Thermal-fluid topology optimization with unconditional energy stability and second-order accuracy via phase-field model

This paper aims to establish a novel and efficient topology optimization method for the thermal-fluid. To adaptively design the fluid–solid coupling structure and make the objective energy to dissipate, the proposed method considers several constraints, such as the volume conservation, inlet and outlet flow velocity field and fluid–solid boundary constraints. The governing system includes the phase-field model, the steady state Darcy equation and the heat transfer equation. Under the constraints of multiple physical fields, we prove the existence of minimal solutions to the optimization problem. We use a Crank–Nicolson (CN) type scheme to discretize the governing system. The multigrid method is used to solve the resulting system of discrete equations. We prove the boundedness and unconditional stability of the original energy, which implies that a large time step can be used. The proposed discrete system is both spatially and temporally second-order accurate. Various computational tests have been performed to demonstrate that the numerical approach is efficient in designing the complicated structures of thermal fluid flows.