Energy Stable Gradient Flow Schemes for Shape and Topology Optimization in Navier-Stokes Flows

We study topology optimization governed by the incompressible Navier-Stokes equations using a phase field model. Unconditional energy stability is shown for the gradient flow in continuous space. The novel generalized stabilized semi-implicit schemes for the gradient flow in first-order time discretization of Allen-Cahn and Cahn-Hilliard types are proposed to solve the resulting optimal control problem. With the Lipschitz continuity for state and adjoint variables, the energy stability for time and full discretization has been proved rigorously on the condition that the stabilized parameters are larger than specific values. The proposed gradient flow scheme can work with large time steps and exhibits a constant coefficient system in full discretization, which can be solved efficiently. Numerical examples in 2d and 3d show the effectiveness and robustness of the optimization algorithms proposed.