Study on Thermal Cloaks for Hexagon and Dodecahedron With Symmetry

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.