A provably efficient monotonic-decreasing algorithm for shape optimization in Stokes flows by phase-field approaches

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