APPLICATION OF TAGUCHI METHODOLOGY FOR OPTIMIZING
TEST PARAMETERS IN MAGNETO-OPTIC IMAGING
Zhiwei Zeng1, Liang Xuan1, William Shih2, Gerald Fitzpatrick3, Lalita Udpa4
1
Dept. of Electrical & Computer Engineering, Iowa State University, Ames, IA 50010
PRI Research & Development Corporation, 25500 Hawthorne Blvd., Torrance, CA 90505
PRI Research & Development Corporation, 12517 131st Court Northeast,
Kirkland, WA 98034
4
Dept. of Electrical & Computer Engineering, Michigan State University,
East Lansing, MI 48824
2
3
ABSTRACT. The optimization of experimental parameters in an NDE test is extremely crucial
making accept/reject decision about a test sample. For instance, in magneto-optic imaging (MOI)
defects are displayed as an analog video image that is interpreted by the inspector. Subtle images such
as for small surface and subsurface defects may be difficult for the inspector to detect. Under these
circumstances, digital image processing methods may assist the inspector to interpret the MOI images.
The accept / reject decision for a test specimen is determined by observing the binary image obtained
by thresholding the magnetic flux density distribution. The coefficient of skewness of the binary
magneto-optic (MO) image can be used for calculating the probability that the image contains a crack.
The larger the skewness value, implies a larger likelihood of the presence of a crack. Several test
parameters affect the skewness of binary MO image and hence the likelihood of the image containing a
crack. The optimal set of test parameters (frequency, threshold, etc.) that generates the maximum
skewness of binary image can be found using an optimization algorithm based on the Taguchi method.
The number of trials necessary for the optimization is significantly reduced with Taguchi's
methodology of experimental design.
INTRODUCTION
The Taguchi method is a statistical analysis technique that is used in quality
improvement and design of experiments. Dr. Genichi Taguchi simplified the conventional
statistical tools by identifying a set of stringent guidelines for experiment layout and
analysis of results. The approach ensures quality by optimizing the design of
product/process and making the design insensitive to influence of uncontrollable factors
(robustness) [1]. The method is especially suitable for engineers who do not possess
sophisticated statistical knowledge.
The most important feature of Taguchi's philosophy of quality improvement is its
definition of loss function. Conventionally, a product is functionally acceptable if the
measure of performance characteristic is within the allowable range. The loss function in
Taguchi method is, however, defined as a quadratic function of the deviation of
performance measure from its target value, as shown in Figure 1. So the basic idea
underlying Taguchi's philosophy of quality control is the minimization of variation around
the target value.
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
651
Taguchi
Conventional
Conventional
100%
LAL
LAL
Target
Target Value
Value (T)
(T)
UAL
UAL
LAL –- Lower
Lower allowable
allowable limit
LAL
limit
UAL –- Upper
Upper allowable
allowable limit
UAL
limit
FIGURE
FIGURE 1.
1. Taguchi
Taguchi and
and conventional
conventionalloss
lossfunction
function
To minimize the number of experiments that need to be performed without losing
losing
much information,
Taguchi
constructed
a
set
of
tables
known
as
orthogonal
arrays
(OA),
in
information,
orthogonal arrays (OA), in
which all interactions of order greater than 2 are neglected [2]. The use of orthogonal
arrays
arrays makes the
the design of experiments easy and consistent. Two orthogonal arrays are
employed
employed in the experimental design using Taguchi method: inner array and outer array,
which handle the controllable and uncontrollable factors respectively. Controllable factors
are the parameters that can be easily controlled under practical conditions. Uncontrollable
factors,
difficult to control in
in practical operations;
operations; but
factors, also referred
referred to as noise factors, are difficult
they are under control in laboratory conditions. For each combination of controllable
factors,
factors, experiments/simulations are performed across all the combinations of noise
factors.
factors. The signal to noise ratio for each
each set of parameters is then calculated and used in
analysis to determine the
the optimal set of parameters.
The Taguchi approach involves (i) selecting the performance characteristics to be
optimized, (ii) listing the controllable and uncontrollable factors and the levels of each
factor
factor that need to be considered, (iii) designing the experiment, (iv) conducting the
experiment, (v) analyzing the results, and (vi) determining the optimal set
set of
of parameters
parameters
and influence
influence of each parameter on performance.
performance.
In this paper, we apply the Taguchi method to Magneto-optic inspection method.
The simulations are performed
performed to validate the feasibility of using the Taguchi method for
optimizing simple experimental parameters such as frequency
frequency and threshold.
MAGNETO-OPTIC IMAGING
Magneto-optic Imaging is a relatively new non-destructive evaluation technique of
of
detecting subsurface
aircraft skin structures. A schematic figure of
subsurface cracks and corrosions in aircraft
the
the MOI system is shown in Figure 2. The magneto-optic sensor used in the MO
instrument consists of a garnet film that has a magnetic anisotropy property with an “easy”
"easy"
axis of magnetization (very sensitive to magnetic field)
field) normal to the sensor surface and a
"hard"
“hard” axis of the magnetization (insensitive to magnetic field) in the plane of the sensor.
Light source
source
Polarizer
Analyzer
Bias coil
Lap joint
Sensor
FIGURE
FIGURE 2.
2. Schematic
Schematic of
of MOI
MOI instrument
instrument [1]
[1]
652
Induction foil
foil
Aluminum
Aluminum
plates
plates
Current foil
foil
Current
Air gap ' 0.35m
0.35m
*————6mm
Sin™————* ^Airpap>
1mm
11st layer
layer X
f 1mm
nd
nd
Rivet
Rivet
1mm
) 1mm
layer
22 layer
rd
rd
1mm
Crack
> ) 1mm
layer
3 layer
Crack |
S7I
y
'
/
x
:
4mm
4mm
(a)
(a)
(b)
(b)
FIGURE 3. Geometry
Geometryunder
underinvestigation
investigationand
andpredicted
predictedbinary
binaryMO
MOimage
image
(a) Geometry
Geometry (b)
(b) binary
binaryimage
image
An induction foil
foil carrying alternating
alternating current
current is
is used
used to
to induce
induce eddy
eddy current
current into
into the
the test
test
specimen. Anomalies in the specimen,
specimen, such
such as fasteners
fasteners and
and cracks,
cracks, generate
generate aa magnetic
magnetic
field component normal to the specimen
field
specimen and sensor
sensor surface.
surface. An
An easy-to-interpret
easy-to-interpret and
and realrealtime binary-valued
binary-valued image
image reflecting
reflecting the
the anomalies
anomalies of
of the
the magnetic
magnetic fields
fields isis then
then generated.
generated.
Details of the principles of the
the magneto-optic
magneto-optic inspection
inspection method
method can
can be
be found
found in
in [3,
[3, 4,
4, 5].
5].
The performance
performance of an MOI system
system under
under given
given measurement
measurement conditions,
conditions, can
can be
be
evaluated using the
the concept
concept of skewness
skewness to
to quantify
quantify the
the strength
strength of
of the
the field/flaw
field/flaw
interaction represented
represented in the binary MO
MO image.
image. The
The larger
larger the
the skewness
skewness value,
value, implies
implies aa
larger likelihood of the presence of a crack.
crack. Several
Several test
test parameters
parameters affect
affect the
the skewness
skewness of
of
binary MO image and hence
hence the
the detectability
detectability of
of flaws.
flaws. In
In this
this experiment,
experiment, we
we try
try toto
optimize the test parameters
parameters simultaneously
simultaneously in
in detecting
detecting aa buried
buried crack
crack in
inthe
thethird
third layer
layerof
of
aluminum plates, as shown in Figure 3 (a).
(a). A
A typical
typical crack
crack has
has height
height 1mm,
1mm, width
width O.lmm
0.1mm
finite element model [6, 7, 8] is utilized to predict the binary MO
and length 5mm.
5mm. A 3-D finite
image for a given
given test
test condition.
condition. Figure
Figure 33 (b)
(b) is
is the
the predicted
predicted image
image corresponding
correspondingtotothe
the
geometry of Figure
Figure 3 (a).
(a).
Parametric studies on the
the performance
performance of
of MOI
MOI system
system are
are usually
usually conducted
conducted to
to
seek the optimum value of a particular
particular test parameter
parameter while
while keeping
keeping other
other parameters
parameters
fixed. The Taguchi method, on the other hand, changes
fixed.
changes parameter
parameter values
values simultaneously
simultaneouslytoto
look for the
optimum
set
of
test
parameters.
Ideally,
one
would
like
to
the optimum set
parameters. Ideally, one would like to have
have maximum
maximum
skewness. At
At the
the same
same time,
time, the
the performance
performance of
of MOI
MOI isis expected
expected toto be
be robust
robust under
under
variations of the parameters.
parameters. The
The Taguchi
Taguchi method
method provides
provides an
an optimal
optimal tradeoff
tradeoff by
by
conducting a signal to noise ratio analysis.
analysis. The
The number
number of
of trials
trials necessary
necessary for
for the
the
optimization is significantly
significantly reduced
reduced with
with Taguchi's
Taguchi’smethodology
methodologyof
ofexperimental
experimentaldesign.
design.
EXPERIMENT DESIGN AND SIMULATION
The
The performance
performance characteristic
characteristic in
in this
this study
study is
is chosen
chosen to
to be
be the
the coefficient
coefficient of
of
skewness (CS)
skewness
(CS) of
of the
the binary
binary MO
MO image.
image. The
The larger
larger the
the coefficient
coefficient of
of skewness
skewness associated
associated
with
critical flaw
with aa critical
flaw the
the better
better the
the performance
performance of
of the
the MOI
MOI system.
system. The
The standard
standard definition
definition
of
CS
is
the
third
moment
of
the
experimental/predicted
data,
which
is
given
of CS is the third moment of the experimental/predicted data, which is givenby
by
N
CS =
∑ (x
i =1
− µ)
3
i
σ3
.
(1)
(1)
where
1, 2,
2, …,
..., TV)
are data
data (x
(x coordinate
coordinate of
of black
where xt(i
xi (i =
= 1,
N) are
black pixel
pixel in
in Figure
Figure 33 (b))
(b)) with
with TV
N the
the
number
data, ju
expectation and
a the
number of
of data,
µ the
the expectation
and σ
the standard
standard deviation.
deviation. This
This is
is valid
valid for
for linear
linear
eddy current
eddy
current excitation.
excitation. The
The skewness
skewness of
of an
an MO
MO image
image should
should be
be measured
measuredwith
withrespect
respecttoto
the
origin
(rivet
center
in
Figure
3
(a))
for
rotating
current
excitation.
The
alternative
the origin (rivet center in Figure 3 (a)) for rotating current excitation. The alternative
definition of
definition
of the
the coefficient
coefficient of
of skewness
skewness of
of an
an MO
MO image
image used
used in
in this
this paper
paper isisthe
thenegative
negative
653
ratio of the sum of the x coordinates of the black pixels with positive x coordinates to the
sum of the x coordinates of black pixels with negative x coordinates, i.e.,
(2)
where
1, x > 0
0, else
(3)
CS is then normalized as
CS =
- 1> (l - mm(CSQ,l/CSQj)
(4)
with
, ;c>0
sign(x)=j 0, * = 0
(5)
such that CSe (-1, l); CS is positive when the image skews in +x direction, negative when
the image skews in -x direction, zero when the image does not skew.
Four test parameters are believed to affect the skewness of MO image. They are,
namely, excitation current value, excitation current frequency, threshold (bias setting) and
liftoff. Since increasing current value is equivalent to reducing liftoff, we do not consider
the effect of liftoff in this paper. These parameters are controllable factors and incorporated
in the inner array. Their variations are noise factors and incorporated in the outer array.
The Taguchi method is utilized to choose the optimum set of parameters that yield an MO
image with maximum coefficient of skewness and are less sensitive to their variations. The
factors and the levels of each factor considered in this paper are listed in Table 1.
TABLE 1. Factors and levels
Factors
Controllable
Uncontrollable
Current value (A)
Excitation frequency (B)
Threshold (C)
Variation of A (D)
Variation of B (E)
Variation of C (F)
Level 1
120
3000
0.2
-24
-40
-0.04
654
Levels
Level 2
Level 3
240
360
4000
5000
0.4
0.6
+24
0
0
+40
+0.04
0
Unit
Amp
Hz
Gauss
Amp
Hz
Gauss
TABLE 2. Orthogonal array L9(34)
L9(34)
1 2 3 4
^^^Q)lumn
Trial^^
1
2
3
4
5
6
7
8
9
1
1
1
2
2
2
3
3
3
1
2
3
1
2
3
1
2
3
1
2
3
2
3
1
3
1
2
1
2
3
3
1
2
2
3
1
TABLE 3. Summary of simulation results
Trial
1
2
3
4
5
6
7
8
9
0.4168
0.3088
0.1417
0.4164
0.3589
0.3852
0.4153
0.3923
0.3692
0.0088
0.0405
0.0336
0.0031
0.0148
0.0204
0.0026
0.0542
0.0061
(S/N),
(S/N)2
33.4877
17.6460
12.5056
42.5668
27.6783
25.5288
44.1593
17.1984
35.6870
4.6829
3.1948
1.3213
4.6771
3.8597
4.2212
4.6611
4.2955
4.0017
ju : Mean Response
a : Standard Deviation
(SM02=-l01og10(MS£>)
The standard OA L9(34), as shown in Table 2, is used for the experiment layouts of
both the inner array and the outer array. Factors A, B and C are assigned to the first three
columns in the inner array. Their variations, i.e., factors D, E and F, are assigned to the
first three columns in the outer array. The last columns of both the inner and outer arrays
are left empty.
For each combination of the simulation conditions in both the inner array and the
outer array, the 3-D finite element model using A-v formulation [6] is performed. The
model has been validated by comparing the simulated image with the corresponding
experimental image [6, 7]. For each row of the inner array, all trials in the outer array are
performed. The resultant coefficients of skewness are then used in calculating statistical
quantities, such as mean response, standard deviation and signal to noise ratio, as shown in
Table 3. The total number of trials using the Taguchi method is 81 (92). This demonstrates
the efficiency of the Taguchi 's experimental design when compared with the full factorial
design where the total number of trials required is 729 (272).
655
ANALYSES
ANALYSES
Main Effects
Main Effects
One is
One is
There are two definitions of signal to noise ratio (S/N) that are commonly used.
There are two definitions of signal to noise ratio (S/N) that are commonly used.
µ
 2 
(S / N )1 = 10 log10(VI

2

(6)
σ 
V
)
and the other one is
and the other one is
(S / N )2 = −10 log(MSD)
10 (MSD )
(S/N)2=-lQlog
w
(7)
(7)
with
deviation of
of the
the image
image skewness
skewnessfrom
fromthe
thetarget
targetvalue
valueexpressed
expressedasas
with mean
mean square
square deviation
MSD =
1 n
2
∑ ( yi − t )
n i =1
= (µ − t ) + σ 2
2
(8)
(8)
In
µ is
response, aσ is
is the
the standard
standard deviation,
deviation,yyt i isis the
the rth
ithresult
resultininthe
the
In (6)
(6) ~~ (8),
(8), ju
is the
the mean
mean response,
outer
array,
n
is
the
number
of
simulation
conditions
in
the
outer
array,
and
t
is
the
target
outer array, n is the number of simulation conditions in the outer array, and t is the target
value
the absolute
absolute value
value of
of the
the coefficient
coefficientof
ofskewness).
skewness).
value (1
(1 for
for the
To
make
the
performance
of
MOI
system
robust to
to variations
variations of
of parameters,
parameters,
To make the performance of MOI system robust
definition
(6)
of
S/N
is
preferred.
Among
the
nine
combinations
of
test
parameters,
the77thth
definition (6) of S/N is preferred. Among the nine combinations of test parameters, the
set
Definition (7)
(7) measures
measures the
the deviation
deviation of
of quality
quality
set has
has the
the highest
highest robustness.
robustness. Definition
characteristic
from the
the target
target value,
value, which
which is
is consistent
consistent with
with Taguchi's
Taguchi’s quadratic
quadratic loss
loss
characteristic from
function
utilized in
in the
the further
further analysis
analysis in
in this
this paper.
paper.
function and
and utilized
The
indicate the
the general
general trend
trend of
of the
the influence
influence of
of the
the factors.
factors.They
Theyare
are
The main
main effects
effects indicate
calculated
by
averaging
the
S/N
ratios
for
each
control
factor
at
the
same
level.
Main
calculated by averaging the S/N ratios for each control factor at the same level. Main
effects
from Table
Table 33 are
are listed
listed in
in Table
Table 44 and
and plotted
plotted in
inFigure
Figure4.4.
effects calculated
calculated from
TABLE
Main effects
effects
TABLE 4.
4. Main
Factors
Factors
Factor
A
Factor A
Factor B
B
Factor
Factor C
C
Factor
Level 11
Level
3.066
3.066
4.674
4.674
4.400
4.400
Level22
Level
4.253
4.253
3.783
3.783
3.958
3.958
FIGURE
4. Main
Main effects
effects
FIGURE 4.
656
Level33
Level
4.319
4.319
3.181
3.181
3.281
3.281
A3
FIGURE 5. Interaction AxC
FIGURE
A×C
Generally the optimal set of parameters
parameters may
may not
not appear
appear ininthose
thosetrial
trialconditions
conditionsinin
the orthogonal
orthogonal array, because the
the
the Taguchi's
Taguchi’s experimental
experimental design
design isis aa partial
partial factorial
factorial
design. Optimal condition is found
found as the
design.
the combination
combination of
of the
the best
best level
level of
of each
each factor.
factor.
Observing Figure 4, the
Observing
the optimal
optimal condition
condition isis Levels
Levels 3,
3, 11 and
and 11 for
forFactors
FactorsA,A,BBand
andCC
respectively, which is not
respectively,
not included
included in
in Table
Table 3.
3. AA confirmation
confirmation run
runisisperformed
performedwith
withthe
the
condition A
A33BBid
condition
signal to
to noise
noise ratio
ratio isis 3.7115,
3.7115, much
much less
lessthan
thanother
other
1C1 and the resultant signal
values of S/N
S/N ratios in Table 3. We should note
values
note that
that we
we have
have neglected
neglectedall
allthe
theinteractions
interactions
between test
test parameters.
parameters.
between
Two factors
factors are
are said
Two
said to
to interact
interact when
when the
the effect
effect of
of changes
changes inin one
one ofof them
them
determines the
the influence
influence of
determines
of the
the other
other factor
factor and
and vice
vice versa
versa [1].
[1].Analyses
Analysesofofresults
resultsshow
show
that the
the interaction
interaction between
that
between current
current value
value and
and threshold,
threshold, denoted
denoted asas AxC,
A×C, cannot
cannot be
be
neglected,
as
represented
graphically
in
Figure
5.
The
combination
A
Ci
should
neglected, as represented graphically in Figure 5. The combination A33C1 should be
be
replaced by
by the
2C2
replaced
the other
other combination
combination that
that yields
yields the
the maximum
maximum S/N
S/Nratio.
ratio.Since
SinceAid,
A1C1,AA
2C2
and
A
Cs
have
almost
the
same
S/N
ratios,
the
optimal
set
of
test
parameters
are
AiBiCi,
3
and A3C3 have almost the same S/N ratios, the optimal set of test parameters are A1B1C1,
A2BiC2
and AsBiCs, i.e., frequency at Level 1 (3000 Hz), current value and threshold at
A
2B1C2 and A3B1C3, i.e., frequency at Level 1 (3000 Hz), current value and threshold at
the
same
level (linearly
(linearly proportional).
the same level
proportional). Further
Further studies
studies reveal
reveal the
the fact
factthat
thatusing
usingthreshold
threshold
proportional
to
the
current
value
will
yield
almost
identical
binary
MO
proportional to the current value will yield almost identical binary MO images.
images. The
The
optimal combinations
combinations appear
appear in
optimal
in Table
Table 3.
3. Hence
Hence no
no further
furtherconfirmation
confirmationruns
runsare
areneeded.
needed.
Analysis of
of Variance
Variance
Analysis
Interaction AxC is assigned to the last column of the inner array. Analysis of
Interaction A×C is assigned to the last column of the inner array. Analysis of
variance
is performed
performed to
variance is
to estimate
estimate the
the percent
percent contribution
contribution of
of each
each factor,
factor,which
whichisishelpful
helpful
in determining which of the factors need to be controlled and which do not.
in determining which of the factors need to be controlled and which do not.
In the left part of Table 5, the degree of freedom of the error term (f ) is negative,
In the left part of Table 5, the degree of freedom of the error term (efe) is negative,
which is not meaningful. In order to calculate other statistical measurements involving
the
which is not meaningful. In order to calculate other statistical measurements involving the
division byfe, the factor that has small variance (Interaction AxC in this case) is pooled to
division by fe, the factor that has small variance (Interaction A×C in this case) is pooled to
obtain new, positive estimate of S (sum of squares of the error term) andf . Pooling is the
obtain new, positive estimate of See (sum of squares of the error term) ande fe. Pooling is the
TABLE 5. ANOVA table
TABLE 5. ANOVA table
Factor
Factor
A
A
B
B
C
C
AxC
A×C
error
error
Total
Total
f
f
2
22
22
24
4
-2
-2
8
8
Before pooling
Before
S pooling V
S
V
2.982
1.491
2.982
1.491
3.382
1.691
3.382
1.691
1.906
0.953
1.906
0.953
0.940
0.235
0.940
0.235
0.000
0.000
9.211
9.211
After pooling
After
f
S
V pooling F
f
S
V
F
2
2.982
1.491
3.171
2
2.982
1.491
3.171
2
3.382
1.691
3.597
3.382
1.691
3.597
22
1.906
0.953
2.028
2
1.906
0.953
2.028
Pooled
Pooled
(4)
(4)
Pooled
Pooled
2
0.940
0.470
0.940
0.470
9.211
82
8
9.211
f: degrees of freedom S: sum of squares V: variance
f: degrees
of freedom
of squares
V: variance
F: variance
ratio S:P:sum
Percent
contribution
F: variance ratio P: Percent contribution
657
P (%)
P (%)
22.166
22.166
26.510
26.510
10.490
10.490
40.834
40.834
100.00
100.00
process of disregarding the contribution of a selected factor and subsequently adjusting the
contributions of other factors [2]. The percent contributions for the factors are listed in the
last column of Table 5. Among the three test parameters considered, frequency has most
influence on the performance on the MOI system in detecting the buried crack as shown in
Figure 2 (a). Other factors or errors, such as interactions between parameters, have a total
contribution of 41%.
CONCLUSIONS
Taguchi's methodology of parameter design has been found to be an effective and
efficient way of optimizing test parameters in NDE systems. This paper investigates the
application of the Taguchi method in choosing optimal and robust test parameters for MOI
inspection in detecting a critical flaw buried in the third layer. The optimal set of
parameters determined for this inspection geometry is frequency at 3000 Hz with current
value and threshold linearly proportionally varied. By performing analysis of variance,
frequency is found to have most influence on the variance of system performance among
the parameters considered. Although much of these findings are as expected this study was
intended as an example of the feasibility of using Taguchi method as a tool for optimizing
the parameters in any nondestructive inspection.
ACKNOWLEDGEMENTS
This material is based upon work supported by the Federal Aviation Administration
under Contract #DTFA03-98-D-00008, Delivery Order #IA013 and performed at Iowa
State University's Center for NDE as part of the Center for Aviation Systems Reliability
program.
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