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

Abstract

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.

References in corpus (9)

• Numerical Approximations for a three components Cahn-Hilliard phase-field Model based on the Invariant Energy Quadratization method
• An efficient iterative thresholding method for image segmentation
• Efficient Second Order Unconditionally Stable Schemes for a Phase-field Moving Contact Line Model Using Invariant Energy Quadratization Approach
• A third order BDF energy stable linear scheme for the no-slope-selection thin film model
• Sharp interface limit for a phase field model in structural optimization
• An extension of the projected gradient method to a Banach space setting with application in structural topology optimization
• A phase field approach to shape optimization in Navier--Stokes flow with integral state constraint
• Numerical analysis of a topology optimization problem for Stokes flow
• An efficient threshold dynamics method for topology optimization for fluids

Cited by in corpus (3)

• An Adaptive Phase-Field Method for Structural Topology Optimization
• Convergence Analysis of an Adaptive Nonconforming FEM for Phase-Field Dependent Topology Optimization in Stokes Flow
• On the Crouzeix-Raviart Finite Element Approximation of Phase-Field Dependent Topology Optimization in Stokes Flow