  In this work, we study shape optimization problems in the Stokes flows. By
phase-field approaches, the resulted total objective function consists of the
dissipation energy of the fluids and the Ginzburg--Landau energy functional as
a regularizing term for the generated diffusive interface, together with
Lagrangian multiplayer for volume constraint. An efficient decoupled scheme is
proposed to implement by the gradient flow approach to decrease the objective
function. In each loop, we first update the velocity field by solving the
Stokes equation with the phase field variable given in the previous iteration,
which is followed by updating the phase field variable by solving an
Allen--Cahn-type equation using a stabilized scheme. We then take a cut-off
post-processing for the phase-field variable to constrain its value in $[0,1]$.
In the last step of each loop, the Lagrangian parameter is updated with an
appropriate artificial time step. We rigorously prove that the proposed scheme
permits an unconditionally monotonic-decreasing property, which allows us to
use the adaptive mesh strategy. To enhance the overall efficiency of the
algorithm, in each loop we update the phase field variable and Lagrangian
parameter several time steps but update the velocity field only one time.
Numerical results for various shape optimizations are presented to validate the
effectiveness of our numerical scheme.
