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Quantitative accuracy of the mass-spring
lattice model in simulating ultrasonic waves
Cite as: AIP Conference Proceedings 615, 152 (2002);
Published Online: 14 May 2002
Hyunjune Yim, and Choon-Jae Lee
AIP Conference Proceedings 615, 152 (2002);
© 2002 American Institute of Physics.
615, 152
Hyunjune Yim and Choon-Jae Lee
Dept. of Mechanical Engineering, Hong-Ik University, 72-1 Sangsu Dong, Mapo Ku,
Seoul, 121-791, Korea
Abstract. In order to assess the quantitative accuracy of the mass-spring lattice model (MSLM)
in simulating various ultrasonic wave phenomena, numerical results for several fundamental
problems, such as reflection, refraction, and crack-tip diffraction, have been obtained and
compared with the analytical results. As a result, it has been found that the MSLM provides
reasonably accurate numerical results, particularly when its relatively short time of computation is
The mass-spring lattice model (MSLM), which models an elastic solid by mass
points and springs interconnecting them, has been studied actively by the group of present
authors [1-3]. The model has demonstrated its capability of predicting all secondarily
generated waves (due to reflection, diffraction, etc.) exactly where they are expected to
exist by the wave physics [1-3]. The propagation directions and amplitudes of the waves,
however, have not been examined closely in terms of their quantitative values.
The objective of this paper, is to investigate the quantitative accuracy of the MSLM
in three fundamental wave phenomena: (1) reflection at a free boundary, (2) refraction at a
material interface, and (3) diffraction at a crack tip. To this end, the numerical results
obtained by using the MSLM are compared with the analytical results provided by the
wave physics.
Reflections of plane incident waves at free boundaries are examined with respect to
the angle of reflection and the reflected wave amplitudes. The numerical model for a free
boundary used in this study is the method used in the conventional MSLM; that is, by
setting to zero the spring constants of all springs outside the free boundary. The idea
underlying this modeling method is that the springs with zero spring constants will exert
no force on the connected mass points and thus approximately satisfies the condition of no
traction there [1].
As an example, Fig. l(b) shows a numerically computed wavefield, generated by
reflection of an incident S wave from a free boundary as schematically shown in Fig. l(a).
The properties of the medium are assumed to be those of steel in this and all the following
numerical results unless otherwise specified. Fig. l(b) shows only a portion of the twodimensional space in Fig. l(a) as delineated by dotted lines. In Fig. l(b), there are two
incident S waves propagating in the opposite directions from the line source, and they
reflect from the vertical and horizontal free boundaries, respectively. As expected by the
wave physics [4], there is no mode-converted P reflected wave as the angle of incidence
exceeds steel's critical angle. Also, edge waves from an end of the line source may be
observed in Fig. l(b). The angles of reflection in all cases numerically computed in this
study, including those in Fig. 1, have agreed well with the analytical results [4].
CP615, Review of Quantitative Nondestructive Evaluation Vol. 21, ed, by D, O, Thompson and D, E. Chimenti
© 2002 American Institute of Physics 0-7354-006l-X/Q2/$ 19,00
The amplitudes of the reflected waves in the numerical results have also been
compared with analytical values [4]. Table 1 shows comparisons of some typical
examples for incident P waves of various angles of incidence. In the numerical results, the
percentage errors are given in parentheses. It may be noted from Table 1 that the
numerical reflection coefficients are quite accurate in some cases, but show significant
errors in other cases. Therefore, it may be concluded that the conventional MSLM
modeling method for free boundaries does not provide very accurate results in general.
Yet, when considering the short time of computation, the accuracy of the MSLM may still
be regarded as good.
Refraction of plane incident waves at an interface between two different materials
(steel and concrete) has been simulated, and the numerical angles of refraction and the
numerical refraction coefficients have been compared with the analytical results. Steel and
concrete have been selected since considerable differences in material properties are
necessary to clearly observe the refraction phenomena. The material interface is modeled
by deriving finite difference equations for the traction boundary conditions requiring
continuity of tractions across the material interface. The displacement continuity across
the interface is automatically satisfied.
As a result, it has been found that the angles of refraction measured from the
numerical results agree almost exactly with the analytical results [4]. The analytical and
numerical refraction coefficients, defined as the ratio of displacement amplitudes of the
refracted and incident waves, are tabulated in Table 2 in the same manner as in Table 1. It
may be seen that the refraction coefficients measured from the numerical results are much
more accurate (i.e. less than 4 % of error) than the numerical reflection coefficients at free
boundaries, tabulated in Table 1. Therefore, no significant modification may be necessary
to the modeling method, described above, of MSLM for material interfaces between two
different media.
FIGURE 1. Reflection of shear incident waves at free boundaries.
Analytical and numerical reflection coefficients for incident P waves.
Angle of incidence
Reflected P wave
Reflected S wave
Reflected P wave
Reflected S wave
(11 %)
(17 %)
(19 %)
TABLE 2. Analytical and numerical refraction coefficients at steel-concrete interface
for incident P waves.
Angle of incidence
Refracted P wave
Refracted S wave
Refracted P wave
Refracted S wave
(0.5 %)
(4.0 %)
(0.2 %)
(3.4 %)
(0.8 %)
(2.5 %)
Now, as the final problem considered in this work, diffraction of a plane incident
wave at a crack tip is considered. Fig. 2 shows a schematic of the problem. The
distribution-in-angle of the displacement amplitudes of numerically computed diffracted P
waves have been compared with the analytical results. The analytical results considered in
this study are the far-field diffraction coefficients, derived by Ogilvy and Temple [5]
based on the geometrical theory of diffraction. The analysis [5] provides the distributionin-angle of the diffracted wave displacement amplitude for harmonic incident waves.
In the numerical analysis, the crack has been modeled in the conventional MSLM
method; that is, by setting to zero spring constants of all springs across the crack faces.
Since the incident wave in the numerical analysis is not harmonic but one-cycle sinusoidal,
a particular calculation scheme has been devised to evaluate the diffracted displacement
amplitudes which can be compared directly with the analytical results. This scheme is
basically to compute the maximum value of the diffracted P wave from the time history of
displacement at observation points along a circle centered at the crack tip, and having a
sufficiently large radius to guarantee the far field. In this work, the radial distance of
observation points in Fig. 2 has been set to 20 times the longitudinal wavelength.
Comparisons between the analytical and numerical results have been made for
various angles of incidence (i.e. for various values of J3 in Fig. 2) and for both incident
P and S waves. Only an example is shown in this paper. Fig. 3(a) shows the analytical
results [5] for amplitudes of radial displacement ur in the far diffracted P wave for an
incident P wave having a propagation direction of J3 = 90° (see Fig. 2 for the definition
of /?), and Fig. 3(b) shows the corresponding numerical results. The huge difference in
numerical values in Figs. 3(a) and 3(b) is not meaningful because the dependence of
analytical and numerical results on the radial distance has not been taken into account
consistently. It may easily be seen from Fig. 3 that the numerical results show a good
agreement with the analytical ones in that the nonzero (or considerable) displacements of
the diffracted P wave
Observation point
Incident plape
Crack ti
Geometrical configuration of crack-tip diffraction problem.
0.0002 -
0.0001 0.2-
0.0000 -
0.20.0001 -
FIGURE 3. Distribution-in-angle of amplitude of radial displacement in far diffracted P wave when a
crack is subjected to an incident P plane wave propagating in the direction of ]8 = 90°.
0.00005 0.00004
0.00003 0.00002
0.00000 0.0000'
0.00002 0.00003
0.00006 J
FIGURE 4. Distribution-in-angle of amplitude of tangential displacement in far diffracted P wave when a
crack is subjected to an incident P plane wave propagating in the direction of j3 = 90°.
occur only in the vicinity of vertical directions, i.e. in the vicinity of 6 = ±90° (see Fig. 2
for the definition of 0). Yet, the width of the two "lobes" in the numerical results is much
greater than that in the analytical results. (Note that the analytical distribution-in-angle
shown in Fig. 3 (a) has been truncated to show the details in the neighborhood of the
coordinates' origin.)
Fig. 4 shows a comparison for the tangential displacement ue in the same problem
as in Fig. 3. As in Fig. 3, it may be seen from Fig. 4 that the tangential displacement of
significant amplitude in both analytical and numerical results occurs in the same (vertical)
directions. Also, as in Fig. 3, the widths of the "lobes" in the analytical and numerical
results in Fig. 4 are not in good agreement. In addition, it may be seen by comparing Figs.
3 and 4 that the tangential displacement (Fig. 4) in the diffracted P wave is much smaller
than the radial displacement (Fig. 3). This is because diffracted P waves are cylindrically
propagating waves from the crack tip, and they accompany particle displacements in the
same direction (that is, the radial direction) as the propagation direction, as its name (that
is, longitudinal or P) indicates. This trend is apparent in both the analytical and numerical
From Figs. 3 and 4, and from all other results not shown here, it has been concluded
that, with respect to the crack-tip diffraction phenomena, the conventional MSLM
provides numerical results of reasonable qualitative accuracy.
The quantitative accuracy of the MSLM has been investigated in this work for three
fundamental problems: reflection at free boundaries, refraction at material interface
between two different media, and diffraction at a crack tip. As a result of examining the
numerical results and comparing them with the corresponding analytical results, it has
been shown that the MSLM does not always provide highly accurate results but, in most
cases considered in this work, yields reasonably accurate results. When considering the
simplicity and short time of computation, however, there will be a broad range of
application of the MSLM.
Efforts will be made to improve the quantitative accuracy of the MSLM by simple
means, that is, without significantly complicating the model or without significantly
lengthening its computation time. Also, transducer models will be developed to complete
a simulator for ultrasonic testing.
This work was supported by the Korea Ministry of Science and Technology as a part
of the Nuclear R&D Program, and by the Safety and Structural Integrity Research Center
sponsored by the Korea Science and Engineering Foundation.
1. Yim, H., and Sohn, Y, IEEE Trans. UFFC 47, 549-558 (2000).
2. Yim, H., and Choi, Y, Mat. Eval 58, 889-896 (2000).
3. Yim, H., and Choi, Y, in Review of Progress in QNDE, Vol. 20, eds. D. O. Thompson
and D. E. Chimenti (American Institute of Physics, New York, 2001), p. 59.
4. Graff, K. E, Wave Motion in Elastic Solids, Dover, New York, 1991, pp. 311-393.
5. Ogilvy, J. A. and Temple, J. A. G, Ultrasonics 21, 259-269 (1983).
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