E-Journal of Advanced Maintenance Vol.3, No.1 (2011) 1-10
Japan Society of Maintenology

Evaluation of Defect Shape Based on Inverse Analysis Considering the
Resolution of Magnetic Sensor
Shogo NAKASUMI1,* and Takayuki SUZUKI11National Institute of Advanced Industrial Science and Technology (AIST), East 1-2-1 Namiki,
Tsukuba, Ibaraki 305-8564, Japan
ABSTRACTThis paper presents a technique for improving the accuracy of non-destructive evaluation using magnetic flux
leakage. Response function that associates magnetic dipoles and magnetic flux densities in terms of geometricalcoordinate is modified so as to capture the average magnetic flux value along the sensor length, that is consistent
with experimentally measured magnetic flux. Inverse analysis of a semi-elliptical surface defect is conducted
through numerical simulation. By using the modified response function, the restored defect shape becomes more
proper and remarkable improvements are indicated, especially in low lift-off cases.
KEYWORDS

ARTICLE INFORMATION

magnetic flux leakage, flux gate sensor, inverse analysis, non-destructive
evaluation, magnetic dipole, Tikhonov regularization

Article history:
Received 28 March 2011
Accepted 24 May 2011

1. Introduction
Nondestructive evaluation that identifies structural defects such as cracks and pitting is importantfor safety and reliable structure evaluation.
Structural materials are often magnetic substances, making non-destructive evaluation using
electromagnetism effective. The magnetic flux leakage density method is an inspection method bywhich defects are detected by measuring the leakage flux density caused by the magnetic charge on
crack surfaces. However, quantitative evaluation of defect size, shape, and other features cannot be
performed solely by measuring leakage flux density. To evaluate these defects quantitatively, it isnecessary to restore the distribution of the magnetic charges by formulating a response function thatindicates a geometrical relation between magnetic charges on the defect and the measured magnetic
flux leakage density, and by conducting inverse analysis [1-4].
The flux gate sensor (FG sensor, below) is a highly sensitive magnetic sensor of about 10-7 T thathas attracted recent attention because can be used from a low lift-off area (the distance between thesample and the observation plane) of about 1 mm to high lift-off area of about 10 mm [5-7].
FG sensor measurements give high accuracy from a low lift-off area. Generally, however, thespatial change rate of the magnetic flux density becomes very large in low lift-off areas, and suchsteep changes within a small area cannot be accurately captured if measured using an FG sensor of
about 3 mm in length. As a result, there is a possibility that the high sensitivity of the FG sensor will
not lead to a highly accurate evaluation of the defect shape. To the authors’ knowledge, there is noprevious research that conducts inverse analysis taking into consideration the influence of sensor
length.
With a high lift-off area, on the other hand, attenuation of the magnetic flux density becomessignificant, and measurement noise greatly influences the inverse analysis.
This paper assumes measurement of magnetic leakage flux density using an FG sensor. The
response function is improved in order to consider the influence of sensor length, and the results ofverifying the magnetic flux density distribution through forward analysis and the defect shape*

Corresponding author, E-mail: nakasumi.shogo@aist.go.jp

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S. Nakasumi, et al. / Evaluation of Defect Shape Based on Inverse Analysis
Considering the Resolution of Magnetic Sensors

3mm
1mm
Flux gate sensor

Lift off

Measurement linezyx

DefectMagnetic dipole
moment vector

30mm

Fig. 1. Measurement of magnetic flux leakage

restoration through inverse analysis are described.

2. Deriving a response function that considers sensor resolution
2.1. Formulation of magnetic charge-flux density
Figure 1 shows the appearance of a specimen of electromagnetic material with a surface slit
crack. The direction of the normal of the structure surface is taken as the z-axis, and a verticaldirection to the crack surface is taken as the y-axis. Positive and negative magnetic charges existingon the crack surfaces are assumed to be magnetic dipoles. A magnetic dipole has a non-zero elementm y only in the direction of the y-axis. As the flux density has non-zero y-axis value elements B y ,we will use it in this study and “flux density” below should be taken to mean B y . When theTcoordinates of the magnetic dipole are described as x′ = ( x′, y ′, z ′ ) , the flux density observed at theTpoint x = ( x, y, z ) is given by Equation (1)[8].
B y ( x ) = ∫ Fy ( x − x′) m y dx′(1)ΩHere, Ω is the total area from which the magnetic dipoles exit, and Fy ( x ) is a function whose
input variables are the relative coordinates of the observation point to the magnetic dipole, as
described by Equation (2)Fy ( x, y, z ) =()
4π ( x + y + z )μ0 x 2 − 2 y 2 + z 2
2

2

2 52

(2)

Here, μ0 is the space permeability. Note that Equation (2) is applicable in cases where magnetic
dipoles exist at the origin. The sum of the discrete magnetic dipoles is substituted for the integrationin Equation (1):N()

B y ( x ) = ∑ Fy x − x′ j m yj
j =1

(3)

Here, the superscript j in terms x′ j and m yj is the index of the magnetic dipole, and N is the numberof magnetic dipoles. Equation (3) holds at every observation point on a plane with a given constant
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Normalized magnetic flux density [a.u.]

S. Nakasumi, et al. / Evaluation of Defect Shape Based on Inverse Analysis
Considering the Resolution of Magnetic Sensors
1

Liftoff = 1mm
Liftoff = 5mm
Liftoff =10mm

0.8
0.6
0.4
0.2
0
-0.2
-20

-15

-10

-5
0
5
Y position [mm]

10

15

20

Fig. 2. Distribution of normalized magnetic flux (x = 0 measurement line)

lift-off value. After some manipulation, Equation (4) is obtained.
⎧ B1y ⎫ ⎡ Fy11⎪
⎪ ⎢⎪⎪ B y2 ⎪⎪ ⎢ Fy21⎨⎬=⎢
⎪ # ⎪ ⎢ #⎪ M ⎪ ⎢ M1⎩⎪ B y ⎭⎪ ⎣⎢ Fy

Fy1N ⎤ ⎧ m1y ⎫⎥⎪⎪⎥ ⎪⎪ m 2y ⎪⎪⎥⎨⎬%⎥⎪ # ⎪⎥⎪⎪
FyMN ⎦⎥ ⎪⎩m yN ⎪⎭Fy12 "%(4)

Here, the superscript i in the term B iy is the index, and M is the number of observation points. Fyij
is the response function of Equation (2), which relates an observation point i to a magnetic dipole j.
Equation (4) can be rewritten as Equation (5). In the following, we call F of Equation (5) as a
response function matrix.

B = Fm(5)

2.2. Influence of lift-off on flux density distribution
Equation (4) gives the magnetic flux density of the configured magnetic dipoles as observed at anarbitrary point, and is considered a forward analysis. In the following, we verify how the lift-off value
influences the flux density distribution obtained by this forward analysis.
For the test piece in Fig. 1, Fig. 2 shows the y-axis distributions of the flux density obtained on
the line x = 0 with lift-off set to 1 mm, 5 mm, and 10 mm. The graph line corresponding to eachlift-off is normalized by dividing by each maximum value so that the maximum value becomes 1.
Note that along the z-axis, the flux density decreases in proportion to approximately the -3 power ofthe z coordinate.
A basic tendency of the distribution where lift-off is 1 mm is confirmed below. The flux density
is 0 at infinite distance. However, it decreases when approaching the crack walls (y = 0 mm), and thenbecomes a negative value. After reaching a minimum value in the vicinity of |y| = 2 mm, it increases
steeply, attaining a peak-like distribution of maximum value on the crack surface.
The FG sensor assumed in this study is of cylindrical shape, 3 mm in length and about 1 mm indiameter (Fig. 1). The measured value of the magnetic flux is an average value of the magnetic flux
penetrating through the full internal length of the sensor. There is a possibility that flux densities
showing rapid increases and decreases cannot be measured accurately in the vicinity of |y| < 2 mm,
which is almost the same as the length of the sensor. On the other hand, in the case where lift-off is 10mm, spatial changes of the magnetic flux are dulled, making the influence of the length of the sensor
relatively small as compared with the case where lift-off is 1 mm.

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S. Nakasumi, et al. / Evaluation of Defect Shape Based on Inverse Analysis
Considering the Resolution of Magnetic Sensors
FG sensor

L

By

By

L

y

2L

y0 − Ly0

y0 + L

Fig. 3. Conceptual image of trapezoidal integration in the FG sensor length interval

2.3.

Method of correcting the response function

As mentioned above, there is a possibility that the FG sensor will not accurately capture steep
changes around the crack surface for a low lift-off area of about 1 mm. In consideration of such, inthis section we modify the response function matrix of Equation (5). It is realistic to assume that
distribution of the flux density in the sensor is weighted towards the vicinity of the sensor’s center. In
this study, however, the simple mean value is used to modify the observed flux density as aT
simplification. Flux density B ymeasure ( x0 ) measured by the sensor at the point x 0 = ( x0 , y0 , z0 ) is
given as follows:B ymeasure ( x0 ) =1 y0 + L
B y ( x0 , t , z0 ) dt
2 L ∫y0 − L(6)Here, 2L is the length of the sensor. The averaging operator acts on the response function. As aresult, after manipulation similar to Equations (1)–(3), Equation (6) leads to Equation (7) andEquation (8).
N()

B ymeasure ( x0 ) = ∑ Fy′ x0 − x′ j m yj

(7)

1 y0 + L
Fy ( x0 , t , z0 ) dt
2 L ∫y0 − L

(8)

j =1

Fy′ =

The right-side term Fy in Equation (8) is given by Equation (2). Since it is difficult to perform theintegration in Equation (8) analytically within the range of real numbers, it is modified according to
the composite trapezoid rule, thus giving Equation (9).

Fy′ ≈1 m 1 2L
∑ ⋅ Fy ( x0 , yi , z0 ) + Fy ( x0 , yi +1 , z0 )2 L i =1 2 m

{

}

(9)

Here, m is the trapezoid number (the number of partitions), and yi (i = 1, 2," , m + 1) is they-coordinate of the division point that divides the y-direction section y0 − L < y < y0 + L into m
sections. Using Fy′ from Equation (9), we obtain the modified response function matrix F′ in
Equation (10).

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S. Nakasumi, et al. / Evaluation of Defect Shape Based on Inverse Analysis
Considering the Resolution of Magnetic Sensors
0.025

BB' (m=2)
B' (m=5)6
5Magnetic flux density [a.u.]

Magnetic flux density [a.u.]

7

4
3
2
1
0
-1
-10

-7

-5

-3-2-1 0 1 2 3
Y position [mm]

5

7

10

B
B' (m=2)
B' (m=5)

0.02
0.015
0.01
0.005
0
-0.005-10-7-5-3-2-1 0 1 2 3
Y position [mm]

5

7

10

(a) z=1mm(b) z=10mm
Fig. 4. Effect of response function revision for magnetic flux density distribution⎡ Fy′11⎢⎢ F ′21F′ = ⎢ y⎢ #⎢ M1⎣⎢ Fy′

Fy′1N ⎤
⎥
⎥
⎥
%
⎥
MN ⎥
Fy′ ⎦⎥

Fy′12 "
%

(10)

2.4.
Flux density distribution by modified response function
This section verifies the effect of the modified response function matrix. Below, the flux density
obtained using F′ from Equation (10) in the forward analysis is described as B′ . That is to say,Equation (11) holds.
B′ = F′m(11)

Figure 4 shows the magnetic flux density distributions obtained by forward analysis in the casewhere lift-off is 1 mm and 10 mm in the specimen shown in Fig. 1. Here, B and B′ are the flux
densities obtained by the forward analysis shown in Equation (5) and Equation (11), respectively, andm of B′ is the number of trapezoidal divisions.
In Fig. 4(a), the distribution of B′ (m=2) is different from that of B because the approximation
shown in Equation (9) and Fig.3 is not enough. As m in increases, the distribution of B′ converges
gradually on a certain curve, and our trials resulted in that m = 5 is suitable value in terms of accuracy
and computational cost. Compared with the distribution of B , that of B′ (m=5) loses the steepness at
y = 0 mm and it rounds smoothly. Moreover, the position where the sign reverses moves about 1 mm
outward. On the other hand, B and B′ (m=5) showed almost the same behavior in Fig. 4(b).
In summary, the distribution of the flux density observed in a low lift-off area is greatly influenced
by modifying the response function, and the influence is smaller in a high lift-off area.

3. Evaluation of defect shape by inverse analysis
3.1.

Deriving a numerical analysis model and simulated flux density

In the preceding section we developed a response function that considers the influence of the lengthof the sensor, and forward analysis confirmed the effect. In this section, we conduct inverse analysis
to develop a practicable model and verify the effect. The model uses a semi-elliptical surface crack on
a test piece. Figure 5 shows an external view. The radii of the long and short axes are 5 mm and 2.5mm, respectively.
Flux density, which becomes input data for the inverse analysis, is originally obtained throughexperimental measurement. In this paper, however, mathematical calculation of the forward analysis
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S. Nakasumi, et al. / Evaluation of Defect Shape Based on Inverse Analysis
Considering the Resolution of Magnetic Sensors

Measurement plane

F.G. Sensor
Magnetic dipole
0.25mmz

0.25mm

y

z

x

x

2.5mm

Lift off

10mm

y=0 crosssection

5.0mm

30mm

Magnetic flux density [a.u.]

Magnetic flux density [a.u.]

Fig. 5. Magnetic dipole configuration for a semi-elliptical surface defecty [mm]y [mm](a) Lift-off = 1 mm(b) Lift-off = 10 mm
Fig. 6. Magnetic flux density distribution of a semi-elliptical surface defect by forward analysis
including 1% error

is used as a substitute for experimental measurement. We will therefore call this the simulated flux
density. The derivation process is explained below.
The magnetic dipoles are configured on the section y = 0 of Fig. 5 in a grid. The configurationinterval is set to 0.25 mm in both the x and z direction, and the value of the magnetic dipole locatedwithin the semi-elliptical area was set to be -1. Moreover, observational points for the flux density are
configured in a grid along the x and y axis at 0.5 mm intervals in the observation plane (the xy planewhere z equals the lift-off) within the area -15 < x <15 (mm), -20 < y < 20 (mm). As a result, thenumber of observation points M = 4941 and the number of magnetic dipoles N = 1701.
From the discussion of sections 2.2 and 2.3, the magnetic flux density that becomes the input data
for the inverse analysis should not be B of Equation (5), but rather B′ of Equation (11). In
addition, measurement noise must also be considered. The noise level is set to 1% of the maximum
value of B′ , and the distribution is assumed to be a uniform random number with a mean value of 0.
The simulated magnetic flux density is therefore given by B′e in Equation (12).

B′e = B′ + R= F′m + R

(12)Here, R is the noise vector mentioned above, and m is a vector indicating the distribution ofmagnetic dipoles shown in Fig. 5. Figure 6 shows the distribution of B′e obtained by Equation (12)when lift-off is 1 mm and 10 mm. This magnetic flux density is called the simulated flux density, andthe elicitation process is explained below.
3.2.

Application of Tikhonov regularization

Using F′ of Equation (10) and B′e of Equation (12), the distribution of the magnetic dipoles that
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S. Nakasumi, et al. / Evaluation of Defect Shape Based on Inverse Analysis
Considering the Resolution of Magnetic Sensors
600

F
F'

Rank

500

Forward analysis

Inverse analysis

・Magnetic dipole
configuration

・Magnetic dipole
restration

400

m

300

・Magnetic flux density

B′ = F ′m

200
100
1

2

3

4

5
6
7
Lift off [mm]

8

9

(

m′ = F ′T F ′ + α I) F ′ B′−1Te・ Tikhonov Regularization
min F ′m ′ − B ′e + α m ′T Im ′{}・Add Error (1%)

10B′e = B′ + R = F ′m + R
Fig. 8. Inverse analysis procedure

Fig. 7. Response function matrix ranks

should be obtained by the inverse analysis becomes the m′ in Equation (13) that minimizes theevaluation function J1 .

J1 = F′m′ − B′e

2

(13)

Note that m′ in Equation (13) is an output value of the inverse analysis, while m of Equation(12) is an input value, which is obtained from the forward analysis. The properties of matrix F′
determine whether the appropriate m′ is obtained by minimizing J1 . In the following, F′ and F
of Equation (4) are verified from a viewpoint of linear independency.
Figure 7 shows the rank (the number of independent linear equations) of F and F′ with lift-offvarying from 1 mm to 10 mm. The number of observation points M = 4941, thus it would seem that M
equations exist. The number of independent equations found in nature, however, are 600 or less, fewer
than the number of dipoles N that should be requested.
Moreover, it can be seen that the rank decreases as lift-off grows, and the rank of F′ becomes
lower than that of F for all lift-off values. Two factors contribute to increasing the number ofsimultaneous equations, one being the dullness of the magnetic flux density for high lift-off values asshown in Fig. 2, the other being the averaging of the magnetic flux density.
If the solution is obtained from simultaneous equations in which linear independency decreases, theinfluence of noise becomes excessively predominant, and as a result so-called vibration solutions areoften obtained. To obtain an appropriate solution, therefore, we will modify Equation (13) using the
Tikhonov regularization method [9]. Namely, we minimize J 2 in Equation (14) instead of J1 in
Equation (13).

J 2 = F′m′ − B′e2+ α m′T Im′(14)

Here, α is a regularization parameter and I is a unit matrix. The first and second terms on theright-hand side are called the residual term and the regularization term, respectively. When J 2 isminimized, and if M > N, m′ is obtained by Equation (15).

(m′ = F′T F′ + α I) F ′ B′
−1

T

e

(15)

Equation (15) is a solution in the case where the modified response function matrix F′ is used forthe inverse analysis. As mentioned later in the numerical examples in next section, m in Equation(16) becomes the solution, when the existing response function F in Equation (5) is used for the
inverse analysis.

(m = FT F + α I) F B′
−1

T

e

(16)

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S. Nakasumi, et al. / Evaluation of Defect Shape Based on Inverse Analysis
Considering the Resolution of Magnetic Sensors
3

10

2

Fm − B ′e + α mT Im+++++++++++++++++
2
F ′m ′ − B ′e + α m ′T Im ′+++++++++++++++++

Regularized term210

α=1×10-11

10

α=1×10-2
0

10

-1

10

1

10
Residuals term

102Fig. 9. L-curve (lift-off = 1 mm)(a) Distribution of m ( α = 10-1)(b) Distribution of m ′ ( α = 10-2)
Fig. 10. Distribution of restored magnetic dipoles (lift-off = 1 mm)

A flowchart of the forward analysis and the inverse analysis described above is shown in Fig. 8.
3.3.

Evaluation of defect shape

Figure 9 shows the curve described by the plot points when the regularization parameter α varies
from 10−7 to 103 and lift-off is 1 mm. The horizontal and vertical axes are the residual term norm
and the regularization term norm, respectively. This curve is called the L-curve, and it is thought theα that corresponds to the vicinity of the bend location of the L-curve gives the most appropriate
distribution of magnetic dipoles [10,11]. Examining the appropriate regularization parameter locatedin this vicinity based on the result image of the inverse analysis, α = 10−1 was obtained for m andα = 10−2 was obtained for m′ . The α giving the best magnetic distribution for m tends to be
somewhat larger than that of the bend point of the L-curve.
Figure 10 shows the distribution of magnetic dipoles obtained by the inverse analysis. Figures
10(a) and 10(b) show the distribution of m in Equation (16) and m′ in Equation (15), respectively.
This notation is the same for Fig. 12, which is mentioned later. Black solid lines in these figures show
the boundary (crack front line) of the assumed crack surface.
The magnetic dipoles of m in Equation (12) are negative value. However, m in Fig. 10(a) has
positive value in the vicinity of the surface (z = 0 mm), which obviously differs from m . Thistendency was seen even if α was somewhat changed. It seems that this phenomenon occurs because
the position at which the sign of the magnetic flux density reverses around the crack is different
according to differences in the response function, as shown in Fig. 2. On the other hand, in m′ ofFig. 10(b), the sign of the magnetic dipole does not reverse like m in Fig. 10(a), and the magnetic
dipole is reproduced accurately in the surface vicinity. As a result, a profile almost corresponding to
the assumed crack front was obtained.
Figures 11 and 12 show the L-curve and magnetic dipole distribution, respectively, where lift-off is
10 mm. In Fig. 11, m and m′ show almost the same behavior. This means that the difference
between m and m′ in Fig. 11 is reduced from the viewpoint of the regularization as comparedwith Fig. 9. Similarly, the difference between Figs. 12(a) and 12(b) is also reduced as compared with
Fig. 10. If both are compared in detail, however, the crack shape of Fig. 12(b) is more accurate thanthat of Fig. 12(a), which signifies that the effect of the modified response function is still effective.
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Regularized term

S. Nakasumi, et al. / Evaluation of Defect Shape Based on Inverse Analysis
Considering the Resolution of Magnetic Sensors
10

3

10

2

10

1

10

0

10

-1

2

Fm − Be′ + α mT Im+++++++++++++++++
2
F ′m ′ − B′e + α m ′T Im′+++++++++++++++++

α=1×10-5

α=1×10-6

-1

10

0

10
Residuals term

1

10Fig. 11. L-curve (lift-off = 10 mm)(a) Distribution of m ( α = 10-5)(b) Distribution of m ′ ( α = 10-6)
Fig. 12. Distribution of restored magnetic dipoles (lift-off = 10 mm)

From low lift-off area to high lift-off area, therefore, the effectiveness of modifying the response
function matrix is confirmed for reproducing the crack shape more accurately. Furthermore, thiseffectiveness becomes especially remarkable with low lift-off areas.

4. Conclusions
The following conclusions were obtained about the magnetic flux leakage density method andevaluation of defect shapes using inverse analysis.
1. The influence of the length of the flux gate sensor was considered, and a new response
function that outputs the average value of the magnetic flux density that penetrates the sensor
was proposed.
2. Inverse analysis of a semi-elliptical surface crack using the Tikhonov regularization method
was conducted using the proposed response function. As a result, a remarkable improvement
effect was obtained at the vicinity of the surface of crack for low lift-off areas. The
improvement was also seen for high lift-off areas, though not so remarkably.

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Considering the Resolution of Magnetic Sensors
and Applications -，p.171, CORONA Publishing Co., LTD，1999[6] T. Suzuki, A. Terasaki, A. Sasamoto, Y. Nishimura and T. Teramoto, Nondestructive evaluation offerromagnetic structural materials using FG sensor, Electromagnetic Nondestructive Evaluation (XII), IOS
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