characterising the shape and material properties ledger...

Characterising the shape and material propertiesof hidden targets from magnetic induction data

Ledger, Paul D and Lionheart, William RB

2013

MIMS EPrint: 2013.57

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IMA Journal of Applied Mathematics (2015) 1´23doi: 10.1093/imamat/**

Characterising the shape and material properties of hidden targets frommagnetic induction data

PAUL D. LEDGERCollege of Engineering, Swansea University,

Singleton Park, Swansea, SA2 8PP, UK.

WILLIAM R.B. LIONHEARTSchool of Mathematics, University of Manchester,

Alan Turing Building, Oxford Road, Manchester, M13 9PL, UK.

[Received on 5th October 2014, Revised on 13th April 2015]The aim of this paper is to show that, for the eddy current model, the leading order term for the per-turbation in the magnetic field, due to the presence of a small conducting magnetic inclusion, can beexpressed in terms of a symmetric rank 2 polarization tensor. This tensor contains information about theshape and material properties of the object and is independent of position. We apply a recently derivedasymptotic formula for the perturbed magnetic field, due to the presence of a conducting inclusion, whichis expressed in terms of a new class of rank 4 polarization tensors (H. Ammari, J. Chen, Z. Chen, J. Gar-nier and D. Volkov (2014) Target detection and characterization from electromagnetic induction data,Journal de Mathematiques Pures et Appliquees 101, 54-75.) and show that their result can be written inan alternative form involving a symmetric rank 2 tensor involving 6 instead of 81 complex componentsin an orthonormal coordinate frame. For objects with rotational and mirror symmetries we show that thenumber of coefficients is still smaller. We include numerical examples to demonstrate that the new polar-ization tensors can be accurately computed by solving a vector valued transmission problem by hp–finiteelements and include examples to illustrate the agreement between the asymptotic formula describing theperturbed fields and the numerical predictions.

Keywords:Polarization tensors, Asymptotic expansions, Eddy currents, hp-Finite elements, Metal detectors, Landmine detection.

1. Introduction

There is considerable interest in being able to locate and characterise conducting objects from measure-ments of mutual impedance between a transmitting and a receiving coil, where the coupling is inductiverather than due to the propagation of radio waves. The most obvious examples are in metal detection,where the goal is to identify and locate a highly conducting object in a low conducting backgroundand applications include security screening, archaeological searches, maintaining food safety as well asfor land mine clearance and the detection of unexploded ordnance (UXO). There is also considerableinterest in being able to produce conductivity images from multiple magnetic induction measurements,most notably in magnetic induction tomography for medical applications (Griffiths 2005, Zolgharni,Ledger, Armitage, Holder, & Griffiths 2009) and industrial applications (Gaydecki, Silva, Ferandes, &Yu 2000, Soleimani, Lionheart, & Peyton 2007). Furthermore, eddy current sensing techniques are alsocommonly used for the monitoring and defect detection in steel structures such as oil pipe lines andcontainment vessels as well as monitoring corrosion of steel reinforcement bars in concrete structures

c Institute of Mathematics and its Applications 2012; all rights reserved.

2 of 23 P.D. LEDGER and W.R.B. LIONHEART

such as bridges and buildings (Simmonds & Gaydecki 1999).The detection of land mines presents a huge challenge, the United Nations estimates that “there

are more than 110 million active mines [...] scattered in 68 countries with an equal number stockpiledaround the world waiting to be planted” and that “every month over 2,000 people are killed or maimedby mine explosions United Nations (1997). Although metal detectors offer a portable means of detectioncurrent techniques are often not able to distinguish between benign and dangerous targets and, therefore,there is great interest in technological advancements that might increase the speed at which mines couldbe detected and increase safety.

By considering the time harmonic regime, and denoting the electric field intensity vector by E

and the corresponding magnetic field intensity vector by H , Somersalo, Isaacson, & Cheney (1992)compute the Frechet derivative of the Dirichet to Neumann (DtN) map with respect to small changesin permeability, permittivity and conductivity, δµ, δ and δµ, respectively, in a bounded domain Ω andobtain

ż

BΩn ¨ pE ˆ H

˚0 ´ E

˚0 ˆ HqdS “

ż

ΩpiωδµH0 ¨ H˚

0 ´ iωδE0 ¨ E˚0 ` δσE0 ¨ E˚

0 qdΩ`

Oppδµ, δ, δσq2q,as the parameter perturbations go to zero, where ω is the angular frequency, pE0,H0q and pE˚

0 ,H˚0 q

are pairs of unique solutions to the unperturbed problem (i.e. plane wave solutions) and E :“ E0 `δE,H :“ H0 ` δH denote the solutions to the Maxwell system using the perturbed parameters. However,this is different to the situation of interest in the present work, which concerns the field perturbationsdue to presence of an inclusion with a large conductivity contrast compared to the background medium.

For a bounded domain, Ammari, Vogelius, & Volkov (2001) have obtained the correspondingasymptotic formula that describes the perturbation of the DtN map, which, when particularised forthe case of single small inclusion with parameters µ˚, ˚ and σ˚ located at z, has a leading order termthat can be expressed in terms of wpzq ¨ pT pcrqE0pzqq and ∇ ˆ wpzq ¨ pT pµrqH0pzqq as the objectsize, α, tends to zero where pE0,H0q denote the solutions in absence of the inclusion and w denotesa solution to the free space vector wave equation. Here, µr ´ 1 “ µ˚µ0 ´ 1 denotes the relativepermeability perturbation, cr ´ 1 “ 10p˚ ´ iσ˚ωq ´ 1 the complex permittivity perturbation andT pcq the rank 2 Polya & Szego (1951) polarization tensor parameterised by a contrast c. This tensorcontains information about the shape and material properties of the inclusion, simplifies to a multiple ofthe identity tensor for a sphere and is diagonalizable for an ellipsoid (Ammari & Kang 2007). Althoughthe perturbation in complex permittivity does include the possibility of describing a conducting objectin a non–conducting background this result is only applicable to wave propagation (electromagneticscattering) problems at fixed wavenumber k :“ ω

?0µ0, where 0, µ0 denote the free space values of

permittivity and permeability, respectively.Asymptopic formulae that describe the perturbation in the electromagnetic fields in unbounded do-

mains due to the presence of a conducting dielectric (and/or magnetic) inclusion have also been obtainedfor the situation where α Ñ 0 (for fixed k) (Ammari & Volkov 2005, Ledger & Lionheart 2014) andalso for a non-conducting dielectric (magnetic) object where k Ñ 0 (for fixed α) (Baum 1971, Dassios& Kleinman 2000, Keller et al. 1972, Kleinman 1967, 1973, Kleinman & Senior 1982, Ledger & Lion-heart 2014). The leading order terms in these results can also also expressed in terms of the rank 2 Polya& Szego (1951) polarization tensor parameterised by µr and cr (µr and r :“ ˚0 for the latter). Theformer category applies when the inclusion is small and conducting, but not to the low-frequency caseof k Ñ 0, while the latter does apply when k Ñ 0, but only for a non–conducting object. Henceneither category describes the situation in magnetic induction where the inclusion is conducting and the

Characterising the shape and material properties of hidden targets from magnetic induction data 3 of 23

frequency is small.Ammari, Chen, Chen, Garnier, & Volkov (2014b) have recently obtained an asymptotic expansion,

which, for the first time, correctly describes the perturbed magnetic field as α Ñ 0 for a conducting(possibly magnetic and multiply connected) object in the presence of a low–frequency backgroundmagnetic field, generated by a coil with an alternating current. Rather than consider the limit as k Ñ 0for the full time–harmonic Maxwell system, they instead consider the eddy current model where thedisplacement currents are neglected. The leading order term they obtain is written in terms of two newpolarization tensors, called the permeability and conductivity tensors, which have different ranks, andthe background magnetic field evaluated at the position of the centre of the inclusion. An algorithmfor identifying conducting objects from induction data based on a process of classification by matchingagainst a library of precomputed polarization tensors has also recently been proposed by Ammari, Chen,Chen, Vollkov, & Wang (2014a).

Our new contributions include considering, in detail, the properties of the conductivity and perme-ability tensors introduced byAmmari et al. (2014b). These studies enable us to show that, in practice,each of the tensors can be represented by just 9 components in an orthonormal coordinate frame. Weshow that it is also possible to express the perturbed magnetic field in terms of a reduced rank 2 con-ductivity tensor and introduce a new symmetric rank 2 tensor, different from the Polya & Szego (1951)tensor, which describes the inclusion in terms of just 6 independent components. For a simple object,with rotational or mirror symmetries, we show that the number of independent coefficients is still fewer.These results have important consequences for characterising conducting objects of different shapes and,in particular, target identification using dictionary based inverse algorithms. We include a descriptionof an efficient numerical approach for accurately computing these new tensors, which is based on theregularised eddy current formulation using hp-finite elements presented in Ledger & Zaglmayr (2010).We also present simulations, which indicate an excellent agreement between the asymptotic formula ofAmmari et al. using the numerically computed polarization tensors and the fields obtained from solvingthe full eddy current problem, for a range of case studies.

The presentation of the paper proceeds as follows: In Section 2. we describe the mathematicalmodel and simplifying assumptions made about the problem under consideration. Then, in Section 3.,we quote the asymptotic formula of Ammari et al. (2014b) and choose to express it in an alternativeform containing a single rank 4 tensor. In this section, we also state our main result, which showsthat their asymptotic formula can instead be written in terms of a new symmetric rank 2 polarisationtensor, then, in Section 4., we prove a series of lemmas required for the proof of our main result. InSection 5., we discuss how further reductions in the number of independent components can be obtainedif the object is rotationally symmetric or has mirror symmetries and the additional simplifications fora sphere. Then, in Section 6., we describe how the independent components of the tensors can berecovered from practical measurement data. In Section 7., we describe an approach to the numericalcomputation of the polarization tensors based on hp–finite elements, in Section 8., we present a seriesof numerical results to validate this approach and finish with our concluding remarks in 9..

2. Mathematical Model

We follow Ammari et al. (2014b) and consider an electromagnetic inclusion in R3 of the form Bα “z ` αB, where B Ă R3 is a bounded, smooth domain containing the origin. Let Γ and Γα demote theboundary of B and Bα, respectively, and µ0 the permeability of free space. We continue to follow their


notation and writeµα “

"µ˚ in Bα

µ0 in R3zBασα “

"σ˚ in Bα

0 in R3zBα, (2..1)

where µ˚ and σ˚ denote the object’s permeability and conductivity, respectively, which we assume tobe constant.

The time harmonic fields Eα and Hα that result from a time varying current source, J0, located apositive distance from Bα and satisfying ∇ ¨ J0 “ 0 in R3, and their interaction with the object Bα,satisfy the eddy current equations (Ammari, Buffa, & Nedelec 2000)

∇ ˆ Eα “ iωµαHα in R3, (2..2a)∇ ˆ Hα “ σαEα ` J0 in R3, (2..2b)

Eαpxq “ Op|x|´1q,Hαpxq “ Op|x|´1q as |x| Ñ 8, (2..2c)

where ω denotes the angular frequency. Letting α “ 0 in (2..2) we obtain the corresponding fields,E0 and H0, that result from time varying current source in the absence of an object. As describedby Ammari et al. (2000), Hiptmair (2002), the eddy current model is completed by ∇ ¨ Eα “ 0 inBc

α. Furthermore, Ammari et al. (2000) show that uniqueness of Eα in Bcα is achieved by additionally

specifying ż

Γα

n ¨ Eα|`dx “ 0, (2..3)

and that, in practice, the decay of the fields is actually faster than the |x|´1 stated in the original eddycurrent model.

The task is to describe the leading order term for the perturbation Hαpxq ´ H0pxq, caused by thepresence of the object Bα, in terms of a polarization tensor, which is independent of position, with allpositional dependence appearing only through the evaluation of background fields.

3. Main result

For the eddy current model described above, Theorem 3.2 in Ammari et al. (2014b) describes the per-turbed magnetic field at positions x away from z, due to the presence of object Bα, as α Ñ 0 whenν :“ α2ωµ0σ˚ “ Op1q, such that ω grows asymptotically as fast as 1α2, which we choose to write inthe alternative form stated below 1:

THEOREM 3..1 Let ν “ Op1q and α Ñ 0 then, for x away from the location of the inclusion z,

pHα ´ H0qpxqj “pD2xGpx, zqqmMmjipH0pzqqi ` pRpxqqj , (3..1)

where Gpx, zq “ 14π|x´z| is the free space Laplace Green’s function, |Rpxq| ď Cα4H0W 2,8pBαq

such that Rpxq “ Opα4q is a small remainder term and C is as defined in (Ammari et al. 2014b). Inthe above, Einstein summation convention is used,

Mmji :“ Pmji ` xxNmji,xxNkmji :“ δkjδmNi,

1Note that I3 on page 10 of Ammari et al. (2014b) requires that ∇xGpx,αξ`zqq´∇xGpx,zq´αD2xGpx,zqξ “ Opα2q,

but instead the correct choice is ∇xGpx,αξ ` zqq ´ ∇xGpx,zq ` αD2xGpx,zqξ “ Opα2q, which adds a minus sign to the

definition of P in (3..2).


are components of rank 4 tensors in an orthonormal coordinate system and

Pmji :“ ´ iνα3

2ej ¨

ˆe ˆ

ż

Bξmpθi ` ei ˆ ξqdξ

˙, (3..2)

Ni :“ α3

ˆ1 ´ µ0

µ˚

˙ ż

B

ˆe ¨ ei ` 1

2e ¨ ∇ ˆ θi

˙dξ. (3..3)

Furthermore , ei is a unit vector for the ith Cartesian coordinate direction, pξqm “ ξm is the mthcomponent of ξ 2 and θi, i “ 1, 2, 3, are the solutions to the transmission problem

∇ξ ˆ µ´1∇ξ ˆ θi ´ iωσα2θi “ iωσα2

ei ˆ ξ in B Y Bc, (3..4a)∇ξ ¨ θi “ 0 in Bc, (3..4b)

rθi ˆ nsΓ “ 0,“µ´1∇ξ ˆ θi ˆ n

‰Γ

“ ´2“µ´1

‰Γei ˆ n on Γ , (3..4c)

θipξq “ Op|ξ|´1q as |ξ| Ñ 8, (3..4d)

where ξ is measured from the centre of B. By additionally specifying an analogues condition to (2..3)the uniqueness of θi in Bc can be achieved.

Proof. Considering Theorem 3.2 in Ammari et al. (2014b), rewriting it in tensoral notation and then

extending N “ Njiej b ei to xxN “ xxNmjie b em b ej b ei the result (3..1) immediately follows.However, it remains to show that P “ Pmjiie b em b ej b ei and N are indeed rank 4 and rank2 tensors in an orthonormal coordinate frame, respectively. To do so it suffices to show that theircomponents obey appropriate transformation rules. In Proposition 4.3 in Ammari et al. (2014a) anequivalent statement to

P 1mki “ RrRmsRktRiuPrstu, (3..5)

for an orthogonal rotation R with determinate |R| “ 1 is shown. Repeating similar steps to the proofof Proposition 4.3 in Ammari et al. (2014a), but now tracking |R|, we see that (3..5) also holds for thecase where R “ Rjiej b ei is orthogonal with |R| “ ˘1 and hence the components of P transform asa rank 4 tensor. Introducing µr “ µr “ µ˚µ0 in B and µr “ 1 in Bc then, by writing the componentsof N in the alternative equivalent form

Nji “ α3rµ´1r sΓ

ˆż

Bei ¨ ejdξ ` 1

2

ż

Γej ˆ n

´ ¨ θiˇ´ dξ

˙,

following similar steps and using a notation similar to that in Proposition 4.3 in Ammari et al. (2014a)

2We will use the notation ξm for the mth component of ξ, but retain the notation pH0pzqqi in (3..1) for the ith element ofH0pzq to ensure consistency with Ammari et al. (2014b).


we obtain for |R| “ ˘1 that

NjirRpBqs “α3rµ´1r sBpRpBqq

˜ż

RpBqδijdξ ` 1

2

ż

BRpBqej ˆ npξq ¨ FRpBq,ei

pξqdξ¸

“α3rµ´1r sΓ

ˆż

Bδijdξ ` 1

2

ż

Γej ˆ pRnpξqq ¨

`|R|RFB,RT ei

pξq˘dξ

˙

“α3rµ´1r sΓ

ˆż

BRipRjqδpqdξ ` |R|2

2

ż

Γ

`RppRT

ejq ˆ npξqq˘

¨`RFB,RT ei

pξq˘dξ

˙

“α3rµ´1r sΓ

ˆż

BRipRjqδpqdξ ` 1

2

ż

ΓpRT

ejq ˆ npξq ¨ FB,RT eipξqdξ

˙

“α3rµ´1r sΓRipRjq

ˆż

Bδpqdξ ` 1

2

ż

Γeq ˆ npξq ¨ FB,eppξqdξ

˙

“RipRjqNpqrBs,

where δpq is the Kronecker delta, FRpBq,eipRξq “ |R|RFB,RT ei

pξq has been applied in the first step

and FB,RT ei“

3ÿ

p“1

RipFB,ep has been applied in the second to last step. Note that the former follows

from Proposition 4.3 in Ammari et al. (2014a), but without assuming that |R| “ 1, and the latter alsofollows from the same proposition. Thus, in abbreviated notation, it follows that

N 1ji “ RjpRiqNpq,

so that the components of N transform as a rank 2 tensor. It immediately follows that xxN and M arerank 4 tensors. Note that in this theorem, and throughout the remainder, we use the notation of a hat todenote the extension of a tensor to a higher rank and a check to denote its corresponding contraction. ˝

REMARK 3..1 Notice that in (3..1) the jth component of the perturbed field is an index of the rank 4tensor M “ Mmjie b em b ej b ei and that D2

xGpx, zq “ pD2xGpx, zqqme b em appears as

a double contraction with M with summation over and m. This is in contrast to other related resultsfor perturbed electromagnetic fields. For instance, in Ammari et al. (2001) the perturbed magnetic fieldfor small conducting object for the full Maxwell system is described in terms of the rank 2 Polya &Szego (1951) polarization tensor, which only appears as a single contraction with the background fieldsolution. We address this apparent contradiction in the following theorem, which is our main result.

THEOREM 3..2 Let ν “ Op1q and α Ñ 0 then, for x away from the location of the inclusion z,

pHα ´ H0qpxqj “pD2xGpx, zqqjmMmipH0pzqqi ` pRpxqqj , (3..6)

where Mmi :“ ´ qCmi ` Nmi are the components of the symmetric rank 2 tensor M in an orthonormalcoordinate frame. Furthermore, Nmi is defined in (3..3),

qCmi :“ ´ iνα3

4em ¨

ż

Bξ ˆ pθi ` ei ˆ ξqdξ. (3..7)

and θi, i “ 1, 2, 3, is the solution to the transmission problem (3..4).


Proof. The proof immediately follows from Lemmas 4..3 and 4..4 stated in Section 4. where Lemma 4..3

establishes that pD2xGpx, zqqmMmji “ pD2

xGpx, zqqjmMmi and Lemma 4..4 establishes that Mis complex symmetric. ˝

REMARK 3..2 Evaluating (3..6) in the direction q at location s we obtain

q ¨ pHα ´ H0qpsq “ H0pzq ¨ pMH0pzqq ` ∆, (3..8)

where, in particular, we have choosen H0pzq :“ D2xGpt, zqm as the (idealised) background magnetic

field generated by a coil at position t with dipole moment m, set H0pzq :“ D2xGps, zqq, as the

background magnetic field that would be generated by a coil at position s with dipole moment q, and∆ :“ q ¨ Rpxq. Written in this form, (3..8) agrees with the prediction in the engineering literature,e.g. (Baum 1971, Das, McFee, Toews, & Stuart 1990, Marsh, Ktisis, Jarvi, Armitage, & Peyton 2013,Norton & Won 2001), which postulate that the magnetic field perturbation can be described in termsof a complex symmetric rank 2 polarisation tensor. However, to the best of our knowledge, this isthe first time that a rigorous mathematical justification and explicit formula is now available for the 6

components of the complex symmetric rank 2 tensor M in an orthonormal coordinate frame.

REMARK 3..3 If the object is non conducting and magnetic so that σ˚ “ 0 and µ˚ ‰ µ0, then, providedthat Bα is a simply connected smooth object, the tensor N reduces to a symmetric polarization tensorparameterised by a contrast µr :“ µ˚µ0 in the object (Ammari et al. 2014b). This, in turn, also agreeswith the first order generalised polarization tensor of Ammari & Kang (2007) and the Polya & Szego(1951) polarization tensor whose (real) components in an orthonormal coordinate frame are

pN qij “T pµrqij “ α3pµr ´ 1q|B|δij ` α3pµr ´ 1q2ż

Γn ¨ ∇φiξjdξ, (3..9)

where φi, i “ 1, 2, 3, satisfies the transmission problem

∇2φi “ 0 in B Y Bc, (3..10a)

rφisΓ “ 0,Bφi

Bn

ˇˇ`

´ µrBφi

Bn

ˇˇ´

“ BξiBn on Γ , (3..10b)

φi Ñ 0 as |ξ| Ñ 8. (3..10c)

4. Proof of lemmas for the main result

We prove a number of lemmas that will be useful for the proof of our main result. Throughout thefollowing the components of the tensors are taken to be with respect to an orthonormal (although notnecessarily right handed) coordinate frame.

LEMMA 4..1 The components of the conductivity tensor P can be expressed as

Pmji “ εjs qPmsi “ εjsCmsi, (4..1)

where ε is the alternating tensor with components

εijk :“

$&

%

1 ijk even permutation of 123´1 ijk odd permutation of 1230 otherwise

,


and C is rank 3 tensor density. The components of C are

Cmsi : “ β

ż

Bξmpθsi ` εspqδpiξqqdξ “ ´ ikα3

2es ¨

ż

Bξmpθi ` ei ˆ ξqdξ,

where β :“ ´ iνα3

2 .

Proof. It suffices to write the components of P in the alternative form

Pmji “ βδtjεts

ż

Bξmpθsi ` εspqδpiξqqdξ “ βεjs

ż

Bξmpθsi ` εsiqξqqdξ

“ εjs qPmsi “ εjsCmsi, (4..2)

where θsi are the coefficents of the rank 2 tensor whose columns are θi. Furthermore, C is a rank 3tensor density with components Cmsi “ qPmsi “ 1

2εsjPmji . ˝

COROLLARY 4..1 It follows from Lemma 4..1 that

Pmji “ εjsCmsi “ ´εjsCmsi “ ´Pjmi,

which means that Pmji is skew-symmetric with respect to the indices and j.

LEMMA 4..2 The rank 3 tensor density C is skew symmetric with respect to the first two indices.

Proof. Starting from

ei ˆ ξ “ ´θi ` µ0

iν∇ ˆ µ´1

˚ ∇ ˆ θi, (4..3)

in B where the subscript ξ on ∇ has been dropped (here and subsequently) for simplicity of notation, itfollows by application of the alternating tensor that

ξm “ 1

2εkpmek ¨

´´θp ` µ0

iν∇ ˆ µ´1

˚ ∇ ˆ θp

¯, (4..4)

in B. It is useful to define χm :“ 12εkpmek ¨

`´θp ` 1

iν∇ ˆ µ´1r ∇ ˆ θp

˘where µr :“ µµ0 such

that µr “ µr “ µ˚µ0 in B and µr “ 1 in Bc. Note also that ∇ ˆ ∇ ˆ θi “ 0 in Bc so thatχm “ ´1

2εkpmek ¨ θp in Bc. Taking this in to consideration then we can write

Cmsi “β

ż

Bχm∇χs ¨ pθi ` ei ˆ ξqdξ

“ ´ α3

2

ˆż

B∇ ˆ pesξmq ¨ µ´1

r ∇ ˆ θidξ `ż

B∇ ¨ pµ´1

r ∇ ˆ θi ˆ χm∇χsqdξ˙

“ ´ α3

2

ˆż

Bem ˆ es ¨ µ´1

r ∇ ˆ θidξ ´ż

Γχm∇χs ¨ µ´1

r p∇ ˆ θi ˆ n´q

ˇ´ dξ

˙.

By using the transmission conditions in (3..4), rχm∇χs ˆ nsΓ “ 0 and the fact that the integrand inthe last integral can alternatively be written in terms of a tangential trace and a twisted tangential trace,we obtain

ż

Γχm∇χs ¨ pµ´1

r ∇ ˆ θi ˆ n´q

ˇ´ dξ “

´ż

Γχm∇χs ¨ ∇ ˆ θi ˆ n

` ˇ` dξ ` 2rµ´1

r sΓż

Γχm∇χs ¨ ei ˆ n

´dξ. (4..5)


First consider,

2rµ´1r sΓ

ż

Γχm∇χs ¨ ei ˆ n

´dξ “ 2rµ´1r sΓ

ż

B∇ ¨ pχm∇χs ˆ eiqdξ

“ 2rµ´1r sΓ εksi

ż

B∇ξm ¨ ∇ξkdξ “ 2rµ´1

r sΓ |B|εmsi, (4..6)

by the properties of χm in B. Secondly, noting that θp “ Op|ξ|´2q and ∇ ˆ θp “ Op|ξ|´3q as|ξ| Ñ 8 (Ammari et al. 2014b) it follows that χm “ Op|ξ|´2q and ∇χm “ Op|ξ|´3q (since θp solvesa Laplace equation with appropriate decay conditions in an unbounded domain exterior to a sufficientlylarge sphere that encloses B in a similar way to Proposition 3.1 in Ammari et al. (2000)). Then, we canapply integration by parts to

ż

Bc

χm∇χs ¨ ∇ ˆ ∇ ˆ θidξ “ 0

“ż

Bc

∇χm ˆ ∇χs ¨ ∇ ˆ θidξ ´ż

Γχm∇χs ¨ ∇ ˆ θi ˆ n

` ˇ` dξ, (4..7)

where the aforementioned decay conditions imply that the far field integral drops out. By rearrangementand inserting (4..7) and (4..6) into (4..5) we have

ż

Γχm∇χs ¨ p∇ ˆ θi ˆ n

´qˇ´ dξ “ ´

ż

Bc

∇χm ˆ ∇χs ¨ ∇ ˆ θidξ ` 2rµ´1r sΓ |B|εmsi,

so that

Cmsi “ ´α3

2

ˆż

Bem ˆ es ¨ µ´1

r ∇ ˆ θidξ `ż

Bc

∇χm ˆ ∇χs ¨ ∇ ˆ θidξ ´ 2rµ´1r sΓ |B|εmsi

˙.

Thus Cmsi “ ´Csmi as required. ˝

COROLLARY 4..2 The skew symmetry of C proved in Lemma 4..2 implies the tensor density has only9 independent components. It immediately follows that a rank 2 tensor qC “ qqP with components

qqPni :“1

2εnms

qPmsi “ 1

4εnmsεsjlPmji “ 1

2εnmsCmsi

“β

2en ¨

ż

Bξ ˆ pθi ` ei ˆ ξqdξ “: qCni, (4..8)

can be introduced such that Cmsi “ εmskqCki.

LEMMA 4..3 The components of double contraction between D2xGpx, zq and M can be expressed as

pD2xGpx, zqqmMmji “ pD2

xGpx, zqqjkMki, (4..9)

where M is a rank 2 tensor with components Mki :“ ´ qCki ` Nki.

Proof. By writing

pD2xGpx, zqqm “ 1

4πr3p3pr b rqm ´ δlmq “ 1

4πr3p3rrm ´ δlmq ,


where r “ x ´ z, r “ |r| and r “ rr, it is clear that D2xGpx, zq has the properties

pD2xGpx, zqqm “ pD2

xGpx, zqqm, trpD2xGpx, zqq “ pD2

xGpx, zqq “ 0.

Using this in combination with Lemma 4..1, Corollary 4..2 and the definition of Mmji the resultimmediately follows. ˝

A consequence of Lemma 4..3 is that the components Pmji and xxNmji of Mmji form the disjointskew-symmetric and symmetric parts of the total tensor with respect to indices and j, respectively.

LEMMA 4..4 The tensor qC is complex symmetric if µ˚ “ µ0 and the tensor M “ ´ qC ` N is complexsymmetric for a general conducting magnetic object.

Proof. The proof builds on a result stated in a preprint of Ammari et al. (2014b). We begin by rewritingthe components of qC in an alternative form, to do so we consider

ż

Bξ ˆ pθi ` ei ˆ ξqdξ ¨ ej “

ż

Bpθi ` ei ˆ ξq ¨ ej ˆ ξdξ, (4..10)

by properties of the scalar triple product. Thus, by using (3..4),ż

Bpθi ` ei ˆ ξq ¨ ej ˆ ξdξ “ µ0

iνµ˚

ż

B∇ ˆ ∇ ˆ θi ¨ ej ˆ ξdξ

“ 1

iν

ż

B

1

µr∇ ˆ ∇ ˆ θi ¨

ˆ1

iνµr∇ ˆ ∇ ˆ θj ´ θj

˙dξ, (4..11)

where µr is as defined in Lemma 4..2. Performing integration by partsż

B

1

µr∇ ˆ ∇ ˆ θi ¨ θjdξ “

ż

B

1

µr∇ ˆ θj ¨ ∇ ˆ θidξ ´

ż

Γθj ¨ 1

µr∇ ˆ θi ˆ n

´ˇˇ´dξ. (4..12)

Next, using the transmission conditions on Γ in (3..4), thenż

Bpθi ` ei ˆ ξq ¨ ej ˆ ξdξ “ 1

piνq2ż

B

1

µ2r

∇ ˆ ∇ ˆ θi ¨ ∇ ˆ ∇ ˆ θjdξ

´ 1

iν

ż

B

1

µr∇ ˆ θi ¨ ∇ ˆ θjdξ ` 1

iν

ż

Γθj ¨ ∇ ˆ θi ˆ n

´ ˇ` dξ

` 2rµ´1r sΓiν

ż

Γei ˆ n

´ ¨ θjdξ. (4..13)

Then, by performing integration by parts in Bc,ż

Bc

θj ¨ ∇ ˆ ∇ ˆ θidξ “ż

Bc

∇ ˆ θi ¨ ∇ ˆ θj `ż

Γθj ¨ ∇ ˆ θi ˆ n

´ ˇ` dξ “ 0, (4..14)

which is valid given the decay conditions on θi and ∇ ˆ θi as |ξ| Ñ 8 (Ammari et al. 2014b). Using(4..14) in (4..13) and recalling (4..8) and (4..10), then we have that the components

qCji “ ´ α3

4iν

ż

B

1

µ2r

∇ ˆ ∇ ˆ θi ¨ ∇ ˆ ∇ ˆ θjdξ

` α3

4

ż

BYBc

1

µr∇ ˆ θi ¨ ∇ ˆ θjdξ ´ α3rµ´1

r sΓ2

ż

Γei ˆ n

´ ¨ θjˇ´ dξ,


are symmetric when µ˚ “ µ0. We rewrite Nji in the following form

Nji “ α3

ˆ1 ´ µ0

µ˚

˙ ż

B

ˆei ` 1

2∇ ˆ θi

˙dξ ¨ ej

“ α3rµ´1r sΓ

ˆż

Bei ¨ ejdξ ` 1

2

ż

Γej ˆ n

´ ¨ θiˇ´ dξ

˙.

It then follows that the components of M “ ´ qC ` N can be written as

Mji “ α3

4iν

ż

B

1

µ2r

∇ ˆ ∇ ˆ θi ¨ ∇ ˆ ∇ ˆ θjdξ ´ α3

4

ż

BYBc

1

µr∇ ˆ θi ¨ ∇ ˆ θjdξ

` α3rµ´1r sΓ

ˆ1

2

ż

Γei ˆ n

´ ¨ θjˇ´ dξ `

ż

Bei ¨ ejdξ ` 1

2

ż

Γej ˆ n

´ ¨ θiˇ´ dξ

˙, (4..15)

which are symmetric. ˝

5. Simplified polarization tensors for classes of geometries

The number of independent components that are required to define C and N (and hence qC, P , xxN , Mand ||M ) in an orthonormal coordinate frame for an object with either a rotational or mirror symmetry (ormultiple symmetries, or both) are often fewer than those required to define a general object. To explainthis, we show how the number of independent components can be reduced for an object which has auniaxial symmetry and an object which has mirror symmetries. We then apply similar techniques toa range of simple objects and consider the further simplification that results in the case of a sphericalobject.

5.1. Polarization tensors for objects with uniaxial symmetry

The number of independent components for an object, which has uniaxial symmetry, in a given direction,can either be determined by a counting argument or by considering the coefficents of the tensor thatshould remain invariant under a rotation. Here we apply the latter and first consider the conductivitytensor P , which can be expressed in terms of the rank 3 tensor density C. We remark that one could alsoexpress this in terms of the rank 2 tensor qC, but instead we apply the former and use the skew symmetryof C. Under the transformation by an orthogonal matrix R the components of the rank 3 tensor densitybecome

C1ijk “ |R|RiRjmRknCmn, (5..1)

where |R| “ 1 for a proper transformation. For example, for a rotation of angle ψ about e3 then

R “¨

˝cosψ sinψ 0

´ sinψ cosψ 00 0 1

˛

‚.

If an object has uniaxial symmetry in the e3 coordinate direction then this means that the components ofthe tensor C should be invariant under a ψ “ π2 rotation about the e3 axis : e1 Ñ e2 and e2 Ñ ´e1.Under this transformation C1 “ C and applying (5..1) to C1 gives C2 “ C1 “ C. By considering a general


C it follows that it’s 7 independent components are C333, C311 “ C322, C131 “ C232, C113 “ C223,C312 “ Ć321, C132 “ Ć231 and C123 “ Ć213. But, we also know that C is skew symmetric withrespect to the first two indices and this reduces the number to just 3: C312 “ Ć321 “ Ć132 “ C231,C123 “ Ć213 and C311 “ Ć131 “ C322 “ Ć232.

Secondly, we consider the permeability tensor N “ Nie b ei which transforms as

N 1ij “ RiRjmNm. (5..2)

Then, by proceeding in a similar manner to above, for an object with uniaxial symmetry in the e3

direction we have N11 “ N22, N12 “ Ń21 and N33.Finally, we know that the total reduced tensor M “ ´ qC ` N is symmetric, so that M12 “ M21,

but rotational symmetry tell us N12 “ Ń21 and qC12 “ ´ qC21 “ C232 so that M12 “ M21 “Ć232 `N12 “ C232 Ń12 “ 0. Thus M is diagonal in this case with just 2 independent components:M11 “ M22 “ N11 ´ C231 and M33 “ N33 ´ C123.

5.2. Polarization tensors for objects with both uniaxial and mirror symmetries

The number of independent components that are required to define C and N can also be reduced if theobject has mirror symmetries. For an object with a mirror symmetry associated with the plane with unitnormal vector n then

Rij “ δij ´ 2ninj . (5..3)

The components of the rank 3 tensor density C and the rank 2 tensor N remain invariant under a reflec-tion provided that the last index remains unchanged under the transformation. By following similar stepsas outlined in Section 5.1. the independent components of C and N for objects with mirror symmetriescan be obtained.

5.3. Examples of symmetries in polarization tensors

By applying similar arguments to those described above, the entries in Table 1, which lists the inde-pendent components for some simple shapes, can be identified. The rotational symmetries of an objectabout an angle π in a given coordinate direction, which are equivalent to an appropriate mirror symme-try, have been omitted. Note that the independent components that define a sphere and a cube are thesame, as are the cases of a cylinder and a cone when their axes are aligned. Furthermore, for all the

simple objects in Table 1, we observe that the rank 2 tensor M “ ´ qC ` N is diagonal.

5.4. Polarization tensor for a spherical geometry

The polarization tensor for a spherical object, which has been obtained by Ammari et al. (2014b), is a

further simplification of (3..6) with components Mi “ ´ qCi ` Ni “ pĆ ` Nqδi such that M is a


scalar multiple of the identity tensor. Here, it can be shown that

N :“1

3Npp,

C :“1

6εmsiCmsi “ β

6εmsi

ż

Bξmpθsi ` εsiqξqqdξ

“β

6

ż

Bpεmsiξmθsi ` 2ξqξqqdξ “ ´β

ż

Bpξ1θ2 ¨ e3 ´ ξ21qdξ,

by noting thatşB ξ21dξ “ ş

B ξ22dξ “ şB ξ23dξ for a sphere and using integration by parts to show thatş

B ξ1θ3 ¨ e2dξ “ ´ şB ξ1θ2 ¨ e3dξ.

Using the analytical solution (Smythe 1968) for the eddy currents generated in a conducting (mag-netic) sphere of radius α with conductivity σ˚, permeability µ˚ at angular frequency ω, when placedin a uniform field H0, we can show that the form of the perturbed field is identical to (3..6), if we set

R “ 0 and Mi “ Mδi. It can also be identified that

M “ 2πα3 pp2µ˚ ` µ0qvI´12 ´ pµ0p1 ` v2q ` 2µ˚qI12qpµ˚ ´ µ0qI´12 ` pµ0p1 ` v2q ´ µ˚qI12

, (5..4)

where v “ ?iσµ˚ωα, I12pvq “

b2πv sinh v and I´12pvq “

b2πv cosh v. The overline indicates the

complex conjugate, which appears due to the eiωt time variation in Smythe (1968) rather than the e´iωt

assumed here. Note that the asymptotic formula (3..6) implies that |N ´ C ´ M | “ Opα4q as α Ñ 0for fixed x and z.

An analytical solution (Wait 1953) is also available for case where the same sphere is now illumi-nated by an incident field generated by a circular coil of radius γ carrying an alternating current I . For acoil centred at s the incident magnetic field at position z can be described in terms of a magnetic dipolein the form

H0pzq “ D2xGpz, sqm, (5..5)

provided that the length of coil, L “ 2πγ, is small compared to the distance from the coil to the object,|z ´ s|. In the above, m is the magnetic dipole moment of the current source, which, for a circular coil,has |m| “ Iπγ2 (Jackson 1967). If the coil is chosen to lie in the pe1, e2q plane then m “ Iπγ2

e3 and

H0pzq “ Iπγ2D

2xGpz, sqe3. (5..6)

Furthermore, we can show that the leading order term for pHα ´ H0qpxq that Wait (1953) obtains isidentical to that described by (3..6).


Tabl

e1.

Non

-zer

oin

depe

nden

tcoe

ffici

ents

that

are

requ

ired

tore

pres

entC

andN

fora

rang

eof

sim

ple

obje

cts.

Obj

ect

Rot

atio

nal

Mirr

orIn

depe

nden

tIn

depe

nden

tSh

ape

Sym

met

ries

Sym

met

ries

Coe

ffici

ents

inC

Coe

ffici

ents

inN

Sphe

re

Isot

ropi

cIn

finite

num

ber

ofpl

anes

C 123

“Ć

132

“Ć

213

“C 2

31

“C 3

12

“Ć

321

N11

“N

22

“N

33

Cub

e:A

ligne

dw

ithax

es

Uni

axia

labo

ute1,e

2,e

3

Plan

esw

ithno

rmal

se1,e

2,e

3,

pe1

è2q

?2,

pe1

è3q

?2,

pe2

è3q

?2

C 123

“Ć

132

“Ć

213

“C 2

31

“C 3

12

“Ć

321

N11

“N

22

“N

33

Blo

ck:

` ´w 2,´

d 2,´

h 2

˘ ˆ` w 2

,d 2,h 2

˘

Alig

ned

with

axes

Non

e

Plan

esw

ithno

rmal

se1,e

2,e

3,

pde1

`we2q

?d2

`w

2

pwe3

`he1q

?w

2`h2

phe2

`de

3q

?d2

`h2

C 123

“Ć

213

C 321

“Ć

231

C 132

“Ć

312

N11

N22

N33

Con

e:A

xis

alig

ned

with

e1

Rot

atio

nally

inva

riant

abou

te1

Plan

esw

ithno

rmal

se2,e

3

any

plan

eK

base

&pa

ssin

gth

roug

hth

eve

rtex

C 123

“Ć

132

“Ć

213

“C 3

12

C 231

“Ć

321

N11

N22

“N

33

Cyl

inde

r:A

xis

alig

ned

with

e1

Rot

atio

nally

inva

riant

abou

te1

Plan

esw

ithno

rmal

se1,e

2,e

3

any

plan

eK

base

&pa

ralle

ltoe1

C 123

“Ć

132

“Ć

213

“C 3

12

C 231

“Ć

321

N11

N22

“N

33

Cub

ew

ithho

le:

Hol

eal

igne

dw

ithe3

Uni

axia

labo

ute3

Plan

esw

ithno

rmal

se1,e

2,e

3,

pe1

è2q

?2,

pe1

è3q

?2,

pe2

è3q

?2

C 123

“Ć

213

C 132

“Ć

231

“C 3

21

“Ć

312

N11

“N

22

N33

Ellip

soid

:G

ener

alca

se

Non

ePl

anes

with

norm

als

e1,e

2,e

3

C 123

“Ć

213

C 132

“Ć

231

C 321

“Ć

312

N11

N22

N33


6. Determining M from field measurements

In the next section we describe a numerical approach for computing θi, i “ 1, 2, 3, which can be used forthe accurate calculation of the polarization tensors. However, there may also be situations (e.g. as partof an inverse algorithm or an experimental validation proceedure) where the independent componentsthat define the polarization tensors should be determined from field measurements of pHα ´ H0qpxq.In particular, by taking sufficient measurements of qpiq ¨ pHα ´ H0qpxpiqq at different positions x

piq

and orientations qpiq an over-determined system can be built from which the 6 independent complex

components of M in an orthonormal coordinate system can be found by solving this system in a leastsquares sense.

7. hp-Finite element methodology for the computation of M

The computation of M “ ´ qC`N requires the approximate solution of the transmission problem (3..4),which in turn has similarities to the A–based formulation of eddy current problems e.g. (Ledger &Zaglmayr 2010). We therefore advocate that the regularised formulation previously developed for eddycurrent problems on multiply connected domains be adapted for it’s approximate solution. For thispurpose, we truncate the otherwise unbounded domain Bc at a finite distance from the object and createthe finite domain Ω “ Bc Y B and on the truncated boundary BΩ we impose ∇ξ ˆ θi ˆ n “ 0.

7.1. Regularised formulation

Let ϑi “ θi. The transmission problem for ϑi on the finite (computational) domain can then be writtenin the form

∇ξ ˆ µ´1r ∇ξ ˆ ϑi ` iµ0ωσα

2ϑi “ ´iµ0ωσα

2ei ˆ ξ in B Y Bc,

∇ξ ¨ ϑi “ 0 in Bc,

rϑi ˆ nsΓ “ 0,“µ´1r ∇ξ ˆ ϑi ˆ n

‰Γ

“ ´2“µ´1r

‰Γei ˆ n on Γ ,

∇ξ ˆ ϑi ˆ n “ 0 on BΩ,

where, µr is as defined in Lemma 4..2.Following Bachlinger, Langer, & Schoberl (2005, 2006), Ledger & Zaglmayr (2010), Zaglmayr

(2006), Zaglmayr & Schoberl (2005) we introduce the perturbed weak problem: Let τ ą 0 be a smallperturbation parameter, then: find ϑ

τi P Hpcurl,Ωq such that

pµ´1r ∇ξ ˆ ϑ

τi ,∇ξ ˆ vqΩ ` pκϑτ

i ,vqΩ “ ´pκei ˆ ξ,vqB ´ 2

ż

Γrµ´1

r sei ˆ n ¨ vdξ, (7..1)

for all v P Hpcurl,Ωq where κ “"

iµ0ωσα2 in Bτ in Bc . This regularised formulation circumvents the

need to enforce the divergence constraint ∇ξ ¨ ϑi “ 0 in Bc (and an analogous integral condition to(2..3) on Γ ) to enforce the uniqueness of θi in Bc. Furthermore, the previous analysis of Bachlingeret al. (2005), Zaglmayr (2006), which considers the error associated with the solution of the perturbededdy current problem, carries over to (7..1).


7.2. Discrete approximation

In this work we use the basis functions of Zaglmayr (2006), Zaglmayr & Schoberl (2005) and we recallthat for a tetrahedral triangulation consisting of vertices Vh, edges Eh, faces Fh and cells Th their(reduced) hierarchic Hpcurlq and H1pΩq conforming finite element basis can be expressed in terms ofthe splitting

V redh,p :“V N0

h ‘ÿ

EPEBh

∇WEp`1 ‘

ÿ

FPFBh

∇WFp`1 ‘

ÿ

FPFh

rV Fp ‘

ÿ

IPT Bh

∇W Ip`1 ‘

ÿ

IPTh

rV Ip ,

and

Wh,p`1 :“Wh,1 ‘ÿ

EPEh

WEp`1 ‘

ÿ

FPFh

WFp`1 `

ÿ

IPTh

W Ip`1 Ă H1pΩq.

In the above V N0h and Wh,1 denotes the set of lowest order Nedelec (edge element) basis functions

and the standard lowest order hat functions, respectively, the former being associated with the edges ofthe element and the latter with the vertices of the element. The extension to arbitrary high polynomialdegree order consists of the enrichment of the finite element space through the addition of higher orderedge, face and interior based basis functions, WE

p`1, WFp`1 and W I

p`1, respectively, for H1 and theaddition of higher order edge, face and interior functions for Hpcurlq, where, in this case, the higherorder edge and some of the higher order face and interior functions are constructed from the gradients oftheir H1 conforming counterparts. Furthermore, the superscript B on Eh, Fh and Th is used to denotethose edges, faces and cells associated with subdomain B, only, as the gradient basis functions in Bc

are omitted. It then follows that the approximate weak formulation is: Find ϑihp P V red

h,p X Hpcurl,Ωqsuch that

pµ´1r ∇ξ ˆ ϑ

ihp,∇ξ ˆ vhpqΩ ` pκϑi

hp,vhpqΩ “ ´

pκei ˆ ξ,vhpqB ´ 2

ż

Γrµ´1

r sei ˆ n ¨ vhpdξ @vhp P V redh,p X Hpcurl,Ωq. (7..2)

The structure of the left hand side of (7..2) is analogous to the gauged A–based formulation of eddycurrent problems and, therefore, the preconditioning technique described in Ledger & Zaglmayr (2010)can be immediately applied to the complex symmetric linear system that results ensuring a robust solverthat is capable of coping with the large contrasts in κ.

For problems with curved geometry the approach described in Ledger & Coyle (2005) is employed,in which, for a degree of the polynomial correction, g, the coefficients of the edge and face corrections,cE P WE

g`1 and cF P WF

g`1 respectively, are determined by solving local L2 minimisation problemson the edges and faces of those elements lying on the curved boundary.

8. Numerical examples

8.1. Polarization tensor for a spherical object

For the case where Bα is a sphere of radius α “ 0.01m, with σ˚ “ 5.96ˆ107S m´1, µ˚ “ µ0 and ω “133.5rad s´1, so that ν “ 1, we present results , for the convergence of the error M ´ Mhp2M2with increasing numbers of degrees of freedom (Ndof) in Fig. 1. Note that ¨ 2 denotes the entry-wise

norm for a rank 2 tensor, M2 “ přm |Mm|2q12, and Mhp is the approximate polarization tensor


1e-05

0.0001

0.001

0.01

0.1

100 1000 10000 100000

Rela

tive E

rror

Ndof

g=2g=3g=4g=5g=6g=7

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1000 10000 100000 1e+06

Rela

tive E

rror

Ndof

g=2g=3g=4g=5g=6g=7

paq pbqFIG. 1. Polarization tensor for a spherical object with α “ 0.01m, σ˚ “ 5.96ˆ107S m´1, µ˚ “ µ0 and 133.5rad s´1 showing

convergence of M ´ Mhp2M2 with p “ 0, 1, 2, 3, 4, 5 and g “ 2, 3, 4, 5, 6, 7 when paq the domain is truncated at aradius of 10|B| and pbq the domain is truncated at a radius of 100|B|.

computed using ϑihp, i “ 1, 2, 3. To compute the numerical tensor, we consider the unit sphere B

and choose Ω to be a sphere which is 10, and then 100, times the radius of B. For these geometrieswe generate (coarse) meshes of 880 and 2425 unstructured tetrahedra, respectively, for discretising thetwo cases. These and subsequent meshes were generated using the NETGEN mesh generator Schoberl(1997). In order to represent the curved geometry of the sphere polynomial representations using degreesg “ 2, 3, 4, 5, 6, 7 are considered and finite elements of order p “ 0, 1, 2, 3, 4, 5 are used to obrain theapproximate solutions ϑi

hp, i “ 1, 2, 3. In each case use a regularisation parameter, τ , that is 8 orders ofmagnitude smaller than ωσµ0α2.In each case, the lines represent the different choices of g and the points on the line represent increas-

ing p. The convergence behaviour resembles that previously described in Ledger & Coyle (2005): InFig. 1 paq we observe that, for truncation at a radius of 10|B|, the error associated with the geometrydominates for low g as indicated by stagnation of the convergence curves for g “ 2, 3, 4 with increasingp. For g “ 2, 3, 4 we see improvements in accuracy as g is increased since the resolution of the geom-etry is improved with each increment of g. However, for g ě 4, no further reduction in error can beachieved by increasing p alone, which means that the error associated with the artificial truncation of theboundary dominates over the error in resolving the curved boundary. On the other hand, by truncating ata radius of 100|B| and repeating the computations, as shown in Fig. 1 pbq, we now see that the boundarytruncation error no longer dominates and, for g ě 6, exponential convergence of the polarization tensordown to relative errors of 10´7 results by performing p–refinement.

The corresponding results for the same sized object with fictional parameters as before, except µ˚ “1.5µ0, are shown in Fig. 2. For low g and truncation at either 10|B| or 100|B|, the geometry errordominates but, for sufficiently high g and high p, the error can be reduced to less than 10´6 by truncatingthe domain at a radius of 100|B|. In particular, for g ě 6 and performing p–refinement, exponentialconvergence of the polarization tensor is achieved.

8.2. Objects in a uniform background field

For uniform H0 we compare pHα ´ H0qpxq predicted by (3..6), when hp–finite elements are used to

numerically compute the rank 2 polarization tensor Mhp, with the results obtained by solving the fulleddy current problem, using hp–finite elements and the formulation in Ledger & Zaglmayr (2010). Weundertake this comparison for a series of different shaped objects taken from Table 1 including when Bα


1e-05

0.0001

0.001

0.01

0.1

100 1000 10000 100000

Rela

tive E

rror

Ndof

g=2g=3g=4g=5g=6g=7

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1000 10000 100000 1e+06

Rela

tive E

rror

Ndof

g=2g=3g=4g=5g=6g=7

paq pbqFIG. 2. Polarization tensor for a spherical object with α “ 0.01m, σ˚ “ 5.96 ˆ 107S m´1, µ˚ “ 1.5µ0 and 133.5rad s´1

showing convergence of M´ Mhp2M2 with p “ 0, 1, 2, 3, 4, 5 and g “ 2, 3, 4, 5, 6, 7 when paq the domain is truncatedat a radius of 10|B| and pbq the domain is truncated at a radius of 100|B|.

is a sphere of radius 0.01m, a 0.0075m ˆ0.015m ˆ0.01m rectangular block, a cone with height 0.01mand maximum radius 0.005m and, finally, a cube of side length 0.01m with a 0.005mˆ0.005mˆ0.01mhole removed.

In each case, we select the far field boundary to be located at distance a 100 times the size of theobject, for cases with curved geometries we use g “ 4 and for approximating the solution to (7..2),we use p “ 4 elements and meshes of 2425, 3433, 19 851 and 7377 unstructured tetrahedra for thecases of a sphere, block, cone and the cube with hole, respectively. We fix the material parametersas σ˚ “ 5.96 ˆ 107S m´1, µ˚ “ 1.5µ0 and the angular frequency as 133.5rad s´1 for all objects.

The polarization tensor Mhp for each object is computed by considering an appropriate unit–sizedobject B, which, when an appropriate scaling is applied, results in the physical object Bα. Then, byassuming a uniform incident field H0pxq “ H0 “ e3 we compare |pHα ´ H0qpxq||H0pxq| whenx “ r “ rei P Bc

α and r ď 0.1m and i “ 1, 2, 3 in turn.The results of this investigation are shown in Fig. 3. For all cases, the perturbed field predicted by

the numerically computed polarization tensor is in excellent agreement with that obtained by solvingthe full eddy current problem.

As a further illustration of the validity of the asymptotic formula (3..6) we compute |pHα´H0qpxq´pD2Gpx, zqpMhp

H0pzqqq||H0pxq| “ |Rpxq||H0pxq| for uniform H0 “ e3, fixed x, z and vary-ing α for the cases of a sphere and a spheroid with material parameters as in Fig. 3. We choose the radiusof the sphere to be α and the minor and major axes of the spheroid to be α and 2α, respectively. For

these objects we numerically compute Mhp for α “ 0.1, 0.075, 0.05, 0.025, 0.1m, in turn, using highfidelity discretisations consisting of uniform p “ 5, p “ 7 elements, geometry resolutions of g “ 7 andg “ 4 and meshes of 2742 and 15 493 unstructured tetrahedra for the cases of the sphere and spheroid,respectively, with truncation at 200|B| in both cases. For Hαpxq we employ the aforementioned ana-lytical solution for the sphere and a known result for a conducting permeable spheroid (Ao et al. 2002).The results of this investigation are shown in Fig. 4, which confirm that Rpxq “ Opα4q in the asymp-totic formula (3..6). Note that in the case of the sphere |Rpxq| is converging faster than α4, which isconsistent with the Landau notation.


1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

||H

_alp

ha-H

_0||

r

hp eddy xhp eddy yhp eddy z

PT Rank 2 xPT Rank 2 yPT Rank 2 z

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

||H

_alp

ha-H

_0||

r



paq pbq

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

||H

_alp

ha-H

_0||

r



1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

||H

_alp

ha-H

_0||

r



pcq pdqFIG. 3. Comparison of |pHα ´ H0qpxq||H0pxq| for uniform H0pxq “ H0 when x “ r “ rei P Bc

α and r ď 0.1m

for i “ 1, 2, 3, in turn, showing the results obtained by using the numerically computed rank 2 polarization tensor Mhp and bysolving the full eddy current problem when Bα is : paq a sphere of radius 0.01m, pbq a 0.0075m ˆ0.015m ˆ0.01m rectangularblock, pcq a cone with height 0.01m and maximum radius 0.005m and pdq a cube of side length 0.01m with a 0.005m ˆ0.005mˆ0.01m hole removed.

1e-15

1e-14

1e-13

1e-12

1e-11

1e-10

1e-09

1e-08

1e-07

10 100

Resi

dual

1/alpha

|R|Calpha^4

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

10 100

Resi

dual

1/alpha

|R|Calpha^4

paq pbq

FIG. 4. Comparison of |pHα ´ H0qpxq ´ pD2Gpx,zqpMhpH0pzqqq||H0pxq| “ |Rpxq||H0pxq| and Cα4 for uniform

H0 “ e3, fixed x and z with varying α for the cases of a paq a conducting permeable sphere and pbq a conducting permeable

spheroid where Mhp has been computed using high–fidelity discretisations.

8.3. Objects in a rotational background field

In this section, we perform a similar comparison to that undertaken in Section 8.2. but now with H0pxqgenerated by a coil carrying a current such that |J0| “ 1 ˆ 106 Am´2. The coil is taken to be a torus ofinner radius 0.005m and outer radius 0.01m and has position 0.4e3 m relative to the centre of the object.The shape and material properties of the different objects are as described in Section 8.2.. We undertake


1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

|H_alp

ha-H

_0|/|H

_0|

r

hp eddy L1hp eddy L2hp eddy L3

PT Rank 2 L1PT Rank 2 L2PT Rank 2 L3

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

|H_alp

ha-H

_0|/|H

_0|

r



paq pbq

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

|H_alp

ha-H

_0|/|H

_0|

r



1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

|H_alp

ha-H

_0|/|H

_0|

r



pcq pdqFIG. 5. Comparison of |pHα ´ H0qpxq||H0pxq| for rotational H0 along the lines L1, L2, and L3, in turn, showing the results

obtained by using the numerically computed rank 2 polarization tensor Mhp and by solving the full eddy current problem whenBα is : paq a sphere of radius 0.01m, pbq a 0.0075m ˆ0.015m ˆ0.01m rectangular block, pcq a cone with height 0.01m andmaximum radius 0.005m and pdq a cube of side length 0.01m with a 0.005m ˆ0.005m ˆ0.01m hole removed.

comparisons of |pHα´H0qpxq||H0pxq| when x “ r “ x3e3 P Bcα (L1), x “ r “ x1e1`x3e3 P Bc

α

(L2) and x “ r “ x2e2 ` x3e3 P Bcα (L3) for r “ |x| ď 1m using p “ 4 elements and g “ 4 when

the geometry is curved. The meshes for computing the solution to the full eddy current problem consistof 36 012, 35 347, 49 086 and 53 743 unstructured tetrahedra for the cases of a sphere, block, cone andthe cube with hole, respectively, where the coil has also been discretised in each case.

The results of this investigation are shown in Fig. 5. In particular, we show how the normalised fieldchanges with distance along a line directly above the object (L1) and along two diagonal lines extendingupwards from the object (L2 and L3). For all objects considered, the agreement between the perturbed

field predicted by the asymptotic expansion and (3..6) using the numerically computed Mhp and thoseobtained by solving the full eddy current problem is excellent.

The small differences between the results predicted by (3..6) and solving the full eddy current prob-lem for large r are attributed to the artificial truncation boundary used for solving the full eddy currentproblem, which, for these examples, has been placed at r “ 2m. These differences are not noticeablein Fig. 3 as the results are shown for small r, significantly further away from the artificial truncationboundary.

9. Conclusion

In this article we have shown that the rank 4 conductivity tensor, P , obtained by Ammari et al. (2014b)that appears in the leading order term for the perturbed magnetic field, in the presence of a conducting

REFERENCES 21 of 23

object at low frequencies as α Ñ 0 when ν “ 1, can be expressed in terms of the rank 3 tensor densityC and this in turn can be expressed in terms of rank 2 tensor qC. By using properties of these tensors,we have shown that at most 9 independent components are required for defining qC in an orthonormalcoordinate frame, and hence P , for a conducting object. A further 9 are required for the rank 2 tensorN if the object is magnetic. Furthermore, we have shown that the perturbed field for a general object

is influenced by a reduced rank 2 symmetric tensor M :“ ´ qC ` N with just 6 complex independentcomponents in an orthornormal coordinate frame. If the object has rotational or mirror symmetries wehave shown that the number of independent components can be reduced further.

We have included results to illustrate how the tensors can be computed accurately by using thehp-finite element method. These results indicate that, for smooth objects, exponential convergenceof the computed tensor can be achieved by performing p–refinement on a coarse grid with accurategeometry and a far field boundary placed sufficiently far from the object. We have also used thesethe hp–finite element approach to numerically verify the perturbed fields predicted by the asymptoticformula for a range of objects and illuminations and all show excellent agreement when comparedwith solving the full eddy current problem. In future work we intend to develop an alternative coupledfinite element–boundary integral approach for solving the transmission problem (3..4), without the needto introduce the artificial regularisation parameter τ in (7..1), and imposing a constraint of the formşΓ n ¨ θi|`dξ “ 0 (Hiptmair 2002).

Acknowledgement

The authors are very grateful to Professors H. Ammari, J. Chen and D. Volkov for their helpful discus-sions and comments and would also like to thank EPSRC for the financial support received from thegrants EP/K00428X/1 and EP/K039865/1 and the support received from the land mine research charityFind A Better Way.

References

AMMARI, H., BUFFA, A. & NEDELEC, J. C. (2000) A justification of eddy currents model for theMaxwell equations. SIAM Journal of Applied Mathematics, 60, 1805–1823.

AMMARI, H., VOGELIUS, M. S. & VOLKOV, D. (2001) Asymptotic formulas for the perturbations inthe electromagnetic fields due to the presence of inhomogeneities of small diamater ii. the full Maxwellequations. J. Math. Pures Appl., 80, 769–814.

AMMARI, H., CHEN, J., CHEN, Z., VOLLKOV, D. & WANG, H. (2014a) Detection and classificationfrom electromagnetic induction data. Journal of Computational Physics. Submitted.

AMMARI, H., CHEN, J., CHEN, Z., GARNIER, J. & VOLKOV, D. (2014b) Target detection andcharacterization from electromagnetic induction data. Journal de Mathematiques Pures et Appliques,201, 54–75.

AMMARI, H. & KANG, H. (2007) Polarization and Moment Tensors: With Applications to InverseProblems. Springer.

AMMARI, H. & VOLKOV, D. (2005) The leading-order term in the asymptotic expansion of thescattering amplitude of a collection of finite number dielectric inhomogeneities of small diameter.International Journal for Multiscale Computational Engineering, 3, 149–160.

22 of 23 REFERENCES

AO, C., BRAUNISCH, H., O’NEILL, K. & KONG, J. (2002) Quasi-magnetostatic solution for aconducting and permeable spheroid with arbitrary excitation. IEEE Transactions on Geoscience andRemote Sensing, 40, 887–897.

BACHLINGER, F., LANGER, U. & SCHOBERL, J. (2005) Numerical analysis of non–linear multihar-monic eddy current problems. Numerische Mathematik, 100, 594–516.

BACHLINGER, F., LANGER, U. & SCHOBERL, J. (2006) Efficent solvers for nonlinear time-periodiceddy current problems. Computing and Visualization in Science, 9, 197–207.

BAUM, C. E. (1971) Some characteristics of electric and magnetic diopole antennas for radiatingtransient pulses. Technical Report 125. Air Force Weapons Laboratory.

DAS, Y., MCFEE, J. E., TOEWS, J. & STUART, G. C. (1990) Analysis of an electromagnetic induc-tion detector for real–time location of buired objects. IEEE Transactions of Geoscience and RemoteSensing, 28, 278–288.

DASSIOS, G. & KLEINMAN, R. E. (2000) Low Frequency Scattering. Oxford Science Publications.

GAYDECKI, P., SILVA, I., FERANDES, B. & YU, Z. Z. (2000) A portable inductive scanning systemfor imaging steel-reinforcing bars embedded within concrete. Sensors and Actuators, 84, 25–32.

GRIFFITHS, H. (2005) Magnetic induction tomography. Electrical Impedance Tomography - Methods,History and Applications (D. Holder ed.). IOP Publishing, pp. 213–238.

HIPTMAIR, R. (2002) Symmetric coupling for eddy current equations. SIAM Journal for NumericalAnalysis, 40, 41–65.

JACKSON, J. D. (1967) Classical Electrodynamics. John Wiley and Sons.

KELLER, J. B., KLEINMAN, R. E. & SENIOR, T. B. A. (1972) Dipole moments in Rayleigh scatter-ing. Journal of the Institute of Mathematical Applications, 9, 14–22.

KLEINMAN, R. E. (1967) Far field scattering at low frequencies. Applied Scientific Research, 18, 1–8.

KLEINMAN, R. E. (1973) Dipole moments and near field potentials. Applied Scientific Research, 27,335–340.

KLEINMAN, R. E. & SENIOR, T. B. A. (1982) Rayleigh scattering. Technical Report RL 728.University of Michigian Radiation Laboratory.

LEDGER, P. D. & COYLE, J. (2005) Evidence of exponetial convergence in the computation ofMaxwell eigenvalues. Computer Methods in Applied Mechanics and Engineering, 587–604.

LEDGER, P. D. & LIONHEART, W. R. B. (2014) The perturbation of electromagnetics fields atdistances that are large compared to the object’s size. IMA Journal of Applied Mathematics. Accepted.

LEDGER, P. D. & ZAGLMAYR, S. (2010) hp finite element simulation of three–dimensional eddycurrent problems on multiply connected domains. Computer Methods in Applied Mechanics and En-gineering, 199, 3386–3401.

REFERENCES 23 of 23

MARSH, L. A., KTISIS, C., JARVI, A., ARMITAGE, D. & PEYTON, A. J. (2013) Three-dimensionalobject location and inversion of the magnetic polarisability tensor at a single frequency using a walk-through metal detector. Measurement Science and Technology, 24, 045102.

NORTON, S. J. & WON, I. J. (2001) Identificiation of buried unexploded ordance from broadbandelectromagnetic induction data. IEEE Transaction on Geoscience and Remote Sensing, 39, 2253–2261.

POLYA, G. & SZEGO, G. (1951) Isopermetric Inequalities in Mathematical Physics. Annals ofMathematical Studies Number 27. Princeton University Press, Princeton, NJ.

SCHOBERL, J. (1997) NETGEN - an advancing fron 2D/3D mesh generator based on abstract rules.Computing and Visualisation in Science, 1, 41–52.

SIMMONDS, J. & GAYDECKI, P. (1999) Defect detection in embedded reinforcing bars and cablesusing a multi-coil magnetic field sensor. Measurement Science and Technology, 10, 640–648.

SMYTHE, W. R. (1968) Static and Dynamic Electricity. McGraw Hill, New York.

SOLEIMANI, M., LIONHEART, W. R. B. & PEYTON, A. J. (2007) Image reconstruction for highcontrast conductivity imaging in mutual induction tomography for industrial applications. IEEE Trans-actions on Instrumentation and Measurement, 56, 2024 – 2032.

SOMERSALO, E., ISAACSON, D. & CHENEY, M. (1992) A linearised inverse boundary value problemfor Maxwell’s equations. Journal of Computational and Applied Mathematics, 42, 123–136.

UNITED NATIONS (1997). http://www.un.org/cyberschoolbus/banmines/facts.asp. [accessed 26thSeptember 2014].

WAIT, R. J. (1953) A conducting permeable sphere in the presence of a coil carrying an oscillatingcurrent. Canadian Journal of Physics, 31, 670–678.

ZAGLMAYR, S. (2006) High order finite elements for electromagnetic field computation. Ph.D. thesis,Institut fur Numerische Mathematik, Johannes Kepler Universitat, Linz, Austria.

ZAGLMAYR, S. & SCHOBERL, J. (2005) High order Nedelec elements with local complete sequenceproperties. COMPEL: The International Jounral for Computation and Mathematics in Electrical andElectronic Engineering, 24, 374–384.

ZOLGHARNI, M., LEDGER, P. D., ARMITAGE, D. W., HOLDER, D. & GRIFFITHS, H. (2009) Imag-ing cerebral harmorrhage with magnetic induction tomography: numerical modelling. PhysiologicalMeasurement, 30, 187–200.

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