Shape sensitivity of a 2D Fluid-Structure Interaction problem between a viscous incompressible fluid and an incompressible elastic structure

TL;DR

This paper investigates how the shape of an elastic structure immersed in a viscous fluid affects the solution of a 2D fluid-structure interaction problem, focusing on shape differentiability and derivative calculations.

Contribution

It introduces a method to compute shape derivatives for a 2D fluid-structure interaction system using the velocity and adjoint methods.

Findings

Derived the shape derivative involving material derivatives of the FSI solution.

Applied the velocity method to compute shape sensitivities.

Provided a simplified expression for the shape derivative using the adjoint method.

Abstract

We study the shape differentiability of a general functional depending on the solution of a bidimensional stationary Stokes-Elasticity system, with respect to the reference domain of the elastic structure immersed in a viscous fluid. The differentiability with respect to reference elastic domain variations are considered under shape perturbations with diffeomorphisms. The shape-derivative is then calculated with the use of the velocity method. This derivative involves the material derivatives of the solution of this Fluid-Structure Interaction problem. The adjoint method is used to obtain a simplified expression for the shape derivative.