Kohn-Vogelius formulation and high-order topological asymptotic formula for identifying small obstacles in a fluid medium

Kohn-Vogelius formulation and high-order topological asymptotic formula for identifying

small obstacles in a fluid medium

Montassar Barhoumi

Abstract

This paper concerns the identification of a small obstacle immersed in a Stokes flow from boundary measurements. The proposed approach is based on the Kohn-Vogelius formulation and the topological sensitivity analysis method. We derive a high order asymptotic formula describing the variation of a Kohn-Vogelius type functional with respect to the insertion of a small obstacle inside the fluid flow domain. The obtained asymptotic formula will serve as very useful tools for developing accurate and robust numerical reconstruction algorithms.

1 Introduction

The development and application of the nondestructive inspection technique has recently received considerable interest in the engineering as well as in the applied mathematics. Such a technique leads to determine interior information about medium from boundary measurements data.

For this purpose, several approaches and methods have been developed, one can cite: the identification of cracks by boundary measurements [1, 2, 3, 4, 5], the reconstruction of electromagnetic imperfections of a small diameter from measurements on the boundary [6, 7], the determination of small conductiv- ity imperfections inside a conductor have been established in [8, 9, 10], the

Key Words: Kohn-Vogelius formulation, Topological sensitivity analysis, Stokes system, Geometric inverse problem, Calculus of variations, Asymptotic formula.

2010 Mathematics Subject Classification: Primary 35N25; Secondary 49K40.

Received: 19.02.2019.

Accepted: 25.04.2019.

35

FLUID MEDIUM 36 reconstruction of imperfections of small diameter in an elastic medium using boundary integral formula [11, 12], the identification of a finite number of in- terior particles of small diameter related to the full Maxwell equations [13], ...

etc.

This paper is concerned with the fluid mechanics area. The objective of this work is to derive a high-order asymptotic formula leading to determine the location, shape and size of an obstacle of a small volume immersed in a fluid medium from boundary measurements.

The proposed method is based on the Kohn-Vogelius formulation [14, 15]

and the topological sensitivity analysis method [16, 17, 18, 19, 20, 21, 22, 23].

We derive a high-order topological asymptotic expansion for a Kohn-Vogelius type functional with respect to the presence of a small obstacle inside the fluid flow domain. An asymptotic formula is derived giving the relation between the known boundary data and the unknown obstacle properties; location, size and shape.

The obtained asymptotic formula will serve as very useful tools for devel- oping very effective reconstruction algorithms for identifying small obstacles from boundary measurements. Such algorithms can be applied in various ap- plications like fiber-reinforced polymers [24, 25], colloid [26, 27, 28] and casting or injection filling [29, 30, 31] where the design of the mixing of liquid metallic should be optimized.

The rest of this paper is organized as follows. In Section 2, we present the formulation of the considered problem. We introduce the Kohn-Vogelius concept and the topological sensitivity technique. The obstacle reconstruction problem is formulated as a topology optimization one minimizing a Kohn- Vogelius type functional. Section 3 is devoted to a preliminary estimate de- scribing the variation of the considered Kohn-Vogelius functional with respect to the presence of a small obstacle. In Section 4, we establish an asymptotic expansion for the perturbed velocity and pressure fields. In Section 5, we extend the topological derivative notion for the high-order case. We derive a high-order topological asymptotic formula for the Stokes operator. The paper ends by a conclusion and perspective works.

2 Formulation of the problem

We consider a viscous and incompressible fluid flow in a bounded domain Ω ofR³,with smooth boundary ∂Ω. Let Γ and Σ be two parts of∂Ω such that

∂Ω = Γ∪Σ and Γ∩Σ = ∅. We assume that the fluid flow is in a laminar regime, in such way that the convection term can be neglected and the Navier- Stokes equations may be approximated by the non-stationary Stokes system.

Inside the fluid flow domain, we assume the existence of a small obstacle Oz, ε =z+εOthat is characterized by its centerz, its sizeεand its shapeO, withOis a bounded domain ofR³containing the origin, whose boundary∂O is connected and piecewise of classC¹.

The aim of this work is to develop an efficient approach for identifying the unknown parameters (location, size and shape) of the obstacleOz, ε from boundary measurement of the velocity field on a part of the boundary. This issue can be formulated as a geometric inverse problem as follow:

- Given a Neumann and Dirichlet data on the accessible boundary Σ: an imposed forceGand a measured velocity fieldWm.

- Determine the locationz, size εand shape Oof the unknown obstacle Oz, εsuch that the solution (w_{Oz, ε}, q_{Oz, ε}) of the Stokes equations satisfies the following over-determined boundary value problem

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−ν∆w_{Oz, ε}+∇q_{Oz, ε} =F in Ω\Oz, ε, div w_{Oz, ε} = 0 in Ω\Oz, ε, σ(w_{Oz, ε}, q_{Oz, ε})n =G on Σ,

w_{Oz, ε} =Wm on Σ, w_{Oz, ε} = 0 on Γ, w_{Oz, ε} = 0 on ∂Oz, ε.

wherew_{Oz, ε} is the velocity field,q_{Oz, ε} is the pressure field,ν is the kinematic viscosity coefficient of the fluid andF is the gravitational force.

In order to examine the considered geometric inverse problem, we propose in this paper a new approach combining the advantages of the Kohn-Vogelius formulation and the topological sensitivity analysis method.

2.1 Kohn-Vogelius formulation

The first step of our approach is based on the Kohn-Vogelius formulation which rephrase the considered geometric inverse problem into a topology optimiza- tion one. It leads to define for any given obstacleOz, εtwo forward problems:

The first one which be called Neumann problem is associated to the Neumann datumF:

(P^Nε)

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−ν∆w^N_ε +∇q_ε^N =G in Ω\Oz, ε, div w_ε^N = 0 in Ω\Oz, ε, σ(w^N_ε, q^N_ε )n =F on Σ,

w_ε^N = 0 on Γ, w_ε^N = 0 on ∂Oz, ε.

FLUID MEDIUM 38 The second one is associated to the measured velocityWmwhich will be called as the Dirichlet problem:

(P^Dε)

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−ν∆w^D_ε +∇q^D_ε =G in Ω\Oz, ε, divw^D_ε = 0 in Ω\Oz, ε,

w^D_ε =Wm on Σ, w^D_ε = 0 on Γ, w^D_ε = 0 on ∂Oz, ε.

As one can remark here, ifOz, εcoincides with the exact obstacleOz^∗, ε^∗= z^∗+ε^∗O^∗, the misfit between the solutions become zero (w_ε^N =w^D_ε). Starting from this remark, the inverse problem can be formulated as a topology opti- mization one. The unknown obstacle will be characterized as the minimum of the following Kohn-Vogelius type functional [15]

K(Ω\Oz, ε) =ν Z

Ω\Oz,ε

∇w_ε^N − ∇w^D_ε

2 dx.

More precisely, the identification problem can be formulated as follow:

(Popt)

( findOz^∗,ε^∗⊂Ω, such that K(Ω\Oz^∗,ε^∗) = min

Oz,ε∈Dad

K(Ω\Oz, ε), whereDad is a given set of admissible domains.

It is interesting to note that, in the absence of obstacles the function K reads

K(Ω) =ν Z

Ω

∇w₀^N− ∇w₀^D

2dx,

where (w^N₀, q^N₀ ) and (w^D₀, q₀^D) satisfy the Stokes equations in the non perturbed domain

(P^N0)

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−ν∆w^N₀ +∇q₀^N =G in Ω, div w₀^N = 0 in Ω, σ(w^N₀, q^N₀ )n =F on Σ, w_ε^N = 0 on Γ,

(P^D0)

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−ν∆w₀^D+∇q^D₀ =G in Ω, divw^D₀ = 0 in Ω, w^D₀ =Wm on Σ, w^D₀ = 0 on Γ.

2.2 Sensitivity analysis method

To solve the topological optimization problem (Popt) and identify the unknown obstacle, we will propose a simplified approach based on the topological sen- sitivity analysis for the Kohn-Vogelius functionK.

The classical topological sensitivity analysis method is based on a first order asymptotic expansion of the form

K(Ω\Oz,ε)−K(Ω) =ρ(ε)δK(z) +o ρ(ε)

, ∀z∈Ω, where

ε 7→ ρ(ε) is a positive scalar function going to zero withε.

z 7→ δK(z) is the first order topological gradient.

In the context of fluid mechanics, the first order topological sensitivity anal- ysis has been derived for the Stokes operator in [19, 21, 32], the quasi-Stokes in [22] and for the stationary Navier-Stokes equations in [20]. In [33], Caubet and Dambrine have used the Kohn-Vogelius formulation combined with the topological sensitivity analysis to locate obstacles in a fluid flow domain.

In this work, we extend the topological sensitivity notion for the high-order case. We will derive a high-order topological asymptotic expansion for the Kohn-Vogelius functionalK with respect to the presence of the Dirichlet ge- ometric perturbation Oz, ε inside the domain Ω. It consists in studying the variationK(Ω\Oz,ε)−K(Ω) with respect toεand establishing an asymptotic formula on the form

K(Ω\Oz,ε)−K(Ω) =

m

X

q=1

ρq(ε)δK^q(z,O) +o(ρm(ε)) (1) where

ε 7→ ρq(ε), 1 ≤ q ≤ m are positive scalar functions defined on R⁺ verifyingρq+1(ε) =o(ρq(ε)) and lim

ε→0ρq(ε) = 0.

δK^q denotes theq^thtopological derivative of the functional K. m∈N^∗ an arbitrary order of the asymptotic expansion.

To this end, we start our analysis by deriving a preliminary result showing the relationship between the quantityK(Ω\Oz,ε)−K(Ω) and the variations of the Neumann and Dirichlet perturbed solutions (w_ε^N −w^N₀, q^N_ε −q₀^N) and (w_ε^D−w₀^D, q^D_ε −q₀^D).

FLUID MEDIUM 40

3 The Kohn-Vogelius functional variation

In this section, we discuss the sensitivity of the considered Kohn-Vogelius functionalK with respect to the presence of a small obstacle Oz, ε inside the fluid flow domain. We will prove that the variationK(Ω\Oz,ε)−K(Ω) can be only described by four integral terms, involving the discrepancy between the initial and the perturbed Stokes problem solutions.

Theorem 3.1. Let Oz,ε be a small obstacle, strictly embedded into the fluid flow domainΩ. In the presence ofOz,ε, the Kohn-Vogelius functionKadmits the following variation

K(Ω\Oz,ε)−K(Ω) = Z

∂Oz,ε

σ(w_ε^N −w₀^N, q_ε^N−q₀^N)nw^N₀ ds

− Z

∂Oz,ε

σ(w^D_ε −w^D₀, q_ε^D−q^D₀)nw^D₀ds +ν

Z

Oz,ε

∇w₀^D

2dx−ν Z

Oz,ε

∇w^N₀

2dx.

(2)

Proof. The variation of the functionK is given by K(Ω\Oz,ε)−K(Ω) = ν

Z

Ω\Oz,ε

∇w^Nε − ∇w^Dε

2

dx−ν Z

Ω

∇w0^N− ∇w0^D

2

dx

=ν Z

Ω\Oz,ε

∇w^N_ε

2

dx+ν Z

Ω\Oz,ε

∇w^D_ε

2

dx−2ν Z

Ω\Oz,ε

∇w^D_ε∇w^N_ε dx

−ν Z

Ω

∇w^N0

2

dx−ν Z

Ω

∇w0^D

2

dx+ 2ν Z

Ω\Oz,ε

∇w0^D∇w^N0 dx.

As one can observeK(Ω\Oz,ε)−K(Ω) can be decomposed as K(Ω\Oz,ε)−K(Ω) =TN(ε) +TD(ε)−2TM(ε), withTN is the Neumann term

TN(ε) =ν Z

Ω\Oz,ε

∇w^N_ε

2dx−ν Z

Ω

∇w₀^N

2dx, TD is the Dirichlet term

TD(ε) =ν Z

Ω\Oz,ε

∇w_ε^D

2dx−ν Z

Ω

∇w₀^D

2dx, andTM is the mixed term

TM(ε) =ν Z

Ω\Oz,ε

∇w^D_ε∇w_ε^Ndx−ν Z

Ω\Oz,ε

∇w₀^D∇w^N₀dx.

Next, we will examine each term separately. We start our analysis by studying the termTN.

−Variation of the Neumann term: The term TN reads TN(ε) = ν

Z

Ω\Oz,ε

∇w^N_ε

2dx−ν Z

Ω

∇w₀^N

2dx

= ν

Z

Ω\Oz,ε

∇(w^N_ε −w₀^N)∇w^N_ε dx+ν Z

Ω\Oz,ε

∇(w^N_ε −w^N₀)∇w₀^Ndx

− ν Z

Oz,ε

∇w₀^N

2dx.

Using Green formula, from the Stokes system satisfied by (w^N_ε −w^N₀ , q^N_ε −q^N₀ ), we obtain

ν Z

Ω\Oz,ε

∇(w^N_ε −w₀^N)∇w₀^Ndx= Z

∂Oz,ε

σ(w^N_ε −w^N₀, q^N_ε −q₀^N)nw₀^Nds.

In addition,

ν Z

Ω\Oz,ε

∇(w^N_ε −w₀^N)∇w^N_ε dx dt= 0.

Then, the termTN can be written as TN(ε) =

Z

∂Oz,ε

σ(w_ε^N−w^N₀ , q^N_ε −q^N₀ )nw^N₀ds−ν Z

Oz,ε

∇w^N₀

2dx. (3)

−Variation of the Dirichlet term: We have TD(ε) = ν

Z

Ω\Oz,ε

∇w^Dε

2

dx−ν Z

Ω

∇w^D0

2

dx

=ν Z

Ω\Oz,ε

∇(wε^D−w0^D)

2

dx+ 2ν Z

Ω\Oz,ε

∇(w^Dε −w^D0)∇w^D0dx−ν Z

Oz,ε

∇w0^D

2

dx.

Using Green formula, the system (P^D0) implies ν

Z

Ω\Oz,ε

∇(w^Dε −w₀^D)∇w0^Ddx= Z

Ω\Oz,ε

F(w^D_ε −w^D₀)dx− Z

∂Oz,ε

σ(w^D₀, q^D₀)nw₀^Dds.

Then, the Dirichlet term admits the following variation TD(ε) = ν

Z

Ω\Oz,ε

∇(w_ε^D−w^D₀)

2dx+ 2 Z

Ω\Oz,ε

F(w_ε^D−w^D₀)dx

−ν Z

Oz,ε

∇w^D₀

2dx−2 Z

∂Oz,ε

σ(w₀^D, q₀^D)nw^D₀ds.

FLUID MEDIUM 42 Using Green formula, from the system verified by (w_ε^D−w₀^D, q^D_ε −q^D₀), we deduce

TD(ε) = − Z

∂Oz,ε

σ(w_ε^D−w₀^D, q_ε^D−q₀^D)nw^D₀ds+ 2 Z

Ω\Oz,ε

F(w^D_ε −w₀^D)dx

−2 Z

∂Oz,ε

σ(w^D₀, q^D₀)nw₀^Dds−ν Z

Oz,ε

∇w^D₀

2dx. (4)

−Variation of the mixed term: We have TM(ε) =ν

Z

Ω\Oz,ε

∇w^N_ε ∇w^D_εdx dt−ν Z

Ω

∇w₀^N∇w^D₀dx.

From the weak formulation of problems (P^Nε ) and (P^N0 ), it follows that TM(ε) =

Z

Ω\Oz,ε

F(w^D_ε −w^D₀)dx− Z

Oz,ε

F w₀^Ddx.

From the fact that−ν∆w^D₀ +∇q^D₀ =F inOz,ε and divw^D₀ = 0 in Oz,ε, one can deduce

Z

Oz,ε

F w^D₀dx=ν Z

Oz,ε

∇w₀^D

2dx+ Z

∂Oz,ε

σ(w₀^D, q₀^D)nw^D₀ds.

Consequently, the termTM can be written as TM(ε) =

Z

Ω\Oz,ε

F(w_ε^D−w^D₀)dx−ν Z

Oz,ε

∇w^D₀

2dx−

Z

∂Oz,ε

σ(w^D₀, q₀^D)nw^D₀ds.

(5)

−Variation of the Kohn-Vogelius function K: Combining the variations (3), (4) and (5), one can deduce that the variation ofK can be written as

K(Ω\Oz,ε)−K(Ω) = Z

∂Oz,ε

σ(w_ε^N−w^N₀ , q^N_ε −q^N₀ )nw^N₀ ds−ν Z

Oz,ε

∇w₀^N

2

dx +ν

Z

Oz,ε

∇w0^D

2

dx− Z

∂Oz,ε

σ(wε^D−w^D0, qε^D−q^D0)nw0^Dds.

4 Asymptotic behavior of the perturbed solutions

In this section, we discuss the influence of the geometric perturbationOz,ε on the solutions of the Stokes problems (P^Nε ) and (P^Dε). More precisely, we derive an asymptotic formula describing the variations of the velocity and pressure fields with respect to the perturbation sizeε. We start our analysis by the perturbed Stokes Neumann problem.

4.1 The Neumann perturbed solution

This section is devoted to an asymptotic formula describing the variation of the Neumann solution (w^N_ε −w₀^N) with respectε. We begin our analysis by the following first order estimate.

Lemma 4.1. LetOz,ε be a small geometric perturbation strictly included into Ω. Then, the perturbed Stokes solution (w^N_ε , q_ε^N)satisfies the behavior

w^N_ε (x)−w^N₀ (x) = E₀^N((x−z)/ε) + O(ε) inΩ\Oz,ε, q^N_ε (x)−q^N₀ (x) = 1

εΠ^N₀((x−z)/ε) + O(ε)inΩ\Oz,ε

where the leading term(E₀^N,Π^N₀ )is solution to the following Stokes exterior problem

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−∆E₀^N +∇Π^N₀ = 0 in R³\O,

∇.E^N₀ = 0 in R³\O,

E^N₀ −→0 at ∞

E^N₀ =−w^N₀(z) on ∂O.

(6)

Proof: The existence of (E₀^N,Π^N₀) can be established with the help of a single layer potential [33]. One can derive

E₀^N(y) = Z

∂O

U(y−t)η^N₀ (t)ds(t),∀y∈R³\O, Π^N₀(y) =

Z

∂O

P(y−t).η₀^N(t)ds(t),∀y∈R³\O, where (U, P) is the fundamental solution of the Stokes system inR³

U(y) = 1 8πνr

I+ere^T_r

, P(y) = y

4πr³, ∀y∈R³, withr=kyk,er= y

r ande^T_r is the transposed vector ofer.

Hereη₀^N is the solution to the following boundary integral equation Z

∂O

U(y−t)η^N₀ (t)ds(t) =−w^N₀(z), ,∀y∈∂O. Setting

z_ε^N(x) = w_ε^N(x)−w^N₀ (x)−E₀^N((x−z)/ε), s^N_ε(x) = q_ε^N(x)−q₀^N(x)−1

εΠ^N₀((x−z)/ε).

FLUID MEDIUM 44 As one can observe, (z^N_ε , s^N_ε ) satisfies the system

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−ν∆z^N_ε +∇s^N_ε = 0 in Ω\Oz, ε,

div z^N_ε = 0 in Ω\Oz, ε,

σ(z_ε^N, s^N_ε)n =−1

εσ(E₀^N,Π^N₀)((x−z)/ε)n on Σ, w^N_ε =−E₀^N((x−z)/ε) on Γ, w^N_ε =−w^N₀ +w₀^N(z) on ∂Oz, ε. Using the change of variable x = z+εy and the standard energy estimate for the Stokes problem, one can derive that there exists a constant c > 0, independent ofε, such that

z_ε^N _H1

(Ω\Oz, ε)+ s^N_ε

_L2

(Ω\Oz, ε)≤ c ε.

One can see ([19], Proposition 3.1) for similar proof.

Next, we extend this estimate to the high-order case. The obtained asymptotic behavior is illustrated by the following theorem.

Theorem 4.2. Let Oz,ε =z+εω be a small geometric perturbation, strictly embedded in the fluid flow domain Ω. Then, the velocity and pressure fields satisfy the following asymptotic behavior

w^N_ε (x)−w^N₀ (x) =E^N₀ ((x−z)/ε)+

m

X

k=1

ε^k[V_k^N(x)+E_k^N((x−z)/ε))]+o(ε^m), (7)

q_ε^N(x)−q₀^N(x) = 1

εΠ^N₀ ((x−z)/ε) +

m

X

k=1

ε^k[S_k^N(x) +1

εΠ^N_k((x−z)/ε))] +o(ε^m), (8) where(V_k^N, S_k^N)_0≤k≤mis a set of smooth functions satisfying the Stokes system in Ωand (E_k^N,Π^N_k)_0≤k≤m is a set of smooth functions verifying the exterior Stokes problem inR³\O.

Proof: The terms of the derived asymptotic expansion are built iteratively.

Initialization: We start our construction process by the terms associated with k = 0. The two sequences (V_k^N, S_k^N)0≤k≤m and (E_k^N,Π^N_k)0≤k≤m are initialized as follows:

−(V₀^N, S₀^N) = (w₀^N, q₀^N) which is the solution to the Stokes problem (P^N0 ), defined in

the non perturbed domain Ω.

−(E₀^N,Π^N₀) is the solution to the exterior Stokes problem (6), defined in R³\O.

The k^th term : Let k∈ {1, ..., m}. Assume that we have already derived the terms (V_i^N, S_i^N) and (E_i^N,Π^N_i ) for all 0≤i≤k−1,and we want to derive the terms (V_k^N, S_k^N) and (E_k^N,Π^N_k).

In order to define the desired terms, we need to establish a preliminary calculus. It concerns the asymptotic behavior of the functionsE_i^N and Π^N_i with respect toε.

Recalling that (E_i^N,Π^N_i ) is constructed as a solution to an exterior Stokes problem defined inR³\O. Then, due to a single layer potential [33], (E_i^N,Π^N_i ) can be written as

E_i^N(y) = Z

∂O

U(y−t)η^N_i (t)ds(t),∀y∈R³\O, Π^N_i (y) =

Z

∂O

P(y−t).η_i^N(t)ds(t),∀y∈R³\O,

whereη_i^N is the solution to a boundary integral equation defined on∂O. From the fact thatU(y/ε) = εU(y) it follows that for each x∈R³\Oz,ε we have

E_i^N((x−z)/ε) = Z

∂O

U((x−z)/ε−t)η_i^N(t)ds(t)

=ε Z

∂O

U((x−z)−ε t)η_i^N(t)ds(t).

From the fact thatOz,ε is not close to the boundary∂Ω, one can remark that for allt∈∂O, the functionε7−→U_x−z,t(ε) =ε U((x−z)−ε t) is smooth with respect toεand admits the following asymptotic expansion

U_x−z,t(ε) =

m

X

p=1

ε^p

(p−1)!U_x−z,t^(p−1)(0) +o(ε^m),

whereU_x−z,t^(p) (0) is thepth derivative ofU_x−z,tatε= 0. It depends on thepth derivative of the function U at the point x−z. Consequently, the function

FLUID MEDIUM 46 ε7−→E_i^N((x−z)/ε) satisfies the following asymptotic behavior

E_i^N((x−z)/ε) =

m

X

p=1

ε^pE_N,i^(p)(x−z) +o(ε^m), (9)

withE_N,i^(p) is the smooth function defined inR³\Oby E_N,i^(p)(y) = 1

(p−1)!

Z

∂O

U_y,t^(p−1)(0)η^N_i (t)ds(t),∀y∈R³\O. (10) Similarly, we applied the same analysis for the pressure field. Then, using the fact thatP(y/ε) =ε²P(y) one can obtain

Π^N_i ((x−z)/ε) =ε² Z

∂O

P((x−z)−ε t).η^N_i (t)ds(t).

From the smoothness of the functionε7−→P_x−z,t(ε) =ε²P((x−z)−ε t) with respect toε, it follows

P_x−z,t(ε) =

m−1

X

q=1

ε^q+1

(q−1)!P_x−z,t^(q−1)(0) +o(ε^m),

whereP_x−z,t^(q) (0) is theqth derivative ofP_x−z,t atε= 0. Hence, we have Π^N_i ((x−z)/ε) =

m−1

X

q=1

ε^q+1Π^(q)_N,i(x−z) +o(ε^m), (11)

with Π^(q)_N,iis the smooth function defined inR³\Oby Π^(q)_N,i(y) = 1

(q−1)!

Z

∂O

P_y,t^(q−1)(0).η_i^N(t)ds(t),∀y∈R³\O. (12)

−Determining the term (V_k^N, S_k^N). It is constructed using the functions E_i^N,0 ≤ i ≤ k−1. It is defined as the solution to the following Stokes system























−∆V_k^N +∇S_k^N = 0 in Ω,

∇.V_k^N = 0 in Ω,

σ(V_k^N, S_k^N)n =−

k

X

p=1

σ(E_N,k−p^(p) ,Π^(p)_N,k−p)(x−z)n on Σ, V_k^N =−

k

X

p=1

E^(p)_N,k−p(x−z) on Γ,

(13)

where the functionsE_N,j^(p) and Π^(p)_N,j are defined by (10) and (12).

−Determining the term (E_k^N,Π^N_k). It is constructed using the functions V_i^N,0 ≤ i ≤k. This term is defined as a solution to the following exterior problem















−∆E_k^N+∇Π^N_k = 0 in R³\O,

∇.E_k^N = 0 in R³\O,

E_k^N −→0 at ∞

E_k^N =−V_k^N(z)−

k

X

p=1

1

p!D^pV_k−p^N (z)(y^p) on ∂O, (14) whereD^pV_k−p^N (z) is thepth derivative of the functionV_k−p^N at the pointzand y^p= (y, ..., y)∈(R³)^p.

Justification of the asymptotic formulas: Here we will prove that the con- structed sequences (V_k^N, S_k^N)_0≤k≤mand (E_k^N,Π^N_k )_0≤k≤m permit us to derive the expected asymptotic formulas.

Posing

R^N_m,ε(x) = w^N₀ (x) +E₀^N((x−z)/ε)) +

m

X

k=1

ε^k[V_k^N(x) +E_k^N((x−z)/ε))]−w_ε^N,

ξ_m,ε^N (x) = S₀^N(x) +1

εΠ^N₀ ((x−z)/ε)) +

m

X

k=1

ε^k[S_k^N(x) +1

εΠ^N_k((x−z)/ε))]−q^N_ε . One can easily verify that (R^N_m,ε, ξ^N_m,ε) solves the Stokes system in Ω\Oz,ε

−∆R^N_m,ε+∇ξ_m,ε^N = 0 in Ω\Oz,ε,

∇.R^N_m,ε = 0 in Ω\Oz,ε, (15) and satisfies the following boundaries conditions:

- On∂Oz,ε: Using the systems (13)-(14), the multi-linearity ofD^pV_k−p^N (z), Taylor’s Theorem and the fact that kx−zk= O(ε) on ∂Oz,ε, one can derive

R^N_m,ε(x) =

m

X

k=0

ε^kh

V_k^N(x)−

m−k

X

p=0

1

p!D^pV_k^N(z)((x−z)^p)i

=o(ε^m).

FLUID MEDIUM 48 - On Γ: the Dirichlet boundary condition in (13) and the asymptotic

expansions (9) and (11) imply R^N_m,ε(x) =

m

X

k=0

ε^kE_k^N((x−z)/ε)) +

m

X

k=1

ε^kV_k^N(x)

=

m

X

k=0

ε^khX^m

p=1

ε^pE_N,k^(p)(x−z)i

−

m

X

k=1

ε^khX^k

p=1

E_N,k−p^(p) (x−z)i

+o(ε^m)

= o(ε^m). (16)

- On Σ: by the change of variablex=z+εy, we have σ(R_m,ε, ξ_m,ε) =1

ε

m

X

k=0

ε^kσ_y(E_k^N,Π^N_k)((x−z)/ε) +

m

X

k=1

ε^kσ(V_k^N, S_k^N)(x).

Using again the change of variable, from (9) and (11) one can deduce σ_y(E_k^N,Π^N_k )((x−z)/ε) =ε

m

X

p=1

ε^pσ(E_N,k^(p),Π^(p)_N,k)(x−z) +o(ε^m).

The two last relations combined with the Neumann condition used in (13) imply

σ(Rm,ε, ξm,ε)n=o(ε^m) on Σ.

4.2 The Dirichlet perturbed solution

This section is concerned with brief analysis of the Dirichlet case. Here, we consider (V_k^D, S_k^D)_0≤k≤ma set of smooth functions satisfying the Stokes system in Ω and (E_k^D,Π^D_k)_0≤k≤m a set of smooth functions verifying the exterior Stokes problem inR³\O. The two considered sequences (V_k^D, S_k^D)_0≤k≤mand (E_k^D,Π^D_k)_0≤k≤mare initialized as follows:

−(V₀^D, S₀^D) = (w^D₀, q₀^D) which is the solution to the Stokes problem (P^D0).

− (E₀^D,Π^D₀) is defined as the solution to the following exterior Stokes problem











−∆E₀^D+∇Π^D₀ = 0 in R³\O,

∇.E₀^D = 0 in R³\O,

E₀^D −→0 at ∞

E₀^D =−w^D₀(z) on ∂O.

•The term (V_k^D, S^D_k). It is defined as the unique solution to the following Stokes system











−∆V_k^D+∇S_k^D = 0 in Ω,

∇.V_k^D = 0 in Ω, V_k^D =−

k

X

p=1

E^(p)_D,k−p(x−z) on Γ∪Σ,

(17)

where the functionsE_D,j^(p),0≤j≤kare defined by E^(p)_D,j(y) = 1

(p−1)!

Z

∂O

U_y,t^(p−1)(0)η^D_j (t)ds(t),∀y∈R³\O. (18)

•The term (E_k^D,Π^D_k). It is constructed using the functionsV_j^D,0 ≤j ≤k.

This term is defined as a solution to the following exterior problem















−∆E_k^D+∇Π^D_k = 0 in R³\O,

∇.E_k^D = 0 in R³\O,

E_k^D −→0 at ∞

E_k^D =−V_k^D(z)−

k

X

p=1

1

p!D^pV_k−p^D (z)(y^p) on ∂O. (19) Theorem 4.3. In the presence of a small geometric perturbationOz,ε=z+εω inside the fluid flow domainΩ, the solution(w^D_ε, q_ε^D)of the perturbed Dirichlet Stokes problem admits the asymptotic behavior

w^D_ε(x)−w₀^D(x) =E^D₀((x−z)/ε) +

m

X

k=1

ε^k[V_k^D(x) +E_k^D((x−z)/ε))] +o(ε^m), (20) q_ε^D(x)−q₀^D(x) =1

εΠ^D₀((x−z)/ε) +

m

X

k=1

ε^k[S^D_k(x) +1

εΠ^D_k((x−z)/ε))] +o(ε^m).

(21)

5 High-order topological asymptotic expansion

In this section we extend the topological derivative notion [16, 17, 18, 19, 20, 21, 34] for the high-order case. We derive a high-order terms in the topological asymptotic expansion for the Stokes operator. We will derive an asymptotic

FLUID MEDIUM 50 formula describing the variation of the Kohn-Vogelius functional K with re- spect to the insertion of a small obstacle inside the fluid flow domain Ω.

From Theorem 3.1, the variation caused by the presence of the geometric perturbationOz,ε=z+εOcan be decomposed as

K(Ω\Oz,ε)−K(Ω) = J^N(ε)−J^D(ε), where the Neumann and Dirichlet terms are defined by

J^N(ε) = Z

∂Oz,ε

σ(w_ε^N−w^N₀ , q_ε^N−q^N₀ )nw^N₀ ds−ν Z

Oz,ε

∇w^N₀

2dx,(22) J^D(ε) =

Z

∂Oz,ε

σ(w_ε^D−w^D₀, q_ε^D−q₀^D)nw^D₀ds−ν Z

Oz,ε

∇w₀^D

2dx.(23)

To derive the expected high-order asymptotic expansion for Kohn-Vogelius functionalKwe will examine the termsJ^N andJ^D separately.

5.1 Estimate of the Neumann terms

Here, we derive a sensitivity analysis for each term in J^N(ε) with respect to the parameterε. We will establish a high-order asymptotic expansion for each term. Our mathematical analysis is based on the asymptotic behavior of the perturbed solution (w^N_ε, q^N_ε).

Lemma 5.1. The first term in (22) admits the estimate Z

∂Oz,ε

σ(w^N_ε −w₀^N, q_ε^N −q₀^N)n.w₀^Nds

=

m−1

X

k=0

ε^k+1 Z

∂O

σ_y(E_k^N,Π^N_k )(y)n(y).w^N₀ (z+εy)ds(y) +

m−2

X

k=1

ε^k+2 Z

∂O

σ(V_k^N, S_k^N)(z+εy)n.w^N₀ (z+εy)ds(y) +o(ε^m).

(24) Proof: From the change of variablex=z+εy, it follows

∇x

E_k^N((x−z)/ε))

=1

ε∇yE_k^N((x−z)/ε)), 0≤k≤m, where∇ξ denotes the gradient with respect to the variableξ(ξ=xory).

Using the previous relation and the decomposition presented in Theorem 4.2, one can derive

σ(w_ε^N−w^N₀ , q_ε^N−q^N₀ ) = 1

εσy(E^N₀ ,Π^N₀)((x−z)/ε)) +

m

X

k=1

ε^kσ(V_k^N, S^N_k )(x)

+

m+1

X

k=1

ε^k−1σ_y(E^N_k ,Π^N_k)((x−z)/ε) +o(ε^m).

Then, the first term in (22) satisfies the estimate Z

∂Oz,ε

σ(w^Nε −w0^N, q^Nε −q0^N)n.w0^Nds = 1 ε Z

∂Oz,ε

σy(E0^N,Π^N0 )((x−z)/ε))n.w^N0 ds

+

m+1

X

k=1

ε^k−1 Z

∂Oz,ε

σy(Ek^N,Π^Nk)((x−z)/ε)n.w^N0ds

+

m

X

k=1

ε^k Z

∂Oz,ε

σ(V_k^N, S_k^N)(x)n.w₀^Nds+o(ε^m).

Making use of the change of variablex=z+εy, one can deduce Z

∂Oz,ε

σy(Ek^N,Π^Nk)((x−z)/ε)n.w^N0 ds = ε² Z

∂O

σy(Ek^N,Π^Nk)(y)n(y).w0^N(z+εy)ds(y),

Z

∂Oz,ε

σ(Vk^N, S^Nk)(x)n.w0^Nds = ε² Z

∂O

σ(Vk^N, Sk^N)(z+εy)n(y).w^N0(z+εy)ds(y).

Consequently, we obtain Z

∂Oz,ε

σ(w_ε^N−w^N₀ , q^N_ε −q^N₀ )n.w^N₀ds=

m−1

X

k=0

ε^k+1 Z

∂O

σy(E_k^N,Π^N_k)(y)n(y).w₀^N(z+εy)ds(y)

+

m−2

X

k=1

ε^k+2 Z

∂O

σ(Vk^N, S^Nk)(z+εy)n.w^N0 (z+εy)ds(y) +o(ε^m).

Next, we will examine the two integral terms in the right hand side of (24).

Lemma 5.2. We have

m−1

X

k=0

ε^k+1 Z

∂O

σ_y(E_k^N,Π^N_k)(y)n(y).w^N₀ (z+εy)ds(y) =

m−1

X

q=0

ε^q+1K^1,Nq (z,O) + o(ε^m),

FLUID MEDIUM 52 where the functionsz7−→K^1,Nq (z,O),0≤q≤mare defined inΩ by

K^1,Nq (z,O) =

q

X

p=0

1 p!

Z

∂O

σ_y(E_q−p^N ,Π^N_q−p)(y)n(y).[D^(p)w₀^N(z)(y^p)]ds(y)

with D^(p)w₀^N(z) denotes the pth derivative of the function w^N₀ at the point z∈Ωandy^p= (y, ..., y)∈(R³)^p.

Proof: Due to the smoothness of the velocity fieldw^N₀, by Taylor’s theorem one can derive

w₀^N(z+εy) =

m

X

p=0

ε^p

p!D^(p)w^N₀(z)(y^p) +o(ε^m). (25) It follows

m−1

X

k=0

ε^k+1 Z

∂O

σy(E_k^N,Π^N_k)(y)n(y).w^N₀(z+εy)ds(y)

=

m−1

X

k=0

ε^k+1X^m

p=0

ε^p p!

Z

∂O

σy(E^N_k ,Π^N_k)(y)n(y).[D^(p)w₀^N(z)(y^p)]ds(y)

+o(ε^m)

=

m−1

X

q=0

ε^q+1X^q

p=0

1 p!

Z

∂O

σy(E^N_q−p,Π^N_q−p)(y)n(y).[D^(p)w^N₀ (z)(y^p)]ds(y)

+o(ε^m) Lemma 5.3. The second integral term in the right hand side of (24) satisfies

m−2

X

k=1

ε^k+2 Z

∂O

σ(V_k^N, S^N_k )(z+εy)n.w₀^N(z+εy)ds(y) =

m−2

X

q=1

ε^q+2K^2,Nq (z,O) + o(ε^m).

where the functionsz7−→K^2,Nq (z,O),1≤q≤m−2 are defined inΩby K^2,Nq (z,O) =

q−1

X

p=0 p

X

l=0

1 l!(p−l)!

Z

∂O

σ^(l)(V_q−p^N , S^N_q−p)(z)n.D^(p−l)w₀^N(z)(y^p−l)ds(y)

Proof: Here we exploit the smoothness of the functions V_q^N and S_q^N. This follows from the fact that (V_q^N, S_q^N) is solution to the Stokes problem (13),