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Inverse Problems 26 (2010) 085011 (19pp) doi:10.1088/0266-5611/26/8/085011

An application of the reciprocity gap functional toinverse mixed impedance problems in elasticity

Christodoulos E Athanasiadis1, David Natroshvili2,Vassilios Sevroglou3,4 and Ioannis G Stratis1

1 Department of Mathematics, National and Kapodistrian University of Athens,Panepistimioupolis, GR 157 84, Athens, Greece2 Department of Mathematics, Georgian Technical University, 0175 Tbilisi, Georgia3 Department of Statistics & Insurance Science, University of Piraeus, GR 185 34 Piraeus, Greece

E-mail: cathan@math.uoa.gr, natrosh@hotmail.com, bsevro@unipi.gr and istratis@math.uoa.gr

Received 18 November 2009, in final form 4 May 2010Published 2 July 2010Online at stacks.iop.org/IP/26/085011

AbstractIn this paper, a mixed impedance scattering problem for partially coatedobstacles is studied. We formulate the direct scattering problem for theNavier equation and the mathematical setting for the description of the inverseproblem. The scattered field satisfies mixed Dirichlet-impedance boundaryconditions on the smooth boundary of the scatterer. A uniqueness theorem forthe determination of the partially coated obstacle is proved. A method for thereconstruction of the shape of the obstacle is presented, for which no a prioriinformation of the physical parameters of the scattering obstacle is needed.The inverse problem is treated using ideas from the linear sampling methodand by the use of the reciprocity gap functional. Further, a characterization ofthe surface impedance is established.

1. Introduction

In this paper we treat an inverse scattering problem in two–dimensional linear elasticity. Weconsider a bounded domain (rigid obstacle) and assume that its (simply connected) boundary isdivided into two parts, a Dirichlet (rigid) one and a Robin (impedance) one. On the Dirichletpart of the boundary, the displacement vector is given, while on the Robin part, a specificcombination of the displacement vector and the surface stress vector is given. The latterpart is due to a coating on the Robin part of the boundary with a material of constant surfaceimpedance. As usual, impedance expresses (intensity× stiffness) in relation towave reflection,diffraction, etc, and measures the contrast between two media. A complete discussion aboutthe notion of impedance in elasticity can be found e.g. in [41]. Obstacles characterized byboundary conditions of the aforementioned type are often called partially coated, and there

4 Author to whom any correspondence should be addressed.

0266-5611/10/085011+19$30.00 © 2010 IOP Publishing Ltd Printed in the UK & the USA 1

Inverse Problems 26 (2010) 085011 C E Athanasiadis et al

is very active current research on inverse problems for elastic, acoustic and electromagneticmedia.

The applications on elastic materials or environments concerning coated obstacles arevery extensive. Some examples playing a very important role include the following. (i) Non-destructive tests for integrity of coating imperfections due to corrosion or thermal shock;notable examples are found in the several types of high-temperature tests developed byengine manufacturers to simulate the corrosive behaviour found in specific engines operatingunder specific conditions. Burner rigs of various designs have evolved at each manufacturerthat, through experience, can generate data for corrosion and thermal-shock resistance. Themeasurement of coating propertiesmust be viewed in the context of a coating/substrate system.Coating compositions andmicrostructures are complex and becomemore so during the serviceat high temperature in corrosive environments, see, e.g., [15]. (ii) Non-destructive evaluationfor isotropic or anisotropic materials: use of elastic waves to measure elastic properties as wellas flaws in solid specimens. Comparison with elastic behaviour of composites with coatedinclusions: the micromechanical approach and applications, see, e.g. [14]. (iii) Materialsscience and civil and mechanical engineering: (A) detection of partial coating: carbon fibrestone is produced by evenly applying a thin carbon fibre matrix coating to form a solidbond with a thin natural stone slab and by stabilizing it so thoroughly that hairline fracturescannot occur in the stone, see, e.g., [47]. (B) Test of tribological properties of thermallysprayed composite coating, using a high-velocity oxy-fuel thermally sprayed wear-resistantTiAl/Al2O3 coating on tool steel which is used for making dies for aluminium high-pressuredie casting, see, e.g., [36]. (C) Partial coating by construction: methods for producing asteel component integrally having a rubber portion without necessity to remove by a latheturning a film formed on the surface of a steel component by a chemical conversion treatmentand without deterioration of the water-resistant adhesive property and anti-corrosion propertyduring production steps due to omission of the chemical conversion treatment; the method forproducing a steel component integrally having a rubber portion includes the steps of subjectinga surface of a steel component to a chemical conversion treatment with iron phosphate as aground treatment, partially coating an adhesive on a portion to which a rubber portion is to bebonded and bonding the rubber portion via baking to the steel component, see, e.g., [48]. (D)Composites: coated nuclear fuel particles are bonded in a matrix, or where duplex cylindricalfibres are employed in fibre-reinforcement, or wherever chemical action induces a protectivefinish either intentionally or inevitably as when an oxide coating covers a metallic surface,see, e.g., [44]. (D) Alloys: fabrication of friction-reducing coatings onto different substrates,see, e.g., [24], [28]. (iv) Applications of engineering mechanics in medicine: stents, see, e.g.,[43]: a drug-eluting stent is coated with a drug that is designed to control the release of a druginto surrounding tissue. The intention of this time-release process is to slow down the growthof unwanted cells (restenosis) and allow the vessel to heal. Improving the biocompatibilityof stents using a phosphorylcholine coated stent as a form of biomimicry, see, e.g., [45].(v)Nanomechanics: the combination of a polymer and single-wall carbon nanotubes to formelectrically conductive composites with desirable rheological and mechanical properties, see,e.g., [33]. (vi) Sound produced by a source adjacent to a partially coated, finite elastic plate(scattering of bending waves at the edge of an unbaffled, thin elastic plate in the presence ofarbitrary fluid loading), see, e.g., [26].

Applications regarding coated obstacles in different settings, as in acoustics andelectromagnetics include e.g. the detection of an object in the earth from measurements of thetotal electric and magnetic field, the problem of a coated cable or pipe which is partially coatedby a dielectric, and many others (see [6]). Clearly, such boundary value problems describe(from a mathematical point of view) direct and inverse scattering problems [9]. The study

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Inverse Problems 26 (2010) 085011 C E Athanasiadis et al

of the solvability of direct mixed problems in elasticity has been performed by the presentauthors using a boundary integral equations approach in [4]. Based on [4] and [27, 31], weinvestigate here the corresponding inverse problem.

In recent years, several quantitative methods in inverse scattering theory have beenestablished. The linear sampling method (LSM) [10, 18], and the factorization method [29]approximate the boundary of an obstacle without using a priori information of the geometry orthe physical properties (boundary conditions) of the scatterer. Excellent sources for acousticsand electromagnetics, concerning these methods, are [11, 16, 21], whereas for elasticity see[3, 8, 13, 37–40] (see also references therein).

A related inverse problem is that of determining the impedance part, which is assumedto be inaccessible, of the boundary of a scatterer, by measurements on the complement of theboundary, which is assumed to be accessible, and on which either a Dirichlet or a Neumanncondition is imposed; this is a classical inverse problem for harmonic functions or for solutionsof uniformly elliptic partial differential equations, see e.g. [5] and references therein. Suchproblems arise in nondestructive testing in corrosion detection.

In this work we present a method for solving an elastic inverse problem without relyingon any a priori knowledge of the physical properties of the scatterer. The inverse problemwill be solved using ideas of the LSM and the use of the reciprocity gap functional (RGF)[2, 20, 23, 25, 39, 40]. The RGF has recently been employed in a variety of areas in inverseproblems, starting from the pioneering work of Calderon, [12]. The incident field we considerwill be given by a point source located outside the scattering object.

This paper is organized as follows. In section 2, we formulate the direct and inversescattering problems for partially coated obstacles. Uniqueness theorems concerning thepartially coated obstacle are proved in section 3. In particular, the properties of the RGFare proved and also a linear integral equation arising from the LSM via an application of theRGF and the fundamental solution is solved in section 4. In section 5 an inversion algorithmfor the determination of the scattering object is established. Finally, in section 6 an expressionof the surface impedance of the scatterer is established.

2. Formulation of the mixed direct and inverse scattering problems

In this section we formulate the scattering problem which is described by a mixed boundaryvalue problem. Assume that a domain D (rigid obstacle) is embedded in a piecewise-constantbackground medium � ⊂ R2. In addition, we assume that D is partially coated, and itsboundary �, which is assumed to be Lipschitz, is divided into two parts, the so-called Dirichletpart �D and the partially coated (impedance) part �I : � = �D ∪�I . The setDwill be referredto as the scatterer, and D = D ∪ �. We also consider a bounded domain B (a ‘box’) with thesmooth boundary ∂B such that � ⊂ B. Throughout this paper ν = ν(x) denotes the outwardunit normal vector at the point x ∈ �.

From the mathematical point of view a scattering problem is described by a boundaryvalue problem. Hence, for our case the mixed impedance boundary value problem is statedas follows. For a given incident point-source field at x = x0 ∈ B \ D find a solutionu = (u1, u2)

� ∈ H 1loc

([[ (� \ D) ∪ (B \ �) ] \ {x0}

] ∪ (R2 \ B)), such that

μ(x)�u(x) + (λ(x) + μ(x)) grad div u(x)

+ �(x) ω2u(x) = 0, x ∈ [(� \ D) ∪ (B \ �)] \ {x0} ∪ (R2 \ B), (1)

u(x) = 0, x ∈ �D, (2)

3

Inverse Problems 26 (2010) 085011 C E Athanasiadis et al

T u(x) + iω c u(x) = 0, x ∈ �I , (3)

limr→∞ usct(β)(x) = 0, β = p, s, (4)

limr→∞

√r

(∂usct(β)(x)

∂r− ikβ usct(β)(x)

)= 0, β = p, s, r := |x|, (5)

where

u(x) = uincx0(x) + usctx0

(x) (6)

with uincx0being the incident field due to the point source x0 ∈ B \D and usctx0

the correspondingscattered one. Here, and in what follows, by the ‘incident field’ we will understand a Green-type function which will be defined later explicitly.

Throughout this paper c will be considered to be a positive constant, while

λ(x) =⎧⎨⎩

λ0, x ∈ R2 \ B

λ1, x ∈ B \ �

λ2, x ∈ � \ D

μ(x) =⎧⎨⎩

μ0, x ∈ R2 \ B

μ1, x ∈ B \ �

μ2, x ∈ � \ D

(7)

and

�(x) =⎧⎨⎩

�0, x ∈ R2 \ B

�1, x ∈ B \ �

�2, x ∈ � \ D.

(8)

For the above piecewise-constant functions, we assume that

μj > 0, λj + 2μj > 0, �j > 0, j = 0, 1, 2. (9)

It is clear that on the discontinuity interfaces ∂� and ∂B we have to impose transmissionconditions, requiring the continuity of the displacement and stress vectors across theseinterfaces, i.e. the one-sided interior and exterior limits of the corresponding vectors oneach interface should be equal to each other:

{u}+ = {u}−, {T u}+ = {T u}− on ∂�, (10)

where {·}+ denotes the limit from �, while {·}− denotes the limit from B, and

{u}+ = {u}−, {T u}+ = {T u}−, on ∂B (11)

where, now, {·}+ denotes the limit from B, while {·}− denotes the limit from R2 \ B.Relations (2) and (3) constitute the mixed impedance boundary conditions on the

scatterer’s surface �. The Sommerfeld–Kupradze-type radiation conditions (4) and (5) areassumed to hold uniformly in x = x/r . In addition, the vector u is representable as a sum ofmetaharmonic vectors, the so-called longitudinal u(p) and transverse parts u(s) [30]:

u(x) = u(p) + u(s) with �u(p) + k2j,p u(p) = 0, �u(s) + k2j,s u(s) = 0,

kj,p = ω

√�j

λj + 2μj

, kj,s = ω

√�j

μj

, j = 0, 1, 2.(12)

Note that in (5) kβ = k0, β , β = p, s.We employ the usual notation for the differential operator �∗:

�∗ = [μδkj � + (λ + μ) ∂k ∂j ]2×2, (13)

4

Inverse Problems 26 (2010) 085011 C E Athanasiadis et al

with � and δkj being the Laplace operator and Kronecker delta, respectively, whereas∂j = ∂/∂xj denotes partial differentiation with respect to the variable xj.

In addition, the stress operatorT = [Tkj ]2×2, with outward normal unit vector ν = (ν1, ν2),is defined as follows:

Tkj = λ νk ∂j + μνj ∂k + μδkj ∂ν, k, j = 1, 2, (14)

∂ν being the normal derivative.

Remark. The solvability of the above direct scattering problem (mixed boundary-transmissionproblem) can be treated by the same boundary integral equation approach used in [4] forhomogeneous bodies: the uniqueness, existence and regularity results are exactly the same(for both smooth and Lipschitz boundaries and interfaces); since our aim in the present paperis the study of the corresponding inverse problem, we do not include this proof for the sake ofbrevity.

We now present the free-space Green’s dyadic G(x, y) of the Navier equation; it satisfiesthe following equation:

�∗G(x, y) + � ω2 G(x, y) = −I δ(x − y), x, y ∈ R2, (15)

and is given, [7, 32], by

G(x, y) = i

4

{1

μI H

(1)0 (ks |x − y|)

− 1

ω2∇x ⊗ ∇x

[H

(1)0 (kp |x − y|) − H

(1)0 (ks |x − y|)

]}, (16)

where H(1)0 (z) is the Hankel function of first kind and zero order, δ(x − y) represents the

Dirac measure concentrated at the point x and I is the 2× 2 identity matrix. Further, as usual,‘∇x’ denotes the action of the gradient operator with respect to the variable x, and ‘⊗’ is thejuxtaposition between two vectors (this gives a dyadic).

In order to avoid the dyadic nature of the fundamental solution, we consider S1 = {d ∈R2 : |d| = 1} as the unit circle in R2. If p ∈ S1 denotes the polarization of an elastic pointsource at any y0 ∈ R2, then uy0(x) = G(x, y0) · p ≡ G(x, y0; p), x ∈ R2 \ {y0}. In particularwe consider uy0(x) = p1 G(x, y0)·e1+p2 G(x, y0)·e2 with arbitraryp1, p2 ∈ R, p21 +p22 = 1,e1 = (1, 0) and e2 = (0, 1). Hence, uy0(x) is a column vector, a linear combination of columnsof the fundamental matrix G(x, y0) with the scalar coefficients p1 and p2. For p1 = 1 andp2 = 0 we have the first column of the fundamental matrix, while for p1 = 0 and p2 = 1 wehave the second column. Concerning the dyadic approach for scattering problems, we referto [42].

Using now asymptotic analysis for G(x, y0; p) we have

G(x, y0; p) = Gp∞(x, y0; p) x

eik0,pr

√r

+ Gs∞(x, y0; p) x⊥ e

ik0,s r

√r+O(r−3/2), (17)

for r → ∞, where x⊥ denotes the perpendicular unit vector to x. The far-field patternsG∞(·, y0; p) = (G

p∞(·, y0; p),Gs

∞(·, y0; p)) of this point source of the longitudinal andtransverse parts of G(x, y0; p) are given by

Gp∞(x, y0; p) = 1

λ0 + 2μ0

i + 1

4√

πk0,pe−ik0,p x·y0 x · p, (18)

5

Inverse Problems 26 (2010) 085011 C E Athanasiadis et al

Gs∞(x, y0; p) = 1

μ0

i + 1

4√

πk0,se−ik0,s x·y− x⊥ · p, (19)

respectively.Furthermore, an asymptotic formula for the surface traction of the fundamental solution

for x ∈ SR = {d ∈ R2 : |d| = R} (clearly x = ν(x), x ∈ SR, ∀R) is obtained in the form

TG(x, y0; p) = TGp∞(x, y0; p) x

eik0,pr

√r

+ TGs∞(x, y0; p) x⊥ eik0,s r√

r+O(r−3/2), x ∈ S1,

for r → ∞, where

TGp∞(x, y0; p) = i− 1

4

√k0,p

πe−ik0,p x·y0 x · p, (20)

TGs∞(x, y0; p) = i− 1

4

√k0,s

πe−ik0,s x·y0 x⊥ · p. (21)

Now in order to obtain the asymptotic form of the radiating solution usctx0of the Navier equation

(1), we exploit Betti’s formulae and through asymptotic analysis, with the aid of (17)–(21),we arrive at [38]

usctx0(x) = u∞

x0,p(x) x

eik0,pr

√r+u∞

x0,s(x) x⊥ eik0,s r√

r+ O(r−3/2) , r → ∞, (22)

uniformly with respect to x = xr

∈ S1. The coefficients of the terms eik0,β r

√r, β = p, s, are the

corresponding far-field patterns of usctx0defined on the unit circle S1 in R2 and are known as

the longitudinal and the transverse far-field patterns, respectively. The functions u∞x0,p

, u∞x0,s

are analytic functions on S1 and the pair (u∞x0,p

, u∞x0,s

) will be considered as an element of[L2(S1)]2.

We consider that our scatterer D is irradiated by an incident field uincx0due to a source

located at a point with the position vector x0 ∈ B \ (D ∪ ∂�), given, for p ∈ S1, by

uincx0(x) = G(x, x0; p) + sct(x, x0; p), x ∈ R2, x �= x0. (23)

As we can see the incident field is defined almost everywhere in R2, sct(x, x0; p) is thescattered field due to the piecewise-constant background medium inR2; it solves the followingboundary value problem:

�∗ sct(x, x0; p) + �(x) ω2 sct(x, x0; p) = 0, x ∈ R2 \ (∂B ∪ ∂�), (24){ sct(x, x0; p) +G(x, x0; p)

}− = { sct(x, x0; p) +G(x, x0; p)

}+, (25){

T sct(x, x0; p) + TG(x, x0; p)}− = {

T sct(x, x0; p) + TG(x, x0; p)}+

, (26){ sct(x, x0; p) +G(x, x0; p)

}− = { sct(x, x0; p) +G(x, x0; p)

}+, (27){

T sct(x, x0; p) + TG(x, x0; p)}− = {

T sct(x, x0; p) + TG(x, x0; p)}+

, (28)

with x ∈ ∂B for relations (25) and (26) and x ∈ ∂� for (27) and (28). sct(x, x0; p)

also satisfies the Kupradze radiation conditions at infinity, and is clearly, uniquely defined inR2 \ {∂B ∪ ∂�}.6

Inverse Problems 26 (2010) 085011 C E Athanasiadis et al

Let us note that the scattered field usctx0, as defined in (6), is due to the scattering of

the incident field uincx0by the partially coated obstacle D. Alternatively, the incident field

uincx0can be considered as the free space Green’s function G(x, x0; p), with x0 ∈ B \ D;

then the corresponding scattered field due to the background medium will be given by sct(x, x0; p)+usctx0

(x). Throughout this paper, we assume that the incident field is defined by(23).

In addition, uincx0(the incident field) and usctx0

(the scattered field) have the reciprocityproperties of the free space Green’s dyadic; hence the following reciprocity relations are true:

uincx0(x) = uincx (x0) and usctx0

(x) = usctx (x0), (29)

for x and x0 in B \ D.In what follows we need the following function spaces: let H 1(D), H 1

loc(R2 \ D) be the

classical Sobolev spaces and H 1/2(�) be their trace space. For a proper subset �0 ⊂ �, let

H 1/2(�0) := {u|�0 : u ∈ H 1/2(�)}, (30)

H 1/2(�0) := {u ∈ H 1/2(�) : supp u ⊆ �0}, (31)

H−1/2(�0) :=(H 1/2(�0)

)′ (the dual space of H 1/2(�0)

), (32)

H−1/2(�0) :=(H 1/2(�0)

)′ (the dual space of H 1/2(�0)

). (33)

We will also need corresponding to (1)–(11) the interior mixed impedance boundary valueproblem in D, which now reads: find a vector u ∈ H 1(D) such that

�∗u(x) + �2 ω2 u(x) = 0, x ∈ D, (34)

u(x) = f (x), x ∈ �D, (35)

T u(x) + iωc u(x) = h(x), x ∈ �I , (36)

where f ∈ H 1/2(�D), h ∈ H−1/2(�I ) and c > 0. Integral representations of solutions of(1)–(5) and (34)–(36) have been derived, and basic uniqueness, existence and regularity resultsfor these mixed impedance problems have been established as well (for details see [4]).

We next deal with the inverse scattering problem concerning obstacles situated ina piecewise homogeneous background medium. In particular, the inverse problemcorresponding to the direct scattering problem (1)–(11) consists (i) in the determination of theunknown boundary� of the scattererD, from the knowledge of the Cauchy data of the scatteredfield measured on the boundary of a domain � containing the object in its interior, and (ii) inthe determination of the surface impedance c. For such inverse problems, a combination ofthe LSM and the use of the so-called reciprocity gap functional (RGF) [2, 20] is needed, andin particular an inversion algorithm for reconstruction of partially coated obstacles from theknowledge of the scattered field will be established.

7

Inverse Problems 26 (2010) 085011 C E Athanasiadis et al

3. Uniqueness theorem for the inverse problem

In this section we establish that if for two scatterers the corresponding scattered fields, for afixed frequency, coincide on ∂� for all point sources x0 lying on any smooth closed curve inB \�, surroundingD, then the scatterers, the Dirichlet and impedance parts of their boundariesand the impedances on those parts of the boundaries coincide as well.

Given the scattered wave usctx0due to the incident field uincx0

given in (23), we want todetermine uniquely the partially coated obstacle D. We first need the following.

Proposition 1. Let A:= {uincx0(·)|∂� : x0 ∈ C}, where uincx0

(x) is the incident point-source fieldin (23) and C is any smooth curve in B \ �, surrounding D. If ω2 is not a Dirichlet eigenvaluefor −�∗ in �, then

spanA = [H 1/2(∂�)

]2 ; (37)

i.e. the restriction of A to the boundary ∂� is complete in[H 1/2(∂�)

]2.

This is the elastic analogue of a corresponding result in acoustics [11, 23]. The proof isomitted for the sake of brevity, since it can be performed with minor modifications as that of[14, lemma 4.4 ] or/and [ 26, lemma 3.1].

The following uniqueness result holds.

Theorem 1. Assume that D1 ⊂ � and D2 ⊂ � with boundaries �(1) and �(2), respectively,are two scattering partially coated obstacles embedded in an elastic piecewise-constantbackground medium in R2, with c1 > 0 and c2 > 0 being the corresponding surfaceimpedance constants. If the corresponding scattered fields coincide on ∂� for all pointsources x0 ∈ B \ � and one fixed frequency, then (i) D1 = D2, (ii) �

(1)D = �

(2)D and

�(1)I = �

(2)I , (iii) c1 = c2.

Proof. (i) Let D1 �= D2 and u(1), u(2) be the solutions corresponding to D1 and D2, for theproblem (1)–(11) with the scattered fields usctx0,1

and usctx0,2, respectively. Assume now that �

denotes the unbounded component of the complement of D1 ∪ D2. Clearly ∂� ⊂ �. Weassume without loss of generality that D1 � D2. Choose an arbitrary x∗ ∈ � such thatx∗ ∈ �(1) and x∗ /∈ D2. We can now choose a sequence xn ∈ � with xn → x∗ for n → ∞given by

xn = x∗ +1

nν(x∗), (38)

such that

infx∈D2

|x − xn| > 0, n = 1, 2, ..., (39)

with ν(x∗) being the outward normal vector to �(1) at x∗. Consider the solutions uxn,1 anduxn,2 of (1)–(5) for the partially coated obstaclesD1,D2, respectively, produced by the incidentpoint-source field uincxn

(·) having corresponding scattered fields usctxn,1and usctxn,2

. Taking intoaccount that, by proposition 1, there exists a sequence of linear combinations of uincxm

0(·) such

that for every ε > 0,∥∥∥∥∥uincxn(·) −

N(n, ε)∑m=1

am(n, ε) uincxm0(·)

∥∥∥∥∥H 1/2(∂�)

< ε, (40)

8

Inverse Problems 26 (2010) 085011 C E Athanasiadis et al

am ≡ am(n, ε) ∈ C, xm0 ∈ B \ �, we can consider the scattered fields usctxm

0 ,1 and usctxm0 ,2

generated by the incident field∑N(n, ε)

m=1 am uincxm0(·). An analogous relation of (40) for T uincxm

0(·)

holds, i.e. ∥∥∥∥∥T uincxn(·) −

N(n, ε)∑m=1

am(n, ε) T uincxm0(·)

∥∥∥∥∥H−1/2(∂�)

< C0 ε, (41)

where C0 is a constant independent of ε and n. Hence, via the assumption usctx0,1= usctx0,2

on∂� for every point source x0 ∈ B \ �, implies that the two scattered waves usctxm

0 ,1 and usctxm0 ,2

coincide outside D1 ∪ D2. The above approximation properties (40) and (41) together withthe uniqueness of the direct scattering problem as well as its well-posedness [4] yield that thescattered waves usctxn,1

and usctxn,2coincide in � and obviously the same result holds for T usctxn,1

and T usctxn,2. Finally, considering the boundary conditions

usctxn,j(x) = −uincxn

(x), x ∈ �(j)

D , (42)

T usctxn,j(x) + iω cj usctxn,j

(x)

= −T uincxn(x) − iω cj uincxn

(x), x ∈ �(j)

I , (43)

j = 1, 2, and letting n → ∞, i.e. xn → x∗ (recall here that x∗ ∈ �(1)D , x∗ /∈ D2), we arrive at a

contradiction since the scattered field usctxn,2corresponding toD2 is bounded while the scattered

one usctxn,1, corresponding to D1, is not. Hence, the assertion (i) of the theorem follows.

(ii) For the second part of the theorem, assume �(1)D �= �

(2)D and without loss of generality

that �(1)D ∩ �

(2)D �= �

(1)D . Recall that by (i) D1 = D2 := D and (thus) �(1) = �(2) := �. Let an

open arc �∗ ⊂ � be such that �∗ ⊂(�

(1)D ∩ �

(2)I

). Concerning the scattered field due to the

incident point source x0 ∈ B \ �, we have

usctx0(x) := usctx0,1

(x) = usctx0,2(x), x ∈ R2 \ D. (44)

It is easy to see that the following boundary conditions hold:{usctx0

(x)}−

�∗ = −uincx0(x) on �∗, (45)

{T usctx0

(x) + iω c2 usctx0(x)

}−�∗

= −T uincx0(x) − iω c2 uincx0

(x) on �∗, (46)

since the scattered field usctx0,j, j = 1, 2, satisfies the boundary conditions{

usctx0,j(x)

}−�∗ = −uincx0

(x), x ∈ �(j)

D , (47)

{T usctx0,j

(x) + iω cjusctx0,j

(x)}−

�∗

= −T uincx0(x) − iω cj uincx0

(x), x ∈ �(j)

I . (48)

Following similar arguments as those of the proof of the corresponding part of theorem 5.1 in[34, 35], we finally obtain that

usctx0(x) = −uincx0

(x), x ∈ � \ D. (49)

This leads to a contradiction, since usctx0(x) is regular for x = x0, while G(x, x0; p) at this

point is singular, resulting, via (23), to uincx0possessing a singularity at x = x0.

9

Inverse Problems 26 (2010) 085011 C E Athanasiadis et al

(iii) Now we prove that c1 = c2 for x ∈ � = �1 = �2 (these last equalities hold by (i)).Let us consider that u(1), u(2) are two solutions of the problem (1)–(11) forD1, D2 and c1, c2,respectively. Since D1 = D2 we have that u(1)(x) = u(2)(x) := u(x), x ∈ R2 \ D0, whereD0 = D ∪ {x0}. Therefore, u(1)(x) = u(2)(x) = u(x) and T u(1)(x) = T u(2)(x) = T u(x) forx ∈ �. Hence, subtracting the impedance boundary conditions for u(j), j = 1, 2, i.e.

T u(j)(x) + iω cj u(j)(x) = 0, x ∈ �(j)

I , (50)

and taking into account that, by (ii), �(1)I = �

(2)I = �I , we arrive at

(c1 − c2) u(x) = 0, x ∈ �I . (51)

Suppose now that u = 0 on an arc � ⊂ �I . By Holmgren’s theorem we arrive at acontradiction, and therefore c1 = c2.

�

4. The elastic reciprocity gap functional and its properties

In what follows, we will establish the basic ideas of finding approximations of the solutionto the inverse elastic scattering partially coated problem, the so-called LSM, which has beensuccessfully employed into a large variety of inverse acoustic and electromagnetic scatteringproblems [17]. In addition, for inverse elastic scattering theory the method has recently beenwidely exploited; theoretical results as well as numerical ones have been presented, and inparticular, various inverse boundary value problems concerning the shape reconstruction ofthe scatterer have been established, (see [1, 3], [13, 38, 40] and references therein). A newversion of the LSM based on the RGF which has been employed in a variety of other inverseproblems [2] will be established.

We now employ the notion of the RGF defined as

RGF(u, υ) =∫

∂�

(u · T υ − υ · T u) ds, (52)

where u and υ are the solutions of the Navier equation in� \D. Further, we will use the RGFin order to construct an integral equation and our aim is to find an approximate solution to theintegral equation

RGF(u, υ) = RGF(u, G(·, y0; p)), y0 ∈ �, (53)

for which a characterization of the scatterer D can be obtained. We continue our analysis byintroducing the set

N(�) := {υ ∈ H1(�), such that �∗υ(x) + �2 ω2υ(x) = 0, x ∈ �}, (54)

as well as the set

Y := {u ≡ u(·, x0) ∈ H1loc(R2 \ D0

)satisfying (1)–(11), x0 ∈ B \ �}.

(55)

Concerning now (52)we consideru ∈ Y, andυ ∈ N(�) to be an elasticHerglotzwavefunction[22, 39, 40]. A key result for elastic Herglotz functions (that finds important applications ininverse scattering) gives υ in the form [40]

υ ≡ υg(x) = ω− 12 e−

iπ4

∫S1

√k2,p gp(d) d eik2,p x·d ds(d)

+ ω− 12 e−

iπ4

∫S1

√k2,s gs(d) d⊥ eik2,s x·d ds(d). (56)

10

Inverse Problems 26 (2010) 085011 C E Athanasiadis et al

In particular, this relation shows that any elastic Herglotz function with the vector densityg = (gp, gs) ∈ [L2(S1)]2 can be represented as a superposition of plane waves over the unitcircle propagating in every direction. In (56), gp and gs are the longitudinal and transverseHerglotz kernels, respectively, with d ∈ S1 being the direction of propagation.

Taking into account that u ∈ Y, the elastic RGF can be considered as an integral operatordefined as

RGF : N(�) → L2(C), (57)

where C is any smooth curve in B \ �, with

RGF(υ)(x0) = RGF(u(·, x0), υ), x0 ∈ C. (58)

In particular, via (53), we are now seeking a solution g ∈ [L2(S1)]2 that satisfiesRGF

(u, υg

) = RGF(u(·, x0), G(·, y0; p)), (59)

where υg is a Herglotz wavefunction (see (56)), y0 ∈ � and u is a solution to (1)–(11)for x0 ∈ C . In general, a solution of (59) does not exist; however, we can always getan approximate solution: in order to find it, let y0 be a fixed point in D and denote byuy0 ∈ H 1(D) the unique solution of the following interior boundary value problem [4]:

�∗uy0(x) + �2 ω2uy0(x) = 0, x ∈ D, (60)

uy0(x) = −G(·, y0; p), x ∈ �D, (61)

T uy0(x) + iωc uy0(x) = − ( T G(·, y0; p) + iωc G(·, y0; p) ) , x ∈ �I . (62)

The above interior boundary value problem plays a central role for the determination of theapproximate shape of an obstacle, related to the LSM. For details concerning the connectionbetween interior boundary value problems and LSM the authors refer to [19, 37, 40]. Recallhere that the set of vector elastic Herglotz functions is dense in the set of solutions to theNavier equation in D, with respect to the H 1(D)-norm for any bounded smooth domain D. Inparticular we state the following property.

Theorem 2. Let D a subset of R2 be a bounded domain with the Lipschitz boundary �. IfR2 \ D is connected, then the set of Herglotz functions is dense in the set of the variationalsolutions of the Navier equation, with respect to the H 1(D)-norm.

The proof can be performed along the same lines of the corresponding result in [3].In what follows, we will study the inverse mixed boundary value problem described, and

will extend the ideas of [20] in 2D linear elasticity. In particular, a new version of the LSMbased on the RGF which has been employed in a variety of other inverse problems [2] will beestablished.

But first, let us see some properties of the RGF.

Theorem 3. The operator RGF : N(�) → L2(C) defined by

RGF(u, υ) =∫

∂�

(u · T υ − υ · T u) ds (63)

is injective, where υ ∈ N(�) and u ∈ Y.

11

Inverse Problems 26 (2010) 085011 C E Athanasiadis et al

Proof. Let υ be a solution of the Navier equation in � with υ ∈ H1(�), which satisfies(RGF)(υ)(x0) = RGF(u(·, x0), υ) = 0 for all x0 ∈ C. Then from Betti’s integral theorem,the value of the integral

∫∂�

(u · T υ − υ · T u) ds remains the same when we replace ∂� bythe boundary of the scatterer D, and via (2) and (3) we get

RGF(u, υ) =∫

�

(u · T υ − υ · T u) ds

= −∫

�D

υ · T u ds +∫

�I

u · (T υ + iωc υ) ds. (64)

The integrals in (64) are understood in the sense of dualities between the corresponding spaces.In what follows we will prove that the set {T u|�D

, u|�I} is dense in H−1/2(�D) × H 1/2(�I ).

Considering the usual duality pairing, we assume that there exist functions f ∈ H 1/2(�D) andh ∈ H−1/2(�I ), such that∫

�D

f · T u ds +∫

�I

h · u ds = 0 for all u ∈ Y. (65)

Denote by w the unique solution of the boundary value problem

�∗ w(x) + �(x) ω2w(x) = 0, x ∈ [(� \ D) ∪ (B \ �)] ∪ (R2 \ B), (66)

w(x) = f (x), x ∈ �D, (67)

T w(x) + iωc w(x) = h(x), x ∈ �I , (68)

limr→∞

√r

(∂w(β)(x)

∂x− ik0,β w(β)(x)

)= 0, β = p, s, (69)

with w also satisfying homogeneous transmission-rigid contact conditions on ∂�, ∂B. Thenequation (65) yields

0 = −∫

�D

w · T u ds +∫

�I

u · (T w + iωc w) ds. (70)

Since now u ∈ Y, we arrive at0 =

∫�

(u · T w − w · T u) ds. (71)

Taking into account that u(x) = uincx0(x) + usctx0

(x) (recall here that uincx0(x) = G(x, x0; p) +

sct(x, x0; p), x0 ∈ B \ � ), Betti’s formula and the representation theorem, we arrive at theequation

0 =∫

�

(G(x, x0; p) · T w(x) − w(x) · G(x, x0; p)) ds(x) = w(x0), (72)

and therefore w|C = 0, where C is a smooth curve in B \ �. By analyticity of w in B \ � weconclude that w = 0 in B \ �. Then it follows that the Cauchy data of w are equal to zero on∂�, i.e.

w+|∂� = w−|∂� = 0, (T w)+ |∂� = (T w)− |∂�= 0, (73)

and consequently w = 0 in � \ D. By the uniqueness theorem we conclude that w = 0 in(R2 \ B

)∪BC, where BC is the domain between the boundary ∂B and the curve C. Now withthe aid of (67) and (68) we obtain f = 0, h = 0. The assertion of the theorem now easilyfollows, since υ(x) = 0 for x ∈ D (recall here the uniqueness of solutions of the interiormixed boundary value problem (34)–(36)). �

12

Inverse Problems 26 (2010) 085011 C E Athanasiadis et al

In what follows, we prove the following property.

Theorem 4. The operator RGF : N(�) → L2(C) defined by (58) has a dense range.

Proof. We consider a square integrable function γ over the smooth curve C in B \ �, i.e.γ ∈ L2(C), such that (γ, RGF(υ)) = 0, for all υ ∈ N(�). If we define

G(x) =∫

C

γ (x0) u(x, x0) ds(x0), (74)

then using the linearity of the RGF, we have (see (58))

(γ, RGF(υ)) = RGF(G, υ) =∫

C

γ (x0)RGF(u(·, x0), υ) ds(x0). (75)

With the aid of the boundary conditions on � and Betti’s integral theorem we arrive at

RGF(G, υ) = −∫

�D

υ · T G ds +∫

�I

(T υ + iωc υ) · G ds, (76)

with υ ∈ N(�). We also recall that the set of vector elastic Herglotz functions is dense in theset of solutions to the Navier equation in D, with respect to the H 1(D)-norm for any boundedLipchitz domain D (see theorem 1). Hence, using the well-posedness of the interior mixedimpedance problem (34)–(36), the trace theorem and the density property, one concludes thatthe set {υg, T υg + iωc υg} is dense inH 1/2(�D) × H−1/2(�I ). Following a similar procedureas before, we obtain G(x) = 0, for x ∈ �I and T G(x) = 0, for x ∈ �D . It is obviousnow that G has zero Cauchy data on ∂D and therefore G = 0, in B \ D. Recalling thatu(x) = uincx0

(x) + usctx0(x), the integral operator (74) is continuous from both sides of the

boundary C. Finally, taking into account the jump relation for T G on the curve C, we arriveat γ = 0, which in view of the assumption (γ, RGF(υ)) = 0, for all υ ∈ N(�), establish theassertion.

�

Combining the ideas of the LSM which arises from an application of the RGF tothe fundamental solution of the Navier equation, as well as density properties of Hergoltzfunctions, we establish the main result of the present paper.

5. An application of the elastic RGF to the LSM

In this section we consider the problem of the determination of the boundary of a partiallycoated obstacle. This is based on the following idea: if υg is a solution of (59), thenthe boundary of the scatterer � can be found from those points y0 having the property‖g(·, y0)‖L2(S1) → ∞. We establish the following.Theorem 5. Assume that � = �D ∪ �I is Lipschitz and that c > 0. Then

(i) for y0 ∈ D, and a given ε > 0, there exists gε(·, y0) ∈ L2(S1) such that

‖RGF (u(·, x0), υgε (·, y0)

) − RGF (u(·, x0), G(·, y0; p)) ‖L2(C) < ε, (77)

where the elastic Herglotz wavefunction υgε (·, y0) converges to −u(·, y0), in H 1(D) as ε → 0,u(·, y0) being the solution of

�∗u(·, y0) + �1 ω2u(·, y0) = 0 in D, (78)

satisfying the boundary conditions

u(·, y0) = −G(·, y0; p) on �D, (79)

T u(·, y0) + iωc u(·, y0) = −T G(·, y0; p) − iωc G(·, y0; p), on �I . (80)

13

Inverse Problems 26 (2010) 085011 C E Athanasiadis et al

(ii) For a fixed ε > 0, we get

limdist(y0, �)→0

‖υgε (·, y0)‖H 1(D) = +∞, (81)

limdist(y0, �)→0

‖gε(·, y0)‖L2(S1) = +∞. (82)

(iii) For y0 ∈ R2 \ D and ε > 0, if gε(·, y0) ∈ L2(S1) satisfies

‖RGF(u(·, x0), υgε (·, y0)) − RGF (u, G(·, y0; p)) ‖L2(C) < ε, (83)

then

limε→0

‖υgε (·, y0)‖H 1(D) = +∞, (84)

and

limε→0

‖gε(·, y0)‖L2(S1) = +∞. (85)

Proof. (i) Let us consider the case where y0 ∈ D. Then

RGF(u(·, x0), υg − G(·, y0; p)

) = −∫

�D

(υg − G(·, y0; p)

) · T u ds

+∫

�I

u · [ T (υ − G(·, y0; p)) + iωc (υg − G(·, y0; p)) ] ds. (86)

Since elastic Herglotz functions are dense with respect to the H 1(D)-norm in the space ofsolutions to the Navier equation in D, then by the well-posedeness of the interior mixedboundary value problem (34)–(36), the trace theorem and (86), we get

‖RGF (u, υgε (·, y0)

) − RGF (u(·, x0), G(·, y0; p)) ‖L2(C) < ε, (87)

for all u ∈ Y.(ii) Further, taking into account that the fundamental solutionG(·, y0; p), with respect to

the H 1(D)-norm blows up as y0 approaches � from inside, we obtain that for a fixed ε > 0,limdist(y0, �)→0 ‖υgε (·, y0)‖H 1(�) = +∞. Hence, the same property holds for the Herglotzkernel, with respect to the L2-norm.

(iii) Let us now consider the case where y0 ∈ � \ D. We set

u(x) = G(x, x0; p) + sct(x, x0; p) + usctx0(x), (88)

where u(x) is the total field for (1)–(11) (due to the incident point source (23)) and usctx0(x) the

corresponding scattered field. Taking into account (52) and (88) we obtain

RGF (u(x),G(x, y0; p)) = J1 + J2 , (89)

where

J1 :=∫

∂�

[ sct(x, x0; p) + usctx0

(x)] · TG(x, y0; p) ds(x)

−∫

∂�

G(x, y0; p) · T[ sct(x, x0; p) + usctx0

(x)]ds(x)

and

J2 :=∫

∂�

(G(x, x0; p) · T G(x, y0; p) − G(x, y0; p) · TG(x, x0; p)) ds(x).

14

Inverse Problems 26 (2010) 085011 C E Athanasiadis et al

Using the reciprocity principle (29) for x0 ∈ B \ D, usctx0is a solution of the Navier equation

with respect to x0. Hence J1 and J2 are the integral representations for υ(x0) andG(y0, x0; p),respectively, and therefore

RGF (u(x),G(x, y0; p)) = υ(x0) − G(y0, x0; p), x0 ∈ C, (90)

υ being a solution of the Navier equation in B \ D (note here that G(y0, x0; p) becomessingular for x0 → y0). We also have that

RGF(u(x), υg

) =∫

�I

u(x) · T υg(x)ds(x) −∫

�D

υg(x) · T u(x) ds(x). (91)

Consider now that gε(·, y0) ∈ L2(S1) is a density of the Herglotz wavefunction υgε (·, y0),such that

‖RGF(u, υgε (·, y0)) − RGF (u, G(·, y0; p)) ‖L2(C) < ε, (92)

for every u ∈ Y. We now assume that the set Xε of all Herglotz wavefunctions with kernelgε(·, y0) is bounded, i.e.

‖υgε (·, y0)‖H 1/2(�) < δ, (93)

for every υgε (·, y0) ∈ Xε, with δ being a positive constant (independent of ε). Usingthe trace theorem we have that υgε (·, y0)|�D

and T υgε (·, y0) + iω c υgε (·, y0)|�Iare also

bounded in H 1/2(�D) and H−1/2(�I ), respectively. Therefore, there exists a weaklyconvergent subsequence

(υgε (·, y0)|�D

, T υgε (·, y0) + iω cυgε (·, y0)|�I

), converging to (f, g) ∈

H 1/2(�D) × H−1/2(�I ) for ε → 0. We now set

u(x0) =∫

�D

f (x) · T u(x) ds(x) +∫

�I

g(x) · u(x) ds(x), (94)

for x0 ∈ C. From (91) we obtain that RGF(u(·, x0), υε

g

)converges to u(x0) with x0 ∈ C.

Hence,

u(x0) ≡ υ(x0) − G(x0, y0; p), x0 ∈ C. (95)

The functions u(x0) and υ(x0) − G(x0, y0; p) coincide as radiating solutions of the Navierequationμ1 �u(x)+(λ1 +μ1) grad divu(x)+�1 ω2u(x) = 0, inB \�, and furthermore (by theunique continuation principle) also coincide for x0 ∈ B \ (

D ∪ {y0}). If we now let y0 → x0,

we arrive at a contradiction. Hence by (93) and the trace theorem we get that ‖υgε (·, y0)‖H 1(D)

is unbounded. Since, clearly, a similar relation for the L2(S1)-norm of the correspondingHerglotz kernel gε(·, y0) is also satisfied, the proof of the theorem is complete.

�

6. A characterization for the surface impedance

In this section an expression for the (positive) surface impedance c will be established.

Theorem 6. For y0 ∈ D consider uy0 to be a solution of the interior mixed boundary valueproblem (60)–(62). Then for the surface impedance c we have

c =−Im uy0(y0) − 1

8

(1

λ0+2μ0+ 1

μ0

)ω ‖ G(·, y0; p) + uy0(·) ‖2

L2(�I )

. (96)

Proof. Let us consider a point y0 ∈ D, and uy0 to be a solution of the interior mixed boundaryvalue problem (60)–(62). If we now define the function

wy0(x) = G(x, y0; p) + uy0(x), x ∈ D, (97)

15

Inverse Problems 26 (2010) 085011 C E Athanasiadis et al

where uy0 is the solution of (60)–(62), then taking into account that uy0 ∈ H 1(D), in the senseof trace theorem, we obtain

wy0 |�D= 0,

(T wy0 + iω c wy0

) |�I= 0 (98)

with wy0 |� ∈ H 1/2(�), T wy0 |� ∈ H−1/2(�).We now evaluate the following boundary integral:∫

�

(wy0 · T wy0 − wy0 · T wy0

)ds = I1 + I2 + I3 + I4, (99)

where

I1 :=∫

�

(uy0 · T uy0 − uy0 · T uy0

)ds(x)

I2 :=∫

�

(uy0 · T G(x, y0; p) − G(x, y0; p) · T uy0

)ds(x)

I3 :=∫

�

(G(x, y0; p) · T G(x, y0; p) − G(x, y0; p) · T G(x, y0; p)

)ds(x)

I4 :=∫

�

(G(x, y0; p) · T uy0 − uy0 · T G(x, y0; p)

)ds(x). (100)

Using the third Betti’s formula and the fact that uy0 is a regular solution of the Navier equationin D, we get that I1 = 0. Furthermore, via the representation theorem for interior solutions,we get I2 = −uy0 and I4 = uy0 . Concerning now I3, with the aid of Betti’s theorem we have

I3=∫

�\D

(G(x, y0; p) · �∗G(x, y0; p) − G(x, y0; p) · �∗G(x, y0; p)

)ds(x)

+∫

∂�

(G(x, y0; p) · T G(x, y0; p) − G(x, y0; p) · T G(x, y0; p)

)ds(x).

(101)

Clearly the integral in � \ D vanishes. We still have to calculate the integral on the boundary∂� on the right-hand side of (101). From Betti’s integral theorem we can write∫

∂�

(G(x, y0; p) · T G(x, y0; p) − G(x, y0; p) · T G(x, y0; p)

)ds(x)

=∫

∂BR

(G(x, y0; p) · T G(x, y0; p) − G(x, y0; p) · T G(x, y0; p)

)ds(x),

(102)

for every circle BR with a large enough radius R containing � in its interior. Using relations(17)–(21) and the asymptotic behaviour of the fundamental solution for r → ∞, we obtain

G(x, y0; p) · T G(x, y0; p) − G(x, y0; p) · T G(x, y0; p)

= − i

4π�0 ω2R

[k20,p (x · p)2 + k20,s (x⊥ · p)2

]+O(R−2) (103)

on the boundary ∂BR . From relations (101), (102) and (103) and letting R → ∞, we getI3 = −

∫S1

i

4π�0 ω2

[k20,p (x · p)2 + k20,s (x⊥ · p)2

]ds(x), (104)

with p being, as before, a polarization unit vector of S1. In addition, for the left-hand side of(99), taking into account the boundary conditions (61) and (62), we obtain∫

�

(wy0 · T wy0 − wy0 · T wy0

)ds = 2 iω c ‖ wy0 ‖2L2(�I )

. (105)

16

Inverse Problems 26 (2010) 085011 C E Athanasiadis et al

Straightforward calculations for (104), and combining relations (99) and (105), lead to

2ω c ‖ wy0(·) ‖2L2(�I )+2 Im uy0(y0) = −1

4

(1

λ0 + 2μ0+1

μ0

). (106)

Taking into account (97) the assertion easily follows. �

Remark. (i) Theorem 6 can be used as a starting point for numerical experiments: even if Dis known (for example, by using the RGF as discussed in [20]), uy0 cannot be computed sinceit depends on c. Nevertheless, an approximation to uy0 , namely −υg(·, y0), can be obtainedby theorem 6. An approximation for c is then given by

c =8ω2 �0 Imυg(y0) −

(k20,p + k20,s

)8ω3 �0 ‖ G(·, y0; p) − υg(·, y0) ‖2

L2(�I )

, (107)

where υg(·, y0) is the Herglotz wavefunction with kernel g(·, y0) being the regularized solutionof (59).

(ii) The proposed method (the combination of the LSM and the use of the RGF)establishes an inversion algorithm for reconstruction of elastic partially coated obstacles fromthe knowledge of the Cauchy data of the total field measured on the boundary of a domaincontaining the object in its interior. Its main advantage is that it does not need any a prioriinformation on the scattering obstacle, and furthermore (from the point of view of numericalcomputations) it does not require the evaluation of the Green’s function for the backgroundmedium.

Acknowledgments

The authors thank one of the referees for substantial remarks that led to the present formof this paper. They are indebted to Professor Nicolas Charalambakis (Aristotle Universityof Thessaloniki, Greece) for useful discussions and his valuable assistance on the literaturefor engineering applications. Finally, they acknowledge financial support from the SpecialAccount for Research Grants of the National and Kapodistrian University of Athens (grantnos 70/4/5643 and 70/4/6412).

References

[1] Alves C J S and Kress R 2002 On the far field operator in elastic obstacle scattering Inst. Math. Appl. J. Appl.Math. 67 1–21

[2] Andrieux S and Ben Abda A 1996 Identification of planar cracks by complete over-determined data: inversionformulae Inverse Problems 12 553–63

[3] Arens T 2002 Linear sampling methods for 2d inverse elastic wave scattering Inverse Problems 17 1445–64[4] Athanasiadis C E, Natroshvili D, Sevroglou V and Stratis I G A boundary integral equations approach for direct

mixed impedance problems in elasticity J. Integral Eqns. Appl. (at press)[5] Bacchelli V 2009Uniqueness for the determination of unknown boundary and impedancewith the homogeneous

Robin condition Inverse Problems 25 1–4[6] Baum C 1999 Detection and Identification of Visually Obscured Targets (London: Taylor and Francis)[7] Bencheikh L 1990 Modified fundamental solutions for the scattering of elastic waves by a cavity Q. J. Mech.

Appl. Math. 43 57–73[8] Bonnet M and Constantinescu A 2005 Inverse problems in elasticity Inverse Problems 21 R1–50[9] Cakoni F, Colton D and Monk P 2001 The direct and inverse scattering problems for partially coated obstacles

Inverse Problems 17 1997–2015[10] Cakoni F and Colton D 2003 On the mathematical basis of the linear sampling method Georgian Math. J.

10 411–25[11] Cakoni F and Colton D 2006 Qualitative Methods in Inverse Scattering Theory (Berlin: Springer)

17

Inverse Problems 26 (2010) 085011 C E Athanasiadis et al

[12] Calderon A P 1980 On an inverse boundary problem Seminar on Numerical Analysis and its Applications toContinuum Physics (Soc. Brazil. Mat., Rio de Janeiro) ed W H Meyer and M A Raupp pp 65–73 (reprintedin Comput. Appl. Math. 25 133–138, 2006)

[13] Charalambopoulos A, Kirsch A, Anagnostopoulos KA, Gintides D andKiriaki K 2007 The factorizationmethodin inverse elastic scattering from penetrable bodies Inverse Problems 23 27–51

[14] Cherkaoui M, Sabar H and Berveiller M 1994 Micromechanical approach of the coated inclusion problem andapplications to composite problems J. Eng. Mater. Technol. 116 274–8

[15] Committee of Coatings for High-Temperature Structural Materials, National Materials Advisory Board,Commission on Engineering and Technical Systems National Research Council 1996 Coatings for High-Temperature Structural Materials: Trends and Opportunities (Washington, DC: National Academy Press)

[16] Colton D and Kress R 1992 Inverse Acoustic and Electromagnetic Scattering Theory (New York: Springer)[17] Colton D and Kirsch A 1996 A simple method for solving the inverse scattering problems in the resonance

region Inverse Problems 12 383–93[18] Colton D, Piana M and Potthast R 1997 A simple method using Morozov’s discrepancy principle for solving

inverse scattering problems Inverse Problems 13 1477–93[19] Colton D, Coyle J and Monk P 2000 Recent developments in inverse acoustic scattering theory SIAM

Rev. 42 369–414[20] Colton D and Haddar H 2005 An application of the reciprocity gap functional to inverse scattering theory Inverse

Problems 21 383–98[21] Colton D and Kress R 2006 Using fundamental solutions in inverse scattering Inverse Problems: Topical

Rev. 22 R49–66[22] Dassios G and Rigou Z 1995 Elastic Herglotz functions SIAM J. Appl. Math. 55 1345–61[23] Di Cristo M and Sun J 2006 An inverse scattering problem for a partially coated buried obstacle Inverse

Problems 22 2331–50[24] Fitzer E and Manocha L M 1998 Carbon Reinforcement and Carbon/Carbon Composites (Berlin: Springer)[25] Griesmaier R 2009 Reciprocity gap music imaging for an inverse scattering problem in two-layered media

Inverse Probl. Imaging 3 389–403[26] Howe M S 1992 Sound produced by an aerodynamic source adjacent to a partly coated, finite elastic plate Proc.

R. Soc. A 436 351–72[27] Hsiao G and Wendland W 1979 On integral equation method for the plane mixed boundary value problem for

the Laplacian Math. Methods Appl. Sci. 1 265–321[28] Hu Z, Li H, Fu Q, Xue H and Sun G 2007 Fabrication and tribological properties of B2O3 as friction reducing

coatings for carbon–carbon composites New Carbon Mater. 22 131–4[29] Kirsch A 1996 An Introduction to the Mathematical Theory of Inverse Problems (Berlin: Springer)[30] Kupradze V D, Gegelia T G, Basheleishvili M O and Burchuladze T V 1979 Three-Dimensional Problems of

the Mathematical Theory of Elasticity and Thermoelasticity (North-Holland Series in Applied Mathematicsand Mechanics vol 25) (Amsterdam: North-Holland)

[31] McLeanW2000 Strongly Elliptic Systems and Boundary Integral Equations (Cambridge: CambridgeUniversityPress)

[32] Martin P A 1990 On the scattering of elastic waves by an elastic inclusion in two dimensions Q. J. Mech. Appl.Math. 43 275–92

[33] Mu M, Walker A M, Torkelson J M and Winey K I 2008 Cellular structures of carbon nanotubes in a polymermatrix improve properties relative to composites with dispersed nanotubes Polymer 49 1332–7

[34] Natroshvili D, Kharibegashvili S and Tediashvili Z 2000 Direct and inverse fluid-structure interaction problemsRediconti di Matematica, Ser. VII 20 57–92

[35] Natroshvili D and Tediashvili Z 2001 Mixed type direct and inverse scattering problems Problems and Methodsin Mathematical Physics (Operator Theory: Advances and Applications vol 121) ed J Elschner, I Gohbergand B Silbermann (Basel: Birkhauser) pp 366–89

[36] Salman A, Gabbitas B, Li J and Zhang D 2009 Tribological properties of thermally sprayed TiAl–Al2O3composite coating Process. Microstruct. Perf. Mater. 4 012006

[37] Pelekanos G and Sevroglou V 2003 Inverse scattering by penetrable objects in two-dimensional elastodynamicsJ. Comput. Appl. Math. 151 129–40

[38] Sevroglou V and Pelekanos G 2001 An inversion algorithm in two-dimensional elasticity J. Math. Anal.Appl. 263 277–93

[39] Sevroglou V and Pelekanos G 2002 Two dimensional elastic Herglotz functions and their applications in inversescattering J. Elast. 68 123–44

[40] Sevroglou V 2005 The far-field operator for penetrable and absorbing obstacles in 2D inverse elastic scatteringInverse Problems 17 717–38

[41] Tanuma K 2007 Stroh formalism and Rayleigh waves J. Elast. 89 5–154

18

Inverse Problems 26 (2010) 085011 C E Athanasiadis et al

[42] Twersky V 1967 Multiple scattering of electromagnetic waves by arbitary configurations J. Math. Phys.8 589–610

[43] Venkatraman S, Tan L P, Joso J F, Boey F Y C and Wang X 2006 Biodegradable stents with elastic memoryBiomaterials 27 1573–8

[44] Walpole L J 1978 A coated inclusion in an elastic mediumMath. Proc. Camb. Phil. Soc. 83 495–506[45] Whelam D M, van der Giessen W J, Krabbendam S C, van Vliet E A, Verdouw P D, Serruys P W and van

Beusekom H M 2000 Biocompatibility of phosphorylcholine coated stents in normal porcine coronaryarteries Heart 83 338–45

[46] Wu T T 1999 Elastic wave propagation and nondestructive evaluation of materials Proc. Natl Sci. Counc. ROCA 23 703–15

[47] www.technocarbon.com/fileadmin/Articles/SD CM 02 08 technocarbon.pdf[48] www.faqs.org/patents/app/20090056871

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