Topology Optimization in Thermal-Fluid Coupling by a Phase-Field Method

We consider topology optimization in thermal-fluid coupling by using a phase-field model. The existence of a solution to the optimization problem is proved. The Fréchet differentiability of the state variables w.r.t. the phase-field function is shown. The gradient flow formulations are presented for both smooth and obstacle potentials. Then we propose a stabilized semi-implicit scheme for the gradient flow equation and use a primal-dual active set for gradient flow in variational inequality to solve the resulting optimal control problems of the phase-field model. Conforming mixed finite elements are used to discretize both the state and adjoint variables. The existence and uniqueness of the governing coupled thermal-fluid equations in finite element discretization are proved. Moreover, the existence of a solution to the discrete optimization problem is discussed. Numerical experiments, such as radiator design, show the effectiveness of the algorithm proposed.