Applying a phase field approach for shape optimization of a stationary Navier-Stokes flow

TL;DR

This paper introduces a phase field method for shape optimization in stationary Navier-Stokes flows, incorporating Ginzburg-Landau regularization and deriving optimality conditions for diffuse and sharp interface limits.

Contribution

It develops a well-posed diffuse interface formulation for shape optimization in fluid flows and derives optimality conditions without extra regularity assumptions.

Findings

Well-posed diffuse interface problem established

Optimality conditions derived for diffuse and sharp interfaces

Necessary optimality system obtained via geometric variations

Abstract

We apply a phase field approach for a general shape optimization problem of a stationary Navier-Stokes flow. To be precise we add a multiple of the Ginzburg--Landau energy as a regularization to the objective functional and relax the non-permeability of the medium outside the fluid region. The resulting diffuse interface problem can be shown to be well-posed and optimality conditions are derived. We state suitable assumptions on the problem in order to derive a sharp interface limit for the minimizers and the optimality conditions. Additionally, we can derive a necessary optimality system for the sharp interface problem by geometric variations without stating additional regularity assumptions on the minimizing set.