An Effective 3-D Finite Element Scheme for Computing ... - IEEE Xplore

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 33, NO. 2, MARCH 1997

An Effective 3-D Finite Element Scheme for Computing Electromagnetic Field Distortions Due to Defects in Eddy-Current Nondestructive Evaluation Zsolt Badics, Yoshihiro Matsumoto, Kazuhiko Aoki, Fumio Nakayasu, Mitsuru Uesaka, and Kenzo Miya

Abstract—A new three-dimensional (3-D) finite element scheme for eddy-current nondestructive evaluation (NDE) problems is described that calculates directly the perturbation of the electromagnetic field due to defects in metallic specimens. The computational costs of such problems are usually very high using available finite element schemes, and the new scheme is supposed to lower these costs. The basic concept, the direct calculation of the field distortion due to the flaw, is provided for rather general defects, but the detailed finite element scheme is discussed for zero-conductivity flaws. The source terms of the formulation are determined from the unperturbed field, and the impedance change due to a defect can be calculated as an integral over the flaw. A finite element scheme for solving problems with cracktype defects is also presented as a limiting case of the formulation for zero-conductivity flaws. Solutions of a benchmark problem from the testing electromagnetic analysis methods (TEAM) workshop series (problem number 15/2) and of tube problems with artificial slots are presented and compared to experimental data. Index Terms— Eddy current nondestructive evaluation, finite element method, flaw field calculation, three-dimensional scheme.

I. INTRODUCTION

I

N eddy-current nondestructive evaluation (NDE), usually a probe coil generates eddy currents in metallic components. The size of the defects in the conductor is small, and their material parameters, e.g., the electric conductivity or the permeability, are different from the parameters of the host material. Therefore, the impedance of the exciting coil or the induced electromotive force (EMF) of a receptive coil changes, and this change indicates the presence and some characteristics of the defects. The change of the observed quantity, however, is usually under 1% of the total value or even smaller in practically important cases. Efficient computational schemes for the problem with plate or tube specimens are presented in [1] and [2] using volume integral techniques. These techniques, however, become extremely expensive computationally in the case of more complicated arrangements than infinite slabs or tubes with homogeneous and linear material parameters. Our main goal is to simulate probe-defect interactions in steam generator tube inspection where the diameter of the tube specimen may vary Manuscript received August 18, 1994; revised December 12, 1995. Z. Badics, Y. Matsumoto, K. Aoki, and F. Nakayasu are with Nuclear Fuel Industries Ltd., Sennan-gun, Osaka-fu, 562 Japan. M. Uesaka and K. Miya are with the University of Tokyo, Bunkyo-ku, Tokyo, 112 Japan. Publisher Item Identifier S 0018-9464(97)00664-X.

in some regions, e.g., in the expansion region at the tube sheet. Besides, the probe signal is often distorted in some zones of the tubing by large massive ferromagnetic components, e.g., in the tube support plate zones or by thin metallic shells with significantly higher conductivity than the conductivity of the tube, e.g., in zones with copper deposits. Because of these requirements, we have chosen the finite element technique to analyze such problems. Since the state variables of finite element schemes (field or potential quantities of the governing equations) represent the total electromagnetic field or the induced eddy-current field, the impedance change due to a defect is calculated by building the difference of the impedance values with the flaw present and absent. To obtain an acceptable accuracy in impedance change, extremely high accuracy has to be reached in the finite element computation. This is the main reason why the analysis of such problems is unusually difficult. The other difficulty that arises in computing such finite element models is the very different discretization requirements for the defect, its surrounding, and the unperturbed field. The defect-free arrangement usually requires larger element sizes and a larger discretized region than the perturbed field because the defect size is usually smaller than the mean diameter of the exciting coil and the dimensions of the host metallic specimen. Or at least one of the extents of the defect is very small as in case of crack-type defects. Because of these different subdivision requirements such problems require very large three-dimensional (3-D) mesh structures that approach the capacity limits of the modern computing tools. Furthermore, the unperturbed model usually shows plane or axial symmetries, and sometimes the unperturbed field can be calculated analytically. But when a defect is present, the perturbed field calculation requires almost always full 3-D computation [3], [4]. It is straightforward to split the computation into a computation without the flaw and a computation of the field distortion due to the flaw. Similar approaches are established in [1] and [2] using integral equation techniques. In this paper we present a new 3-D finite element scheme that calculates directly the distortion of the eddy-current field due to a flaw. The source terms of the formulation are determined from the eddy-current distribution with the defect absent so any simplification in calculating the unperturbed field, such as taking into account symmetries or using analytical solution, can be applied. The values of the unflawed field have to be known only in the flaw

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BADICS et al.: COMPUTING ELECTROMAGNETIC FIELD DISTORTIONS

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where the prime denotes the field with the flaw present. is outside the flaw and continuous on the interface, S . Similarly, is outside the flaw and continuous on the interface. Suppose that and are linear and anisotropic. Subtracting (1) from (3), and adding the terms and to the right-hand side, one can obtain (5) where the field quantities without superscript are the perturbations ( and ), and is an incident electric current density that generates the perturbation of the field due to the change of . After performing similar manipulations on (2) and (4), we obtain Fig. 1. Typical probe-flaw problem.

(6) to calculate the perturbation. Furthermore, the flaw impedance, the impedance change due to the defect, can be determined by performing integrals only over the flaw region. First, we explain the basic concept of the new scheme, and then we describe the algorithmic details at flaw problems with zero conductivity. Next, we derive a formulation for crack-type defects as a limiting case of the scheme for zero-conductivity flaws. Finally, we present solutions of representative test problems with reliable experimental results.

is an incident magnetic current where density. The flaw field, the distortion of the electromagnetic field due to the defect, is available directly solving the equations system (5), (6). When the excitation is time harmonic, the observed quantity is usually the flaw impedance, impedance change due to the flaw. This could be calculated from the computed flaw field and the known unflawed field values, as (7)

II. BASIC CONCEPT The basic idea of the new scheme is explained here through a typical probe-flaw arrangement (see Fig.1). The figure shows a flaw, R , in a homogeneous conducting material, R . Suppose that the eddy-current pattern is known in the conductor without flaw and satisfies Maxwell‘s equations with the displacement currents neglected (1) (2) where the superscript, , denotes the unflawed quantities, and and are the conductivity and the permeability of the unflawed host material, respectively. It is assumed that they are linear and anisotropic. This unflawed field is generated by an outer probe coil, and it can be taken as given now. We concentrate here on sinusoidal excitation but the results can be extended to transient flaw field calculation as long as the assumptions on material parameters are valid. The flaw has a different conductivity and permeability than the host material but we temporarily assume that they are continuous on the interface, S . This assumption is valid only in this section, and from Section III on, the discussion also includes the case of discontinuous conductivity. The flaw with different permeability is not discussed here in detail, but the basic idea for treating this case is hinted at in this section. In the flawed arrangement, the electromagnetic field in the conductor is also described by Maxwell‘s equations (3) (4)

is the flaw where is the current of the probe coil, and impedance. Equation (7) can be derived from the reciprocity theorem applying a routine procedure [5] on Maxwell’s equations of the whole flawed arrangement and the equations (5) and (6) (see the Appendix). A formula obtained in a similar way has already been applied in eddy-current NDE [2], [6]. Advantageously, (7) is determined by performing a volume integral only over the flaw region. Another benefit is that a relatively small region has to be discretized for the defect field calculation because the field distortion is concentrated only around the flaw. This can be seen easily from the fact that the source quantities and are in a conducting matter where the modulus of a field quantity attenuates rapidly moving away from the source. So the field distortion can be taken as zero beyond 5 or 6 ( is the penetration depth). Thus, the mesh structure is usually smaller for the flaw field calculation using the present method than for calculating the total perturbed field. Summing up, we can first determine the unflawed field taking advantage of any kind of symmetries or analytical solutions. Then the flaw field can be obtained using a mesh structure whose size depends on the defect shape and on the applied frequency only and is almost independent of the dimensions of the probe coil and of the host specimen. III. FINITE ELEMENT SCHEME ZERO-CONDUCTIVITY FLAWS

FOR

For the sake of simplicity, a complete discussion of a finite element scheme is only presented here for a zero-conductivity

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flaw with


everywhere in the arrangement, so in in

(8)

This case includes a discontinuous conductivity that causes difficulties in implementing the formulation. The spatially varying conductivity in the flaw is straightforward to introduce based on the following discussion, but examples are presented here for zero-conductivity flaws only. The finite element scheme of a flaw with different permeability is not discussed, only the basic concept has been hinted in Section II. A. Field Equations of Zero-Conductivity Flaw The field equations in the flaw region, placement current density neglected are

Fig. 2. Schematic arrangement for flaw field calculation.

, with the dis(9) (10)

variable, although the conductivity is zero. So the magnetic field can be expressed, as in

and

(17)

in

where

(18)

and the electric field is described as (11) In the conductor, equations

in

(19)

in

(20)

, the field distortion satisfies the (12) (13)

The conductivity is discontinuous on the interface, S the interface conditions has to be considered, as

In the air region, as usual, we do not calculate the electric field. After performing the routine mathematical manipulations [9], we arrive at the governing equations

, so (14) (15) (16)

in

(21)

in

(22)

where [ ] denotes the jump of a quantity on the interface, suffix, , denotes quantities in the conducting region, , and n denotes the unit normal of the interface, . Interface condition (16) is necessary to ensure the divergence-free property of the current density distribution. To prescribe this surface current flux is not straightforward in a finite element analysis. Usually, zero current flux is set on the interfaces between the conducting and nonconducting regions.

in

(23)

in

(24)

B. Potential Formulation To specify the potential formulation of the finite element scheme, consider the whole region depicted in Fig. 2. is the air region, and and are surfaces where the flaw field can be taken to be zero. Surfaces and are chosen at a distance where the field perturbation can be neglected. They can be usually much closer in this case than in unflawed field calculation. The magnetic vector potential, A, and the electric scalar potential, , are introduced in the conductor, and the total magnetic scalar potential, , is used in the air [7], [8]. For the sake of simplicity, we consider singly connected regions only. In the flaw, , the magnetic vector potential, A, is the state

The second term of (21) and (23) is appended to the equations for ensuring the satisfaction of the Coulomb-gauge, but in the conductor then (22) has to be considered to get a divergence-free current density distribution [9]. To couple the governing equations and to ensure the uniqueness of the potentials, interface and boundary conditions are imposed, as on

and (25)

on

and (26)

on

and

(27)

on

(28)

on

(29)

on

(30)


on on on on

and

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(31) (32) (33) (34)

Conditions (25) and (26) yield the continuity of the tangential magnetic field and the normal magnetic flux density on the interface between regions of magnetic vector and scalar potentials, and (27) erases the normal component of the eddy current on conductor surfaces not connected to the flaw. From (28)–(30), the satisfaction of the conditions (14)–(16) follows. To ensure the vanishing flaw field far from the flaw, conditions (31) and (32) are prescribed. Finally, the additional conditions (33) and (34) are necessary to obtain unique potentials satisfying the Coulomb-gauge.

where is the outer normal of . The surface terms in (38) and (39) serve to incorporate the boundary and interface conditions (26), (27), (29), (31), and (34) as discussed in [7]–[9]. The only new item here is the consideration of the natural boundary condition (30) by substituting into the surface term of (39) on the surface, . Similar mathematical manipulations can be done on equations (23) and (24). After these transformations we arrive at the equation system

C. Finite Element Scheme In the finite element scheme the paper applies, the whole region is discretized into tetrahedrons, and the nodal shape functions defined on this mesh are the basis functions. denotes a scalar basis function corresponding to a node, and or or defines vector basis functions at that node. The vector and scalar functions are approximated by the sets of these basis functions as (35) (36) (37) , and are the nodal values of the potentials. where , One can use the Galerkin weighting residual technique to set up the finite element equations. In this case, the basis functions satisfy the essential boundary and interface conditions (28), (32), and (33). First the approximations (35)–(37) are substituted into the governing equations (21)–(24), then the error is minimized by taking the scalar product by the basis functions. After the usual mathematical manipulations for (21), (22) we obtain

(40) (41) It is easy to prove that the system is symmetric if we introduce the time integral of instead of [7]–[9]. The solution of (40) and (41) provides the required flaw field. D. Impedance Calculation from Potentials Since the conductivity is discontinuous on the flaw surface, we have to consider this while calculating the impedance. From the general formula (7) we obtain in our case (42) where (43) If in the flaw, is also zero inside the flaw but jumps on the boundary, . Thus, the gradient of can be written as a surface delta function in

(44)

where is the scalar potential value on surface, , and is the surface delta function [10]. After substituting (43) and (44) into (42), we obtain (38)

(39)

(45) This formula is still easy to evaluate and shows the advantage of the discussed finite element scheme. This concept of impedance calculation can be extended to calculate induced EMF of various types of receptive coils at harmonic excitation, but this extension is discussed elsewhere [11].

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Fig. 3. Crack-type defect as a limiting case of zero-conductivity flaw.

IV. DEFECT

WITH

NEGLIGIBLE THICKNESS

The crack-type defects, e.g., stress corrosion crack (SCC) or fatigue cracks, have an extreme feature: one of their dimensions (width) is much smaller then the other extents (length and depth). A method has been presented to decrease the numerical costs of solving such problems in [12]. The idea can be incorporated into the finite element scheme discussed here. If the volume of the flaw tends to zero in a manner that the flaw becomes a thin sheet, as depicted in Fig. 3, the electromagnetic conditions on the crack surface are (46) (47) (48) It is easy to see that the tangential component of the electric field jumps on the crack surface, and the jump can be described by a surface gradient of a scalar [6], [12] as (49) where subscript denotes the tangential component of E, and denotes the surface gradient. Using the above potential formulation we can describe the surface scalar, , by the jump of the electric scalar potential [12]. Hence (50) The finite element scheme remains (41) in the air. In the conducting and flaw regions, the scheme can be obtained from , and . (40) by considering vanishing Thus, (40) becomes

(51)

Fig. 4. Arrangement of the TEAM Workshop Problem number 15.

In the same way, the flaw impedance can be derived from (45) as (52)

V. RESULTS A. Plate with a Rectangular Slot First, we present the solution of a testing electromagnetic analysis methods (TEAM) workshop benchmark problem (problem number 15/2) and ACES benchmark problem [13], [17], [18]. Experimental results have been published in [14] and [19]. The arrangement of the problem is shown schematically in Fig. 4 where a circular air-cored coil is scanned, parallel to the axis along the length of a rectangular slot in an aluminum plate. The frequency and the lift-off are fixed, and is measured as a function of coil-center position. The parameters for the test experiment are listed in Table I. The penetration depth is about one-fifth of the crack depth, and the mean radius of the coil is about the crack length. At this parameter combination, the solution by the finite element method becomes very difficult and time consuming. On a workstation, HP Apollo 730 with 120 megabyte memory, the authors could not reach a solution with acceptable accuracy by using an ordinary finite element scheme [12]. Applying the new scheme, we have obtained good agreement with the experimental data as shown in Fig. 5, and the agreement is within the range of the reported measurement error. Some characteristics of the computation can be seen in Table I. We could choose the discretized conducting region as small as 6 around the flaw that is much smaller than the discretized region needed for the total perturbed field calculation. As a further advantage here, the unperturbed field could be calculated analytically applying the solution published in [15]. We have also solved the problem by using the thin sheet crack model discussed in Section IV. Fig. 6 shows the results of the fully discretized and the thin sheet crack models. The latter yields smaller impedance values than the full discretization. The largest difference is about 11%. The reason


PARAMETERS

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TABLE I PLATE PROBLEM

OF THE

Fig. 6. Modulus of the flaw impedance vs. coil-center position for the TEAM workshop problem number 15/2. (a) Results via the fully discretized defect model (40), (41), and (45). (b) Results via the thin sheet crack model (51), (41), and (52).

(a)

(b) Fig. 5. Flaw impedance vs. coil-center position for the TEAM workshop problem number 15/2 via the scheme (40), (41), and (45). (a) Real part. (b) Imaginary part.

for this difference is that the width of the slot is not negligible compared to the penetration depth. However, if the width is small enough compared to the penetration depth, the sheet model requires lower computational efforts than the discretization and yields acceptable accuracy [12]. The same problem was solved among other plate benchmark problems by a volume integral technique in [19]. Comparing the computational resources for this problem required by the volume integral scheme and our finite element scheme, we can conclude that the volume integral solution needs a much smaller amount of central processing unit (CPU) time and memory than our finite element solution. For example, a “medium precision” solution in [19] requires 92 min CPU time and 624 kbytes memory on a PC using a conjugate gradient technique for the solution of the equations system. The terms “medium precision” and “high precision” solutions are used in [19] to label different discretization levels. Furthermore, the CPU time of this medium precision calculation can be decreased to 3.8 min on a PC using a fast direct solution technique. However, the medium precision solution means that the error is about 7.5% and 25% in the modulus and in the phase, respectively. Considering our main goal, the simulation of steam generator inspection devices, such a big error, especially in phase, is not acceptable in our finite element analysis because the phase contains very important information about the defects. The “high precision” solution where the same errors are about 7.5% and 9.5% requires 345 min CPU time and about 2.5 Mbytes memory on a PC by the conjugate gradient solver. There is no solution in [19] for the high resolution case by the direct solver. We have two comments on these results. First, if we allow the same error in the finite element computation as the error in the high precision volume integral solution, the required computational resources drop to around 16% of the values in Table I. The main reason why such a fine mesh is necessary to reach an accurate solution is the approximately singular behavior of the field near the slot edges. Second, if we examine the results in Fig. 6, we can see that the error of the solution by the thin sheet finite element crack model is about 11%,

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Fig. 7. Long axial slot in a half tube.

i.e., even the high precision volume integral solution is closer to the thin-sheet approximation than to the full volumetric approximation. This is because in [19] the subdivision in slot width direction is always one, i.e., constant field is assumed inside the slot, while in the fine finite element solution the width is subdivided into four layers to reach the accurate solution. Reference [19] does not contain results with finer resolution in the width direction.

(a)

B. Slots in a Half Tube The other two test problems that we present here are depicted in Figs. 7 and 9. The host specimen in both cases is a stainless steel tube cut in half along its axis, and a long axial and a short circumferential slot is investigated by a circular air-cored coil. The electrical discharge machined (EDM) slots are located at the center of the specimen. The impedance of the coil has been measured by an HP4284A precision LCR meter. The values of and were reproducible within 0.003 at the applied frequency. Further details of the experiment and the probe model calibration are discussed elsewhere in [11]. Table II lists the parameters of the problems. Since the tube is long enough to neglect the effect of the tube ends at any meaningful coil location, one unflawed field calculation could be used for all coil locations, and only one-fourth of the arrangement may be considered due to the symmetries. Otherwise, at the flaw field calculation half of the arrangement has to be meshed up. 1) Long axial slot: The coil is scanned along the slot, parallel to the axis. The flaw field here has also been considered zero beyond the distance of six skin depth from the slot. The comparison between the experimental and calculated impedance trajectories are shown in Fig. 8. The agreement is again within the range of the experimental error. 2) Short circumferential slot: The impedance trajectory has been measured and calculated parallel to the axis, i.e., perpendicular to the slot (see Fig. 9). The results are plotted in Fig. 10. Even the largest error is within the range of the reported experimental error, but the agreement is worse than at the axial slot. The reason for the larger error, on one hand, comes from the simulation side. There are uncertainties in the modeling, especially in the coil and defect shapes. We considered the coil as a perfect circular coil, but the real outer diameter varies within 0.2 mm that can cause itself impedance change

(b) Fig. 8. Flaw impedance vs. coil-center position for the tube problem with an axial slot via the scheme (40), (41), and (45). (a) Real part. (b) Imaginary part.

Fig. 9. Through short circumferential slot in a half tube.

of such a magnitude. On the other hand, the impedance change is closer to the sensitivity limit of the impedance analyzer than at the long slot measurement. Since the flaw impedance value is about one ten-thousandth of the total impedance, the result is a good approximation.


PARAMETERS

TABLE II OF THE TUBE PROBLEMS

(a)

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VI. CONCLUSION The basic idea of the finite element scheme presented in the paper is well known in solving wave scattering problems, i.e., the incident field (the unflawed field here) can be considered as given, and the scattered field (the flaw field here) is to be determined. A finite element approach of field calculation based on a similar idea has been published for magnetostatic problems in [16]. In eddy-current NDE, however, a similar concept is used only in volume integral techniques [1], [2], but in finite element modeling it has not been previously introduced. The paper describes the basic idea of the flaw field calculation in finite element modeling and discusses the algorithm for zero-conductivity flaws in detail. Since the scheme considers discontinuous material properties, it is straightforward to generalize it for flaws with spatially varying conductivity. While calculating the unflawed field, we can use an analytical solution if it is available, e.g., at the plate problem or consider any kind of symmetries of the unflawed problem as it has been shown at the tube problems. Another benefit of this flaw-field finite element scheme is that the discretized region can be chosen independently of the dimensions of the coil and the host specimen, and it is usually smaller than the unflawed mesh because the attenuation of the flaw field is strong in the conductor. A further merit is that the flaw impedance can be calculated by performing an integral over the flaw region only, and no total impedance calculation is required. The scheme of crack-type defects has been applied for the plate problem only. Although the width is not negligible with respect to the skin depth, the model yields an acceptable approximation. Additional investigations of the thin-sheet approximation can be found in [11]. Finally, the solutions of the tube problems, which are very similar to important practical arrangements, demonstrate well the applicability and efficiency of the new scheme. APPENDIX The basic formula for computing the impedance change is (A1) where is the given current density distribution of the coil [2]. In order to obtain (7) from (A1), we must consider Maxwell’s equations (A2) (A3)

(b) Fig. 10. Flaw impedance vs. coil-center position for the tube problem with a circumferential slot via the scheme (40), (41), and (45). (a) Real part. (b) Imaginary part.

for the total perturbed field in the whole arrangement with time-harmonic excitation and Maxwell’s equations (5) and (6) of the field distortions with harmonic time variation. Then, we must multiply (A2) and (A3), and the time-harmonic forms of ), and ( ), respectively. Finally, (5) and (6) by E, H , ( we must add up the four equations and integrate both sides of the sum over the whole arrangement. The left-hand side of the result is identically zero via the reciprocity relation [5], and

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the right hand side becomes

(A4)

Upon considering the distributions of obtain (7).

,

, and

[18] H. A. Sabbagh, and S. Burke, "Benchmark problems in eddy-current NDE," in Review of Progress in Quantitative Nondestructive Evaluation, vol. 11. New York: Plenum, 1992, pp. 217-224. [19] S. A. Jenkins, J. C. Treece, R. K. Murphy, L. D. Sabbagh, and H. A. Sabbagh, "Solutions of benchmark problems in eddy-current NDE," in Review of Progress in Quantitative Nondestructive Evaluation, vol. 12. New York: Plenum, 1993, pp. 227-234.

, we

REFERENCES [1] H. A. Sabbagh and L. D. Sabbagh, "An eddy-current model for threedimensional inversion," IEEE Trans. Magn., vol. MAG-22, no. 4, pp. 282-291, 1986. [2] J. R. Bowler, "Three-dimensional eddy current probe-flaw calculation using volume elements," Electrosoft, vol. 2, no. 2/3, pp. 142-156, 1991. [3] N. Ida and W. Lord, "3-D finite element modeling of eddy current NDT phenomena," IEEE Trans. Magn., vol. MAG-21, no. 6, p. 2635, 1985. [4] Y. Sun, H. Lin, Y. K. Shin, Z. You, S. Nath, and W. Lord, "3-D finite element modeling of the remote field eddy current effect," in Review of Progress in Quantitative Nondestructive Evaluation, vol. 9. New York: Plenum, 1990, pp. 319-326. [5] R. F. Harrington, Time Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961. [6] J. R. Bowler, "Eddy current field theory for a flawed conducting halfspace,” in Review of Quantitative Nondestructive Evaluation, vol. 5A. New York: Plenum, 1986, pp.149-155. [7] D. Roger and J. F. Eastham, “A formulation for low frequency eddy current solutions," IEEE Trans. on Magn., vol. MAG-19, pp. 2443-2446, 1983. [8] C. R. I. Emson and J. Simkin, "An optimal method for 3-D eddy currents," IEEE Trans. on Magn., vol. MAG-19, pp. 2450-2452, 1983. [9] O. Biro and K. Preis, "On the use of the magnetic vector potential in the finite element analysis of three-dimensional eddy currents," IEEE Trans. Magn., vol. 25, no. 4, pp. 3145-3159, 1989. [10] J. Van Bladel, Singular Electromagnetic Fields and Sources. Oxford, U.K.: Clarendon Press, 1991. [11] Z. Badics, H. Komatsu, Y. Matsumoto, K. Aoki, F. Nakayasu, M. Uesaka, and K. Miya, “Accurate impedance and EMF calculation in eddy-current nondestructive evaluation by finite element method," J. Nondestructive Evaluation, vol. 14, no. 3, Sept. 1995. [12] Z. Badics, H. Komatsu, Y. Matsumoto, K. Aoki, F. Nakayasu, and K. Miya, “A thin sheet finite element crack model in eddy current NDE," IEEE Trans. on Magn., vol. 30, no. 5, pp. 3080-3083, 1994. [13] L. R. Turner, “TEAM workshop problems,” Aragon Nat. Lab., Apr. 1988. [14] D. McA. McKirdy, “Recent improvements to the application of the volume integral method of eddy current modeling," J. Nondestructive Evaluation, vol. 8, no. 1, p. 45, 1989. [15] C. V. Dodd and W. E. Deeds, “Analytical solutions to eddy-current probe-coil problems," J. Appl. Phys., vol. 39, pp. 2829-2838, 1968. [16] M. Gyimesi and D. Lavers, "General potential formulation for 3-D magnetostatic problems," IEEE Trans. Magn., vol. MAG-28, no. 4, pp. 2450-2452, 1983. [17] S. Burke, "Eddy current NDE additional benchmark problems," Appl. Computational Electromagnetics Soc. Newslett., vol. 6, no. 1, pp. 17-34, Mar. 1991.

Zsolt Badics was born in Hungary in 1959. He received the Dipl. Ing. degree in electrical engineering in 1984 and the Dr. Tech. degree in 1992, both from the Technical University of Budapest, Budapest, Hungary. He received the degree of Candidate of Technical Sciences in electromagnetic computation from the Hungarian Academy of Sciences, Budapest, Hungary, in 1992. From 1988 to 1992 he was at the Department of Electromagnetic Theory, Technical University of Budapest. Since 1992 he has been a Chief Engineer at the Nuclear Fuel Industries Ltd., Osaka, Japan. He has performed research on the numerical methods of electromagnetic field computations and the numerical modeling of eddy-current nondestructive evaluation devices. Dr. Badics is a member of the International Compumag Society and the Japan Society of Applied Electromagnetics.

Yoshihiro Matsumoto was born in Tokushima, Japan, in 1965. He received the M.S. degree in engineering from Osaka University, Osaka, Japan, in 1990. He is a member of the engineering staff in the Engineering Service Division at the Nuclear Fuel Industries Ltd., Osaka, Japan. His current research activity is nondestructive testing. Dr. Matsumoto is a member of the Japan Society of Applied Electromagnetics.

Kazuhiko Aoki was born in Osaka, Japan, in 1956. He received the doctor degree of engineering from Kyoto University, Kyoto, Japan, in 1986. He is a Section Manager in the Engineering Service Division at Nuclear Fuel Industries Ltd., Osaka, Japan. His current research activities include nondestructive testing and high temperature corrosion engineering. Dr. Aoki is a member of the Atomic Energy Society of Japan.

Mitsuru Uesaka was born in Yokohama, Japan, in 1957. He received the doctor degree of engineering from The University of Tokyo, Tokyo, Japan, in 1985. He is an Associate Professor of The Nuclear Engineering Research Laboratory, University of Tokyo, Tokyo, Japan. His research activities include electromagnetics in micro sensors, nondestructive testing, and accelerator technology. Dr. Uesaka is a member of the Atomic Energy Society of Japan, the Institute of Electrical Engineers of Japan, the Japan Society of Mechanical Engineering, and the Japan Society of Applied Electromagnetics.