HYBRID MODELING OF ELASTIC WAVE SCATTERING IN A
WELDED CYLINDER
A. Mahmoud, A. H. Shah, N. Popplewell
Faculty of Engineering, University of Manitoba, Winnipeg, MB R3T 2N2, Canada
ABSTRACT. In the present study, a 3D hybrid method, which couples the finite element region with
guided elastic wave modes, is formulated to investigate the scattering by a non-axisymmetric crack in a
welded steel pipe. The algorithm is implemented on a parallel computing platform. Implementation is
facilitated by the dynamic memory allocation capabilities of Fortran 90T and the parallel processing
directives of OpenMp™. The algorithm is validated against available numerical results. The agreement with
a previous 2D hybrid model is excellent. Novel results are presented for the scattering of the first
longitudinal mode from different non-axisymmetric cracks. The trend of the new results is consistent with
the previous findings for the axisymmetric case. The developed model has potential application in
ultrasonic nondestructive evaluation of welded steel pipes.
INTRODUCTION
Welded steel pipes are used widely in engineering applications. The pipes may
fail due to severe loading and/or environmental conditions, so that there is an imperative
need to nondestructively inspect their structural integrity. The use of ultrasonic guided
elastic waves has proved promising in this endeavor [1]. The scattered field of an incident
wave by an irregularity, e.g. a crack, carries substantial information about the irregularity.
This information might be inferred by comparing test results with theoretical predictions.
There are few theoretical studies on the scattering of guided elastic waves from
irregularities in welded steel pipes. Zhuang et al. [2] investigated the scattering of guided
elastic waves from axisymmetric cracks in an infinitely long, welded steel pipe by using the
hybrid method. The hybrid method is a technique that combines a finite element
idealization of a bounded region containing irregularities and a wave function expansion in
the exterior region. Continuity conditions of displacements and interactive forces are
applied at discrete points on the common boundaries between the two regions. This results
in a system of linear equations in the unknown wave function amplitudes. These
amplitudes are used to obtain the reflection and transmission coefficients. Since welding
defects are usually localized in the vicinity of the weld, the hybrid method is ideal for
investigating elastic wave scattering in welded pipes. Zhuang et al.'s 2D hybrid model
proved very efficient in handling 2D geometric and material irregularities [2]. However,
the most general and most practical case of elastic wave scattering by a non-axisymmetric
crack was not addressed. This generalization will be the subject of this paper. First, a
modified algorithm of the hybrid method is formulated in 3D that compromises the
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
71
Finite Element
.R
Region J\
B
~
\
\
A i.
R
!
i
B
J ™
i.
Rf
/£
+
h unctions
Functions
i Li
z-
i
j
\
!
+
i
!
Cylinder's Axis
II
i ^r
i
i
!
FIGURE 1. Lateral section in a cylinder having a surface breaking, symmetrical crack.
B-
4-
3-
2-
Crack I
1
\
z=z
2+
3+
1+
z=0
Cylinder's Axis
FIGURE 2. An illustration of the nodal numbering in the finite element mesh.
conflicting requirements of computational efficiency and numerical accuracy. Next, the
algorithm is implemented and validated. Then, new results are presented for the scattering
by a non-axisymmetric crack.
FORMULATION
A lateral section of a cylinder having a mathematical crack (i.e. no width) is
represented in Figure 1. However, for the sake of illustration, a hypothetical thickness is
shown. The nodal points in the finite element mesh are grouped in vertical planes. The
planes are ordered as shown in Figure 2. When an incident wave mode (/?,#),
corresponding to a wavenumber p in the 0-direction and a wavenumber ^pq in the zdirection, strikes the crack, a scattered wave field is generated. The displacement vector of
the scattered field on the boundaries B+ and B~ can be written in the matrix form:
Uf=G±A±
(1)
72
Moreover, a "+" superscript refers to B+ , a "-" superscript refers to B , and a "S"
superscript refers to the scattered wave field. The quantities on the right side of equation
(1) are defined as:
.(2)
where
(4)
M is the number of circumferential wavenumbers; Wm (m-0,1,2, ... A/) is the number of
axial modes corresponding to a particular m ; awn is the unknown amplitude of the
(m9ri) scattered mode; and
\_U\mn
U mn ~
U
\mn
U
\mn
~'Uimn
U
imn
U
imn
'"UNBmn
U
NBmn
U
NBmn J '
(?)
omn
The urimn, ufmn, and w^ are the displacement components in the r, 0 and z directions,
respectively, at the /th nodal point. They correspond to the (m,ri)th mode in the
undamaged cylinder [3,4]. The normalizing factor, gmn , is given by:
(6)
9"
=
own
where NB is the number of nodal points on each boundary, B+ and B'. Similarly, the
force vectors at the boundaries B+ and B' due to the scattered field can be written as:
Pf=F^*
(7)
where
j7*_|j>±
^ -\^-Ml
"
D±
-WM"
^m»=———[Aw,
omn
The primn, peimn, and
p±
-ll
r
^L
p±
i>±
" ^-INj^Ol
Plmn
-
-PL
i>±
^OA^
PL
o±
o±
" ^ 7 " ^^-
Pimn
'"P^mn
o±
" ^M7
P^mn
ra±
" ^M
P^nn]'
(9)
/>^w are the force components in the r, #andz directions that
correspond to the (m,n) mode at the /th nodal point. The boundary displacement and
force vectors can be constructed for the incident wave as:
73
(10)
exp(-y
where a "7" superscript indicates the incident wave field. G*g and F^ are the columns of
the G± and F1 matrices corresponding to the incident mode (p,q) and a*pg is the
amplitude of the incident mode (p,q).
The bounded region that contains the irregularity is idealized by finite elements. 20node brick elements are used throughout the entire finite element mesh. A conventional
assembly over the finite elements gives the total energy of the bounded region, ER, as
[5,6]:
= \{URTSUR-UBTPB -UBTPB]
(U)
where a "T" superscript denotes a transpose and
T
UK' =(U!
I//), S = K-
513
S23
0
0
$33
*^34
0
0
(12)
0
0
Sy (i, j = 1,2,3,4, B) are the dynamic stiffness sub-matrices between the degrees of
freedom associated with the nodes in plane "z" and the degrees of freedom associated with
the nodes on the plane "j". (See Figure 2.) K and M are the global stiffness and mass
matrices, respectively [5,6].
Minimizing the total energy given by equation (11) results in the following equation
of motion for the interior region [5,6]:
SER = SURTSUR - 8UBTPB = 0.
(13)
S implies a first variation and a bar denotes the complex conjugate.
The general solution is obtained by applying the displacement and force continuity
conditions at the common boundaries B+ and B~ between the bounded and unbounded
regions, as follows:
1
s
U
= ^UB~^
+ IIB^
^B
where
74
P
=rB
P1 ^+rBPs
B
r
(14)
=(u]B-T UJ;-T) USBT=(USB-T t/
(15)
Using equations (1) and (7) in equation (14) and, in turn, in equation (13) results in:
S»Ut + SUU2
STU} + S22U2
(16a)
(16b)
= 0
= 0
34t/4
T
S aU2 + S33U}
+ S3BUB = 0
S3BUB = 0
T
+ SBBUB ) = G PB
(16c)
(16d)
(16e)
where
G=
G'
0
0
G+
(16f)
Elimination of U1 and I72 in equations (16a and 16b) and substituting for the resulting
U3 and U4 into equation (16e) gives:
[GT(S;BG-F)]A=GT(PB-S;BU'B)
(17)
where S^B is detailed in [4], and
F
0
0
F+
(18)
The size of the matrix of coefficients in equation (17) is 2NT by 2NT where NT is the
total number of modes in the wave function expansion. The amplitudes <arn are obtained
by solving the linear equation (17) for A+ and A~, and making use of (4). The reflection
coefficients, Rpq>mn, and the transmission coefficient, Tpg>mn, of the (m,n) mode due to the
incident (p,q) mode are calculated as:
-y2/ > *pq,mn
pq
m ^ p or n * q
(19)
U
m — p and n — q.
The elastodynamic reciprocity relations and the principal of energy conservation are used
to check the accuracy of the numerical computations [7].
75
IMPLEMENTATION
The dynamic memory allocation capabilities of Fortran 90 [8] facilitated the
numerical implementation of the new condensation procedure. The directives of OpenMP
[9] were employed to implement the code on a parallel computing platform. Four threads,
each with a 700 MHz clock speed and a \GB hard drive, save about 67% of the CPU time
compared with the use of only one thread.
In the following numerical results, a non-dimensional frequency, Q, and a nondimensional wavenumber, 7 are defined respectively as:
• * '• •
<20)
H is the cylinder's thickness. // andp are the shear modulus and mass density of the pipe
material. The crack's geometry is described by two parameters: the non-dimensional crack
depth, Z), and the non-dimensional crack length L. They are given by
D=— andL=-^-,
H
In
(21)
where d is the crack depth and #c(rads) is the crack angle. Furthermore, the nondimensional width of the bounded region, W9 and the non-dimensional breadth of the weld,
B, are defined as:
=
:,
(22)
where w is the width of the bounded region, i.e. w = z+ — z~ , and b is the weld's
(constant) thickness. The notation | RUtnm \ refers to the reflection coefficient for the
( m, n ) mode from an incident ( &, / ) mode. Similarly, \Tklmn \ is the corresponding
transmission coefficient. To illustrate the validity of the algorithm, the following example
is presented for which previously and independently obtained results exist.
Example 1
Two steel pipes are welded edgewise by a vertical weldment, with a surfacebreaking crack on one of the weldment's interfaces. The non-dimensional width, W9 of the
bounded region is 0.1875 for a non-dimensional thickness of the weldment of 0.09375.
The thickness-to-mean radius ratio, H I R, and the total thickness, H , of the steel pipes are
0.1 and 5.08mm, respectively. The cylinder's material has a modulus of elasticity, E , and
Poisson ratio, v, of 211.7 GPa and 0.29 respectively. The longitudinal and torsional
wave speeds are cp =35.6xl0 3 mis and cs=5.88xlQ3 m/ s . The weld material is
orthotropic. Its density is 780 kg/m3 and the other elastic constants (in Gpd) are
summarized in Table 1. Figure 3 shows that the results from the 2D [2] and 3D hybrid
models are identical. This example also shows the sensitivity of the reflection coefficient,
76
TABLE 1. Elastic constants of the weld material.
C
ll> C22
231.83
C33
265.23
c12
c13 , c23
67.83
161.97
C
44 9 C55 •> C66
82.00
0.8
•D=0.5, Zhuang et al.
D=0.5, Hybrid Model
•0=0.25, Zhuang et al.
0=0.25, Hybrid model
0 = 0.125, Zhuang et al.
Q — 0=0.125, Hybrid model
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
1.0
FIGURE 3. Reflection coefficients versus normalized frequency for variously sized
axisymmetric cracks in a welded steel pipe. H/R=Q. I, V=0.29, JF=0.28,5=0.14
1.0
2.0
3.0
4.0
Q 5.0
FIGURE 4. Reflection coefficients versus normalized frequency for variously sized nonaxisymmetric vertical cracks in a welded steel pipe. H/R=QA, v=0.29, FF=0.28,5=0.14
^oioi> to a variation in the non-dimensional crack depth, D. In the next example, new
results are presented that demonstrate the ability of the current 3D hybrid model to handle
non-axisymmetric cracks.
Example 2
The geometric and material properties are identical to those used in Example 1. Results are
presented in Figure 4 for a vertical crack having non-dimensional lengths, L, of 0.5 or 0.1
and a constant non-dimensional crack depth, D, of 0.5. The corresponding axisymmetric
77
curve is superimposed on Figure 4. A comparison of the three curves shows that the
reflection coefficient 17? 0101 1 is very sensitive to the variation of the non-dimensional crack
length, L.
CONCLUSIONS
In the present study a 3D hybrid method, involving finite elements and a wave
function expansion, was extended to analyze the scattering of guided elastic waves by nonaxisymmetric cracks in a welded steel pipe. A modified algorithm of the hybrid method
that circumvents the extraordinarily large computer storage and prolonged computational
time through a new condensation technique was formulated and implemented successfully.
Numerical results were compared with available numerical data. Results from the current
model and a previous 2D hybrid model are identical. The trend of the new results is
consistent with previous findings for an axisymmetrically cracked pipe.
ACKNOWLEDGMENT
The first author would like to acknowledge the financial support from the University of
Manitoba through the graduate fellowship award. AHS and NP acknowledge the financial
support of the Natural Science of Engineering Research Council of Canada. The help and
cooperation of the staff at the University of Manitoba High Performance Computing (HPC)
Unit is appreciated highly.
REFERENCES
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(1999).
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3. Bai, H., "Elastic Wave Scattering in Cracked Cylinders," Ph. D. thesis, University of
Manitoba, Canada (2001).
4. Mahmoud, A., "Hybrid Formulation of Wave Scattering in a Cracked Circular
Cylinder," M. Sc. thesis, University of Manitoba, Canada (2002).
5. Bathe, K. J., Finite Element Procedures in Engineering Analysis. Prentice-Hall Inc.,
Englewood Cliffs, New York, (1982).
6. Zienkiewicz, O.C., The Finite Element Method. McGraw-Hill Book Co., London,
(1977).
7. Karunasena, W. M., Shah, A. H., and Data, S. K., J. Eng. Mech. 117(8), 1738-1753
(1991).
8. Schick, W. and Silverman G., Fortran 90 and Engineering Computations. John Wiley
and Sons, New York, (1995).
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Programming in OpenMP. Morgan Kaufmann Publishers, San Francisco, CA, (2001).
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