Shape optimization for surface functionals in Navier--Stokes flow using a phase field approach

TL;DR

This paper introduces a phase field method for shape and topology optimization in Navier--Stokes flows, incorporating a Ginzburg--Landau functional for perimeter penalization, and demonstrates its effectiveness through numerical experiments.

Contribution

It develops a novel phase field approach for fluid shape optimization, including existence proofs, optimality conditions, and a new modeling variant for surface functionals.

Findings

Existence of minimizers established.

Derived first order necessary conditions.

Numerical results confirm the method's applicability.

Abstract

We consider shape and topology optimization for fluids which are governed by the Navier--Stokes equations. Shapes are modelled with the help of a phase field approach and the solid body is relaxed to be a porous medium. The phase field method uses a Ginzburg--Landau functional in order to approximate a perimeter penalization. We focus on surface functionals and carefully introduce a new modelling variant, show existence of minimizers and derive first order necessary conditions. These conditions are related to classical shape derivatives by identifying the sharp interface limit with the help of formally matched asymptotic expansions. Finally, we present numerical computations based on a Cahn--Hilliard type gradient descent which demonstrate that the method can be used to solve shape optimization problems for fluids with the help of the new approach.