  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.
