Artificially designed composite materials consist of microstructures, that exhibit various thermal properties depending on their shapes, such as anisotropic thermal conductivity. One of the representative applications of such composite materials for thermal control is the thermal cloak. This study proposed a topology optimization method based on a level set method for a heat conduction problem to optimally design composite materials that achieve a thermal cloak. The homogenization method was introduced to evaluate its effective thermal conductivity coefficient. Then, we formulated a multiscale topology optimization method for the composite materials in the framework of the homogenization method, where the microstructures were optimized to minimize objective functions defined using the macroscopic temperature field. We presented examples of optimal structures in a two-dimensional problem and discussed the validity of the obtained structures.

翻译：人工设计的复合材料由微观结构组成，根据其形状表现出各向异性热导率等多种热学特性。此类复合材料在热控制领域的代表性应用之一即为热斗篷。本研究针对热传导问题，提出了一种基于水平集法的拓扑优化方法，用于优化设计可实现热斗篷功能的复合材料。引入均匀化方法来评估其等效热导率系数。在此基础上，我们在均匀化方法的框架下提出了一种针对复合材料的多尺度拓扑优化方法，其中通过优化微观结构来最小化由宏观温度场定义的目标函数。我们给出了二维问题中优化结构的实例，并讨论了所得结构的有效性。