PREDICTION OF INSONIFYING VELOCITY FIELDS AND FLAW
SIGNALS OF THE 2002 ULTRASONIC BENCHMARK PROBLEMS
Sung-Jin Song1, Joon Soo Park1, Hak Joon Kirn1'2
1 School of Mechanical Engineering, Sungkyunkwan University, Suwon, Korea
2 Currently, Center for NDE, Iowa State University, Ames, IA, 50011, USA
ABSTRACT. In the present study, the radiation issues related to the ultrasonic benchmark
problems of year 2002 are explored by adopting two ultrasonic measurement models based on 1) the
multi-Gaussian beams and 2) the Rayleigh-Sommerfeld integral with high frequency approximation,
while keeping the choice of scattering model (the plane wave far-field scattering amplitude
estimated by the Kirchhoff approximation) unchanged. The insonitying beam fields and flaw signals
calculated by two models showed very good agreement in most of the cases. However, they showed
significant difference at the near-critical angle (which is corresponding to the cases of refracted Swaves with the refracted angle of 30°) due to the rapid variation in the transmission coefficient in
that region.
INTRODUCTION
This paper describes our approach and results of predicting the insonifying
velocity fields and flaw signals of a set of ultrasonic benchmark problems [1] proposed by
the World Federation of Nondestructive Evaluation Centers. This particular set of the
benchmark problems is considering the measurement of ultrasonic pulse-echo signals from
two types of standard scatters (spherical voids and side drilled holes) in an aluminum
specimen immersed in water, with two types of refracted (longitudinal and shear) waves
generated by two kinds of transducers (with planar and focused surfaces). Thus, the present
problem is, in essence, quite similar to the ultrasonic benchmark problem of the year 2001
[2], so that the prediction of flaw signals can be done by adopting a similar approach to the
last year [3].
Unlike the problem of last year, however, the problem of this year considers the
incidence of insonifying beams at several different refracted angles, so that a variety of
interesting issues are involved in both scattering and radiation. For example, there are
issues related to accuracy and computational time in the calculation of scattering fields
from 3-D scatters (spherical voids) and 2-D scatters (side drilled holes). In the prediction of
radiation fields, there would also be interesting problems to be considered, especially at the
near-critical and high-refracted angles.
hi the present study, the focus of our research endeavor was placed on the
radiation issues. Specifically, we investigated the accuracy of the multi-Gaussian beam
models [4] (which have been developed under the paraxial approximation and has a great
advantage in computation time) in the prediction of the insonifying velocity fields and flaw
signals, especially at near-critical and high-refracted angles. For this purpose, we adopted
another beam model, that is the generalized Rayleigh-Sommerfeld integrals with high
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
1784
frequency assumption [5], for the bilateral comparison, while keeping the plane wave farfield scattering amplitude obtained by the KirchhorY approximation [5] as the only one
scattering model for the calculation of scattered wave fields from scatters.
ULTRASONIC MEASUREMENT MODELS
In the present study, we implemented two ultrasonic measurement models for the
calculation of responses from two standard scatters. The first one is the ultrasonic
measurement model based on the multi-Gaussian beams, and the second is the model
adopting the generalized Rayleigh-Sommerfeld integral with high frequency approximation.
Since the details of the adopted models can be found in several references [3,4,5], here a
brief description will be given for the continuity of our discussion.
In the ultrasonic measurement model using the multi-Gaussian beams, the
received ultrasonic signal from a small, isolated scatter in a solid specimen immersed in a
fluid medium can be calculated by use of Eq. (1) for both planar and focused transducers.
FM = £Mexp(2^^^
\-i7tfa
Aci )
(1)
where, pl9 p2 are densities of the fluid and solid, respectively, q, c2a are the velocities
of the fluid and solid, respectively, and ^, k% are wave numbers in the fluid and solid,
respectively. T£P is the transmission coefficient from fluid to solid, H2 is the distance
from the interface to the flaw, and A(co) is the far-filed scattering amplitude (of which the
explicit expressions for a spherical void and a cylindrical void will be given in the next
section). And, the diffraction correction, C(o)) is given by Eq. (2).
rexp
(2)
where, da(a = P,SV) is the polarization vector, and the definitions of other terms
including 6^(0) and G 2 (7/ 2 ) matrices can be found in reference [4].
For the generalized Rayleigh-Sommerfeld integral with high frequency
approximation, however, the received ultrasonic response can be calculated by Eq. (3), for
both planar and focused transducers.
V(<o) = p(to*£<L f
S
-
where, 7^;P is the transmission coefficient from solid to fluid, D%(m = l,2) are distances
traveled through the interface along the ray path. The incident displacement amplitude, U0 ,
is given by Eq. (4).
1785
3
U0 =
− ρ1ν 0
1
å
ρ 2 cα 2 α =P ,S 2π òS
(
T12α ; p (cosθ1α )dα exp ik p1 D1α + ikα 2 D2α
α
1
α
2
α
1
)
D + cα 2 D / c p1 D + cα 2 cos θ1 D / c p1 cos 2 θ 2α
2
α
2
f
i
*
«
"
cos
/ cpl cos2
The detailed description of the terms appeared can be found in reference [5].
dS (y )
(4)
(4)
The detailedSCATTERING
description of theAMPLITUDES
terms appeared can be found in reference [5],
FAR-FIELD
FAR-FIELD SCATTERING AMPLITUDES
The plane wave far-field scattering amplitude, denoted as A(ω ) in Eq. (1), is
chosen to describe
thewave
scattered
field
from aamplitude,
flaw under
consideration.
Based
on is
the
The plane
far-field
scattering
denoted
as A(co) in
Eq. (1),
(
A
ω
Kirchhoff
approximation,
the
far-field
scattering
amplitude
from
a
spherical
void,
chosen to describe the scattered field from a flaw under consideration. Based onSV the) ,
can
be obtained
by Eq. (5). the far-field scattering amplitude from a spherical void, Asv (CD) ,
Kirchhoff
approximation,
can be obtained by Eq. (5).
sin (k α 2 a ) ù
é
−a
exp(− ik α 2 a )êexp(− ik α 2 a ) − sin
A SV (ω ) =
ú
(*«2«)"
2
k α 2 a ûú
ëê
(5)
(5)
where, a is the radius of the spherical void.
The
far-filedvoid.
scattering amplitude for a 3-dimensional side-drilled
where, a is
thecorresponding
radius of the spherical
corresponding
far-filed
scattering
a 3-dimensional
side-drilled
hole of the The
length
of ∆L , A
(
ω
)
, can be amplitude
given byforEq.
(6) using the
Kirchhoff
SDH
hole
of
the
length
of
AL,
A
(co),
can
be
given
by
Eq.
(6)
using
the
Kirchhoff
SDH
approximation with the assumption of small size in its diameter. The detailed discussion on
with in
theSchmerr
assumption
small[6].
size in its diameter. The detailed discussion on
Eq.approximation
(6) can be found
and of
Sedov
Eq. (6) can be found in Schmerr and Sedov [6].
− ikα 2 a ì
2ü
(6)
A(ω ) = ∆L
í H 1 (2kα 2 a ) + iJ 1 (2kα 2 a ) − ý
πþ
2 î
(6)
where, a is the radius of the side-drilled hole, and H1 and J1 is the Struve and Bessel
where, of
a the
is the
the side-drilled hole, and HI and Ji is the Struve and Bessel
functions
firstradius
order,ofrespectively.
functions of the first order, respectively.
INSONIFYING VELOCITY FIELDS: RESULTS AND DISCUSSION
INSONIFYING VELOCITY FIELDS: RESULTS AND DISCUSSION
Figure
of the
the insonifying
insonifyingvelocity
velocityfields
fieldsforforthethe
Figure1 1shows
showsthe
thecalculation
calculation results
results of
refracted
S-wave
(with
the
refracted
angle
of
30
°
)
produced
in
two
media
thegiven
given
refracted S-wave (with the refracted angle of 30°) produced in two media bybythe
planar
transducer
using
a)
the
multi-Gaussian
beam
models,
and
b)
the
Rayleighplanar transducer using a) the multi-Gaussian beam models, and b) the RayleighSommerfeld
Sommerfeldintegral
integralwith
withhigh-frequency
high-frequency approximation.
approximation.
(a)
(b)
(a)the insonifying velocity fields for the refracted
(b) S-wave (with the refracted
FIGURE 1. The prediction of
FIGURE 1. The prediction of the insonifying velocity fields for the refracted S-wave (with the refracted
angle of 30°) produced by the planar transducer using (a) the multi-Gaussian beam model, (b) the Rayleighangle
of 30°) produced
by the
planar transducer
using (a) the multi-Gaussian beam model, (b) the RayleighSommerfeld
integral with
high-frequency
approximation.
Sommerfeld integral with high-frequency approximation.
1786
Transmission coefficient
3
2.5
2
1.5
1
0.5
0
0
0
10
10
20
30
20
30Refracted
40 50 Angle
60 70
Refracted Angle
40
50
60
70
80
80
90
90
FIGURE 2. The transmission coefficient of the refracted S-wave through the water/aluminum interface.
FIGURE 2. The transmission coefficient of the refracted S-wave through the water/aluminum interface.
The refracted beam patterns calculated by two beam models showed significant
difference inThe
the refracted
solid specimen,
as shown
in Figure
1. The
transmission
coefficient
of the
beam patterns
calculated
by two
beam
models showed
significant
refracted
S-wave
very rapidas(according
to the angle
incidence) coefficient
around theofcritical
difference
in thevaries
solid specimen,
shown in Figure
1. Theoftransmission
the
angle,
whereS-wave
the beam
patterns
in Figure
1 wastoconsidered,
asincidence)
shown in Figure
rapid
refracted
varies
very rapid
(according
the angle of
around 2.
theThis
critical
variation
in thethe
transmission
coefficient
in fact,
the major
source in
ofFigure
the difference,
since
angle, where
beam patterns
in Figureis,1 was
considered,
as shown
2. This rapid
in the transmission
coefficient
fact, the major
source of the
difference,
since
thevariation
multi-Gaussian
beam models
does is,
notin account
this variation,
while
the Rayleighthe multi-Gaussian
beam Thus,
modelsone
does
not to
account
thiscareful
variation,
while
the RayleighSommerfeld
integral does.
needs
be very
when
adopting
the multiSommerfeld
Thus,
needs
to beangles,
very careful
adopting the multiGaussian
beamintegral
modelsdoes.
around
theone
near
critical
wherewhen
the multi-Gaussian
beam
Gaussian
beam
models around the near critical angles, where the multi-Gaussian beam
models
break
down.
models break
down.
Figure
3 shows the similar calculation results of the insonifying velocity fields
Figure
3 showswith
the similar
calculation
results
of°.the
insonifying
fields
for the refracted S-wave
the refracted
angle
of 75
The
refractedvelocity
beam patterns
for
the
refracted
S-wave
with
the
refracted
angle
of
75°.
The
refracted
beam
patterns
calculated by two beam models showed difference to the some extent. However, it is hard
by two
models
to the
some extent.
However,
it is hard
to calculated
justify which
one beam
is correct,
andshowed
which difference
is not, at this
moment,
since in
this particular
case
to justify which one is correct, and which is not, at this moment, since in this particular case
the angle of refraction is quite high so that the refracted angle can also vary very rapidly
the angle of refraction is quite high so that the refracted angle can also vary very rapidly
even with a small change in the angle of incidence. In fact, there is a possibility that both
even with a small change in the angle of incidence. In fact, there is a possibility that both
models
could
to have
have further
furtherinvestigation
investigationtotoclarify
clarify
models
couldbreak
breakdown.
down.Thus,
Thus,ititisisstrongly
strongly needed
needed to
thisthis
issue.
issue.
(a)
(b)
(b) S-wave (with the refracted
FIGURE 3. The prediction (a)
of the insonifying velocity fields for the refracted
FIGURE
3.
The
prediction
of
the
insonifying
velocity
fields
for
the
refracted
refracted
angle of 75°) produced by the planar transducer using (a) the multi-Gaussian beamS-wave
model, (with
(b) thethe
Rayleighangle
of 75°) produced
by the
planar transducer
using (a) the multi-Gaussian beam model, (b) the RayleighSommerfeld
integral with
high-frequency
approximation.
Sommerfeld integral with high-frequency approximation.
1787
TABLE 1. Insonifying velocities at the center of the flaw
(unit: mm/s)
TR and Reft Wave
Angle (degree)
Model
RSI
0°
0.3073
0.3035
0
0.0578
Focused transducer
(Refracted P-wave)
MGB
1.6777
RSI
1.5760
Focused transducer
(Refracted S-wave)
MGB
0
RSI
0.3055
Planar transducer
(Refracted P-wave)
Planar transducer
(Refracted S-wave)
MGB
RSI
MGB
30°
45°
60°
75°
0.2817
0.2303
0.1471
0.0552
0.2774
0.2269
0.1452
0.0544
0.2695
0.5907
0.4864
0.2297
0.3955
0.5829
0.4803
0.2264
1.0027
0.9472
0.3119
0.0755
0.0342
0.3269
2.3564
0.0720
0.9656
0.1417
2.3729
0.9435
0.1379
1.2179
1.7366
0.0343
Table 1 summarizes the prediction of the insonifying velocities at the center of
the flaws by two beam models considered. The predictions of the multi-Gaussian beam
agreed very well with those of the Rayleigh-Sommerfeld integral, except at the near critical
angle (which is corresponding to the case of the refracted S-wave with the refracted angle
of 30°).
TIME-DOMAIN WAVEFORMS: RESULTS AND DISCUSSION
The calculation of the time-domain waveforms requires the estimation of the
system efficiency factors for the given transducers, which was done by adopting the wellknown front-surface reflection model given by Rogers and Van Buren [7]. By taking the
inverse Fourier transform to the frequency-domain responses from the standard reflectors
(calculated by Eq. (1) or Eq. (3)), we obtained the time-domain flaw signals. (Here, it is
worthwhile to note that for the case of the side-drilled holes Eq. (3) were integrated along
the axis of the hole in the lit region of the insonifying beam.)
Figure 4 shows the predicted time domain waveforms for (a) a spherical void
and (b) a side-drilled hole with the diameter of 0.125 mm located at the depth of 25.4 mm
from the specimen surface by use of the refracted shear wave (with the refracted angle of
30°) produced by the planar transducer. The solid line is the prediction made by the multiGaussian beam model, while the dotted line by the Rayleigh-Sommerfeld integral with
high-frequency approximation. The waveforms predicted by two models agreed very well
in their shape. However, there were significant differences in their amplitudes. As discussed
above, this discrepancy resulted from the rapid variation in the transmission coefficient
around the near-critical angle, where these particular waveforms were predicted.
We have calculated all of the time-domain waveforms for the complete
combination of transducer, refracted wave, and scatter type and size. However, due to the
space limitation the entire waveforms cannot be presented here. From the set of complete
waveforms, it was observed that two beam models showed very good agreement in the
prediction of the shape of waveforms. In addition, the peak-to-peak amplitudes of the
waveforms, which is summarized in Tables 2 and 3 for spherical voids and side-drilled
holes, respectively, showed very good agreement for most of the cases, except at the nearcritical angle (which is corresponding to the case of the refracted S-wave with the refracted
angle of 30°). As pointed out earlier, even though the results of two model predictions at the
high-refracted angle of 75° agreed each other, it would be necessary to have further
investigation for the verification of prediction.
1788
TABLE 2 The predicted peak-to-peak voltages of waveforms for the spherical voids with various size in
diameter, located at the depth of 25.4 mm from the specimen surface.
(unit: volt)
Diameter
(mm)
Refr.
Refracted
Model
Wave
Angle
1
4
2
0.125
0.25
0.5
MGB
0.00044
0.0017
0.0059
0.0134
0.0237
0.0472
RSI
0.000438
0.0017
0.0058
0.0133
0.0234
0.0466
MGB
0.0003
0.0012
0.0041
0.0091
0.0162
0.032
RSI
0.00030
0.0012
0.004
0.0089
0.0159
0.0318
MGB
0.000129
0.00050
0.0017
0.0038
0.0067
0.0134
RSI
0.000127
0.000490
0.0017
0.0037
0.0066
0.0131
MGB
0.000018
0.000071
0.00024
0.00053
0.00096
0.0019
RSI
0.000018
0.000070
0.00024
0.00052
0.00094
0.0019
MGB
0.000204
0.000718
0.0017
0.003
0.0061
0.0122
RSI
0.000476
0.0017
0.0039
0.0065
0.013
0.0258
MGB
8=30°
0-45o
L-wave,
Planar
Tr.
6=60°
— /j
— j(j
S-wave
Planar
Tr.
o 45°
0.001
0.0036
0.0084
0.0146
0.0291
0.0576
RSI
0.001
0.0036
0.0083
0.0146
0.029
0.0578
MGB
0.000750
0.0026
0.0056
0.0098
0.0195
0.0388
RSI
0.000744
0.0025
0.0056
0.0097
0,0193
0.0386
MGB
0.000179
0.000612
0.0013
0.0022
0.0044
0.0089
RSI
0.000177
0.00060
0.0013
0.0022
0.0043
0.0087
MGB
0.0013
0.005
0.0172
0.0384
0.0693
0.1394
RSI
0.0012
0.0048
0.0165
0.0367
0.0656
0.1315
MGB
1.389E-04
5.414E-04
0.0019
0.0052
0.0083
0.0171
RSI
1.57E-04
0.000612
0.0022
0.0059
0.0091
0.0187
MGB
9.92E-06
3.88E-05
1.382E-04
3.653E-04
5.82 IE-04
0.0012
RSI
l.OOE-05
3.86E-05
0.000138
0.00036
5.857E-04
0.0012
MGB
9.18E-07
3.52E-06
1.2 IE-05
2.779E-05
4.753E-05
9.223E-05
RSI
9.92E-07
3.82E-06
1.3 IE-05
3.24E-05
5.353E-05
1.053E-04
MGB
0.0011
0.0038
0.0087
0.0152
0.0304
0.0609
RSI
0.0025
0.0086
0.0197
0.0347
0.0691
0.1379
MGB
0.004
0.0137
0.0314
0.0564
0.113
0.2261
0.0593
0.1186
0.2376
9=60°
9=75°
9=30°
L-wave,
Focused
Tr.
9=45°
9=60°
9=75°
9=30°
S-wave
Focused
Tr.
9=45°
9=60°
RSI
0.0042
0.0145
0.0329
MGB
0.00079897
0.0027
0.0058
0.0101
0.0199
0.0395
RSI
7.57E-04
0.0026
0.0056
0.0098
0.0195
0.039
MGB
1.3496E-05
4.92E-05
1.363E-04
2.295E-04
4.676E-04
9.245E-04
RSI
1.49E-05
5.37E-05
1.45E-04
2.41E-04
4.8 IE-04
9.56E-04
9=75°
1789
TABLE 3 The predicted peak-to-peak voltages of waveforms for the side-drilled holes with various size in
diameter, located at the depth of 25.4 mm from the specimen surface.
(unit: volt)
Diameter (mm)
Refr.
Refracted
Models
Wave
Angle
1
2
4
0.125
0.25
0.5
MGB
0.0236
0.0456
0.081
0.1122
0.1538
0.2194
RSI
0.0239
0.0466
0.0824
0.1133
0.1551
0.2179
MGB
0.0181
0.0346
0.0614
0.0843
0.1175
0.1695
RSI
0.0181
0.0346
0.0612
0.0838
0.1141
0.1622
MGB
0.0087
0.0166
0.0294
0.0403
0.0561
0.0811
RSI
0.0086
0.0165
0.0289
0.0396
0.054
0.0769
MGB
0.0016
0.0031
0.0055
0.0074
0.0102
0.0146
RSI
0.0016
0.003
0.0053
0.0073
0.0099
0.0141
MGB
0.0099
0.0178
0.0249
0.0334
0.0469
0.0652
0.021
0.0374
0.0506
0.0689
0.0976
0.1372
MGB
0.053
0.0936
0.1282
0.174
0.2458
0.3473
RSI
0.053
0.0933
0.1268
0.1727
0.2435.
0.3429
MGB
0.042
0.0737
0.0984
0.1358
0.1934
0.2772
RSI
0.0419
0.074
0.0993
0.1363
0.1922
0.2719
MGB
0.0116
0.0201
0.0264
0.037
0.0521
0.0753
RSI
0.0115
0.0259
0.0361
0.0512
0.0727
MGB
0.027
0.0527
0.0954
0.1388
0.2021
0.3216
6=30°
L-wave,
Planar
Tr.
0=45°
6=60°
6=75°
0=30°
S-wave,
Planar
Tr.
0=45°
RSI
6=60°
0=75°
L-wave,
Focused
Tr.
0.02
RSI
0.026
0.0509
0.0913
0.1277
0.1691
0.2388
MGB
0.0037
0.0073
0.0135
0.0208
0.0291
0.0456
RSI
0.0044
0.0087
0.0159
0.0235
0.0307
0.0435
MGB
7.12E-04
0.0014
0.0025
0.0037
0.005
0.0072
RSI
8.98E-04
0.0017
0.0031
0.0046
0.0061
0.0086
MGB
9.21E-05
1.79E-04
3.25E-04
4.68E-04
6.35E-04
9.08E-04
RSI
1.16E-04
2.24E-04
4.05E-04
5.80E-04
7.79E-04
0.0011
MGB
0.0173
0.0305
0.0419
0.0561
0.0748
0.092
RSI
0.0606
0.1117
0.1501
0.2085
0.2944
0.4141
MGB
0.0655
0.1171
0.1704
0.2367
0.3439
0.5133
RSI
0.0846
0.1514
0.2102
0.2906
0.4104
0.578
MGB
0.0139
0.0246
0.034
0.0463
0.0665
0.097
RSI
0.0104
0.0185
0.0271
0.0361
0.0513
0.0724
MGB
5.79E-04
0.001
0.0014
0.002
0.0028
0.0041
RSI
8.46E-04
0.0013
0.0018
0.0026
0.0037
0.0052
0=45°
0=60°
— /j
0=30°
S-wave,
Focused
Tr.
0=45°
0=60°
0=75°
1790
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(a)
(b)
FIGURE 4. Predicted time domain waveform from (a) a spherical void and (b) a side-drilled hole with the
diameter of 0.125 mm located at the depth of 25.4 mm from the specimen surface by use of the refracted shear
wave (with the refracted angle 30°) produced by the planar transducer.
SUMMARY
In the present study, we have investigated the radiation issues related to the
ultrasonic benchmark problems of year 2002, by adopting two ultrasonic measurement
models based on 1) the multi-Gaussian beams and 2) the Rayleigh-Sommerfeld integral
with high frequency approximation. The insonifying beam fields and flaw signals
calculated by two models showed very good agreement in most of the cases. However, they
showed significant difference at the near-critical angle (which is corresponding to the cases
with refracted S-waves with the refracted angle of 30°) due to the rapid variation in the
transmission coefficient in that region.
REFERENCES
1. Thompson, R. B., A review paper on the ultrasonic benchmark problems of year 2002, in
Recview of Progress in Quantitative NDE, eds. D. O. Thompson andD. E. Chimenti (AIP,
New York 2003), (in this volume)
2. Thompson, R. B., An ultrasonic benchmark problem: overview and discussion of results,
in Recview of Progress in Quantitative NDE, eds. D. O. Thompson and D. E. Chimenti
(AIP, New York, 2002), Vol 21, pp. 1917-1924
3. Song, S. J., Kim, H. J., Kim, C-H, Prediction of flaw signals of the ultrasonic benchmark
problems by Sungkyunkwan University, in Recview of Progress in Quantitative NDE, eds.
D. O. Thompson andD. E. Chimenti (AIP, New York, 2002), Vol 21, pp. 1941-1948
4. Schmerr, L. W., Lecture Note on Ultrasonic NDE Systems - Models and Measurements,
Sungkyunkwan University, Suwon, Korea, 2000.
5. Schmerr, L. W., Fundamentals of ultrasonic nondestructive evaluation - A Modeling
Approach, Plenum, New York, 1998.
6. Schmerr, L. W, and Sedov, A., Modeling ultrasonic problems for the 2002 benchmark
session, in Recview of Progress in Quantitative NDE, eds. D. O. Thompson and D. E.
Chimenti (AIP, New York, 2003), (in this volume)
7. Rogers, P. H and Van Buren, A. L., An Exact Expression for the Lommel Diffraction
Correction Integral, J. Acoust. Soc. Am., 55, pp. 724-728, 1974
1791