# Numerical solution to a free boundary problem for the Stokes equation using the coupled complex boundary method in shape optimization settings

## TL;DR

This paper introduces a novel boundary reformulation for free boundary Stokes problems using complex Robin boundary conditions, enabling shape optimization via a domain variation method with numerical validation in 2D and 3D.

## Contribution

It proposes a new complex boundary reformulation and cost functional for free boundary Stokes problems, with explicit shape derivatives and a shape optimization algorithm.

## Key findings

- Effective in 2D and 3D numerical tests
- Outperforms classical Dirichlet tracking methods
- Provides explicit shape derivatives for free boundary identification

## Abstract

A new reformulation of a free boundary problem for the Stokes equations governing a viscous flow with overdetermined condition on the free boundary is proposed. The idea of the method is to transform the governing equations to a boundary value problem with a complex Robin boundary condition coupling the two boundary conditions on the free boundary. The proposed formulation give rise to a new cost functional that apparently has not been exploited yet in the literature, specifically, and at least, in the context of free surface problems. The shape derivatives of the cost function constructed by the imaginary part of the solution in the whole domain in order to identify the free boundary is explicitly determined. Using the computed shape gradient information, a domain variation method from a preconditioned steepest descent algorithm is applied to solve the shape optimization problem. Numerical results illustrating the applicability of the method is then provided both in two and three spatial dimensions. For validation and evaluation of the method, the numerical results are compared with the ones obtained via the classical tracking Dirichlet data.