E-Journal of Advanced Maintenance Vol.5-1 (2013) 42-50
Japan Society of Maintenology

Simplified Electromagnetic Inverse Analysis Technique about the Inclined
Defect for Welded Zone of High Chromium Steel
Shogo NAKASUMI1,*, Takayuki SUZUKI1, Naoyuki TAKASHIMA2 and Yuji TSUCHIDA3
1

National Institute of Advanced Industrial Science and Technology, East 1-2-1 Namiki, Tsukuba,
Ibaraki 305-8564, Japan
2
University of Tsukuba, Tennodai 1-1-1, Tsukuba, Ibaraki, 305-8573, Japan
3
Oita University, 700, Dannoharu, Oita-shi, Oita, 870-1192, Japan
ABSTRACT
A simplified technique of obtaining the shape of inclined defect at welded zone of high chromium steel was
proposed. Magnetic flux leakage measurement and the inverse analysis using singular value decomposition wereused. From the magnetic flux density distribution obtained under the high magnetic field, adequate defect shape
was obtained without the influence of welded zone. Under the low magnetic field, adequate defect shape wasobtained by subtracting the magnetic field of welded zone from measured magnetic flux distribution.
KEYWORDS

ARTICLE INFORMATION

inclined defect, magnetic flux leakage, welded zone, inverse analysis,
magnetic dipole moment

Article history:
Received 17Nobember 2012
Accepted 12 March 2013

1.

Introduction

In recent years, the high accuracy evaluation technique of degradation and damages in industrial
plants such as fast breeder reactor “Monju” has been studied for safety and reliability. In case of usinga ferromagnetic material like high chromium steel for structural members of nuclear instrument as
weld joint, inclined defects and damages often occur in the Heat Affected Zone (HAZ) around weldedzone as shown in Fig.1. Difference of material and the shape of inclined defects have complicatedinfluence on the distribution of magnetic flux density.
To detect the defect, magnetic flux leakage (MFL) technique is one of the non-destructive
evaluation (NDE) methods for ferromagnetic structural materials [1]. In MFL, defects are detected bymeasuring the magnetic flux leakage caused by the magnetic charges on the defect surfaces. Sensors
for measuring MFL are Hall sensors, flux gate sensors [2], SQUID and so on.
In numerical simulation of MFL, defects can be represented by using magnetic dipoles on thedefect plane, and the defect shape is estimated by conducting the inverse analysis. This technique has
been studied by several researchers. For example, Minkov et al. used the analytically integrated
expression of magnetic field of a parallelepiped surface crack, and they estimated the width and depthof the crack [3,4]. Baskaran and Janawadkar detected bar shape defects [5], and they identified thepositions and magnitude of the magnetic dipoles using the orthogonally projected MUSIC (MUltiple
SIgnal Classification) algorithm based on pseudo-inverse technique [6,7,8]．Nakasumi and Suzuki
improved the function which associates the magnetic dipoles and magnetic flux so as to capture theaverage value of magnetic flux along the sensor length, and they showed the effect by inverse analysis
of Tikhonov regularization technique using single crack models [9] and multiple crack models [10].
However, targets of these researches are simple shape defects in homogeneous material, and theinverse analysis for inclined defects in unhomogeneous material such as welded parts has not been
conducted.
In this study, therefore, the inclined defect of Type-III around HAZ of high chromium steel
welded zone is focused on as practical structural parts. Although the Type-IV defect is also important*

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

ISSN-1883-9894/10 © 2010 – JSM and the authors. All rights reserved.

42

S. Nakasumi, et al./ Simplified Electromagnetic Inverse Analysis Technique
about the Inclined Defect for Welded Zone of High Chromium Steelfor practical purpose, NDE using magnetic flux leakage is considered to be suited for Type-III better
than Type-IV, because the magnetic flux decreases steeply as the distance between the defect and thesensor becomes large. Then, we will present the simple approach for defect shape evaluation using
inverse analysis.

Type‐III

Type‐I

Type‐II
Type‐IV

WeldedHAZ Base metal
Base metal HAZ zone
Fig. 1. Type of defects around HAZ of weld joint

2.

Theoretical background of inverse analysis for inclined defect2.1. Representation of defects using magnetic dipole moments
In this section, we will set the relationship between defect shape and magnetic flux distribution.
Figure 2 shows the conceptual model with an inclined defect. The defect is rectangular shape and it isinclined to angle  from the perpendicular line. Magnetic dipoles are configured in lattice pattern onthat inclined plane. We assumed that the magnitude of magnetic density on the defect is uniform.
Therefore, magnetic dipoles are configured at regular intervals, and the magnitude of them on thedefect is constant. As for this, some researchers assume that the magnitude of magnetic dipoleschange linearly in the depth direction [3]. However, the purpose of this inverse analysis is to estimate
the defect shape. Therefore, for inverse analysis, it is not necessary to obtain the physically precise
distribution of magnet.
The defect is represented by magnetic dipole moments as shown in Figure 2, and we assume thatthe direction of magnetic dipole is fixed to be that of y-axis, regardless of the inclination angle of thedefect. The magnetic flux density observed at the point is given by Eq. (1) [9, 10].

By  x    Fy  x  x my(1)Here, the point x   x, y, z  is the coordinate of magnetic dipole on the defect plane, N is thetotal number of the magnetic dipoles, my is density of magnet, and Fy  x  is the response function
which is denoted by Eq. (2).
TFy  x, y, z   0 x 2  y 2  2 z 2 4  x  y  z 
2

2

2 52

(2)

Here, 0 is the magnetic permeability of vacuum. Eq. (1) holds at every observation point when theobservation point varies on the plane that maintains a certain constant liftoff value. Finally, we obtaina matrix equations of Eq. (3). Here, B and m are vectors which are obtained by setting the observation
points and magnet dipoles, respectively. F is a matrix, which is composed of Fy in Eq. (2).

B  Fm(3)

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E-Journal of Advanced Maintenance Vol.5-1 (2013) 42-50
Japan Society of Maintenology

2.2. Procedure of inverse analysis using singular value decomposition technique
The defect shape, namely the value of magnetic dipoles m of equation (3) is derived from theMagnetic flux densityzyMeasurement plane

Magnetic dipolez

2.5 mmx
x

Defect plane

5.0 mmFig. 2. Defect and coordinate system

magnetic flux distribution B by the inverse analysis. However, F of equation (3) often
becomes ill-posed condition matrix, and so-called vibration solutions are often obtained.
To avoid this difficulty, Tikhonov regularization technique [11] or Singular Value Decomposition(SVD) technique are often used. As for the Tikhonov regularization technique, however, arbitrarinessof the formulation of regularization term occurs, and this makes the interpretation of the solution of
inverse analysis unclear. In this work, therefore, we use the SVD technique, and thus we can avoid
that arbitrariness. We will briefly explain the formulation below. The matrix F is generally
decomposed like following formula.
r

F   i ui vTi

(4)

i 1

Here, i is the ith singular value, ui and vTi are the left and the right singular vectors
corresponding to i respectively. And, r is the total number of singular values. From equation (3)and equation (4), we can obtain the solution m as equation (5).
r

m
i 1

vi uTii(5)

B

Usually, noises, such as errors of measurement, are contained in magnetic flux density. Therefore,
equation (5) means that the noises contained in the measured magnetic flux density is amplified by the
singular value of quite small size, and this reduces remarkably the accuracy of inverse analysis. To
avoid this, therefore, we set the threshold value k 1  k  r  to i , and we truncate the singularvalues smaller than k . Namely, m is represented like equation (6).
k

m
i 1

ui vTii

B 1  k  r (6)

By this operation, the information of high-frequency component which corresponds to more than(k+1) th singular value is lost from the whole solution space. However, since the noise effect containedin this area is also removed, we can expect to obtain the stable solution.

44

S. Nakasumi, et al./ Simplified Electromagnetic Inverse Analysis Technique
about the Inclined Defect for Welded Zone of High Chromium Steel

However, since the important information is lost if the threshold value k is too large, we need
to be cautious of determining it. We used so-called L-curve approach [12]. Namely, as k enlarges
from zero, we can obtain a L-shape curve by plotting points to the plane whose horizontal and verticalaxes are the residual term norm ( Fm  B ) and the solution vector norm ( m ), respectively. Generally
speaking, it is thought the k that corresponds to the kink of this curve gives the most appropriate
distribution of magnetic dipoles. That is to say, the threshold value k which makes both the residual
norm and the solution norm simultaneously minimized gives the most suitable solution.

Base metal： Mod.9Cr‐1Mo
Welding material： TGS9Cb

Welded
zone
Weld
metal

15

35

2.5

5

0.3

Base metal

HAZ

20°

80
Fig. 3. Specimen with an inclined defect

Unit mm

NeodymiummagnetNeodymiummagnet

20.0

45.0

20.0

(a) Overall view

3.

(b) Enlarged view of inclination defectand welded zone
Fig. 4. Magnetization setup

Inverse analysis using experimental magnetic flux data of inclined defect3.1. Specimen and magnetization

Figure 3 shows the specimen used in this study. It was obtained from the high chromium(Mod.9Cr-1Mo) steel welded specimen. On the boundary of a welded zone layer and HAZ, the defectof 2.5 mm depth and inclined angle of 20 degree was prepared in parallel with weld line by wire
cutting. In order to compare the magnetic flux density distribution by the difference of magnetization
conditions, the specimen was magnetized using two kinds of magnets, neodymium magnets (surface-inductive-flux 4.88  10-1T) and ferrite magnets (the 1.33  10-1T).
Figure 4 shows the setup for magnetization of the specimen. The specimen was magnetized in alongitudinal direction using the yoke with two neodymium magnets. The magnetic flux density
around the tip of the yoke was 4.88  10-1T when the neodymium magnets were attached to the yoke.

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E-Journal of Advanced Maintenance Vol.5-1 (2013) 42-50
Japan Society of Maintenology

3.2. Measurement system

Figure 5 shows the experimental setup for magnetic flux density measurement. A hall sensor isused to capture the magnetic flux density at the distance of liftoff. Resolution of this hall sensor is
1.0  10-5T . Magnetic flux leakage was measured along on six layers in the direction of longitudinal.
Liftoff is set to be 1 mm and the interval of measurement is 0.1 mm in x-axis direction, and 1 mm in
y-axis direction.
PC
20.0

20.0

Hall sensor

1.0 1.0 1.0 1.0 1.0

Scanning line

A/D
converter

Liftoff
Specimenzx

Pedestal

Start line

Inclined defectyyUnit mm(a) Experimental setup(b) Scanning line
Fig. 5. Measurement system

Magnetic
Magneticflux
fluxdensity
density TG

Magnetic flux
flux density
density TG
Magnetic

×10‐4

250

Welded metal
zone

200
150
100
50
0
‐20

‐10

0

10

20

25

×10‐4

zone
Welded metal

20
15
10
5
0
‐20

Positionmm
mmPosition

‐10

0

10

20

Position mm
Position(b) Applied magnetic field 1.33  10-1T(a) Applied magnetic field 4.88  10-1T
Fig. 6. Magnetic flux leakage distribution

3.3. Inverse analysis using original experimental data

Figure 6 shows the distribution of magnetic flux density along y-axis. The value is averaged in
x-axis direction. In Fig. 6.(a), we can see that the magnetic flux density becomes asymmetrical
distribution to the inclined defect. Furthermore, in Fig. 6.(b), we can observe the change derivingfrom the welded zone or HAZ.
Then, defect shape evaluation for the inclined defect was conducted by using magnetic dipole
model and the inverse analysis based on the SVD technique. In this study, we assumed that the
inclination angle of the defect is known in advance, because as for the defects like Type III around
welded zone, they often generate along with the welded zone in many cases, and that inclined angle is46

S. Nakasumi, et al./ Simplified Electromagnetic Inverse Analysis Technique
about the Inclined Defect for Welded Zone of High Chromium Steelknown in most cases. Therefore, the inverse analysis is conducted for the magnetic dipoles configuredon the plane of inclination angle   20 ( x  z  plane of Fig.2).
Figure 7 shows the magnitude of reconstructed magnetic dipoles on the x  z  plane obtainedby the inverse analysis. In this figure, black color corresponds to the high density area of magnetic
dipoles, thus this means this area is defected. Similarly, white area corresponds to non-defected area.
The boundary of the real defect is indicated as dotted lines (The depth is 2.5 mm as shown in Fig.2).
In Fig. 7(a), the magnitude in the area of 0.5 mm < z < 0.0 mm indicates almost zero, andthis does not represent the characteristic of the surface defect. In our experience, this phenomenon
often happens when the magnetic flux data for inverse analysis is obtained by experimental
measurement, although this phenomenon does not happen when the magnetic flux data is obtained
theoretically by numerical calculations as in equation (1)～(3). However, this phenomenon happens
only on the area of near z   0 mm. Thus, this does not have a bad influence on defect evaluation.
On the contrary, in Fig. 7(b), the unsuitableness of the magnetic dipole magnitude abovementioned
spreads far more widely. The reason is considered to be the influence of the welded zone. As a result,the black area crosses over the boundary of defect extensively. In contrast, the black area is above theboundary adequately in Fig.7(a). For these reasons, acceptable solution is obtained for case of appliedmagnetic field 4.88  10-1T. Whereas, adequate result was not obtained in case of applied magnetic
field 1.33  10-1T.
0[a.u.]

-2

100

-4
50

-6
-8

-10

0

0

1

2

3

4

[a.u.]

-2

10

-4

0

-6

-10

-8

-20

mm
mm mm
DefectZ’Zlength

mm mm
mm
DefectZ’Zlength

0

- 100

5

-30
1

3

4

5

mm
XY mm

Y mm

X mm

2

(a) Applied magnetic field 4.88  10 T(b) Applied magnetic field 1.33  10-1T
Fig. 7. Magnitude of reconstructed magnet dipoles of inverse analysis on the inclined plane(Dotted lines indicate the boundary of real defect)-1

Base metal HAZ Welded zone HAZ Base metal

Fig. 8. Welded zone of high chrome steel

3.4. Result of inverse analysis using modified experimental data

In order to evaluate the influence of the welded zone on the magnetic flux distribution, the47

E-Journal of Advanced Maintenance Vol.5-1 (2013) 42-50
Japan Society of Maintenology

magnetic property of that area was investigated. The magnetic property of the specimen which was
cut out from the welded zone of the high chrome steel plate (shown in Fig. 8.) was measured by aVibrating Sample Magnetometer (VSM), and the B-H curves for each material were obtained asshown in Fig.9. In the range of low magnetic field, permeability of the welded zone and HAZ werelower than that of base metal. Therefore, it seems that this discrepancy of permeability affected the
distribution of magnetic flux density and this made the result of the inverse analysis unsuitable. On
the other hand, in the range of saturated magnetization, there was almost no difference between bothmaterials, and it seems that the inverse analysis was hardly affected.
Magnetic flux density distribution of the specimen in Fig. 8. was measured, and we subtracted
this value from the distribution of Fig. 6.(b) as shown in Fig. 10. By this operation, we could remove
the influence of welded zone, and we could obtain only the influence of the inclined defect.
Figure 11 shows the magnitude of reconstructed magnetic dipoles of the inverse analysis
obtained from the magnetic flux distribution of Fig.10. In this case, adequate defect shape wasobtained.

4. Conclusion

2
Magnetic flux density T

Magnetic flux density T

Magnetic flux density distribution of inclined defect around the welded zone of high chromium
steel welded specimen was measured, and inverse analysis was conducted. Defect shape was obtained
adequately from the magnetic flux density under the high magnetic field, without being subject to the
influence of a welded zone. Under the low magnetic field, influence of welded zone became large
relatively. However, by carrying out subtraction operation of the magnetic property, magnetic flux
density distribution without influence of welded zone was extracted, and adequate defect shape wasobtained by inverse analysis.
Base metal
HAZ

1

Weld metal
Welded
zone

0
-1
-2
-0.6

-0.4 -0.2
0
0.2
0.4
6
6
Magneticx10 A/mA/mMagnetic
fieldfield
×10
Fig. 9. B-H Curve

48

0.6

Magnetic flux density

T

S. Nakasumi, et al./ Simplified Electromagnetic Inverse Analysis Technique
about the Inclined Defect for Welded Zone of High Chromium Steel

30

×10‐4
With defectWithout defect25

Subtracted
Welded zone

20
15
10
5
0
‐5
‐10
‐20 ‐15 ‐10‐50
5Position mm

10

15

20

Fig. 10. Magnetic flux leakage distribution
0[a.u.]

6

Z’ mm

1

-2

5

3

4

-4

3

5

2

-6

1

7

0

-8

-1

9

-100

-2
1

2

3

4

5

X mm

X mm
Fig. 11. Result of inverse analysis using subtracted magnetic flux distribution

Acknowledgement
Present study includes the result of “Core R&D program for commercialization of the fast
breeder reactor by utilizing Monju”, entrusted to Universitiy of Fukui by the Ministry of Education,
Culture, Sports, Science and Technology of Japan(MEXT), and reentrusted to National Institute ofAdvanced Industrial Science and Technology by University of Fukui at FY 2011.

References[1] T. Suzuki, A. Terasaki, A. Sasamoto, Y. Nishimura and T. Teramoto, Nondestructive evaluation offerromagnetic structural materials using FG sensor, Electromagnetic Nondestructive Evaluation, (XII)(2009), 271-278.
[2] T. Suzuki, K. Hasumi, T. Kurota and T. Teramoto, Analysis and evaluation of cross-sectional shape for
defect of SS400 using fluxgate sensor, Journal of JSAEM, 19 – 2 (2011), 342-347 (in Japanese).
[3] D. Minkov, J. Lee and T. Shoji, Study of crack inversions utilizing dipole model of a crack and hallelement measurements, Journal of Magnetism and Magnetic Materials, 217 (2000), 207-215.
[4] D. Minkov and T. Shoji, Method for sizing of 3-D surface breaking flaws by leakage flux, NDT&E
international, 31-5 (1998), 317-324.
[5] R. Baskaran and M. P. Janawadkar, Imaging defects with reduced space inversion of magnetic flux leakage
fields, NDT&E international, 40 (2007), 451-454.
[6] R. Baskaran and M. P. Janawadkar, Defect localization by orthogonally projected multiple signal
classification approach for magnetic flux leakage fields, NDT&E international, 41 (2008), 416-419.
[7] R. Baskaran and M. P. Janawadkar, Localization of defects in magnetic permeable bar by multiple signal

49

E-Journal of Advanced Maintenance Vol.5-1 (2013) 42-50
Japan Society of Maintenology
classification approach using rotating dipole model, NDT&E international, 42 (2009), 664-667.
[8] R. Baskaran and M. P. Janawadkar, Dipole identification and localization using pseudo-inverse techniques
for magnetic flux leakage experiments, Electromagnetic Nondestructive Evaluation, (XV) (2012), 167-174.
[9] S. Nakasumi and T. Suzuki, Evaluation of defects based on inverse analysis considering the resolution ofmagnetic sensors, E-Journal of Advanced Maintenance, 3-1 (2011), 1-10.
[10] S. Nakasumi and T. Suzuki, Improvement of restoration accuracy of semi-elliptical surface crack
considering the resolution of flux gate sensor, Electromagnetic Nondestructive Evaluation, (XV) (2012),
151-158.
[11] A. N. Tikhonov and V. Y. Arseine, Solutions of ill-posed problems, Halsted Press, 1977.
[12] C. Hansen, Analysis of discrete ill-posed problems by means of the L-Curve, Society for Industrial andApplied Mathematics, 34 - 4 (1992), 561-580.

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