Hierarchical topology optimization for transient heat conduction in porous media with microstructure-dependent property

A topology optimization framework for designing both micro- and macrostructures is developed for transient heat transfer in porous media. The proposed micro-macro optimization model incorporates the size effect of the microstructural surface area within a density-based and homogenization framework. As a density-based approach, the adjacent design variable is used to interpolate the heat transfer occurring at the microstructure’s interface, while other properties are determined using the standard power-law function. Optimal topologies for both micro- and macrostructures are obtained by solving two-scale optimization problems, which are addressed using a gradient-based optimizer and the proposed micro- and macro analytical sensitivity formulations. Numerical results demonstrate that the designed topologies from the proposed framework are consistent with steady-state benchmarks, while varying heating time leads to different optimized results. Furthermore, as the size effect increases, the optimized macrostructure in the steady-state condition begins to resemble the topology obtained in the unsteady-state condition.