Asymptotic Imaging of Perfectly Conducting Cracks∗
Habib Ammari†
Hyeonbae Kang‡
Hyundae Lee‡
Won-Kwang Park§
Abstract
In this paper, we consider cracks with Dirichlet boundary conditions. We
first derive an asymptotic expansion of the boundary perturbations that are
due to the presence of a small crack. Based on this formula, we design a noniterative approach for locating a collection of small cracks. In order to do so, we
construct a response matrix from the boundary measurements. The location
and the length of the crack are estimated, respectively, from the projection onto
the noise space and the first significant singular value of the response matrix.
Indeed, the direction of the crack is estimated from the second singular vector.
We then consider an extended crack with Dirichlet boundary conditions. We
rigorously derive an asymptotic expansion for the boundary perturbations that
are due to a shape deformation of the crack. To reconstruct an extended crack
from many boundary measurements, we develop two methods for obtaining a
good guess. Several numerical experiments show how the proposed techniques
for imaging small cracks as well as those for obtaining good initial guesses
toward reconstructing an extended crack behave.
AMS subject classifications. 35R30, 35B25
Key words. crack detection, asymptotic imaging, reconstruction algorithm
1
Introduction and Problem Formulations
The purpose of this paper is to design new efficient methods to detect perfectly conducting cracks, linear or nonlinear, inside a conductor from the boundary measurements. We also perform some numerical experiments using the proposed algorithms
to test their performance and efficiency.
∗
This work was partially supported by the ANR projects EchoScan (AN-06-Blan-0089) and
SISTAE (TLOG-004), the STAR project 190117RD, the KICOS project K20803000001-08B120000110, and research grants at the Inha University.
†
Institut Langevin, CNRS UMR 7587, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France
(habib.ammari@espci.fr).
‡
Department of Mathematics, Inha University, Incheon 402-751, Korea (hbkang@inha.ac.kr,
hdlee@inha.ac.kr).
§
Laboratoire des Signaux et Systèmes, École Supérieure d’Électricité, Plateau de Moulon, 3 rue
Joliot-Curie 91192, Gif-sur-Yvette Cedex, France (park@lss.supelec.fr).
1
Let Σ be a crack inside a bounded conductor Ω ⊂ R2 , which is a C 2 curve. We
assume that the crack is at some distance from ∂Ω, i.e., there is a positive constant
c such that
dist(Σ, ∂Ω) ≥ c.
(1.1)
With the crack inside, the relevant boundary value problem is

2

 ∆u + ω u = 0 in Ω \ Σ,


u = 0 on Σ,



 ∂u = g ∈ L2 (∂Ω) on ∂Ω.
∂ν
(1.2)
Here ν is the outward normal to ∂Ω. We will assume that ω 2 is not an eigenvalue of
(1.2) so that the problem is well-posed. Note that the Dirichlet boundary condition
on the crack yields a good mathematical model for scattering of electromagnetic
waves from perfectly conducting cracks or patches.
We first deal with a perfectly conducting linear crack of small size. Suppose that
Σ is a line segment (or a collection of them) of length 2. In this case, let Σ denote
the crack and let u be the solution to (1.2). We shall emphasize that, if ω 2 is not a
Neumann eigenvalue of −∆ in Ω, it follows from [5, Chapter 5] that problem (1.2)
has a unique solution for small enough.
The problem we will consider in the first part of this paper is to detect Σ from
the boundary measurement u |∂Ω for several Neumann data g. For doing so, we
first derive an asymptotic expansion formula of u on ∂Ω as goes to zero. Our
derivations are rigorous and based on layer potential techniques on open curves.
Our asymptotic formula proves not only a point source type approximation of the
small crack but also gives a second-order approximation which involves the direction
of the crack. Higher-order terms can be derived following the same approach. These
results are new, as far as we know, and are expected to have important applications
in practice.
In a second step, using our asymptotic formula, we design a non-iterative approach for locating a collection of cracks. Our approach is of MUSIC-type (MUSIC
stands for MUltiple Signal Classification). We start with constructing a response
matrix from the boundary measurements. The singular value decomposition of such
response matrix yields an estimate of the location and the size of the cracks. We
show how the point source type approximation can be used to locate a collection
of cracks and estimate their sizes. Using the second-order term in our asymptotic
formula, we provide a method to reconstruct the directions of the cracks.
We perform several numerical tests to show the validity of our algorithm. In
particular, we show that the size estimate of the crack obtained from the first significant singular eigenvalue of the response matrix is in a good match with the actual
value. It is also shown that the significant singular values can not yield an estimation of the direction of the crack. However, using the second singular vector we
2
can estimate the direction of the crack with a good accuracy. The resolution limit
of our algorithm is also discussed. Our numerical results clearly demonstrate that
in order to distinguish between two closely located cracks we have to use waves of
wavelength not more than twice the distance between them.
For related works on MUSIC-type imaging we refer the reader, for instance, to
[3], [4], [15], [30], and [35].
In the second part of the paper, we consider an extended crack with Dirichlet
boundary conditions. We first develop and test numerically two different ways of
constructing a good initial guess from many boundary measurements. The first
method is based on the concept of topological derivative. The second one is to use
a standard MUSIC-type projection approach. To do so, we construct a response
matrix as in the case of cracks of small size. We prove that this matrix describes in
fact the far field-behavior of the field radiated by the jump of the normal derivative
of the solution on the crack. Dividing the crack into segments of size of order half
the wavelength, we show that each of these points can be imaged using the standard
MUSIC imaging function. The resolution of the image provided by this technique
is therefore of order the half-wavelength. Since the measurements are done at the
boundary of the background domain, we then discuss optimization procedures for
the purpose of refining the reconstruction and obtaining better image of the crack.
To compute the shape derivative of the associated cost functionals, we rigorously
derive an asymptotic expansion for the boundary perturbations that are due to a
shape deformation of the crack using a layer potential technique.
In connection with our work, we refer the reader to [7] where the sensitivity
of a class of cost functionals with respect to the insertion of a perfectly insulating
small crack (i.e. with Neumann boundary conditions). The approximation is in
this case a dipole-type one. The method of derivation and the detection algorithm
also significantly differ from ours. For other related works on crack identification,
we refer to [8], [11], [12], [13], [14], [16], [23], [24], [31], [33], [36], and the numerous
references therein.
2
Preliminary results
Let Σ = Σ be a narrow crack inside a bounded conductor Ω ⊂ R2 . Assume for the
sake of simplicity that
Σ = (x, 0) : − ≤ x ≤ .
Set ϕ to be the jump of the normal derivative of u , the solution to (1.2) on the
crack Σ :
∂u ∂u ϕ :=
−
on Σ .
(2.1)
∂ν +
∂ν −
Here the subscripts + and − indicate the limits from above and below, respectively.
3
Suppose that ω 2 is not a Neumann eigenvalue on Ω and let u0 be the background
solution, that is, the solution to

2
 ∆u0 + ω u0 = 0 in Ω,
(2.2)
 ∂u0 = g on ∂Ω.
∂ν
Let NΩω (x, z), for x, z ∈ Ω, be the Neumann function for Ω, namely, the solution to

2
ω

in Ω,
 (∆x + ω )NΩ (x, z) = −δz (x)
(2.3)
∂NΩω

(x, z) = 0
x ∈ ∂Ω.

∂νx
By Green’s formula, one can easily see that
Z
u (x) − u0 (x) =
NΩω (x, y) ϕ (y) dy,
x ∈ Ω,
(2.4)
Σ
where ϕ defined by (2.1). In particular, the following integral equation holds:
Z
ω A [ϕ ](x) :=
NΩω (x, y) ϕ (y) dy = −u0 (x), (x, 0) ∈ Σ .
(2.5)
Σ
Note that we write NΩω (x, y) for NΩω ((x, 0), (y, 0)) and ϕ (y) for ϕ (y, 0) for the sake
of simplicity.
Endowed with the norm
Z p
1/2
2
2
2
kϕkX =
− x |ϕ(x)| dx
,
−
the set
X ,
defined by
X =
Z
p
2
ϕ:
−
x2 |ϕ(x)|2 dx
< +∞ ,
(2.6)
−
is a Hilbert space. Define
Y
=
0
0
ψ ∈ C ( [ −, ] ) : ψ ∈ X
,
(2.7)
where ψ 0 is the distributional derivative of ψ. The set Y is a Hilbert space with the
norm
1/2
kψkY =
kψk2X + kψ 0 k2X .
Writing
NΩω (x, y) = −
1
ω
ln |x − y| + RN
(x, y),
2π
4
(2.8)
ω (x, y) is C 1,α in both x and y for some α with 0 < α < 1, the
where (x, y) →
7 RN
operator Aω can be decomposed as
Aω = −
where
1
L + Rω ,
2π
(2.9)
Z
ln |x − y| ϕ(y) dy,
L [ϕ](x) =
−
and
Rω [ϕ](x) =
Z
−
ω
RN
(x, y) ϕ(y) dy.
We recall from, for instance, [5, Chapter 5] the following lemmas.
Lemma 2.1 For all 0 < < 2, the integral operator L : X 7→ Y is invertible. For
a given function ψ ∈ Y , L−1
[ψ] ∈ X is given by
Z p 2
1
− y 2 ψ 0 (y)
a(ψ)
−1
L [ψ](x) = − √
dy +
(2.10)
p
2
2
2
x−y
π − x −
π(ln ) 2 − x2
2
for x ∈ ] − , [, where a(ψ) is constant and defined by
Z √ 2
− z 2 ψ 0 (z)
1
p
a(ψ) = ψ(x) + L
dz (x),
y−z
π 2 2 − y 2 −
(2.11)
(the right-hand side is actually constant).
In view of (2.10) and (2.11), we have
L−1
[1](x) =
and
π(ln
1
√
2)
2 − x2
,
x
L−1
.
[y](x) = − √ 2
π − x2
(2.12)
(2.13)
Lemma 2.2 There exists a positive constant C, independent of , such that
ω
kL−1
R kL(X ,X ) ≤
C
,
| ln |
where
ω
kL−1
R kL(X ,X ) =
ω
kL−1
R [ϕ]kX .
sup
ϕ∈X ,kϕk
5
X =1
(2.14)
3
Asymptotic expansion
Because of (2.5) and (2.9), ϕ is the solution in X of the integral equation (2.5) or
equivalently
1
− L [ϕ ] + Rω [ϕ ] = −u0 on Σ .
(3.1)
2π
Thus we have
ω −1
ϕ − 2πL−1
R [ϕ ] = 2πL [u0 ],
and hence
ω −1
kϕ kX ≤ C kL−1
R [ϕ ]kX + kL [u0 ]kX .
Note that
kL−1
[u0 ]kX ≤ Cku0 kY ≤ C.
It then follows from (2.14) that if is sufficiently small, then
kϕ kX ≤ C
for some constant C independent of .
Set
ϕ̃ (x) := ϕ (x),
(3.2)
x ∈] − 1, 1[.
One can easily see that
kϕ̃ kX 1 = kϕ kX ≤ C,
and that (3.1) reads now
Z 1
Z 1
1
ω
−
(ln + ln |x − y|)ϕ̃ (y) dy +
RN
(x, y)ϕ̃ (y) dy = −u0 (x),
2π −1
−1
Observe that
Z
x ∈] − 1, 1[.
1
|ϕ̃ (y)| dy ≤ Ckϕ̃ kX 1 ≤ C.
−1
By Taylor expansions, we obtain
1
ln ∂u0
ω
− L1 [ϕ̃ ] + C −
+ RN (0, 0) = −u0 (0) − x
(0) + O(2 ),
2π
2π
∂T
where
Z
1
C =
ϕ̃ (y) dy.
−1
Here O(2 ) is in the X 1 -norm and ∂/∂T denotes the tangential derivative on Σ . In
view of (2.12) and (2.13), we now have
ϕ̃ =
ω (0, 0)
− ln + 2πRN
2u0 (0)
∂u0
x
√
√
+ C
− 2
(0) √
+ O(2 ),
2
2
∂T
ln(1/2) 1 − x
π ln(1/2) 1 − x
1 − x2
6
where O(2 ) is in the Y 1 -norm.
By integrating both sides of the above identity, one can see that
C =
2πu0 (0)
2
ω (0, 0)) + O( ).
ln(1/2) + ln − 2πRN
(3.3)
Therefore we get
1
∂u0
x
2u0 (0)
+ O(2 ),
− 2
(0) √
ϕ̃ (x) = √
ω
∂T
1 − x2 ln(/2) − 2πRN (0, 0)
1 − x2
By scaling back, we obtain the following lemma.
Lemma 3.1 The following asymptotic expansion holds:
ϕ (x) =
2u0 (0)
√
ln(/2) 2 − x2 1 −
1
ω (0,0)
2πRN
ln(/2)
−2
∂u0
x
(0) √
+ O(2 ).
2
∂T
− x2
Observe that the first and second terms in (3.4) are of order
norms, respectively.
Substituting (3.4) into (2.4), we obtain
u (x)−u0 (x) =
2πu0 (0)
ln(/2) 1 −
1
ω (0,0)
2πRN
NΩω (x, 0)−π2
ln(/2)
1
| ln |
(3.4)
and in X -
∂N ω
∂u0
(0) Ω (x, 0)+O(3 ), (3.5)
∂T
∂T
uniformly on x ∈ ∂Ω.
We finally obtain the following theorem:
Theorem 3.2 Let Σ be a linear crack of length 2 whose center is at z satisfying
(1.1). Then the following expansion holds
u (x) − u0 (x) =
2πu0 (z)
ln(/2) 1 −
1
ω (0,0)
2πRN
NΩω (x, z) − π2
ln(/2)
∂N ω
∂u0
(z) Ω (x, z) + O(3 ),
∂T
∂T
(3.6)
uniformly on x ∈ ∂Ω.
We also have an asymptotic expansion for the Dirichlet problem:

∆v + ω 2 v = 0 in Ω \ Σ ,



v = 0 on Σ ,



v = f ∈ H 1/2 (∂Ω) on ∂Ω.
Here, H 1/2 (∂Ω) is the standard trace space.
7
(3.7)
Define v0 to be the background solution, that is, the solution to
(
∆v0 + ω 2 v0 = 0 in Ω,
v0 = f
(3.8)
on ∂Ω,
and introduce GωΩ (x, z), for x, z ∈ Ω, to be the Dirichlet function for Ω, namely, the
solution to
(
(∆x + ω 2 )GωΩ (x, z) = −δz (x)
in Ω,
(3.9)
GωΩ (x, z) = 0
x ∈ ∂Ω.
Analogously to (2.8), we have
GωΩ (x, y) = −
1
ω
ln |x − y| + RD
(x, y),
2π
ω (x, y) is C 1,α in both x and y for some α with 0 < α < 1. The
where (x, y) 7→ RD
following asymptotic formula can be proved exactly as in Theorem 3.2.
Theorem 3.3 Let Σ be a linear crack of length 2 whose center is at z satisfying
(1.1). Then the following expansion holds
∂(v − v0 )
2πv0 (z)
(x) =
∂ν
ln(/2) 1 −
1
ω (0,0)
2πRD
ln(/2)
∂GωΩ
(x, z)
∂ν(x)
(3.10)
∂ 2 GωΩ
∂v0
(z)
(x, z) + O(3 ),
−π2
∂T
∂T (z)∂ν(x)
uniformly on x ∈ ∂Ω.
4
MUSIC-type imaging
Suppose that Ω contains m small cracks Σs located at zs , s = 1, . . . , m. The cracks
are well-separated from each other and from the boundary ∂Ω. Let 2s = |Σs |.
Suppose that s = αs , where αs = O(1) and 2 is the order of magnitude of the
lengths of the cracks.
Since ln(s /2) = ln(/2)(1 + O(| ln |−1 )), one can see from Theorem 3.2 that
m
Z
(u − u0 )(x)
∂Ω
∂v
2π X
1
(x) dσ(x) =
u0 (zs )v(zs ) + O(
)
∂ν
ln(/2)
| ln |2
s=1
for any smooth V satisfying
(∆ + ω 2 )v = 0
8
in Ω.
(4.1)
Let {θ1 , . . . , θN } be a set of N unit directions. Throughout this paper, we suppose
that they are equi-distributed on the unit circle. Choosing
u0 (x) = eiωθ·x
0
and v(x) = e−iωθ ·x ,
(4.2)
where θ, θ0 ∈ {θ1 , . . . , θN }, leads to a standard MUSIC-type reconstruction of the
locations {zs }m
s=1 from the response matrix
Z
iω
N
iωθl ·x −iωθl0 ·x
A = (All0 )l,l0 =1 , All0 =
θl0 · ν(x) (u(l)
)e
dσ(x), (4.3)
(x) − e
N ∂Ω
(l)
where u is the solution to (1.2) with g = iω(ν · θl )eiωθl ·x and the division by N is
for normalization. The approximation (4.1) shows
m
All0
X
2π
≈−
eiω(θl −θl0 )·zs .
N ln(/2)
(4.4)
s=1
If we define
g(z) = (eiωθ1 ·z , . . . , eiωθN ·z )t ,
z ∈ Ω,
where t denotes the transpose, then the formula (4.4) clearly shows that the range
of A is the space spanned by {g(zs ) | s = 1, . . . , m }. Therefore, in order to find the
location zs of cracks, we look for z such that
g(z) ∈ Range(A),
which is considered as the MUSIC characterization of the locations of cracks. The
MUSIC algorithm is then to plot the imaging function
W (z) :=
1
||(I − P )g(z)||
for z ∈ Ω,
(4.5)
where P is the orthogonal projection onto Range(A). The function W (z) has large
peaks at the locations of the cracks.
It is worth noticing that, in view of (4.4), the direction of the crack (the tangent
vectors to Σs ) can not be obtained from the leading-order approximation in (3.5).
Moreover, only the order of magnitude , not s , can be estimated.
One may think that the higher-order terms in (3.5) may lead to an estimate of
the directions of the cracks since they contain tangential derivatives of the solutions.
In what follows, we show that it is the case.
Using such higher-order terms in (3.5), we obtain that
Z
m
X
∂v
2π
1
(u − u0 )(x) (x) dσ(x) =
u0 (zs )v(zs )
ω
2πR
∂ν
ln(s /2) 1 −
N (0,0)
∂Ω
s=1
ln(/2)
− π2s
9
∂v
∂u0
(zs )
(zs ) + O(3 ).
∂Ts
∂Ts
Therefore, if we choose u0 and v as in (4.2), it follows that
Z
∂v
(u − u0 )(x) (x) dσ(x)
∂ν
∂Ω
m
X
1
2π
2 2
0
iω(θ−θ0 )·zs
−
π
ω
(θ
·
T
)(θ
·
T
)
+ O(3 ).
=
s
s e
ω (0,0)
s
ln(s /2) 1 − 2πRN
s=1
ln(/2)
(4.6)
Suppose for simplicity that m = 1, and let z be the center of the crack. From
(4.6), one can see that A defined in (4.3) is approximated by
A≈−
2π
ln(/2)N 1 −
1
ω (0,0)
2πRN
w1 ⊗ w1 +
ln(/2)
π2 ω 2
w2 ⊗ w2 ,
N
(4.7)
where
w1 = (eiωθ1 ·z , . . . , eiωθN ·z )t
and w2 = (θ1 · T eiωθ1 ·z , . . . , θN · T eiωθ1 ·z )t .
(4.8)
Suppose that N is even. Then, since θ1 , . . . , θN are equi-distributed on the unit
circle, we have
N
X
w2 · w1 = w1 · w2 =
θl · T = 0.
l=1
Therefore, (4.7) shows that A has only two significant singular values (eigenvalues
in this case) given by
τ1 = −
2π
ln(/2) 1 −
1
, τ2 = π2 ω 2
2πRω (0,0)
N
N
1 X
(θl · T )2 ,
N
(4.9)
l=1
ln(/2)
whose normalized
are given by w1 /kw1 k and w2 /kw2 k, respectively.
P eigenvectors
2 ≈ 1/2 regardless of T if N is sufficiently large. In fact,
Observe that N
(θ
·
T
)
l
l=1
one can easily see that
N
1 X
1
lim
(θl · T )2 = ,
N →∞ N
2
uniformly in T.
l=1
We observe that since the eigenvector w2 contains information on θj · T for j =
1, . . . , N , it is easy to find T from w2 .
5
Numerical simulations for imaging of small cracks
In this section, results of numerical simulations for imaging of small cracks are
presented. In order to perform the numerical simulation, we choose the homogeneous
domain Ω, which contains small cracks, as a unit disk centered at (0, 0) in R2 .
Throughout this section, the length of all cracks is set to 0.02, i.e., = 0.01.
10
Figure 1: Distribution of singular values of the response A (left) and plots of W (z)
2π
for Σ1, using N = 12 incident waves at the frequency ω = 0.5
. The data set was
generated using (4.6).
5.1
Imaging of a single small crack
For illustration we choose the following two cracks:
Σ1, = (x − 0.2, 0.2) : − ≤ x ≤ ,
Σ2, = Rπ/6 (x, 0) : − ≤ x ≤ ,
where Rπ/6 is the rotation by π/6.
For simulation we take the number of directions N = 12 and the wavelength
λ = 0.5. The data set of the matrix A in (4.3) is collected in two different ways: by
calculating (4.6) and by solving the forward problem (1.2). Figures 1 and 2 show the
distributions of the singular values of the response matrix and plots of W (z) defined
in (4.5). Both of them show that the location can be detected pretty accurately
Figure 3 shows the results when we add a noise. A noise is added as follows: for
a numerical value ξ
unoise (x) = 1 + ξ × (rnd1 (−1, 1) + irnd2 (−1, 1)) u(x)
where rnd1 (−1, 1) and rnd2 (−1, 1) are arbitrary real values between −1 and 1. For
this example we take ξ = 0.3. Although the distribution of singular values is quite
different from the noiseless one and a few ghost replicas appear in the image, the
location of Σ1, is successfully identified.
Figures 4 and 5 show results of reconstruction for Σ2, . Like Σ1, , the location is
well identified.
11
Figure 2: Results for Σ1, with the data set generated by solving forward problems,
without noise.
Figure 3: Results for Σ1, with the data set generated by solving forward problems,
with 30% random noise (ξ = 0.3).
12
Figure 4: Results for Σ2, with the data set generated using (4.6).
Figure 5: Results for Σ2, with the data set generated by solving forward problems.
13
Figure
1
2
3
4
5
true location
(−0.2, 0.2)
(−0.2, 0.2)
(−0.2, 0.2)
(0, 0)
(0, 0)
reconstructed location
(−0.1933, 0.2105)
(−0.1933, 0.2105)
(−0.1899, 0.2174)
(0.00001506, 0.00008568)
(0.001378, −0.003054)
Table 1: True and reconstructed locations.
Table 1 shows numerical values of reconstructed locations.
5.2
Size estimation
In the reconstruction of Σ1, in subsection 5.1, the first significant singular value
of the response matrix is 1.451 when the data were collected by solving forward
problems, and 1.528 when we add noise. In view of (4.4), the first significant singular
value is given by
2π
τ1 ≈ −
.
ln(/2)
Therefore, the estimated is 0.0132 in the noiseless case and 0.0164 in the noisy
case. In both cases, the estimated value of is close to the actual value, 0.01.
5.3
Direction estimation
Set, as before, N = 12. Thus the directions θl are given by
2l
2l
θl = cos π, sin π .
12
12
Since
w2 (l)
,
eiωθl ·z
where w2 (l) is the l−component of the second singular vector w2 , then the direction
T can be recovered by looking at the maximal (or minimal) values of dl := |w2 (l)|.
Consider first the small crack Σ1, . Then by using the dataset generated by
solving forward problem, we find that dl attains its maximal for l = 6 and 12, which
shows that T is parallel to (1, 0)t . If we consider Σ2, , then the maximum is attained
for l = 1 and l = 7. See Table 2.
θl · T ≈
14
l
1
2
3
4
5
6
7
8
9
10
11
12
value of dl for Σ1,
0.8658
0.5002
0.0024
0.5008
0.8663
0.9998
0.8663
0.5008
0.0024
0.5002
0.8658
0.9992
value of dl for Σ2,
0.9994
0.8660
0.5004
0.0001
0.5006
0.8661
0.9996
0.8661
0.5006
0.0001
0.5004
0.8660
Table 2: Recovering the directions of the small cracks Σ1, and Σ2, .
5.4
Imaging of multiple small cracks
We now consider imaging of multiple small cracks. Again two examples of multiple
(1)
(2)
cracks, Σ and Σ , are chosen for illustration as follows:
(1)
(1)
Σ(1)
= Σ1, ∪ Σ2, := {(x − 0.5, 0) : − ≤ x ≤ } ∪ {(x + 0.5, 0) : − ≤ x ≤ }
(2)
(2)
Σ(2)
= Σ1, ∪ Σ2, := {(x − 0.5, −0.2) : − ≤ x ≤ } ∪ Rπ/4 (x + 0.5, 0.3) : − ≤ x ≤ ,
(j)
(j)
where Rπ/4 is the rotation by π/4. It should be noted that cracks, Σ1, and Σ2, , in
both examples are well-separated. We use N = 20 for the number of directions of
the incident waves and λ = 0.4 for the wave length. The data set for the response
matrix was generated in two different ways as in subsection 5.1: by using (4.6) and
by solving forward problems.
(1)
(2)
Figure 6 and 7 are for Σ and Figures 8 and 9 for Σ , without adding noise.
One can see that the locations of multiple cracks are also successfully identified. It
(1)
is interesting to observe that unlike the case of Σ where two peaks of W (z) have
(2)
almost the same magnitudes, W (z) for Σ has a peak of much smaller magnitude
at the rotated crack (the one on the right) than at the other one. Figure 10 shows
results when we add 30% (ξ = 0.3) random noise. The location is well identified
even under noise. Table 3 shows numerical values of reconstructed locations.
5.5
Imaging of two closely located cracks
The purpose of numerical experiments in this section is to consider the resolution of
the image. According to the Rayleigh resolution limit, any detail less than one-half
15
Figure 6: Distribution of singular values of the response matrix A (left) and plots
(1)
2π
, with the data
of W (z) for Σ using N = 20 incidences at the frequency ω = 0.4
set generated using (4.6).
(1)
Figure 7: Results for Σ
lems.
Figure
6
7
8
9
10
11 (on the left)
11 (on the right)
with the data set generated by solving the forward prob-
true locations
(−0.5, 0), (0.5, 0)
(−0.5, 0), (0.5, 0)
(−0.5, −0.2), (0.5, 0.3)
(−0.5, −0.2), (0.5, 0.3)
(−0.5, −0.2), (0.5, 0.3)
(−0.1, 0), (0.1, 0)
(−0.05, 0), (0.05, 0)
reconstructed locations
(−0.5023, 0.00846), (0.4946, −0.005384)
(−0.5023, 0.00846), (0.4946, −0.005384)
(−0.4912, −0.1998), (0.5048, 0.3159)
(−0.4912, −0.1998), (0.5048, 0.3159)
(−0.4912, −0.1998), (0.5048, 0.3159)
(−0.03198, −0.0001345)
(−0.06345, −0.001372), (0.06473, 0.002353)
Table 3: True and reconstructed locations.
16
(2)
Figure 8: Results for Σ
(2)
Figure 9: Results for Σ
without noise.
with data set generated using (4.6).
with data set generated by solving the forward problems,
17
(2)
Figure 10: Results for Σ
with noise ξ = 0.3.
with data set generated by solving the forward problems,
of the wavelength can not be seen. See, for instance, [1].
In order to explore a resolution limit, we consider the following examples:
(3)
Σ = (x − 0.1, 0) : − ≤ x ≤ ∪ (x + 0.1, 0) : − ≤ x ≤ ,
(4)
Σ = (x − 0.05, 0) : − ≤ x ≤ ∪ (x + 0.05, 0) : − ≤ x ≤ .
(3)
(4)
Both Σ and Σ consist of two small cracks, whose distances are 0.2 and 0.1,
(3)
respectively. We illuminate the medium containing Σ with plane waves of N =
(4)
20 directions and of the wavelength λ = 1, while the medium containing Σ is
illuminated with waves of the wavelength λ = 0.05.
Figure 11 clearly shows that the waves of wavelength 1 can not distinguish two
(3)
cracks of distance 0.2 in Σ , while the waves of wavelength 0.05 can distinguish
(4)
two cracks in Σ , whose distance is even shorter than other ones.
To conclude this section, let us make two remarks. From the derivations of our
MUSIC-type imaging functional, it follows that the number N of measurements has
to be at least twice the number of cracks inside the medium in order to detect them
and estimate their sizes and directions. On the other hand, we point out that if
the crack is very close to the boundary (at a distance of order ), then formula
(3.2) on which our imaging approach is based is not valid anymore. Because of the
singularity of the Neumann function, the assumption that the crack is well-separated
from the boundary is essential in our derivations in order to estimate the remainder
in (3.2). However, since local multiple scattering occurs between the crack and
the boundary, one can prove that the perturbation u − u0 takes values of order
larger than O(1/| ln |). We refer, for instance, to [6] for similar observations in the
18
(3)
(4)
Figure 11: Left: image of Σ with the wavelength λ = 1. Right: image of Σ
with λ = 0.05. The data set was generated by solving the forward problems.
case where the background medium contains an inclusion near the boundary. The
derivation of an expansion to describe the crack-boundary interaction and make an
extrapolation from the approximation (3.2) to the crack signature in the case where
it is near the boundary is a nontrivial task and beyond the scope of this paper.
6
Deformation of an extended crack
We now consider the problem of reconstructing an extended crack Σ which is not
necessarily linear. By ‘extended’ we mean that the size of the crack is of order few
wavelengths.
For that purpose we first derive pointwise asymptotic expansion formula for the
perturbation of the solution due to the shape perturbation of a perfectly conducting
crack. Again, our derivations are based on layer potential techniques. Moreover, it
seems to be no difficulty in continuing the calculations to higher orders. As far as
we know, we are not aware of such derivations from available literature.
Let a, b ∈ R, with a < b, and let X(t) : [a, b] → R2 be the arc length parametrization of Σ, namely, X is a C 2 -function satisfying |X 0 (t)| = 1 for all t ∈ [a, b] and
X := x = X(t), t ∈ [a, b] .
Then X(a) and X(b) are the endpoints. The outward unit normal to Σ, ν(x), is
given by ν(x) = R−π/2 X 0 (t), where R−π/2 is the rotation by −π/2. Note that the
unit tangential vector at x is given by T (x) = X 0 (t) and X 0 (t) ⊥ X 00 (t). Set the
curvature κ(x) to be defined by
X 00 (t) = κ(x)ν(x).
19
Let X (t) = X(t) + h(t)ν(x) = X(t) + h(t)R−π/2 X 0 (t) be a parametrization of
Σ which is a perturbation of Σ.
By ν (x ), we denote the outward unit normal to Σ at x = x + h(t)ν(x) ∈ Σ .
Let us now invoke from [5, Section 3.3] the following expansions: ν (x ) can be
expanded as
ν (x ) = ν(x) − h0 (t)T (x) + O(2 ), x ∈ Σ.
(6.1)
Likewise, we get an expansion for the length element dσ (x ):
dσ (x ) = (1 − κ(x)h(t) + O(2 )) dσ(x).
(1)
Let H0
(6.2)
be the Hankel function of first kind and order zero. We have
(1)
(1)
H0 (ω|x − y |) = H0 (ω|x − y|)
(1) 0
+ωH0 (ω|x − y|)
hx − y, h(t)ν(x) − h(s)ν(y)i
+ O(2 ),
|x − y|
(6.3)
where x = X(t) and y = X(s). The term O(2 ) is uniform in both x and y on Σ.
Set
Z
p
2
|X(b) − x||x − X(a)| |ϕ(x)| dσ(x) < +∞ ,
X (Σ) = ϕ :
Σ
and
Z
kϕkχ(Σ) :=
1
p
2
2
|X(b) − x||x − X(a)| |ϕ(x)| dσ(x)
.
Σ
(1)
We now introduce SΣ and SΣ,h , defined for any φ ∈ X (Σ), by
i
SΣ [φ](x) =
4
Z
Σ
(1)
H0 (ω|x − y|) φ(y) dσ(y),
and
(1)
SΣ,h [φ](x) =
Z hx − y, h(t)ν(x) − h(s)ν(y)i
(1) 0
ωH0 (ω|x − y|)
|x − y|
Σ
(1)
−H0 (ω|x − y|)κ(y)h(s) φ(y) dσ(y).
i
4
(6.4)
Let Ψ be the mapping from Σ onto Σ given by
Ψ (x) = x + h(t)ν(x),
x = X(t).
Then the following lemma is easily derived from (6.2) and (6.3) (see also [5, Section
3.3].
20
Lemma 6.1 For any φ̃ ∈ X (Σ ),
(1)
(SΣ [φ̃]) ◦ Ψ − SΣ [φ] − SΣ,h [φ]
X (Σ)
≤ C2 kφkX (Σ) ,
(6.5)
where φ := φ̃ ◦ Ψ .
Let us assume that the problem

∆uΣ + ω 2 uΣ = 0 in Ω \ Σ,




uΣ = 0 on Σ,



 ∂uΣ = g ∈ L2 (∂Ω) on ∂Ω,
∂ν
(6.6)
is well-posed, namely, ω 2 is not an eigenvalue of the problem. Then, for small
enough, the following problem is also well-posed:

∆uΣ + ω 2 uΣ = 0 in Ω \ Σ ,




uΣ = 0 on Σ ,
(6.7)

 ∂uΣ


= g on ∂Ω.
∂ν
Let uΣ and uΣ be the respective solutions to (6.6) and (6.7). Let NΩω (x, y) be
the Neumann function defined by (2.3) and u0 be the solution when there is no
crack, i.e., the solution to (2.2). As before, we have by Green’s formula,
Z
(uΣ − u0 )(x) =
NΩω (x, y) φ(y) dσ(y), x ∈ ∂Ω,
(6.8)
Σ
where φ ∈ X (Σ) is the solution to the integral equation
Z
AΣ [φ](x) :=
NΩω (x, y)φ(y) dσ(y) = −u0 (x)
on Σ.
(6.9)
on ∂Ω,
(6.10)
Σ
Likewise, we have
Z
(uΣ − u0 )(x) =
NΩω (x, y ) φ̃ (y ) dσ (y )
Σ
where φ̃ ∈ X (Σ ) is the solution to
Z
NΩω (x , y ) φ̃ (y ) dσ (y ) = −u0 (x )
Σ
21
on Σ .
(6.11)
Let us prove that AΣ is invertible on X (Σ). Because of (2.9), one can see that AΣ
is a compact operator on X (Σ). Suppose that AΣ [φ] = 0 on Σ for some φ ∈ X (Σ).
Define
Z
NΩω (x, y)φ(y) dσ(y), x ∈ Ω.
v(x) =
Σ
Then v satisfies (6.6) with g = 0 and therefore, v ≡ 0 in Ω. But, φ is equal to
the jump of the normal derivative of v on Σ, and hence φ = 0 on Σ. Thus AΣ is
invertible on X (Σ).
Writing
i (1)
ω
NΩω (x, y) = H0 (ω|x − y|) + RN
(x, y),
4
ω (x, y) is of class C 1,α , 0 < α < 1, one can see that (6.11) can be written as
where RN
Z
ω
SΣ [φ̃ ](x ) +
RN
(x , y )φ̃ (y ) dσ (y ) = −u0 (x ) on Σ .
Σ
Since
u0 (x ) = u0 (x) + h(x)ν(x) · ∇u0 (x) + O(2 ),
we use Lemma 6.1 to deduce that
φ̃ = φ ◦ Ψ + O(2 ),
where φ satisfies
(1)
(1)
SΣ + RΣ + SΣ,h + R
[φ ](x) = −u0 (x) − h(x)ν(x) · ∇u0 (x)
on Σ.
Here the operators RΣ and R(1) are defined by
Z
ω
RΣ [φ](x) =
RN
(x, y)φ(y) dσ(y),
Σ
and
Z ω
ω
∂RN
∂RN
ω
(x, y)h(x)+
(x, y)h(y) φ(y) dσ(y).
R [φ](x) =
−κ(y)h(y)RN (x, y)+
∂ν(x)
∂ν(y)
Σ
(6.12)
(1)
(1)
(1)
Put AΣ = SΣ,h + R . Since SΣ + RΣ = AΣ , one can easily see that
(1)
φ = φ0 + φ1 + . . . ,
where φ0 and φ1 are the unique solution to

 AΣ [φ0 ]
(6.13)
= −u0
(6.14)
 AΣ [φ1 ] + A(1) [φ0 ] = −h ∂u0 .
Σ
∂ν
By inserting (6.2) and (6.13) into (6.10), we obtain the following theorem, which
is our main result in this section.
22
Theorem 6.2 Let uΣ and uΣ denote the solutions to (6.7) and (6.6), respectively.
For x ∈ ∂Ω,
Z ∂NΩω
ω
ω
(uΣ − uΣ )(x) = (x, y)h(y) − κ(y)h(y)NΩ (x, y) φ0 (y) + NΩ (x, y)φ1 (y) dσ(y)
∂ν(y)
Σ
+O(2 ),
where φ0 , φ1 ∈ X (Σ) are given by (6.14).
One can try to express uΣ −uΣ in terms of the Neumann function in the presence
of the crack Σ, that is, the solution to

2
ω
in Ω \ Σ,

 (∆x + ω )NΣ,Ω (x, z) = −δz (x)


 ω
NΣ,Ω = 0
on Σ,

ω

∂NΣ,Ω



(x, z) = 0
x ∈ ∂Ω.
∂νx
However, even though such derivation is possible, it would be useless in practice
ω
are more costly and more involved than those for
since the computations of NΣ,Ω
ω
NΩ where there is no crack in the medium. Moreover, if one looks to reconstruct
the shape of the crack from discrete measurements using any iterative inversion
ω at each iteration.
procedure, one has to repeat the calculations of NΣ,Ω
If the measurements are done on the whole boundary, then a formula similar to
(4.1) can be derived. For any f ∈ L2 (∂Ω), one can see from Theorem 6.2 that
Z
(uΣ − uΣ )(x)f (x) dσ(x)
∂Ω Z ∂v
=
(y)h(y) − κ(y)h(y)v(y) φ0 (y) + v(y)φ1 (y) dσ(y) + O(2 ),
∂ν
Σ
(6.15)
where v is the solution to

2
 (∆ + ω )v = 0 in Ω,
 ∂v = f on ∂Ω.
∂ν
R
Note that v plays the role of an adjoint state. One shall now express Σ v(y)φ1 (y) dσ(y)
in terms of h. Define w ∈ X (Σ) as the solution to AΣ [w] = v on Σ. Using (6.14),
one computes
Z
Z
Z
Z
∂u0
(1)
vφ1 dσ =
wAΣ [φ1 ] dσ = − h
w dσ −
AΣ [φ0 ]w dσ.
∂ν
Σ
Σ
Σ
Σ
23
Therefore,
Z
Z (uΣ
∂Ω
∂v
− uΣ )(x)f (x) dσ(x) = (y)h(y) − κ(y)h(y)v(y) φ0 (y)
∂ν
Σ
∂u0
(1)
(y)w(y) − AΣ [φ0 ](y)w(y) dσ(y) + O(2 ).
−h(y)
∂ν
Since
(1)
AΣ [φ0 ](x) =
Z hx − y, h(t)ν(x) − h(s)ν(y)i
(1) 0
ωH0 (ω|x − y|)
|x − y|
Σ
(1)
−H0 (ω|x − y|)κ(y)h(s) φ0 (y) dσ(y)
Z ω
ω
∂RN
∂RN
ω
− κ(y)h(y)RN (x, y) +
+
(x, y)h(x) +
(x, y)h(y) φ0 (y) dσ(y),
∂ν(x)
∂ν(y)
Σ
i
4
then changing the order of integrations yields
Z
Z
(uΣ − uΣ )(x)f (x) dσ(x) = h(x)dS uΣ (x) dσ(x) + O(2 ),
∂Ω
(6.16)
Σ
for some function dS uΣ (x) which is independent of h and can be computed in an
explicit form.
Formula (6.16) is of use to us. It immediately gives the shape derivative of the
cost functional introduced in (9.1). It is particularly appropriate for an integral
equation based computer code since it is written only in terms of the density φ0 on
the crack.
In this connection, we refer the reader to, for instance, [32] and [38] for classical
results on the concept of shape derivative. We shall also mention that the shape
derivative of crack deformation attracted the attention of many authors in various
and more general situations. See [18], [19],[20],[25], [26], [27], [28], [29], and the
references therein.
7
Initial guess for an extended crack
There are many possible ways to get a good initial guess for an extended crack. One
of them is to use the concept of topological derivative. The topological derivative
measures the influence of creating a small crack at a certain point inside the domain
Ω. Mathematically speaking, the topological derivative dT J(z) of a functional J at
a point z inside Ω can be defined by
J(Ω \ Σ ) = J(Ω) + ρ()dT J(z) + o(ρ()),
where Σ is a small crack of size 2 placed at z and the function ρ() → 0 as → 0.
See, for instance, [17], [21], and [37].
24
Another way of getting a good initial guess is to use a standard MUSIC-type
projection approach. The aim of this section is to show how these algorithms perform
in the case of perfectly conducting crack and to compare them.
Suppose that Ω contains an extended perfectly conducting crack, denoted by
Σtrue . Let g (l) , l = 1, . . . , N be N given functions on ∂Ω. The problem we consider
in this section is to construct a good initial guess for Σtrue from the boundary mea(l)
(l)
surements (umeas |∂Ω )N
l=1 , where the function umeas for l = 1, . . . , N is the solution
to

(l)
2 (l)

in Ω \ Σtrue ,
 ∆umeas + ω umeas = 0
(l)

 ∂umeas = g (l)
∂ν
7.1
on ∂Ω,
Use of the concept of topological derivative
In this section, we follow an idea of Auroux and Masmoudi; see [9] and [10]. We
(l)
(l)
first construct uD and uN as the solutions to

 ∆u(l) + ω 2 u(l) = 0
in Ω,
D
D
and
 u(l) = u(l)
meas
D
on ∂Ω,

(l)
2 (l)

 ∆uN + ω uN = 0
in Ω,
(l)

 ∂uN = g (l)
∂ν
(l)
on ∂Ω.
(l)
Note that the functions uD and uN are constructed in the absence of any crack.
Then, using Theorems 3.2 and 3.3 we compute the topological derivative of the
discrepancy functional
N
1X
J(Ω) :=
2
l=1
(l)
2 Z 2
∂uD
(l)
(l)
(l)
+
−
g
u
−
u
meas .
N
∂ν
∂Ω
∂Ω
Z
Suppose that Ω contains a small crack Σ at the point z inside Ω and of size 2.
(l)
(l)
Denote by uD, and uN, the solutions of the following problems:

(l)
(l)

∆uD, + ω 2 uD, = 0 in Ω \ Σ ,



(l)
uD, = 0 on Σ ,



 (l)
(l)
uD, = umeas on ∂Ω,
25
and

(l)
(l)

∆uN, + ω 2 uN, = 0 in Ω \ Σ ,




 (l)
uN, = 0 on Σ ,


(l)


∂uN,


= g (l) on ∂Ω.
∂ν
It then follows from Theorems 3.2 and 3.3 that
(l)
(l)
2 Z 2
N Z
N Z
2 1 X
∂uD,
∂uD
1X
(l)
(l)
(l)
(l)
J(Ω \ Σ ) =
− g +
− g uN, − umeas =
2
∂ν
2
∂ν
∂Ω
l=1 ∂Ω
l=1 ∂Ω
Z 2
1
2π
(l)
(l)
dT J(z) + o(
).
+
uN − umeas +
| ln(/2)|
| ln |
∂Ω
where, for small enough,
dT J(z) = <e
X
(l)
(l)
pD (z)uD (z)
−
(l)
(l)
pN (z)uN (z)
,
(7.1)
l
(l)
(l)
and the adjoint states pD and pN are defined as the solutions to

(l)
2 (l)

 ∆pD + ω pD = 0 in Ω,
(l)

 p(l) = ∂uD − g (l)
D
∂ν
and

(l)
2 (l)

 ∆pN + ω pN = 0
on ∂Ω,
in Ω,
(l)

 ∂pN = u(l) − u(l)
meas on ∂Ω.
N
∂ν
The points where the topological derivative is the most negative are expected
to be approximately on Σtrue . This would give an initial guess for Σtrue . Roughly
speaking, inserting small cracks in the most negative gradient regions minimizes the
(l)
discrepancy between the boundary measurements umeas and the solutions in the
(l)
presence of these cracks with the Neumann data g , for l = 1, . . . , N . As shown in
[10], the negative gradient regions are obtained in only one iteration.
7.2
A MUSIC-type approach
An alternative way of getting a good initial guess is to use a standard MUSIC-type
projection approach. See, for instance, [22] and [34].
Let {θ1 , . . . , θN } be a set of N unit directions and let
g (l) (x) = iωθl · ν(x)eiωθl ·x
26
on ∂Ω, l = 1, . . . , N.
(7.2)
Let
v (l) (x) = e−iωθl ·x
in Ω, l = 1, . . . , N.
Construct the response matrix Ameas = ((Ameas )ll0 ) given by
Z
Z
0
∂v (l )
0
(l)
(Ameas )ll0 =
umeas
dσ −
g (l) v (l ) dσ.
∂ν
∂Ω
∂Ω
Note that an integration by parts shows that
(l) Z
∂umeas (l0 )
v dσ,
(Ameas )ll0 = −
∂ν
Σtrue
where
(7.3)
(l)
(l)
(l)
∂umeas
∂umeas ∂umeas :=
−
.
∂ν
∂ν +
∂ν −
We want to reconstruct an initial guess for Σtrue from Ameas = ((Ameas )ll0 ).
We plot the imaging function
W (z) :=
1
||(I − P )g(z)||
for z ∈ Ω,
where P is the orthogonal projection onto the range of the response matrix Ameas .
As shown by the numerical examples, the function W (z) has large peaks at points
on the crack. This can be explained using (7.3). The matrix Ameas describes in
fact the far-field behavior (|y| → +∞, y/|y| = θl0 ) of
Z
(l)
∂u
−
[ meas ](x)Γω (x, y) dσ(x),
∂ν
Σ
true
where Γω (x, y) is the outgoing Green function of the Helmholtz equation. Divide
the crack into segments of size of order half the wavelength. Having in mind the
resolution limit, only one point at each segment will contribute at the image space of
the response matrix Ameas . Each of these points can in principle be imaged using
the standard MUSIC imaging function. The resolution of the image provided by
this technique is of order the half-wavelength. Since the measurements are done at
the boundary of Ω, higher-resolution in imaging the extended crack can be achieved
using the optimization algorithm briefly described in the next section.
8
Numerical simulations for imaging extended cracks
In this section, we present results of numerical simulations using the two approaches
we described in the previous subsections to image extended cracks. For simulation,
we choose the homogeneous domain Ω as a unit disk centered at (0, 0) in R2 . Here
the data sets are generated by solving forward problems.
27
Figure 12: Image of dT J(z) when applied frequency is ω =
is the true crack.
8.1
2π
0.8 .
Black colored line
Initial guess through the topological derivative
We first implement the method of deriving a good initial guess described in the
previous section. An extended crack chosen for illustration is
sπ
sπ
3sπ
Σ(5) =
0.6s, 0.5 cos
+ 0.5 sin
− 0.1 cos
: s ∈ [−1, 1] .
(8.1)
2
2
2
The operating wavelengths are 0.8 in Figure 12 and 0.5 in Figures 13 and 14.
Figure 12 is the plot of values of dT J(z) for all z ∈ Ω. One can easily notice
that the points where the topological derivative is the most negative value appears
in the neighborhood of Σtrue . A smooth curve connecting most negative values can
be taken as an initial guess. One can notice from Figure 13 that if we increase the
frequency of the waves, then the region of most negative values shrinks and becomes
close to the actual crack, and hence provides a better initial guess. Figure 14 shows
the result when we add 30% (ξ = 0.3) noise. The result is still reasonably good,
which tells that this method is robust even under presence of noise.
8.2
Simulation of the MUSIC-type algorithm
We now implement the MUSIC-type projection algorithm proposed in the previous
section. We use Σ(5) given in (8.1) as our crack.
In order to obtain a data set of the response matrix Ameas in (4.3), we solve
forward problems for N = 24 incident directions at the wavelength λ = 0.6. Figure
15 and Figure 16 show the reconstructed images without noise and with noise, ξ =
0.03, respectively. With this noise level, the reconstruction is satisfactory. Figure
17 is to show what happens if we increase the wavelength. There the wavelength is
2π and the image is clearly deteriorated.
28
Figure 13: Image of dT J(z) when ω =
Figure 14: Image of dT J(z) when ω =
2π
0.5 ,
2π
0.5 ,
without noise
with noise ξ = 0.3.
Figure 15: The distribution of singular values of the Ameas (left) and the plot of
W (z) for Σ(5) . N = 24 and λ = 0.6. Without noise.
29
Figure 16: Same as Figure 15, with noise ξ = 0.03.
Figure 17: Same as Figure 15, but λ = 2π.
30
Figure 18: The distribution of singular values of Ameas (left) and the plot of W (z)
for Σ(6) . N = 24 and λ = 0.5. Without noise.
We then apply the algorithm for imaging of multiple extended cracks,
(6)
(6)
Σ(6) = Σ1 ∪ Σ2 ,
where
(6)
Σ1
(6)
Σ2
(s, −0.5(s − 0.2) + 0.5) : s ∈ (−0.7, 0.3) ,
(s, (s − 0.2) + (s − 0.2) ) − 0.4 : s ∈ (−0.3, 0.7) .
=
=
2
3
2
The data set of matrix Ameas is collected with N = 24 and λ = 0.5. Figure 18
and Figure 19 show the reconstructed images of Σ(6) without noise and with noise,
ξ = 0.05, respectively. Figure 20 shows the image when the wavelength is increase
to 2π.
9
An optimization approach for extended cracks
Let, as before, {θ1 , . . . , θN } be a set of N unit directions and let g (l) be given by
(7.2). In this last section, we briefly mention about a new optimal control approach
in order to achieve better reconstruction than the MUSIC-type algorithm.
(l)
Let umeas be the solution to

(l)
(l)
∆umeas + ω 2 umeas = 0 in Ω \ Σtrue ,




 (l)
umeas = 0 on Σtrue ,


(l)


 ∂umeas = g (l) on ∂Ω.
∂ν
31
Figure 19: Same as Figure 18, but with noise ξ = 0.05.
Figure 20: Same as Figure 18, but with λ = 2π.
32
Suppose now that Ω contains a crack Σ away from the boundary. Denote u(l)
the solution to


∆u(l) + ω 2 u(l) = 0 in Ω \ Σ,



 (l)
u = 0 on Σ,

 ∂u(l)



= g (l) on ∂Ω.
∂ν
A standard approach would be to minimize over Σ the functional
N
1 X (l)
(l)
||u − umeas ||2L2 (∂Ω) .
2
l=1
In [2], we have suggested a new approach toward inclusion reconstruction which
applies here. In the context of extended crack reconstruction, our approach would
be to minimize over Σ at the step (n + 1)
N
M
1 XX
J(Σ) :=
|
2
l=1 j=1
Z
∂Ω
(l)
(u(l) − umeas )fj |2 ,
(9.1)
where {fj }M
j=1 are chosen as the basis of the image space of a certain operator
involving the crack reconstructed at the step n. The MUSIC-type algorithm we
designed in the last section for obtaining an initial guess corresponds to the particular
case of such optimization algorithm where the functions fj are freezed and simply
chosen to be plane waves.
The shape derivative dS J(Σ), which measures the sensitivity of shape perturbations, can be directly computed from (6.16). The implementation of this optimization algorithm would be the subject of a forthcoming work.
10
Conclusion
In this paper, we have provided a MUSIC-type algorithm for locating and estimating the size of small perfectly conducting cracks. Our algorithm is based on a
new asymptotic formula that describes the effect of a small crack on the boundary
measurements and the construction of a response matrix from these boundary measurements. This algorithm has also been applied to get a very good initial guess in
the case of extended cracks. Using the concept of topological derivative, a second algorithm has been tested. To achieve a better reconstruction with higher resolution,
an original optimization approach has been proposed. It is based on a updated cost
functional. Its shape derivative has been computed. The numerical implementation
of the algorithm would be the subject of a forthcoming work.
33
Acknowledgements
W.K. Park would like to thank D. Lesselier for several useful comments and discussions. The authors are very grateful to the referees for their most helpful and
detailed comments.
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