# A phase field approach to shape optimization in Navier--Stokes flow with integral state constraint

## TL;DR

This paper develops a phase field method for shape optimization in Navier--Stokes flow, incorporating integral state constraints and regularization, with numerical demonstrations on topology optimization and lift maximization.

## Contribution

It extends existing shape optimization frameworks by integrating phase field, porous medium models, and multiple constraints, including center of mass, volume, and drag, with numerical validation.

## Key findings

- Successful implementation of the phase field approach for shape optimization.
- Numerical results demonstrate effectiveness in topology optimization and lift maximization.
- Comparison shows advantages over previous drag minimization methods.

## Abstract

We consider the shape optimization of an object in Navier--Stokes flow by employing a combined phase field and porous medium approach, along with additional perimeter regularization. By considering integral control and state constraints, we extend the results of earlier works concerning the existence of optimal shapes and the derivation of first order optimality conditions. The control variable is a phase field function that prescribes the shape and topology of the object, while the state variables are the velocity and the pressure of the fluid. In our analysis, we cover a multitude of constraints which include constraints on the center of mass, the volume of the fluid region, and the drag of the object. Finally, we present numerical results of the optimization problem that is solved using the variable metric projection type (VMPT) method proposed by Blank and Rupprecht, where we consider one example of topology optimization without constraints and one example of maximizing the lift of the object with a state constraint, as well as a comparison with earlier results for the drag minimization.