2002 ULTRASONIC BENCHMARK PROBLEM: OVERVIEW AND
DISCUSSION OF RESULTS
R. Bruce Thompson
Center for Nondestructive Evaluation
Departments of Materials Science and Engineering & Aerospace
Engineering and Engineering Mechanics
Iowa State University
Ames, IA USA 50011
ABSTRACT. The predictions of four measurement models are compared for the 2002 Ultrasonic
Benchmark Problem. The problem involves the pulse-echo responses of spherical and cylindrical
cavities. The ultrasonic waves are considered to be generated and detected by either a planar or
spherically focused probe, each of finite diameter and positioned to produce normally incident or
refracted waves (longitudinal or shear) of the desired angle. Among the results are a new expression
for the response of the cylindrical cavity and a quantitative comparison of the various models.
Noteworthy is the differences in the predictions of the beam models, for refracted angles near critical
angles and for focused probes.
INTRODUCTION
In the previous Review of Progress in Quantitative Nondestructive Evaluation,
solutions to an initial ultrasonic benchmark problem were presented by four sets of
investigators, one from Korea, one from Germany and two from the United States [1-4].
That problem involved the prediction of the pulse-echo response of spherical cavities and
circular cracks placed below planar, liquid-solid interface. The measurements were
assumed to be made with both a planar or a spherically focused probe, illuminating the
surface at normal incidence. In solving this problem, it was necessary to use an
ultrasonic beam model (to determine the fields illuminating the flaw) and a scattering
model (to determine the response of the flaw) with a reciprocity relation used to predict
the observed signal. In the course of investigation, four beam models and five scattering
models were examined. A discussion of the philosophy of the benchmark comparison
and an overview of these initial results was presented by the author [5]. Among the
conclusions of that paper was the observation that the models were in good general
agreement with one another with a few cases being observed where further work was
needed to clarify differences that were observed.
A second benchmark problem was considered by the same groups during the
current Review of Progress in Quantitative Nondestructive Evaluation. This problem
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
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introduced two new features. The list of defects considered was extended to consider
cylindrical cavities, or side-drilled holes. In addition, the case of oblique incidence was
considered. An overview of the results obtained by the four groups [6-9] follows.
PROBLEM DEFINITION
The problem is to predict the signals observed, in a pulse-echo measurement,
from spherical and cylindrical cavities in a solid. Both normally and obliquely incident
waves are considered in an immersion geometry, as illustrated by Figure 1 in the papers
of both Schmerr and Sedov [8] and Gray [9]. The waves were excited and detected by
both planar and spherically focused probes.
The material is an aluminum block with a planar surface: (density = 2.71 gm/cc,
longitudinal wave speed = 6374 m/sec, shear wave speed = 3111 m/sec). The block is
placed in a water bath (density = 1 gm/cc, wave speed = 1470 m/sec).
Two sets of six scattering objects are considered. Each member of the first set is
a spherical cavity, centered at a distance of 25.4 mm (1 inch) into the material. They
should have respective diameters of 0.125 mm, 0.25 mm 0.5 mm, 1 mm, 2 mm and 4mm.
Each member of the second set is a cylindrical cavity. They should have the same depths
and diameters as the pores. The diameters have been selected to range from the regime in
which the defect is small with respect to the wavelength to the case in which it is large
with respect to the spot size of the focused beam.
Two transducers are considered. The first is a circular, unfocussed transducer of
12.7 mm (1A inch) diameter and of 5 MHz center frequency: The transducer is assumed
to act as a piston source, with all points on its surface moving with equal amplitude. The
second is a circular, focused transducer of 25.4 mm (1 inch) diameter, of 5 MHz center
frequency: The transducer is assumed to have a curved element with a radius of
curvature of 152.4 mm (6 inches) and also act as a piston source. (The radius of
curvature is equal to the geometrical focal length, i.e. the point of maximum amplitude in
the absence of diffraction.)
The ability of the driving electronics and transducer to generate and detect
ultrasonic waves is defined by a reference signal, which is taken to be the signal observed
during the normal incidence reflection from the front surface of the aluminum block.
This contains information about the bandwidth of the ultrasonic signal and its absolute
level, as influenced by transducer efficiency and driving/receiving electronics. The
proper interpretation of the reference waveform requires a specification of the distance of
the transducer to the part surface. For the unfocused probe, the transducer to frontsurface distance is taken to be 50.8 mm (2 inches). For the focused probe, the transducer
to front-surface distance is taken to be 152.4 mm (6 inches). This is equivalent to placing
the geometrical focal plane on the part surface.
In either case, the reference waveform is taken to have the following, timedomain waveform:
1-cos
fl/f),
forQ<t< ——
otherwise
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Here t is in microseconds and f is the center frequency of the transducer, in MHz.
Given the specification of this reference waveform in a particular reference
experiment, models are used to predict signals from the various scattering objects and
experiments. The specific goals are twofold.
• Predict the amplitude of the insonifying velocity field at the location of the flaw
center, normalized by that of the transducer face, for the 5 MHz center frequency
of the transducer.
• Predict the time domain response (i.e. the radio frequency waveform as a function
of time) of the signal scattered from each combination of scattering objects,
transducers and angles.
The transducers should be positioned such that they excite longitudinal waves
propagating normal to the surface, and refracted waves, both longitudinal and shear,
propagating at angles of 30°, 45°, 60°, 75° with respect to the surface normal. For all
calculations, it is to be assumed that the transducer is positioned such that the water path
along the central ray is 50.8 mm (2 inches) before that ray enters the solid. At normal
incidence, this should place the flaw in the far field of the radiation of the unfocused
probe and near the focal point for the focused probe.
OVERVIEW OF RESULTS
The four groups adopted approaches that differed in the way that both the
radiation from the transducer to into the material and the scattering from the flaw were
treated. Table 1 summarizes the approaches. References to the individual beam radiation
and flaw scattering techniques may be found in the citations in the individual papers. In
each of the contributions, the electromagnetic reciprocity relationship of Auld [10] was
used to predict the flaw response based on the beam radiation and crack scattering
models.
Detailed results may be found by consulting the individual manuscripts. Here,
some comparisons will be made. Note that there are multiple problems being considered,
pulse-echo scattering from spherical and cylindrical cavities as observed with focused
(planar) and unfocused probes, with both normal and refracted incidence (longitudinal
and shear). For each of these cases, insonifying field and flaw response predictions have
been made for various combinations of beam and scattering models. This benchmark
problem allows these models to be compared to one another. In addition, it contains
several important insights into the differences in the scattering from spherical and
cylindrical cavities and of the effects of incident angle on the predictions of various beam
models.
Response of a Cylindrical Cavity
The cylindrical cavity, a model for the side-drilled hole commonly used as a
calibration reflector, was not considered in the previous benchmark problem and
introduces new scattering issues. Because of the axially symmetric geometry of this
reflector, the scattering issues are the same for all angles of incidence. Schmerr and
Sedov [8] make a number of assumptions that greatly facilitate the modeling of the
response of this reflector and provide insight into the underlying wave physics. Starting
with Auld's electromechanical reciprocity relation [10] and describing the incident wave
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TABLE 1. Comparison of the modeling approaches of the four groups.
Cylindrical Cavity Model
Beam
Pore
Radiation
Scattering
K
PSS
Spies
7
Spherical Cavity Model
Pore
Beam
Scattering
Radiation
K
PSS
Song et al
6
MG
Kl
MG
K3
RS
Kl
RS
K3
8
MG
Kl
MG
K2,K3
9
GH
sov
GH
K3
Author
Schmerr and
Sedov
Gray
Ref
Beam Model Legend:
PSS-Point Source Superposition
RS-Rayleigh-Summerfeld Integral (high frequency limit)
MG-Multi-Gaussian (paraxial approximation)
GH-Gaussian-Hermite (paraxial approximation)
Scattering Model Legend:
K-Kirchhoff Approximation
Kl-Kirchhoff Approximation for front surface response of sphere in quasi-plane
wave limit
K2-Kirchoff Approximation for front surface response of cylinder in quasi-plane
wave limit
K3-Kirchoff Approximation for front surface response of cylinder when
variations in velocity over curved surface neglected.
SOV-Separation of Variables
fields in a quasi-plane wave approximation, they develop a series of approximate
measurement models in a systematic way. The first, appropriate to the spherical cavity,
reduces to the Thompson-Gray measurement model [11]. The second and third develop
the analogous results for the case of the cylinder, with the essential issue being how to
treat the fact that the amplitudes of the insonifying fields vary along the length of the
cylinder. Two forms of the Kirchhoff approximation are derived and shown to give very
similar results for the cases studied. In applying these measurement models, the authors
show that the response of the cylindrical cavity is always greater than that of a spherical
cavity of the same radius, a result that is not surprising. They also note that, when
compared to the response of a crack of the same radius, the cylindrical cavity produces a
stronger pulse echo response for smaller radii, but a weaker response for greater radii.
Effects of Oblique Incidence on Ultrasonic Radiation Pattern
Four different forms of beam models were used in the solution of the benchmark
problems. The paraxial approximation greatly speeds the calculation of the beam pattern,
and various forms of this model were used in the papers of Song et al [6], Schmerr and
Sedov [8], and Gray [9]. One way of describing this approximation is to note that the
radiation of a transducer can be described by an angular spectrum of plane waves. In a
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rigorous description of the propagation of a beam through an interface, each plane wave
component must be multiplied by the appropriate interface transmission coefficient. The
beam propagating away from the interface is still described by an angular spectrum of
plane waves, by the modification of their relative amplitudes and phases by the interface
transmission coefficients leads to new structures in the beam pattern. Within the paraxial
approximation, one essentially makes two assumptions. The transmission coefficient is
taken to be the same for all plane wave components. In addition, small angle (with
respect to the central ray) approximations are used to relate the angle of refraction of the
plane wave components to the angle of the incident plane wave component. These
approximations are considered to be good when the range of angles of propagation within
the incident beam is not too large and when one is not too close to a critical angle, where
the interface transmission coefficient can change rapidly with angle. A consequence of
the paraxial approximation is that, within the plane of refraction, the beam profile will be
symmetric about the central ray. Song et al [6] and Schmerr and Sedov [8] use what is
know as the Multi-Gaussian (MG) beam model whereas Gray [9] uses the Gauss-Hermite
(GH) beam model. Each makes use of the paraxial approximation. They differ in that
they decompose the beam into different sets of elementary solutions to the wave
equation.
More accurate, but computationally slower, beam models were also considered.
Song et al [6] utilize a generalized Rayleigh-Sommerfeld (RS) model with high
frequency approximations to treat the propagation through the interface. Spies uses a
closely related point-source-synthesis method (PSS).
Song et al [6] present a systematic study of the differences between the MG and
RS models. Table 1 of their paper presents a comparison of the insonifying velocities at
the center of the flaw, as predicted by these two models. The predictions of the two
models agree very well for most of the cases studied. The two exceptions are for shear
waves when the angle of shear-wave refraction is 30°, close to the longitudinal wave
critical angle, for both planar and focused probe. Figure 1 of that paper shows a
comparison of the beam patterns predicted for the two beam models for the case of the
planar probe and the 30° refracted shear-wave angle. They also noted differences in the
beam pattern for the planar transducer at the 75° refracted shear-wave angle. Although
the velocity insonifying the flaw is similar for the two cases, there are noticeable
differences in the radiation fields off of the axis. The authors note, however, that neither
model may be accurate in this limit because of the rapid variation of the refracted angle
with the angle of incidence. They also note that, in predicting the time domain
waveforms after introducing a scattering model and using Auld's electromechanical
reciprocity relation, they obtained very good agreement on the shape of the waveforms.
Based on the point-source-synthesis technique, Spies makes another interesting
observation [7]. He examines plots of the radiation pattern for various incident angles.
For the case of planar probes, the plots shown for the case of 45° refracted longitudinal
and shear waves indicate that the beam is relatively symmetric in the refracted plane and
that the center of the beam is close to the direction of the central refracted ray. However,
this result is not achieved for the case of focused probes. Considering again the case of
45° refracted longitudinal and shear waves, he observes significant asymmetry. For
example, he finds that, when the intended refracted wave is a longitudinal wave at 45°,
the actual fields are centered as an angle closer to 35°. Thus, to achieve the desired
refracted angle, it would be necessary to make a correction in the nominal refracted
angle.
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Prediction of Ultrasonic Signals
The predicted behavior of the ultrasonic signals follows closely the expectations
based on the behavior of the illuminating fields discussed above. Regarding the angular
dependence of the signal, three of the investigators [6,8,9] predicted that the longitudinal
wave signals will decrease monotonically as the refracted angle increases for all flaw
sizes. Likewise, they predict that, as the refracted angle increases, the shear wave signals
will increase from a value of zero at 0° to a peak at 45°, followed by a decrease at higher
angles. This result is quantified in Figure 1 of [8] and Figures 3-6 of [9], as well as in
Tables 2 and 3 of [6] and Tables 3-6 of [9]. The strength of the peak at 30° depends on
which beam model is used, consistent with the discussion above.
Since Spies [7] compares his signals to those of a reference reflector at the same
angle, his results are not in a form that can be compared to the above predictions. He
does, however, note an interesting point. Application of the Kirchhoff theory in the point
source superposition technique leads to deviations from the expected amplitude behavior
when ka is equal to 5 or less, where a is the radius of the scatter. He notes that this
behavior is consistent with results reported by Gray and Thompson in the study of the
previous benchmark problem [4]. This is because, as the flaw becomes small with
respect to the wavelength, the Kirchhoff approximation, that each point on the flaw
scatters as if it were on a planar interface, breaks down.
Both Song et al [6] and Gray [9] provide detailed tables of the numerical values of
the predicted signals, as was noted in the first paragraph of this sub-section. Based on a
casual comparison of these numbers, the general trends follow one another quite closely.
Differences between the outputs of the two models are generally on the order of 20% or
less (1.6 dB). The reasons for these differences have not been explored.
Relationship to More Approximate Engineering Models
Under additional assumptions (transducer and reflector in the mutual far-field and
reflector smaller than the beam width in at least one dimension), approximate expressions
for the amplitude dynamic curves measured in a contact, pulse-echo mode have been
derived by Werneyer et al. [12]. Spies [7] compares the predictions of his more exact
calculations to those engineering approximations for a number of cases.
CONCLUSIONS
A number of interesting points were revealed by the study of the 2002 Benchmark
Ultrasonic Problem. These can be summarized as follows.
• New approximations were derived for the response of the cylindrical cavity (sidedrilled hole).
• Several beam models were compared. A direct numerical comparison showed the
predictions of the paraxial beam models to be in good agreement with those of the
generalized Rayleigh-Sommerfeld model in the high frequency limit. Differences
were observed when the refracted wave was near a critical angle. The Point
Source Superposition method predicted, for the case of focused probes, that the
actual angle of the beam could deviate from the intended angle (as would be
computed by applying SnelFs Law to the central ray of the beam). Further work
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would appear to be desirable to explore the various cases where the predictions of
the models do not agree.
• Numerical comparisons were made for a number of models. For the case of the
models of Gray [8] and Song et al [6], systematic differences on the order of 20%
(1.6 dB) were observed. Since this is on the order or less than the error in typical
industrial NDE measurements, this difference is of little practical significance.
However, it should be explored to improve our fundamental understanding of the
models and their behavior.
Future benchmark studies, which will involve experimental validation, are planned.
ACKNOWLEDGEMENTS
This work was supported by the National Science Foundation, Industry/University
Cooperative Research Program at the Center for Nondestructive Evaluation, Iowa State
University.
REFERENCES
1.
S.-J. Song, H.-J. Kim, and C.-H. Kim, "Prediction of Flaw Signals of the Ultrasonic
Benchmark Problem by Sungkyunkwan University", in Review of Progress in
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Eds. (AIP, New York, 2002), pp. 1941-1948.
2.
M. Spies, "Simulating a Standard Type Problem of Ultrasonic Testing-A
Contribution to the Comparison of Models", in Review of Progress in Quantitative
Nondestructive Evaluation 21B, D. O. Thompson and D. E. Chimenti, Eds. (AIP,
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3.
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4.
T. A. Gray and R. B. Thompson, "Solution of an Ultrasonic Benchmark Problem
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5.
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Results", in Review of Progress in Quantitative Nondestructive Evaluation 21B, D.
O. Thompson and D. E. Chimenti, Eds. (AIP, New York, 2002), pp. 1917-1924.
6.
S.-J. Song, J. S. Park, and H.-J. Kim, "Prediction of the Insonifying Fields and Flaw
Signals of the 2002 Ultrasonic Benchmark Problem", these proceedings
7.
M. Spies, "Prediction of the Transient Flaw Signals of the Ultrasonic Benchmark
Problem", these proceedings
8. L. W. Schmerr and A. Sedov, "Modeling Ultrasonic Problems for the 2002
Benchmark Session", these proceedings
9.
T. A. Gray, "Ultrasonic Benchmark Problem: Application of a Paraxial Model to
Side-Drilled Holes and Oblique Incidence", these proceedings.
10. B. A. Auld, "General Electromechanical Reciprocity Relations Applied to the
Calculations of Elastic Wave Scattering Coefficients," Wave Motion 1, 3 (1979).
11. R. B. Thompson and T. A. Gray, "A Model Relating Ultrasonic Scattering
Measurements Through Liquid-Solid Interfaces to Unbounded Medium Scattering
Amplitudes", /. Acoust. Soc. Am, 74, 1279-1290 (1983).
12. R. Werneyer, F. Walte, and W. Muller., Technical Report No. 78052-TW (in
German), IZFP Saarbrucken, Germany (1978).
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