# Optimal Shape Design for Stokes Flow Via Minimax Differentiability

## TL;DR

This paper develops a shape sensitivity analysis for optimizing the shape of domains in Stokes flow, using minimax differentiability and gradient algorithms, with practical numerical validation.

## Contribution

It introduces a novel shape gradient derivation for Stokes flow problems via minimax differentiability and demonstrates an effective gradient-based optimization method.

## Key findings

- The shape gradient formula is derived using minimax differentiability.
- A gradient algorithm effectively optimizes shape in Stokes flow.
- Numerical examples validate the theory and algorithm feasibility.

## Abstract

This paper is concerned with a shape sensitivity analysis of a viscous incompressible fluid driven by Stokes equations with nonhomogeneous boundary condition. The structure of shape gradient with respect to the shape of the variable domain for a given cost function is established by using the differentiability of a minimax formulation involving a Lagrangian functional combining with function space parametrization technique or function space embedding technique. We apply an gradient type algorithm to our problem. Numerical examples show that our theory is useful for practical purpose and the proposed algorithm is feasible.