A COUPLING METHOD OF BOUNDARY ELEMENT METHOD AND
GENERALIZED RAY THEORY FOR ELASTIC WAVE SCATTERING
IN A THICK PLATE
K. Kimoto1 and S. Hirose1
1
Department of Mechanical and Environmental Informatics, Graduate School of Information Science and Engineering, Tokyo Institute of Technology 2-12-1 O-okayama, Meguroku, Tokyo 152-8552 Japan
ABSTRACT. This paper presents a method for 3D elastic wave scattering analysis by an obstacle in a
thick plate. The method is a coupling method of the boundary element method (BEM) and the generalized
ray theory (GRT). For computational efficiency,the free field displacement in a plate is calculated by the
GRT, and the solution is used as an incident field in the successive BEM analysis. The displacements or the
tractions on the surface of the scattering object are obtained by the BEM. The boundary values obtained
thus are then substituted into the integral expression of the scattered waves, in which the Green's functions
are evaluated by the GRT to reduce computational cost further. As a numerical example, the scattering
by a spherical cavity embedded in a plate is solved. The numerical results are shown for displacement
waveforms as a function of time at points on the cavity and plate surface. Through the numerical example,
benefit of the proposed method is discussed.
INTRODUCTION
Wave propagation and scattering in plates are important problems in NDE of materials
and have been investigated extensively. For a plate of an infinite lateral extent, it is well
known that the wave field can be expressed as a superposition of Lamb modes. However,
if the plate has an inhomogeneous part like cavities or inclusions, no analytical solution is
available and numerical approach is needed. The finite and the boundary element methods
are often used for this purpose because the both are applicable to arbitrarily shaped bodies.
Unfortunately, when wave response are necessary for a wide region or a long time range of the
plate, the numerical analysis becomes practically intractable due to huge computational cost.
Therefore, we have been trying to develop a technique that couples the boundary element
method (BEM) and the ray theory to reduce computational cost for scattering analysis in a
thick plate. The coupling method has been applied so far to the 2D SH-waves and numerically
implemented in the frequency domain. [1]
In the present paper, we extend the method to 3D elastic waves in the time domain.
However, since the simple ray theory breaks down in 3D case due to the strong interaction
between the body and the inhomogeneous waves in a plate like head or surface waves, we
couple the Generalized Ray Theory (GRT) with the BEM. Although normal mode expansion
technique may also be possible to use with the BEM, we prefer the GRT because interpreting
the wave field in terms of rays is essential not only in estimating the flaw location but also
in applying the shape reconstruction methods such as a synthetic open aperture or an inverse
scattering method.
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
41
point load
B
S: cavity
B
FIGURE 1. A plate with an embedded cavity S. The plate is subjected to a time dependent normal point load.
In the fallowings, we first state the problem to be solved, and then the coupling method
of the BEM and the GRT is described. As a numerical example, the scattering by a spherical
cavity located at the mid-plate is solved by the coupling method. Finally, the present study is
summarized and some comments on our future works is addressed.
PROBLEM STATEMENT
Let's consider an isotropic linearly elastic plate of infinite lateral extent. Although the
plate can contain either voids or elastic inclusions, we consider here a single cavity with
a smooth surface. As shown in Fig.l, a concentrated normal load with the time variation
f ( t ) is applied on the upper plate surface. In the fallowings, the scattering problem by the
cavity S subjected to the incident wave field excited by the normal load is investigated. The
observation points can be located inside the plate or on the plate surface. The governing
equation for the displacement vector u is expressed as follows:
(X + fi)VV - u(x, t) = pu(x, t)
(1)
where A and p, are the Lame constants, and p is the density. The traction free boundary
conditions hold on the cavity surface as well as on the plate surfaces B± except for the point
load given by
[ TZT — TzO
—
U.
Since the plate is motionless prior to the time t = 0, we set the initial conditions of
M(OJ,O) = w(x,0) = 0.
(3)
METHOD OF SOLUTION
Boundary Element Method
In order to solve the initial-boundary value problem stated above, we use the boundary
element method (BEM) in time domain. The boundary integral equation for this problem can
be derived from eqs.(l)-(3) as follows: [2]
\u**(y,t) + f
Zi
J B~T~{-)B
+ f {r;(x, y, t) * «f (x, t) - U}(x, y, t) * tf\x, t)} dS = u{(y, t) (y e S) (4)
JS
42
where C/j(x,t/,£) is the fundamental solution of elastodynamics, Tj(x,y,£) is its traction
component, and * means a time convolution integral. In eq.(4), the superscripts 'tot','sc' and
T denote the total , scattered and free fields, respectively. The free field means the wave field
without the cavity, while the scattered field is defined by
usc(x,t)
tsc(x,i)
= utot(x,t)-uf(x,t)
= ttot(x,t)-tf(x,t).
(5)
(6)
Although part of the scattered waves propagate back and forth due to multiple scattering
between B± and 5, most of scattered waves are outgoing waves without multiple interaction
with the scatterer in the plate. This implies that the first integral in eq.(4) does not contribute
much to the generation of the scattered field on S unless the scatterer is very close to the
boundaries B±. Therefore, in solving eq.(4) by the BEM, the integration is required over S
and a small portion of B± in the vicinity of S. Furthermore, if multiple scattering effect is
negligible or not of interest, we can discard the first integral on the left hand side of eq.(4)
altogether. This may be an advantage of using the time domain integral equation (4) since the
reduction of the domain of integration directly leads to the decrease of the size of the problem.
However, it is not easy to obtain the exact form of the free field u{ on the right hand side of
eq.(4) for all times. Therefore, we calculate the free field displacement ray by ray using the
GRT, and then solve the integral equation (4) by the BEM to obtain the displacement on S.
Once the surface displacement of the cavity is given, the scattered displacement at an
arbitrary point can be obtained from the following integral equation.
=0
(7)
s
where Gj(x,y,t) and #j(x,y,£) are the displacement and traction Green's function for
the plate, respectively. Evaluating GJ and Hj rigorously for all time is laborious and time
consuming. We use the GRT again for the evaluation of the Green's functions ray by ray.
Generalized Ray Theory [3]
The GRT described below can calculate the response to a point load applied at any point.
Therefore, we use the method here to calculate the free field displacement in eq.(4) as well as
Glj and H^ in eq.(7). For simplicity, we describe the theory exclusively for the axisymmetric
case where a vertical point load is applied to the plate. Similar discussion is also possible for
an obliquely applied load. Firstly, the displacement field due to the point load is written in
terms of the scalar and the vector potentials 0 and ^
u = V^-f V x t/>, ( V - ^ = 0).
(8)
In polar coordinate(r, 0, z)9 the vector potential can be expressed by two scalar potentials ij)
It can be shown that the functions 0, ip and x satisfy the 3D wave equations. When the
medium is unbounded, the solution is the fundamental solutions, which can be given in the
Laplace domain as follows. [3]
(10)
$o(r,z,s)
= -sV(«)j»*re-"*l*IJoKr)£de
Xo = 0
°
43
(11)
(12)
where (•) means the Laplace transform of a function (•), s is the transform parameter, and
rip = Jg2 -f c^2 (/3 = L or T). Since the integrands of eqs.(10)-(l 1) are the plane waves in
Laplace-Hankel domain, the reflected waves from B± can be obtained by multiplying them
with reflection coefficients of corresponding canonical problems. Although there is no clear
image of 'rays', each integral obtained thus is called the 'generalized ray' or 'ray integral'. It
is obvious that the generalized rays of multiple reflected waves can be calculated similarly,
and the wave field in the plate is expressed as a sum of those generalized rays. The solution
obtained thus by the GRT is exact until the (n + l)st ray arrives at an observation point if we
take into account up to nth ray.
Since reflection coefficients are the functions of £, we find that any ray integral in the
Laplace domain can be written in the following forms.
_
I0(r,z,s)
(13)
=
(14)
where zp the vertical component of the traveling paths of the wave of type /3. It is noted
that the integral /i in eq.(14) appear in the differentiation of the potentials with respect to r
for calculation of displacement or traction. The time domain solutions can be obtained by
taking the inverse Laplace transform of these integrals. To perform the inverse transform, we
substitute the integral representation of the Bessel functions
Jn(z) - —
/* eiscos"cos(nuj)duj
n
l 7T Jo
(15)
into eqs.(13) and (14) and then have
2
i"71"/2
r°°
/o(r,z,s) - -J^e /
duj
E0
7T
JO
71 2
JO
(16)
00
2
r "/
A
/i(r, z,s) = —Im I
duo I EI
7T
Jo
(17)
Jo
Application of the Cagniard's method to eqs.(16) and (17), and change of variables gives the
ray integrals in time domain.
I0(r,z,t)
= H(t-tA)-Im fl(t) EO(£)-——^——-
(18)
Ii(r,z,t)
= H(t-tA)—Imf1
(19)
TT
TIT
Jo
'K(£;r,z,t)
E0
Jo
where //(t) is the Heaviside step function, and the path of integration is shown in Fig.2.
and K(£, r, z, t) are defined by
(20)
(21)
As indicated in Fig.2, there is no singularity between the contour and the positive real axis.
44
1/c,
1/c.
1/c,
o
: branch cut
FIGURE 2. Path of integration in a complex £ plane.
+
B
h=30a
30a
\
<*-•> S: spherical cavity
2a
B
*2.
fft)<M
0
-0.2
-0.4
0
4
8
12
time (cLt/a)
16
FIGURE 3. Dimensions of numerical model and time history/ (t) of the applied normal load.
We can change the integration contour as we want in the first quadrant. This enables us
to evaluate eqs.(18) and (19) numerically by the standard quadrature technique. Since the
solutions obtained by the GRT satisfy the boundary conditions on B±, the solutions can be
substituted into the Green's functions GTj and Hj in eq.(7). Also the free field u{ in eq.(4)
can be obtained by taking the convolution of eqs.(18) and (19) with the source function f ( t ) .
NUMERICAL EXAMPLE
As a numerical example, we consider the scattering by a spherical cavity embedded in a
plate. Dimensions of the numerical model are shown in Fig.3 together with the time variation
f ( i ) of the applied load. All quantities used in the numerical analysis are normalized by p,
CL, a and their combinations. For example, length is normalized by the unit length a and
traction force by pc2L. Also, the material constant of the plate is set to be CT/CL = 3.1/5.7.
In evaluating the free field displacement in eq.(4) and Green's function in eq.(7) by the GRT,
the direct waves and waves reflected once at the top or the bottom surfaces are taken into
account.
Numerical Results
Fig.4 shows snapshots of the radial component u{ of the free field at several time steps.
It is shown that in addition to the body waves, inhomogeneous waves like head and surface
waves are evaluated by the GRT. To see the free field displacement at a point on the cavity sur-
45
1:11
FIGURE 4. Snapshots of the free field displacement uf.
0.002
L
0 —*\f* —
ri
^~*\ ~~^s~~-^*
V
\y^
I
•
40
1 V
" \ y r*
-0.002 •
0 004
1
60
u
Ujf
1 \ \i\
-/
time (erf/a)
V.
1
:
2 —"-
U/ ———
i
1
80
FIGURE 5. Free field displacement observed at a;s=(30.2a, -0.03a, 14.0a).
46
u
Uf
0.001 -
Uj ———
^~\
0
/°*V
—
u2 --.
U3 ———
-0.001
n nm
40
50
time (cLt/a)
FIGURE 6. Time variations of the total displacements at the point xs=(30.2a, -0.03a, 14.0a) on the cavity
surface subjected to the incident L-wave.
0.005
•
•
•
7
0
•
•
-
_
:
\/7v\
Uj
U2
......
U3 --
-0.005
60
70
time (cLt/a)
80
90
FIGURE 7. Time variations of the total displacements at the point xs=(30.2a, -0.03a, 14.0a) on the cavity
surface subjected to the incident Head and T-wave.
face S, the time variations of the displacement components at xs = (30.2a, — 0.03a, 14.0a)
are shown in Fig.5. The longitudinal, head and transverse waves are indicated in the graph
by 'L', 'H' and 'T', respectively. As explained in the previous section, the GRT gives the
solutions in the form of superposition of the generalized rays. It is, therefore, possible to
calculate the scattered field ray by ray and afterward add the results together to yield the
complete solution. This enables us to identify easily where a particular wave comes from. In
what follows, scattered waves due to the incident L-waves are calculated separately from the
ones due to the incident head and T-waves.
Figs.6 and 7 show the total displacements at x$ due to the incident L-waves and the
incident head and T-waves, respectively. In either case, the initial waveforms are similar to
the free field and are followed by very small tails that decrease gradually. The total displacements on the cavity surface S as shown in Figs. 6 and 7 are substituted into eq.(7) to obtain
the scattered waves at an arbitrary point. Fig. 8 shows numerical examples for the scattered
waves calculated at the point XB+ = (60a, 0,0) on the top surface B+. The top and the
middle graphs are the displacements due to the incident L-wave, and the incident head and
T-waves, respectively. The bottom is all scattered waves which are a sum of results shown
in the top and the middle graphs. 'L','T' and 'H' in the figure again denote the longitudinal,
transverse and head waves, respectively. The letters on the left hand side of arrows mean the
incident wave types whereas the ones on the right hand side are the types of the scattered
waves radiated from the cavity. Comparing the three graphs in Fig.8, we can understand how
the scattered waves interact one another to form the total response, which is not clear from
the graph at the bottom alone.
SUMMARY AND FUTURE WORKS
A coupling method of the BEM and the GRT was developed for 3D elastic wave scattering problems in a thick plate. The GRT was used to reduce the computational cost for the
47
0.0001 •
0
I
1
.
,
l
Response to the incident L wave
Uj
T ~^T—T
/VVNj
A/
L^L
_~^
~
\jk^
'
L->T
l
SC
'.
sc
u2 -—
SC
-0.0001
100
0.0001 ~ Response to the incident head and T-wave
150
r
0
-0.000 Ih
0.0001
100
150
100
time (cjt/d)
150
Total response
0
-0.0001
FIGURE 8. Time variations of the scattered waves observed at xB+=(60a, 0, 0) on the top surface.
BEM by calculating the free field displacement in the plate as well as the scattered field from
the scattering object. As a numerical example, scattered waves from a spherical cavity were
calculated for the incident wave generated by a normal point load. The results were shown for
time variations of displacement waveforms given at the points on the cavity and on the plate
surfaces. Identifying types of waves is easy with the present method since the GRT gives the
solutions as a superposition of rays. This will be an advantageous feature in developing a
simulation code for ultrasonic flaw detection in a plate. However, the GRT analysis involves
the contour integral in a complex plane and sometimes consumes much computational time.
Improving computational efficiency of the GRT is required in our future work.
REFERENCES
1. K. Kimoto and S. Hirose, "Reflection and scattering analysis of SH-wave using a combined method of BEM and ray theory," in Review of Progress in QNDE, vol. 19, AIP Conference Proceedings 509, New York, 1999, pp.65-72.
2. A. C. Eringen and E. S. Suhubi, Elastodynamics vol. //, (Academic Press,New York,
1975).
3. Y. H. Pao and R. R. Gajewski, "The generalized ray theory and transient responses of
layered elastic solids" in Physical Acoustics: principles and methods, vol. 13, (Academic
Press, New York, 1977).
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