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Reconstruction of complexcracks by far fieldmeasurements

P. Krutitskii, J. Liu, M. Sini

RICAM-Report 2008-15

Reconstruction of complex cracks by far field measurements

P. Krutitskii∗ J.J. Liu† M. Sini‡

January 30, 2008

Abstract

In this paper, we deal with the acoustic inverse scattering problem for reconstructing com-plex cracks from the far field map. The scattering problem models the diffraction of wavesby thin two-sided cylindrical screens. A complex crack is characterized by its shape, the typeof boundary conditions and the boundary coefficients (surface impedance). We give explicitformulas which can be used to reconstruct the shape of the crack, distinguish its type ofboundary conditions, its two faces and reconstruct the possible material coefficients on it byusing the far-field map. To test the validity of these formulas, we present some numericalimplementations, which show the efficiency of the proposed method for suitably distributedsurface impedance. The difficulties for numerically recovering the properties of the crack inthe concave side as well as near the tips are presented and some explanations are given.

Key words. Inverse scattering, cracks, far-field, numerics.

AMS subject classifications. 35P25, 35R30, 78A45.

1 Statement of the problem

To describe the diffraction of acoustic waves by thin two-side cylindrical screens, the scatteringproblems are governed by the Helmholtz equation by a crack Γ in R

2. Let Γ be a two dimensionalopen curve of class C3 with a parameterization representation Γ = x := x(s), s ∈ [a, b] wherex : [a, b] → R

2 is locally of class C3 . We set P = x(a) and Q = x(b) to be the two tips of Γ, and fixthe orientation of Γ as follows. Traveling on Γ from P to Q, we associate to the right side the sign+, i.e. Γ+, and to the left side the sign −, i.e. Γ− and we set ν to be the unit normal on Γ orientedtowards Γ+. The different boundary conditions specified on Γ represent the acoustic properties ofthe crack. For given incident waves ui(x) = eiκd·x, we consider the following scattering problemsfor total waves u(x) = ui(x) + us(x):

(1.1) ( Dirichlet )

(∆ + κ2)u = 0, in R2 \ Γ,

u = 0 on Γ±,u = us + eiκd·x,

limr→∞√

r(∂us

∂r − iκus) = 0,

∗KIAM, Miussqaya Sq.4, Moscow, 125047, Russia. email: krutitsk@math.phys.msu.su†Department of Mathematics, Southeast University, Nanjing, 210096, email: jjliu@seu.edu.cn, P.R.China.‡RICAM, Austrian Academy of Science, Linz, A-4040, Austria. email: mourad.sini@oeaw.ac.at

1

(1.2) ( Mixed )

(∆ + κ2)u = 0, in R2 \ Γ,

u = 0 on Γ+,∂us

∂ν − iκσ−us = 0, on Γ−,u = us + eiκd·x,

limr→∞√

r(∂us

∂r − iκus) = 0,

(1.3) ( Robin )

(∆ + κ2)u = 0, in R2 \ Γ,

∂u∂ν ± iκσ±u = 0, on Γ±,u = us + eiκd·x,

limr→∞√

r(∂us

∂r − iκus) = 0,

where us(x) is the scattered wave outside of Γ, while κ > 0 is the wave number and d is the directionof incidence of plane wave ui(x). We assume that σ± are complex valued Holder continuousfunctions of order β ∈ (0, 1], σ± = σr

± + iσi±, and their real parts have positive uniform lower

bounds.The problems (1.1), (1.2) and (1.3) are well posed, see [9], [10], [11], [13], and [2] for more details.

These models describe the scattering problems of the cracks with different acoustic properties.Using the asymptotic behavior of the fundamental solution , as in [3], we can show that thescattered wave has the asymptotic behavior:

us(x, d) =eiκr

√r

u∞(x, d) + O(r−3/2), r := |x| → ∞,(1.4)

where the function u∞(·, d) defined on the unit circle S1 is called the far-field of the scattered

wave us corresponding to incident direction d. We introduce a constant γ2 := eiπ/4√

8πκand Φ(x, y) :=

i4H

(1)0 (κ|x − y|), x 6= y, x, y ∈ R

2, the fundamental solution to the Helmholtz equation in R2,

where H(1)0 is the Hankel function of the first kind of order zero. In this paper, we will consider

the following:Complex crack reconstruction problem. Given u∞(·, ·) on S1 × S1 for the scattering

problems (1.1) or (1.2) or (1.3), reconstruct the shape of the crack Γ, distinguish the two faces andreconstruct the eventual surface impedances σ±(x).

Remark 1.1 We do not know a priori to which problem is associated the data u∞(x, d) on (x, d) ∈S1 × S1.

The inverse problems for cracks detection have been studied by many authors. We refer to[1] for some results concerning, in particular, detection of piecewise linear cracks from one or fewexterior measurements. We are interested by detection of cracks of general shapes but using manymeasurements. Precisely, we use the far field map and our aim is to reconstruct the whole crack.There were several works devoted to the detection of cracks from many measurements. Amongothers, we shall cite [7], [6] and [2], and the references there, where the authors gave reconstructionmethods to detect the shape of the cracks. In this paper, we shall be concerned by reconstructingcomplex cracks by giving the shapes, the type of boundary conditions, distinguishing the two facesand computing the pointwise values of the complex valued surface impedances distributed alongthem. Precisely, we provide direct formulas which link the far field map to the unknowns. Weshow that these formulas contain the information on the cracks throuth an asymptotic expansion

2

with respect to the point sources used. Hence, we should be careful to what extent these indicatorscan approximate the crack properties such as its shape, the acoustic properties and the possiblesurface impedance. Our numerical realizations presented in the last section show that the propertyof the crack in the convex side can be reconstructed well, but the reconstruction in the concaveside of the crack is quite limited. This phenomenon can be explained physically by the multiplereflections of the waves within the cavity, which lead to a relatively less information about theconcave side in the far field pattern. In addition appropriate types of variations of the surfaceimpedance can improve the reconstruction in the concave parts but also can destroy the one nearthe convex parts. This shows how difficult it is to reconstruct numerically the shape of the crackwithout knowing its properties or some a-priori informations. We wish also to point out that ourformulas are valid on the points of the crack away from the tips. We believe that the indicatorfunctions near the tips should be more singular. But such an assersion needs to be justified.

The plan of the paper is as follows. In section 2, we present the theoretical results related tothe asymptotic formulas by adding some comments on how one can use them. In section 3, we givethe justifications of these results and in section 4, we present extensive numerical results followedby some detailed explanations.

2 Presentation of the results

It is well known, see [3], that the scattered field associated with the Herglotz incident fieldvi

g := vg defined by vg(x) :=∫

S1 eiκx·dg(d) ds(d), x ∈ R2 with g ∈ L2(S1) is given by vs

g(x) :=∫

S1 us(x, d)g(d) ds(d), x ∈ R2 \ Γ, and its far field is v∞

g (x) :=∫

S1 u∞(x, d)g(d) ds(d), x ∈ S1.

We will need the following identity, see [3],

(2.1) u∞(x, d) = −γ2

∫

∂D

∂us(y, d)

∂νe−iκx·y − ∂e−iκx·y

∂νus(y, d)

ds(y)

where ∂D is a closed curve containing a part of Γ and avoiding the tips (P,Q). In addition, weassume that the bounded domain surrounded by ∂D, i.e D, is such that Γ ⊂ D. In particular Dcontains the tips (P,Q).

Assume that Γ ⊂⊂ Ω for some known Ω with smooth boundary. For a ∈ Ω \ Γ, denote byzp ⊂ Ω \ D a sequence tending to a. For any zp, set Dp

a to be a C2− regular domain such that

Γ ⊂ Dpa with zq ∈ Ω \ Dp

a for every q = 1, 2, · · · , p and that the Dirichlet interior problem on Dpa

for the Helmholtz equation is uniquely solvable. In this case, the Herglotz wave operator H definedfrom L2(S1) to L2(∂Dp

a) by

(2.2) H[g](x) := vg(x) =

∫

S1

eiκx·dg(d) ds(d)

is injective, compact with dense range, see [3]. Now we consider the sequence of point sourcesΦ(·, zp). For every p fixed, we construct two density sequences gp

n and f j,pm in L2(S1) by the

Tikhonov regularization such that

(2.3) ||vgpn− Φ(·, zp)||L2(∂Dp

a) → 0, n → ∞

(2.4) ||vfj,pm

− ∂

∂xjΦ(·, zp)||L2(∂Dp

a) → 0, m → ∞.

3

We choose ∂D to contain a part of Γ surrounding the fixed point a, such that zp ⊂ Ω \ D (for p large enough ) and D ⊂ Dp

a. Since both vgpn

and Φ(·, zp) satisfy the same Helmholtz equationin Dp

a, (2.3) implies that

(2.5) ||vgpn− Φ(·, zp)||

H12 (∂D)

→ 0, n → ∞

and

(2.6) || ∂

∂νvgp

n− ∂

∂νΦ(·, zp)||

H− 12 (∂D)

→ 0, n → ∞

Similarly, it follows from (2.4) that

(2.7) ||vfj,pm

− ∂

∂xjΦ(·, zp)||

H12 (∂D)

→ 0, m → ∞

and

(2.8) || ∂

∂νvfj,p

m− ∂

∂ν(

∂

∂xjΦ(·, zp))||

H− 12 (∂D)

→ 0, m → ∞

Multiplying (2.1) by f j,pm (d)gp

n(x) and integrating over S1 × S1, we have

−∫

S1

∫

S1

u∞(−x, d)f j,pm (d)gp

n(x) ds(x)ds(d)

= γ2

∫

∂D

∫

S1

∂us(y, d)

∂νf j,p

m (d) ds(d) ·∫

S1

eiκx·ygpn(x) ds(x) −

∫

S1

∂eiκx·y

∂νgp

n(x) ds(x) ·∫

S1

us(y, d)f j,pm (d) ds(d)

ds(y)

= γ2

∫

∂D

∂vsfj,p

m

∂ν(y)vi

gpn(y) −

∂vigp

n

∂ν(y)vs

fj,pm

(y)

ds(y).(2.9)

From (2.7), (2.8) and (2.9), we have

limn→∞

∫

S1

∫

S1

u∞(−x, d) f j,pm (d) gp

n(x) ds(x)ds(d)

= γ2

∫

∂D

vsfj,p

m

∂Φ(y, zp)

∂ν(y)−

∂vsfj,p

m

∂ν(y)Φ(y, zp)

ds(y)

= γ2vsfj,p

m(zp)(2.10)

from the Green formula, where vsfj,p

m(·) is the scattered wave corresponding to incident wave

vifj,p

m(x) = H[f j,p

m ](x).

Denote by Esj (x, zp) the scattered wave corresponding to the incident wave

∂Φ(x,zp)∂xj

, which is

well defined for every x ∈ R2 \Γ. Then it follows from (2.6), (2.7), the well posedness of the direct

scattering problem and the use of interior estimate that

(2.11) Esj (x, zp) = lim

m→∞vs

fj,pm

(x), x ∈ R2 \ D.

4

Finally, it follows from (2.10) that

limm→∞

limn→∞

∫

S1

∫

S1

u∞(−x, d) f j,pm (d) gp

n(x) ds(x)ds(d) = γ2Esj (zp, zp).(2.12)

We set

(2.13) Ij(zp) :=1

γ2lim

m→∞lim

n→∞

∫

S1

∫

S1

u∞(−x, d) f j,pm (d) gp

n(x) ds(x)ds(d).

Let us mention that the construction of f j,pm and gp

m is independent on the unknown crack.Hence Ij(zp) is computable from our data only.

The reconstruction of the Γ as well as its eventual surface impedance is established by analysingthe behavior of (2.12) when zp approaches a. For this, we need the C3 smoothness assumption onthe regularity of Γ. Precisely, for every point a ∈ Γ \ P,Q, there exists a rigid transformation ofcoordinates under which the image of a is 0 and a function f ∈ C3(−r, r) such that

(2.14) f(0) =df

dx(0) = 0, D ∩ B(0, r) = (x, y) ∈ B(0, r); y > f(x)

in terms of the new coordinates where B(0, r) is the 2-dimensional ball of center 0 with radiusr. For the points a ∈ Γ, we choose the sequence zpp∈N included in Ca,θ, where Ca,θ is a conewith center a, angle θ ∈ [0, π

2 ) and axis ν(a). The answer to the inverse problem is based on thefollowing theorem.

theorem 2.1 Assume that Γ is of class C3 and σ± := σr± + iσi

± are complex valued Holdercontinuous functions with positive lower bounds for their real parts σr

±. Then we have the followingformulas:

1. Re(Ij(zp)) =

(2.15)

±νj(a)4π|(zp−a)·ν(a)| ± νj(a)κσi

±(a) 1π ln(|(zp − a) · ν(a)|) + O(1), a ∈ Γ± \ P,Q

( for the impedance boundary conditions) ,

∓νj(a)2π|(zp−a)·ν(a)| + O(1), a ∈ Γ± \ P,Q

( for the Dirichlet boundary conditions) .

2.

(2.16) Im(Ij(zp)) =

∓νj(a)π κσr

± ln(|(zp − a) · ν(a)|) + O(1), a ∈ Γ± \ P,Q

(for the impedance boundary conditions),

O(1), a ∈ Γ± \ P,Q

(for the Dirichlet boundary conditions) .

The notation a ∈ Γ± means that the sequence zp tends to a from the right (+) (or, the left (−))side of Γ.

5

2.1 Comments

The formulas (2.15) and (2.16) can be used to provide the following information on the crack:

• A sample of points on the curve and the normals on these points. The points can be givenby numerically solving |Re Ij(z)| = C for constants C large. The normals are obtained asfollows

ν(a) = ±(t

√

1

1 + t2,

√

1

1 + t2) where t := lim

zp→a

ReI1(zp)

ReI2(zp).

• Distinguish the parts where we have Dirichlet or Impedance type of boundary conditions.This is a consequence of the following identities for a ∈ Γ± \ P,Q and any given s ∈ (0, 1):

limzp→a

|Im Ij(zp)|| ln(|(zp − a) · ν(a)|)|s =

∞, Impedance boundary

0, Dirichlet boundary(2.17)

• In addition, in case of impedance type boundary conditions, we can reconstruct the real andthe imaginary parts of the surface impedance σ±:

(2.18) σr±(a) = lim

zp→a

π∑2

j=1 ±νj(a)Im Ij(zp)

κ ln(|(zp − a) · ν(a)|)

and

(2.19) σi±(a) = − lim

zp→a

π∑2

j=1 ±νj(a)Re Ij(zp) + 14|(zp−a)·ν(a)|

κ ln(|(zp − a) · ν(a)|) .

• The formulas (2.18), rewritten as

(2.20) ±σr±(a) = lim

zp→a

π∑2

j=1 ±νj(a)Im Ij(zp)

κ ln(|(zp − a) · ν(a)|)

enables us to know if a ∈ Γ+ or a ∈ Γ−, i.e to distinguish between the two faces of the crack.Indeed, since σr(a) > 0 then if the right hand side of (2.20) is positive then a ∈ Γ+ and if itis negative then a ∈ Γ−.

• We need to point out a misprint in our previous paper [15]. That is in the equality (2.17)(of that paper) we need to change the sign ± by ∓. However, such a misprint does not haveserious effects on the other results since later we used the absolute value.

3 Justification of the results

In this section, we give the proof of Theorem 2.1 for the impedance case since for the Dirichletcase the proof is similar and easier. We consider only the case j = 2. The case j = 1 can behandled in a similar way with the appropriate changes. We start by some preparations. For anygiven point a ∈ Γ, we firstly take the rotation Ra and the translation Ma such that Ra(ν(a)) =(0, 1), Ra(a) + Ma = 0 in the new coordinate system x. Under the transform x := T(x) :=Ra(x) + Ma, it follows that T(ν(a)) = (0, 1), T(a) = 0. The justification of the results is basedon the following propositions which give the dominant part of E2(x, z) for x, z near a.

6

Proposition 3.1 1. Impedance boundary condition case. Let a ∈ Γ± \ P,Q , then there existδ(a) > 0 and C > 0 such that

(3.1) |Es2(z, z) − w±

σ(a)(z, z)| ≤ C, for z ∈ B+(a, δ(a)) ∩ Ca,θ,

where B+(a, δ(a)) := B(a, δ(a)) ∩ (R2 \ D) and B(a, δ(a)) is the ball of center a and radius δ(a).2. Dirichlet boundary conditions. If a ∈ Γ+ \ P,Q, we obtain (3.1) by replacing w±

σ(a) by wD.

The functions w±σ(a)(x, z) and wD are given by w±

σ(a)(x, z) := w±σ(a)(x, z) and wD(x, z) :=

wD(x, z) where w±σ(a)(x, z) and wD(x, z) satisfy the folowing properties.

Proposition 3.2 The function w±σ(a)(x, z) has the following explicit form

w±σ(a)(x, z) =

−ν2(a)

4π

∫

R

ei(x1−z1)ξ1e−(x2+z2)|ξ1| |ξ1| ± iκσ(a)

|ξ1| ∓ iκσ(a)dξ1

(3.2) +iν1(a)

4π

∫

R

ei(x1−z1)·ξ1e−(x2+z2)|ξ1| ξ1

|ξ1||ξ1| ± iκσ(a)

|ξ1| ∓ iκσ(a)dξ1,

while wD(x, z) has the form

wD(x, z) = +ν2(a)

4π

∫

R

ei(x1−z1)ξ1e−(x2+z2)|ξ1|dξ1

(3.3) −iν1(a)

4π

∫

R

ei(x1−z1)ξ1e−(x2+z2)|ξ1| ξ1

|ξ1|dξ1.

In addition,

(3.4) w±σ(a)(x, z) = − ν2(a)

2π(x2 + z2)± iκν2(a)σ(a)

πln(x2 + z2) + O(1)

and

(3.5) wD(x, z) = +ν2(a)

2π(x2 + z2)+ O(1).

We consider the following two problems in the coordinate x = (x1, x2) for any given z =(z1, z2) ∈ R

2+. Then w±

σ(a)(x, z) and wD(x, z) are two functions satisfying

(3.6)

∆w±σ(a) = 0, x ∈ R

2+

( ∂∂x2

w±σ(a) ± iκσ(a)w±

σ(a))(x, z)|x2=0 = −( ∂∂x2

± iκσ(a))∇Γ(x, z) · τ |x2=0,

(3.7)

∆wD = 0, x ∈ R2+,

wD(x, z)|x2=0 = −∇Γ(x, z) · τ |x2=0

respectively, where Γ(x, z) = 12π ln 1

|x−z| and the subscript D in wD(x, z) refers to the Dirichlet

boundary condition in (3.7). The vector τ is given by τ := Ra(0, 1) = (−ν1(a), ν2(a)).This can be proven by expressing

w±σ(a)(x, z) = (U+[x2]φ±)(x1), wD(x, z) = (U+[x2]φ)(x1)

in R2+ with (U+[x2]φ)(x1) := 1

2π

∫

Reix1ξ1+x2|ξ1|φ(ξ1, z)dξ1 and computing the density functions

φ± and φ from the boundary value problems (3.6), (3.7), where φ is the 1-dimensional Fouriertransform of φ, see [17] for explicit computations.

7

3.1 End of the proof of Theorem 2.1

We recall the notation x = (x1, x2), z = (z1, z2) and assume that x1 = z1 and x2, z2 > 0. We dothe computations for w+

σ(a). We get the formulas for w−σ(0) by replacing ν(a) by −ν(a).

w+σ(a)(x, z) = −ν2(a)

4π

∫

R

e−(x2+z2)|ξ1| |ξ1| + iκσ(a)

|ξ1| − iκσ(a)dξ1

= −ν2(a)

4π

∫

R

e−(x2+z2)|ξ1|dξ1 −ν2(a)

4π

∫

R

e−(x2+z2)|ξ1| 2iκσ(a)

|ξ1| − iκσ(a)dξ1

= − ν2(a)

2π(x2 + z2)− ν2(a)

2π(iκσ(a))

∫

R

e−(x2−z2)|ξ1|

|ξ1| − iκσ(a)dξ1

However∫

R

e−(x2+z2)|ξ1|

|ξ1| − iκσ(a)dξ1 =

∫

R

e−(x2+z2)|ξ1|

|ξ1| + κσi(a) − iκσr(a)dξ1

=

∫

R

e−(x2+z2)|ξ1| |ξ1| + κσi(a) + iκσr(a)

(|ξ1| + κσi(a))2 + (κσr(a))2dξ1

=

∫

R

e−(x2+z2)|ξ1| |ξ1| + κσi(a)

(|ξ1| + κσi(a))2 + (κσr(a))2dξ1 + O(1).

Now we have∫

R

e−(x2+z2)|ξ1| |ξ1| + κσi(a)

(|ξ1| + κσi(a))2 + (κσr(a))2dξ1

= 2

∫ ∞

0

e−(x2+z2)ξ1ξ1 + κσi(a)

(ξ1 + κσi(a))2 + (κσr(a))2dξ1

= 2

∫ ∞

κσi(a)

e−(x2+z2)(r−κσi(a)) r

r2 + (κσr(a))2dξ1

And∫ ∞

κσi(a)

e−(x2+z2)rr

r2 + (κσr(a))2dξ1 = − ln(x2 + z2)e

−(x2+z2)κσi(a) + O(1).

Hence

w+σ(a)(x, z) =

−ν2(a)

2π(x2 + z2)+

iκσ(a)ν2(a)

πln(x2 + z2) + O(1).

It is clear that for x1 = z1, we have wD(x, z) = 0.

Coming back to the original coordinates (x, z), we have:

w+(z, z) = − ν2(a)

4π|(z − a) · ν(a)| +iκσ(a)ν2(a)

πln(|(z − a) · ν(a)|) + O(1).

We end the proof by using Proposition 3.1.

8

3.2 Proof of Proposition 3.1.

We consider the case where a ∈ Γ+ \ P,Q, i.e. the sequence (zp)p∈N, tending to a, is in thepositive side of Γ. The case a ∈ Γ− can be done by replacing ν and −ν. We show the details ofthe case j = 2.

Let Es(x, zp) be the solution of

(3.8)

(∆ + κ2)Es(x, zp) = 0 in R2 \ D,

( ∂∂ν + iκσ(x))Es(x, zp) = −(∂ν + iκσ(x)) ∂

∂x2Φ(x, zp) on ∂D

Es(·, z) satisfies the Sommerfeld radiation condition.

We state Hσ(x, z) := E(x, z) + ∂∂x2

Φ(x, z). Hence Hσ(a) satisfies

(3.9)

(∆ + κ2)Hσ(x, z) = −∇δ(x, z) · (0, 1) in R2 \ D,

( ∂∂ν + iκσ)Hσ(x, z) = 0, on ∂D

Hσ(·, z) satisfies the Sommerfeld radiation condition.

Similarly, we set Hσ(x, z) := E(x, z) + ∂∂x2

Φ(x, z). Hence W := Hσ − Hσ satisfies the properties:

(3.10)

(∆ + κ2)W (x, z) = 0 in R2 \ D,

( ∂∂ν + iκσ)W (x, z) = 0, on Γ ∩ ∂D

Let B be a ball of center a and radius r > 0. The arguments in [5], see also [4], show that Hσ(a) and

Hσ(a) satisfy the estimates |Hσ(a)(x, z)|, |Hσ(a)(x, z)| ≤ C|x−z| and |∇xHσ(a)(x, z)|, |∇xHσ(a)(x, z)| ≤

C|x−z|2 , for x, z ∈ B\D where C is a positive constant. Hence, in particular, ‖W (·, z)‖H1/2(∂(B\D)\Γ∩∂D)

is bounded for z near a. With this property and (3.10), we see that W (·, z) satifies the Helmholtzequation in B \ D and has bounded H−1/2(∂D ∩ Γ) and H1/2(∂(B \ D) \ Γ ∩ ∂D) norms of themixed boundary conditions. We choose B small enough so that κ2 is not an eingenvalue for themixed problem. From the well posedness of this problem in Sobolev spaces, see [16] or [2], wededuce in particular that ‖W (·, z)‖H1(B\D) is bounded with respect to z near a.

We choose G to be the Green’s function for (∆ + κ2) in B \ D with homogeneous Dirichletcondition on ∂(B \ D). An integration by parts shows that

(3.11) W (x, zp) =

∫

∂(B\D)

∂

∂νG(y, x)W (y, zp)ds(y), for x, zp in B \ D.

Taking x tending to ∂(B \ D, then using the discontinuity relations of the double layer potential,we get

(3.12)1

2W (x, zp) =

∫

∂(B\D)\(Γ\∂D)

∂

∂νG(y, x)W (y, zp)ds(y), for x ∈ ∂(B \ D), zp ∈ B \ D.

By a perturbation argument, writting G = Γ+ + (G − Γ+), we prove that | ∂∂ν G(y, x)| is bounded

for y, x ∈ ∂(B \ D). Hence (3.12) implies that necessary |W (x, zp)| is bounded for x ∈ ∂(B \ D)and zp near a. We take a smaller B if necessary to insure that κ2 is not a Dirichlet eingenvalue ofthe Laplacian. Then the maxmum principle implies that |W (x, zp)| is bounded for x and zp neara.This means that we can replace in Proposition 3.1 Es by Es. Hence, we will analyse Es near thepoint a.

9

We introduce wsσ(·, zp) as the solution of

(3.13)

(∆ + κ2)wsσ(x, zp) = 0 in R

2 \ D,( ∂

∂ν + iκσ(a))wsσ(x, zp) = −(∂ν + iσ(a)) ∂

∂x2Φ(·, zp) on ∂D

wsσ(·, z) satisfies the Sommerfeld radiation condition.

We have the following lemma:

Lemma 3.3 There exist δ(a) > 0 and C(R) > 0 such that

|(Es − wsσ)(x, zp)| ≤ C(R),

for zp ∈ B(a, δ(a)) ∩ Ca,θ and x ∈ (R2 \ D) ∩ B(0, R), for any R > 0 fixed.

Let wsσ(a),Φ(·, z) be the solution of

(3.14)

(∆ + κ2)wsσ(a),Φ(x, z) = 0 in Ω \ D,

( ∂∂ν + iκσ(a))ws

σ(a),Φ(x, z) = −( ∂∂ν + iκσ(a)) ∂

∂x2Φ(x, zp) on ∂D

wsσ(a),Φ(·, z) = − ∂

∂x2Φ(x, zp) on ∂Ω

and wsσ(a),Γ(·, z) be the solution of (3.14) replacing Φ by Γ. Then we have

Lemma 3.4 There exists C > 0 such that

|(wsσ(a) − ws

σ(a),Φ)(x, z)| ≤ C, |(wsσ(a),Φ − ws

σ(a),Γ)(x, z)| ≤ C

for z ∈ Ω \ D near D and x ∈ Ω \ D.

We define ws,0σ(a) to be the solution of (3.14) replacing Φ by Γ and the Helmholtz equation by

the Laplace equation. Then we have

Lemma 3.5 There exists C > 0 such that |(wsσ(a),Γ − ws,0

σ(a))(x, z)| ≤ C, for z ∈ Ω \ D near D and

x ∈ Ω \ D.

Finally, we have

Lemma 3.6 There exist C > 0, δ(a) > 0 such that |(ws,0σ(a) − w+

σ(a))(z, z)| ≤ C for z ∈ B(a, δ(a))∩Ca,θ.

By combining all the lemmas stated above, we end the proof of Proposition 3.1.

In the next section, we justify the lemmas we used to prove Proposition3.1

3.3 Proof of the auxiliary lemmas

Proof of Lemma 3.3.

We set R(x, z) := Esσ(x, z) − ws

σ(a)(x, z). Then it satisfies

(3.15)

(∆ + κ2)R(x, z) = 0 in R2 \ D,

∂R(x,z)∂ν + iκσ(a)R(x, z) = −iκ(σ(x) − σ(a))(Es(x, z) + ∂

∂x2Φ(x, z)) on ∂D,

R(·, z) satisfies the Sommerfeld radiation condition.

10

From (3.15), we have the representation:

(3.16) R(x, z) = −∫

∂D

iκ(σ(y) − σ(a))Gσ(a)(y, x)(Es +∂

∂x2Φ)(y, z)ds(y), for (x, z) ∈ R

2 \ D.

We know that (Es + ∂∂x2

Φ)(y, z) = Hσ has the estimate |Hσ(y, z)| ≤ C|y−z| , then from (3.16) and

the Holder regularity of σ(x), we deduce that

|R(x, z)| ≤ c

∫

∂D

|y − a|β ln(|y − x|)||z − y|−1ds(y).

From the inequality |y − a| ≤ c(θ)|y − z| for y ∈ ∂D and z ∈ Ca,θ ∩ B(a, δ(a)), we have

|y − a|β|y − z| ≤ c(θ)βC

|y − z|1−β,

which implies

|R(x, z)| ≤∫

∂D

c(θ)βC| ln |y − x|||y − z|1−β

dy

and therefore |R(x, z)| = O(1) for x ∈ R2 \ D and z ∈ Ca,θ ∩ B(0, R).

With similar arguments as for the proof of Lemma 3.3, we prove Lemma 3.4 and Lemma 3.5.Proof of Lemma 3.6. Since ws,0

σ(a) satisfies

(3.17)

∆ws,0σ(a)(x, z) = 0 in Ω \ D,

( ∂∂ν + iκσ(a))(ws,0

σ(a)(·, z)) = −( ∂∂ν + iκσ(a)) ∂

∂x2Γ on ∂D,

ws,0σ(a)(·, z) = − ∂

∂x2(Γ) on ∂Ω,

then it is clear that G0σ(a) := ws,0

σ(a)(x, y) + ∂∂x2

Γ(x, z) satisfies

(3.18)

∆(G0σ(a))(x, z) = − ∂

∂x2δ(x − z) in Ω \ D,

( ∂∂ν + iκσ(a))(G0

σ(a))(·, z) = 0 on ∂D,

(G0σ(a))(·, z) = 0 on ∂Ω.

We can assume that a = (0, 0) and ν(a) = (0, 1) by using the rigid transformation of coordinatesT := Ra(ν(a)) + Ma with which (3.18) needs to be replaced by:

(3.19)

∆(G0σ(a)oT

T )(x, z) = −∇δ(x − z) · τ2 in Ω \ D,

( ∂∂ν + iκσ(a))(G0

σ(a)oTT )(·, z) = 0 on ∂D,

(G0σ(a)oT

T )(·, z) = 0 on ∂Ω,

where τ2 := Ra

[

01

]

= (−ν1(a), ν2(a)).

Let ξ = F (x) be the local change of variables

(3.20) ξ1 = x1, ξ2 = x2 − f(x1),

where f is the function defined in the introduction. We have the following properties:

(3.21)

c1|x − z| ≤ |F (x) − F (z)| ≤ c2|x − z|,|F (x) − x| ≤ c3|x|2,|DF (x) − I| ≤ c4|x|

11

for x, z near the point a, where ci(i = 1, 2, 3, 4) are positive constants, which is due to the hypothesison the regularity of ∂D.

Let x, z be points near a. From (3.18), we deduce that G0σ(a)(ξ, η) = G0

σ(a)oTT (x, z) satisfies:

(3.22)

∇ξ · B(ξ)∇ξG0σ(a) = −JT (ξ)∇ξδ(ξ − η) · τ2 near F (a),

|J−T ν|B(ξ)∇ξG0σ(a) · ν + iκσ(a)G0

σ(a) = 0 on ∂R2+ near F (a),

where ξ := F (x), η := F (z), B := JJT , J := ∂ξ∂x (F−1(ξ)) and ν := (0, 1) is the unit normal to

∂R2+. We denoted by J−T the adjoint of J−1. We have from (3.21) that

|JT (ξ) − JT (0)| ≤ c|ξ|, |B(ξ) − B(0)| ≤ c|ξ|

and J(0) = B(0) = I.We set Γσ(a)(x, z) := (w+

σ(a)oTT + ∂

∂x2ΓoTT )(x, z) = (w+

σ(a)oTT + ∇Γ · τ2)(x, z) and write

R(ξ, η) := G0σ(a)(ξ, η) − Γσ(a)(ξ, η). Then the function R(·, η) satisfies

∇ξ · B(ξ)∇ξR = ∇ξ · (I − B)∇ξω+σ(a)oT

T ,

B(ξ)∇ξR · ν + iκσ(a)R = (I − B)∇ξΓσ(a) · ν + iκσ(a)(1 − |J−T ν|−1)G0σ(a),

(3.23)

where the first relation holds in R2+ near F (a), while the second one is satisfied on ∂R

2+ near F (a).

Let G be the Neumann Green’s function associated to the expression ∇ξ ·B(ξ)∇ξ on the circleB(0, r) of radius r.

We set B+r := B(0, r) ∩ R

2+ and write ∂B+

r = Sr ∪ Scr with Sr := ∂B+

r ∩ ∂F (D).Integrating by parts in (3.23), we obtain:

R(ξ, η) =

∫

B+r

(I − B)∇G(z, ξ) · ∇ω+σ(a)(z, η)dz +

∫

Sr

(I − B)∇ω+σ(a) · νGds(z)

−iκσ(a)

∫

Sr

R(z, η)G(z, ξ)ds(z) + iκσ(a)

∫

Sr

|(1 − J−T ν)|Gσ(a)(z, η)G(z, ξ)ds(z)

(3.24) −∫

Scr

(I − B)∇ωσ(a) · νGds(z) +

∫

Scr

B∇R · νGds(z).

Similar arguments as in ([17], [15]) show that R(ξ, η) = O(ln |ξ − η|). Now, we are going to provethat actually R(ξ, η) = O(1). We start by the asymptotic:

(3.25) G(z, ξ) = Γ(z, ξ) + Γ(z, ξ∗) + O(1)

(3.26)∂

∂zjG(z, ξ) =

∂

∂zjΓ(z, ξ) +

∂

∂zjΓ(z, ξ∗) + O(ln(z − ξ))

which can also be obtained as in [17], where ξ∗ = (ξ1,−ξ2).Hence the third and the forth terms of (3.24) are bounded for ξ and η near the point a. Since a isaway from Sc

r then the fifth and the sixth terms are also bounded.The next step is to prove that

(3.27)

∫

B+r

(I − B)∇G(z, ξ) · ∇ω+σ(a)(z, η)dz = O(1)

12

(3.28)

∫

Sr

(I − B)∇ω+σ(a)(z, η) · νG(z, ξ)dz = O(1)

for ξ = η with η ∈ CF (a),θ.For this, we use the explicit form of B, i.e.

B(z) :=

[

1 −f ′(z1)−f ′(z1) 1 + (f ′)2(z1)

]

to write

B(z) − B(η) =

[

0 −f ′′(η1)(z1 − η1)−f ′′(η1)(z1 − η1) 2f ′(η1)f

′′(η1)(z1 − η1)

]

+ O(z1 − η1)2

and then

B(z) − B(η) =

[

0 −1−1 0

]

f ′′(η1)(z1 − η1) + O(η1)O(z1 − η1) + O(z1 − η1)2

Hence, we obtain:∫

B+r

(B(η) − B(z))∇G(z, ξ) · ∇Γσ(a)(z, η)dz =

(3.29) f ′′(η1)

∫

B+r

(z1−η1)[∂z2G(z, ξ)∂z1

(∂z2Γσ(a))(z, η)+∂z1

G(z, ξ)∂z2(∂z2

Γσ(a))(z, η)]dz +O(1).

From Proposition 3.2, we get the estimate:

ω+σ(a)(z, η) = ∇Γ(z, η∗) · τ2 + O((ln |z − η|)).

From this property and (3.25), it is enough to compute the integrals of (3.29) for Γ(z, ξ) and∂zj

Γ(z, η∗). Hence we have

∫

B+r

(B(η) − B(z))∇G(z, ξ) · ∇ω+σ(a)(z, η)dz =

2f ′′(η1)

∫

B+r

(z1 − η1)[∂

∂z2

Γ(z, ξ)∂

∂z1

∇Γ)(z, η∗) · τ2 +∂

∂ z1

Γ(z, ξ)∂

∂ z2

(∇Γ)(z, η∗) · τ2]dz + O(1)

= 2f ′′(η1)[−ν1(a)

∫

B+r

(z1 − η1)[∂

∂z2

Γ(z, ξ)∂

∂z1

(∂

∂z1

Γ)(z, η∗) +∂

∂z1

Γ(z, ξ)∂

∂z2

(∂

∂z1

Γ)(z, η∗)]dz+

(3.30)

2f ′′(η1)[ν2(a)

∫

B+r

(z1 − η1)[∂

∂z2

Γ(z, ξ)∂

∂z1

(∂

∂z2

Γ)(z, η∗) +∂

∂z1

Γ(z, ξ)∂

∂z2

(∂

∂z2

Γ)(z, η∗)]dz + O(1)

Let us compute the second integral of (3.30). We have:

∂z2Γ(z, ξ)∂z1

(∂z2Γ(z, η∗)) + ∂z1

Γ(z, ξ)∂z2(∂z2

Γ(z, η∗)) =

= − 1

4π2[[−2

(z − ξ∗) · (0, 1)

|z − ξ|2 [(z − η∗) · (0, 1)(z − η∗) · (1, 0)

|z − η∗|4 ]+

(3.31)(z − ξ) · (1, 0)

|z − ξ|2 [1

|z − η∗|2 − 2[(z − η∗) · (0, 1)]2

|z − η∗|4 ]]

13

which we write as

∂z2Γ0(z, ξ)∂z1

(∂z2Γ0(z, η∗)) + ∂z1

Γ0(z, ξ)∂z2(∂z2

Γ0(z, η∗)) = − 1

4π2[−I + II − III]

where

I := 2(z2 − ξ2)(z2 + η2)(z1 − η1)

|z − ξ|2|z − η∗|4 , II :=(z1 − ξ1)

|z − ξ|2|z − η∗|2and

III := 2(z2 + η2)

2(z1 − ξ1)

|z − ξ|2|z − η∗|4In the following, we estimate the integrals for I, II and III.

Estimating the integral of I

Let us estimate∫

B+r

2(z1 − η1)(z2 − ξ2)(z2 + η2)(z1 − η1)

|z − ξ|2|z − η∗|4 dz.

Let r1 > 0 and r2 > 0 be such that (−r1, r1) × (0, r2) ⊂ B+r . Since we are interested in ξ ∈ Sr

and η ∈ CF (a),θ, then∫

B+r

2(z1 − η1)(z2 − ξ2)(z2 + η2)(z1 − η1)

|z − ξ∗|2|z − η∗|4 dz

=

∫ r2

0

∫ r1

−r1

2(z1 − η1)(z2 − ξ2)(z2 + η2)(z1 − η1)

|z − ξ∗|2|z − η∗|4 dz1dz2 + O(1).

Hence∫

B+r

(z1 − η1)(z2 − ξ2)(z2 + η2)(z1 − η1)

|z − ξ|2|z − η∗|4 dz

=

∫ r2

0

(z2 − ξ2)(z2 + η2)[

∫ r1

−r1

(z1 − η1)2

|z − ξ|2|z − η∗|4 dz1]dz2 + 0(1).

However∫ r1

−r1

(z1 − η1)2

|z − ξ|2|z − η∗|4 dz1 =

∫ r1

−r1

(z1 − η1)2|z − η∗|2

|z − ξ|2|z − η∗|6 dz1

=

∫ r1

−r1

(z1 − η1)2|z − ξ|2

|z − ξ|2|z − η∗|6 dz1 +

∫ r1

−r1

(z1 − η1)2[|ξ − η∗|2 + 2(ξ − η∗) · (z − ξ)]

|z − ξ|2|z − η∗|6 dz1

Remark that

|∫ r2

0

∫ r1

−r1

(z2 − ξ2)(z2 + η2)(z1 − η1)2[|ξ − η∗|2 + 2(ξ − η∗) · (z − ξ)]

|z − ξ|2|z − η∗|6 dz1dz2| ≤

∫ r2

0

∫ r1

−r1

|ξ − η ∗ |2|z − ξ||z − η∗|3 +

|ξ − η ∗ ||z − η∗|3 dz1dz2 = O(

|ξ − η∗|2η22

) + O(|ξ − η∗|

η2),

which is bounded for ξ = η.Now,

∫ r1

−r1

(z1 − η1)2

|z − η∗|6 dz1 =

∫ r1

−r1

1

|z − η∗|4 dz1 − (z2 + η2)2

∫ r1

−r1

1

|z − η∗|6 dz1

First we have after a change of variables

∫ r1

−r1

1

|z − η∗|4 dz1 =

∫ arctanr1−η1|z2+η2|

arctan−r1−η1|z2+η2|

(cos θ)2

|z2 + η2|3dθ.

14

Similarly∫ r1

−r1

1

|z − η∗|6 dz1 =

∫ arctanr1−η1|z2+η2|

arctan−r1−η1|z2+η2|

(cos θ)4

|z2 + η2|3dθ.

We use the formula (cos θ)4 = (cosθ)2 − 18 + 1

8cos(4θ). Then we have:

∫ r1

−r1

(z1 − η1)2

|z − η∗|6 dz1 =1

8|z2 + η2|3[[arctan

r1 − η1

|z2 + η2|− arctan

−r1 − η1

|z2 + η2|]−

1

4[sin 4(arctan

r1 − η1

|z2 + η2|) − sin 4(arctan

−r1 − η1

|z2 + η2|)]].

Then∫

B+r

(z1 − η1)(z2 + ξ2)(z2 + η2)(z1 − η1)

|z − ξ∗|2|z − η∗|4 dz =

∫ r2

0

1

8|z2 + η2|[arctan

r1 − η1

|z2 + η2|− arctan

−r1 − η1

|z2 + η2|]dz2−

∫ r2

0

1

32|z2 + η2|[[sin 4(arctan

r1 − η1

|z2 + η2|) − sin 4(arctan

−r1 − η1

|z2 + η2|)]]dz2 + O(1).

Hence∫

B+r

(z1 − η1)(z2 − ξ2)(z2 + η2)(z1 − η1)

|z − ξ|2|z − η∗|4 dz = −π

8ln(η2) + O(1).

Similar computations give

∫

B+r

(z2 − ξ2)(z2 + η2)(z1 − η1)2

|z − ξ|2|z − η∗|4 dz = −π

8ln(η2) + O(1)

and∫

B+r

(z1 − ξ1)(z1 − η1)

|z − ξ∗||z − η∗|2 dz = −π

2ln(η2) + O(1).

Then, replacing these values in (3.30) by using (3.31), we obtain:

∫

B+r

(B(z) − B(η))∇G(z, ξ) · ∇w+σ(a)(z, η)dz = O(1)

for ξ = η with η ∈ CF (a),θ. Now,

|∫

B+r

(I − B(η))∇G(z, ξ) · ∇ω+σ(a)(z, η∗)dz| ≤ C|η|

∫

B+r

1

|z − ξ|1

|z − η∗|2 = O(1).

for ξ and η near 0. We conclude then that

|∫

B+r

(I − B(z))∇G(z, ξ) · ∇ω+σ(a)(z, η∗)dz| ≤ C|η|

∫

B+r

1

|z − ξ|1

|z − η∗|2 = O(1),

for ξ, η ∈ CF (a),θ and ξ = η.With similar computations, the second integral of (3.24) is also bounded for ξ, η ∈ CF (a),θ such

that ξ = η.Gathering these estimates, we have R(ξ, η) = O(1) for ξ = η, η ∈ CF (a),θ.

15

We go back to R(x, z) := G0σ(a)(x, z) − Γσ(a)(x, z) and we write it as

R(x, z) = G0σ(a)(x, z) − Γσ(a)(F (x), F (z)) + Γσ(a)(F (x), F (z)) − Γσ(a)(x, z),

and thenR(x, z) = R(F (x), F (z)) + [Γσ(a)(F (x), F (z)) − Γσ(a)(F (x), z)]+

(3.32) [Γσ(a)(F (x), z) − Γσ(a)(x, z)].

From previous computations, we have:

R(F (z), F (z)) = O(1). for z ∈ Ca,θ near a.

Arguing as in ([17], pages 835-836), we show that the second and the third terms in (3.32) arebounded.

Finally, going back to the original coordinates, we have:

R(z, z) = O(1) for z ∈ B(a, δ(a)) ∩ Ca,θ

for some δ(a) > 0.

4 Numerical tests

In our model problem we take the crack as a half semi-circle with the representation

(4.1) Γ = x : x = (x1(s), x2(s)) = 1.2 × (cos s, sin s), s ∈ [0, π].

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Γ

1.2

a

zp

1.2+l*δ0

δ1

O

Dap

∂ Dap

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−1

−0.5

0

0.5

1

1.5

2

2.5

Γ

Dap

1.2 2.0

a

zp

1.2−l*δ0

δ1

2.0

∂ Dap

O

Figure 1: Construction of two Dpa’s. zp can approach Γ from its convex side using Dp

a in theleft-hand side, while Dp

a in the right-hand side is used for zp approaching Γ from its concave side.When δ1 + l × δ0 → 0 with l the approaching step at each direction, zp → a ∈ Γ± from both sidesof Γ along radius direction, respectively.

We check the reconstruction formula for the scattering problem (1.3) which is the most difficult.The other two problems (1.1) and (1.2) can be checked in a similar way. In order to detect thecrack Γ as well as its surface impedance σ±, we need to construct some domain Dp

a such thatΓ ∈ Dp

a and zp outside of Dpa can approach Γ from its two sides. For Γ given by (4.1), we construct

two kinds of Dpa shown in Figure 1. Notice that the parameter δ1 > 0 represents the singularities

16

of Φ(x, zp), ∂xiΦ(x, zp) for x ∈ ∂Dp

a near zp and l × δ0 > 0 is the distance between Γ and ∂Dpa.

The sum δ1 + l× δ0 determines the approximation accuracy |zp − a| for a ∈ Γ. On the other hand,it is easy to see that the domain Dp

a in the right-hand side can also be used for zp → a ∈ Γ fromthe convex part by translating Dp

a along x2 direction.We test our inversion method by showing the reconstruction effect for all unknown ingredients

in the model: the crack shape, crack type and surface impedance σ± in two sides of the crack. Wewill consider different configurations to show the validity of the method and reveal the physicalproperties behind the numerical behavior. In fact, we will see that the crack property in the convexside can be distinguished efficiently, while the crack property in the concave side is relatively difficultto be reconstructed numerically. This phenomena comes from the multiple reflection for scatteredwaves in the concave parts of crack and therefore more energy is absorbed.

Firstly, we use the blowing-up property of the indicator

(4.2) Loc(zp) := |Re(I1(zp))| + |Re(I2(zp))| as zp → Γ

to detect the location of crack due to (2.15). That is, when Loc(zp) is large enough, we considerzp to be almost on Γ.

Secondly, we use the following equivalent form of the reconstruction formulas (2.18) and (2.19)

limzp→a

π∑2

j=1 νj(a)Im(Ij(zp))

κ ln(|(zp − a) · ν(a)|) =

−σr+, if zp → a from Γ+

σr−, if zp → a from Γ−(4.3)

limzp→a

πP2

j=1 νj(a)Re(Ij(zp))− 14|(zp−a)·ν(a)|

κ ln(|(zp−a)·ν(a)|) = σi+, if zp → a from Γ+

limzp→a

πP2

j=1 νj(a)Re(Ij(zp))+ 14|(zp−a)·ν(a)|

κ ln(|(zp−a)·ν(a)|) = −σi−, if zp → a from Γ−

(4.4)

for the surface impedance reconstruction as well as for distinguishing Γ+ from Γ−. This lastproperty follows from (4.3) since σr is assumed to be positive. Hence if the left hand side ispositive then we are on the side Γ− and if not we are on the side Γ+.

Finally, the crack type is shown by considering the blowing-up property of the function

(4.5) Type(zp) :=|Im(I1(zp))| + |Im(I2(zp))|

| ln |(zl − a) · ν(a)||1/2as zp → Γ

using the formula (2.17). That is, Type(zp) should increase up to some value (theoretically ∞) forthe impedance crack.

In all the formulas (4.2)-(4.5), zp are taken along direction tj to approach the point a =R0(cos tj , sin tj) ∈ Γ for all tj ’s. In this way, the property of the crack is detected.

In our model problems we take the wave number κ = 1.2. The far-field pattern data for ourinversion are synthesized by solving the direct problem using the combined angular potential andsingle-layer potential developed in [12].

Example 1 We take the surface impedance as the complex functions of the forms

(4.6) κσ−(x) ≡ 1 + 1.5i, κσ+(x) ≡ 2 + 1.2i

and use incident plan waves along 64 directions distributed uniformly in [0, 2π].Let zp = z(j, l) approach to Γ from its convex side. By convex and concave side of crake, we

mean the upper side and lower side of Γ given by (4.1), respectively. The crack Γ is detectedfrom 33 directions tj = π/32 × j with j = 0, 1, · · · , 32. The radius for reconstruction at eachdirection tj is determined by the following way. For given blowing-up criterion CB, Let z(j, l) =

17

(δ1 + l × δ0)(cos tj , sin tj) for l = 16, 15, · · · , 1. Here we take δ1 = 0.01, δ0 = 0.02. For any fixedj = 0, 1, · · · , 32, compute the indicator value Loc(z(j, l)) defined by (4.2). If this value is largerthan CB the first time, we record this step l(j) and the value Cj := Loc(z(j, l(j))). Then we goto the next direction. Using this way, we get the data l(j), Cj32

j=0. Compute the average value

C :=∑32

j=0 Cj/33. Finally, we compare the value Cj and C and use this perturbation to correctthe radius R0 + l(j) × δ0 by linear interpolation as R0 + l(j) × δ0 − (Cj − C)/C × δ0. This is thefinal radius at direction tj .

Take the blow-up parameter CB = 0.8, 1.0, 1.2, 1.3. The reconstruction results for crack loca-tion are given in the left-hand side of Figure 4.2 using the above procedure. Using the techniquein [15], we can improve the reconstruction results by combining the reconstructions for differentblowing-up criteria together. That is, when the concave closure for the reconstructions with dif-ferent blowing-up values are taken, the crack will be detected from the convex side with a highaccuracy. The only a-priori information about the crack is that we know that zp is in the upper-sideof the crack. Notice, in our setting here σi

+ > 0. To use our reconstruction formula (2.17) moreefficiently, we expect that the reconstruction will be much improved for the case σi

+ < 0. However,we need to clarify the physical meaning of this condition. To our knowledge, all the reconstructionproblems for surface impedance up to now always consider the case σi

+ ≡ 0.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.5

0

0.5

1

1.5

2trueCB=0.8CB=1.0CB=1.2CB=1.3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.5

1

1.5

2

2.5

3

3.5

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Figure 2: Construction of Γ from the convex side of Γ (left). It can be seen that the tips are noteasy to identify with satisfactory accuracy. The crack type detection is shown in the right-handside. The blowing-up property for the boundary type is obvious, except at the tips.

The crack type detection are checked using (4.5) with the numerical performance given in theright-hand side of Figure 2. The blowing-up property are shown obviously, except at the tips ofthe crack. Notice, here we use the same singularity to identify the arc shape and the boundarytype. It can be shown that the increasing property is quite weak when zp → Γ along directionθ ≈ 1.57 ≈ π/2. This is reasonable since ν1(a) = 0 for a = 1.2(cos π

2 , sin π2 ). Therefore |ImI1(zp)|

in (4.5) is almost a constant as zp → a along the direction t = π/2 and then the numerator in (4.5)is relatively small, compared with those along other directions.

Now let us recover the boundary impedances σ±. We need to apply different singularity todetect its real part and imaginary part, respectively.

We take δ0 = 0.1, δ1 = 0.1 for recovering the imaginary part of σ+. The reconstructionresults for l = 7, 6, 5, 1 are shown in the left hand side of Figure 3. Noticing δ0 = 0.1 and theradius of crack is 1.2. For recovering real part of σ+, we need a strong singularity. Here wetake δ0 = 0.01, δ1 = 0.003. The reconstruction for l = 20, 10, 5, 2, 1 are given in the right handside of Figure 3. Notice that the numerical performance is not monotone with respect to l forboth real part and imaginary part. Although there are some oscillations for real part of σ+, the

18

0 0.5 1 1.5 2 2.5 3 3.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

truel=7l=6l=5l=1

0 0.5 1 1.5 2 2.5 3 3.5−3

−2.5

−2

−1.5

−1

−0.5

0

truel=20l=10l=5l=2l=1

Figure 3: Construction σ+ from the convex side of Γ: Imaginary part κσi+ (left) and real part

−κσr+ (right).

reconstruction is satisfactory.

Next we will check the reconstruction of σ−. Since σ− is defined in the concave part of Γ,we use Dp

a shown in the right hand side of Figure 1. For recovering its imaginary part, we takeδ0 = 0.045, δ1 = 0.010. The results for l = 6, 4, 2, 1 are shown in the left-hand side of Figure 4.

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5truel=6l=4l=2l=1

0 0.5 1 1.5 2 2.5 3 3.5−10

−8

−6

−4

−2

0

2

truel=20l=6l=3l=1

Figure 4: Construction σ− from the concave side of Γ: Imaginary part κσi− (left) and real part

−κσr− (right). The imaginary part is reconstructed well. For real part, we can only get the

distribution behavior, rather than the exact value.

Huge jumps appear at two points j = 1, 2. To show thess huge jumps, we list the values nearthe two tips as follows.

Tab.4.1 Numerical behavior of reconstructing σ− near the tips.

j = 1 j = 2 j = 3 j = 30 j = 31l = 6 10.61619 41.76799 1.320316 0.3975598 2.470905l = 4 6.977529 26.65242 1.012647 0.4274354 1.673494l = 2 4.947188 17.07493 1.112924 0.7361127 1.462406l = 1 4.462630 13.42291 1.539952 1.256604 1.776302

For recovering the real part of σ−, we take δ0 = 0.002, δ1 = 0.001, a stronger singularity. Theresults for l = 10, 6, 3, 1 are shown in Figure 4 (right). It can be seen that distribution behavior

19

of σr− is well detected, but the exact value can not be reconstructed well. This phenomena for

recovering σr− can also be shown from the other sets of (δ0, δ1) with

δ0 = δ1 = 10−i, i = 1, 2, 3, 4.

The results as well as its refinement are given in Figure 5.

0 0.5 1 1.5 2 2.5 3 3.5−45

−40

−35

−30

−25

−20

−15

−10

−5

0

5

true(δ

0,δ

1)=(1e−1,1e−1)

(δ0,δ

1)=(1e−2,1e−2)

(δ0,δ

1)=(1e−3,1e−3)

(δ0,δ

1)=(1e−4,1e−4)

0 0.5 1 1.5 2 2.5 3 3.5

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

true(δ

0,δ

1)=(1e−1,1e−1)

(δ0,δ

1)=(1e−2,1e−2)

(δ0,δ

1)=(1e−3,1e−3)

(δ0,δ

1)=(1e−4,1e−4)

Figure 5: Indicator value for real part of σr− for different sets of (δ0, δ1) (left), the right-hand side is

its refinement. It can be seen that σr− can not be recovered numerically. However, the distribution

behavior of σr− in the interior part of crack is well detected.

−2 −1 0 1 2−0.2

0

0.2

0.4

0.6

0.8

1

1.2

−2 −1 0 1 2−0.2

0

0.2

0.4

0.6

0.8

1

1.2

−2 −1 0 1 2−0.2

0

0.2

0.4

0.6

0.8

1

1.2

−2 −1 0 1 2−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Figure 6: Reconstruction of crack from concave side using the singularity δ0 = 0.02, δ1 = 0.001,where the blowing-up criteria are token as CB = 0.2, 0.4, 0.9, 1.0.

The physical background behind this numerical uncertainty for recovering σr− is the multiple

reflection of scattered wave in the cavity of the crack Γ in its concave side. Near these concavepoints, the incident wave will be multi-reflected. For our impedance crack with the energy absorbingcoefficient σr

− in the concave side, the energy of scattered wave is decreased by each reflection.Therefore the information about the concave side of the crack contained in the far-field pattern isalso relatively decreased. We observe also that the multiple reflection of the concave side of thewhole crack for a given σr

− has the same effects as the other non-concave part of the crack with ahigher σr

− distribution. Such an energy absorbing phenomena is also studied by engineering, see[8] and the references therein. Due to these reasons, it is understandable that we can not expecttoo much about the shape reconstruction from the concave side of the crack. Two reconstructionresults using formula (4.2) using zp → a ∈ Γ from the concave side with δ0 = 0.02, δ1 = 0.001 and

20

−2 −1 0 1 2−0.2

0

0.2

0.4

0.6

0.8

1

1.2

−2 −1 0 1 2−0.2

0

0.2

0.4

0.6

0.8

1

1.2

−2 −1 0 1 2−0.2

0

0.2

0.4

0.6

0.8

1

1.2

−2 −1 0 1 2−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Figure 7: Reconstruction of crack from concave side using δ0 = 0.02, δ1 = 0.01, where the blowing-up criteria are token as CB = 0.2, 0.4, 0.6, 0.8.

δ0 = 0.02, δ1 = 0.01 are shown in the following Figure 6 and Figure 7, respectively. In these twofigures, we also take l = 16, · · · , 1.

It can be seen that the parts near the two tips can not be identified in the same way as theinterior points of the crack. Actually, there si more scattering on these tips than on the interiorpoints on the crack. Notice that the formulas given in section 2 are valid just on the points awayfrom the tips. Also the best approximation accuracy can not be improved. What we can expectis that more part of the crack will be visible using a larger CB but with a finite accuracy. In thisconfiguration, we can not get the blowing-up property of the indicator value numerically since forδ0 = 0.02 and large l, the distance between zp and the crack is still large, while the blowing-upproperty is established theoretically for l × δ0 + δ1 → 0.

Our next example is to consider the reconstruction problem where the surface impedance distri-bution is not constant in the surface, which shows the practical applicability of our reconstructionmethod.

−1 −0.5 0 0.5 10

0.5

1

1.5

2

−1 −0.5 0 0.5 10

0.5

1

1.5

2

Real part of κσ+ Imaginary part of κσ

+

Figure 8: The non-constant distribution of κσ+ with respect to variable x1: real part(left) andimaginary part (right). The half cycle represents the crack.

Example 2. The configuration is taken as follows:

(4.7) κσ+(x) = (cos2(x1 + x2) + 1.2) + i(sinx1 + x2

10+ 1.5), κσ−(x) = (x1 + 2) + ix2

for x = (x1, x2) ∈ Γ. The real parts of σ± are positive. The distribution of κσ+ are shown in

21

Figure 8. Since we can not expect satisfactory results for the information about Γ− as explainedin Example 1, here we focus on the reconstruction of σ+ as well as the shape detection from theconvex side of crack. We will reveal the effect of the variable surface impedance on the crack shapedetection.

We use singularities δ0 = 0.02, 0.01 to detect the crack shape. The reconstruction result isshown in Figure 9 (left). It can be seen that the reconstruction in domain A is relatively poor.This domain is the part where we have a large value of Re σ+, compare with Figure 8. The largevalue of σr

+ in this domain means a large energy absorbing of the scattered wave. So we can notdetect this part using the same singularity as that for the other part. Since σr

+ is relatively smallin the domain B, the the shape detection is well in this part.

As it is done in Example 1, we use the same singularity to detect the boundary type. Thenumerical behavior of (4.5) in this case is shown in the right-hand side of Figure 9. The verticalvariable is the detection direction tj = π/32× j for j = 0, 1, · · · , 32. It can be seen that absorbingproperty of σr

+ helps us to detect the boundary type obviously. That is, in domain A with largevalue of σr

+, the blowing-up property is obvious. This phenomena is consistent with the detectingformula (2.16).

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

trueCB=0.6CB=0.8CB=1.0CB=1.2

large σ+r

A B

small σ+r

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.5

1

1.5

2

2.5

3

3.5

0.05

0.1

0.15

0.2

0.25

0.3

0.35

A

B

Figure 9: Reconstruction of Γ using (δ0, δ1) = (0.02, 0.01) with CB = 0.6, 0.8, 1.0, 1.2 (left). Thecrack shape in domain A is not easy to detect due to the large σr

+ in this part, see the recovery forCB = 1.0, 1.2. In the right-hand side, the impedance type is revealed clearly. That is, the largerσr

+ is, the clearer the impedance type is.

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

truel=20l=4l=2l=1

0 0.5 1 1.5 2 2.5 3 3.5−3

−2.5

−2

−1.5

−1

−0.5

0

truel=20l=10l=5l=1

Figure 10: Construction σ+ from the convex side of Γ: Imaginary part κσi+ (left) and real part

−κσr+ (right).

22

Finally, we reconstruct σ+. To recover the imaginary part of σ+, we use the singularity δ0 =0.02, δ1 = 0.05. The results for l = 20, 4, 2, 1 are given in the left-hand side of Figure 10. Also we usethe singularity δ0 = 0.003, δ1 = 0.002 to detect the real part of σ+. The results for l = 20, 10, 5, 1are given in the right-hand side of Figure 10.

Example 3: This example is for showing the importance of introducting the artificial coefficientImσ± 6= 0 in the surface impedance. Keep other parameters in Example 1 unchanged, just replacethe impedance by

σ−(x) ≡ 1 + 2i, σ+(x) ≡ 2 + 5i.

Then the arc Γ is reconstructed in Figure 11 (left). The blow-up values are CB = 1.0, 1.2, 1.5, 1.7,where we take z(l) = l × 0.05 for l = 16, 15, · · · , 1 to approach a ∈ Γ with δ1 = 0.1. Thereconstruction for crack shape is much better than Example 1 due to the large imaginary part ofσ±. Notice, this picture shows that the points near (0, 1.2) can not be reconstructed well. Thereason is that the normal direction is ν = (0, 1), which means ν1 = 0. The figure in the right-handside of Figure 11 shows the boundary detection.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.5

0

0.5

1

1.5

2ExactCB=1.0CB=1.2CB=1.5CB=1.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.5

1

1.5

2

2.5

3

3.5

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Figure 11: Boundary shape reconstruction and boundary type using the same singularity. It canbe seen that the boundary shape is reconstructed well (left). Since σ± are large, the crack behaveslike a Dirichlet crack., i.e. Type(zp) ≈ 0. This phenomena is shown in the right-hand Figure.

Conclusions. In this paper, we consider an inverse scattering problem caused by an opencrack Γ. Compared with the inverse scattering problem caused by an impenetrable obstacle withsmooth closed boundary, the crack detection problems are much more complicated, due to the jointeffects of the tips of crack, the concave side of crack and the inhomogeneous surface impedancedistributions. We propose theoretical formulas to detect the property of the crack such as its shape,the boundary type and the surface impedance. The numerical realizations are presented, whichshow the validity of this method and also some difficulties arising in the detection of concave sideof crack. Such difficulties can be explained physically from the multiple reflection of waves in thecavity.

23

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