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

TL;DR

This paper studies the long-term behavior of solutions in a fluid shape optimization problem using phase-field methods, showing convergence to stationary solutions as time approaches infinity.

Contribution

It extends previous work by analyzing the convergence of time-dependent solutions to stationary minimizers in a phase-field shape optimization framework.

Findings

Minima of the time-dependent problem converge to stationary minima as time tends to infinity.

A convergence rate for the objective functional with respect to the time horizon is derived.

Numerical results validate the theoretical convergence results.

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.