THREE DIMENSIONAL MODELING OF THE DC POTENTIAL DROP METHOD USING
FINITE ELEMENT ANDBOUNDARYELEMENT ANALYSIS
S. Nath, T. J. Rudolphi and W. Lord
College ofEngineering
Iowa State University
Ames, Iowa 50011
INTRODUCTION
Finite element (FE) and boundary element (BE) methods ofnumerical analysis have been
utilized in the solution ofnumerous electromagnetic nondestructive evaluation (NDE) phenomena.
Investigations using these methods have been undertaken by Lord [1,2,3], Palanisamy [4],
Atherton [5], Beissner [6], Burais [7] and others.
The finite element method (FEM) is a domain technique of solving the underlying governing
equation in the region. The solution obtained in the total region is ideal to study energy/defect
interactions, but the extensive discretization demands vast computer resources. On the other hand,
the potential savings in computation resources, due to a limited surface or boundary discretization,
is the primary motivation in using the boundary element method (BEM). The numerical solution
of the boundary integral representation of the governing equation is the BEM. This is based on the
two point kernel function (or fundamental solution) which is a distinctive feature ofthe BEM. The
kernel exists only for linear operators, which poses a limitation.
This research is part of an endeavor [8, 9] to compare and contrast FE and BE methods as
applied to electromagnetic NDE. In particular, this paper presents a three dimensional FE and BE
model ofthe DC Potential drop method to characterize fatigue cracks. The next few sections
describe the principle ofthe DC Potential drop (DCPD) method and its applications, the FE and
BE formulations, data obtained from modeling a compact tension specimen, and finally, some
conclusions.
DC POTENTIAL DROP METHOD
The operating principle of the potential drop method is that any surface defect or crack in a
conducting specimen will perturb the ftow of electric current around it, causing a measurable
potential difference across the crack. In the DCPD method, a constant current is passed through a
specimen and a probe straddles the crack with a fixed spacing. As the crack increases in length,
the uncracked cross-sectional area ofthe specimen decreases with an increase in the current path
resistance and potential. In practice, for a particular geometry, calibration curves are presented in
the form of JL as a function of A, where Uo is the reference potential drop, U is the potential
Uo
W
drop as the crack length increases and
~
is the crack length to specimen width ratio (Fig. 1).
Review 01 Progress in Quantitative Nondestructive Evaluation, Vol. 11
Edited by D.O. Thompson and D.E. Chimenti, Plenum Press, New York, 1992
553
DC Source
r------®---
-,,
Probe Leads
q
\
I
'---__t
r
/
-----
~
........
Figure 1
~w
1
_-.-
DC potential drop experimental setup
Applications for the DCPD method include monitoring fatigue, stress corrosion and creep
cracks. Additionally, the technique has been used to evaluate the extent of crack closure in fatigue
crack propagation studies and others [10, 11]. To date, calibration curves have been determined
experimentally [12, 13], analytically (Johnson's formula) [14] and numerically [10, 11]. These
have been accepted for use by the fracture mechanics community. Recently, Iwamura and Miya
[15], and Kubo and others [16], have used the measurement ofpotential in the specimen to predict
the shape and size of cracks in planar and tubular geometries.
The goveming partial differential equation (pde) for the DCPD method is Laplace's equation
V2V = 0
(1)
where V is the steady state potential in the geometry with a constant current in the plane of the
geometry. The next section briefly describes the FE and BE procedures used to solve Laplace's
equation with specific boundary conditions.
FORMULATION
Finite Element
The three dimensional FEM uses variational principles to solve Laplace's equation. The
problem can be stated as
F(V) =
JJ J[( dV
)2 + (dV )2 + (dV )2] dv
dX
dy
dZ
v
(2)
with boundary conditions given as,
V = 0 over surface SI
V = V I at point E
where F(V) is the functional for Laplace's equation, which is minimized over the entire domain.
The known potential at point E and at surface SI gives the desired uniform field around the crack
(Fig.2). By varying the surface SI, the crack length is changed. Eight node isoparametric
elements are used to discretize the geometry over which the potential V varies linearly.
Minimizing the functionalleads to a matrix equation
554
A
Typical
62 .5 mm
four node
boundary
elemen,
w =50mm
Typical
eigh' node
finite element
SI
I
l __ _ _ _
~30mm~
Figure 2
Compaet tension speeimen with boundary eonditions
[S] {P} = {Q}
(3)
where [S] is banded, symmetrie and sparse, {Q} is a veetor due to the boundary eonditions and {P}
is a veetor ofunknowns V to be determined at every node point in the domain.
Boundary Element
The boundary integral equation (BIE) for Laplaee's equation is
e(S)V(s) =
JI [G(t,S) ~~ ctJ - ~~ (t,S)vctJ] dS
(4)
where S is the entire surfaee sUITounding the domain, G is the two point kernel funetion or the
fundamental solution given by
G(t,S)
=--s
41tr
where ris the veetor from Sto X. e(s) is the free term determined from the loeal geometry at point
Sand eorresponds to the prineipal part of the SIE as extraeted from a limit analysis as S
approaehes the boundary. The fundamental solution has a weak singularity
normal derivative has a strong singularity O(
~).
?
O(~), while its
r
This strong singularity is regularized before it is
integrated. Applying the homogenous Neumann boundary eondition over the total surfaee, exeept
over surface SI and point E, the boundary value problem is solved by diseretizing the surfaee into
four node isoparametrie elements, varying the potential over eaeh element linearly, and then
forming a matrix equation:
[T]{P} - [H] {
~~ } = {O}
The eoeffieient matriees [T] and [H] are non-symmetrie and fully populated, while {P} and
(5)
{~Pn}
0
are veetors ofthe unknown potential and its normal derivative at the nodes on the surfaee.
555
RESULTS
A quarter of the compact tension specimen is modeled for the simulations (Fig. 2). The
discretization for the FE model has 1548 nodes as compared to 260 nodes for the BE model. These
meshes are optimized by trial and error. The FE algorithm exploits the symmetric and sparsity
pattern of the global matrix and stores only its lower tri angular portion. A preconditioned
Incomplete Cholesky Conjugate Gradient (ICCG) technique is used to solve the set oflinear
equations, making it quick and efficient. The essence of the BEM technique is the regular and
singular integration ofthe kernel and its normal derivative. The bulk ofthe processing time is
devoted to the integration, and assembly ofthe stiffuess matrix. The cpu time forthe FE and BE
model for predicting each point on the calibration curve is 225.5 and 501 seconds respectively .
2.20
-
2. 10
-
2.00
-
I
./
111
0
w_
o
0
0
1.90 1.80
-
Y. 1.70
J-
Y\I
0
1
--
0
4 ___ __
- JO.,._
f
i
UD
I
1/: ,',
j..•• ".,, ,
I"
J//
1.40
1.30
I.
2.
3.
4.
" li
;~"
1.20
I~'
FEM (t=25 .4mm)
BEM (t=25.4m m)
FEM (t=6.25mm)
BEM (t=6.25mm)
/
0.40
Figure 3
"
, ,
.,','
1.50
1.00
I
l/~2,~.3, /'4
1
1.60
1.10
/
0.60
A
W
0. 0
1.00
FE and BE simulated calibration curves
Figure 3 compares the calibration curves for two different thicknesses for the two models. The
results indicate less than three percent error. Compared to the data obtained from 10hnson's
formula (analytical) and experimental data [101, results from the three dimensional numerical
models for a 25.4 mm specimen (Fig. 4) confirms the validity ofthese codes. Figure 5 is the
calibration curves using FE for varying specimen thickness. In the next two plots one can
visualize the effect ofthe excitation and the boundary conditions in the compact tension specimen.
Figure 6 compares the potential distribution on the top layer for the FE and BE models, which are
nearly identical. Aseries ofpotential distributions in the different layers is plotted in Fig. 7 from
the solution obtained by the FEM.
556
3.40
3.20
3.00
2.80
2.60
-
I
I
3.60
I G
- llIIl
--
-- ---
J--
=,u_~
-,
.1'
..
l.....
1
--
1
:
- ; 0 ...-
~
uo
2
,
,
,)
/ :. .' - .,'
" ,'
, ~/'~ 4
, .,'
2.20
,~:...
2.00
""
~
--
V
~.
1.80
1. lohn on' formu la
..... .? ~ 2. 2 dimensional
./
1.60
1.40
.S
1.00
,,
,,
, "... ,'
.Y. 2.40
1.20
J
.J~
P V
3. Experimental (25.4 mm)
4. 3 dimensional (25.4 mm)
?
400.00
500.00
600.00
0.25. 1 x 10. 3
&-
800.00
700.00
30,20, experimental and Johnson's forrnula comparison
Figure4
UNO
2. 0 I-- ~
2.60 I - f-
-Gil
I
:
<w_
2.00 I - I1.80
I 1
I1
I
11
,
,
2.40 I - I2.20 I - I-
J-
./
3.00 I - ~
i
I
l,.
-- lO_~-----
1.60
:;.
" .'
,~. .,:>#
,/:"
;((:' .
, " ...:.'
-p:L
V'
;-' "
" ."
1.20
___tE
1.00
)
,', J..:'" .
- -, " . ' -. ,,'- ' -
1.40
2
1. Expl (25.4mm)
2. t=25.4mm (FE)
3. t= 12.5mm (FE)
4. t=6 .2Smm (FE)
0.80
0.60
0.40
0.20
0.00
0.00
Figure 5
0.20
0.40
0.60
0.80
1.00
AfW
Calibration curves for varying thickness
557
o
c '"
o
,>
,
..J
.D
.J
l ~
J ~
~
.J
1l
...J
o
.:!~
c ...
~
J
o
Q.
o
0 ,,,___
30~
,
\
"\ '\
I
,
,
,
I
,
62..5 mm
0
c '"
\
"
!
,
w
\
'\
'\
~J.
\
____
0
.>
,
J
--;Omm---
JJ
.J
1l
...J
0
J
o
Q.
.018 02
X-GRIO '
1.0 21. 0 27.0
Figure 6
558
.
.\
"-
1.0 60
. 9.0 12.0 15
I
,
/\
,
I
Potential distribution
I
Figure 7
Potential distribution at different layers
CONCLUSIONS
According to the authors' knowledge, this is the first time a finite thickness compact tension
specimen is modeled for predicting fatigue crack depth. Plots indicate that the potential
distribution is constant throughout the thickness except elose to the excitation. This confirms that
calibration curves are independent of thickness, leading one to conelude that a two dimensional
model is sufficient to predict fatigue crack lengths by the DCPD method. Computing times show
the FE algorithm is faster which can be attributed to the efficient preconditioning and iterative
solver; ICCG. In general, for problems involving infinite boundaries, or problems with more
degrees of freedom, BEM will be faster. For smaller problems, the FEM is efficient and attractive
if one exploits the properties of the global matrix. The BEM needs only surface discretization,
which reduces the problem dimension and, correspondingly, the number of unknowns.
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4.
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5.
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559
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560