A spacecraft structure designed by composite phase change thermal control and digital intelligence

By adopting a combined design of composite phase change layer, thermal conductivity layer and reflective coating in the spacecraft thermal control structure, combined with hollowing of Schwarz P-type and Gyroid-type three-period extremely small curved surface structure, the problem of low laser thermal shock and phase change thermal storage rate of the spacecraft thermal control structure is solved, and efficient heat management and load bearing capacity are achieved.

CN119590646BActive Publication Date: 2025-05-16BEIJING INST OF TECH
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Patent Information

Application Number
CN202510142481.8
Authority / Receiving Office
CN · China
Patent Type
Patents(China)
Current Assignee / Owner
Filing Date
2025-02-10
Publication Date
2025-05-16
Estimated Expiration
2045-02-10

AI Technical Summary

Technical Problem

The thermal control structure of the spacecraft is difficult to resist laser thermal shock, and the phase change heat storage rate is low.

Method used

A spacecraft structure with composite phase change thermal control digital intelligence design is designed, including composite phase change layer, thermal conductivity layer and reflective coating. The composite phase change layer adopts a combination of Schwarz P-type and Gyroid-type three-period extremely small curved surface structures, and the thermal conductivity is improved through hollowing out treatment.

Benefits of technology

The spacecraft thermal control structure has improved the laser thermal shock resistance and phase change thermal storage efficiency, enhanced the load bearing capacity, and achieved lightweight.

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Abstract

The present invention provides a composite phase change thermal control digital intelligent design spacecraft structure, which relates to the field of aerospace technology and solves the technical problems that the thermal control structure of the spacecraft is difficult to resist laser thermal shock and the phase change heat storage rate is low. The spacecraft thermal control structure includes a composite phase change layer, a heat conductive layer and a reflective coating in sequence along the direction away from the spacecraft body; the composite phase change layer includes a first skeleton structure and a second skeleton structure, wherein the first skeleton structure is a Schwarz P-type three-period minimal surface structure obtained by hollowing treatment, and the second skeleton structure is a Gyroid-type three-period minimal surface structure obtained by hollowing treatment; the composite phase change layer also includes a phase change structure, which is embedded in the first pore and the first hollow of the first skeleton structure, and the second pore and the second hollow of the second skeleton structure. The spacecraft thermal control structure has high resistance to laser thermal shock and improves the phase change heat storage rate.
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Description

Technical Field

[0001] The present invention relates to the field of aerospace technology, and in particular to a spacecraft structure with composite phase change thermal control digital intelligent design. Background Art

[0002] Aerospace equipment, such as orbiting satellites, rovers, and space telescopes, usually use phase change thermal storage technology to eliminate the temperature shock caused by fluctuating thermal load environments, while making efficient use of thermal energy. In related engineering practices, the thermal control structure of spacecraft is not resistant to laser thermal shock, the thermal conductivity of the phase change material itself is low, and the phase change thermal storage rate is difficult to increase. Summary of the invention

[0003] The purpose of the present invention is to provide a spacecraft structure with a composite phase change thermal control digital intelligent design to solve the technical problems that the thermal control structure of the spacecraft is difficult to resist laser thermal shock and the phase change heat storage rate is low.

[0004] In order to achieve the above object, the present invention provides the following technical solutions:

[0005] In a first aspect, the present invention provides a composite phase change thermal control digital intelligent design spacecraft structure, in which the thermal control structure includes a composite phase change layer, a heat conductive layer and a reflective coating in sequence along a direction away from the spacecraft body;

[0006] The composite phase change layer includes a first skeleton structure and a second skeleton structure, wherein the first skeleton structure is a Schwarz P-type three-period minimal surface structure obtained by hollowing, and the second skeleton structure is a Gyroid-type three-period minimal surface structure obtained by hollowing;

[0007] The following relationship is satisfied between the first skeleton structure and the Schwarz P-type three-period minimal surface structure, and between the second skeleton structure and the Gyroid-type three-period minimal surface structure:

[0008] ;

[0009] ;

[0010] ;

[0011] in, The governing equations used to characterize the hollowed-out skeleton structure are, The governing equations used to characterize the unhollowed skeleton structure are, The governing equation for the wall thickness of the hollowed-out skeleton structure, R, is used to characterize the gradient of the hollowed-out skeleton structure. It is used to characterize the relative wall thickness, dimensionless, where x, y, and z are the coordinates in the X, Y, and Z directions of the three-dimensional space, respectively, and t is the offset of the controlled minimal surface along the normal direction;

[0012] The composite phase-change layer further includes a phase-change structure, and the phase-change structure is embedded in the first pores and the first hollows of the first skeleton structure and the second pores and the second hollows of the second skeleton structure.

[0013] According to at least one embodiment of the present invention, the governing equation of the Schwarz P-type three-periodic minimal surface structure is The following relationship is satisfied:

[0014] ;

[0015] Among them, x, y, z are the coordinates in the three-dimensional space in the X, Y, and Z directions, respectively, a is the unit cell size, n is the neutral plane offset value, and C is the wall thickness of the skeleton structure.

[0016] According to at least one embodiment of the present invention, the governing equation of the Gyroid-type three-periodic minimal surface structure is The following relationship is satisfied:

[0017] ;

[0018] Among them, x, y, z are the coordinates in the three-dimensional space in the X, Y, and Z directions, respectively, a is the unit cell size, n is the neutral plane offset value, and C is the wall thickness of the skeleton structure.

[0019] According to at least one embodiment of the present invention, the phase change material includes one of paraffin, a gallium-based alloy, and a bismuth-based alloy.

[0020] According to at least one embodiment of the present invention, the heat conductive layer includes a working medium accommodated in an aluminum alloy shell, and the working medium includes one of water, methanol, acetone and methane.

[0021] According to at least one embodiment of the present invention, the reflective coating includes an aluminum coating or a silver coating.

[0022] According to at least one embodiment of the present invention, the materials of the first skeleton structure and the second skeleton structure include one of AlSi10Mg type aluminum alloy and 2B50 type aluminum alloy respectively.

[0023] According to at least one embodiment of the present invention, the connection between the first skeleton structure and the second skeleton structure is a smooth transition connection.

[0024] According to at least one embodiment of the present invention, the control equation of the connection between the first skeleton structure and the second skeleton structure satisfies the following relationship:

[0025] ;

[0026] in, is the governing equation of the skeleton structure of the composite phase change layer, is the governing equation of the Gyroid type three-periodic minimal surface structure, is the governing equation of the Schwarz P-type three-periodic minimal surface structure, is the spatial weight function, satisfying = (x, y, z) ∈ [0, 1];

[0027] ;

[0028] in, is the surface of the transition boundary, and k is a constant of the transition gradient.

[0029] In a second aspect, the present invention further provides a spacecraft, comprising the spacecraft structure described in the first aspect.

[0030] Among the one or more technical solutions provided in the exemplary embodiments of the present invention, at least one of the following beneficial effects can be achieved.

[0031] The composite phase change thermal control digital intelligent design spacecraft structure of the exemplary embodiment of the present invention includes a composite phase change layer, a heat conducting layer and a reflective coating in sequence along the direction away from the body of the spacecraft, wherein the reflective coating has a high reflectivity and can be used as a barrier in laser protection; the composite phase change layer forms an effective heat transfer path with the reflective coating through the heat conducting layer. The composite phase change layer adopts a combination of two three-periodic minimal surface structures (Schwarz P type and Gyroid type), wherein the first skeleton structure is farther away from the body of the spacecraft than the second skeleton structure. Furthermore, based on the high stiffness and high strength of the hollow form of bamboo, a hollow structure is also adopted on the first skeleton structure and the second skeleton structure. Specifically, a first hollow is formed on the first skeleton structure and a second hollow is formed on the second skeleton structure, so that the entire skeleton structure of the composite phase change layer can withstand a larger load under the same mass, which is also conducive to lightweight.

[0032] Furthermore, by placing the second skeleton structure close to the body of the spacecraft, the deformation resistance of the thermal control structure can be improved. By placing the first skeleton structure on the side of the second skeleton structure away from the body of the spacecraft, the first skeleton structure has good thermal conductivity, and the first skeleton structure and the second skeleton structure are connected by a smooth transition, so that when the phase change material flows in the pores of the above two, the flow direction can be continuously changed, thereby enhancing the convective heat transfer capacity of the phase change material. Experiments show that the disadvantage of the traditional phase change material's own low thermal conductivity can be overcome by the smooth transition connection of the hollowed first skeleton structure and the hollowed second skeleton structure, so that the phase change heat storage efficiency is improved. Based on this, in the composite phase change thermal control digital intelligent design of the spacecraft structure of the exemplary embodiment of the present invention, when the reflective coating is irradiated by laser, the heat energy will be transferred to the composite phase change layer through the heat conductive layer, and the composite phase change layer will quickly transfer the heat in the phase change material and store it for standby use, for example, it can be used for temperature control of the spacecraft body in a low temperature environment. From the above, it can be seen that the thermal control structure can have the characteristics of high heat absorption efficiency, resistance to laser thermal shock, strong load bearing capacity and lightweight. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] The accompanying drawings illustrate exemplary embodiments of the present invention and together with the description serve to explain the principles of the present invention, and these drawings are included to provide a further understanding of the present invention and are incorporated in and constitute a part of this specification;

[0034] Figure 1 is an isometric structural schematic diagram of a thermal control structure according to an embodiment of the present invention;

[0035] Figure 2 is an axonometric structural schematic diagram of a skeleton structure according to an embodiment of the present invention;

[0036] Figure 3 is a schematic diagram of an axonometric structure of a skeleton structure (with a hollow core) according to an embodiment of the present invention;

[0037] Figure 4A is an axonometric structural diagram of a skeleton structure from another viewing angle according to an embodiment of the present invention;

[0038] Figure 4B is a schematic diagram of the axonometric structure of a composite phase change layer according to an embodiment of the present invention;

[0039] Figure 5 is a simulation schematic diagram of a second skeleton structure according to an embodiment of the present invention;

[0040] Figure 6 is a simulation schematic diagram of a second skeleton structure (with a hollow core) according to an embodiment of the present invention;

[0041] Figure 7 is a schematic diagram of a deformation of a skeleton structure (with a hollow core) according to an embodiment of the present invention;

[0042] Figure 8 is a force-displacement graph of a skeleton structure (with a hollow core) according to an embodiment of the present invention.

[0043] Figure numerals: 10, composite phase change layer; 11, first skeleton structure; 111, first hollow; 12, second skeleton structure; 122, second hollow; 20, phase change structure; 30, heat conductive layer; 40, reflective coating. DETAILED DESCRIPTION

[0044] In order to make the technical problems, technical solutions and beneficial effects to be solved by the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0045] Figure 1 Schematic diagram of the isometric structure of the thermal control structure according to an embodiment of the present invention. Figure 1 As shown, the composite phase change thermal control digital intelligent design spacecraft structure provided by an exemplary embodiment of the present invention includes a composite phase change layer 10, a heat conductive layer 30 and a reflective coating 40 in sequence along the direction away from the spacecraft body.

[0046] The reflective coating 40 includes an aluminum coating or a silver coating. Aluminum or silver, which is a highly reflective material, is disposed on the thermal control structure in the form of a coating. Compared with a thin film, it is not limited by volume and shape, and has a mature preparation process and low cost, which is more conducive to laser protection. Compared with ceramic and polymer materials, metal materials have a higher reflectivity.

[0047] The heat conducting layer 30 can be designed by heat transfer using the heat pipe principle. Specifically, the heat pipe is based on the gas-liquid phase change heat transfer principle and has an ultra-high heat transfer capacity. Its equivalent heat transfer coefficient is much greater than that of common high thermal conductivity materials such as metals and graphene. The heat pipe consists of a closed cavity, a capillary wick and a working fluid. It is a device that achieves rapid heat transfer by means of the phase change process of evaporation and condensation of the working fluid under a vacuum state.

[0048] The heat pipe provided by the exemplary embodiment of the present invention has an aluminum alloy shell forming a closed cavity, and the working fluid may include one of water, methanol, acetone and methane.

[0049] The composite phase change layer 10 is formed by a skeleton structure and a phase change structure, wherein the material of the phase change structure can be one of paraffin, gallium-based alloy and bismuth-based alloy.

[0050] The following uses paraffin as an example to explain that paraffin can be filled into the pores and hollows of the skeleton structure by vacuum impregnation. Specifically, at a temperature higher than the melting point of paraffin, the composite phase change layer 10 is prepared by infiltration into the skeleton structure. In this process, the paraffin is adsorbed into the skeleton structure under capillary force to form the composite phase change layer 10.

[0051] The connection between the first skeleton structure 11 and the second skeleton structure 12 in the skeleton structure of the composite phase change layer 10 is a smooth transition connection. The transition structure of the two skeleton structures allows the internal flow direction of the fluid to change continuously, which can significantly enhance the convective heat transfer capacity of the paraffin fluid.

[0052] Figure 2 Schematic diagram of the isometric structure of the skeleton structure according to an embodiment of the present invention. Figure 2 As shown, the first skeleton structure 11 is obtained by hollowing out the Schwarz P-type three-periodic minimal surface structure, and the second skeleton structure 12 is obtained by hollowing out the Gyroid-type three-periodic minimal surface structure. The three-periodic minimal surface (Triplyperiodic minimal surface, TPMS) used in the skeleton structure is a minimal surface that is periodic in the three-dimensional xyz direction and has the characteristics of being smooth and highly connected in holes. It should be noted that Figure 2 and Figure 3 The top surfaces are all cross-sections.

[0053] The skeleton structure constructed using two three-periodic minimal surface structures not only has the characteristics of high specific strength and high specific stiffness of traditional porous structures, but also has the characteristics of large specific surface area, high porosity and controllable structure.

[0054] Specifically, the governing equations of the Schwarz P-type three-periodic minimal surface structure are The following relationship is satisfied:

[0055] ;Formula 1,

[0056] Among them, x, y, z are the coordinates in the three-dimensional space in the X, Y, and Z directions, respectively, a is the unit cell size, n is the neutral plane offset value, and C is the wall thickness of the skeleton structure.

[0057] Governing equations of Gyroid-type three-periodic minimal surface structures The following relationship is satisfied:

[0058] ; Formula 2,

[0059] Among them, x, y, z are the coordinates in the three-dimensional space in the X, Y, and Z directions, respectively, a is the unit cell size, n is the neutral plane offset value, and C is the wall thickness of the skeleton structure.

[0060] Figure 3 is a schematic diagram of an axonometric structure of a skeleton structure (with a hollow core) according to an embodiment of the present invention. Figure 3 As shown, the skeleton structure provided by the exemplary embodiment of the present invention is based on the principle that bamboo has a hollow form with high rigidity and strength. The first skeleton structure 11 and the second skeleton structure 12 are both hollowed out, so that the skeleton structure has higher strength and energy absorption capacity. The hollow structure can withstand a larger load under the same mass and reduce the overall weight.

[0061] For example, in order to obtain a hollow porous skeleton of a minimal surface, a wall thickness control function is introduced into the four-dimensional implicit level set function of the minimal surface: , to adjust the wall thickness of the hollow porous skeleton.

[0062] Specifically, the first skeleton structure 11 having the first hollow 111 and the original first skeleton structure 11 (not hollowed out), and the second skeleton structure 12 having the second hollow 122 and the original second skeleton structure 12 (not hollowed out) both satisfy the following relationship:

[0063] ; Formula 3,

[0064] ; Formula 4,

[0065] ; Formula 5,

[0066] in, The governing equations used to characterize the hollowed-out skeleton structure are, The governing equations used to characterize the unhollowed skeleton structure are, The governing equation for the wall thickness of the hollowed skeleton structure is used to characterize the gradient of the hollowed skeleton structure. It is used to characterize the relative wall thickness. It is dimensionless. x, y, and z are the coordinates in the X, Y, and Z directions of the three-dimensional space respectively. t is the offset of the controlled minimal surface along the normal direction.

[0067] In some embodiments, the governing equation of the transition connection between the first skeleton structure and the second skeleton structure satisfies the following relationship:

[0068] ; Formula six,

[0069] in, is the governing equation of the skeleton structure of the composite phase change layer, is the governing equation of the Gyroid type three-periodic minimal surface structure, is the governing equation of the Schwarz P-type three-periodic minimal surface structure, is the spatial weight function, satisfying = (x, y, z) ∈ [0, 1];

[0070] ; Formula 7,

[0071] in, is the surface of the transition boundary, and k is the constant of the transition gradient. The larger the k value, the narrower the transition zone.

[0072] Exemplarily, the value range of k is 1 to 4, and can be 2, 2.5 or 3.

[0073] Figure 4A is an axonometric structural diagram of a skeleton structure from another viewing angle according to an embodiment of the present invention; Figure 4B Schematic diagram of the isometric structure of the composite phase change layer according to an embodiment of the present invention. Figure 4A-4B As shown, the composite phase change layer provided by the exemplary embodiment of the present invention is formed into a phase change structure 20 in the first hollow 111 and the first pore of the first skeleton structure 11 and the second hollow 122 and the second pore of the second skeleton structure 12 by vacuum impregnation of paraffin.

[0074] In a specific implementation, the modeling software is Ntopology 3D modeling software, the second skeleton structure 12 uses a Schwarz P type TPMS structure to ensure heat transfer efficiency, and the first skeleton structure 11 uses a Gyroid type TPMS structure to enhance anti-deformation capability.

[0075] The unit cell size is set to 5 mm, the volume fraction of the two skeleton structures is 30%, and the three-dimensional space size is Then it is hollowed out, and the wall thickness of the hollow structure is 0.5mm.

[0076] The hollow TPMS structure is printed in an integrated manner using an additive manufacturing process. The material of the hollow TPMS structure is AlSi10Mg, and the phase change material is RT50 paraffin.

[0077] The paraffin wax and the hollow TPMS structure are formed into a composite phase change layer 10 by vacuum impregnation. The heat conductive layer 30 uses a flat heat pipe to accelerate the heat transfer process. Then the reflective coating 40 (aluminum coating) is formed on the surface of the thermal control structure. The hollow TPMS structure formed by the above integrated printing has high heat absorption capacity, load-bearing capacity and lightweight characteristics.

[0078] Taking the second skeleton structure 12 as an example, the evolution process of the paraffin volume fraction is simulated. Figure 5 and hollowed out Figure 6 As shown, Figure 5is a simulation schematic diagram of a second skeleton structure according to an embodiment of the present invention; Figure 6 is a schematic simulation diagram of a second skeleton structure (with a hollow core) according to an embodiment of the present invention.

[0079] Depend on Figure 5 and Figure 6 As shown, at the same heating time point, the liquid phase ratio of the hollow-treated second skeleton structure 12 is significantly higher than that of the non-hollow-treated second skeleton structure 12. This is because the hollow-treated second skeleton structure 12 has a higher thermal conductivity, which can improve the heat transfer inside the paraffin.

[0080] Table 1 Thermal storage performance of composite phase change layer

[0081]

[0082] It can be seen from Table 1 that the heat storage rate of the composite phase change layer of the hollowed second skeleton structure 12 is 1.18 times higher than that of the composite phase change layer of the second skeleton structure 12 without hollowing; the heat storage rate of the composite phase change layer of the hollowed first skeleton structure 11 is 0.9 times higher than that of the composite phase change layer of the first skeleton structure 11 without hollowing. Therefore, the hollowed skeleton structure can facilitate the rapid diffusion of heat when the spacecraft is irradiated by laser and be absorbed and stored by the phase change material for temperature control of the spacecraft body, effectively eliminating the temperature shock caused by the fluctuating heat load environment, and making efficient use of heat energy.

[0083] Figure 7 is a schematic diagram of a deformation of a skeleton structure (with a hollow core) according to an embodiment of the present invention; Figure 8 is a force-displacement curve diagram of a skeleton structure (with a hollow core) according to an embodiment of the present invention. Figure 7 and Figure 8 As shown in FIG. 1 , the deformation process of the skeleton structure formed by connecting the hollow first skeleton structure 11 and the hollow second skeleton structure 12 when being compressed. Figure 8 In the force-displacement curve shown, the horizontal axis is the strain, the vertical axis is the ratio of stress to mass, and the area enclosed by the curve and the X-axis represents the energy absorption capacity of the skeleton structure.

[0084] Specifically, PG represents a mixed structure of the hollowed-out first skeleton structure 11 and the hollowed-out second skeleton structure 12; G represents a single structure of the hollowed-out second skeleton structure 12; and P represents a single structure of the hollowed-out first skeleton structure 11. Figure 8As shown, the hollowed-out second skeleton structure 12 has a higher yield strength and energy absorption capacity; the hybrid structure of the hollowed-out first skeleton structure 11 and the hollowed-out second skeleton structure 12 inherits the advantages of the hollowed-out second skeleton structure 12, has a higher yield strength and energy absorption capacity, and improves the disadvantages of the hollowed-out first skeleton structure 11 (low yield strength and insufficient energy absorption capacity).

[0085] An exemplary embodiment of the present invention further provides a spacecraft, comprising the spacecraft structure in the above embodiment.

[0086] The technical advantages of the above-mentioned spacecraft over the prior art are the same as the technical advantages of the spacecraft structure of the above-mentioned embodiment, which will not be repeated here.

[0087] It should be understood by those skilled in the art that the above embodiments are only for the purpose of clearly illustrating the present invention, and are not intended to limit the scope of the present invention. For those skilled in the art, other changes or modifications may be made based on the above disclosure, and these changes or modifications are still within the scope of the present invention.

Claims

1. A composite phase change thermal control digital intelligent design of a spacecraft structure, characterized in that: Along the direction away from the spacecraft body, the thermal control structure includes a composite phase change layer, a heat conductive layer and a reflective coating in sequence; The composite phase change layer includes a first skeleton structure and a second skeleton structure, wherein the first skeleton structure is a Schwarz P-type three-period minimal surface structure obtained by hollowing, and the second skeleton structure is a Gyroid-type three-period minimal surface structure obtained by hollowing; The following relationship is satisfied between the first skeleton structure and the Schwarz P-type three-period minimal surface structure, and between the second skeleton structure and the Gyroid-type three-period minimal surface structure: ; ; ; in, The governing equations used to characterize the hollowed-out skeleton structure are, The governing equations used to characterize the unhollowed skeleton structure are, The governing equation for the wall thickness of the hollowed-out skeleton structure, R, is used to characterize the gradient of the hollowed-out skeleton structure. It is used to characterize the relative wall thickness, dimensionless, where x, y, and z are the coordinates in the X, Y, and Z directions of the three-dimensional space, respectively, and t is the offset of the controlled minimal surface along the normal direction; The composite phase-change layer further includes a phase-change structure, and the phase-change structure is embedded in the first pores and the first hollows of the first skeleton structure and the second pores and the second hollows of the second skeleton structure.

2. The spacecraft structure according to claim 1, characterized in that: The governing equations of the Schwarz P-type three-periodic minimal surface structure The following relationship is satisfied: ; Among them, x, y, z are the coordinates in the three-dimensional space in the X, Y, and Z directions, respectively, a is the unit cell size, n is the neutral plane offset value, and C is the wall thickness of the skeleton structure.

3. The spacecraft structure according to claim 1, characterized in that: The governing equations of the Gyroid-type three-periodic minimal surface structure The following relationship is satisfied: ; Among them, x, y, z are the coordinates in the three-dimensional space in the X, Y, and Z directions, respectively, a is the unit cell size, n is the neutral plane offset value, and C is the wall thickness of the skeleton structure.

4. The spacecraft structure according to any one of claims 1 to 3, characterized in that: The material of the phase change structure includes one of paraffin, gallium-based alloy and bismuth-based alloy.

5. The spacecraft structure according to claim 4, characterized in that: The heat-conducting layer includes a working medium accommodated in an aluminum alloy shell, and the working medium includes one of water, methanol, acetone and methane.

6. The spacecraft structure according to claim 4, characterized in that: The reflective coating includes an aluminum coating or a silver coating.

7. The spacecraft structure according to claim 4, characterized in that: The materials of the first skeleton structure and the second skeleton structure include one of AlSi10Mg aluminum alloy and 2B50 aluminum alloy respectively.

8. The spacecraft structure according to any one of claims 1 to 3, characterized in that: The connection between the first skeleton structure and the second skeleton structure is a smooth transition connection.

9. The spacecraft structure according to claim 8, characterized in that: The control equation of the connection between the first skeleton structure and the second skeleton structure satisfies the following relationship: ; in, is the governing equation of the skeleton structure of the composite phase change layer, is the governing equation of the Gyroid type three-periodic minimal surface structure, is the governing equation of the Schwarz P-type three-periodic minimal surface structure, is the spatial weight function, satisfying = (x, y, z) ∈ [0, 1]; , in, is the surface of the transition boundary, and k is a constant of the transition gradient.

Citation Information

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