω1” that really exists. The “region Ω1” of the sub-region Ωi could be set as a dummy region with the same shape of Ωi, and its position should be perpendicular or parallel to the coordinate axis to establish the transformation equations of Ω1 easily.
Thermal Flux Insulation Numerical Verification of the Dodecahedral Cloak
The dodecahedral cloak with a=50m,b=30m, embedding in a cube background, is established in the heat transfer module of the finite element software as Figure 9A shown. Taking the thermal conductivity of the original space λ0, the middle stealth zone and the background as 1 W·m‐1K‐1, the calculated anisotropic thermal conductivity tensors λ′ of the dodecahedral cloak shell are set for the case of dodecahedron with cloak effect. For comparison, the dodecahedron without the cloak effects is considered by setting the thermal conductivity coefficient of the whole region is 1 W·m‐1K‐1. Set the upper and lower boundary of the background as 400 K and 300 K, respectively. Other boundaries are set to thermal insulation.
(A,B) The schematic diagram of the whole and half section of the rhombic dodecahedron thermal cloak with the background. (C,D) The half section’s temperature distribution of the background and the rhombic dodecahedron with the cloak or not.
Diagram illustrating the concept of thermal cloaking with four panels labeled A to D. Panel A shows a cube with internal structures, highlighting a section parallel to the YZ plane. Panel B displays a labeled half-section indicating the background, cloak shell, and thermal stealth region. Panels C and D depict temperature distribution, transitioning from cooler (blue) to warmer (red) colors, with annotations for uniform and gradient temperature. A color bar at the bottom represents temperatures from 300K to 400K.
The temperature distribution of the half section, which is taken parallel to YZ plane as Figure 9B, is given to facilitate the observation of the temperature distribution inside the rhombic dodecahedron thermal cloak. The uniform temperature filed is illustrated with the cloak effects in Figure 9C, which is different from the central gradient temperature without the cloak effects in Figure 9D.
The isothermal surface distribution of the rhombic dodecahedron with and without cloak effects is given in Figures 10A,B, respectively. Besides, Figures 10C,D show the temperature distribution of the rhombic dodecahedron’s tangent plane with and without the cloak. The temperature distribution on the center line of the rhombic dodecahedron and the background is illustrated in Figure 11. Without the heat cloak, it could be seen in Figure 10B that all isothermal surfaces are parallel to each other and the whole region is a gradient temperature filed from high to low. Figure 10A illustrates that the isothermal surfaces changes and buckle around the central invisibility region when the heat cloak is applied. The temperature gradient is concentrated at the cloak shell so the slope of the temperature line in the cloak setting region is greater than the absence of the cloak in Figure 11, and the central region in the dodecahedron is a uniform temperature field as Figure 10C shown. Moreover, in Figures 10, 11, the temperature filed of the background is identical whether the rhombic dodecahedron has the cloak effect or not, which proves that the dodecahedral thermal invisibility cloak doesn’t affect the background’s temperature filed. This proves that the dodecahedral thermal invisibility cloak plays a role in avoiding heat flow while the background filed isn’t disturbed, and the calculated thermal conductivity tensors are correct.
(A,B) The isothermal surface distribution of the rhombic dodecahedron with the thermal cloak effect or not. (C,D) Temperature distribution of the rhombic dodecahedron’s tangent plane with and without the cloak effect.
Four 3D geometric models labeled A to D, each displaying color gradients representing temperature from 300K (blue) to 400K (red). Models A and B are symmetrical with layered structure, while C and D are more angular and solid, showing temperature variation from cooler blues to warmer reds. An axis symbol and a temperature color scale accompany the models.
The temperature distribution on the center line of the background and the rhombic dodecahedron with and without the thermal cloak effect.
Graph showing temperature in Kelvin versus Z-axis position in meters. Blue circles represent data with the cloak effect, and green triangles represent data without it. Colored regions are labeled: gray for background, yellow for cloak shell, and pink for inside cloak. Temperature ranges from 300 to 420 Kelvin.Conclusion and Perspectives
In summary, the thermal conductivity tensors of hexagonal and dodecahedral thermal invisibility cloaks are derived in this work. The numerical verification shows that both hexagonal and dodecahedral thermal invisibility cloaks can avoid heat flow, and the temperature gradient in their thermal invisibility region is zero. This proves the correctness of the derivation method and the thermal conductivity tensors calculated and lays the foundation for adjusting thermal fields with complex structures.
3D morphologic mechanical metamaterials with specific functions have extraordinary properties due to their 3D building blocks. Researchers have been studying exotic static mechanical properties such as stiffness and strength close to theoretical limits [25], high mechanical elasticity or energy absorption [35–37] or negative Poisson ratios [38, 39]. However, the mechanical properties of the combination with the thermal field are still limited. The thermal conductivity tensors of hexagonal and dodecahedral thermal cloaks calculated in this paper aim to provide a uniform temperature field for the applications and prevent the mechanical properties from being affected by the non-uniform stress and deformation caused by the heat flow in the actual work. Furthermore, geometric configurations and deformation effects may change the thermal pathway [40–42], which could provide an effective way to adjust the anisotropic thermal conductivities of the thermal cloak.
Data Availability Statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
Author Contributions
YS conducted major research work and wrote this paper. JQ contributed to calculation and participated in the discussion of the research. YC and XY directed and participated in the discussion and analysis of the research work. YL directed the overall research work and guided the writing of the thesis. All authors contributed to the article and approved the submitted version.
Funding
The author(s) declare that financial support was received for the research and/or publication of this article. This work was supported by the National Science Foundation of China (Grant No. 52192633).
Conflict of Interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Generative AI Statement
The author(s) declare that no Generative AI was used in the creation of this manuscript.
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