Triply periodic minimal surfaces based topology optimization for the hydrodynamic and convective heat transfer

Triply periodic minimal surfaces (TPMS), possessing diverse and crucial properties for thermal convection analysis, hold significant promise for topology optimization in convective and hydrodynamic systems. This study aims to develop an innovative topology optimization method based on TPMS for channel shape design. We focus on maximizing heat dissipation and mass flux while maintaining a constant mean curvature. The channel structures undergo free evolution during the optimization process, resulting in efficient and intricate geometries based on the TPMS assumption. To incorporate TPMS properties, we modify the original energy formulation, which comprises kinetic energy, thermal energy, and Ginzburg–Landau energy. The governing system encompasses the phase field model, the Darcy–Stokes model, and the reaction–diffusion heat transfer model. The hydrodynamic response within the phase change structure is computed by solving the transient Darcy–Stokes equation augmented with a temperature-dependent diffusion term. To achieve second-order temporal and spatial accuracy, we employ the Crank–Nicolson method for the time scale and the central difference method for the spatial scale. Through the utilization of the Lagrange multiplier method, we establish the unconditional decrease of the discretized original energy. This implies that the proposed scheme can utilize a large time step. To verify the robustness of our proposed method, we conduct several numerical tests from both qualitative and quantitative perspectives. The results demonstrate the effectiveness of our method in enhancing heat and mass transfer capabilities.