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In this paper, we consider topology and shape optimization problem related to the nonstationary Navier-Stokes system. The minimization of dissipated energy in the fluid flow domain is discussed. The proposed approach is based on a sensitivity analysis of a design function with respect to the insertion of a small obstacle in the fluid flow domain. Some numerical results show the efficiency and accurate of the proposed approach.

Topological shape optimization in fluid mechanics has wide and valuable applications in hydrodynamic and aerodynamic problems such as the design of car hoods, airplane wings, and inlet shapes for jet engines. Various optimization methods are introduced to determine the optimal design of minimum drag bodies [

It is only recently that topological optimization has been introduced and used in fluid shape optimization problems. It can be used to design features within the domain allowing new boundaries to be introduced into the design. In this context, one of the first approaches is proposed by Borrvall and Petersson in [

In order to minimize

Starting with this observation, topological optimization algorithm can be constructed. The optimal domain is obtained using an iterative process building a sequence of geometries

To our knowledge, the topological sensitivity analysis for the nonstationary Navier-Stokes equations has not been studied so far. The most contributions have been focused on the stationary regime.

The aim of this work is to extend the topological gradient method for the nonlinear unsteady flow environment. The main difficulty comes from the nonlinearity of the operator and the treatment of the associated adjoint problem. To overcome such a difficulty, we have used the discrete adjoint approach. This alternative takes a discretization of the Navier-Stokes equations, linearizes the discrete equations, and then uses the transpose of the linear operator to form the adjoint problem. The discrete adjoint approach has been developed by Elliott and Peraire [

The rest of this paper is organized as follows. In Section

We consider a viscous and incompressible fluid flow in a bounded domain

We assume that the fluid flow domain

Cavity\Omega.

Here

To solve the considered topology optimization problem

Let

Our aim is to determine the optimal location of the obstacle

From the asymptotic (

Starting with this observation, the topological gradient

To this end, we will derive a topological sensitivity analysis for the Navier-Stokes equations in the next section. The obtained results are valid for a large class of cost functions

We start our analysis by the time discretization of the Navier-Stokes problem. It leads to solving steady state generalized Stokes equations at each time step. The topological sensitivity analysis for the Navier-Stokes equations is derived in Section

We remark that the convective term in the first equation of system (

Then, at each time step, we have to solve a steady state generalized Stokes problem having the following generic form:

In this section, we give the topological sensitivity analysis for the generalized Stokes equations when creating a small hole

The function

There exists a real number

Under the assumptions of Hypothesis

In the particular case where

Let

Let

The cost function

Consider a shape function

For all

There exist a real number

In this section we consider the nonstationary Navier-Stokes equations and we compute the variation of the cost function

The function

If

Then, we deduce the following corollary.

Let

This section is devoted to some numerical investigations for our shape optimization problem

As already mentioned, the optimal domain is obtained using an iterative process building a sequence of geometries

Initialization: choose

Repeat until

compute

compute

compute the topological gradient

determine the obstacle

get the new domain

The function

We consider a cavity with one inlet and two outlets having the same section (see Figure

The initial domain and Figureelocity field.

The initial domain

The initial velocity field

First case: cavity with one inlet and two outlets.

The optimal design

The obtained velocity field

Variation of the function

The obtained geometries during the optimization process.

Iteration 2

Iteration 3

Iteration 6

Iteration 8

Iteration 13

Iteration 19

In this case, we use a cavity with one inlet

The initial domain and velocity field.

The initial domain

The initial velocity field

The initial domain and the initial velocity field are given in Figure

Second case: a cavity with one inlet and three outlets.

The optimal shape design

The obtained velocity field

The obtained geometries during the optimization process.

Iteration 2

Iteration 3

Iteration 5

Iteration 8

Iteration 11

Iteration 17

In this work, we have extended the topological gradient method for the nonstationary case. The discrete adjoint approach is introduced to overcome the difficulty coming from the nonlinearity of the operator. The proposed algorithm is applied to determine the optimal shape of tubes in a cavity. The optimal domain is obtained iteratively by inserting some obstacles in the initial one. The location and size of the obstacles are described by the topological gradient.

The proposed approach has two main features. The first one concerns the adaptation for other nonstationary problems. The derived analysis is general and can be adapted for various operators like elasticity, Helmholtz, Maxwell, and so forth

The second interesting feature concerns the efficiency and the simplicity of the numerical algorithm. It is easy to be implemented and can be used for many applications. Only a few iterations are needed to construct the final domain. At each iteration, we only need to solve the direct and the adjoint problems on a fixed grid.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This project was supported by the King Saud University, Deanship of Scientific Research, College of Sciences Research Center.