Key Engineering Materials Vols. 270-273 (2004) pp. 453-460
online at http://www.scientific.net
© 2004 Trans Tech Publications, Switzerland
Citation &
Copyright (to be inserted by the publisher)
Three-Dimensional Defect in a Plate Boundary Element
Modeling for Guided Wave Scattering
Xiaoliang Zhao1, Joseph L. Rose2
1
2
Intelligent Automation, Inc., 7519 Standish Place, Suite 200, Rockville, MD 20855, USA
Department of Engineering Science and Mechanics, The Pennsylvania State University, 212
Earth-Engineering Science Building, University Park, PA 16802, USA
Keywords: Ultrasonic Guided Waves, Boundary Element Methods, Wave Scattering
Abstract. A hybrid three-dimensional boundary element normal mode expansion technique is
developed to study the guided wave scattering from an arbitrary shape defect in a plate. Lamb wave
incident into a circular through plate hole was studied as an example. The scattered wave
displacement field and its normal mode component are given. Mode conversion from Lamb wave to
SH wave at the defect boundary were observed.
Introduction
Ultrasonic guided waves have demonstrated great potential for detecting many kinds of defects that
occur in tubes, pipes or plate structures. However, defect classification and sizing by using guided
waves is still a difficult problem to be resolved due to the complexity of wave propagation
characteristics. Analytical studies of guided wave scattering from a defect had limited success for
defects of simple geometries [1, 2]. Numerical tools are becoming absolutely essential to study the
guided wave scattering problems in complex-shaped waveguides or defects. Real world defects are in
three dimensions, and the defect widths are often narrower than the inspection wave beam-width.
Therefore, a three-dimensional (3-D) model of guided waves scattering from those defects is required.
Some pioneer work in this field can be found in [3-7].
Boundary Element Method (BEM), as one of the engineering analysis techniques, has the
advantages such as reduction of the dimensionality, less computer time and storage, easy management
of unbounded domains, and more targeted calculations over the popular Finite Element Method
(FEM). The Normal Mode Expansion (NME) technique has also been used in wave field analysis and
synthesis. In this paper, we present a hybrid model of three dimensional (3-D) BEM combined with
the NME technique to calculate the guided wave scattering field from a 3-D defect in a plate. The
angular profile of the displacement field as well as its guided wave mode component, scattered from a
circular through plate hole with S0 mode Lamb wave incidence, is presented. The mode conversion
phenomenon from Lamb wave to SH wave was also observed.
Licensed to Penn State University - University Park - USAJoseph L. Rose ([email protected]) - Penn State University USA
All rights reserved. No part of the contents of this paper may be reproduced or transmitted in any form or by any means without the
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Theory
θ
r
Incident guided
wave
Fig. 1. A schematic problem of a plane guided wave incident into a three-dimensional defect in an
infinite plate.
Problem Setup and Concept of the Approach
Consider a 3-D guided wave scattering problem as shown in Fig. 1, where an infinitely large flat plate
in vacuum has a 3-D arbitrary shaped defect (e.g. a circular through plate hole). A time-harmonic
guided plane wave (either Lamb wave or SH wave) is incident in the positive x direction. The guide
wave mode incident into the defects will result in scattered waves of all orders of the propagating as
well as non-propagating modes (both Lamb wave and SH waves) at the vicinity of the defect. The
evanescent modes die out quickly within several wavelengths; only propagating modes survive and
carry out energy flux at the far-field from the defect [8]. At different directions, the wave
displacement and stress fields are different in general and form an angular pattern. For different
defects, this pattern will also be different, which could in turn be used for defect shape and sizing
analysis.
In order to calculate the scattered wave field, the hybrid BEM normal mode expansion technique
that was successfully applied to the 2-D defect characterization problem [9] was extended to the 3-D
cases. In specific, a virtual cylinder enclosing the region with defects in the plate was defined for
BEM meshing. The radius of the disc region is chosen to be larger than the largest wavelength of all
possible propagating guided wave modes. A cylindrical coordinate system was established at the
center of the disc region, with the z=0 plane coinciding with the mid-plane of the plate. Outside the
defined disc region, the total displacement and traction field are the superposition of all the incident
and scattered propagating wave modes. Within the defect region, the 3-D elastodynamic BEM was
used to calculate the wave field. Once the boundary conditions are specified at the virtual cylindrical
boundary, the BEM calculates and outputs the displacement and traction value for each boundary
element, from which the scatted wave field is determined and so as each wave mode component.
Three Dimensional Guided Wave Normal Mode Expansion
The Lamb wave and SH waves in a three dimensional infinite plate can be expressed in the form [3, 5]
1
∂φ
 n
u r = k Vn ( z ) ∂r (r ,θ )
n

1
1 ∂φ
 n
(r ,θ ) (Lamb wave)
uθ = Vn ( z )
kn
r ∂θ

u n = W ( z )φ (r ,θ )
n
 z

1 ∂ψ
 n 1
u r = l U n ( z ) r ∂θ (r ,θ )
n

1
∂ψ
 n
(r ,θ ) (SH wave)
uθ = − U n ( z )
ln
∂r

u n = 0
 z

(1)
Key Engineering Materials Vols. 270-273
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where k n is the nth mode Lamb wave number and can be determined by the 2-D Rayleigh-Lamb wave
dispersion equations; l n is the wave number of nth mode SH wave. Vn and Wn are the Lamb modal
function in the plate thickness direction as in the 2-D cases; U n (z ) is the SH mode shapes. The
potential functions φ ( r , θ ) and ψ ( r , θ ) satisfies Helmholtz equation, i.e.:
∂ 2φ 1 ∂φ 1 ∂ 2φ
2
+
+ 2
+ kn φ = 0 ,
2
2
∂r
r ∂r r ∂θ
(2)
and
∂ 2ψ 1 ∂ψ
1 ∂ 2ψ
2
+
+
+ ln ψ = 0 .
2
2
2
∂r
r ∂r r ∂θ
(3)
Consequently, solutions of the potentials functions have to be found in order to describe the wave
field. They can be given in the following form
φ ( r , θ ) = Φ ( r )e imθ ,

imθ
ψ ( r , θ ) = Ψ ( r )e ,
for Lamb modes
(4)
for SH modes
where Φ (r ) and Ψ (r ) are the solutions of Bessel’s differential equations, i.e.
 d 2 Φ 1 dΦ
m2
2
+
+
(
k
−
)Φ = 0,
n
 dr 2 r dr
r2
 2
2
 d Ψ + 1 dΨ + (l 2 − m )Φ = 0,
n
 dr 2
r dr
r2
for Lamb modes
(5)
for SH modes
The suitable solutions to equations (5) for the scattered wave field can be expressed as Hankel
functions of the first kind, i.e.
Φ m (k n r ) = H m ( k n r ),
Ψm (l n r ) = H m (l n r )
(6)
Thus the scattered wave displacement field can be expressed as:
N
u rscat = ∑
∞
L
(7)
L
∞
H m (k n r ) imθ
e − ∑ ∑ BmnU n ( z ) H m ' (l n r )e imθ
kn r
n = 0 m = −∞
(8)
n = 0 m = −∞
N
uθscat = ∑
imAmnVn ( z )
n = 0 m = −∞
N
u zscat = ∑
∞
∑A
∑ imB
n = 0 m = −∞
∞
∑
∞
H m (l n r ) imθ
e
ln r
∑ AmnVn ( z) H m '(k n r )e imθ + ∑
Wn ( z )H m (k n r )e imθ
mn
U n ( z)
mn
(9)
n = 0 m = −∞
where the coefficients Amn and Bmn relate to the contribution from the Lamb modes and SH modes,
respectively. N and L are the total number of Lamb and SH modes that can propagate in the plate. The
relevant stresses for the scattered wave field can also be expanded by using Hooke’s law,
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1
m2
~n
n
A
[
Σ
(
z
)
H
(
k
r
)
−
Σ
(
z
)[
H
'
(
k
r
)
−
H m (k n r )]e imθ
∑
mn
rr
m
n
rr
m
n
2
r
k
r
n = 0 m = −∞
n
∞
N
σ rrscat = ∑
(10)
∞
L
2
2
+ ∑ ∑ imBmn µU n ( z )[ H m (l n r ) − 2 H m (l n r )]e imθ
r
ln r
n =0 m= −∞
N
σ rscat
θ = ∑
∞
∑ imA
mn
n =0 m = −∞
1
1
Σ nrθ ( z )[ H m ' ( k n r ) −
H m ( k n r )]e imθ +
r
kn r 2
(11)
2
2m 2
µ
B
U
(
z
)
[
H
'
(
l
r
)
+
(
l
−
) H m (l n r )]e imθ
∑
∑
mn
n
m
n
n
2
r
l
r
n =0 m = −∞
n
L
∞
N
∞
σ rzscat = −∑
L
∑ AmnΣ nrz ( z)H m ' (k n r)e imθ +∑
n =0 m= −∞
∞
∑ imµB
U n ' ( z)
mn
n =0 m= −∞
1
H m (l n r )e imθ
ln r
(12)
where functions Σ ijn (z ) are the coefficient related to the wave structure of the nth mode. The total
displacement and stress field in the plate are actually the sum of the incoming and scattered fields, i.e.
u = u inc + u scat ; σ = σ inc + σ scat .
(13)
Three-Dimensional BEM Formulation of the Elastodynamic Problem
The boundary integral formulation for the time-harmonic elastodynamic state with zero body force,
defined over the region Ω with boundary Γ is
clki uki = ∫ ulk* pk dΓ − ∫ plk* uk dΓ
Γ
(14)
Γ
in which the displacement components uki at any boundary point “ i ” are related to the displacement
uk and traction pk components over the whole boundary Γ , and the fundamental solution
displacement ulk* and plk* on the boundary Γ due to a unit load at point “ i ”. The coefficient clki
depends on the smoothness of the boundary if point “ i ” is a boundary point. clki = δ lk / 2 for smooth
boundary points and clki = δ lk for internal points. After discretizing the boundary of the disc region
into NE elements. Equation (14) can be transformed into
NE
NE
c i u i + ∑ ∫ p * dΓ u j = ∑ ∫ u * dΓ p j
Γ
Γ


j =1  j
j =1  j
(15)
Here vector notion is used, i.e.
*
u11
u1 
 p1 
 *
u = u 2  ; p =  p 2  ; u * = u 21
*
u 31
u 3 
 p3 

u12*
*
u 22
*
u 32
*

 p11*
u13
 *
* 
*
u 23
 ; p =  p 21
* 
*
 p31
u 33


p13* 
* 
p 23

* 
p 33

p12*
*
p 22
*
p 32
)
 H ij
) ij
*
ij
By defining G ij = u * d Γ , H = ∫ p dΓ , and H =  )
∫Γ
Γj
 H ij + c i
The system equation for node “ i ” becomes
j
(16)
when i ≠ j
when i = j
Key Engineering Materials Vols. 270-273
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NE
NE
j =1
j =1
∑ H ij u j = ∑ G ij p j
or in a matrix form
HU = GP
(17)
(18)
The fundamental solutions for a three-dimensional elastodynamic problem can be written as [10]
ulk* =
1
4πµ
)
)
[U 1δ lk − U 2 r,l r,k ]
)
1 dU 1 1 )
∂r
2 )
∂r
p =
[(
− U 2 )(δ lk
+ r,k nl ) − U 2 (n k r,l − 2rl r,k ) −
4π dr
r
∂n
r
∂n
)
)
)
2
dU 2
∂r
c
dU
dU 2 2 )
2
r,l r,k
+ ( L2 − 2)( 1 −
− U 2 )r,l nk ]
dr
∂n
cT
dr
dr
r
(19)
*
lk
(20)
where
)
exp(− k 2 r )
1
1 exp(−k 2 r ) c 22 1
1 exp(− k1 r )
U1 =
+( 2 2 +
)
− 2( 2 2 +
)
r
k2 r
r
k1 r
r
k2 r
c1 k1 r
(21)
)
exp( −k 2 r ) c22 3
exp( − k1r )
3
3
3
U2 = ( 2 2 +
+ 1)
− 2( 2 2 +
+ 1)
k2 r
k2r
r
c1 k1 r
k1 r
r
(22)
and k1 = ik L ; k 2 = ikT .
Consider the mesh geometry as shown in Fig. 1 and separate the boundary elements in four groups
“Top”, “Cyl”, “Bot” and “Def”, which stands for the elements on the top plate surface (excludes the
defect), virtual cylindrical boundary, bottom surface and those on the defect. The equation (23) can be
written as
 uTop 
 0 
u 
t 
Cyl 

[H ]  = [G ] Cyl 
u Bot
 0 


 
u Def 
 0 
(23)
Here the traction free boundary conditions are already implemented into the equations. The only
non-zero traction term is from the virtual cylindrical boundary, where the incident wave and scattered
waves superimpose.
Assume that there are k elements in the z direction and l elements in the θ direction. Apply
equation (13) to each node, and truncate the Fourier transform to term M (instead of ∞ ). The
displacement and traction matrix on the virtual boundary can be written as
{u}3Γkl×1 = [u ]3I kl×1 + [u ]3BSkl×JV {A}JV ×1
(24)
{t}Γ3kl×1 = [t ]3I kl×1 + [t ]3BSkl× JV {A}JV ×1
(25)
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where J is the maximum number of modes that can propagate in the plate, and V is the truncation
BS
BS
parameter for the angular frequency, i.e. J = N + L +2, V = 2 M + 1 . [u ]3kl× JV and [t ]3kl× JV are the
eigen mode wave structure matrix in which each column represents one eigen mode and each row
gives one component of the displacement or traction at one virtual element. Eliminate {A}JV ×1 in
equation (24) and (25), we can get
{t}3Γkl×1 = [t ]3I kl×1 − [t ]3BSkl× JV [u −1 ]BSJV ×3kl [u ]3I kl×1 + [t ]3BSkl× JV [u −1 ]BSJV ×3kl {u}Γ3kl×1 ,
(26)
thus in equation (23), tCyl can be expressed as uCyl , which in turn can be solved from the linear
algebraic equation. Once uCyl is obtained, the scattered wave displacement field and traction field can
be calculated based on equation (13). Furthermore, the scattering coefficients for each wave mode can
be computed based on equations (24), (7) ~ (9).
To check the validity of the BEM solution, the energy conservation method was used. It states that
in an enclosed wave field region with no source or sink, the wave energy flows into the region should
be equal to that flows out. The average power P radiated across a closed surface S can be evaluated as
[11]
P=
ω
2
(
)
)
Im ∫ u(r ) ⋅ (σ (r ) ⋅ n)dS ,
s
(27)
)
where n is the normal vector of the surface S and the symbol ‘Im’ denotes the imaginary part of a
complex number.
Numerical Results
Consider a numerical experiment as schematically shown in Fig. 1, where the plate thickness is 1mm,
the radius of the through-plate hole is 1mm, and the radius of the virtual cylindrical boundary is
10mm. There are 200 elements uniformly meshed on the virtual boundary. The total number of mesh
elements is around 1100. Consider a plane S0 mode Lamb wave of frequency 0.8 MHz propagating in
the positive x direction. The calculated r , θ and z component of the scattered wave particle
displacement amplitude on the virtual cylindrical surface is shown in Fig. 2 (a), (b) and (c),
respectively. Fig. 3 (a), (b) and (c) shows the model decomposition values for the S0, A0 and SH0 mode
of different angular Fourier components. In this specific case, we truncated the angular expansion
terms to M =12 instead of ∞ and discarded higher order items since they are negligible in amplitude,
so m=-M,…,M components are displayed from 1 to 25 in those figures.
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
50 2
48 49
47
1
2 3
4
1.5
6
7
8
9
1
10
11
12
13
0.5
0
30
29 28
27
5
26
(a)
14
15
16
17
18
19
20
21
22
23
25 24
x
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
50
48 49 2.0
1
2 3
4
1.5
0.5
0.0
27
6
7
8
9
1.0
29 28
1
5
26
25 24
x
(b)
23
10
11
12
13
14
15
16
17
18
19
20
21
22
50
48 49 0.4
47
46
45
0.3
44
43
0.2
42
41
0.1
40
39
0.0
38
37
36
35
34
33
32
31
30
29 28
27
2 3
4
5
6
7
8
9
26
25
10
11
12
13
14
15
16
17
18
19
20
21
22
24 23
x
(c)
Fig. 2. (a) r component, (b) θ component, and (c) z component of the scattered wave displacement
amplitude on the virtual cylindrical boundary.
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It is seen that all the three displacement components are symmetric with respective to the x axis.
Numerical experiments with various numbers of elements for mesh generation and different size virtual
circular region showed similar scatter profile patterns, which indicate good stability and convergence of
the code developed. The numerical error for the calculation from equation (27) is found around 2.5%.
From Fig. 3 (a), (b) and (c), we could see that the scattered wave has two major components, one is the S0
mode Lamb wave and the other is the n0 mode SH wave. It is thus clear that mode conversion occurs at
the circular through plate hole from the S0 mode Lamb wave to n0 mode SH wave, but not to A0 mode
Lamb wave although all and only these three modes are admissible at that frequency.
Figure 4 (a) and (b) draw the angular amplitude profile for the displacement of S0 mode Lamb
wave and n0 mode SH wave. Notice that in those figures, the wave vector direction is not necessarily
in the radial direction due to the geometry of the defect and possible wave grating around the
through-plate hole. Nonetheless, the scattered Lamb wave amplitude is seen maximized along the x
axis and the SH wave is around 45 degrees away from negative x direction.
1.2
1.20
1.0
1.00
0.8
0.80
0.6
0.60
0.4
0.40
0.2
0.20
0.0
0.00
1
3
5
7
9
11
13
15
17
19
21
23
25
1
(a)
3
5
7
9
11
13
(b)
15
17
19
21
23
25
1.2
1.0
0.8
0.6
0.4
0.2
0.0
1
3
5
7
9
11
13
15
17
19
21
23
25
(c)
Fig. 3. (a) S0 mode Lamb wave, (b) A0 mode Lamb wave, and (c) n0 mode SH wave angular Fourier
component amplitude of the scattered wave particle displacement on the virtual cylindrical boundary.
1
46
47
48
45
44
49 50 2
1
2
3
4
5
6
46
7
1.5
45
44
8
43
42
41
40
9
1
0.5
39
0
38
37
36
35
34
40
13
39
14
38
15
37
36
35
34
33
32
19
20
31
30
29
28 27
25 24
26
23
22
21
49 50 2
2
4
5
6
7
8
9
1
10
11
0.5
12
13
0
14
15
16
17
18
19
20
31
30
29
28 27
25 24
26
x
3
1.5
43
42
41
10
11
12
16
17
18
33
32
47
48
23
22
21
x
Fig. 4. The scattered (a) S0 mode Lamb wave and (b) n0 mode SH wave particle displacement
amplitude on the virtual cylindrical boundary.
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Conclusions
Hybrid boundary element normal mode expansion technique was used to model the guided wave
scattering from a 3-D defect. The developed code was applied to study the scattered wave field from a
circular through-plate hole with a plane S0 mode Lamb wave incidence. The scattered displacement
field was calculated and plotted in amplitude angular profile. Model decomposition of the scattered
wave field gave the amplitude of each wave mode that was admissible in the plate. Mode conversion
was observed from Lamb wave to the SH waves at the through-plate hole.
Acknowledgments
This work is partially funded by the Oak Ridge National Lab, Oak Ridge, TN.
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