Topology optimisation of Stokes flow

Abstract

In this paper we will investigate topology optimization of Stokes flow. We will construct two different algorithms for solving the problem. We will first construct the algorithms with using sobolev spaces. We will then discretize the algorithms using finite volume methods. The paper is organized into four parts. In the first part we give an introduction to the Stokes equation, and the Darcy-Stokes equation. We will make use of a Lax-Milgram theorem to show existence and uniqueness of solution to Stokes and Darcy-Stokes. In the second part we present a methodology for solving the topolgy optimization problem of stokes flow, where we introduce a control function which effects permeability of the flow. This methodology is described in the paper (Topology optimization of fluids in Stokes flow,Borrvall and Petersson, 2003).We end up with an objective function on this control function, where we wish to find a control function which minimizes the objective function. We will show some continuoity resulsts for the objective function to the resulting minimization problem, and we will show exictence of solution to the minimization problem. In the third part we will construct two algorithmes for solving the minimization problem. In the last part we will discretize the algorithms using a finite volume methods scheme. We solve a few example problems numerically, and compare the performance of the algorithmes.