International Journal of Analysis and Applications

Volume 19, Number 6 (2021), 949-969

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-19-2021-949

TOPOLOGICAL SENSITIVITY ANALYSIS FOR THE ANISOTROPIC LAPLACE

PROBLEM

IMEN KALLEL1,2,∗

1Northern Border University, College of Science, Arar, P.O. Box 1631, Saudi Arabia

2UR Analysis and Control of PDE’s, UR 13E64, Department of Mathematics, Faculty of Sciences of

Monastir, University of Monastir, 5019 Monastir, Tunisie

∗Corresponding author: imenkallel16@gmail.com

Abstract. This paper is concerned with the reconstruction of objects immersed in anisotropic media from

boundary measurements. The aim of this paper is to propose an alternative approach based on the Kohn-

Vogelius formulation and the topological sensitivity analysis method. The idea is to formulate the reconstruc-

tion problem as a topology optimization one minimizing an energy-like function. We derive a topological

asymptotic expansion for the anisotropic Laplace operator. The unknown object is reconstructed using

level-set curve of the topological gradient. We make finally some numerical examples proving the efficiency

and accuracy of the proposed algorithm.

1. Introduction

In this work we will establish a topological sensitivity analysis for the anisotropic Laplace operator. The

topological sensitivity analysis consists of studying the variation of a given cost functional with respect to the

presence of a small domain perturbation, such as the insertion of inclusions, cavities, cracks or source-terms.

In our paper we concentrate in a small Dirichlet geometric perturbation.

Let us briefly discuss the history of this method. Its main idea was originally introduced by Schumacher

[22] in the context of compliance minimization in linear elasticity. In the same context Sokolowski and

Received August 28th, 2021; accepted October 12th, 2021; published November 16th, 2021.

2010 Mathematics Subject Classification. 49Q12, 65N21, 35N10.

Key words and phrases. topological gradient; Kohn-Vogelius formulation; Laplace problem; reconstruction problem.

©2021 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

949

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-19-2021-949


Int. J. Anal. Appl. 19 (6) (2021) 950

Zochowski [13], who studied the effect of an extract infinitesimal part of the material in structural mechanics.

Then in [17] Masmoudi worked out a topological sensitivity analysis framework based on a generalization

of the adjoint method and on the use of a truncation technique. By using this framework the topological

sensitivity is obtained for several equations [8, 18, 20, 21]. For other works on the topological sensitivity

concept, we refer to the book by Novotny and Sokolowski [19].

The general idea of the proposed method is to rephrase the inverse problem as a topology optimization

problem, where the object immersed in the anisotropic media is the unknown variable. Our aim is to detect

this unknown object immersed from over-determined boundary data.

Let H be an unknown object immersed inside the background domain Ω and having a smooth boundary

Σ = ∂H. The geometric inverse problem that we consider here can be formulated as follows:

• Giving two boundary data on Γ ; an imposed flux Φ ∈ H−1/2(Γ) and a measured datum ϕd ∈

H1/2(Γ).

• Find the unknown location of the object H inside the domain Ω such that the solution ϕ of the

anisotropic Laplace equation satisfies the following over-determined boundary value problem




−div(A∇ϕ) = F in Ω\H,

A∇ϕ · n = Φ on Γ,

ϕ = ϕd on Γ,

ϕ = 0 on ∂H.

where A is a symmetric positive definite matrix, n is the exterior unit normal vector and F ∈ L2(Ω) is a

given source term.

In this formulation the domain Ω\H is unknown since the free boundary
∑

is unknown. This problem is

ill-posed in the sense of Hadamard [10]. The majority of works dealing with this kind of problems fall into

the category of shape optimization and based on the shape differentiation technics. It is proved in [3, 7] that

the studied inverse problems, treated as a shape optimization problems, are severely ill-posed (i.e. unstable),

for both Dirichlet and Neumann conditions on the boundary
∑

. Thus they have to use some regularization

methods to solve them numerically.

To solve this inverse problem, we extend the topological sensitivity analysis notion to the anisotropic

case and we suggest an alternative approach based on the Kohn-Vogelius formulation [6] and the topological

gradient method [1, 2, 5, 8, 9, 16, 17]. We combine here the advantages of the Kohn-Vogelius formulation as a

self regularization technique and the topological gradient approach as an accurate and fast method.

The first step of our approach is based on the Kohn-Vogelius formulation which rephrase the considered

geometrical inverse problem into a topology optimization one. It leads to define for any given permutation



Int. J. Anal. Appl. 19 (6) (2021) 951

H two forward problems: The first one, called Neumann problem, is associated to the Neumann datum Φ

(PN)



−div(A∇ϕN ) = F in Ω\H,

A∇ϕN · n = Φ on Γ,

ϕN = 0 on ∂H.

The second one is associated to the measured data ϕd, it is called the Dirichlet problem:

(PD)



−div(A∇ϕD) = F in Ω\H,

ϕD = ϕd on Γ,

ϕD = 0 on ∂H.

One can observe that if H coincides with the exact obstacle H∗ , the misfit between the solutions of(PN)

and (PD) vanishes: ϕN = ϕD . Starting from this observation, the inverse problem can be formulated as

a topological optimization one. The unknown object will be characterized as the minimum of the following

Kohn-Vogelius type functional [6]

J(Ω\H) =
∫

Ω\H
A|∇ϕN (x) −∇ϕD(x)|2dx

Where the Kohn-Vogelius function is exactly

J(Ω\H) =
∫

Ω\H
[∇ϕN (x) −∇ϕD(x)]T ·A[∇ϕN (x) −∇ϕD(x)]dx

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

(P)




Find H∗ ⊂ Ω such that

J(Ω\H∗) = min
H∈Dad

J(Ω\H)

where Dad is a given set of admissible domains.

To solve the topological optimization problem (P) and detect the location of the unknown object we

will derive a topological sensitivity analysis for the Kohn-Vogelius function J which gives the variation of a

criterion with respect to the presence of a small Dirichlet geometric perturbation in the domain. A one-shot

reconstruction algorithm is proposed. The main advantage of this algorithm is that, it provides fast and

accurate results for detection.

The paper is organized as follows. In the next Section, we present the perturbed Neumann and Dirichlet

problems. In Section 3 we study the topological sensitivity analysis for the function J . The obtained results

are based on a preliminary estimate describing the perturbation caused by the presence of a small geometry

modification of the background domain Ω. A simplified formulation of the shape function variation with

respect to the creation of the hole Hz,ε in Ω is derived in Section 4. The Section 5 is devoted to the Kohn-

Vogelius type function variation. The proposed numerical algorithm and the detection results are described

in Section 6.



Int. J. Anal. Appl. 19 (6) (2021) 952

2. The perturbed problems

In this section, we present the Neumann and Dirichlet problems in the perturbed domain. In the presence

of a small geometry perturbation Hz,ε inside the domain Ω, the Neumann problem consists in finding

ϕεN ∈ H
1(Ω\Hz,ε) solution to 


−div(A∇ϕεN ) = F in Ω\Hz,ε,

A∇ϕεN · n = Φ on Γn,

ϕεN = 0 on ∂Hz,ε.

The Neumann problem in the non perturbed domain is: find ϕ0N ∈ H
1(Ω) solution to

 −div(A∇ϕ
0
N ) = F in Ω,

A∇ϕ0N · n = Φ on Γn.

Similarly, the perturbed Dirichlet problem consists in finding ϕεD ∈ H
1(Ω\Hz,ε) solution to


−div(A∇ϕεD) = F in Ω\Hz,ε,

ϕεD = ϕd on Γd,

ϕεD = 0 on ∂Hz,ε.

In the absence of any perturbation (i.e ε = 0), the Dirichlet problem is: find ϕ0D ∈ H
1(Ω) solution to

 −div(A∇ϕ
0
D) = F in Ω,

ϕ0D = ϕd on Γd.

We introduce the considered shape functional J . Given a small geometrical perturbation Hz,ε inside the

initial domain Ω, the function J measures the difference between the Neumann and Dirichlet perturbed

solutions. We define J as

J(Ω\Hz,ε) =
∫

Ω\Hz,ε
A|∇ϕεN (x) −∇ϕ

ε
D(x)|

2dx, ∀Hz,ε ⊂ Ω

In the non perturbed domain (ε = 0), the function J is expressed as

J(Ω) =
∫

Ω

A|∇ϕ0N (x) −∇ϕ
0
D(x)|

2dx.

Our aim is to derive an asymptotic expansion for the function J and calculate the topological sensitivity

function δJ .

The variation of the function J with respect to the presence of a small perturbation is given by

J(Ω\Hz,ε) −J(Ω) =
∫

Ω\Hz,ε
A|∇ϕεN (x) −∇ϕ

ε
D(x)|

2dx

−
∫

Ω
A|∇ϕ0N (x) −∇ϕ

0
D(x)|

2dx.

In the following, we will derive a topological sensitivity analysis valid for all function Jε verifying the

following hypothesis:



Int. J. Anal. Appl. 19 (6) (2021) 953

Hypothesis 2.1. - The function J0 is differentiable in H1(Ω).

- There exist a real number δJ ∈ R, independent of ε and a scalar function ρ : R+ → R+ such that

∀ε ≥ 0

Jε(ϕε) −J0(ϕ0) = DJ0(ϕ0)(ϕε −ϕ0) + ρ(ε)δJ + o(ρ(ε)),

lim
ε→0

ρ(ε) = 0

with ϕε is the solution of the perturbed anisotropic Laplace problem inside the perturbed domain Ω\Hz,ε

(2.1) (Pε)




−div(A∇ϕε) = F in Ω\Hz,ε,

A∇ϕε · n = Φ on Γ,

ϕε = ϕd on Γ,

ϕε = 0 on ∂Hz,ε.

Here δJ is called the topological sensitivity function.

3. Topological asymptotic expansion

In this section, we derive a topological asymptotic expansion for the anisotropic Laplace operator. We

start our analysis by establishing a variational formulation associated to the anisotropic Laplace system.

From the weak formulation of 2.1, we deduce that ϕε ∈Vε is the unique solution to

aε(ϕε,v) = lε(v) ∀v ∈Vε,

where the function space Vε, the bilinear form aε and the linear form lε are defined by

Vε
{
w ∈ H1(Ω\Hz,ε); w = 0 in Γd ∪∂Hz,ε

}
and for all ϕ,v ∈Vε 


aε(v,w) =

∫
Ω\Hz,ε

[∇w]T ·A∇vdx,

lε(w) =
∫

Ω\Hz,ε
Fwdx +

∫
Γn

Φwds.

Under the hypothesis 2.1, the variation of the shape function J reads

J(Ω\Hz,ε) −J(Ω) = Jε(ϕε) −J0(ϕ0)

= DJ0(ϕ0)(ϕε −ϕ0) + ρ(ε)δJ + o(ρ(ε))

Let v0 ∈V0 be the solution to the associated adjoint problem

a0(w,v0) = DJ0(ϕ0)(w),∀w ∈V0.



Int. J. Anal. Appl. 19 (6) (2021) 954

Then, the shape function variation rewritten as

J(Ω\Hz,ε) −J(Ω) = a0(ϕ0 −ϕε,v0) + ρ(ε)δJ + o(ρ(ε)).

Aiming to derive an asymptotic expansion for J , we examine in the next section the asymptotic behavior

with respect to ε of the term a0(ϕ0 −ϕε,v0).

3.1. Asymptotic formula for the anisotropic Laplace problem. This section is devoted to the main

theoretical result. A topological asymptotic expansion is derived for the anisotropic Laplace operator with

respect to the presence of a small topological perturbation Hz,ε in the initial domain. The obtained results

are general and valid for large class of cost functions. More precisely, the derived asymptotic expansion is

valid for all cost function J satisfying the assumption 2.1.

The main result of this section is summarized in the following theorem.

Theorem 3.1. Let j a design function of the form j(Ω\Hz,ε) = J(ϕε). If J satisfies the assumption 2.1 ,

then j has the following asymptotic expansion

j(Ω\Hz,ε) − j(Ω) = ρ(ε)(δa + δJ) + o(ρ(ε))

where


 ρ(ε) =

−1
log(ε)

, δa = 2π
√
|A|ϕ0(z)v0(z) if d = 2,

ρ(ε) = ε, δa = −4π
√
|A|ϕ0(z)v0(z) if d = 3.

The term δJ is the variation of the considered cost function J .

In order to check the hypothesis 2.1, we derive an asymptotic expansion the variation of the bilinear form.

We have

a0(ϕ0 −ϕε,v0) =
∫

Ω

[∇(ϕ0 −ϕε)]T ·A∇v0dx.(3.1)

Using the Green formula, we obtain

(3.2) a0(ϕ0 −ϕε,v0) =
∫
Hz,ε

[∇ϕ0]T ·A∇v0dx +
∫
∂Hz,ε

[A∇(ϕ0 −ϕε)]T · nv0dx

Next, we shall examine each term on the right hand side of 3.2 separately. The following lemma gives an

estimate for the first term.

Lemma 3.1. We have ∫
Hz,ε

[∇ϕ0]T ·A∇v0dx = εd|H|[∇ϕ0(z)]T ·A∇v0(z) + o(εd).



Int. J. Anal. Appl. 19 (6) (2021) 955

Proof

Using the change of variable x = z + εy one obtains

∫
Hz,ε

[∇ϕ0]T ·A∇v0dx = εd|H|[∇ϕ0(z)]T ·A∇v0(z)

+ εd
∫
H

{
[∇ϕ0(x + εy)]T ·A∇v0(z + εy) − [∇ϕ0(z)]T ·A∇v0(z)

}
dy

Due the the smoothness of ϕ0 and v0 in Hz,ε, we derive

lim
ε→0

∫
H

{
[∇ϕ0(x + εy)]T ·A∇v0(z + εy) − [∇ϕ0(z)]T ·A∇v0(z)

}
dy = 0

Then it follows ∫
Hz,ε

[∇ϕ0]T ·A∇v0dx = εd|H|[∇ϕ0(z)]T ·A∇v0(z) + o(εd).

Next to examine the second term of 3.2 we introduce the variable

χε = ϕ0 −ϕε.

it is easily to show that χε satisfies the following system

(3.3)




−div(A∇χε) = 0 in Ω\Hz,ε,

A∇χε · n = 0 on Γn,

χε = 0 sur Γd,

χε = ϕ0 on ∂Hz,ε.

We can write χε as

χε = hε + rε

where

hε(x) =
E(x−z)
E(z)

ϕ0(z), x ∈ Ω,

with E is the fundamental solution of the anisotropic Laplace operator [14]:

E(x) =




1

2π
√
|A|

log(||A∗x||), d = 2,

−1
4π
√
|A|||A∗x||

, d = 3.



Int. J. Anal. Appl. 19 (6) (2021) 956

where |A| is the determinant of A, A∗ be the positive-definite symmetric matrix such that A2∗ = A−1.

Then rε is solution of

(3.4)




−div(A∇rε) = div(A∇hε) in Ω\Hz,ε,

A∇rε · n = −A∇hε · n on Γn,

rε = −hε sur Γd,

rε = ϕ0 −hε on ∂Hz,ε.

We set

R1(ε) =

∫
∂Hz,ε

[A∇hε]T · n(v0 −v0(z))ds,

R2(ε) =

∫
∂Hz,ε

[A∇rε]T · nv0ds.

Lemma 3.2. We have∫
∂Hz,ε

[A∇(ϕ0 −ϕε)]T · nv0ds = −
ϕ0(z)v0(z)

E(ε)
+ R1(ε) + R2(ε)

Proof

Using the Green formulation, we obtain

∫
∂Hz,ε

[A∇(ϕ0 −ϕε)]T · nv0ds =
∫
∂Hz,ε

[A∇χε]T · nv0ds

=

∫
∂Hz,ε

[A∇hε]T · nv0ds +
∫
∂Hz,ε

[A∇rε]T · nv0ds

=

∫
∂Hz,ε

[A∇hε]T · n(v0 −v0(z))ds

+

∫
∂ωε

[A∇hε]T · nv0(z)ds + R2(ε)

=

∫
∂Hz,ε

[A∇hε]T · nv0(z)ds + R1(ε) + R2(ε)

=
ϕ0(z)v0(z)

E(ε)

∫
∂ωε

[A∇E(x−z)]T · nds + R1(ε) + R2(ε).

Such that ∫
∂Hz,ε

[A∇E(x−z)]T · nds = −1

Hence ∫
Hz,ε

[∇χε]T ·A∇v0dx = −
ϕ0(z)v0(z)

E(ε)
+ R1(ε) + R2(ε)

Assuming that Ri(ε) = o (ρ(ε)) , i = 1, 2. We will give the proof for the two dimensional case in section 4.3.

Besides thanks to the fundamental solution, we obtain the main result presented in the following sections

concerns the topological asymptotic expansion of an arbitrary design function j. Some cost function examples

are presented in section 4.



Int. J. Anal. Appl. 19 (6) (2021) 957

4. A particular class of cost function

In this section, we present some useful examples of shape functions and we gives their variations δJ .

4.1. First example. This example is concerned with the L2−norm. We consider the shape function defined

by

J(Ω\Hz,ε) =
∫

Ω\Hz,ε
|ϕε −ϕd|2dx, ∀ϕε ∈ H1(Ω\Hz,ε).

where ϕd ∈ H1(Ω) is a given desired (objective) function.

Proposition 4.1. The cost function Jε defined by

Jε(ϕ) =
∫

Ω\Hz,ε
|ϕ−ϕd|2dx, ∀ϕ ∈ H1(Ω\Hz,ε).

satisfies the hypothesis 2.1 with

DJ0(ϕ0) = 2
∫

Ω

(ϕ0 −ϕd) w dx, ∀w ∈ H1(Ω) and δJ(x) = 0,∀x ∈ Ω.

4.2. Second example. Here, we are dealing with the H1−semi-norm. We consider the shape function

J(Ω\Hz,ε) =
∫

Ω\Hz,ε
A|∇ϕε −∇ϕd|2dx

with ϕd ∈ H2(Ω) is a given desired function.

Proposition 4.2. The cost function Jε defined by

Jε(ϕ) =
∫

Ω\Hz,ε
A|∇ϕ−∇ϕd|2dx, ∀ϕ ∈ H1(Ω\Hz,ε),

satisfies the hypothesis 2.1 with

DJ0(ϕ0) = 2
∫

Ω

(∇ϕ0 −∇ϕd)T ·A∇w dx, ∀w ∈ H1(Ω)

and

∀x ∈ Ω, δJ(x) =




2π
√
|A||ϕ0(x)|2, d = 2

−4π
√
|A||ϕ0(x)|2, d = 3.

4.3. Proofs.



Int. J. Anal. Appl. 19 (6) (2021) 958

4.3.1. Preliminary results. The aim of this section is to give some technical results which will be used in

section 4.3.

Lemma 4.1. [4] Consider ψ ∈ H1/2(∂Hz,ε),gd ∈ H1/2(Γd) and gn ∈ H−1/2(Γn)

If X is solution of the following system

(4.1)




−∆X = 0 in Ω\Hz,ε,

∇X ·n = gn on Γn,

X = gd on Γd,

X = ψ on ∂Hz,ε.

There exists a non negative constant c such that

||X||1,Ω\Hz,ε ≤ c

[
1√
− log(ε)

||ψ(z + εy)||1/2,∂H + ||gd||1/2,Γd + ||gn||−1/2,Γn

]

Lemma 4.2. The function defined by hε(x) =
E(x−z)
E(ε)

ϕ0(z), ∀x ∈ Ω\Hz,ε admits the following estimates

||hε||1,ΩR ≤
−c

log ε
||hε||1,Ω\Hz,ε ≤

c
√
− log ε

Where R is a strictly positive real number such that ωε ⊂ B(z,R) ⊂ Ω, we set ΩR = Ω\B(z,R).

Proof of Lemma 4.2

Using the definition of hε, we have

||hε||1,ΩR =
−1

log ε
||ϕ0(z) log(|x−z|)||1,ΩR

Since that z /∈ ΩR, we deduce that

||hε||1,ΩR ≤
−c

log ε
.

Then we have hε is solution of

(4.2)




−∆hε = 0 in Ω\Hz,ε,

∇hε ·n =
ϕ0(z)

E(ε)
∇E(x−z) ·n on Γn,

hε =
E(x−z)
E(ε)

ϕ0(z) on Γd,

hε =
E(x−z)
E(ε)

ϕ0(z) on ∂Hz,ε.



Int. J. Anal. Appl. 19 (6) (2021) 959

Note that, by Lemma 4.1,

‖hε‖1,Ω\Hz,ε ≤ c
{ 1√
− log(ε)

‖
E(εy)

E(ε)
ϕ0(z)‖1

2
,∂H + ‖

ϕ0(z)

E(ε)
∇E(x−z).n‖−1

2
,Γn

+||
E(x−z)
E(ε)

ϕ0(z)||1
2
,Γd

}

≤
c1√
− log(ε)

+
c2

− log(ε)
.

Then,

‖hε‖1,Ω\Hz,ε ≤
c√

− log(ε)

This completes the proof.

�

Lemma 4.3. There exists a constant C strictly positive such that

‖ϕ0 −hε‖1
2
,∂Hz,ε ≤

−c
log(ε)

.

Proof of Lemma 4.3

Using the change of variables x = z + εy, we obtain

ϕ0(x) −hε(x) = ϕ0(z + εy) −hε(z + εy)

= ϕ0(z + εy) −
E(εy)

E(ε)
ϕ0(z)

= ϕ0(z + εy) −
(

1 +
E(y)

E(ε)

)
ϕ0(z).

The smoothness of ϕ0 and E in Hz,ε and H gives that

ϕ0(z + εy) −ϕ0(z) = O(ε)

and

E(y)ϕ0(z)

E(ε)
= O

(
−1

log(ε)

)
.

Hence

ϕ0(x) −hε(x) = O
(
−1

log(ε)

)
.

Thus the proof is complete.

�

Lemma 4.4. We have the following estimation

‖rε‖1,Ω\Hz,ε ≤
c√

− log(ε)
.



Int. J. Anal. Appl. 19 (6) (2021) 960

Proof of Lemma 4.4

We can write rε solution of the system 3.4 as follows rε = r
1
ε + r

2
ε where r

1
ε satisfies

(4.3)




∆r1ε = 0 in Ω\Hz,ε,

∇r1ε · n = 0 on Γ,

r1ε = 0 sur Γ,

r1ε = ϕ0 −hε on ∂Hz,ε,

and r2ε is solution of

(4.4)




∆r2ε = −∆hε in Ω\Hz,ε,

∇r2ε · n = −∇hε · n on Γ,

r2ε = −hε sur Γ,

r2ε = 0 on ∂Hz,ε,

It then follows from 4.3 and the Green formulation that∫
Ω\Hz,ε

|∇r1ε|
2dx =< ∇r1ε · n,ϕ0 −hε >−1/2,1/2,∂H,

Note that, by the theorem of the normal trace, we obtain∫
Ω\Hz,ε

|∇r1ε|2dx ≤ ||∇r1ε||−1/2,∂Hz,ε||(ϕ0 −hε)||1/2,∂Hz,ε

≤ ||r1ε||1,Ω\Hz,ε||ϕ0 −hε||1/2,∂Hz,ε

≤ −c
log ε
||r1ε||1,Ω\Hz,ε

The Poincare inequality gives us

||r1ε||1,Ω\Hz,ε ≤ c
∫

Ω\Hz,ε
|∇r1ε|

2dx

||r2ε||1,Ω\Hz,ε ≤ c
{
||∇hε.n||−1/2,Γn + ||hε||1/2,Γd + ||∆hε||0,Ω\Hz,ε

}

≤ ||hε||1,Ω\Hz,ε
Then,

||r2ε||1,Ω\Hz,ε ≤
c

√
− log ε

Since

||rε||1,Ω\Hz,ε ≤ ||r
1
ε||1,Ω\Hz,ε + ||r

2
ε||1,Ω\Hz,ε

We conclude that

||rε||1,Ω\Hz,ε ≤
c

√
− log ε

.

�



Int. J. Anal. Appl. 19 (6) (2021) 961

Proof of theorem 3.1

We only need to prove that

R1(ε) = o

(
−1

log(ε)

)
and R2(ε) = o

(
−1

log(ε)

)
.

Remember that

R1(ε) =

∫
∂Hz,ε

[A∇hε]T · n(v0 −v0(z))ds

We have

|R1(ε)| ≤ ||A||∞||∇hε · n||−1
2
,∂Hz,ε||v0 −v0(z)||12 ,∂Hz,ε

≤ ||A||∞||hε||1,Ω\Hz,ε||v0 −v0(z)||12 ,∂Hz,ε

Changing variables x = z + εy and using the lemma 4.2

|R1(ε)| ≤
Cε√
− log(ε)

= o

(
−1

log(ε)

)
.

On the other hand

R2(ε) =

∫
∂Hz,ε

[A∇rε]T · nv0ds.

We have

|R2(ε)| ≤ C||∇rε||−1
2
,∂Hz,ε||v0||12 ,∂Hz,ε

≤ C||rε||1,Ω\Hz,ε||v0||1,Hz,ε.

Likewise, using the same change of variables and due to lemma 4.4, it follows that

|R2(ε)| ≤
Cε√
− log(ε)

= o

(
−1

log(ε)

)
.

which completes the proof.

�

Proof of proposition 4.1

The function J is differentiable with respect to ϕ and we have

DJ0(ϕ0)(w) = 2
∫

Ω

(ϕ0 −ϕd)w dx, ∀w ∈V0.



Int. J. Anal. Appl. 19 (6) (2021) 962

Computing the variation J(Ω\Hz,ε) −J(Ω)

J(Ω\Hz,ε) −J(Ω) =
∫

Ω\Hz,ε
|ϕε −ϕd|2dx−

∫
Ω

|ϕ0 −ϕd|2dx

=

∫
Ω

[|ϕε −ϕd|2 −|ϕ0 −ϕd|2]dx−
∫
Hz,ε
|ϕd|2dx

=

∫
Ω

[|ϕ0 −ϕε|2 − 2(ϕ0 −ϕε)(ϕ0 −ϕd)]dx−
∫

Ω\Hz,ε
|ϕd|2dx

= DJ0(ϕ0)(ϕε −ϕ0) +
∫

Ω\Hz,ε
|ϕ0 −ϕε|2dx

+

∫
Hz,ε
|ϕ0|2dx−

∫
Hz,ε
|ϕd|2dx.

By the divergence formula and the system 3.3, we have∫
Ω\Hz,ε

|ϕ0 −ϕε|2dx = o
(
−1

log(ε)

)
.

A change of variable and the fact that ϕ0 and ϕd are of class C
2 in a neighborhood of the origin yield∫

Hz,ε
|ϕd|2dx ≤ cε2 and

∫
Hz,ε
|ϕ0|2dx ≤ cε2.

Hence

J(Ω\Hz,ε) −J(Ω) = DJ0(ϕ0)(ϕε −ϕ0) + o
(
−1

log(ε)

)
.

Finally, by theorem 3.1, we deduce

j(Ω\Hz,ε) − j(Ω) =
−1

log(ε)
2π
√
|A|ϕ0(x0)v0(x0) + o

(
−1

log(ε)

)
.

�

Proof of proposition 4.2

The function J is differentiable and we have

DJ0(ϕ0)(w) = 2
∫

Ω

A(∇ϕ0 −∇ϕd)T .∇w dx, ∀w ∈V0.

Moreover, we have

J(Ω\Hz,ε) −J(Ω) =
∫

Ω\Hz,ε
A|∇ϕε −∇ϕd|2dx−

∫
Ω

A|∇ϕ0 −∇ϕd|2dx

= DJ0(ϕ0)(ϕε −ϕ0) +
∫
Hz,ε

A|∇ϕ0|2dx−
∫
Hz,ε

A|∇ϕd|2dx

+

∫
Ω\Hz,ε

A|∇ϕ0 −∇ϕε|2dx

As ϕ0 and ϕd are sufficiently regular in Hz,ε, we obtain∫
Hz,ε

A|∇ϕ0|2dx = O(ε2) = o
(
−1

log(ε)

)



Int. J. Anal. Appl. 19 (6) (2021) 963

and ∫
Hz,ε

A|∇ϕd|2dx = O(ε2) = o
(
−1

log(ε)

)
.

By the divergence formula, we have∫
Ω\Hz,ε

A|∇ϕ0 −∇ϕε|2dx =
∫
∂Hz,ε

[A(∇ϕ0 −∇ϕε)]T .nϕ0dx

=
−2π

log(ε)

√
|A||ϕ0(x0)|2 + o

(
−1

log(ε)

)
.

Hence

J(Ω\Hz,ε) −J(Ω) = DJ0(ϕ0)(ϕε −ϕ0) +
−2π

√
|A|

log(ε)
|ϕ0(z)|2 + o

(
−1

log(ε)

)
.

Finally, by theorem 3.1, we deduce

J(Ω\Hz,ε) −J(Ω) =
−1

log(ε)
2π
√
|A|
[
ϕ0(z)v0(z) + |ϕ0(z)|2

]
+ o

(
−1

log(ε)

)
.

�

5. The Kohn-Vogelius norms

The Kohn-Vogelius criterion [15] is used like a cost functional. Since the boundary conditions (ϕd, Φ) are

overspecified, one can define for any hole H two forward problems:

• the ”Dirichlet” problem: 

−div(A∇ϕD) = 0 in Ω\H,

ϕD = ϕd on Γd,

ϕD = 0 on ∂H.

• the ”Neumann” problem:



−div(A∇ϕN ) = 0 in Ω\H,

A∇ϕN · n = Φ on Γn,

ϕN = 0 on ∂H.

The optimal hole H∗ coincides with the actual boundary H when the misfit between the solutions vanishes:

ϕD = ϕN . Therefore, we propose an identification process based on the minimization of the following energy

functional

J(Ω\H) =
∫

Ω\H
A|∇ϕN −∇ϕD|2dx.

This is the so-called Kohn-Vogelius criterion [15]. Our approach concerns the derived topological optimization

problem:

min
H⊂Ω

J(Ω\H).



Int. J. Anal. Appl. 19 (6) (2021) 964

We will use the topological gradient method to solve this problem. It provides an asymptotic expansion of

the function J with respect to a small topological perturbation of the domain Ω.

5.1. Asymptotic expansion of the cost functional. The following Theorem describes the variation of

the function J when creating a small hole Hz,ε inside the domain Ω with a Dirichlet boundary condition on

∂Hz,ε.

For all ε ≥ 0,J(Ω\Hz,ε) =
∫

Ω\Hz,ε
A|∇ϕεN −ϕ

ε
D|

2dx

where ϕεN and ϕ
ε
D are the solutions to the systems


−div(A∇ϕεN ) = 0 in Ω\Hz,ε,

A∇ϕεN · n = Φ on Γn,

ϕεN = 0 on ∂Hz,ε.

;



−div(A∇ϕεD) = 0 in Ω\Hz,ε,

ϕεD = ϕd on Γd,

ϕεD = 0 on ∂Hz,ε.

5.1.1. The three dimensional case. In this paragraph, we present the topological asymptotic expansion for

the Anisotropic Laplace equations in the three dimensional case. In this case the fundamental solution of

the anisotropic Laplace operator E is given by

E(x) =
−1

4π
√
|A|||A∗x||

, ∀x ∈ Ω.

Theorem 5.1. Under the same hypotheses of theorem 3.1, the function J has the following asymptotic

expansion

J(Ω\Hz,ε) −J(Ω) = ε
{

4π
√
|A|
[
ϕ0N (z)v

0
N (z) + ϕ

0
D(z)v

0
D(z)

]
+ δJ(z)

}
+ o(ε),

with

δJ(x) = 4π
√
|A|
{
|ϕ0N (x)|

2 + |ϕ0D(x)|
2
}
, ∀x ∈ Ω.

5.1.2. The two dimensional case. In this paragraph, the result is obtained using the same technique described

in the previous paragraph. The unique difference comes from the expression of the fundamental solution of

the Anisotropic Laplace equations. In this case E is given by

E(x) =
1

2π
√
|A|

log(||A∗x||), ∀x ∈ Ω.

Theorem 5.2. Under the same hypotheses of theorem 3.1, the function J has the following asymptotic

expansion

J(Ω\Hz,ε) −J(Ω) =
−1

log(ε)

{
2π
√
|A|
[
ϕ

0
N (z)v

0
N (z) + ϕ

0
D(z)v

0
D(z)

]
+ δJ(z)

}
+ o

(
−1

log(ε)

)
,

with

δJ(x) = 2π
√
|A|
{
|ϕ0N (x)|

2
+ |ϕ0D(x)|

2
}
, ∀x ∈ Ω.



Int. J. Anal. Appl. 19 (6) (2021) 965

6. Numerical Result

This section is concerned with some numerical investigations. We consider the bidimentional case and we

present a fast and simple one-iteration identification algorithm. The unknown object H is identified using a

level set curve of the topological gradient δJ . More precisely, the unknown object H is likely to be located

at zone where the topological gradient δJ is more negative.

One-iteration algorithm: [11, 12]

• Solve the two problems (P0N ) and (P
0
D) in initial domain Ω,

• Compute the topological gradient function δJ(x), x ∈ Ω,

• Determine the unknown object

H = {x ∈ Ω, such that δJ(x) < cmin},

where cmin is a negative constant chosen in such a way that the cost function J has the most negative

value.

Next, we will present some numerical simulations using the proposed algorithm. In Figure 1, we test our

algorithm on circular shape. In Figure 2, we consider the case of an elliptical shape. In Figure 3, we can

notice that, when the shape is non-regular, the reconstruction is quite efficient. In the case of non trivial

shape, yet we applied a one-iteration algorithm, we obtain an interesting reconstruction result (see Figure

4).

The obtained result can serve as a good initial estimate for an iterative optimization process based on the

shape derivative.

The considered model can be viewed as a prototype of a geometric inverse problem valid in many appli-

cations.

6.0.1. Reconstruction of circular-shaped objects. In this case, we test our procedure to detect an object having

circular-shaped. We reconstruct in this case the object H described by a disk centered at z = (2, 0) with

different radius: r ∈ {0.2, 0.4, 0.6}. The obtained results are illustrated in Figure 1. One can easily observe

in Figure 1, the unknown object is located in the region where the topological sensitivity function δJ is the

most negative (red zone). The boundary of H∗ is approximated by an iso-value curve. Our one-iteration

algorithm gives an efficient reconstruction results for the different chosen sizes.

6.0.2. Reconstruction of ellipse-shaped objects. In this example, we reconstruct an object described by an

ellipse inserted in the disc D = B((2, 0), 1) and centered at (2, 0). We represent the results in Figure 2. In

this case, we examine the numerical reconstruction of various ellipses having different directions and sizes.

As one can observe in Figure 2, the boundary of the object is again detected and located in the zone where



Int. J. Anal. Appl. 19 (6) (2021) 966

the topological gradient is most negative (red lines). Also, our one-iteration algorithm gives quite effecient

reconstruction results for different chosen ellipse-shaped objects.

6.0.3. Reconstruction of geometry with corners. We tried to apply our proposed algorithm to detect more

complicated geometry. Our objective is to reconstruct an object with corners. More precisely, we are trying

to detect a square and rectangle shape . We can see in the Figure 3 (a), that the unknown square H∗ is

located in the zone where the topological gradient function δJ is the most negative (red zone) and also its

boundary is approximated by an iso-value curve. So here, our one-iteration algorithm detects the location

and the shape of the square. But in the case of a rectangle shape (see Figure 3 (b)) the boundary H∗ cannot

be approached by any iso-value curve of the topological gradient function. We can remark in this case, that

the one-iteration algorithm detects the zone containing the unknown geometry but the reconstruction result

is not good.

6.0.4. Reconstruction of a non trivial-shaped objects. We apply now our proposed algorithm to detect a non

trivial shapes. We can see in Figure 4 that the unknown shape H∗ is located in the zone where the topological

gradient function δJ is the most negative (red iso-values) but we cannot approximate the boundary of H∗ by

any iso-value curve of the topological sensitivity function δJ . We can improve these reconstruction results

by suggesting an iterative algorithm.

r=0.2:negative zone(red zone) iso-value of δJ zoom showing the iso-value of δJ

approximating δH∗(black line)

r=0.4:negative zone(red zone) iso-value of δJ zoom showing the iso-value of δJ

approximating δH∗(black line)



Int. J. Anal. Appl. 19 (6) (2021) 967

r=0.6:negative zone(red zone) iso-value of δJ zoom showing the iso-value of δJ

approximating δH∗(black line)

Figure 1. Reconstruction of circle shaped objects

Figure 2. Reconstruction of an ellipse shaped objects

(a) square shape (b) rectangle shape

Figure 3. Reconstruction of objects with corners.

Acknowledgement

The author gratefully acknowledge the approval and the support of this research study by the grant no.

7772-SCI-2018-3-9-F from the Deanship of Scientific Research at Northern Border University, Arar, Saudi

Arabia.



Int. J. Anal. Appl. 19 (6) (2021) 968

Figure 4. Reconstruction of a non trivial shape.

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.

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	1. Introduction
	2. The perturbed problems
	3. Topological asymptotic expansion
	3.1. Asymptotic formula for the anisotropic Laplace problem

	4. A particular class of cost function
	4.1. First example
	4.2. Second example
	4.3. Proofs

	5. The Kohn-Vogelius norms
	5.1. Asymptotic expansion of the cost functional.

	6. Numerical Result
	Acknowledgement
	References