SIMULATION OF THE WORLD FEDERATION'S SECOND
EDDY CURRENT BENCHMARK PROBLEM
Y. Tian, Y. Li, L. Udpa, and S. S. Udpa
Department of Electrical and Computer Engineering
Michigan State University
2120 Engineering Building, East Lansing, MI 48824
ABSTRACT. This paper presents computational results of the second eddy current benchmark
problem proposed by the World Federation of Nondestructive Evaluation Centers. The problem
involves the prediction of the change in impedance of a pancake coil as it scans the inner surface of
a metal tube along both the axial and circumferential directions for detection of flaws on the outer
surface. A finite element scheme is carried. The real and imaginary part of the impedance is
obtained by observing the total stored energy and the dissipated energy, respectively.
INTRODUCTION
Eddy current tests (ECT) are carried out in many industrial settings to ascertain the
structural integrity of critical components. Nuclear power plants employ such tests, for
example, to inspect steam generator tubes. Numerical models simulating the inspection
process can be employed, among other things, to optimize test parameters, design new and
improved probes, enhance defect detection capabilities and estimate the probability of
detection. The utility of these models has motivated research groups worldwide to develop
a number of different modeling tools and approaches. Each of these models offers a
unique set of advantages and limitations. Comparison of these models can be frustrating
due to the lack of a common set of bases for evaluating their performance. The World
Federation of Nondestructive Evaluation Centers (WFNDEC) is addressing this issue by
proposing a series of benchmark problems for validating NDE simulation models.
In this paper, we focus attention on the second eddy current benchmark problem
proposed by the WFNDEC. This problem involves the simulation of a pancake type eddy
current coil that scans a defect contained in a tube, a scenario that is commonly
encountered in the inspection of nuclear power plant tubes. Since the coil has to move in
scanning, a key challenge is to avoid having to remesh the geometry each time the coil
moves relative to the defect. Existing approaches include a hybrid boundary element/finite
element method and an edge element based FEM employing the reduced magnetic vector
potential [1]. In this work, we model the defect in a virtual sense. This approach avoids the
complexity of reformulating the problem for each scan position. We use a MATLAB
based commercial software FEMLAB® package. The software package employs the
CP657, Review of Quantitative Nondestructive Evaluation Vol. 22, ed. by D. O. Thompson and D. E. Chimenti
© 2003 American Institute of Physics 0-7354-0117-9/03/S20.00
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A - V formulation based finite element method (FEM) to solve the problem. The energy
method [2] is used to compute the impedance of the coil.
A-V FORMULATION AND IMPEDANCE CALCULATION
A-V Formulation
The three-dimensional quasi-static electromagnetic problem is modeled using a potential
formulation based on a magnetic vector potential A and electric scalar potential V . The
time-harmonic eddy current problem is expressed by
(1)
and
r,
(2)
where Je is the external current density, // is the permeability, a is the conductivity and
co is the angular frequency of the excitation source. We specify zero electric and magnetic
potential on the boundaries of the FEM solution domain, i.e.
F = 0, nxA = 0,
(3)
where n is the outward normal vector on the boundaries.
We set the conductivity of air surrounding the tube at a very low value instead of zero to
avoid problems that are typically encountered with using FEMLAB when the dependent
variable vanishes. Fortunately, the numerical error is trivial when the value of a is small
compared to the conductivity of the conducting region.
Impedance Calculation
After A and V are solved for , we can calculate the impedance of the pancake coil
(4)
where R is the coil resistance and L is the coil inductance.
The resistance is linked with the dissipated energy P in the conductor region in the
form of
R = P/I2l
(5)
while the inductance is linked with the total stored energy W in the whole solution domain
by
L = 2W/I2,
where / is the external current intensity. P and W can be expressed by
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(6)
p=
(7)
and
(8)
respectively. Here E = -vT-y&>A is the electric field intensity, B = V x A is the
magnetic flux density, H = B / // is the magnetic field intensity, T refers to the solution
domain, and ( j denotes the complex conjugate operator. Formula (4)-(8) represents a
general scheme for calculating the coil impedance with no distinction between 2D and 3D
problems.
PROBLEM DESCRIPTION
The basic geometry of the second eddy current benchmark problem is shown in Fig 1.
The specimen is an Inconel tube with an inner diameter Di =19.69 mm and an outer
diameter D0 = 22.23 mm. The conductivity and relative permeability of the tube is
<7 = 106 S/m and 1, respectively. The tube contains a defect (void) whose depth h can be
20%, 40% and 60% of the tube wall thickness. The other two dimensions of the defect are
fixed at w = 3 mm and t = 1 mm, respectively.
FIGURE 1. Basic Geometry, (upper) Pancake coil, (lower left) X-Z cross-section of the tube, (lower right)
Y-Z cross-section of the tube. Defect is shown in black color.
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L
0.05
0,04
0.03
0.02
0.01
0
-0.01
-0.02
-0.03
-0.04
-0.05
-0.08
FIGURE 2. Y-Z cross-section of the FEMLAB solution domain.
The moving pancake coil with TV = 400 turns is energized by an alternating current
source with excitation current intensity / = 100 mA and frequency / = 100, 150 or 200
KHz. The coil has an inner diameter di — 1 mm, an outer diameter d0 = 3 mm and height
h0 =0.8 mm. The coil also moves 10 steps along the axial direction with a step size of 1
mm and 4 steps along the circumferential direction with a step size of 10°. The lift-off
between the coil and the inner surface of the tube is 0.8 mm
The object of the problem is to predict the change in the impedance of the pancake coil,
AZ, as it moves past the defect. This kind of information could be further investigated to
determine the position and size of the defect.
NUMERICAL SIMULATION AND RESULTS
Numerical simulations are carried out using a PC equipped with a 2.2 GHz INTEL®
Pentium 4 processor and 1 GB physical memory. As indicated earlier, FEMLAB®'s
electromagnetic module was used to model the three-dimensional qausi-static problem.
Fig 2 shows the geometry of the problem. The length of the tube is set at 100 mm,
which approximates a tube of 'infinite' length. The tube occupies a small portion of the
cylindrical FEM solution domain whose diameter is 80 mm and length equals 120 mm. We
use the Krylov subspace method [3] to solve the problem. The convergence criterion is
based on the residual and the convergence tolerance is set atlO~ 8 . The mesh contains about
30,000 nodes and 170,000 linear tetrahedrons. A preconditioning technique, called the
symmetric successive over-relaxation (SSOR) method is employed to minimize
computation time and memory.
Self-Inductance
In order to validate the code, we calculated the inductance of the coil inside an infinite
region of air and compared the result with those obtained using a well-known empirical
formula for calculating the impedance of pancake coils [4].
Expressed in the metric system, the empirical formula takes the form of
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_
x 1A
10 7
rT =
__ .
Henries ,
(9)
4/20
where,
1+ 0.225|^^^| +0.64| ^^
I AO
The FEMLAB results obtained with different meshes are given in Table 1 and compared
with the result obtained using the empirical formula.
TABLE 1. Self-inductance calculations, (unit: H)
FEMLAB Results
Empirical Formula
2.2013e-4
1003 nodes
4072 nodes
13452 nodes
2.4393e-4
2.1959e-4
2.2154e-4
Mesh Noise Control
The geometry involves stepping the coil through 40 positions within the tube. In
addition, three different defect sizes have to be modeled. It is not only cumbersome to build
a new geometry for each case, but also difficult to avoid error introduced in the result when
the geometry is remeshed. In order to avoid this difficulty, we model the defect in virtual
sense.
Coil
Defect
60%
20%
axial
direction
FIGURE 3. (left) a prototype is build to model different sizes of the defect at certain axial position; (right);
VDA structure for axial scan.
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It is observed that the changes in impedance of a coil moving around a fixed defect can
be equivalently invoked by a fixed coil transferring electromagnetic energy to a virtually
moving defect. This idea can be implemented by constructing a virtual defect array (VDA)
as illustrated in Fig 3. At first, a prototype virtual defect at a certain position is divided into
three parts to model three different sizes of the defect. Replicas of such prototypes are
stacked together to describe relative positions between the coil and the defect. The Fig 3
also includes a VDA structure describing the relative axial positions. The conductivity of
each cell of such a structure can be adjusted to form the real defect with a specific position
and size while the material properties of other cells are the same as the metal tube. Four
geometries are constructed corresponding to each of the four circumferential movement of
the coil while each one includes a VDA structure for simulating each axial scan. The error
associated with the axial scan process is completely eliminated while the error due to the
circumferential scan is also reduced due to the very high mesh density around the defect
region. An alternative procedure is to describe the region around the defect in a manner that
allows circumferential scans along all four value of a (0, 10, 20, 30) without remeshing
for each case.
Simulation Results
A total 360 calculations are carried out to account for 10 axial positions, 4
circumferential positions, 3 defect sizes, and 3 excitation frequencies. Figure 4 shows the
effect of different defect sizes when the defect is scanned axially with the excitation
frequency / = 150 KHz and the angle of rotation a - 0° . Figure 5 shows the similar
results obtained when the angle of rotation a - 20°. Figure 6 shows the variation in
impedance as the defect is scanned along the circumferential direction. The excitation
frequency / = 150 KHz and the axial position of the coil is d = 0 mm. Figure 7 shows the
variations in impedance when a 60% deep defect is scanned along the axial and
circumferential direction. The excitation frequency / = 150 KHz. Figure 8 shows the
effect of variation in the excitation frequency when a 60% deep defect is scanned along the
axial direction (a - 0°).
u.^o -
0.2-
S.
0.15 -
«
0.1 -
D>
~
VN
n
* V^J\
—+—20% Defect
-•—40% Defect
-A— 60% Defect
r\ f\c
0-
-0.05
0
n
n4
A-v
/\
• \
f Vs.
fi]*
w^
0.051
r-3**.... 1
00
_n ne; .
-0.1
0.2a>
a 0.15 i
'E
0.05
5
10
Axial Distance (mm)
Real Part
FIGURE 4 Change in impedance for an axial scan. OC = 0° , f = 150 KHz. (left) locus of change of
impedance, (right) magnitude of impedance change along Y-axis.
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A
0.12 -
&
r
0.12 -
^
1
0.08 -
J
r\ nc
•
-4— 20% Defect
0.1 -
-•—40% Defect
0.08-
\
"
\
^
f\ f\c
y
—A— 60% Defect
B 0.04E 0.02 0.0.02 - ————,————,————,————,
-0.01
0
0.01 0.02 0.03
W^/
-**^ —I
0.04 00? -
^\
XX
n -
*^^nh* •
10
Axial Distance (mm)
Real Part
FIGURE 5 Change in impedance for an axial scan, a — 20° . (left) locus of change of impedance, (right)
magnitude of impedance change along Y-axis.
Imaginary Part
n o*\
\\
0.2 -
\
0.15-
X
Pv/
\J
0.1 r\ HR
0 -
-n DF; -0.1
-0.05
0.05
0
10
20
Angle of Rotation (degree)
Real Part
FIGURE 6 Change in impedance for a circumferential scan, (left) locus of change of impedance, (right)
magnitude of impedance change along Y-axis.
—4— rotation angle 0
A
0.3 3^^
1
•
rotation angle 1 0
rotation angle 20 ——— rotation angle 30
—— ... .
_
———— . _
.. _
—— __ _,
Magnitude
n OK
0.2 -
^\
0.15, /
•-—^^
0.1 -
»\
\\
^A^
0
2
"^
4
6
8
1
Axial Distance (mm)
FIGURE 7 Change in impedance for both axial and circumferential scan, (left) locus of change of
impedance, (right) magnitude of impedance change along Y-axis.
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0.3-
u.o •
t:
0.25 (0
t
°- 2 <5 0.15£
0.05 0-0.05 -0 .1
\^^*
\
\
\^
im\V
-+- f=100KHz
-•— f=150KHz
\
>.
^^Vx^
\
A———
r^y——
-*— f=200KHz
0)0.15!
——————'
s 0.1 <- y*"-^^
1
o nc
iti^*
-0.05 0
« °-25 ;
1
a2
'
3
\
/
±.
0 - —————^^f""«
C)
5
0.05 0 1
Real Part
m m m
1
Axial Distance (mm)
FIGURE 8 Effect of different excitation frequency (60% defect a = 0° ). (left) locus of change of
impedance, (right) magnitude of impedance change along Y-axis.
SUMMARY
The second eddy current benchmark problem proposed by the World Federation of
Nondestructive Evaluation Centers has been studied using the finite element method. The
approach avoids errors that are typically introduced when the geometry has to be
remeshed to model the scanning process. The reasonable numerical results are achieved.
REFERENCES
1. Fukutomi, H., Takagi, T., Tani, J. and Kojima, F., Electromagnetic Nondestructive
Evaluation II, eds. R. Albanese, G. Rubinacci, T. Takagi and S. S. Udpa, IOS Press,
1998,p305.
2. Ida, Nathan, Three Dimensional Finite Element Modeling of Electromagnetic
Nondestructive Testing Phenomena, Ph.D dissertation, Colorado State University, 1983.
3. Saad, Y., Schultz, M. H., SIAMJ. of Set. Statist. Comp., 7, pp856-869,1986.
4. Palanisamy, P., Ph.D dissertation, Colorado State University, 1980.
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