APPLICATION OF FAST MULTIPOLE BOUNDARY
ELEMENT METHOD TO SCATTERING ANALYSIS
OF SH WAVES BY A LAP JOINT
T. Saitoh1, A. Gunawan2, and S. Hirose2
1
Department of Civil Engineering,
Graduate School of Science and Engineering,
Tokyo Institute of Technology
2-12-1 O-okayama, Meguro-ku, Tokyo 152-8552 Japan
2
Department of Mechanical and Environmental Informatics,
Graduate School of Information Science and Engineering,
Tokyo Institute of Technology
2-12-1 O-okayama, Meguro-ku, Tokyo 152-8552 Japan
ABSTRACT. A fast multipole boundary element method (FMBEM) in conjunction with a mode
exciting method is developed for scattering analysis of SH waves by a lap joint. In the numerical
calculation, reflection and transmission coefficients are obtained as a function of the wave number
for the lap joint having a debonding part with various lengths. Numerical results show that due
to the mode conversion, the reflection coefficients are not always proportional to the length of the
debonding.
INTRODUCTION
A lap joint is an important structural component in aircraft, which is an adhesively
bonded and riveted part between two thin plates. Because of complicated mechanical
behaviors at the joint, a lap joint structure often has hidden defects which are made by
debonding, fatigue, corrosion and so on. Therefore, it is required to develop an effective
nondestructive technique for the detection of hidden defects at a lap joint.
Lamb waves may have the potential for cheap and convenient inspection over a
wide range of a plate. Therefore, several numerical studies on scattering, reflection
and transmission of Lamb waves have been done to obtain quantitative information
on defects in a plain plate. The finite element method (FEM) for Lamb and SH wave
problems in a plate has been analyzed by several researchers [1, 2, 3, 4]. Rose et al.
[5, 6] have developed a hybrid method of the boundary element method and the mode
expansion technique for Lamb wave scattering analysis. Gunawan and Hirose [7] have
recently proposed the mode exciting method in conjunction with the BEM. For a lap
joint problem, a hybrid method, called the global local finite element method (GLFEM),
has been used to study the characteristics of Lamb wave propagation[8]. Lowe et al.[9]
has also used the finite element analysis to investigate the propagation of Lamb waves
in the overlap region.
In this paper, the mode exciting method, proposed by Gunawan and Hirose[7],
is applied to the reflection and transmission of SH waves across a bonded lap joint.
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
1103
transmitted wave
I <————> / \^^
* —————————————^•^^•F——————
2h ,
_____ _
X
""'"
^^
~'
x 7'
debonding
reflected wave \Xv ,
bonding
FIGURE 1. Reflection and transmission at a lap joint.
Since the lap joint structure with complex geometry requires large scale analysis, the
boundary element analysis with a fast multipole algorithm is adopted in the present
study. Reflection and transmission coefficients for SH wave modes are calculated as a
function of the wave number for debonding with various lengths.
PROBLEM
Figure 1 shows the geometry of reflection and transmission problem of SH waves
across the bonded lap joint. Two semi-infinite plates with the thickness 2h are lapped
over the region of the length s having partial debonding parts. When an SH wave is
incident from the left plate to the overlapped region, a part of wave energy is reflected
and the other energy is transmitted through the lap joint. The problem is to determine
the amplitudes, r\ and t\, of reflected and transmitted waves of the mode i due to the
incident SH wave of the mode j with the unit amplitude.
Let the semi-infinite plates be an isotropic homogeneous elastic solid with the shear
modulus IJL and the mass density p. The antiplane displacement u with time harmonic
variation of exp(iut) satisfies the equation of motion
/A72i/(x) + pu2u(x) = 0.
(1)
The traction free condition is given as a boundary condition on the boundaries except
for the bonded interface:
t(x)=^(x)=0
(2)
where d/dn denotes the normal derivation on the boundary. For the bonded interface,
it is assumed that the bonding layer is so thin that the inertia effect of the bond can
be neglected and the bond is represented by the distribution of springs. Then the
boundary conditions on the bonding area are expressed as
t(x) = -t(x) = K(u(x] - u(x))
(3)
where K is the spring constant and the variables with and without the bar ~ indicate
the quantities with respect to one plate and the opposite plate, respectively.
MODE EXCITING METHOD
We now consider the boundary value problem in the finite regions, Q! and H 2 ,
bounded by the boundaries 5i U 5 l5 ^2 U 52, respectively, as shown in Fig. 2. On
the edge boundaries B\ and B^ arbitrary boundary conditions are given to excite SH
1104
X2
incident
c
s
incident
a&+a£^/^+
b
i2
b0L-+^ ^\^
scattering
L
S2
+c L
^\^ o o
i i
Q
V^rf n £+4Zr
scattering
xt
St
FIGURE 2. Wave field in plates with finite lengths.
wave modes in the domains ^i and ^2. For the other boundaries Si and 52, the same
boundary conditions as shown in Fig. 1 are specified.
In general, there are nonpropagating wave modes as well as propagating modes
in a plate. The nonpropagating modes, however, decay rapidly with the distance from
the disturbances of source and scatterer. Therefore only propagating wave modes can
propagate in the intermediate portions of the plates as shown in Fig. 2, if the distances
between the lap region and the boundaries, BI and £ 2) are large enough. In such a
situation, the wave field around the lap joint between finite plates is almost equivalent
to the wave field around the lap joint of infinite plates subjected to the incident waves
from both right and left sides. Without the loss of generality, we suppose that the
debonding occurs at the center part of |#i < b/2 on the overlapped boundary and
thus the problem is symmetric with respect to the origin. Also it is assumed that
four propagating modes of L^ and L^ are excited as shown in Fig. 2, where the sign
± denotes the SH wave propagating in the ±£1-direction, respectively. In Fig. 2, the
terms of a^L^ -f- ai
and col/o" + ciLj~ are considered as incident waves and the terms
are scattered waves. Therefore the scattering process
"+
of bQLQ + fri^r and
can be written as
+
+
(4)
It is noted that after solving the boundary value problem defined in the regions Cti and
n2 as shown in Fig. 2, the coefficients a^ b^ q and di (i — 0,1) can be obtained by
decomposing the solutions into SH wave modes of LQ and L^ [11].
The objective of this study is to determine the coefficients of reflection and transmission in infinite plates subjected to the incident wave with the unit amplitude as
shown in Fig. 1. For the incident waves propagating from the left hand side, we have
the following expressions:
1x
(5)
r}Lj- + tl0L+
1 X
(6)
Similarly, we have the expressions for the incident waves from the right hand side as
follows:
lx
1x
-»•
+ r\Ll + tl0Lo +
1105
(7)
(8)
where the symmetric property of the problem is used.
Multiplying eqs.(5), (6), (7) and (8) by ao, ai, CQ and GI, respectively, taking
summation of these equations, and comparing the right hand side with eq.(4) yield the
following equations:
cQt° + ci*J - b0
+ airj + cot? + cjtj
a
c r
(9)
= 61
(10)
c r
ao*o + i*o + o o + i o = d0
aot? + aitj + cor? + c^r} = d^.
(11)
(12)
Since the coefficients, a*, 6;, q and ^ (i = 0,1), are already known by the mode
decomposition of the solution of the problem as shown in Fig. 2, eight variables, r\ and
£? (i^j — 1,2), are unknown in the above equations. As seen in eqs.(9)-(12), only four
equations are obtained for a set of boundary values given on B\ and B^- Therefore
another set of boundary values are given on B\ and BI to obtain more four equations,
which give a sufficient number of equations to determine eight coefficients , r\ and
FAST MULTIPOLE BOUNDARY ELEMENT METHOD
In order to obtain the coefficients, a^ b^ Q and di (i — 0, 1) in eq.(9)-(12), the
boundary value problem defined in the finite regions Q! and O2 is necessary to be solved.
Because of the complex geometry of the lap joint, a large matrix calculation is required
to solve the boundary value problem. Since in the large-scale computation, a matrix
solver with iterative scheme is generally used to reduce the computational memory, it is
necessary to assemble the matrix components quickly at each iterative time. The fast
multipole boundary element method (FMBEM)[10] has been developed to meet such
demand for solving large-scale problems. A brief description on the FMBEM is shown
in the sequel.
The boundary integral equation for the internal domain Q& (k = 1,2) is written
as
c(x)u(x) = j^ U(x, v)t(y)dT(y) - j^ T(x, y)«(y)dT(y),
for x e rk
(13)
where Ffc = Sk U Bk, c(x) is the free term at the point x. Here U(x,y) and T(x,y)
are fundamental solutions given by
U(x,y) =
\kTr),
T(x,y) =
U(x,y) = -H?\kTr}
(14)
where r = \x — y|, kT is the wave number of SH wave and H^ is the first kind Hankel
function of the n-th order.
Suppose that the boundary Fk is divided into the Nk elements of rk(j = 1, . . . , 7V fc ),
i.e., Tk = SJLiTj, and the displacement and traction are approximated by constant
values, uk and tk, on the element Tk. Eq.(13) is then expressed in the discretized form
Nk
Nk
(x^ y}dT(y)uk
(i = 1, . . . , Nk, k = 1, 2).
(15)
1106
near field: II
far field: D
field pointy
FIGURE 3. The concept of fast multipole boundary element method.
In the FMBEM, the elements are divided into near-field and far-field groups depending
on the distance between the field point a^ and the element F*. For example, the first
term of the RHS in eq.(15) can be written as
ffk;Near
ux
dr
jyk;Far
,y)dr(y)*£= E L,N^ ( ^y^ (y^+ E /^..tfte.tfWv)**(16)
The integral over the element in near field is carried out in the same manner as the
conventional BEM, while for the elements in far field, the integrals are evaluated using
the multipole expansion of the fundamental solution. Using the Graf's addition theorem
of the Hankel function, the integral with respect to the element r*>Far may be expressed
,/cjFa
3
i9
y}dT(y)tk =
(17)
Applying the Graf s addition theorem to the term M&J again, we have the following
recursive form
Maj = UaZ(UZY(UYX . . . (UBAUAj) . . .))
(18)
where UBA denotes the operator that relates the series expansion at the point A to
the expansion at the point B. If the points for expansions can be taken in hierarchy
structures as shown in Fig. 3, the computational memory and time for the recursive
calculation in eq.(18) can be reduced drastically. Similar procedure can be applied to
the evaluation for Li0i(Xi) with respect to the field point av
NUMERICAL RESULTS
In the following, all numerical calculations are made for the overlapped boundary
with the length s/h — 1 and the debonding part with various lengths of b/h = 0, 0.25, 0.5
and 0.8. Also, the spring constant K is taken as hK/fi = I in the calculation. Figure 4
shows the reflection and transmission coefficients rf and t® (i = 0, 1,2) as a function
of the nondimensional wave number fc^/i for the incident wave of the zeroth SH mode.
In all figures, the reflection and transmission coefficients vary irregularly near the wave
numbers fc^/i = ?r/2 and TT, which correspond to the cut-off frequencies of the first and
second SH modes, respectively. For the wave number of &r/i > 7T/2 and except for
the cut-off frequency, the reflection coefficient r$ shows a larger value as the debonding
length b becomes longer. Also the transmission coefficient t® becomes smaller with the
increase of the debonding length b. These tendencies match our intuition. It is, however,
interesting to see that the reflection coefficient rj shows a smaller value with the increase
1107
0 1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0 i
t,
0.8
0 1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
(
1
o .
kTh
0
o
b/h
0.8
2 0.8
-o.o
-0.25
0.6
1
0.6
0.4
0.4
0.2
0.2
0
0
I
kTh
FIGURE 4. Reflection and transmission coefficients versus
Z.
O
kTh
t
for the SHo mode incidence.
of the value b due to the mode conversion, which phenomenon is contradictory to our
expectation.
Figure 5 shows the reflection and transmission coefficients r\ and t\ (i — 0,1,2)
for the incident wave of the first SH mode. As expected, the coefficients r\ and t\ show
larger and smaller values, respectively, for the larger debonding length. However, the
reflection coefficient TQ shows a different tendency against our expectation. It is noted
that TO and t\ are identical to rj and t®, respectively, due to the reciprocal relationship.
CONCLUSIONS
The FMBEM in conjunction with the mode exciting method was applied to the
scattering analysis of SH waves by debonding defects at the lap joint. The reflection and
transmission coefficients were calculated as a function of the wave number for various
lengths of debonding and wave modes. The numerical results showed that due to the
1108
0.8
0.8
0.6
0.4
0.2
1
"kxli
2
3
kTh
4
5
4
5
b/h
-0.0
0.6
-0.25
-0.5
-0.8
-0.8
0.4
0.4
0.2
0.2
0
2
3
4
5
0
kTh
kTh
t,"
b/h
0.8
-0.0
0.6
-0.25
-0.5
-0.8
0.4
0.2
0.2
1
-0.25
-0.5
-0.8
0.6
0.4
0
b/h
-0.0
0.8
2
3
kTh
4
5
0
1
2
3
kTh
FIGURE 5. Reflection and transmission coefficients versus ^h for the SHi mode incidence.
mode conversion, the reflection coefficients are not always proportional to the length
of the debonding. In near future, we will extend our method to inplane problems and
problems with more complex geometries like T-shaped or cross-shaped joints, which
require larger-scale computation.
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1109
4. Moulin, E., Assaad, J. and Delebarre C., J. Acoust. Soc. Am. 107, 87-94 (2000).
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