IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 4, APRIL 2007 1785

Source Field Computation in NDT ApplicationsT. Henneron1, Y. Le Menach1, F. Piriou1, O. Moreau2, S. Clénet3, J.-P. Ducreux2, and J.-C. Vérité2

“Laboratoire de Modélisation du Matériel Electrique” (LAMEL) L2EP, University of Sciences and Technologies of Lille, CitéScientifique 59655 Villeneuve d’Ascq, France

LAMEL-Electricité de France, R&D, 92141 Clamart, FranceLAMEL, “Ecole Nationale des Arts et Métiers” (ENSAM), Lille, 59046 CEDEX, France

Numerical modeling of nondestructive testing (NDT) by eddy currents has been applied for qualification of two monitoring devices.Experimental data have enabled us to validate reliable and relevant finite-element method models. The encountered difficulties due togeometry (size and topology), the nature of the control signal (weak flux differences), and the movement accounting have been success-fully overcome by evaluating theA and T dual formulations with appropriate source fields and applying classical method tosuppress the mesh numerical error and choosing the well-known lock-step technique.

Index Terms—Eddy currents, finite-element method, magnetic fields, nondestructive testing (NDT).

I. INTRODUCTION

NUMERICAL modeling of nondestructive testing (NDT)by eddy currents is being evaluated as a support tool for

the qualification of testing devices used in heat exchanger tubes.However, the nature of the probe signal (differences betweenweak fluxes), the geometry (size and topology of the system),and the movement accounting lead to difficulties for finite-ele-ment software [1]. This paper deals with perfecting reliable fi-nite-element method (FEM) models for NDT by eddy currents.Investigations on feasibility and accuracy have been carried outon two testing probes by comparing with experimental data.This process has consisted in evaluating two dual formulations( and ) after having chosen existing methods formovement accounting and suppression of the numerical errordue to the mesh. Time-harmonic simulations for both formula-tions have been performed using CARMEL, a FEM softwarebased on Whitney’s elements developed at the LAMEL-L2EP.

II. CARMEL FORMULATION

A. Studied Problem

The eddy-current problem involved in NDT applications canbe summarized as shown in Fig. 1. The whole domain con-tains a source current , a conducting region , a region withmagnetic material (ferrite), and an air box. In linear time har-monic state the Maxwell’s equations become

(1a)

(1b)

where represents the electric field, the magnetic flux den-sity, the magnetic field, and the eddy current in .Equations (1a) and (1b) correspond, respectively, to Faraday andAmpere laws.

Digital Object Identifier 10.1109/TMAG.2007.892522

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Fig. 1. Eddy-current problem.

The boundary will be assumed to be split into two comple-mentary parts and so that

(2)

In addition, formulation requires a further boundarycondition on expressing that

(3)

To complete the problem, behavior laws have been introducedunder the following form:

(4)

One way to estimate the accuracy is to solve the two dual po-tential formulations: and . As a matter of fact,both of them converge to the exact solution when sufficientlyrefining the mesh [2], [3]. To undertake these formulations, dif-ferent kinds of source fields have to be derived to take into ac-count the currents flowing through the source and receptive coilsas well as in the conductive regions with holes.

B. Electric Formulation

As the magnetic field density is divergence free, a magneticvector potential can be introduced. Consequently, using (1a),the electric field in can be written, with the electric scalarpotential , as

(5)

A turn density vector is used to impose the currentthrough the coil constituted of stranded conductors [4]. This

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1786 IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 4, APRIL 2007

Fig. 2. Part of D in the neighborhood of flaw.

vector stands for a source field and makes it possible to expressthe current density as . Applying the weightedresidual method with the test functions and and takinginto account the boundary conditions lead to

(6.a)

(6.b)

The flux flowing through a coil “ ” can then be written as [4]

(7)

where represents the source field due to the coil “ .”

C. Magnetic Formulation ( )

As soon as the topology of a conductive region is not simply-connected (i.e., comprising holes), the use of formulations in-volving the magnetic scalar potential requires further condi-tions to ensure the Ampere’s law (1b).In our formulation, the current density source is imposedthrough the magnetic source field which is definedfrom a turn density vector so that

(8)

Moreover, as in NDT problems the flaws may go through thetube wall, it is necessary to introduce also further constraints for

[5]. The adopted approach is to define inside an arbitraryarea noted surrounding the flaw as shown in Fig. 2. issupposed to carry an uniform current density . As previouslyderived for strand conductors, a vector and a vector aredefined so that with .

Then, the current is an unknown of the problem. Consid-ering the area as a short circuit, a further electric equationcan then be derived [6]

(9)

The total current density in can then be written as thesuperposition of with an additional current density noted .The Ampere’s law in domain becomes

(10)

According to the definition of and considering the electricvector potential as , the magnetic field incan, thus, be expressed as follows:

(11)

Since is equal to zero outside , the use of (11)can be extended in all conducting regions. As has the sameproperty outside the coil, the expression of the magnetic field inthe whole studied domain is

(12)

Using the same method as for (6), it becomes the set of (13)whose unknowns are , and

(13a)

(13b)

In order to complete the set of (13), (9) must be considered byrewriting it with the help of (12). The flux through the coil “ ”can then be written under the form [4]

(14)

where represents the source field due to the coil “ .”

D. Discretization

Both potential formulations have been solved using the FEM.In 3-D, four discrete spaces can be introduced: the nodal ele-ment space , the edge element space , the facet elementspace , and the volume element space . According totheir properties, and belong to nodal element space andand to edge element space. In the same way, the vectors

and belong, respectively, to edge and facet ele-ment spaces.

In order to determine the families of vectors and , theclassical procedures based on a tree of facets for and a treeof edges for are applied [4].

The movements involved in our eddy current testing (ECT)studies have 2-degrees-of-freedom (DoF): rotation and transla-tion. The locked-step method [7] has been chosen so that thequality of solution be the same for all positions. Indeed, this ap-proach forces the mesh to remain conform.

III. APPLICATION AND RESULTS

CARMEL modeling applied to NDT by eddy currents hasbeen evaluated as a support tool for the qualification process oftwo testing probes. The works detailed hereafter deal with thefirst step of this process which consists in validating the FEMmodels for standard types of flaw by comparing with experi-mental data. The next step will be the investigations on the effi-ciency of the probes with regard to detection of complex combi-nations of flaws for which very few measurements are available.

The first probe consists in two identical coilswith a ferrite core,playing both the source and the receptive roles. The second onehas a separate source coil and two receptive coils. Both of themare driven helicoidally inside the heat exchanger tube under test.

A. Probe With Two Both Source and Receptive Coils

Simulations performed on this probe are intended to studythe feasibility of FEM modeling with CARMEL applied to ECT

HENNERON et al.: SOURCE FIELD COMPUTATION IN NDT APPLICATIONS 1787

Fig. 3. Studied probe with two identical coils.

Fig. 4. (a) Distribution of N around the flaw and (b) the correspondingvector K .

Fig. 5. (a) Real part of the current density distribution far from the flaw and(b) above.

problem and to identify the most appropriate modeling processwith regard to geometry, mesh, formulation, accuracy, time con-suming, etc.

This first probe has two coils fed by a current source at240 kHz (Fig. 3) and is mainly dedicated to detect longitudinalflaws. A finite length and width (5 mm 0.2 mm) flaw goingthrough the tube wall (1 mm) has been studied.

In the case of magnetic formulation, this problem requires tocompute three source fields and . Two for the coils andone because of the resulting nonsimply-connected topology ofthe tube due to the geometry of the flaw (Fig. 2). The resultingdistribution of vector and the corresponding vector areshown, respectively, in Fig. 4(a) and (b).

Fig. 5 shows the real part of the current density distribution(obtained with the formulation) in the tube when thecoils are far from the flaw Fig. 5(a) and above Fig. 5(b). It canbe noticed how the flaw unbalances the eddy-current density.

The present ECT control signal (Lissajou) is based on the fluxdifferences between the two coils. It quickly appeared that themagnitude of this value has the same order than the numericalnoise generated by the mesh. In order to get rid of this offset,the actual values have been computed by subtracting those ob-tained with and without flaw for problem implying the samemesh. These classical techniques are carried out by consideringa geometry where the flaw area is filled by either air or metaldepending on the simulated case.

In Fig. 6, the magnitude of the flux difference after correctionof the mesh noise is given for both formulations. The results arein good agreement according to the fact that the modulus ofrepresents less than 1% of the total flux flowing one coil.

It is worth noticing that the number of iterations of the conju-gate gradient is greater for the formulation than for ,

Fig. 6. Flux difference with regard to the probe position for both formulations(�T � � �A � ').

TABLE ICOMPUTATION CHARACTERISTICS

Fig. 7. (a) Geometry of the probe and (b) the flaws.

while the computation time is three times as big (Table I). andare defined in the whole domain. Since the number of edges

is approximately six times greater than the number of nodesfor a tetrahedral mesh, the ratio between the number of DoFs(nDoFs) for and the nDoFs for is almost 6. and areonly defined in the conducting part that is much smaller than thewhole domain. Hence, even if the nDoFs for is greater thanthe nDoFs for , the nDoFs for remains much smallerthan the nDoFs for .

B. Probe With One Coil Source and Two Receptive Coils

The second probe has a separate source coil and two receptivecoils Fig. 7(a) and is more dedicated to detect circumferentialdefaults. simulations have been applied to two standardflaws (reference and coherence defaults) at two frequencies (100and 600 kHz). For symmetry reasons, only a translation move-ment has been treated (30 positions).

1) Reference Flaw: The reference default consists in an in-ternal circumferential flaw with a depth equal to half the tubethickness Fig. 7(b). For each frequency, the resulting signal isused to derive the normalization constants which shift the mag-nitude (respectively, the phase) of its maximum to the value of0.5 V (respectively, 10 ). This transformation is then applied toall the following control signals. As the flux difference shouldbe equal to zero, the signal obtained without any flaw standsfor the numerical noise due to the mesh. Control signals ob-tained before and after suppression of the numerical offset are

1788 IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 4, APRIL 2007

Fig. 8. Control signal for reference flaw before numerical error correction.

Fig. 9. Control signal for reference flaw after numerical error correction.

TABLE IICOMPUTATION CHARACTERISTICS AND MESH

Fig. 10. Normalized control signal at 100 kHz for coherence default after nu-merical error correction.

Fig. 11. Normalized control signal at 600 kHz for coherence default after nu-merical error correction.

depicted, respectively, in Figs. 8 and 9. Computation character-istics and mesh are given in Table II.

2) Coherence Default: The coherence default consists in anexternal circumferential flaw with a depth equal to 40% of thetube thickness Fig. 7(b). The probe signal used for monitoring,which is actually a combination of the normalized signals ob-tained at 100 and 600 kHz has been simulated (Figs. 10–12).Table III shows a very good agreement between simulationresults and experimental measurements of the parameters of thecontrol signal which are taken as relevant for the monitoring: Thevalue of the maximum of magnitude and its associated phase.

Fig. 12. Combined control signal for coherence default.

TABLE IIIMAGNITUDE AND PHASE OF THE MAXIMUM OF THE LISSAJOU

IV. CONCLUSION

At EDF Research and Development, numerical modeling ofNDT by eddy currents is being evaluated as a support tool forthe qualification of testing devices. FEM software CARMEL,developed at LAMEL-L2EP, has been applied to simulate twotesting probes. Experimental data have enabled us to validatereliable and relevant FEM models and their associated compu-tation process. Thus, complex geometries (size and topology),the nature of the control signal (weak flux differences), andthe movement accounting have been successfully overcome byusing the formulation with appropriate source fields,much less time consuming than the dual formulation , incombination with the well-known lock-step technique and clas-sical method to suppress the mesh numerical error. The nextstep will deal with the efficiency of the probes with regard todetection of complex combinations of flaws for which very fewmeasurements are available.

REFERENCES

[1] J.-P. Ducreux, S. Lepaul, and J.-C. Vérité, “Modelling of NDT in tubesof heat exchanger,” in Proc. ISEM, 2005, p. 290.

[2] A. Bossavit, “Complementary formulations in steady-state eddy-cur-rent theory,” IEE Proc. A, vol. 139, pp. 265–272, 1992.

[3] C. Li, Z. Ren, and A. Razek, “Complementary between the energy re-sults of H and E formulations in eddy-current problems,” IEE Proc.Sci. Meas. Technol., vol. 141, pp. 25–30, 1994.

[4] Y. Le Menach, S. Clénet, and F. Piriou, “Numerical model to discretizesource fields in the 3D finite element method,” IEEE Trans. Magn., vol.36, no. 4, pp. 676–679, Jul. 2000.

[5] Z. Ren, “T- Formulation for eddy-current problems in multiply con-nected regions,” IEEE Trans. Magn., vol. 38, no. 2, pp. 557–560, Mar.2002.

[6] T. Henneron, S. Clénet, and F. Piriou, “Calculation of extra copperlosses with imposed current magnetodynamic formulations,” IEEETrans. Magn., vol. 42, no. 4, pp. 767–770, Apr. 2006.

[7] T. W. Preston, A. B. J Reece, and P. S. Sangha, “Induction motor anal-ysis by time-stepping techniques,” IEEE Trans. Magn., vol. 24, no. 1,pp. 471–474, Jan. 1988.

Manuscript received April 25, 2006 (e-mail: olivier.moreau@edf.fr).