Shape Optimization Constrained by Time-Dependent Stokes Flow: Modeling, Analysis and Numerical Simulation

This paper focuses on mathematical modeling of shape design, shape sensitivity analysis, and numerical implementation of shape optimization constrained by the time-dependent Stokes flow. The optimization models aim to minimize two classical shape functionals: energy dissipation and least-squares for velocity tracking. A porous medium model is applied to formulate the time-dependent Stokes equations on a fixed domain consisting of both fluid and solid regions. The existence of a solution to the perimeter regularized shape optimization problem is proved. For shape sensitivity analysis, the existence of the material derivative of the PDE solution and the Eulerian derivative of the shape functional have been rigorously established. Convected level set evolution governed by the distributed Eulerian derivative is applied to address shape design problems, such as pipe design, obstacle shape design, and shape reconstruction. Numerical examples demonstrate the effectiveness of the proposed algorithms in various scenarios.