Detecting sound-hard cracks in isotropic inhomogeneities
This work is a contribution to sampling methods in inverse scattering theory when the issue is to determine the shape of an unknown background from xed frequency multi-static data. The Factorization Method (FM) and the Generalized Linear Sampling Method (GLSM), which are methods among the class of the sampling methods, have shown good results in solving this problem [83, 10]. The FM has been justi ed for di erent complex backgrounds, allowing one to implement it in practical applications such as geophysics or nondestructive testing. We mention for instance the papers [86, 15, 122] where backgrounds made of both impenetrable obstacles and inhomogeneous medium are considered. The GLSM implementation mainly requires two complementary factorizations of the far eld operator, one used in the Linear Sampling Method (LSM) and another used in the FM. As a consequence, the GLSM can be used as soon as the use of the FM is valid. Furthermore, the GLSM has provided the possibility to identify changes of the refractive index in a given inhomogeneity. The method is described by the Di erential Linear Sampling Method (DLSM)  which requires the knowledge of far eld measurements collected before and after the occurrence of the degradation. A natural research perspective is to adapt the DLSM for the identi cation of emergence of impenetrable defects in a surveyed material.
We consider the problem of detecting the presence of sound-hard cracks in a non homoge-neous reference medium from the measurement of multi-static far eld data. First, we provide a factorization of the far eld operator in order to implement the Generalized Linear Sampling Method (GLSM). The justi cation of the analysis is also based on the study of a special interior transmission problem. This technique allows us to recover the support of the inhomogeneity of the medium but fails to locate cracks. In a second step, we consider a medium with a multi-ply connected inhomogeneity assuming that we know the far eld data at one given frequency both before and after the appearance of cracks. Using the Di erential Linear Sampling Method (DLSM), we explain how to identify the component(s) of the inhomogeneity where cracks have emerged. The theoretical justi cation of the procedure relies on the comparison of the solutions of the corresponding interior transmission problems without and with cracks. Finally we illus-trate the GLSM and the DLSM providing numerical results in 2D. In particular, we show that our method is reliable for di erent scenarios simulating the appearance of cracks between two measurements campaigns.
The forward scattering problem
We consider an isotropic medium embedded in Rd, d = 2 or 3, containing sound-hard cracks. Following , a crack is de ned as a portion of a smooth nonintersecting curve ( d = 2) or surface (d = 3) that encloses a domain , such that its boundary @ is smooth. We assume that is an open set with respect to the induced topology on @ . The normal vector on is de ned as the outward normal vector to (see Fig. 6.1).
To de ne traces and normal derivatives of functions on we use the following notation for all x 2
f (x) = lim f(x h (x)) and @ f(x) = lim (x): r f(x h (x)):
We shall also work with the jump functions @ := @+f @ f:
[f] := f+ f and @f
We assume that the propagation of waves in time harmonic regime in the reference medium is governed by the Helmholtz equation u+k2u = 0 in Rd where stands for the Laplace operator of Rd and where k is the wave number. We assume that the cracks are embedded in a local perturbation of the reference medium. To model this perturbation, we introduce n 2 L1(Rd) a complex valued coe cient (the refractive index of the medium) such that n = 1 in Rd n D and n 6= 1 in D. Here D Rd is a bounded domain with Lipschitz boundary @D such that Rd n D is connected. We assume that =m(n) 0 in Rd and that D. The scattering of the incident plane wave ui( ; ) := eik x of direction of propagation 2 Sd 1 by the medium is described by the problem
Find u = ui + us such that
u + k2n u = 0 in Rd n
r!+1 r @ u = 0 on (2.1)
lim d 1 @us = 0;
with ui = ui( ; ). The last line of (2.1), where r = jxj, is the Sommerfeld radiation condition which selects the outgoing scattered eld and which is assumed to hold uniformly with respect to xb = x=jxj 2 Sd 1. For all k > 0, Problem (2.1) has a unique solution u belonging to H1(O n for all bounded domain O Rd. The scattered eld us( ; ) has the expansion us( ; x) = deikrr d 1 us1( ; x) + O(1=r) ; (2.2) as r ! +1, uniformly in x = x=jxj 2 Sd 1. In (2.2) the bconstant d is given by d = ei =p 8 k for d = 2 and by = 1=(4 ) for d = 3. The function us1( ; ) : Sd 1 ! C, is called the far eld pattern associated with uib( ; ). From the far eld pattern, we can de ne the far eld operator F : L2(Sd 1) ! L2(Sd 1) such that (F g)(x) = Z Sd 1 g( ) us1( ; x) ds( ): (2.3) b b
By linearity, the function F g corresponds to the far eld pattern of the scattered eld in (2.1) with
Z ui = vg := g( )eik x ds( ) (Herglotz wave function): (2.4) Sd 1
Factorization of the far eld operator
In this section we explain how to factorize the far eld operator F de ned in (2.3). From the Green representation theorem, computing the asymptotic behaviour of the Green’s function as r ! +1 gives
us1(x) = k2 ZD(n(y) 1)u(y)e ikxy dy + Z [u(y)]@+(y)e for the far eld pattern of us in (2.2). A rst step towards the factorization of F is to de ne the Herglotz operator H : L2(Sd 1) ! L2(D) L2( such that Hg = (vg ; @+vg ): (2.6) jD j
We give in Proposition 2.3.1 below a characterization of the closure of the range of H. Set = v L2(D) v + k2v = 0 in D : (2.7) H 2 2 2 j and de ne the map : H ! L (D) L ( such that v = (v jD ; @+v ): (2.8) j
Proposition 2.3.1. The operator H : L2(Sd 1) ! L2(D) L2( de ned in (2.6) is injective and R(H) = (H ).
Proof. The proof of the injectivity of H follows a classical argument based on the Jacobi Anger expansion (apply [29, Lemma 2.1]). To establish the second part of the claim, rst we note that vg (de ned in (2.4)) belongs to H so that R(H) (H ). On the other hand, classical results of interior regularity ensure the existence of some constant C > 0 such that k@ vkL2( CkvkL2(D) for all v 2 H . This in addition to k vkL2(Dn L2( kvkL2(D) allows one to show that (H ) is a closed subspace of L2(D) L2( The regularity result implies that : (H ; k kL2(D)) ! L2(D) L2( is continuous. Since the set of Herglotz wave functions is dense in (H ; k kL2(D)), we deduce that R(H) = (H ).
Next we de ne the operator G : ! L2(Sd 1) such that R(H) G(v; @+v) = us1; (2.9) where us1 is the far eld pattern of us, the outgoing scattered eld which satis es us + k2n us = k2(1 n)v in Rd n (2.10)
@ us = @ v on : Note that if (v; @+v) 2 R(H) then interior regularity implies @+v = @ v on We also de ne the map T : L2(D n L2( ! L2(D n L2( such that T (v; @+v) = (k2(n 1)(v + us); [v + us]): (2.11) Clearly we have F = GH. And one can check using (2.5) that G = H T so that F admits the factorization F =H TH: (2.12)
The justi cation of the techniques we propose below to recover the cracks will depend on the properties of the operators G, T . And the latter are related to the solvability of the so-called interior transmission problem which in our situation states as follows: given f 2 H3=2(@D); g 2
Find (u; v) 2 L2(D) L2(D) such that
u v 2 f’ 2 H1(D n j ’ 2 L2(D n g
u + k2n u = 0 in D n u v = f on @D (2.13)
v + k2 v = 0 in D @ u @ v = g on @D
@ u = 0 on :
We shall say that k > 0 is a transmission eigenvalue if (2.13) with f = g = 0 admits a non zero solution. One can show for example that if the coe cient n is real and satis es 1 < n < n < n for some constants n , n , then the set of transmission eigenvalues is discrete without accumulation point and that Problem (2.13) is uniquely solvable if and only if k is not a transmission eigenvalue (this will be part of a future work). We shall say that (2.13) is well-posed if it admits a unique solution for all f 2 H3=2(@D); g 2 H1=2(@D).
Proposition 2.3.2. Assume that k > 0 is not a transmission eigenvalue. Then the operator G : R(H) ! L2(Sd 1) is compact, injective with dense range.
Proof. First we show the injectivity of G. Let V = (v; @+v) 2 R(H) such that GV = 0. Then from the Rellich lemma, the solution us of (2.10) is zero in Rd n D. Therefore, if we de ne u = v + us, then the pair (u; v) satis es the interior transmission problem (2.13) with f = g = 0. Since we assumed that k > 0 is not a transmission eigenvalue, we deduce that v = 0 and so V = 0.
Now we focus our attention on the denseness of the range of G. First we establish an identity of symmetry. Let V1 = (v1; @+v1); V2 = (v2; @+v2) 2 R(H). Denote w1, w2 the corresponding solutions to Problem (2.10). In particular we have
w1 + k2nw1 = k2(1 n)v1; w2 + k2nw2 = k2(1 n)v2 in Rd n : (2.14)
Multiplying the rst equation by w2 and the second by w1, integrating by parts the di erence over BR, the open ball of radius R centered at O, we obtain k2 (n 1)(v1w2 v2w1) dx
Taking the limit as R ! +1 and using that limR!+1 @BR (@ w1w2 and w2 satisfy the radiation condition), we nd the identity
w1@ w2 ds(x) = 0 (w1
k2 Z D(n 1)v1w2 dx + Z @+v1[w2] ds(x) = k2 ZD(n 1)v2w1 dx + Z @+v2[w1] ds(x): (2.15)
Using (2.15), we deduce that for , g 2 L2(Sd 1), we have hG(H ); iL2(Sd 1)
= k2 ZD(n 1)(H + us( ))Hg dx + Z [H + us( )]@+(Hg) ds(x)s
= k2 ZD(n 1)(Hg + us(g))H dx + Z [Hg + us(g)]@+(H ) ds(x)
= hG(Hg); iL2(Sd 1):
Therefore if g 2 R(G)? then G(Hg) = 0. The injectivity of G and H imply that g = 0 which shows that G has dense range.
Finally, using again the estimate k@ vkL2( CkvkL2(D) for all v 2 H , results of interior regularity and the de nition of H (see (2.6)), one can check that H : L2(Sd 1) ! L2(D) L2 ( is compact. Since G = H T and T is continuous, we deduce that G : L2(D) L2( ! L2(Sd 1) is compact.
Proposition 2.3.3. For all V = (v; @+v) 2 R(H) , we have the energy identity =m (hT V; V iL2(Dn L2( ) = k2 ZD =m (n)jus + vj2 dx + kkGV kL22(Sd 1); (2.16)
where us denotes the solution of (2.10). As a consequence if =m(n) 0 a.e. in D and if k is not a transmission eigenvalue of (2.13), then T is injective.
Proof. Multiplying by the equation us +k2us = k2(n 1)(us +v) and integrating by parts us over the ball BR, we obtain k2 ZD(n 1)(us + v) dx = us (2.17) ZBR (jrusj2 k2jusj2) dx + Z@BR @ us ds(x) Z @+us[ ] ds(x): us us Using (2.17), then we nd hT V; V iL2(Dn L2( = k2 ZD(n 1)jus + vj2 dx ZBR (jrusj2 k2jusj2) dx + Z [v + us]@+ ds(x) Z @+us[ ] ds(x) + Z@BR @ us ds(x): v us us
Since @+us = @+v and [v] = 0 (interior regularity) on we deduce hT V; V iL2(Dn L2( = k2 ZD(n 1)jus + vj2 dx ZBR (jrusj2 k2jusj2) dx (2.18) 2<e Z [us]@+ ds(x) + Z@BR @ us ds(x): us us
The radiation condition (see (2.1)) implies limR!1 @BR @ us
ds = ik 1 jus1j2d = ikkGV kL22(Sd 1).
For z 2 Rd, we denote by (:; z) the outgoing fundamental solution of the homogeneous Helmholtz equation such that i (1) (kjx zj) if d = 2 eikjx zj (x; z) = H0 and if d = 3: (2.19) 4 4 jx zj
Here H0(1) stands for the Hankel function of rst kind of order zero. The far eld of (:; z) is z(xb) = e ikz:xb. The GLSM uses the following theorem whose proof is classical .
Theorem 2.4.1. Assume that the interior transmission problem (2.13) is well-posed. Then z 2 D if and only if z 2 R(G): The particularity of the GLSM is to build an approximate solution (F g ’ z) to the far eld equation by minimizing the functional J ( z; 🙂 : L2(Sd 1) ! R de ned by J ( z; g) = hF ]g; giL2(Sd 1) + kF g zkL22(Sd 1); 8g 2 L2(Sd 1); (2.20)
where F ] := j12 (F + F )j + j21i (F F )j.
Theorem 2.4.2 (GLSM). Assume that the interior transmission problem (2.13) is well-posed, that the index n satis es [ =m(n) 0; <e(n 1) n a.e. in D ] or [ =m(n) 0; <e(1 n) n a.e. in D ] for some constant n > 0. Let gz 2 L2(Sd 1) be a minimizing sequence of J ( z; 🙂 such that
J inf J ( ; g) + p( ); (2.21)
( z; gz ) g z
where p is such that lim 1p( ) = 0. Then
z 2 D !0 0h z z iL 1
if and only if (S ) < + .
lim F ]g ; g 2 d 1
If z 2 D then there exists h 2 R(H) such that z = Gh and Hgz converges strongly to h as ! 0.
Thus the GLSM, justi ed by this theorem, o ers a way to recover D, that is to identify the perturbation in the reference background. Note that the GLSM, contrary to the LSM, provides an exact characterization of D. However it does not give any information on the location of the crack
Proof. We establish this theorem by applying the abstract result of [29, Theorem 2.10]. The latter requires that the following properties hold.
i) F = GH = H T H is injective with dense range and G is compact.
ii) F ] factorizes as F ] = H T ]H where T ] satis es the coercivity property
jhT ]V; V iL2(Dn L2( j kV kL22(Dn L2( ; 9 > 0; 8V 2 R(H); (2.22)
iii) V 7! Tjh]V; V iL2(Dn L2( j1=2 is uniformly convex on R(H) .
Item i) is a consequence of Propositions 2.3.1, 2.3.2 and 2.3.3. Moreover, we deduce iii) from ii) and from the fact that hF ]g; giL2(Dn L2( = k(F ])1=2gk2L2(Sd 1) (see e.g. ). Therefore, it remains to show ii). To proceed, we use [29, Theorem 2.31] which guarantees that it is true if :
T injective on R(H); =m(hT V; V iL2(Dn L2( ) 0 for all V 2 R(H); <e(T ) decomposes as <e(T ) = T0 + C where T0 satis es (2.22) and where C is compact on R(H).
The rst two items have been proved in Proposition 2.3.3. Let us focus our attention on the last one. By de nition, we have T V = (k 2 ~ 2 (n 1)us; [v + (n 1)(v + us); [v + us]). Set CV = (k + ~ us] @ vj ). Using results of interior regularity, one can check that C = <e(C) is compact.
Now, de ne T0 := <e(T ) C = (k2<e(n 1)v; @+vj ). Clearly one has jhT0V; V iL2(Dn L2( j n kV kL22(Dn L2( when <e(n 1) n . The case <e(1 n) n can be dealt in a similar way.
When one has only acces to a noisy version F of F , then F ]; might not have the required factorization and the cost function (2.20) must be regularized. For this aspect, we refer the reader to [10, Section 5.2].
Table of contents :
State of the art
Outline of the thesis
Etat de l’art
Plan de la these
1 Sampling methods
1.2 The basics of acoustic scattering theory
1.3 An overwiew of some sampling methods
1.3.1 The linear sampling method
1.3.2 The factorization method
1.3.3 The generalized linear sampling method
1.4 Explicit computations in a simple case
1.4.1 Expression of the scattered eld for a disk
1.4.2 Spectral properties of the far eld operator
1.4.3 Study of the sampling methods
1.4.4 The interior transmission problem
2 Detecting sound-hard cracks in isotropic inhomogeneities
2.2 The forward scattering problem
2.3 Factorization of the far eld operator
2.4 Reconstruction algorithms
2.5 Numerical results
2.5.1 Reconstruction of the background
2.5.2 Identication of sound hard crack defects in inhomogeneities
3 The interior transmission problem for penetrable obstacles with sound-hard cracks
3.2 Setting of the problem
3.3 Denition and properties of the interior transmission problem
3.3.1 A weak formulation
3.3.2 The Fredholm property
3.4 Transmission eigenvalues
3.4.1 Faber-Krahn inequalities for transmission eigenvalues
3.4.2 Discreteness of transmission eigenvalues
3.4.3 Existence of transmission eigenvalues
4 Detection of sound-hard obstacles in inhomogeneous media
4.2 The forward scattering problem
4.3 The far eld operator
4.3.1 Factorization of the far eld operator
4.3.2 Properties of the involved operators
4.4 Inversion algorithms
4.4.1 Reconstruction of the background
4.4.2 Identication of sound hard defects in inhomogeneities
4.5 Numerical results
5 The interior transmission problem for inhomogeneities with sound hard inclusions
5.2 Setting of the problem and notations
5.3 The fourth order equation approach
5.3.1 Reformulation of the problem
5.3.2 Variational formulation
5.3.3 Fredholm property of the interior transmission problem
5.3.4 Faber-Krahn inequalities for transmission eigenvalues
5.3.5 Discreteness of transmission eigenvalues
5.3.6 On the existence of transmission eigenvalues
5.4 Discreteness of transmission eigenvalues via the Dirichlet-to-Neumann approach
5.4.1 Main analysis
5.4.2 Proof of the intermediate results
6 Local estimates of crack densities in crack networks
6.2 Setting of the problem
6.3 Quantication of crack density using relative transmission eigenvalues
6.3.1 The relative far eld operator for cracks embedded in free space
6.3.2 Relative transmission eigenvalues
6.3.3 Computation of the relative transmission eigenvalues from the data
6.3.4 Description of the inversion algorithm
6.3.5 Numerical validation of the algorithm
6.4 An alternative method using measurements at one xed frequency
6.4.1 Solution of the far eld equation
6.4.2 Comparison of two transmission problems revealing the presence of cracks
6.4.3 Numerical results and comparison with the multiple frequencies approach