3D MODELING OF A MAGNETO-OPTIC IMAGER BY A DYADIC
GREEN'S FUNCTIONS APPROACH
J.M. Decitre1'2, D. Premel1 and M. Lemistre2
^aboratoire Systemes et Applications des Technologies de 1'Information et de
1'Energie (SATIE - CNRS UMR 8029) - Ecole Normale Superieure de Cachan
61 avenue du President Wilson, 94235 Cachan Cedex - France
2
Office National d'Etudes et de Recherches Aerospatiales (ONERA) - BP 72
29 avenue de la Division Leclerc, 92322 Chatillon Cedex - France
ABSTRACT. A 3D semi-analytical model based on dyadic Green's functions is proposed to solve
the forward problem for a magneto-optic imager with a simplified eddy-current excitation. The
normal component of the magnetic field is evaluated in place of the ferrimagnetic garnet. A
comparison between data provided by the model and data computed by a 3D finite element method is
presented. Theoretical results are also validated with experimental data obtained by a Magneto-Optic
Imager designed by Physical Research Incorporation.
INTRODUCTION
Among all techniques using eddy-currents for non destructive testing, the magnetooptic imaging combines simplicity of use and lower investigation time. Thus, the
Magneto-Optic Imager 303 (MOI 303®) designed by G.L. Fitzpatrick and Physical
Research Incorporation (PRI) examines straightforwardly an area of about 50 mm x 60
mm and provides an image from this area which highlights the defects [1]. Compared to a
classical Eddy-Current method, the gain of time is about 8. Nevertheless, interpretation
requires qualified people even when a lot of methods are proposed for helping the
detection of flaws by a complementary image processing and for improving the MOI®
capabilities in order to detect corrosion or cracks close to fasteners [2,3,4].
Models of MOI® using finite-element techniques has already been developed
[5,6,7]. hi this paper, after a brief recall of the principle of MOI system, the authors give a
faster method for 3D modeling, based on integral equations by using a Dyadic Green's
function approach. So, the images provided by MOI can be interpreted and the capabilities
of the imaging system can be improved. Data provided by this integral model are
successively compared to data provided by a finite element (FE) model and experimental
images obtained with MOI 303® of PRI.
OPERATING CYCLE OF MAGNETO OPTIC IMAGER
The sensor used in MOI® is a 75 mm diameter garnet. It features a large Faraday
rotation (about 30000%m): when a polarized beam crosses this garnet, his rotation angle
varies according to the local material magnetization. Its hysteresis loop is given in figure 1.
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
695
Faraday effect
I
ePaK,HaU « M7
Pixels
either white (Mz = -Ms)
or black (Mz = +MS)
FIGURE 1. Hysteresis loop of MOI garnet.
As the garnet magnetization can take only two values (+MS or -Ms), the rotation
angle in relation to the thickness of the garnet can only take two values too. It is the reason
why each pixel on the camera is either white (light) or black (dark). The optic system
which creates a polarized beam includes classically a light source (LEDs), a polarizer, an
analyzer and a CCD camera (see figure 2).
Two sources of magnetic field are added in the MOI®. First, a circular bias coil is
placed around the garnet creating a static magnetic field Hbias- This value can be adjusted
by the user. A disturbance field is produced by the defect itself when the induction foil is
supplied. Therefore, eddy-current lines are generated in the target (see figure 2). In
absence of defect in the investigated conductive target, these lines are parallel and no
normal component magnetic field is created. The pixels of the image remain white. When
a defect is present, these current lines are curved and a normal component of the magnetic
field appears over the surface of the target. Then, the resulting magnetic field becomes
greater than the garnet's threshold Hthreshow; the pixels become black.
Magneto-optic image
F/xefe decome locally
bfack wh&n H. > Hfewsfc^
during th® cyst® and
thBy &a@p this slafe unfit
Camera
Light
source
Normal .magnetic fitlci
Anaiyztr
Polarizer
Garnet
Bias ooli
Mirror
Induction
foil
Target plate in aluminium
Def&ct md EddyCurmni iin&$.
FIGURE 2. Operating cycle of magneto-optic imager (in reality, the MOI 303 has got two transformers
constituted by perpendicular secondary windings in order to create a multidirectional magnetic field).
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FIGURE 3. Example with the PRI's MOI 303®: an experimental image at 50 kHz with vertical monodirectional excitation for a 8 mm diameter fastener with a 2 mm long crack on test side.
iocatmof
the ganiti <£
Fisw
FIGURE 4. (a) Real Eddy-Current excitation system, (b) simplified Eddy-Current excitation geometry, (b-1)
3D view and (b-2) slice view.
So, this system is allowed to detect defects embedded at different depths inside the
structure by changing the excitation frequency. Nevertheless, the depth analysis does not
exceed a few millimeters, due to the limited skin effect. The expression of the skin depth 6
is expressed as follows:
s = ^=
(1)
where f is the frequency, a and jn are respectively the conductivity and the magnetic
permeability of the structure under test.
A lot of phenomena disturb the images such as: the variation of the magnetic field
distribution due to bias coil, the minority garnet magnetic domains (appearance of
"worms" in the images) and the noise (figure 3).
THE FORWARD PROBLEM
The simplified geometry of eddy-current excitation presented in figure 4, is
constituted by several copper foils placed like a vertical large coil which creates parallel
lines of induced currents in a conductive target. The excitation system is 80 mm long, 80
mm wide and 30 mm high and foils are 3 5 um thick, approximately such as in the
commercialized Magneto-Optic Imager MOI 303®.
The purpose of this paragraph is to establish analytical calculus in order to
determine the magnetic field at the location of the garnet (just above the inferior induction
foil), due to the current sources and their interaction with the flaws. The target is
characterized by the constitutive parameters given in table 1.
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TABLE 1. Constitutive parameters of the target.
Conductivity
7
tfo = 3.52x10 S/m
Permittivity
9
*o = 1/(3677l0 )F/m
Permeability
JL/O = 47TlO~7 Him
Calculus of the Incident Electrical Field
The current density, in any point P'(x',y',z'), created by the induction foils is
given by the following expression:
J0(x',y',z')=i0
section 1
u
+i
section 2
(2)
- «o
section 3
u
-1
section 4
To calculate the incident magnetic field in this first step, let us consider an isotropic
and homogeneous aluminum target. Let us use the dyadic Green's function G21 , which
allows to express the electric field E0 in any point P(r) of the target, due to the source
[8]:
EQ(f) = i
jjj G2l(ee\r,f')
(3)
excitation foils
Add of a Defect in the Target
In this second step, let us introduce the normalized contrast function in terms of the
relative conductivity:
(4)
Then, the presence of a flaw is translated like a source of fictitious currents:
(5)
This current density changes the previous electrical field E 0 . So, let us introduce
the dyadic Green's function G22
to calculate the electric field E 2 in the target taking
into account the fictitious currents [8]. The state equation is then given by:
698
flaw
One can note that the internal electric field E 2 is computed by the conjugate
gradient method.
Observation
Finally, the magnetic field Hd at the location of garnet (i.e. above the surface of
the conductive copper plate) due to the disturbance electric field E 2 in medium 2 is
evaluated by:
Hd(r) = /&>//0cr0
G12
e
(r,r')/(r')£' 2 (r f )J r'
(7)
flaw
The dyadic Green's function G12(me) [8] indicates the magnetic field in the air
region (1) due to a point source located in the damaged zone.
Discretization
The discretization of the two coupled equation (6) and (7) is performed by
subdividing the supposed damaged zone into a regular mesh. We obtain a system of two
coupled matrix equations:
yy - F^
12 F e->
2
(8)
-2 ~ ^22 ^ e2 + eO
where eo is the incident electric field vector and 62 is the total electric field vector. The
observation vector y is the magnetic field. F12 et F22 are the matrix derived from the
Green's functions Gi2 and G22- Finally, F is a diagonal matrix which contains the values of
the contrast function f.
VALIDATING THE FORWARD MODEL
Simulation Conditions for the First Test: a Rectangular Defect
A parallelepipedic defect is first considered. It is 8 mm long, 1 mm wide, 2.5 mm
deep (see figure 5). The simulated experiments are carried on with a frequency of 5 kHz. It
is important to note that a 4x16x1 regular mesh is used to discretize the damaged zone
whereas more than 50000 elements are required to perform the same simulations with
Ansys® software (3D model with FE method).
Comparison between Finite-Element Method and Green's Method
The figure 6 presents some results of the comparison between the simulated data provided
by the integral model and FE data. The amplitudes of the normal components of the
magnetic fields are graphically shown. The amplitude of the magnetic field is the most
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i mm
Mesh
Defect zoee
FIGURE 5. Simulations with a parallelepipedic defect (8 x 1 x 2.5 mm) in a 5 mm height aluminum plate.
INTEGRAL EQUATIONS
FINITE ELEMENT
METMQP..(ANSYS)
.METHOD
FIGURE 6. Results with the rectangular defect with both the dyadic Green's functions and the finite element
method.
FIGURE 7. Experimental data obtained with MOI with the rectangular defect.
interesting to consider for the comparison, since the MOI® system is able to provide an
image of the same kind. By many computations, we have noted that the RMS error
between simulated data has been lower than 15% and that this value decreases as the mesh
of the FE method is refined. Moreover, the error on the evaluation of the maxima of the
amplitudes of the magnetic field does not exceed 5%. The biggest advantage of the integral
approach is the speed of computations with regards to a FE method: the computation
requires only a few minutes instead of many hours.
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Experimental Data with MOI303®
Because of the images provided by the MOI® system are binary, no actual
quantitative cartography can be carried out to make a comparison with the captured
images. However, a quantitative approach may be approximated by progressively
decreasing the static bias magnetic field (by action of current in bias coil in figure 2) and
creating a series of qualitative images. For each image, the switching threshold of the
garnet is obtained with a weaker magnetic field due to the flaw. So, the series of images
are obtained for which the contours give an idea of the spatial evolution of magnetic field
due to the defect. The results, presented in figure 7, are in agreement with the results
presented above.
Complementary Results
The integral model was used to simulate several experiments. One of them consists
in the detection of a crack near a hole. The diameter of the hole is 6 mm and the crack is 3
mm long, 0.3 mm wide and 5 mm deep (figure 8, first column). The frequency remains at
5 kHz and the target is the same as above. The simulated data provided by the integral
model (figure 8, second column) are compared to magneto-optic images for a given static
bias magnetic field (figure 8, third column). The results given by the model are in good
agreement with both experimental images and typical results obtained by other authors [5].
(UPPER LAYER)
MTHOD
MOi
§ 8 mm/
FIGURE 8. Experiment with a 2.5 mm height and a 6 mm diameter non emerging hole at 5 kHz, without
any crack (first line), and with a 3 mm long and 0.3 mm wide crack (second line). The first column presents
the defects. The second column presents the results given by the model whereas the third column shows the
images obtained with MOI in similar experimental conditions.
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CONCLUSION AND FUTURE WORKS
In this study, a fast numerical model based on dyadic Green's functions was
developed. Some numerical experiments have lead to the validation of the model by a
comparison to FE data. This approach has numerous interests. First, the computations time
is around few minutes instead of many hours by a finite-element method. This fact is
mainly due to the requirement of only meshing a small anomalous zone instead of a whole
3D geometry. The optimization of the experimental set up may be achieved without using
any other simplification or symmetry to reduce the huge computational time required by
the FE method. Future works will improve this model by taking into account the tilt of the
driving coil and the tilt of the garnet.
Finally, the ultimate goal of this project is to develop a sensitive quantitative
magneto-optic imager. So, this approach of modeling may not only improve the
capabilities of detection of the imaging system by optimizing of the system, but it can also
lead to the resolution of the inverse problem.
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