The topological asymptotic expansion for the Quasi-Stokes problem

October 2004, Vol. 10, 478–504 DOI: 10.1051/cocv:2004016

THE TOPOLOGICAL ASYMPTOTIC EXPANSION FOR THE QUASI-STOKES PROBLEM

Maatoug Hassine

and Mohamed Masmoudi

Abstract. In this paper, we propose a topological sensitivity analysis for the Quasi-Stokes equations.

It consists in an asymptotic expansion of a cost function with respect to the creation of a small hole in the domain. The leading term of this expansion is related to the principal part of the operator. The theoretical part of this work is discussed in both two and three dimensional cases. In the numerical part, we use this approach to optimize the locations of a fixed number of air injectors in an eutrophized lake.

Mathematics Subject Classification. 49Q10, 49Q12, 74P05, 74P10, 74P15.

Received August 28, 2002. Revised November 6, 2003.

1. Introduction

The goal of topological optimization is to find an optimal design even witha prioripoor information on the optimal shape. Unlike the case of classical optimization, the topology of the structure may change during the optimization process, as for example by the inclusion of holes.

Most of the known results in this field concern structural mechanics. In such cases, classical topology optimization involves relaxed formulations or homogenization (see e.g. [2, 3, 5, 9, 26, 30]). This method leads to a Neumann condition on the unknown boundary. This boundary condition is quite natural in structural mechanics but this is not the case in fluid dynamics. In that direction, global optimization techniques like genetic algorithms or simulated annealing, have been proposed (seee.g. [36]). But these methods are very slow and can hardly be applied to industrial problems.

The recently introduced notion of topological sensitivity gives new perspectives on shape optimization. It provides an asymptotic expansion of a cost function with respect to the creation of a small hole in the domain.

To present the basic idea, we consider Ω a domain of R^N, N= 2,3 andj(Ω) =J(Ω, u_Ω) a cost function to be minimized, whereu_Ωis the solution to a given PDE problem defined in Ω. For ε >0, let Ωε= Ω\(x0+εω) be the domain obtained by removing a small part (x₀+εω) from Ω, wherex₀∈Ω andω⊂R^N is a fixed bounded domain containing the origin. Then, an asymptotic expansion of the function j is obtained in the following

Keywords and phrases. Topological optimization, topological sensitivity, Quasi-Stokes equations, topological gradient, shape optimization.

1 ENIT-LAMSIN & FSM, Campus Universitaire, Le Belv´edaire BP 37, 1002 Tunis, Tunisia;

e-mail:maatoug.hassine@enit.rnu.tn

2Math´ematiques pour l’Industrie et la Physique, UMR 5640, Universit´e de Paul Sabatier, 118 route de Narbonne, 31032 Toulouse Cedex 4, France.

c EDP Sciences, SMAI 2004

form:

j(Ωε) =j(Ω) +ρ(ε)δj(x₀) +o(ρ(ε))

εlim→0ρ(ε) = 0, ρ(ε)>0,

whereρ(ε) is a scalar function known explicitly andδj(x₀) is easy to compute at each pointx₀of Ω.

The topological sensitivity δj(x₀) gives us the information where to create holes, in fact ifδj(x₀)<0, then j(Ωε)< j(Ω) for smallε. The functionδjcan be used as a descent direction of the domain optimization process.

The optimality conditionδj(x₀)≥0 in the domain recalls the one obtained by Buttazzo-Dal Maso [5] for the Laplace equation using the homogenization theory. The notion of topological asymptotic gives an interesting alternative to homogenization methods and genetic algorithms: its applications field is very large and using topological sensitivity information, one can build fast algorithms.

The total energy variation with respect to the creation of a small hole is well known [16]. Schumacher intro- duced the bubble method that uses this energy variation for topological optimization in [35]. Then Sokolowski extended this idea to more general cost functions using the adjoint approach in [37] but still with Neumann boundary condition. A topological sensitivity framework using an adaptation of the adjoint method [7, 29] and a truncation technique was introduced in [29] in the case of the Laplace equation with a circular hole and a Dirichlet condition on the boundary of the hole. It was generalized in [15] to the elasticity equations in the case of arbitrary shaped holes. Recently, the same technique is adapted, using a Dirichlet boundary condition and non circular holes, to the Poisson equation in [23] and to Stokes equations in [24].

All these contributions concern operators which symbol is an homogeneous polynomial. The goal of this paper is to address the situations where non homogeneous polynomials arise. We will illustrate our approach by the Quasi-Stokes equations case. The basic idea is to say that the leading term of the topological expansion is given by the elementary solution of the principal part of the operator. The theoretical part of this work is discussed in both two and three dimensional cases. Such an expansion is obtained for a large class of cost functions and arbitrary shaped holes.

In the numerical part, we consider the aeration process of eutrophized lakes. Eutrophication leads to a 3- layer situation, the bottom layer being quite poor in oxygen necessary to aquatic life [1]. The aeration process consists in inserting air by the means of injectors located at the bottom of the lake in order to generate a vertical motion mixing up the water of the bottom with that in the top, thus oxygenating the lower part by bringing it in contact with the surface air. A simplified model based on incompressible Quasi-Stokes equations is used, only considering the liquid phase, which is the dominant one. The injected air is taken into account through local boundary conditions for the velocity on the injectors holes. We aim to optimize the injectors location in order to generate the best motion in the fluid with respect to the aeration purpose. The main idea is to compute the asymptotic topological expansion with respect to the insertion of an injector. The injector is modeled as a small holeωε around a point x₀, having an injection velocityUinj. The best locations and orientations are the one for which the cost function decrease most, i.e. the sensitivity is as negative as possible. Numerical tests clearly indicate the approach to be quite efficient.

An outline of the paper is as follows. In Section 2, we recall briefly the adaptation of adjoint method to the topological optimization. In Section 3, we derive the Quasi-Stokes equations and we give a description of the shape optimization problem that we consider. Next in Section 4 the truncation technique is applied to the problem. The main results are presented in Section 5. An asymptotic expansion is given in a general form, for a large class of cost functions and arbitrary shaped holes. In Section 6, we present, for the two dimensional case, some numerical experiments validating the above analysis. Finally in Section 7, some background materials related to the Stokes and Quasi-Stokes equations are reviewed.

2. The generalized adjoint method

In this section, we recall the fundamental results introduced in [14, 29] which extends the adjoint method [7]

to the topology shape optimization.

Let V be a fixed Hilbert space. For ε ≥ 0, let aε be a bilinear, symmetric, continuous and coercive form on V and lε be a linear and continuous form on V, that is, there exist constants M >0, γ > 0 and L >0, independent ofεsuch that for allε≥0,

aε(u , v) ≤Mu v, ∀u, v∈ V aε(u , u)≥γu², ∀u∈ V lε(v) ≤Lv, ∀v∈ V.

Assume that there exist a bilinear and continuous formδa, a linear and continuous formδl, and a real function ρ(ε)>0 defined onR+ such that

aε−a₀−ρ(ε)δa_L2(V)=o(ρ(ε)), (1) lε−l₀−ρ(ε)δl_L(V)=o(ρ(ε)), (2)

εlim→0ρ(ε) = 0,

whereL(V) (respectivelyL2(V)) denotes the space of continuous and linear (respectively bilinear) forms onV. Forε≥0, letuεbe the solution to the problem: finduε∈ V such that

aε(uε, v) =lε(v), ∀v∈ V. (3)

Lemma 2.1 [15]. Forε≥0, problem (3) has a unique solutionuε, and

uε−u₀=O(ρ(ε)). (4)

Next we consider a cost function of the form j(ε) = Jε(uε), where Jε is defined on V for ε ≥ 0 and J₀ is differentiable with respect tou, its derivative being denoted byDJ₀(u).

Suppose that there exists a function δJ defined onV such that: for allε >0

Jε(v)−J₀(u) =DJ₀(u)(v−u) +ρ(ε)δJ(u) +o(v−u+ρ(ε)) ∀u, v∈ V. (5) Theorem 2.1 [15, 29]. Under the hypotheses (1), (2) and (5) the function j has the following asymptotic expansion

j(ε) =j(0) +ρ(ε) [δa(u₀, v₀)−δl(v₀) +δJ(u₀)] +o(ρ(ε)) (6) wherev₀∈ V is the solution to the adjoint problem: find v₀∈ V such that

a₀(w, v₀) =−DJ₀(u₀)w, ∀w∈ V.

3. Position of the problem 3.1. The Quasi-Stokes equations

We consider Ω a bounded domain of R^N,N = 2,3. We denote by Γ its boundary.

The standard form of the Navier-Stokes equations describing the motion of an incompressible fluid in Ω is

given by: 





∂u_Ω

∂t + (u_Ω.∇)u_Ω−ν∆u_Ω+∇p_Ω=F in Ω

∇.u_Ω= 0 in Ω u_Ω=Ud on Γ

(7) where u_Ω andp_Ω denote respectively the velocity and the pressure fields,F is a given body force per unit of mass,ν denotes the kinematic viscosity of the fluid andUd is a given boundary velocity.

Because of the divergence-free condition onu_Ω,Udmust necessary satisfy the compatibility condition

Γ

Ud.n ds= 0, wherenis the unit normal vector along the boundary Γ.

Denoting by ∆tthe time step and posingtⁿ=n∆t andtⁿ⁺¹=tⁿ+ ∆t.

Using the characteristic method (see,e.g., [12]), an approximation for the convection term is given by ∂u_Ω

∂t (t, x) +u_Ω.∇u_Ω(t, x) ^(t=tn)= du_Ω

dt (t, x(t)) (t=tⁿ)

u_Ω(tⁿ⁺¹, x(tⁿ⁺¹))−u_Ω(tⁿ, x(tⁿ))

∆t ·

By an implicit scheme, a time discretization of the system (7) can be written as



 1

∆tuⁿ_Ω⁺¹−ν∆uⁿ_Ω⁺¹+∇pⁿ_Ω⁺¹ =Fⁿ⁺¹+u_Ω(tⁿ, x(tⁿ))

∆t

∇.uⁿ⁺¹ = 0.

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





αu_Ω−ν∆u_Ω+∇p_Ω =f in Ω

∇.u_Ω = 0 in Ω

u_Ω =Ud on Γ. (8)

3.2. The shape optimization problem

For a given x₀ ∈Ω, consider the modified domain Ωε= Ω\ωε, ωε =x₀+εω, whereω is a given fixed and bounded domain ofR^N, containing the origin, whose boundary∂ωis connected and piecewise of classC¹.

Ω

Γ Γ

Ω

∂ω_ε ω_ε

Figure 1. The initial domain and the same domain after creation of a small holeωε. In the modified domain Ωε, the velocity and pressure fields are required to satisfy









αu_Ωε−ν∆u_Ωε+∇p_Ωε =f in Ωε

∇.u_Ωε = 0 in Ωε

u_Ωε =Ud on Γ u_Ωε =Uinj on∂ωε

(9)

whereUinj is a given velocity on∂ωε.

Note that forε= 0, one hasu_Ω0 =u_Ωandp_Ω0 =p_Ω.

Theorem 3.1[4,17,38]. Letε≥0, for a givenf ∈L²(Ωε)^N,Uinj∈H^1/2(∂ωε)^N and Ud ∈H^1/2(Γ)^N, the prob- lem (9) has a unique solution(u_Ωε, p_Ωε)∈H¹(Ωε)^N×L²₀(Ωε)^N whereL²₀(Ωε)^N = θ∈L²(Ωε)^N,

Ωε

θdx= 0

. Letu_Ωε an extension of the boundary dataUd andUinj in Ωε, (∂Ωε= Γ∪∂ωε), satisfying









αu_Ωε−ν∆u_Ωε+∇p_Ωε = 0 in Ωε

∇.u_Ωε = 0 in Ωε

u_Ωε =Ud on Γ u_Ωε =Uinj on∂ωε.

(10)

The solution u_Ωε can be recuperated as u_Ωε =w_Ωε +u_Ωε, with w_Ωε is the solution of the system (9) with a homogeneous Dirichlet boundary condition on∂Ωε.

Thanks to the previous variable substitution, in the theoretical part of this work, we will consider only a homogeneous boundary condition. Then, we will assume thatUd= 0 on Γ and Uinj = 0 on∂ωε.

We now consider a cost functionj(ε) of the form

j(ε) =Jε(u_Ωε), (11)

withJε being defined onH¹(Ωε)^N forε≥0.

Our aim is to obtain an asymptotic expansion ofj with respect toε. The velocity fieldu_Ωε is defined in the variable domain Ωε, thus it belongs to a functional space which depends onε. Hence, if we want to derive the asymptotic expansion of j we cannot apply directly the results of Section 2, which require a fixed functional space (cf. Th. 2.1).

In classical shape optimization, this condition is satisfied by the mean of a domain parameterization method.

This method involves a fixed domain and a bi-Lipshitz map between the initial domain and the modified one.

In the topology optimization context, such a map does not exist between Ω and Ωε.

However, a functional space independent of ε can be constructed by using a domain truncation technique described in the next paragraph (see also [29] and [15]).

This truncation is needed only for analysis, and will never be used for practical computation. During the optimization process, we have just to solve the system (8) and the adjoint problem associated to the cost function (11).

4. The truncated problem

LetR >0 be such that the closed ballB(x₀, R) is included in Ω andωε⊂B(x₀, R).

Γ

Ω

Γ

R

x0

Γ

Ω

Γ

D_ε

Figure 2. The truncated domain.

We denote by ΓR the boundary of B(x₀, R) and we consider the fixed domain ΩR = Ω\B(x₀, R) and Dε=B(x₀, R)\ωε. Also we use the following space of traces on ΓR

H_V^1/2(ΓR)^N = φ∈H^1/2(ΓR)^N;

ΓR

φ.ndx= 0

(12)

wherenis the unit vector normal to ΓR. Its dual space is denoted byH_V^−1/2(ΓR)^N.

For a givenf ∈L²(Ω)^N, ϕ∈H_V^1/2(ΓR)^N andε >0, let u^f,ϕ_ε , p^f,ϕ_ε be the solution to the problem:

find (u^f,ϕ_ε , p^f,ϕ_ε )∈H¹(Dε)^N×L²₀(Dε)^N such that









αu^f,ϕ_ε −ν∆u^f,ϕ_ε +∇p^f,ϕ_ε =f inDε

div u^f,ϕ_ε = 0 inDε

u^f,ϕ_ε =ϕ on ΓR

u^f,ϕ_ε = 0 on∂ω_ε.

(13)

This problem has a unique solution [4, 17, 38].

Forε= 0, (u^f,ϕ₀ , p^f,ϕ₀ ) is the solution to





αu^f,ϕ₀ −ν∆u^f,ϕ₀ +∇p^f,ϕ₀ =f in B(x₀, R) divu^f,ϕ₀ = 0 inB(x₀, R)

u^f,ϕ₀ =ϕ on ΓR.

(14)

Clearly we have

u^f,ϕ_ε =u^f,_ε⁰+u⁰_ε^,ϕ, p^f,ϕ_ε =p^f,_ε⁰+p⁰_ε^,ϕ, ∀ε≥0. (15) This decomposition will be used to construct the bilinear formaεand the linear formlεpresented in Section 2.

Forε≥0, we consider the Dirichlet-to-Neumann operator

Tε : H_V^1/2(ΓR)^N −→H_V^−1/2(ΓR)^N ϕ−→Tεϕ=σ(u⁰_ε^,ϕ).n

(16)

whereσ(u⁰_ε^,ϕ) = (ν∇u⁰_ε^,ϕ−p⁰_ε^,ϕI) is the stress tensor.

And the functionfε∈H_V^−1/2(ΓR)^N

fε=−σ(u^f,_ε⁰).n=−(ν∇u^f,_ε⁰−p^f,_ε⁰I).n (17) with the normalnis chosen outward toDε on ΓRand ∂ωε.

Hence, for allε≥0 andϕ∈H_V^1/2(ΓR)^N we haveσ(u^f,ϕ_ε ).n=Tεϕ−fε. Finally, we define forε≥0 the solutionuε, pε to the truncated problem









αuε−ν∆uε+∇pε =f in ΩR

div uε = 0 in ΩR

uε = 0 on Γ σ(uε).n+Tεuε =fε on ΓR.

(18)

The variational formulation associated to (18) is: finduε∈ VR such that

aε(uε, v) =lε(v), ∀v∈ VR (19)

where the functional spaceVR, the bilinear formaε and the linear formlεare defined by VR=

u∈H¹(ΩR)^N; divu= 0, u= 0 on Γ ,

aε(u, v) =α

ΩR

u.v dx+ν

ΩR

∇u:∇v dx+

ΓR

Tεuv dγ(x), (20)

lε(v) =

ΩR

fv dx+

ΓR

fεv dγ(x). (21)

Symmetry, continuity and coercivity ofaεand continuity oflεfollow directly from

ΓR

Tεϕψ dγ(x) =α

D_ε

u⁰_ε^,ϕ.u⁰_ε^,ψ dx+ν

D_ε

∇u⁰_ε^,ϕ:∇u⁰_ε^,ψ dx, (22)

ΓR

fεψdγ(x) =

D_ε

fu⁰_ε^,ψ dx, (23)

and the relation

α

D_ε

u^f,_ε⁰.u⁰_ε^,ψ dx+ν

D_ε

∇u^f,_ε⁰:∇u⁰_ε^,ψdx= 0. (24) Proposition 4.1. Let ε≥0. Problems (9) and (18) have unique solution.

Moreover, the restriction toΩR of the solutionu_Ωε, p_Ωε to (9) is the solutionuε, pεto (18), and we have inDε

(u_Ωε)_|D_ε=u^f,ϕ_ε , (p_Ωε)_|D_ε=p^f,ϕ_ε (25) whereϕis the trace ofu_Ωε on ΓR.

Proof. We refer to [4, 17, 38] for the existence and uniqueness of the solutions to both problems (9) and (18).

Recall that we have denoted by (u_Ωε, p_Ωε) the solution of (9) and by (uε, pε) the solution of (18).

Letϕ=u_Ωε|Γ

R anduR=u_Ωε|Ω

R. Clearly (25) holds for thisϕ, and it remains to prove thatuR=uε. Letθ∈ VRandψ=θ_|ΓR. We extendθonDεbyu⁰_ε^,ψ. Its extension is still denoted byθ, and it is divergence free on Dε.

Using (22), (23), (24) and the definition ofu_Ωε, we have α

ΩR

uR.θdx+ν

ΩR

∇uR:∇θ dx+

ΓR

(TεuR−fε)θdγ(x)

=α

ΩR

uR.θ dx+ν

ΩR

∇uR:∇θ dx+

ΓR

TεuRθ dγ(x)−

ΓR

fε.θ dγ(x)

=α

ΩR

uR.θ dx+ν

ΩR

∇uR:∇θ dx+α

D_ε

u⁰_ε^,ϕ.u⁰_ε^,ψ dx + ν

D_ε

∇u⁰_ε^,ϕ:∇u⁰_ε^,ψdx−

D_ε

f.u⁰_ε^,ψ dx

=α

Ωε

uε.θ dx+ν

Ωε

∇uε:∇θdx−

ΓR

f.u⁰_ε^,ψdγ(x)

=α

Ωε

uε.θ dx+ν

Ωε

∇uε:∇θdx−

D_ε

f.θdx=

ΩR

f.θdx.

This proves thatuRis the solution to (19). From uniqueness of the solution, we deduce thatuR=uε. Now we have at our disposal the fixed Hilbert spaceVR required by Section 2. The cost function (11) can be redefined in the following way:

foru∈ VR, letϕ=u_|ΓR, one defines ˜uε∈H¹(Ωε)^N the extension ofuon Ωε as follow u˜ε= u on ΩR

u^f,ϕ_ε onDε. (26)

Then, a functionJεcan be defined onVR by

Jε(u) = ˜Jε(˜uε). (27)

Particularly, we have from the previous proposition that

j(ε) = ˜Jε(u_Ωε) =Jε(uε). (28) Remark thatJε(uε) is independent of the choice ofR. For example, for a given target function Ug, if

J˜_ε(u_Ωε) =

Ωε

|u_Ωε− Ug|² dx (29) then, we have for allu∈ VR

Jε(u) =

ΩR

|u− Ug|² dx+

D_ε

u^f,ϕ_ε − Ug² dx, withϕ=u_|ΓR. (30)

5. Main result: the asymptotic expansion

In this section, we present the main results of this paper, which concern the asymptotic analysis with respect to the parameterε of the functional (27). An asymptotic expansion is obtained for the Quasi-Stokes operator for a large class of cost functions and arbitrary shaped holes.

We begin by the three dimensional case. The principal result is given by Theorem 5.1, it gives the topological sensitivity expressionδj(x₀) if a hole is created atx₀. The proof of Theorem 5.1 is relegated to Section 7.

5.1. The three dimensional case

In order to derive the topological sensitivity of the functionj, we introduce two auxiliary problems.

The first problem, which we call the “exterior problem”, is formulated in R³\ω and consists to find (U, P)

solution to 









−ν∆U+∇P = 0 in R³\ω div U = 0 in R³\ω

U = 0 at infinity U =u_Ω(x₀) on∂ω

(31)

whereu_Ωis the solution to Quasi-Stokes problem (8). Recall thatf ∈L²(Ω)³, so thatu_Ωis continuous inside Ω.

Here, one can remark that just the principal part of the Quasi-Stokes operator is used, which is the Stokes equations. A such approach can be justified by the fact that operators−∆ andI−∆ have the same behavior near the holeωεand give the same leading terms for topological expansion. Moreover, this approach has a good advantage, it avoids calculations with the complicated expression of the Quasi-Stokes fundamental solution, which involves the convolution product of Green functions for both operatorsI−∆ and ∆.

We return now to the system (31). The functionsU, P can be expressed by a simple layer potential on∂ω.

Coordinate system can be changed, in what follows one can suppose for convenience thatx₀= 0.

Posing r = y and er = y

y, the fundamental solution system to the Stokes equations in R³ can be written as

GU(y) = 1

8πνr(I+ere^T_r), GP(y) = y

4πr³ (32)

such that

−ν∆GU j+∇GP j=δej (33) whereGU j denote thejth column ofGU,{ej}j=1,3 is the canonical basis ofR³ andδis the Dirac distribution.

Then, the functions U, P read [11]

U(y) =

∂ω

GU(y−x)T(x) dγ(x), y∈R³\ω P(y) =

∂ω

GP(y−x).T(x) dγ(x), y∈R³\ω (34)

whereT ∈H^−1/2(∂ω)³is a solution to the boundary integral equation (seee.g. [11])

∂ω

GU(y−x)T(x) dγ(x) =u_Ω(x₀), y∈∂ω. (35) One can observe that the function T is determined up to a function proportional to the normal, hence it is unique inH^−1/2(∂ω)³/Rn.

Using the first order Taylor expansion ofGU at the pointy= 0 forxbounded, we have GU(y−x) =GU(y) +O

1 r²

, GP(y−x) =GP(y) +O 1

r³

; (36)

from which follows the asymptotic expansion at infinity ofU andP:

U(y) =SU(y) +LU(y), P(y) =SP(y) +LP(y) (37) whereSU(y) andSP(y) are the dominant part respectively ofU andP

SU(y) =GU(y)A(u_Ω(x₀)), SP(y) =GP(y).A(u_Ω(x₀)) (38) with

A(u_Ω(x₀)) =

∂ω

T(x) dγ(x)∈R³. (39)

Notice thatα−→A(α) is linear onR³ andSU ∈L²_loc(R³).

The last parts ofU andP are respectively given by LU(y) =O

1 r²

, LP(y) =O 1

r³

· (40)

The second problem, which we call “interior problem”, is formulated in D₀ = B(x₀, R) and consists to find (R^h_U, R^h_P) solution to 









αR^h_U−ν∆R_U^h +∇R^h_P = 0 inB(x₀, R) divR^h_U = 0 inB(x₀, R)

R^h_U =SU on ΓR.

(41)

Here, the idea is to consider an interior and exterior problems that gives the asymptotic behavior of u^f,ϕ_ε

−u^f,ϕ₀

|D_ε withϕ=u_Ω|ΓR in a sense which will be stated in Section 7.

It will not be possible to derive the asymptotic behavior ofu^f,ϕ_ε −u^f,ϕ₀ fromR^h_U−SU. We have first to take into account the error due to the simplification of the fundamental solution. We propose to cancel this error by adding a correcting term toR^h_U.

In such a case, we consider the correcting term (R_U^c, R_P^c) as solution to





αR^c_U −ν∆R^c_U+∇R^c_P =αSU inB(x₀, R) divR^c_U = 0 inB(x₀, R)

R^c_U = 0 on ΓR. (42)

Setting RU =R^h_U +R^c_U,RP =R^h_P+R^c_P, then (RU, RP) is solution to





αRU −ν∆RU+∇RP =αSU inB(x₀, R) divRU = 0 inB(x₀, R)

RU =SU on ΓR. (43)

We will prove in Section 7, using the corrected interior problem (43), it will be possible to derive the asymptotic behavior of

u^f,ϕ_ε −u^f,ϕ₀

|D_ε. The main result is the following. It will be proved in Section 7.

Theorem 5.1. Let f ∈ H²(Ω)³, and let j(ε) = Jε(uε). Suppose that Jε satisfy the hypothesis (5): for all v∈ VR and allε >0

Jε(v)−J₀(u₀) =DJ₀(u₀)(v−u₀) +εδJ(u₀) +o

ε+v−u_0VR

, (44)

whereDJ₀(u₀)is linear and continuous on VR, anduε,ε≥0 is the solution to (19).

Let v₀∈ VR be the solution to the adjoint equation

a₀(w, v₀) =−DJ₀(u₀)w, ∀w∈ VR. (45) Then, the functionj has the following asymptotic expansion

j(ε) =j(0) +εδj(x₀) +o(ε) (46)

with

δj(x₀) =

ΓR

(σ(RU −SU)).nv₀dγ(x) +δJ(u₀). (47) The functional δj(x₀) is called the “topological sensitivity” of the Quasi-Stokes operator. It is also called the

“topological gradient”.

The cost function j is independent of Rand δj(x₀) is independent of ε, thenδj(x₀) is also independent of R. This follows from the uniqueness of an asymptotic expansion. As we will observe in Section 7, this is not necessarily true for the terms δa(u₀, v₀), δl(v₀) orδJ(u₀), becausea,l andJ do depend onR.

Practically, we need just to compute the solution u_Ω to (8) and v_Ω the solution to the associated adjoint problem

α

Ω

wv_Ωdx+ν

Ω

∇w:∇v_Ωdx=−DJ˜₀(u_Ω)w, ∀w∈ V0 (48) withV0=

v∈H₀¹(Ω)³; div v= 0 in Ω .

It has been shown in Proposition 4.1 thatu₀ is the restriction to ΩR of u_Ω. Similarly,v₀ is the restriction to ΩRof v_Ω, this can be proved in the same way. Consequently, the function u_Ω (oru₀) and the adjoint state v_Ω (or v₀) do not depend onx₀. Hence, the basic property of an adjoint technique is here satisfied: only two systems have to be solved in order to compute the topological sensitivityδj(x) for allx∈Ω.

Moreover, there exists a uniqueq_Ω∈L²₀(Ω)³ such that α

Ω

wv_Ωdx+ν

Ω

∇w:∇v_Ωdx−

Ω

q_Ω divwdx=−DJ˜₀(u_Ω)w, ∀w∈H₀¹(Ω)³. (49)

Corollary 5.1. Letx₀∈Ω. Under the hypotheses of Theorem 5.1 and that αv_Ω−ν∆v_Ω+∇q_Ω∈L²(D₀)³, we have

δj(x₀) =A(u_Ω(x₀)).v_Ω(x₀) +

D₀

(αv_Ω−ν∆v_Ω+∇q_Ω) (RU −SU) dx+δJ(u₀). (50) Moreover in the particular case;ω is the unit ball B(0,1),U(y),T(y)andA(u_Ω(x₀))are given explicitly by

U(y) =πν(6GU + ∆GU)(y)u_Ω(x₀) T(y) = ³₂^νu_Ω(x₀), ∀y∈∂ω A(u_Ω(x₀)) = 6πνu_Ω(x₀).

(51)

Proof. Thanks to Green’s Formula together with (41) (with SU =RU on ΓR), (47) reads also δj(x₀) =

ΓR

[(ν∇RU −RPI)−(ν∇SU −SPI)].n v_Ωdγ(x) +δJ(u₀)

=

ΓR

(ν∇v_Ω−q_ΩI).n SU dγ(x)−

ΓR

(ν∇SU −SPI).n v_Ω dγ(x) +

D₀

(αv_Ω−ν∆v_Ω+∇q_Ω)RU dx−α

D₀

SUv_Ωdx+δJ(u₀).

(52)

Using a regularizing and localization technique, we derive

ΓR

(ν∇v_Ω−q_ΩI).n SU dγ(x)−

ΓR

(ν∇SU−SPI).n v_Ωdγ(x)

=

D₀

(ν∆v_Ω− ∇q_Ω)SU dx− ν∆SU− ∇SP, ϕv_Ω (53) whereϕ∈ D(D0) satisfiesϕ(x₀) = 1.

Finally, from (38) and (33) one can check that

−ν∆SU+∇SP, ϕv_Ω=−ν∆(GUA(u_Ω(x₀))) +∇(GP.A(u_Ω(x₀))), ϕv_Ω

=

j

Aj(u_Ω(x₀))δej, ϕv_Ω

=A(u_Ω(x₀)).v_Ω(x₀),

(54)

which proves (50).

For the case ω = B(0,1), one can derive the explicit expressions of the terms U(y), T(y) and A(u_Ω(x₀)) from (32), (35) and

∂B(0,1)

GU(y−x) dγ(x) = 2

3νI, ∀y∈∂B(0,1). (55)

For more details concerning the explicitly calculation of this terms, one may consult [24].

Now we discuss briefly the hypothesis (5) used in Theorem 5.1. It concerns the variation of the cost func- tionJε. This question has been examined in [23] for the Dirichlet problem and in [24] for the Stokes problem.

Here we limit ourselves to cost functions of the form J˜ε(u) =

Ωε

g(x, u(x)) dx, u∈H¹(Ωε)³ (56)

withg is a given function defined on Ω×R³.

The case wheregdepends on the pressurepis more complicated and we don’t consider it in this paper.

Next we suppose thatg satisfy the following hypotheses:

• for allx∈Ω, the functions−→g(x, s) is of classC¹ onR³, its gradient being denoted by∇sg(x, s);

•for allx∈Ω, the functions−→ ∇sg(x, s) is Lipschitz continuous and there exists a constantM such that

|∇sg(x, t)− ∇sg(x, s)| ≤M|t−s|, ∀(x, s, t)∈Ω×R³×R³, (57) where|t|denotes the usual norm onRⁿ;

• the functionx−→ ∇sg(x,0) belongs toL²(Ω)³and x−→g(x,0) belongs toL²(Ω)^3/2. These hypotheses imply that for all (x, s)∈Ω×R³

|g(x, s)| ≤ |g(x,0)|+|∇sg(x,0).s|+M 2 |s|²

|∇sg(x, s)| ≤ |∇sg(x,0)|+M|s|, (58) and the functionsx−→g(x, u(x)) andx−→ |∇sg(x, u(x))|² are integrable on Ω for allu∈L²(Ω)³.

A standard example of this functions is given by

g(x, u) =|u(x)− Ug(x)|². (59)

The following result is taken from [24].

Proposition 5.1. Under the previous hypotheses and f ∈L²(Ω)³, we have δJ(u₀) =

D₀

∇sg(x, u_Ω)(RU−SU) dx.

The adjoint state(v_Ω, q_Ω)∈ V0×L²₀(Ω) is the solution to

αv_Ω−ν∆v_Ω+∇q_Ω=−∇sg(x, u_Ω). (60) The functionj has the asymptotic expansion

j(ε) =j(0) +εA(u_Ω(x₀))v_Ω(x₀) +o(ε). (61) If ω is the unit ball B(1,0), then

j(ε) =j(0) + 6πνε u_Ω(x₀).v_Ω(x₀) +o(ε). (62)

5.2. The two dimensional case

In this paragraph, we intend to derive the asymptotic expansion of the function j in the two dimensional case. The technique used is similar to that of the three dimensional case. We use the principal part of the Quasi-Stokes operator to derive the topological sensitivity expression. Next we briefly describe the transposition of the previous results to the two dimensional case. First, let us recall that u_Ω and the adjoint state v_Ω are respectively the solution to (8) and (48).

Let (U, P) be the solution to the Stokes exterior problem









−ν∆U+∇P = 0 in R²\ω div U = 0 in R²\ω U/logy =u_Ω(x₀) at infinity

U = 0 on∂ω.

(63)

In 2D, the fundamental solution system to the Stokes equations is given by GU(y) = 1

4πν(−( logr)I+ere^T_r), GP(y) = y

2πr²· (64)

The functionsU andP are written

U(y) =u_Ω(x₀) logy+SU(y) +LU(y), P(y) =SP(y) +LP(y) (65) where at infinitySU(y) =O(1),LU(y) =O(1/r),SP(y) =O(1/r) andLP(y) =O

1/r² . The associated interior problem consists in finding (RU, RP) such that





αRU −ν∆RU+∇RP =αSU inD₀ div RU = 0 inD₀

RU =SU on ΓR. (66)

In this case, a first order expansion of

u^f,ϕ_ε −u^f,ϕ₀

|D_ε withϕ=u_Ω|ΓR is given by

−1 logε

u_Ω(x₀) logx

R +RU−SU

|D_ε.

Theorem 5.2. Under the same hypotheses of Theorem 5.1, the functionj has the following asymptotic expan- sion

j(ε) =j(0)− 1

logεδj(x₀) +o 1

logε

(67) with

δj(x₀) =

ΓR

σ(RU−SU).n v₀dγ(x) +δJ(u₀) (68) wherev₀∈ VR is the solution to the adjoint equation

a₀(w, v₀) =−DJ₀(u₀)w, ∀w∈ VR. (69) And for a cost function of the form (56), we have:

Proposition 5.2. Let J˜ε a cost function of the form J˜ε(u) =

Ωε

g(x, u(x)) dx, u∈H¹(Ωε)². (70)

Under the same hypotheses of Proposition 5.1, the function j has the following asymptotic expansion j(ε) =j(0)−4πνu_Ω(x₀).v_Ω(x₀)

logε +o

1 logε

· (71)

6. Numerical results

Here, we limit ourselves to the two dimensional case. As application of the previous theoretical results, we present two examples. The first example concerns the identification of locations and orientations of several injectors in a water reserve. In the second example we treat the water eutrophication phenomena in a lakevia dynamic aeration process.

In both cases, we deal with a cost functionJ of the form J(u_Ω) =

Ωm

|u_Ω− Ug|² dx, with Ωm⊂Ω is the measurement domain andUg is a given target flow.

Recall that we consider the Quasi-Stokes equations with a non homogeneous boundary condition on∂ωε (u_Ωε

=Uinj) (see (9)). From (71) we deduce that

δj(x) = (u_Ω(x)− Uinj).v_Ω(x), ∀x∈Ω (72) whereu_Ωandv_Ω are, respectively, solution to

αu_Ω−ν∆u_Ω+∇p_Ω = 0 in Ω

∇.u_Ω = 0 in Ω (73)

αv_Ω−ν∆v_Ω+∇q_Ω=−2 (uΩ− Ug)χ_Ωm in Ω

∇.v_Ω= 0 in Ω (74)

whereχ_Ωm is the characteristic function of the measurement domain.

Our implementation is based on the following optimization algorithm introduced by C´ea et al. [6] and presented in the topological asymptotic context in [8].

The algorithm:

• initialization: choose Ω₀= Ωd, and setk= 0;

• repeat until target is reached:

• solve (73) and (74) in Ωk;

• compute the topological sensitivity δjk;

• set Ωk+1 ={x∈Ωk, δjk(x)≥ck+1} where ck+1 is chosen in such a way that the cost function de- creases;

• k←−k+ 1.

This algorithm can be seen as a descent method where the descent direction is determined by the topological sensitivityδjk and the step length is given by the volume variation.

We propose an adaptation of the previous algorithm to our context. We consider the set {x∈Ωk;δjk(x)<

ck+1}. Each connected component of this set is a hole created by the algorithm. Our idea is to replace each hole by an injector located at the local minimum ofδjk(x).

In the above algorithm, the systems (73) and (74) are discretized by a finite element method. The computation of the approximate solution is achieved by Uzawa algorithm.

6.1. Test 1: identification of some injectors in a reserve water

In this example, the computational domain Ω is a reserve water. Our purpose is to insert some injectors in Ω in order to reach a given target flowUg. Each injector Injk is supposed as a small hole ωk aroundxk ∈Ω,