SEMI ANALYTICAL FINITE ELEMENT ANALYSIS
FOR ULTRASONIC FOCUSING IN A PIPE
Takahiro Hayashi1, Koichiro Kawashima1, Zongqi Sun2 and Joseph L. Rose2
Department of Mechanical Engineering, Nagoya Institute of Technology,
Gokiso Showa Nagoya, 466-8555, Japan
Department of Engineering Science and Mechanics, The Pennsylvania State University
412 Earth & Engineering Building, University Park, Pennsylvania 16802
ABSTRACT. Guided wave focusing has been developed as a promising technique for defect
detection in pipeworks, where non-axisymmetric flexural modes are tuned so that ultrasonic energy
can be focused at a target point in a pipe. If a defect is located at the target point, large amplitude
reflected waves can be observed. In this study, the focusing phenomenon is analyzed using a
semi-analytical finite element method where a region of a pipe is divided in the thickness direction
into the cylindrical subdivisions and is analytically treated in the circumferential and longitudinal
directions. Visualization of the calculation results reveals that focusing occurs gradually in the
vicinity of the target point.
INTRODUCTION
Guided waves can propagate over long distances along bar or plate-like structures
with great potential for the rapid NDE of pipeworks. Recently, guided wave focusing has
been developed as a promising technique for defect detection in pipeworks[1-9]. Guided
wave natural focusing via non-axisymmetric was introduced in [4]. Controlled focusing
via phased array analysis was introduced in [5,9]. The basic idea of the controlled
focusing technique is as follows. First, a displacement distribution in the circumferential
direction (circumferential profile) is predicted theoretically for partial loading by a single
transducer. Circumferential profiles for multiple transducers can be expressed by a
superposition of this single transducer profile. Then, by tuning amplitudes and time
delays of each channel independently, waveforms are enhanced at some predetermined
target in a pipe. Since ultrasonic energy in converged on the target, a large reflected echo
can be detected only when a defect is located at the target. Changing time delays and
amplitudes properly enables us to scan target points over the entire pipe. Analyses of
reflected waves then provide useful information on defects in a pipe, such as location and
size.
The purpose of this study is to reveal this focusing phenomenon by way of
simulation and visualization of guided wave propagation.
Finite element (FE) and boundary element (BE) methods are generally used for
the calculation of ultrasonic wave propagation. Standard FEM and BEM, however, have
serious problems of calculation time and memory to apply to such large structures as
plates and pipe. In this study, we use a computational efficient semi-analytical finite
element technique (SAFE) in which wave propagation in the longitudinal direction is
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/$20.00
250
analytically treated and only the cross-section of a pipe is sub-divided [10-16].
SEMI ANALYTICAL FINITE ELEMENT METHOD
In Lamb wave calculations by the semi-analytical finite element method
assuming plain strain, a cross-section of a plate is divided in the thickness direction into
layered elements, and waves in the propagating direction z are described by the
orthogonal function exp(z^) where £is the wavenumber of the Lamb wave [17]. The
mm eigenvalue %m of the eigensystem derived here denotes the wavenumber of the mth
resonance mode. In the case of three dimensional guided wave calculations in a hollow
cylinder, the two dimensional cross-section is sub-discritized and waves in the
longitudinal direction z are described by the orthogonal function exp(/^). Similarly,
wavenumbers are obtained as the eigenvalues %m of the eigensystem, and then dispersion
curves can be depicted.
For straight pipes, wave propagation in the circumferential direction can be
described by the orthogonal function exp(z>?#), too. Thus, a cross-section of a pipe is
discretized into cylindrical elements. Since the index n stands for the nth circumferential
harmonic (nth family), guided waves of the nth family can be analyzed selectively.
Calculation speed is very fast due to its resulting one-dimensional calculation.
FOCUSING ALGORITHM
Exciting local dynamic stresses in a pipe, received waveforms vary in the
longitudinal direction and also in the circumferential direction. When multiple transducers
are mounted in the circumferential direction, waveforms at a certain location are
described as a superposition of these various waveforms. Tuning time delays and
amplitudes, therefore, provides focusing by the following algorithm.
Suppose that N transmitters and N receivers are located in the circumferential
direction and numbered 1,2...N from the top of the pipe as shown in Fig.l. Then the
complex amplitude of a harmonic wave at the frequency a> traveled from the fth
transmitter to the yth receiver is defined as Hy. Hy is predetermined by experiments or
calculations, containing the information of amplitude and phase. Controlling the output
level (amplitude) At and time delay tt of the /th transmitter, the received signal is
described as afHy9 where
at = Afexp(-ia)ti).
(1)
Tuning all amplitudes and time delays from 1 to N, signals at the yth receiver #7 can be
described as a superposition of a/^y as
1=1
This is represented in the matrix form as
q = Ha
Here letting the received signals be
q = [l 0 0 ••• Of
then we have a complex amplitude
a =H'q.
(4)
(5)
251
Pulser
Receiver
1/2+1
FIGURE 1 Focusing algorithm.
Complex amplitude components a/ are represented by amplitudes and time delays as
shown in Eq.(l).
From the predefined target amplitude q and the matrix H determined by
numerical calculations, necessary amplitudes and time delays for each element are
calculated.
VISUALIZATION OF THE FOCUSING PHENOMENA
In the case of eight channels, time delays and amplitudes for focusing at 1m, 2m,
4m, 6m in a straight pipe are shown in Table 1, which are given by calculating
circumferential profiles from a single transmitter and substituting these values into the
focusing algorithm written above. Fig. 2 shows circumferential profiles (maximum
amplitude distributions in the circumferential direction) at the target when the tone burst
signals with time delays and amplitudes given in Table 1 are emitted from the eight
transmitters. All of them show good focusing at the target (Top, 0°). Fig. 3 shows the
visualization results by using time delays and amplitudes for focusing at 6m. A significant
peak can be seen at the focal point. Fig.4 shows the zoomed results at the focal point.
These figures reveal that the focusing phenomena do not occur abruptly at the focal point.
Complicated waveforms can be seen in the far field from the focal point, while in the
vicinity of the focal point the wave gradually increases as it approaches the focal point
and decreases after the focal point.
TABLE 1 Time delays and amplitudes for focusing at various points.
2m
Focal
1m
4m
6m
0°
0°
0°
0°
Time Amplitude Time Amplitude Time Amplitude Time Amplitude
delay
delay
delay
delay
Ch. \ tN
[fis]
[N
Qw]
0.0
1.00
2.6
0.0
1.00
1(0°)
0.30
1.00
3.9
0.54
6.1
8.9
19.7
2(45°)
0.46
0.53
0.27
2.0
29.9
0.42
2.9
0.99
0.0
3(90°)
0.33
0.25
0.0
8.2
0.54
4(135°) 1.7
0.23
14.0
0.62
0.39
21.7
0.84
0.95
24.2
5(180°) 5.0
8.5
0.99
11.8
1.00
8.2
0.54
6(225°) 1.7
0.23
0.62
14.0
0.39
21.7
0.99
0.0
29.9
0.42
7(270°) 2.9
0.25
0.0
0.33
8.9
19.7
0.54
8(315°) 6.1
0.46
0.53
0.27
2.0
point
252
315'
315°
270°
270°
225°
0°TOP
0° TOP
1.0
0.8
0.6
0.4
0.2
0.0
Calculation
315'
315°
45°
90°
270°
270°
135°
225°
1m
315°
270°
270°
225'
225°
Calculation
Calculation
45°
90°
135°
180° BOTTOM
180° BOTTOM
2m
180° BOTTOM
0° TOP
1.0
0.8
0.6
0.4
0.2
0.0
0°TOP
0° TOP
1.0
0.8
0.6
0.4
0.2
0.0
2m
315'
315°
45°
90°
90°
270°
270°
225'
135°
225°
180° BOTTOM
0° TOP
1.0
0.8
0.6
0.4
0.2
0.0
Calculation
45°
45°
90°
135°
135°
180° BOTTOM
4m
180° BOTTOM
6m
180° BOTTOM
FIGURE 2 Circumferential4m
profiles at various axial distances shown
6m efficient focusing at the selected
FIGURE
2 Circumferential profiles at various axial distances shown efficient focusing at the selected
target
point.
target point.
FIGURE
6m6m
focal
point.
FIGURE33 Focusing
Focusingresults
resultsat ata predetermined
a predetermined
focal
point.
253
focal poin1r^6.0m
focal point 6.0m
Time=1220.
visualization region
5.5m-6.5m
ifocal pointKi.Qm
focal point 6.0m
Time=1300.
visualization region
5.5m-6.5m
ifocal point^S.Om
focal point 6.0m
visualization region
Time-1380.
5.5m-6.5ni
FIGURE 4
Zoomed images at the focal point.
FIGURE 4 Zoomed
images at the focal point.
CONCLUSIONS
CONCLUSIONS
Guided wave propagation in a pipe was studied with a semi-analytical finite
Guided
propagation
in a ofpipe
was isstudied
semi-analytical
finite
element
methodwave
where
a cross-section
a pipe
dividedwith
into acylindrical
elements.
element
method
where
a
cross-section
of
a
pipe
is
divided
into
cylindrical
elements.
Using this calculation technique, time delays and amplitudes for focusing at a given point
Using
calculation
technique,
time delays
and amplitudes
for focusing
at a giventhe
point
were this
obtained.
Moreover,
these focusing
phenomena
were observed
by visualizing
were
obtained.
Moreover,
thesefield
focusing
calculation
results.
In the near
of the phenomena
target point,were
waveobserved
increasedby
andvisualizing
converged the
calculation
In the near
field and
of the
target
point,and
wave
increased
and converged
gradually asresults.
it approached
the target,
then
decreased
spread,
while complicated
gradually
as
it
approached
the
target,
and
then
decreased
and
spread,
while
complicated
wave motions were seen in the far field.
wave motions were seen in the far field.
ACKNOWLEDGEMENTS
ACKNOWLEDGEMENTS
We thank Plant Integrity Ltd., Cambridge, England for their partial support of
this work,
special
thanks
to Peter
Mudge
for technical
discussion.
Wewith
thank
Plant
Integrity
Ltd.,
Cambridge,
England
for their partial support of
this work, with special thanks to Peter Mudge for technical discussion.
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