\ IEEE TRANSACTIONS ON MAGNETICS, VOL. 26, NO. 6, NOVEMBER 1990 3089

Numerical Simulation of Pulsed Eddy-Current Nondestructive Testing Phenomena

Abstract-A general-purpose hybrid finite element and finite difference computational model has been developed for the prediction of pulsed eddy-current distribution in metals for nondestructive testing purposes. The numerical model employs an axisymmetric formulation to study coil configurations suspended over a metallic specimen. As a driving function, a pulsed Maxwell-distributed current density is ap- plied. Resulting eddy-current distributions are discussed as a function of conductivity, permeability, and lift-off. Furthermore, the transient voltage response of a coil over an infinite half-space is numerically computed and compared to the solution obtained by a novel analytical approach. The numerical model is then employed to determine the in- duced voltage due to a surface-breaking flaw.

I. INTRODUCTION HE numerical study of single-frequency eddy-current T phenomena for nondestructive testing (NDT) pur-

poses has successfully been applied to many practical in- spection situations [ 11-[3] for which analytical solutions are either difficult or impossible to obtain [4], [5 ] . Espe- cially, the inherent NDT problem of predicting magnetic field interaction with arbitrarily shaped defects in com- plex material structures precludes closed-form analytical approaches [6] . Nevertheless, in order to check the cor- rectness of a particular numerical modeling approach, a comparative analysis is a prerequisite to gain confidence in applying the numerical method. This can be achieved either by using an existing analytical solution or by mak- ing direct experimental measurements. In the case of sin- gle-frequency eddy-current finite element modeling, for instance, such an approach has resulted in a better under- standing of the capabilities and limitations of the numer- ical approach [7].

The extension frQm single-frequency to multifrequency and pulsed eddy-current applications is inspiring in- creased industrial research [8]-[ 101. Pulsed eddy-current testing, in particular, offers several advantages over the conventional single-frequency technique such as reduced sensitivity to coil lift-off variation and better magnetic field penetration due to wide-band frequency excitation. Surprisingly, few theoretical research efforts have gone into the study of the potential use of this electromagnetic .

Manuscript received March 19, 1990; revised May 17, 1990. This work was supported by the Nondestructive Evaluation and Sensor Technology Department, Timken Company, Canton, OH.

X.-W. Dai and R. Ludwig are with the Department of Electrical Engi- neering, Worcester Polytechnic Institute, Worcester, MA 01609.

R. Palanisamy is with Tirnken Research, Timken Company, Canton, OH 44706.

IEEE Log Number 9038057.

nondestructive method. The numerical solution of the un- derlying parabolic partial differential equation requires more elaborate computer algorithms accompanied by an increase in computer resources. Furthermore, in order to be of practical relevance, the resulting magnetic fields have to be linked to the interrogating coil to yield a mea- surable probe response.

Within the context of modeling, a weighted residual- based finite element model for the computation of tran- sient single-component axisymmetric magnetic vector po- tential and eddy-current distributions was formulated. The model predictions were quantitatively verified against a novel transient analytical model [1 11 of a step function driving current in an infinite wire or in a wire loop sus- pended over a nonmagnetic, conducting half-space. The resulting numerical temporal and spatial eddy-current dis- tributions exhibited excellent agreement with this analyt- ical model.

The purpose of this paper is to extend the numerical modeling of pulsed eddy-current phenomena to more complex, and thus realistic, testing situations. In partic- ular, the magnetic fields and induced eddy -current distri- butions associated with a realistic axisymmetric probe are studied as a function of lift-off and changes in material parameters such as conductivity and permeability. The in- duced transient voltage in the coil due to an impressed current pulse is predicted and placed in context with rig- orous analytical studies. Furthermore, the field interac- tion with a surface-breaking axisymmetric crack is inves- tigated in terms of the resulting induced voltage response in the coil as a function of crack depth and radius.

11. UNDERLYING PHYSICAL PRINCIPLES A. Parabolic System Formulation

The underlying partial differential equation governing pulsed eddy-currents for linear material parameters in axisymmetric geometries can be specified in terms of the single-component magnetic vector potential A, [ 121 :

1 a 4 P at - V2A, - U - = -Js

with p , U , Js denoting, respectively, total magnetic permeability, electric conductivity, and source current density. Equation (1) is discretized as part of a weighted residual finite element approach [ 111 to yield an implicit

!:

0018-9464/90/1100-3089$01 .OO O 1990 IEEE

3090 IEEE TRANSACTIONS ON MAGNETICS, VOL. 26, NO. 6, NOVEMBER 1990

time stepping algorithm:

where [SI , [ C ] , [D] are global matrices whose coeffi- cients involve integrations over linear polynomial shape functions [ 131. The quantities { A , }[, { Js}[ represent the nodal vector potential and source current density at dis- crete time steps of step size A t . { Q } , + , , contains the surface integrals over the normal derivative of the vector potential as part of the Neumann-type boundary condi- tions. The solution for { A , } + is found by inverting the left-hand side of (2) by employing either direct or iterative matrix solvers. This model has been applied to predict A,, and through the relationship

the induced eddy-current density Je due to an impressed step current density Js in a single-wire loop suspended over a conducting half-space. The resulting numerical eddy-current predictions in a conducting half-space as generated by (2) and (3) are in good agreement with a transient analytical model developed for this comparison.

B. Numerical Computation of Probe Response The numerical formulation for the calculation of the in-

duced voltage (emf) in the coil can be based on the eval- uation of Faraday's law which relates the induced emf to the flux density B, and subsequently the line integral of the vector potential. For a finite cross-sectional one-turn coil, we can define an induced voltage Vf

. .

Vf = (4) . j 1 dr dz

with [7]

- - - t j j ( V x A ) - d S = - at IC$ * d l

where Vp indicates the voltage due to a point pick-up coil. Therefore, (4) denotes the total point contributions aver- aged over the cross area of the finite size coil. The finite element evaluation of (4) yields the numerical expression

aAci -2r C rci - ai

; = I at

Fig. 1 . Axisymmetric geometry for pulsed eddy current analysis. R, is the radius of the centroid of a typical triangular element lmn. A, is the mag- netic vector potential at the centroid. Electrical conductivity ( U ) of the test specimen is 3.5 x lo7 S /m, and the relative permeability ( p,) is 2.

Here, Ai, rcj, A,; describe the area of the ith triangular element, its centroidal distance, and centroidal vector PO-

tential value as shown in Fig. 1 . The summation extends over all the source elements comprising the probe. The time derivative in ( 5 ) is approximated by using the same backward difference formula as given in (3).

C. Analytical Calculation of Probe Response As discussed in [ l l ] , [12] the vector potential of a sus-

pended point source wire loop over a half-space (Fig. 2) can be expressed in the Laplace domain s for z 2 0 as

~

a,<,, z , s; ro, ho) CO

-A(ho + Z ) = 1 Pojsro j, J I ( Xr) J d hro) e

* {eZhho + Q ( s ) ) dX (6)

where r,, ho, 1, are the radius, elevation, and transformed current of an elemental loop. J , ( ) and po denote the Bes- se1 function of the first kind and the absolute permeabil- ity; Ap is the point value due to a point source coil. Fur- thermore, (6) contains the dimensionless quantity [ 121

(7) Xpr - JX2 + spopru

Apr + Jh2 + spopru B(s) =

which represents the reflective effect of the specimen in terms of the relative permeability pr and electric conduc- tivity U . It can be shown [14], [15] that (7) has a closed- form expression in the time domain

2 p r ( p ; - 1)X2t 4 ( t ) = E L I 2 - ~

p r + 1 ~ : - l

I

3 I

DAI et al. : SIMULATION OF PULSED EDDY-CURRENT TESTING PHENOMENA 3091

t

Fig. 2. Thin wire loop of radius r, over a conducting and ferromagnetic half-space (lift-off = ho) .

Fig. 3. Cross section through an axisymmetric cup-core probe with fer- romagnetic core ( p, = 100) over a half-space [16].

where erfc ( ) specifies the error function. In the time do- main, (6) transforms into

Ap(r , z , t; ro, ho) m

= 1 poZs(t)ro lo JI( Xro) J1( Xr) dX

where the symbol 63 indicates convolution with a realistic transient current source Is. The value of q ( t ) is specified in (8). The first integral in (9) is the unperturbed vector potential due to an element current loop in air, and the second integral is a result of the influence of the half- space. To account for a finite-size source coil, the ele- mental loop expreeion (6) must be superimposed to yield a combined vector potential

p h o + A h p r g + A r

J, A p ( r , z , t; r’, h ) dr’ dh ho

1 where A f is the point value due to a finite cross-sectional source coil. Therefore, the voltage in a point pick-up coil is given by

( 1 1 )

Substituting ( 1 1) into (4), we obtain an analytic evalua- tion of the induced emf

aAp(r, Z, t; r ’ , h ) at

The two inner integrals represent the total probe response, which in turn requires an additional double integration to determine the induced voltage.

111. SIMULATIONS A. Eddy-Current Distributioii as a Function of Conductivity, Permeability, and L i f -Of

An axisymmetric eddy-current testing arrangement is considered in Fig. 3. The driving probe is a ferromagnetic cup-core probe with dimensions as provided by Bowler et al . [ 161. For the purpose of this paper, a pulsed, Maxwell distributed, discrete current density (Fig. 4)

is applied as the driving function for the excitation coil. Here, At = 0.1 ps and Jo = lo2 A/m2. The integer m is selected to be 100. Therefore, the current density reaches its maximum at 10 ps. Fig. 5(a)-(d) shows, for a constant lift-off of 0.8 mm and a fixed conductivity of U = 3.5 X lo7 S/m, the spatial distribution of the eddy currents as a function of relative permeability p r . The time interval of 80 ps has been chosen such that the driving current den- sity has sufficiently decreased to zero. As can be clearly seen from Fig. 5(b)-(d), even a relatively small increase in permeability reduces the field penetration significantly such that for values of k, > 10, the vector potential is almost completely restricted to the near surface. Fig. 6

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3092 IEEE TRANSACTIONS ON MAGNETICS, VOL. 26, NO. 6, NOVEMBER 1990

0 00 2000 4000 time (11s)

Fig. 4. Sampled (Maxwell-distributed) excitation current density Js.

(C) ( 4 Fig. 5 . Spatial eddy-current distribution for fixed probe lift-off of 0.8 mm

and fixed U = 3.5 x lo7 S/m at discrete time instance of 80 ps as a function of relative permeability values of (a) p, = 2, (b) p, = 4, (c) pr = 6, and (d) p , = 10.

(a) and (b) provides eddy-current distributions for the ex- treme cases of p, = 80 and 100 on an enlarged scale which focuses on the near-surface effects. Alternatively, Fig. 7(a)-(d) discusses the eddy-current penetration due to

multiplicable increments in conductivity. Again, a sub- stantial decrease in field strength is observed as conduc- tivity is increased. The third parameter of interest is the variation in lift-off and its effect on the induced eddy-cur-

DAI et al. : SIMULATION OF PULSED EDDY-CURRENT TESTING PHENOMENA 3093

(a) (b) Fig. 6. Spatial eddy-current density distribution for same lift-off, conduc-

tivity, and time instance as specified in Fig. 5 for (a) p, = 80 and (b) p, = 100 on a ten times enlarged depth scale.

(C) (d) Fig. 7. Spatial eddy-current distribution for fixed probe lift-off of 0.8 mm

and fixed permeability p, = 2 at discrete time instance of 80 ps as a function of conductivity values of (a) U = 3.5 x IO’ S/m, (b) U = 2 x 3.5 X lo7 S/m, (c) U = 4 X 3.5 X 107S/m, and (d) U = 16 x 3.5 x 107s /m.

3094

200.00 3

IEEE TRANSACTIONS ON MAGNETICS, VOL. 26, NO. 6, NOVEMBER 1990

150.00

/--. c\I 100.00 * ,$ 50.00

a: 0.00

v

-5000

1 - 100.00

-150.00 ;I I I I I I ! I I I , , I I , , , , , , , , , , I

0.00 20.00 40b0 time (ps )

Fig. 8. Temporal eddy-current distribution as a function of lift-off at.sur- face location r = 4 mm ( z = 0) based on U = 3.5 x lo7 S/m and p r = 2.

rents in the specimen. In Fig. 8, the eddy-current distri- bution is predicted for the surface location of r = 4 mm based on U = 3.5 X lo7 S/m and p, = 2. It is noteworthy to point out, that although the eddy-currents are weakened for larger lift-off values, the crossover points for all dis- tances occur at the same location of 13.5 ps, a feature characteristic of pulsed eddy-current testing.

B. Numerical Calculation of Probe Response To confirm the accuracy of predicting the probe re-

sponse by the numerical method ( 9 , a comparison to the analytical equation (12) is conducted. The probe geome- try chosen for this comparison is depicted in Fig. 9. A uniform current density is applied to this coil with a time history as given by (13). Fig. 10 shows good agreement between both methods. Especially when compared to step current densities as discussed in [ 1 11, an exponentially distributed excitation function yields better agreement as the continuously increasing pulse is followed with higher fidelity by the numerical method. The discretization of the solution domain into 32 600 triangular elements and a time step size of 0.1 ps is identical to the configurations given in [ 1 13, which also discusses mesh parameters and run times in detail.

C. Probe Response due to a Suflace-Breaking Crack The potential of any numerical method lies in its ability

to handle complex geometric shapes such as a simulated crack as shown in Fig. 11. Here, the probe emf due to a surface-breaking cylindrical crack is studied as a function of depth and radius. The crack volume is assumed to con- sist of air. The impressed excitation current density again is given as in (13). Fig. 12 examines variations in the transient emf due to changes in radii from r = 1 mm to r = 10 mm for a fixed z of - 1.5 mm. Alternatively, Fig. 13 shows the reduction in emf for increasing crack depth from 0.1 to 4 mm at a fixed radius of r = 6 mm.

From a practical point of view, it is important to ex- amine the probe sensitivity to changes in crack size. This sensitivity may be expressed in terms of changes in the

a=3.5*El Slm F t'

r

Fig. 9. Finite area coil over a half-space in an axisymmetric coordinate system.

- 150.00 0.00 b time steps (0.1 us/step)

Fig. 10. Comparison between the analytically determined emf based on (12) and the numerically computed emf based on (5) for the coil given in Fig. 9.

t

A I = Imm

Fig. 1 1 . Coil suspended over an axisymmetric surface-breaking crack.

I

DAI et al.: SIMULATION OF PULSED EDDY-CURRENT TESTING PHENOMENA 3095

100 00

50.00

0.00 A

v >

-50.00

- 100.00

TABLE I CALCULATED PROBE SENSITIVITY ACCORDING TO DEFINITIONS

GIVEN IN (14) FOR (a) FIXED CRACK DEPTH OF 1.5 mm AND (b) FIXED CRACK RADIUS OF 6 mm

0.0002 r < 2 m m A emf 7 ( v / m m ) 10.5 r = 4 m m

0.75 r = 6 m m

(a) at - z = 1.5 mm

45 -z = 0.1 mm A emf - ( v / m m ) 34 -z = 0.5 mm

A z

20 -z = 1.0 mm

1.2 -z = 1.5 mm

-200 00 0 0 0 20000 40000

time steps (0 1 ps/step)

(b) at r = 6 mm

- 1 5 0 0 0 L

Fig. 12. Numerically predicted emf due to change in crack radius at a fixed crack depth of 1.5 mm.

i 150.00

100.00

50.00

A 0.00 > v

-50.00

E -10000

-15000

-200.00

-250.00 0.00 200.00 400.00

t ime steps (0.1 ps/step)

Fig. 13. Numerically predicted emf due to changes in crack depth at a fixed crack radius of 6 mm.

maximum emf voltage as a function of either depth Or radial resolution, i.e. ,

An alternative approach involves the time difference in the signal crossings through the zero voltage line as func- tions of depth and radius, i.e.,

Since both definitions are equivalent and yield the same results, only (14) is used and recorded in Table I. As can be seen for a fixed depth z, the highest sensitivity is ob- tained for a radius of 4 mm which equals the probe radius. On the other hand, if the radius is fixed, the highest sen- sitivity occurs for small depth values or near-surface de- fects. These probe predictions are a direct result of the higher potential field densities at these locations.

IV. CONCLUSIONS This paper discusses the development of a hybrid nu-

merical model for the analysis of pulsed eddy-currents in conducting and ferromagnetic materials and its applica- tion to nondestructive testing. The model is capable of simulating 2D and axisymmetric test geometries by pre- dicting the penetration of the single-component vector po- tential and the associated eddy-current distributions due to variations in probe lift-off and material parameters such as permeability and conductivity. The numerical model predicts a significant reduction in field penetration as the conductivity and permeability increase.

In order to study the induced coil voltage, a novel tran- sient analytical model has been introduced to confirm the correctness and high-fidelity signal reproduction of the numerical code. Subsequently, the model has been em- ployed to investigate an axisymmetric crack as a function of crack depth and radius. The numerical approach re- veals how changes in crack size influence the probe re- sponse as part of a sensitivity analysis of the numerically predicted induced emf.

Having introduced a transient numerical computational model for NDT purposes, and having confirmed its cor- rectness against both line and thin wire loop current ex- citations [ 113 as well as finite size probe analytical models, it now remains to validate this model with the help of practical measurements. However, even in the absence of practical studies, the numerical model provides a wealth of important information regarding parameters such as the magnetic vector potential and the induced eddy-current distributions which are generally difficult, if not impos- sible, to determine experimentally. In particular, the field distributions inside the specimen provide valuable insight into the field/flaw interaction and the factors affecting the probe response.

ACKNOWLEDGMENT The authors are grateful to Dr. D. Cyganski, Vice Pres-

ident for Information Services, WPI, for having provided

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3096 IEEE TRANSACTIONS ON MAGNETICS, VOL. 26, NO. 6, NOVEMBER 1990

the computational Support to generate the numerical and graphical results reported in this paper.

1161 J. R. Bowler, L. D. Sabbagh, and H. A. Sabbagh, “A theoretical and computational model of eddy-current probes incorporating volume in- tegral and coniugate gradient methods.” IEEE Trans. Maan., vol.

REFERENCES R. Palanisamy and W. Lord, “Finite element modeling of electro- magnetic NDT phenomena,” IEEE Trans. Magn., vol. MAG-16, pp. 1083-1085, Sept. 1980. - , “Interaction of support plate and defect eddy-current trajecto- ries in steam generator tubing NDT,” Mater. Evaf . , vol. 39, pp. 651- 655, June 1981. - , “Prediction of eddy-current signals for nondestructive testing of condenser tubing,” IEEE Trans. Magn., vol. MAG-19, pp. 2213- 2215, Sept. 1983. N. N. Zatsepin and V. E. Shcherbinin, “Calculation of the magneto- static field of surface defects, I. Field topography of defect models,” Defektoskopiya, pp. 50-59, Sept.-Oct. 1966. M. L. Burrows, “Theory of eddy-current flaw detection,” Ph.D. dis- sertation, Dep. Elec. Eng., Univ. Michigan, Ann Arbor, 1964. W. Lord, “Application of numerical field modeling to electromag- netic methods of nondestructive testing,” ZEEE Trans. Magn., vol.

- , “Development of a finite element model for eddy-current NDT phenomena,” Elec. Power Res. Inst. (EPRI), Palo Alto, CA, Rep. 1395-2, Sept. 1981. J. L. Fisher and R. E. Beissner, “Pulsed eddy-current crack charac- terization experiments,” in Proc. Rev. Progr. Quantitative Nonde- structive Evaf . , D. 0. Thompson and D. E. Chimenti, Eds., vol. 5A.

MAG-19, pp. 2437-2442, NOV. 1983.

New York: Plenum, 1986, pp. 199-206. [9] G. Wittig and H. M. Thomas, “Ein Beitrag zur Kalibrierung und

Auswertung von Signalen in der Wirbelstrompriifung mit Rechnerun- terstiitzung,” Maferialprufung, vol. 29, pp. 209-213, July-Aug. 1987.

[IO] H. A. Crostack and J. Nehring, “Development and application of pulsed eddy-current testing using CS-technique,” in Proc. 11th World Conj Nondestructive Testing (Las Vegas, NV), Nov. 1985, pp. 208- 215.

[ l l ] R. Ludwig and X. W. Dai, “The numerical and analytical modeling of pulsed eddy-currents in a conducting half-space,” IEEE Trans. Magn., vol. 26, pp. 299-307, Jan. 1990.

[I21 C. V. Dodd and W. E. Deeds, “Analytical solutions to eddy-current probe-coil problems,” .I. Appl. Phys., vol. 39, pp. 2829-2838, May 1968.

[I31 L. J. Segerlind, Applied Finite Element Analysis. New York: Wiley, 1984.

[I41 A. F. Chub, N. L. Bondarenko, and A. P. Stipura, “Nonstationary electromagnetic field of a coil located above a ferromagnetic half- space,” Defektoskopiya, pp. 106-1 13, May-June 1977.

[I51 -, “Density of eddy-currents induced in a ferromagnetic half-space by the transient magnetic field of a loop,” Defektoskopiya, pp. 41- 45, Mar.-Apr. 1977.

25: pp. 2650-;664, Gay 1989.

Xiao-Wei Dai was born in Cheng Du, People’s Republic of China, on May 25, 1960. He received the B.S. and M.S. degrees in electrical engineering from the Changsha Institute of Technology, Changsha, Hunan, People’s Republic of China, in 1982 and 1985, respectively.

He was an instructor of Electrical Engineering at Changsha Institute of Technology. Since January 1989 he has been with Worcester Polytechnic Institute, Worcester, MA, where he is a Ph.D. candidate and Graduate Research Assistant in the Department of Electrical Engineering. His re- search interests are in electromagnetic and ultrasonic system modeling, digital image and signal processing, and microprocessor-based system de- velopment.

Reinhold Ludwig (S’83-M’86-SM’90) was born in Ravensburg, Ger- many, on July 22, 1955. He received the Dipl-Ing. (M.S.E.E.) degree from the University of Wuppertal, Germany, in 1983, and the Ph.D. degree from Colorado State University, Fort Collins, in 1986, both in electrical engi- neering.

He is currently an Associate Professor with the Department of Electrical Engineering, Worcester Polytechnic Institute, Worcester, MA. His re- search interests include the numerical modeling of electromagnetic and ul- trasonic field phenomena for nondestructive material inspection.

Dr. Ludwig is a member of the American Society of Nondestructive Testing (ASNT), Eta Kappa Nu, and Sigma Xi.

R. Palanisamy (S’77-M’79) received the B.S. degree in physics and the B.S. and M.S. degrees in electrical engineering from the University of Madras, Madras, India, the M.E. degree in mechanical engineering from Howard University, Washington, DC, and the Ph.D. degree in electrical engineering from Colorado State University, Fort Collins.

He is a Senior Research Specialist in the Nondestructive Evaluation and Sensor Technology Department, the Timken Company, Canton, OH, and an Adjunct Associate Professor of Electrical Engineering at the University of Akron, Akron, OH. He has conducted research at the Indian Space Re- search organization, Colorado State University, and the Ames Laboratory (USDOE).

Dr. Palanisamy is an Associate Technical Editor for Murerials Evufua- tion, the official journal of the American Society for Nondestructive Test- ing. He is the 1984 recipient of ASNT’s Achievement Award.