Long-time behavior of solutions to fluid dynamic shape optimization problems via phase-field method

TL;DR

This work addresses the long-time behavior of shape and topology optimization for flows governed by the time-dependent Navier–Stokes equations, using a phase-field design within a fixed domain and a porous-media formulation. It develops a rigorous link between time-dependent and stationary optimizations by proving that global minimizers of the time-dependent functional $J_T$ converge in value to a global minimizer of the stationary functional $J_s$ as the horizon $T$ grows, and it provides an explicit convergence rate. The authors establish existence and regularity for both time-dependent and stationary governing equations, derive first-order optimality conditions via adjoint systems, and prove that the time-dependent minimizers converge to stationary ones, with additional results on the turnpike-like behavior. Numerical experiments in FreeFem++ corroborate the theory, showing $|J_T(\varphi^T)-J_s(\varphi^s)| = \mathcal{O}(T^{-1})$ and convergence of $\varphi^T$ toward $\varphi^s$, illustrating practical benefits for large-horizon optimization. The work contributes to the understanding of bilinear phase-field shape optimization under unsteady fluids and provides a framework for turnpike phenomena in such PDE-constrained problems.

Abstract

We investigate the long time behavior of solutions to a shape and topology optimization problem with respect to the time-dependent Navier--Stokes equations. The sought topology is represented by a stationary phase-field that represents a smooth indicator function. The fluid equations are approximated by a porous media approach and are time-dependent. In the latter aspect, the considered problem formulation extends earlier work. We prove that if the time horizon tends to infinity, minima of the time-dependent problem converge towards minima of the corresponding stationary problem. To do so, a convergence rate with respect to the time horizon, of the values of the objective functional, is analytically derived. This allowed us to prove that the solution to the time-dependent problem converges to a phase-field, as the time horizon goes to infinity, which is proven to be a minimizer for the stationary problem. We validate our results by numerical investigation.

Figures (7)

• Figure 1: The geometrical setup using a phase field function $\varphi$.
• Figure 2: Geometric setup of the numerical example. We consider a flow along a channel with Poiseuille-type Dirichlet data $\mathbf{g}_s$ on $\Gamma_{i}$ and $\Gamma_{o}$ and an enclosed obstacle.
• Figure 3: This image shows the desired velocity field $\mathbf{u}_d$ (in magnitude and by streamlines) together with the zero-level set of $\varphi_d$. Despite the porous media approximation of the obstacle, we observe a substantial decrease of velocity inside the region $\{x\in\Omega: \varphi(x) \le 0\}$.
• Figure 4: We show the approximate design function at iteration number $1,8,15,22,29,37$ of the optimization process from top left to bottom right, where the 37$^{th}$ iterate corresponds to the optimal shape. The figures also include the zero-level set of $\varphi_d$, for comparison. The images are cropped to the subset $(0.9,2.0)\times (0.225,0.775)$ of $\Omega$. We observe that after the first iteration we achieve a reasonable guess for the location and the optimal topology which subsequently shrink towards the optimal shape, which also converges towards the target phase-field $\varphi_d$.
• Figure 5: The figure shows the evolution of $J_s$ over the optimization process. The optimization stops after 37 iterations by reaching the stopping criterion. Moreover, we show the evolution of the individual components of $J_s$. Furthermore, we observe a substantial decrease on the tracking part $T_s$, while the regularization terms $P_s$ and $\gamma\mathbb{E}_\varepsilon$ exhibit a minimal decrease and equilibrates early on during the optimization process.

Theorems & Definitions (38)

• Claim 2.1
• Remark 2.2
• Remark 3.6
• Theorem 4.1
• Theorem 4.2
• Remark 4.3
• Proposition 4.4
• proof
• Theorem 4.5
• Proposition 4.6