Article 15

Elastic-wave identification of penetrable obstacles using shape-material sensitivity framework

Marc Bonnet

^a,*

, Bojan B. Guzina

^b

aSolid Mechanics Laboratory (UMR CNRS 7649), Ecole Polytechnique, F-91128 Palaiseau Cedex, France

bDepartment of Civil Engineering, University of Minnesota, Minneapolis, MN 55455, USA

a r t i c l e i n f o

Article history:

Received 14 February 2008

Received in revised form 31 August 2008 Accepted 10 September 2008

Available online 18 September 2008

Keywords:

Shape-material sensitivity Elastodynamics Identification Inclusion

Boundary element method Constrained optimization

a b s t r a c t

This study deals with elastic-wave identification of discrete heterogeneities (inclusions) in an otherwise homogeneous ‘‘reference” solid from limited-aperture waveform measure- ments taken on its surface. On adopting the boundary integral equation (BIE) framework for elastodynamic scattering, the inverse query is cast as a minimization problem involving experimental observations and their simulations for a trial inclusion that is defined through its boundary, elastic moduli, and mass density. For an optimal performance of the gradient-based search methods suited to solve the problem, explicit expressions for the shape (i.e. boundary) and material sensitivities of the misfit functional are obtained via the adjoint field approach and direct differentiation of the governing BIEs. Making use of the message-passing interface, the proposed sensitivity formulas are implemented in a data-parallel code and integrated into a nonlinear optimization framework based on the direct BIE method and an augmented Lagrangian whose inequality constraints are employed to avoid solving forward scattering problems for physically inadmissible (or overly distorted) trial inclusion configurations. Numerical results for the reconstruction of an ellipsoidal defect in a semi-infinite solid show the effectiveness of the proposed shape-material sensitivity formulation, which constitutes an essential computational com- ponent of the defect identification algorithm.

Ó2008 Elsevier Inc. All rights reserved.

1. Introduction

Elastic-wave sensing of penetrable (i.e. deformable) heterogeneities in a solid matrix is a long-standing problem in mechanics with applications to nondestructive material testing, seismic prospecting, medical diagnosis, and underground object identification. In the context of seismic exploration, comprehensive three-dimensional (3D) mapping of subterranean structures commonly entails the interpretation of a large number, often thousands, of motion measurements using elasto- dynamic or acoustic models based on domain discretization, see e.g.[41]. In contrast, this investigation focuses on the map- ping of objects buried in a known reference solid, from only a limited number of remote measurements. In such instances, boundary integral equation (BIE) formulations[11,15]provide a direct mathematical link between the observed waveforms and the geometry and material characteristics of a hidden object, and therefore allow to exploit effectively the limited data.

Although inverse scattering in general has been the subject of intensive mathematical and computational research [40,26,13,16], only limited efforts have so far been devoted to the wave-based reconstruction of homogeneous elastic inclu- sions. Two-dimensional BIE formulations of the inclusion identification problem were proposed in[27,29](elastostatics) and [43](elastodynamics). Volume integral equations of the Lippmann–Schwinger type, which entail domain discretization over

0021-9991/$ - see front matterÓ2008 Elsevier Inc. All rights reserved.

doi:10.1016/j.jcp.2008.09.009

* Corresponding author.

E-mail addresses:bonnet@lms.polytechnique.fr(M. Bonnet),guzina@wave.ce.umn.edu(B.B. Guzina).

Contents lists available atScienceDirect

Journal of Computational Physics

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / j c p

a region in which flaws area prioriexpected, are used in e.g.[46]with seismic waves idealized as acoustic waves and[38,1]

in conjunction with the contrast source inversion concept for elastic and electromagnetic waves, while fast solution methods are proposed in[23]. Mathematical results on identifiability are given in e.g.[24]. More recently, approximate identification methods based on the small-inclusion asymptotics[5,19,6,25], specialized analytical solutions[7], or energy considerations [4]were proposed for preliminary ‘‘scanning” of solid bodies.

The focus of this investigation is the development of a computational platform for the 3D identification of penetrable elas- tic inclusions via an elastodynamic BIE framework. This approach rests upon the full three-dimensional elastodynamic mod- el, with no resort to approximations such as the Born linearization or small-inclusion asymptotics. The inclusions are assumed homogeneous and bonded to the surrounding reference medium; as such, they are characterized by their boundary (i.e. the surface that separates them from the reference medium), elastic tensor, and mass density. For identification pur- poses, the inverse problem is reduced to the minimization of a cost functional representing the misfit between experimental observations (values of displacements at sensor locations) and their simulations for an assumed inclusion configuration. The latter are based on a coupled system of regularized boundary integral equations[11,36]. For computational efficiency of the gradient search technique employed by the inverse solution, theshapesensitivity of the featured cost functional is evaluated via an adjoint field approach which, besides the matter of elegance, is computationally much more efficient than finite-dif- ference evaluations. This is accomplished by generalizing upon the shape sensitivity approach proposed in[21]for elastic- wave void identification and in[10]for the inverse scattering of acoustic waves. To completely characterize penetrable elas- tic defects, thematerialsensitivity of the cost functional is derived using two alternative methodologies: (i) a direct differ- entiation approach based on the material parameter derivative of the governing BIE pair, or (ii) adjoint field approach. A similar BIE-based treatment of shape-material sensitivity has been recently proposed[47]for optical tomography featuring the scalar Helmholtz equation with a complex wavenumber, while e.g.[3,37]present other applications of adjoint-based shape sensitivity analyses.

Making use of the message-passing interface, shape and material sensitivities derived in this study are implemented in a data-parallel code and integrated into a nonlinear optimization algorithm towards the solution of the 3D inverse problem.

Preliminary identification studies on sample inclusion configurations demonstrated that the unconstrained (quasi-Newton) minimization algorithm, that was successfully used in previous studies[21,32]for 3D void reconstruction, performs unsat- isfactorily for the geometric-material identification problem at hand. For this reason, a more robust algorithm based on the augmented Lagrangian cost functional[34]has been developed as a means to deal with physical inequality constraints, with- out increasing the dimension of the parametric space, and ensure that all iterates (in terms of the inclusions’ geometric and material parameters) be physically admissible. The elastodynamic transmission problem, whose repeated solution is re- quired by the minimization, is thus always well-posed. The numerical results, which employ a direct boundary element method (BEM) for the primary, adjoint and material sensitivity solutions, demonstrate the feasibility of identifying the geometry and material characteristics of penetrable (elastic) defects hidden in a semi-infinite solid from only a limited num- ber of waveform measurements taken on the (traction-free) surface.

2. Direct and inverse scattering by elastic inclusions

Consider an inverse scattering problem where the reference homogeneous solidX, containing a bonded inclusionXb^true with boundaryC^true, is probed by elastic waves. The reference medium, whose external boundary (available for testing) is denoted byS, is characterized by its elastic tensorCand mass density

q

; the respective material characteristics of the inclu- sion are denoted asCb^trueand

q

^^true. The ensuing shape and material sensitivity analyses (Sections3–5) are carried out for the reference solid of an arbitrary shape, whereas the computational treatment and numerical results presented thereafter (Sec- tions6, 7) assume a semi-infinite configuration wherebySdenotes the traction-free surface of the half-space, and isotropic elastic properties inXandXb^true.

Inverse problem.To identify the geometryXb^trueand material characteristicsCb^trueand

q

^^trueof the hidden defect, time-har- monic excitations are applied in the form of volume (f) and surface (g) force densities having respective supportsVXand StS, and displacementsu^Dover the complementary surfaceSu¼SnSt. The implicit time-harmonic factor expði

x

tÞwhere

x

denotes the angular frequency of excitation is omitted hereon. Letting Xb denote a trial inclusion bounded by Cand X¼Xn ðC[XbÞbe the region surrounding the obstacle, the prescribed excitationðf;g;u^DÞgives rise to elastodynamic dis- placement fieldsu¼u½Xb;Cb;

q

^inXandu^¼u½^Xb;Cb;

q

^inXb.

For identification purposes, the displacementu^obsinduced in the flawed solid byðf;g;u^DÞis monitored over the measure- ment surfaceSobsSt(other possibilities, e.g. finite sets of measurement points, being also allowed by the ensuing treat- ment). Ideally, a defect configurationðXb^true;Cb^true;

q

^^trueÞsuch that

u½Xb^true;Cb^true;

q

^^true ¼u^obs ðonSobsÞ ð1Þ is sought, whereuonSobsis understood in the sense of the trace. In practice, due to many factors (e.g. incomplete and/or inexact measurements, modelling uncertainties), the inclusion is sought so as to minimize a misfit cost functional

JðXb;bC;

q

^Þ ¼ Z

S_obs

u

ðu½Xb;bC;

q

^ðnÞ;u^obsðnÞ;nÞdS_n; ð2Þ

where function

u

, which quantifies the misfit between the predicted and observed displacements, is assumed to be differ- entiable with respect to its arguments. For example, the usual least-squares measure of misfit is defined through 2

u

ðu;u^obs;nÞ ¼ juu^obsj².

Forward problem.LetLandLbdenote the Navier partial differential operator, respectively associated with the reference solid and inclusion, i.e.

LudivðC:$_{uÞ þ}

qx

²u; L^budivðbC:$_{uÞ þ^}

qx

_^ ²u:^{^} ð3Þ In(3)and thereafter, all quantities defined with reference to the inclusion or its constitutive parameters are indicated with a hat symbol. Moreover, the nabla symbol$denotes the gradient operator, with the convention$_{ðÞ ¼ ðÞ}

;‘e‘(the

comma denoting a partial derivative), while the column symbol ‘:’ indicates a two-fold inner product between tensors, e.g.ðC:$_uÞ

ij¼C_ijk‘u_k;‘.

The predicted displacement featured in cost functional(2)solves the forward problem

ðaÞLuþf¼0ðinXÞ; ðbÞL^bu¼0ðin XbÞ; ð4Þ

u¼u;^ tþ^t¼0 ðonCÞ; ð5Þ

ðaÞt¼gðonStÞ; ðbÞu¼u^DðonSuÞ; ð6Þ

where Eqs.(4)–(6), respectively state the elastodynamic field equations, the perfect-bonding interfacial transmission condi- tions, and the external boundary conditions. In(5) and (6),t ðC:$_{uÞ nand}^t ðCb:$^uÞ n^are the boundary tractions rel- ative to the reference medium and the inclusion, respectively, withn^¼ nandndenoting the unit normal exterior toX. If the reference domainXextends to infinity in any direction, as is the case for the semi-infinite solid examined later,u must in addition satisfy a suitable radiation condition at infinity (this can be relaxed so as to allow e.g. scattering of incident plane waves, seeAppendix A.2). Moreover,(4)implicitly carries the assumptionXb\V¼ ;, i.e. the forward problem(4)–(6)is considered only for a inclusion that is separated from the body force support.

Two alternative formulations of problem(4)–(6)are now summarized: (i) the weak formulation, upon which the general results in terms of sensitivity formulas are established (Sections3–5), and (ii) the BIE formulation used here as a basis for the computational treatment and numerical results (Sections6, 7).

Weak formulation.The forward transmission problem(4)–(6)can be recast in weak form wherebyðu;uÞ 2^ Vðu^DÞmust satisfy

Aððu;^uÞ;ðw;wÞÞ ^ FðwÞ ¼0 8ðw;wÞ 2^ Vð0Þ; ð7Þ

with the function spacesV, the symmetric bilinear formA, and the linear formFdefined by Vðu^DÞ ¼ ðw;wÞjw^ 2 hH¹_locðXÞi3

;w^ 2 ½H¹ðXbÞ³;w¼u^DonSu;w¼w^ onC

; ð8aÞ

Aððu;^uÞ;ðw;wÞÞ ¼^ Z

X

aðu;wÞdVþ Z

b

X

^aðu;^ wÞdV^ ð8bÞ

FðwÞ ¼ Z

V

fwdVþ Z

St

gwdS ð8cÞ

and the bilinear energy densitiesainXand^ainXbgiven by

aðu;wÞ ¼$_u:C:$_w

qx

²uw; ð9aÞ

^aðu;^ wÞ ¼^ $u^:Cb:$w^

qx

^ ^{2^}uw:^ ð9bÞ

BIE formulation (semi-infinite solid).Wave propagation and scattering in an elastic half-space is a suitable idealization for a number of applications such as nondestructive material testing and seismic exploration. BIE formulations, which deal effec- tively with unbounded domains, are a natural framework for such configurations. With reference to the Cartesian frame fO;n₁;n₂;n₃g, let the host domainXbe semi-infiniteðn3P0Þand bounded by the traction-free surfaceS¼ fnjn3¼0g. Let U(x,n) andT(x,n) denote the (half-space) elastodynamic Green’s tensors, defined such thatUi‘andTi‘ði; ‘¼1;2;3Þ, respec- tively denote theith component of the displacement and traction atn2Xresulting from a unit time-harmonic point force applied atx2Xin the‘th direction, withT_i‘vanishing identically onS. Similarly, letUðx;b nÞandTbðx;nÞdenote the (full- space) elastodynamic Green’s tensors corresponding to the material properties of the inclusion. With these definitions, the forward problem(4)–(6)for a semi-infinite solid can be reformulated in terms of a pair of regularized boundary integral equations[11,36]:

uðxÞ þ Z

C

ðuðnÞ uðxÞÞ ½Tðx;nÞ₁dSnþ Z

C

uðnÞ ½Tðx;nÞ₂dSn Z

C

tðnÞ Uðx;nÞdSn¼u^FðxÞ; x2C; ð10Þ Z

C

ðuðnÞ uðxÞÞ ½Tbðx;nÞ₁dSnþ Z

C

uðnÞ ½Tbðx;nÞ₂dSnþ Z

C

tðnÞ Ubðx;nÞdSn¼0; x2C ð11Þ

written, respectively for the ‘‘matrix”Xand inclusionXb, in terms of the traces (u,t) onCof the exterior field. Thefree fieldu^F featured in(10)is the solution to(4), (5)when no inclusion is present, and is explicitly given by

u^FðxÞ ¼ Z

V

fðnÞ Uðx;nÞdVnþ Z

S

gðnÞ Uðx;nÞdSn; x2X: ð12Þ

For regularization purposes, the traction Green’s functions in(10) and (11)are decomposed[21]into the sum of (fre- quency-independent) singular parts½T₁;½bT₁and (frequency-dependent) regular parts½T₂;½Tb₂according to

Tðx;nÞ ¼ ½Tðx;nÞ₁þ ½Tðx;nÞ₂; bTðx;nÞ ¼ ½Tbðx;nÞ₁þ ½Tbðx;nÞ₂:

On solving(10) and (11)foruandt, the displacement in the (reference) solid surrounding the inclusion is given by the integral representation formula

uðxÞ ¼u^FðxÞ þ Z

C

ftðnÞ Uðx;nÞ uðnÞ Tðx;nÞgdSn; x2X: ð13Þ

LettingSRdenote the sphere of radiusRcentered at the origin(10) and (13)rest on the assumption thatuis aradiating elastodynamic state in the semi-infinite solidX, wherebyu,tsatisfy the generalized radiation condition[28]

R!þ1lim Z

S_R\X

fuðnÞ Tðx;nÞ tðnÞ Uðx;nÞgdS_n¼0; x2X: ð14Þ

3. Differentiation with respect to inclusion perturbations

To quantify the effect of the inclusion’s boundary and material parameter perturbations on the cost function(2), the de- fect configurationðXb;Cb;

q

^Þis assumed to depend on a time-like evolution parameter

s

[39,44], with the unperturbed, ‘initial’

configurationðXb;bC;

q

^Þconventionally associated with

s

¼0. In this study, only the first-order infinitesimal perturbations of ðXb;Cb;

q

^Þ, i.e. the first-order ‘‘time” derivatives at

s

¼0, are considered. Perturbations of the inclusion’s constitutive proper- ties can thus be expressed as

b

C_s¼bCþCb⁰

s

;

q

^s¼

q

^þ

q

^⁰

s

; ð15Þ

whereas the shape perturbations ofXb can be synthesized as

x2X!xs¼xþhðxÞ

s

2Xs; ð16Þ

wherehðxÞis a given (i.e. prescribed) transformation velocity field. In the sequel,hðxÞis assumed to vanish outside of a neigh- bourhood ofXb, which postulates the existence of a bounded regionOsatisfying

ðaÞXbOX; ðbÞh¼0inXnO; ðcÞO\S¼ ;; ðdÞO\V¼ ;; ð17Þ with (c) and (d) stemming from the natural assumption that the trial inclusion intersect neither the external surface nor the body force support.

Total and Lagrangian derivatives of field variables.Consider a generic field variablegwhich depends on the inclusion con- figuration ðXbs;bC_s;

q

^_sÞ, e.g. the solution of the forward problem (4)–(6). Such quantity can be represented in the form gðx;Xbs;Cb_s;

q

^sÞ. Let thetotal derivative,g^}, ofgbe defined by

g}

¼lim s&0

1

s

^{½gðxs;Xbs;bCs;}

q

^{^}sÞ gðx;Xb;Cb;

q

^Þ; ð18Þ i.e. by following the evolution ofgat a pointx_smoving according to geometric transformation(16), while the inclusion’s material parameters are perturbed according to(15). The contributions tog^}of the inclusion’s geometric and material per- turbations can be separated through additive decomposition

g}

¼gþg⁰; ð19Þ

with the shape sensitivityg and material sensitivityg⁰, respectively defined by ‘‘freezing” the material parameters and the shape of the inclusion in(18), i.e.

g¼lim s&0

1

s

^{½gðxs;Xbs;bC;}

q

^{^}Þ gðx;Xb;bC;

q

^Þ ð20Þ g⁰¼lim

s&0

1

s

^{½gðx;Xb;Cbs;}

q

^{^}sÞ gðx;Xb;bC;

q

^Þ ¼og

obCbC⁰þog

o^

q q

^{^0}: ð21Þ

The shape sensitivity thus corresponds to the Lagrangian time derivative of continuum kinematics with the physical time variable replaced with a pseudo-time. For the ensuing developments, it is useful to note that the shape sensitivity of a gra- dient is given[39]by

ð$_gÞ¼$_g$_g$_{h: ð22Þ} Differentiation of integrals.Consider a generic domain integralIof the form

I¼ Z

O

gðx;Xbs;Cb_s;

q

^sÞdV;

in Eulerian description, whereOdenotes the support of geometric perturbation(16). Upon noting that the shape sensitivity of dV is given by dV ¼ ½divhdV, the total derivative ofI[39]can be written as

ðaÞ^}I¼IþI⁰; ðbÞI¼ Z

O

fgþgdivhgdV; ðcÞI⁰¼ Z

O

g⁰dV: ð23Þ

4. Shape sensitivity using an adjoint solution

Taking into account assumption(17c)wherebySobsremains invariant under perturbation(16), the shape sensitivity of cost functional(2)takes the form

J

¼Re Z

S_obs

u

_;uðu;u^obs;nÞ udS

( )

; ð24Þ

where the complex-valued function

u

_;uis defined as

u

;u¼o

u

ouRio

u

ouI; ðuR¼ReðuÞ;uI¼ImðuÞÞ: ð25Þ

On parameterizing the geometric perturbation(16)in terms of a selected shape parameteraand setting

s

¼aa0, for- mula(24)yields the derivative ofJwith respect toaevaluated ata0in terms of the derivative solutionu; this is the essence of the so-called direct differentiation approach. One can however circumvent the actual computation of derivativesu (one foreachshape parameter describing the sought inclusion) by resorting to theadjoint field approach, to whose formulation the remainder of this section is devoted.

4.1. Displacement shape sensitivity – weak formulation

Since the weak formulation(7)of the forward problem(4)–(6)holds for all perturbed inclusion configurations, the gov- erning weak formulation for the displacement shape sensitivityðu;^u

Þcan be obtained by exploiting the identity A

ððu;^uÞ;ðw;wÞÞ ^ FðwÞ ¼0 8ðw;wÞ 2^ Vð0Þ: ð26Þ

As shown inAppendix A.1, application of the Lagrangian differentiation formula(23b)to(8b)results in A

ððu;^uÞ;ðw;wÞÞ ¼^ Aððu;u^

Þ;ðw;wÞÞ þ^ Aððu;uÞ;^ ðw;w^

ÞÞ þ Z

X

Lw ð$_uhÞdV þ

Z bX

b

Lw^ ð$u^hÞdVþ Z

C

½nEðu;wÞ þ^nbEðu;^ wÞ ^ hdS; ð27Þ where the bilinear tensorial functionE(u,w), related to the dynamic Eshelby energy-momentum tensor[17], is defined by Eðu;wÞ ¼aðu;wÞI ðC:$_wÞ $_{u ð}C:$_uÞ $_{w: ð28Þ} (I: second-order identity tensor) andEðbu;^ wÞ^ is defined similarly in terms of the inclusion characteristics. Invoking assump- tion(17d), one further finds that the Lagrangian derivative ofFðwÞis given by

FðwÞ ¼FðwÞ: ð29Þ

On substituting(27) and (29)into(26), invoking the equality obtained by settingðw;wÞ ¼ ðw^ ;w^

Þin(7), and noting that u¼0onSu(since the prescribed displacementu^Dis insensitive to the inclusion shape), the displacement shape sensitivity ðu;u^

Þ 2Vð0Þis found to be governed by the weak formulation:

Aððu;^u

Þ;ðw;wÞÞ ¼ ^ Z

X

Lw ð$_uhÞdV Z

b

X

b

Lw^ ð$u^hÞdV Z

C

½nEðu;wÞ þn^Ebð^u;wÞ ^ hdS 8ðw;wÞ 2^ Vð0Þ:

ð30Þ 4.2. Adjoint solution

The main motivation behind the adjoint state approach is to evaluate the shape sensitivity(24)in an indirect, and com- putationally faster, manner by circumventing the actual computation of field sensitivitiesu. Interpreting the integral in(24)

as a virtual work and treating the sensitivityutherein as a test function leads to define the adjoint stateðv;vÞ 2^ Vð0Þby the weak statement

Aððv;^vÞ;ðw;wÞÞ ¼^ Z

S_obs

u

;uðu;u^obs;nÞ wdS 8ðw;wÞ 2^ Vð0Þ; ð31Þ whose solution ðv;^vÞ is theadjoint state. The adjoint transmission problem(31) can equivalently be stated in strong form as:

ðaÞ Lv¼0 ðinXÞ;

ðbÞ L^bv¼0 ðinXbÞ

( ðcÞ p¼

u

_;u ðonSobsÞ

ðdÞ p¼0 ðonStnSobsÞ ðeÞ v¼0 ðonSuÞ ðfÞ v¼^v; pþp^¼0 ðonCÞ 8>

>>

<

>>

>:

ð32Þ

wherep¼ ðC:$_{vÞ nand}p^¼ ðbC:$vÞ ^ ^nare the traction vectors, respectively associated withvandv. Alternatively, traces^ ðv;pÞonCof the solution to problem(32)satisfy integral Eqs.(10) and (11)with the free field given by

v^FðxÞ ¼ Z

S_obs

u

_;uðuðnÞ;u^obsðnÞ;nÞ Uðx;nÞdSn; x2X: ð33Þ

The nature(32c)of the adjoint excitation and the assumed boundedness ofSobsensure that (v,p) satisfies the generalized radiation condition(14).

4.3. Shape sensitivity formula Settingðw;wÞ ¼ ðu^ ;^u

Þin(31), formula(24)for the shape sensitivityJ becomes J ¼Re Aððv;vÞ;^ ðu;^u

ÞÞ

:

Choosingðw;wÞ ¼ ðv;^ ^vÞin(30)then readily yields, by virtue of field equations(32a,b)and the symmetry ofAð;Þ, the following expression forJ, where the displacement shape sensitivity no longer appears:

J

¼ Re Z

C

½nEðu;vÞ þn^bEðu;^ ^vÞ hdS

: ð34Þ

Eq.(34)is, however, not well suited for applications where the free and adjoint solutions are computed by means of the BEM, as it features displacement gradients onC. This is addressed by introducing the decomposition

$_u¼$_{Suþu;nn ð35Þ}

of a gradient in terms of its tangential component$_Suand the normal derivativeu;n, and expressing the latter in terms of$_Su andtby inverting the relationshipt¼ ðC:$_{uÞ nwhereby}

u;n¼DDt: ð36Þ

Here, the second-order tensorDand the combinationDtare, respectively defined byD¼ ½nCn¹(exploiting the minor symmetry ofC), and

Dt¼t ðC:$_{SuÞ n: ð37Þ}

By virtue of(35)–(37), tensor functionEðu;vÞcan be written in terms of the interfacial tractions and tangential displace- ment gradients, so that

nEðu;vÞ h¼ ½$_Su:C:$_SvDtDDp

qx

²uvhn ðt$_Svþp$_{SuÞ h:}

This finally allows to express(34)in terms of quantities directly available from the boundary element solution. Utilizing transmission conditions(5) and (32f), the desired form of the shape sensitivity result is thus established as

J ¼Re Z

C

½$_Su:ðbCCÞ:$_SvD^{^}tDbDp^{^}þDtDDp ð

q

^

q

Þ

x

²uvhndS

; ð38Þ

withD^{^}t;Dp^{^}defined by(37)withC;nreplaced byCb;n, and^ Db¼ ½n^Cbn^¹.

In the case where either material has isotropic elasticity (which is not a prerequisite for the derivation of(38)), tensors C;DorbC;Dbare given in terms of the shear modulus

l

and Poisson’s ratio

m

of the relevant material (withIdenoting the symmetric fourth-order identity tensor) by

C¼2

l m

12

m

^IIþI

h i

; D¼1

l

^I

1

2ð1

m

Þnn

:

5. Material sensitivity

The material sensitivity of the generic cost function(2)isa priorigiven by J⁰¼Re

Z

S_obs

u

;uðu;u^obs;nÞ u⁰dS

( )

: ð39Þ

5.1. Domain integral formulation, adjoint solution approach

Proceeding along the lines of Section4, the domain formulation of the displacement material sensitivityðu⁰;u^⁰Þis found by considering the material sensitivity of weak statement(7), i.e.

Aððu⁰;^u⁰Þ;ðw;wÞÞ ¼ ^ Z

b

X

^a⁰ðu;^ wÞdV;^ 8ðw;wÞ 2^ Vð0Þ ð40Þ where the bilinear density^a⁰is defined by(9b)with the inclusion characteristicsCband

q

^replaced, respectively by their sen- sitivitiesCb⁰and

q

^⁰.

Expression(39)suggests thatJ⁰can be expressed in terms of the previously-defined adjoint problem(31). Indeed, on settingðw;wÞ ¼ ðv;^ ^vÞin(40),ðw;wÞ ¼ ðu^ ⁰;^u⁰Þin(31), and proceeding as before, one finds that

J⁰¼ Re Z

bX

a^⁰ðu;^ ^vÞdV

: ð41Þ

The domain-integral format of formula(41)is obviously impractical for BEM-based applications. For that reason, alterna- tive approaches that facilitate the evaluation ofJ⁰using BEM forward and adjoint solutions are examined next.

Before proceeding to the BIE specialization of(41)and the subsequent numerical results, both focused here on semi-infi- nite host domains, it should be emphasized that the shape sensitivity formula(38)and its material counterpart(41)are gen- eral in the sense that they uniformly apply to finite, semi-infinite or infiniteðR³Þhost domains. The present BIE formulation and its underlying assumption of a semi-infinite ‘‘host” define an illustration, not a limitation, of the proposed shape-mate- rial sensitivity framework.

5.2. Surface integral formulation, direct approach

The direct differentiation approach makes use of(39)with the material sensitivityu⁰onSobsevaluated by differentiating the governing boundary integral equations. On expressing the governing pair(10) and (11)in operator form as

T½uðxÞ U½tðxÞ ¼u^FðxÞ b

T½uðxÞ Ub½tðxÞ ¼0 ðx2CÞ; ð42Þ

and keeping in mind that the full-space Green’s tensorsU,b Tbdepend on the inclusion’s material parameters whereasU,T and the free fieldu^Fdo not, the sensitivitiesu⁰;t⁰onCcan be shown to solve the pair of integral equations

T½u⁰ðxÞ U½t⁰ðxÞ ¼0

Tb½u⁰ðxÞ U½tb ⁰ðxÞ ¼Ub⁰½tðxÞ Tb⁰½uðxÞ ðx2CÞ ð43Þ where integral operatorsUb⁰,Tb⁰featured on the right-hand side are defined in terms of the respective Green’s tensor deriv- ativesUb⁰,Tb⁰, seeAppendix A.3.

Once Eq.(43)are solved foru⁰;t⁰onC, the displacement sensitivityu⁰onSobsfollows by taking the material sensitivity of representation formula(13). On substituting the resulting expression into(39), the material sensitivityJ⁰is finally given, using an operator notation similar to that in(42), by

J⁰¼Re Z

S_obs

u

_;uðu;u^obs;nÞ ðU^obs½t⁰ðxÞ T^obs½u⁰ðxÞÞdS

( )

: ð44Þ

5.3. Surface integral formulation, adjoint solution approach

As an alternative to the above direct differentiation strategy, an[4]adjoint field approach for the evaluation ofJ⁰as a surface integral may be formulated as follows. Let the new adjoint stateðv;^vÞbe defined as the solution of a transposed sys- tem of integral equations, written in weak form as

Z

C

fvðxÞ ðT½wðxÞ U½twðxÞÞ þ^vðxÞ ðTb½wðxÞ Ub½twðxÞÞgdSx¼ Z

S_obs

u

;u ðU^obs½twðxÞ T^obs½wðxÞÞdSx; 8ðw;twÞ ð45Þ

wherewandtware trial (vector) functions onC. Next, multiplying the first and the second equation in(43)respectively by vðxÞand^vðxÞ, integrating the result overx2C, and adding the identities one obtains

Z

C

fvðxÞ ðT½u⁰ðxÞ U½t⁰ðxÞÞ þvðxÞ ð^ Tb½u⁰ðxÞ Ub½t⁰ðxÞÞgdSx¼ Z

C

^vðxÞ ðbU⁰½tðxÞ Tb⁰½uðxÞÞdSx; 8ðw;twÞ: ð46Þ Settingðw;twÞ ¼ ðu⁰;t⁰Þin(45), subtracting the resulting identity from(46)and recalling the material sensitivity formula (44), material sensitivityJ⁰follows in a surface integral, adjoint-based form as

J⁰¼ Z

C

vðxÞ ðb^ U⁰½tðxÞ Tb⁰½uðxÞÞdSx: ð47Þ

The above material sensitivity formula is in particular well suited for use within a Galerkin BEM framework, as it involves two nested surface integrals.

6. Computational treatment

In what follows, an inclusion identification method based on the proposed shape-material sensitivity approach and an augmented Lagrangian cost functional is implemented using a BEM framework and the underlying assumption that the background domainXis semi-infinite.

6.1. Boundary integral approximation

To illustrate the utility of sensitivities(38) and (44)for elastic-wave identification of penetrable defects, let the trial obstacle Xb be isotropic and described in terms of a finite seta¼ ða1;a2;. . .;aD;

l

^;^

m

;

q

^ÞofDþ3 geometric and material parameters, where

l

^,^

m

and

q

^denote respectively the shear modulus, Poisson’s ratio, and mass density of the defect. Intro- ducing an auxiliary notation

J_aðaÞ JðXbða1;. . .;aDÞ;bCð

l

^;^

m

Þ;

q

^Þ ð48Þ

to reflect the defect parametrization and making reference to(20) and (21), the sensitivitiesoJ_a=oakused for minimizingJ_aðaÞ are computable according to

Geometric: oJa

oak

¼Jj_s¼a_k; k¼1;2;. . .;D ð49aÞ

Material: oJ_a oak

¼J⁰j_s¼ak; k¼Dþ1;Dþ2;Dþ3: ð49bÞ

Boundary element discretization.For the evaluation of surface integrals overoXb¼Cthat are involved in the computation ofJ andJ⁰, one may assumeC¼SK

e¼1Ee, wherefEeg^K_e¼1are closed and non-overlapping surface elements. Following the usual approach, each boundary elementEeC is parametrized by a mappingE!Eethat introduces local coordinates, g¼ ð

g

¹;

g

²Þ 2E, whereEis a polygonal domain inR². The approximating boundary surface,C^h¼SK

e¼1E^h_e, can then be gener- ated by interpolating a set of parameter-dependent nodesn^qðaÞ 2Eewith pre-defined mesh connectivity. To this end, theQ- noded approximationE^h_eof a generic surface elementE_eCis written as

nðgÞ ¼X^Q

q¼1

NqðgÞn^qðaÞ; n2E^h_e; n^q2Ee; g2E; ð50Þ

whereNq are the relevant shape functions. By virtue of(50)the normal transformation velocityhnin(38)is, for a given parameterak, approximated as

hnðnÞ h^k_nðnÞ ¼X^Q

q¼1

NqðgÞon^q oak

n; n2E^h_e;n^q2Ee;k¼1;2;. . .;D ð51Þ

wherendenotes the unit normal to the surface. Assuming next the isoparametric representation of elastodynamic quanti- ties, the boundary displacement (u) and traction (t) fields overCare interpolated at a generic pointn2E^h_ein terms of the nodal displacementsu^qand tractionst^qatn^q2E_eas

u¼X^Q

q¼1

NqðgÞu^q; t¼X^Q

q¼1

NqðgÞt^q: ð52Þ

Evaluation of shape sensitivities.The computation ofJ andJ⁰entails solving transmission problems associated, respec- tively with the primary fieldu, the adjoint fieldv, and the material sensitivity fieldu⁰for each material parameter associated with the inclusion (i.e. a total of five problems under the assumption of isotropic inclusions). With reference to(12), (33), (42), (43), (50) and (52), the discrete algebraic systems for these fields can be written as

Primary HUeGeT¼U^F;

HbUeGbTe¼0; ð53aÞ

Adjoint HVeGPe¼V^F;

HbVeGbeP¼0; ð53bÞ

Material sensitivity HUe⁰GeT⁰¼0;

HbUe⁰GbeT⁰¼Gb⁰eTHb⁰Ue ð53cÞ

where vectorsUeandeTcontain the nodal approximations of the primary (displacement and traction) fieldsuandt; coeffi- cient matricesH,G,Hb andbGapproximate the respective integral operatorsT;U;Tb andUbin(42);Ve andPeare associated with the respective adjoint fieldsvandp;U^F andV^Fcollect the nodal values of the free fields(12) and (33); vectorsUe⁰ andeT⁰refer respectively tou⁰andt⁰, whileHb⁰andbG⁰are the matrix discretizations ofTb⁰andUb⁰in(43). The coefficient matri- ces are the same for all three discretized systems, which allows for a computationally-effective solution of(53b) and (53c) once the primary problem(53a)has been solved.

In the context of(38), however, the shape sensitivity computation requires not only the primary and adjoint fields onC, but also their surface gradients. By virtue of(52), the required surface gradient ofuis approximated as

$_Su¼^X

Q

q¼1

$_{SNqðgÞ u}^q _ð54Þ

over each boundary element, with a similar expression applying in terms of$_Sv. To evaluate the tangential derivative$_SNq in(54), let the local companion basisfr1;r2;ngat any pointn2E^h_ebe defined from the (differentiable) boundary element parametrizationn¼nðgÞof(50)by

r1¼X^Q

q¼1

oNq

o

g

¹ⁿ

q; r2¼X^Q

q¼1

oNq

o

g

²ⁿ

q; n¼ r1 r2

kr1 r2k; ð55Þ

so that vectorsr1andr2are tangent toE^h_e. The surface gradient of shape functions$_SNqthen takes an explicit form

$_SNq¼n ^oNq o

g

^2r1

oNq

o

g

^1r2

q¼1;2;. . .;Q: ð56Þ

Since the tripletfr1;r2;ngdefined by(55)forms a positive-oriented basis inR³, reversing the connectivity of a given ele- ment (i.e. listing its nodes in opposite order) leads to the sign-reversal ofnthrough swapping ofr1andr2. Hence, a desired orientation for the approximate surfaceC^hcan be achieved by adequately setting the mesh connectivity. Here, quantities pertaining toX(background) andXb (inclusion) in sensitivity formula(38), and consequently matricesHandHbin(53a–

c), are defined in terms of the inward and outward orientations ofC, respectively. Consistent element orientation is thus ensured by using two opposite mesh connectivity tables forC^h(one ‘‘direct” forX, the other ‘‘reverse” forXb). This method allows for systematic generalization towards multiple-inclusion or nested-inclusion configurations. The surface gradients (54)are insensitive to the choice of mesh connectivity orientation.

6.2. Parallel computation

Owing to the high computational cost commonly associated with 3D inverse scattering, regularized boundary integral treatment[36]of the primary, adjoint, and material sensitivity problems in(53a–c)is implemented, together with formulas (38) and (44)forJ andJ⁰, in adata-parallelcode using the message-passing interface (MPI)[35]. Data-type parallelism nor- mally applies when identical operations are performed concurrently on multiple data items. With reference to(11) and (53a), such is the case with repeated, time-consuming, computation of the elastodynamic Green’s tensorsUandT(under- lying the evaluation of matricesHandG), in situations when the reference domainXis semi-infinite. In contrast to the fun- damental solution for an infinite domainR³which is available in closed form (e.g.UbandTbunderpinning matricesHbandbG, see(A.8a,b)), elastodynamic Green’s tensors for a semi-infinite solid are given as improper integrals[22]whose numerical quadrature entails two to three orders-of-magnitude higher computational effort.

On denoting bynpthe number of processes and byNthe number of BEM degrees of freedom onC, the code is accordingly parallelized by block-distributing the computation ofHandGmatrices (both of dimensionN N) where every participating process is assigned approximatelyN=npcolumns of each array. As vectorsU^FandV^Fin(53a,b)also involve the (displace- ment) Green’s tensorUfor the reference domain, see(12) and (33), their computation is similarly distributed among the participating processes. For consistency, matricesHb andbGare likewise computed in parallel fashion, even though this does not result in a meaningful reduction of the run time owing to the closed form nature ofUb andbT. Once computed for the solution of the primary problem, the LU-factorized ‘‘global” coefficient matrix (combiningH,G,HbandbG) is stored and reused for solving the adjoint and material sensitivity problems.

To illustrate the performance of the parallel code the speed-up ratioQs, i.e. the ratio of the elapsed time of a serial pro- gram over that of its parallel counterpart withnpprocesses, is considered. The computation is performed on the IBM Blade-

Center H cluster equipped with 307 LS21 nodes, each containing two dual-core 2.6 GHz Opteron processors sharing 8 GB of memory. For the purpose of comparison, the inclusion is described as a nine-parameter ellipsoidðDþ3¼9Þcharacterized by its three centroidal coordinates, three semi-axes, and three material constants. The boundary element mesh approximat- ing the surface of the defect has 650 nodes; the testing configuration is comprised of 25 uniaxial sources and 36 triaxial receivers located on the surface of a semi-infinite solid (seeFig. 1).Table 1shows the CPU times per evaluation of the cost functionJ_aand its sensitivitiesoJ_a=oakðk¼1;. . .;9Þ, the speed-up ratiosQ_s, and a measure of the per-processor efficiency Qs=npfor the sample problem in asemi-infinitesolid. As a point of reference, computation of the analogous problem when the host domain isinfiniteðX¼R³Þ, takes 1 minute and 31 seconds on a single processor. In the ensuing examples, the half- space calculations are performed withnp¼48 which represents a reasonable compromise between the speed-up ratio and per-processor efficiency.

6.3. Defect parametrization

The geometry of the trial defectXb is, for the ensuing numerical experiments, described in terms of an ellipsoid whose principal axes are aligned with the reference Cartesian framefO;n1;n2;n3g; its evolution within the host domainXis re- stricted to (i)translationand (ii)stretchalong the principal axes. For problems involving identification of a single isotropic defect, such description entails the use of a nine-dimensional parametric space

a¼ c1

d;c2

d;c3

d;

a

1

d;

a

2

d;

a

3

d;

l

^

l

^{;^}

m

;

q

^

q

ð57Þ (i.e.D¼6), which incorporates the defect’s centroidal motionðci;i¼1;2;3Þ, principal stretchesð

a

i;i¼1;2;3Þ, and material characteristicsð

l

^;

m

^;

q

^Þ. Parametersa_kin(57)are moreover defined in dimensionless fashion using material characteristics ð

l

;

q

Þof the reference solid and an arbitrary length scaled. With such definitions, analytical dependence of the nodal coor- dinates,n^q¼n^qðaÞ, of the surface mesh on the evolving defect boundaryCis introduced as an affine deformation of the boundary element mesh for a reference unit sphereS(described by Lagrange coordinatesðX1;X2;X3Þ) so that

n^q_i ¼ciþ

a

iX^q_i; X^q2S; i¼1;2;3 ð58Þ

assuming no summation over indexi. On the basis of(57) and (58), one finds that the normal transformation velocitiesh^k_n defined by(51)are given by

ðh^k_n;k¼1;. . .;DÞ ¼ n1;n2;n3;n₁c1

a

1

n1;n₂c2

a

2

n2;n₃c3

a

3

n3

d: ð59Þ

Since formulas(38) and (44)are not restricted to simply-connected defects, one could parametrize the subsurface heter- ogeneity as multiple defects, using e.g. description(57)for each. The assumed topology then cannot be altered during the minimization process. As to the correct choice of ‘‘initial” topology (e.g. in terms of the number of defects), such preliminary information could be obtained from the available measurements using for instance the methods of topological sensitivity [18,12,19]or linear sampling[31,8,20].

Fig. 1.Testing grid and sample defects in a semi-infinite solidn3>0ðm¼0:35Þ.

Table 1

Sample CPU times [min:sec] per evaluation the cost function and its sensitivities

np 2 4 8 12 24 48 96

CPU time 46:01 23:27 12:41 9:05 5:52 3:24 2:31

Qs 1.98 3.89 7.20 10.05 17.01 26.85 36.28

Qs=np 0.99 0.97 0.90 0.84 0.70 0.56 0.38

6.4. Minimization

As examined earlier, the goal of this study is the 3D identification of ‘‘penetrable” subsurface defects via the minimization of cost functionalJðXb;bC;

q

^Þgiven by(2). Here it is important to remember, however, that certain arguments ofJ, most notably the material characteristics of the trial defect, are subject to inequality constraints that must be enforced to maintain the physical relevance of the solution. In the context of isotropic elasticity assumed in the ensuing examples, one finds that besides

q

^>0, one must have

l

^>0 and1<^

m

<0:5 to sustain the positive definiteness of the strain energy density[2].

While numerous techniques are available for nonlinear minimization subject to inequality constraints[34], most of such algorithms provide only ‘‘soft” bounds that can be violated during the minimization process. To aid the strict enforcement of the featured inequality constraints on

l

^;

m

^and

q

^, the nine-dimensional defect parametrization in(57)is restated using the transformed variablesb¼ ðb1;b2;. . .;b9Þ 2R⁹where

bk¼ak; k¼1;. . .;6; bk¼logðakÞ k¼7;9; b8¼logð0:5a8Þ ð60Þ

which ensure that

l

>0,

m

<0:5, and

q

>0. The cost function is then expressed naturally in the transformed variables throughJ_bðbÞ ¼J_aðaÞ, withJ_adefined by(48). On the basis of(60), the required sensitivities ofJ_bcan be computed in terms ofoJ_a=oa_kgiven by(38), (44) and (49b)as

oJb

obk

¼oJa

oak

; k66; oJb

obk

¼oJa

oak

ak k¼7;9 oJb

ob8¼oJa

oa8ða80:5Þ:

The minimization problem is posed in a constrained fashion, using parametrization(60), as min

b JbðbÞ; CiðbÞP0 8i2I; ð61Þ

where the ‘‘soft” inequality constraintsCi(Ibeing a set of integers) reflect anyadditionalrestrictions in terms of defect’s centroidal coordinatesðc1;c2;c3Þ, semi-axesð

a

1;

a

2;

a

3Þ, and material propertiesð

l

^;^

m

;

q

^Þ. The inequality constraints used in the ensuing numerical examples are listed inAppendix A.4.

Following[34], optimization problem(61)is for practical implementation reduced, using slack variables, to theuncon- strainedminimization

minb L_Aðb;k^m;

c

_mÞ ð62Þ

of an augmented Lagrangian L_Aðb;k^m;

c

mÞ J_bðbÞ þX

i2I

wðCiðbÞ;k^m_i ;

c

mÞ;

wðC;k;

c

Þ ¼ kCþC²=2

c

C6

c

k;

c

k²=2 C>

c

k

( ð63Þ

whose gradient is computable as rbL_Aðb;k^m;

c

mÞ ¼r^LAðbÞ X

i2IjC_iðbÞ6cmk^m_i

k^m_i CiðbÞ=

c

m

rCiðbÞ: ð64Þ

Given the initial penalty parameter

c

₀>0, tolerance

s

0>0, starting pointb⁰, and reference vector of Lagrange multipliers k⁰, the augmented Lagrangian method introduces a sequenceðm¼1;2;. . .Þof unconstrained minimization problems with explicit Lagrange multiplier estimatesðk^mÞand decreasing penaltiesð

c

_mÞthat produce a good estimate of the local KKT (Kar- ush–Kuhn–Tucker) solution,b^I, of(61)even when

c

is not particularly close to zero[34]. This latter feature is highly desir- able as it reduces the possibility of ill-conditioning that commonly occurs for vanishing values of the penalty parameter

c

. The algorithm terminates whenkrbL_Ak<

s

^I, where

s

^Iis the user-chosen ultimate tolerance.

For any given iteratem, the nonlinear minimization(62)is effected using the BFGS quasi-Newton method[34](with stop- ping criterion defined bykrbL_Ak<

s

m) and an inexact line search based on the strong Wolfe conditions[30]. Due to the unconstrained character of minimization subproblems(62), constraints(A.14)are ‘‘soft” in that they do not by themselves preventbfrom reaching physically inadmissible, or merely undesirable, values (e.g. penny-shaped ellipsoids which may lead to an ill-conditioned BEM solution). To deal with the problem, the line search algorithm embedded in(62)has been aug- mented by a step-reduction feature that preventsbfrom exceeding the soft-bound limits(A.14)by more than 30%. For the numerical examples presented next, the internal parameters were set to

c

₀¼0:025,

s

0¼100, k0¼1,

c

_mþ1=

c

_m¼

s

mþ1=

s

m¼0:2, and

s

^I¼10³. 7. Results

The effectiveness of the proposed shape-material sensitivity approach as a tool for reconstructing buried penetrable ob- jects is now demonstrated on a set of numerical results. In all examples to follow, the buried obstacle is ‘‘illuminated” using N¼4 4¼16 point forces, sequentially applied at locationsxⁿðn¼1;. . .;NÞin then3-direction over the square testing grid

ðn1;n₂Þ 2 ½3d;3d ½3d;3din then₃¼0 plane, wheredis a reference length; for each source, the response of the solid is monitored usingM¼5 5¼25 triaxial receivers with locationsx^mðm¼1. . .MÞarranged (in the same plane) as shown in Fig. 1. The distance function

u

ðu;u^obs;nÞin(2)is taken in the least-squares form and given by

u

ðu;u^obs;nÞ ¼1 2

X^N

n¼1

X^M

m¼1

dðnx^mÞjuu^obsj²:

Problem quantities are normalized using the length scaledtogether with the shear modulus and mass densityð

l

;

q

Þof the reference solid; in particular, the nondimensional angular frequency

x

¼

x

dð

q

=

l

Þ¹⁼²is introduced. In all examples, the reference solid is additionally characterized by

m

¼0:35.

7.1. Sensitivity evaluation

To verify the numerical implementation, geometric and material sensitivitiesoJ_a=oa_kðk¼1;2;. . .;9Þstemming from(38) and (44)are compared with their central difference approximations computed using the boundary integral approach of[36]

and a surface mesh with 650 eight-noded quadrangular elements. The comparison is performed at a¼ ð:1; :1;3;:5; :5; :5;2; :35; :9Þfor a ‘‘true” ellipsoid given bya^true¼ ð0;0;3;:5; :5; :5;5; :25;1Þ, assuming an infinite reference domainX¼R³to ensure maximum accuracy for the Green’s functions and focus on the performance of the proposed com- putation scheme. The frequency of illumination corresponds to

x

¼3. FromTable 2, one can see that the relative discrep- ancy between the sensitivity formulas and their central difference approximations (computed using ±4% perturbation on each parameter) does not exceed 0.4%. It is moreover important to note that the speedup (i.e. the reduction in the compu- tational effort) over the central difference approach is approximately 1=ð2ðDþ3ÞÞwhereDþ3 is the total number of (geo- metric and material) design parameters. This estimate stems from the fact that formulas(38) and (44)essentially revolve around the solution of one forward problem (since the adjoint and material sensitivity problems then exploit the existing matrix factorization), whereas central difference evaluations entail the set-up and solution of two irreducible problems for eachak.

7.2. Obstacle reconstruction

Two examples are now presented to illustrate the reconstruction of penetrable defects in an isotropic, semi-infinite solid via shape sensitivities(38), material sensitivities(44), and the constrained minimization approach described in Section6.4.

In both examples, anticipating the non-convexity characterizing most inverse scattering problems, the initial trial defect is placed relatively close to its target. This assumption is made reasonable by the fact that probing techniques[42,9]based on e.g. topological sensitivity[18,12,19]or linear sampling[31,8,20]provide a reliable preliminary information about the defect location and material characteristics. While these techniques can in principle use the same experimental data exploited by the nonlinear minimization, their explicit coupling with the present scheme is beyond the scope of this study.

Hard obstacle.For this example the testing grid, placed on the surface of the half-spaceðn3¼0Þ, and the true defect (indi- cated as ‘‘Hard”) are shown inFig. 1. The frequency of illumination is again such that

x

¼3, corresponding to a shear wave- length kS¼2

p

d=3. The true defect, centered at ða1;a2;a3Þ^true¼ ð1; :5;3Þ, is described as an ellipsoid with semi-axes ða4;a5;a6Þ^true¼ ð:4; :6; :5Þ and material properties ð

l

^;^

m

;

q

^Þ^true¼ ð4

l

;0:25;1:1

q

Þ; the initial iterate is placed at a⁰¼ ð:4;:2;2:5;:5; :5; :5;1:8; :3;1Þ, seeFig. 1.

To avoid committing the ‘‘inverse crime”[16]whereby the same model is used to synthesize as well as to invert the data in an inverse problem, synthetic observationsu^obsare generated using a boundary element mesh with 1460 nodes, whereas the minimization exploits a coarser mesh with 650 nodes.Fig. 2illustrates the iterative reconstruction process for this exam- ple. To aid the physical insight, the bottom right panel depicts selected iterations in the 3D space, with the surface color of each iterate (i.e. trial defect) corresponding to its shear modulus according to the attached color bar. As can be seen from the display, the solution converges to the global minimum ofJ after approximately 70 iterations. Not surprisingly, the centroi- dal coordinates exhibit the fastest convergence, followed by that in terms of the semi-axes and material properties. It should be noted, however, that the synthetic data in this example contain no extraneous perturbations other than those caused by the use of dissimilar BEM meshes. To examine the effect of measurement uncertainties, synthetic observationsu^obsare next corrupted as

u~^obs:¼ ð1þ

.v

Þu^obs ð65Þ

over all source-receiver pairs, where

.

is the noise amplitude and

v

2 ½1;1is a uniform random variable.Table 3lists the reconstructed defect parameters for the noise amplitude levels of 0, 1, and 2%. The defect reconstruction is seen to be fairly Table 2

Sensitivitieso_a

kJoJ=oak: comparison with central differences

Sensitivity o_c1J o_c2J o_c3J o_a1J o_a2J o_a3J o_^lJ o_^mJ o_^qJ

Formulae 0.3884 0.3884 3.669 8.384 8.384 9.803 3.286 .07205 2.857

Finite diff. 0.3873 0.3873 3.666 8.384 8.384 9.802 3.289 .07231 2.856