FAST TECHNIQUES FOR CALCULATING DISPERSION
RELATIONS OF CIRCUMFERENTIAL WAVES IN ANNULAR
STRUCTURES
J. Fong1, MJ.S. Lowe1, D. Gridin2, R.V. Craster2
Department of Mechanical Engineering, Imperial College, London, SW7 2BX, UK
Department of Mathematics, Imperial College, London, SW7 2BX, UK
ABSTRACT. Dispersion curves are essential for analyzing and understanding the physical behaviors
of guided waves propagating in structures. For circumferential waves in annular structures, the
calculation of dispersion curves can be very time consuming. Furthermore, due to the nature of the
mathematical functions involved in the curved structure geometry, the solutions can become unstable
at high frequencies. In this paper three asymptotic approximation methods are introduced, which
resolve the instability problem and speed up the calculations. Comparison of dispersion relations
evaluated from the exact and asymptotic approximation methods are presented, showing errors to be
typically less than 0.1%, except for the very low frequency region. Results obtained using a two
dimensional finite element technique are also presented for comparison.
INTRODUCTION
Inspection techniques to find defects in annular structures using guided waves are
potentially very efficient in covering a large area rapidly [1,2]. These techniques utilize
Lamb type waves propagating in either the axial or circumferential direction. The physical
behavior of these guided waves can be understood and interpreted with dispersion curves,
which have been thoroughly studied by previous investigators [3,4].
In general, the computation of dispersion relations of the frequency and the
wavenumber requires finding the roots of a complicated determinant [4,6], which is solved
numerically. This is implemented, for example, in the DISPERSE program developed at
Imperial College [3,5] which treats both flat plate and cylindrical geometries. However, in
the case of waves travelling in a cylinder, and at high frequencies, many of the terms within
the determinant become either exponentially very large or very small causing a poor
conditioning of the determinant. This is not due to the well known "large f-d problem",
which may be eliminated by appropriate choice of the matrix formulation [5]. Instead, it is
caused by the nature of the Bessel functions contained in the elements of the eigen-problem
matrix when the order and the argument of the functions are both large. Therefore tracing
dispersion curves at high frequencies can be numerically unstable and time consuming. It
is the authors' intention to address these problems by employing various asymptotic
approximation methods. As an example, circumferential waves are chosen for our
investigation in this paper but similar approximation approaches can be adopted for axial
cylindrical waves.
This paper is divided into two main sections. First, it starts with a brief introduction
to the formulation of the dispersion relations for the circumferential Lamb waves using
exact and various asymptotic approximation methods (refer to Gridin et al [6] for the full
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
213
details of the theory). Then, comparison of each asymptotic method and exact solution will
be presented. In the second section, an alternative method to extract dispersion relations
for annular structures using a two-dimensional axially symmetry finite element (FE)
method [7] is also being employed, providing further results for comparison.
ANALYTICAL METHODS
Exact Dispersion Relations of Circumferential Lamb Waves in an Annulus
We start by considering the Euler equation of motion in the polar coordinates 9 and
r as shown in figure 1 . Since the problem is two-dimensional, it is convenient to introduce
the potentials cp and \|/ to represent the displacement fields, which reduces the equations of
motions to two uncoupled Helmholtz equations. We then assume the modal solutions in
the following form:
<p(r90) = 0(ry»V(r,0) = V(r)etv°
(1)
where v is the angular wavenumber
The modal equations are substituted into the uncoupled equations of motion to form the
Bessel equations for 4> and *¥ where the general solutions are in terms of the Bessel
function of the first kind (J) and second kind (Y):
* = alJv(kLr)+a2Yv(kLr\
¥ = <*3Jv(kTr)+aJv(kTr)
(2)
where kL and kT are the bulk longitudinal and shear wavenumbers (^CO/CL, A/r^co/Ci in
which CL and CT are the longitudinal and shear bulk velocities), and a} 2 3 4 correspond to
the wave field amplitudes
For the case of an annular structure in vacuum, stress free conditions at the inner
and outer surfaces are applied (i.e. (Tee and <sre - 0 at r = rj and r^). The outcome is that the
dispersion relation can be described by an eigen-problem matrix, £>, relating the frequency
(aj) and the angular wavenumber (v). The elements in the eigen-problem matrix, D, are
composed of a combination of 4 pairs of Bessel functions and their derivatives, namely
Jv[A;rz -r 1 2 ] and ^[A: r i -r 1 2 J. For each angular wavenumber (v), there is an infinite
number of frequencies (<2^) that satisfy the eigen-problem corresponding to the mth order
propagating mode.
D(v,a>m).a=0
(3)
where a is column vector of the wave field amplitudes
In the authors' implementation, the dispersion curves are traced numerically first by
specifying an angular wavenumber value (v), which is substituted into Z), where the roots
are extracted by evaluating the determinant of £>, the process is then repeated for a 2nd v
value nearby. The obtained v-CQm pairs are then used to linearly extrapolate an estimated
angular wavenumber value in a small frequency step, 8(O. Convergence is obtained by
iterating over a range of angular wavenumbers as demonstrated in Figure 2. The estimation
points switch to a quadratic extrapolation once a third point is available and the tracing
process is repeated until the assigned limit is reached (full details of the technique are given
inRef[5]).
214
search range
extrapolation
staring points (vj
Angular wavenumber (v)
FIGURE 1. The geometry of the circumferential
waves problem.
FIGURE 2. Typical dispersion curves tracing
procedure.
Asymptotic Approximation Methods
As stated in the introduction, the exact solution is computationally expensive and
prone to instability. Therefore rapid convergent methods based on the asymptotic
approximation are introduced in this paper. Three main asymptotic methods are presented
here. Full details are given in [6].
In the first method (method 1) the Bessel functions and their derivatives in the
eigen-problem matrix, £>, are replaced with the leading terms and the first derivatives of the
uniform asymptotic expansions, expressed in terms of Airy functions (refer to [8]). This
method does not resolve the instability problem at high frequencies since the nature of the
Airy functions is similar to the Bessel functions. However, it significantly reduces the
complication to calculate the Bessel functions, thus the time required for the calculations.
In the second method (method 2), it is assumed that the bulk longitudinal
wavelength is much smaller than the inner radius (i.e. kLr\ » 1). This means that all the
arguments of the Bessel functions in matrix D are large and any higher asymptotic order in
the matrix elements may be neglected. Furthermore, the calculation of the Bessel functions
and their derivatives adopts the leading terms and their first derivatives of a conditional
asymptotic expansions for large-arguments. The approximation divides the calculation into
3 regimes depending on the order and the argument of the Bessel functions in the following
relationship:
Regime I: x < z
Regime II: x « z
Regime III: x > z
(4)
where x and z are the order and the argument of the Bessel functions respectively
By choosing the appropriate regime to represent the Bessel functions in the
dispersion relation, the dispersion curves can be divided into 9 regions as demonstrated in
figure 3. With further mathematical manipulations, the dispersion relation in each region
may be described by a single equation, expressed in basic trigonometric functions.
The physical behavior of the partial waves corresponding to each region was
analyzed as shown in figure 4 where the length of the arrows indicates the magnitude of ki
and kT at radius r/ and r2. Due to the geometry of the annular structures, the wave front of
these partial waves rotates around the central axis. Thus there are infinite numbers of sets
of longitudinal-shear partial waves pairs along the inner and outer surfaces for this
problem. When tracing across a region boundary, one of the partial waves changes from
homogeneous to inhomogeneous (i.e. from real to imaginary wavenumber) or vice versa.
This explains why an appropriate regime is required to represent the Bessel function
correctly in each region. Nonetheless this method eliminates the need to assemble the
215
FIGURE 3. Typical example of regions defined in method 2 for the dispersion curves
matrix D for each iteration, and the instability problem at high frequency, by expressing the
solutions with trigonometric functions.
In general there are many terms in the dispersion relation equation of each region,
some are exponentially large but some are exponentially small. In the last method (method
3), we can take advantage of this property to further increase the speed of the iteration by
neglecting the exponentially small terms in the equations, leaving the average and
exponentially large terms. Since only the roots of the equations are needed, the accuracy of
the solutions should not be affected. In figure 5, a dispersion relation function is plotted
using both method 2 and method 3. It can be observed that a significant reduction in
amplitude and removal of the exponential growth property of the functions were achieved
using method 3, which makes iteration of the roots quicker and more stable.
NUMERICAL RESULTS
As an example, numerical results of the dispersion curves for the circumferential
Lamb waves in an annular structure have been obtained using the exact solution and the
three asymptotic approximation solutions, and are shown in Figure 6. The annular
structure is steel (Q = 5960m/s, CT - 3260m/s and p - 7932kg/m3) and has a geometry of
inner, r/, and outer, r2, radius of 0.02 and 0.021 respectively. In figure 6a and 6b,
corresponding to the exact and uniform asymptotic approximation method (method 1),
there are no solutions found at the large-frequency region, this is due to the instability of
the eigen-matrix D as discussed in the previous section. On top of that, some obvious
inconsistencies at the very low frequencies are observed in all asymptotic methods, which
is apparently caused by the large-arguments assumption in the asymptotic expansions for
the Bessel functions that do not cover the very low frequencies. Despite this, only few
modes exist at these low frequencies and they are often of little practical value, therefore no
amendment is necessary.
By simple observation, there are few other differences between all 4 curves shown
in figure 6. To investigate the actual difference of each method, a more specified analysis
was conducted to measure the relative error for the dispersion curves. The relative error is
defined as the percentage difference in angular wavenumber between the approximation
and the exact solution.
216
(a)
Homogeneous partial wave
Inhomogeneous partia! wave
;:• Longitudinal partia! wave ——» Shear partial wave
FIGURE 4. Partial waves at the boundaries of an annular structure corresponding to different locations in
figure 3. (a) Region 1. (b) v= 0}j. (c) Region 3. (d) Region 5. (e) Region 7. (f) Region 9.
dispersion relation
function , f(v,co) 0
-8
""rr^±lj
_
.. _
/
X
_.-— ——..^
_
DO
750
/-x,
\
J \
/
1
\l
.x^/ ------\ /---------4/
;
8 DO
Aiguiar woven i
jmber
8 50
(\ )
9(
_
_^.
6
dispersion relation o
function , fCvxo)
-6
_-1O
— -^\, |i '
2
/ j.
i
/
Jt
700
750
—\h
il-
800
Ang U ar wa venu m be r (v)
FIGURE 5. Dispersion relation function of region 5 at frequency of 25MHz for an annular structure (r/ =
0.02 and r2 = 0.021) using method 3 (top) and method 2 (bottom) with a "zoom-in" for low angular
wavenumber values.
no roots are found in these regions
2000!
2000
FIGURE 6. First 45 modes of dispersion curves for the circumferential Lamb waves in an annular structure
with a frequency upper limit of 50MHz (ri = 0.02m and r2 = 0.021m). (a) Exact solution, (b) Uniform
asymptotic approximation (Method 1). (c) Conditional asymptotic approximation (Method 2). (d) Simplified
Conditional asymptotic approximation (Method 3).
217
The relative error measures the percentage error in angular wavenumber of the mth
mode at each frequency step, 8co, thus a 3-D plot can be mapped over the area of the
dispersion curves for each method, as demonstrated in figure 7. However, the calculation
is limited by the fact that the exact solution calculations collapse at high frequencies. The
box in the projected dispersion curves of figure 7a indicates the area where the relative
errors are possible for calculation.
Simplification of the dispersion relations retains reasonable accuracy in all 3
methods, with typical error of less than 0.1%, other than at very low frequency region and
near the boundaries of the approximation regions. Generally accuracy increases with
increasing frequency apart from the boundaries. The sharp peaks at very low frequencies
that have been truncated in figure 7 coincide with the observed inconsistency in figure 6,
which are due to the large argument assumption for the Bessel functions. Figure 8 shows
the time taken to trace the first 45 modes of the dispersion curves in figure 6 for various
methods using Matlab™ on a P4 1.6GHz computer, revealing the superior performance of
the asymptotic schemes.
40 35
(a)
30
25
?0
fre
^^(MH2)
FIGURE 7. Measurement of the relative error in angular wavenumber, v, of dispersion curves for
circumferential waves in an annular structure (r2 = 0.02, r2 = 0.021). (a) Relative error plot for method 1,
projecting over the exact dispersion curves area, (b) Relative error plot for method 2. (c) Relative error plot
for method 3.
Solution
1
Method Met'hod "Finite
2
3
Element
FIGURE 8. Time for tracing the dispersion curves as
FIGURE 9. Schematic diagram of an 2D cyclic
in figure 6 using exact, various asymptotic methods and axial symmetry model for extraction of dispersion
finite element method (with element size of 0.5%).
curves.
218
o
increasing n
Radial d isplacement
—
1400
Circumferential displacement
FIGURE 10. Dispersion curves extracted using finite element method with mode shapes at certain points.
FINITE ELEMENT ANALYSIS
Theory
The use of the standard commercial Finite Element (FE) programs to extract
dispersion curves for guided waves has been previously reported by Wilcox et al [7] and so
only a brief description is given here. 2D axisymmetric harmonic elements are employed
to represent the cross section of the structure (see figure 9). The wall thickness, d, is
divided into a desired number of elements, and a symmetry boundary in the z direction is
applied to the top and the bottom of the elements. Now, assuming the waves propagate in
the 9 direction, the circumferential harmonic order, n, of the FE model is identical to the
angular wavenumber, v, in the analytical formulation, except that here n must be integer.
The natural frequencies (cOm) corresponding to a certain harmonic order, n, can be solved
using an "Eigensolver routine" in most FE packages. By solving Cf^ for the first m mode of
a number of models with different harmonic orders, a complete set of dispersion curves can
be calculated (see figure 10). Our results were obtained using the program Finel developed
at Imperial College [9].
FE Results
The accurate calculation of the curves relies on the spatial accuracy with which the
elements represent the mode shapes. Typically at least four elements are required to
represent one displacement harmonic cycle correctly. The mode shape complexity
increases both with increasing mode order m, and with increasing frequency, as illustrated
in figure 10. To demonstrate this, the average error in frequency of the first 20 modes (in
comparison with the exact analytical solutions) are calculated at each small n interval and
also for a number of different element sizes as a percentage of the wall thickness (0.001m
in this case) shown in figure 11. Furthermore, errors are also calculated for each individual
mode order, m, and a number of element sizes at a fixed harmonic order of 1200 in figure
12. These plots clearly suggest that more elements are required in order to obtain accurate
results for high m and high frequency. However, this would also significantly increase the
calculation time.
219
average error
in frequency
of the first
20 modes
average error
in frequency
FIGURE 11. Average error plot of the first 20 modes
as a function of angular wavenumber and element size.
FIGURE 12. Error plot as a function of mode
order and element size at a fixed n of 1200.
CONCLUSIONS
The benefits of using asymptotic methods to calculate circumferential Lamb
dispersion relations for an annular structure have been investigated. The proposed
asymptotic methods have resolved the instability problem at high frequencies and have
significantly sped up the tracing time without compromising the accuracy. The accuracy
has been investigated by comparing with the exact solution and with Finite Element
calculations.
REFERENCES
1. Alleyne D.N., Pavlakovic B., Lowe M. J. S. and Cawley P., "Rapid, Long Range
inspection of Chemical Plant Pipework Using Guided Waves", Review of Progress in
QNDE, Vol 20, eds. D. O. Thompson and D. E. Chimenti, American institute of
Physics, New York, 2000, p. 180 - 187
2. Li Z., Berthelot Y. H., NDT&E International 33, 225 - 232 (2000)
3. Pavlakovic B., Lowe M., Alleyne D., Cawley P., "Disperse: A General Purpose
Program for Creating Dispersion Curves", Review of Progress in QNDE, Vol 16, eds.
D. O. Thompson and D. E. Chimenti, Plenum, New York, 1997, p. 155 - 192
4. Liu G., Qu J., Journal of Applied Mechanics, 65, 424 - 430 (1998)
5. Lowe, M. J. S., IEEE Trans. Ultra. Ferro. Freq. Cont. 42, 525-542 (1995)
6. Gridin D., Craster R.V., Fong J., Lowe M.J.S., Beard M., "The high-frequency
asymptotic analysis of guided waves in a circular elastic annulus", submitted to Wave
Motion, May 2002.
7. Wilcox P., Evans M., Diligent O., Lowe M., Cawley P., "Dispersion and Excitability of
Guided Acoustic Waves in Isotropic Beams with Arbitrary Cross Section", Review of
Progress in QNDE, Vol 21, eds. D. O. Thompson and D. E. Chimenti, American
institute of Physics, New York, 2001, p. 203 - 210
8. Abramowitz M., Stegun I. A., Handbook of Mathematical Functions, National Bureau
of Standards, Washington, 1964
9. Hitchings D., FE77 User Manual, Imperial College internal report, 1995.
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