2D-3D Transformations Models for NDT Systems Studies
AREZKI DICHE 1*, HASSANE MOHELLEBI 1**, MOULOUD FÉLIACHI 2
1
Département d’Electrotechnique
Université de Tizi-Ouzou,
B.P. 17 RP, 15000,
ALGERIA
2
IREENA(LRTI.-IUT), CRTT,
BP 406, 44602 Saint-Nazaire Cedex,
FRANCE
Abstract: - The present work deals with the exploitation of conformal mapping method principles using
established transformations models for eddy current problems in non-destructive testing (NDT) systems. The
interest of such a transformation is the generation of electromagnetic field for 2D or 3D axisymmetric problems
from only the field distribution known in simpler 2D(x-y) or 3D(x-y-z) geometry. The transformation
technique is applied to both geometry and electromagnetic field distribution and validated by comparing the
predicted results to reference data for 2D study case [4]. Then the developed models are considered in their 3D
forms and applied successfully to 3D structure and quantities.
Key-words: Defect analysis; Transformations models; Conformal mapping; Finite element method; NDT.
1 Introduction
Transformation methods are a very powerful tool in
finite element analysis. An adequate mapping
transforms the problem into an easier one or allows
to take advantage of the symmetries [1-3]. In this
paper transformation models developed to predict
electromagnetic behaviour of axisymmetrical 2D or
3D non-destructive testing problems from the
knowledge of 2D(x-y) field distribution are
presented at first.
The procedure resolution converts, at first, the real
cylindrical geometry to a plate by means of a
geometrical transformation ; in a second step, the
electromagnetic problem is solved in terms of vector
potential in the plate using finite element analysis ;
finally, one transforms the vector potential to obtain
its equivalent distribution within the cylindrical
device.
The predicted results are compared in the 2D case
to those given by reference data [4] to obtain the
validity of the proposed transformations; after that
they were extended and applied to 3D axisymmetric
structure and quantities given in the team workshop
N° 13.
2 Transformations Models
2.1 Geometrical Transformation Model
The geometrical model is obtained with the use of
transformation function technique [2]. For its
construction, we have to solve the inverse problem,
which is consisting on the transformation of the
plate into a tube free of deformation with similar
width than the initial plate.
The method based on the use of the punctual
transformations is presented in detail in several
references as mathematical and electromagnetic
non-destructive evaluation domain [2].The punctual
transformation in descriptive geometry is applied to
transforms, as a demonstration of the interest
method, a line A-B of L dimension as shown
following (Fig. 1).
Y
Y’
M’(x’,y’)
R0
A
O
M(x,y)
θ
B
L
X
x
’
X’
Fig.1 Transformation of line A-B
A point M(x, y) in the A-B segment is
transformed into a point M’(x’,y’) in x’,y’, and z’
co-ordinates (Fig.1). The relationship between the
initial and the transformed co-ordinates is given by:
x'  R0 . cos( x ) ; y'  R0 . sin( x ) ;
R0
R0
with:  
z'  z
(1)
x
R0
Where R0 is the circle radius, x the arc length and
knowing that the z component is set invariable. The
geometrical model which transforms a plate having
some depth into a cylinder is obtained seemly by
considering the plate with depth as an infinite
superposition of plates without depth. Expressions
(2) and (3) give the direct and inverse models
respectively.
y'

 x  R0  arctan x'
P: 
2
2
 y  R0  x'  y'

 z  z'


x

 x'  ( R0  y )  cos R

0
P-1: 
x
y
'

(
R

y
)

sin

0
R

0

z
'

z



(2)
(3)
In formula (2) and (3), x’, y’, and z’ are co-ordinates
of a point in the cylindrical geometry and x, y, and z
are co-ordinates of the corresponding point in the
plate. R0 is the exterior radius of the cylindrical
object (Fig. 2).
X
Y’
-1
P
X’
Y
Z’
Z
P
a)
b)
Fig.2 Position of the problem.
2.2 Victorial Transformation Model
The use of the differential geometry theory is unable
necessary in searching the vectors transformation.
This is consisting on the determination of several
derivatives of the transformed components x’, y’
  
and z’ along the three directions i  , j  , k  as
expressed below [3]:

'  y'
i '  x i 
x
x
 x'  y'
j' 
i 
y
y
 z' 
j
k
x
 z' 
j
k
y

'  y'  z' 
k'  x i 
j
k
z
z
z
Unitarians vectors transformation obtained are

then used to deduce a given transformed vector A' ,
which could be in our case a magnetic vector
potential having only one component in the x
direction, generally defined by its three components:




A'  A' x i '  A' y j'  A' z k'
(4)
As the unknown vector studied in our problem is
reduced to having only one component in the x
direction, the derivative of expression (3) yields to
the inverse vectorial model given by formula (6).
   R  y'
0
i 

x' 2  y' 2

V:   R0  y'
j  2
x'  y' 2

 
k  k 




x'  x' 2  y' 2 
.i  
.j
x' 2  y' 2

y'  x' 2  y' 2 
.i  
.j
x' 2  y' 2
x 
x 

i    sin R  i  cos R  j
0
0




x
x 
V-1 :  j    cos  i  sin  j
R0
R0

k   k


(5)
(6)
For the inverse model we
obtain seemly the
  
transformed Unitarians i , j , k vectors as follows:

 y
i  x i ' 
x'
x'

 y
j  x i ' 
y'
y'
;
 z 
j' 
k'
x'
 z 
j' 
k'
y'

 y  z 
k  x i ' 
j' 
k'
z'
z'
z'

Direct model of a given vector transformation A

 
  
from i  , j  , k  to i , j , k base is obtained by:




A  Ax i  A y j  Az k
(7)
In formula (5) is represented the direct model
when considering only the non zero A x component
and the derivatives of relation (2).
3 Numerical Analysis
The non-destructive testing problem considered in
the present work concerns the evaluation of the
transducer’s impedance. This necessitates solving
the electromagnetic equation with harmonic
hypothesis. The numerical approach is given by the
time harmonic problem expressed in terms of vector
potential:


 

  (  A)  j 0 . A   0 J
(8)
 0 = 4π.10-7 H/m ; σ : Electric conductivity.
J: current density;   2f , where f is the frequency.
Expression (8) is solved using finite element
method. The finite element formulation was given
as follow:


a)
b)
Fig.3 Cartesian equivalent problem and constructed
problem with external defect
Fig. 4 shows the vector potential distribution,
once it is obtained in Cartesian case (a), its
distribution in the tube is obtained just by applying
the proposed transformations models (b). A
comparison between the predicted distribution and
the actual one obtained with finite element analysis
(c) show a good agreement.
 


 
 i   A    A   j Adxdy

x

x
0
y  y 


(6)

  0 J dxdy

i


A  (0,0, Az ) and J  (0,0, J z )
z
Az and J z are components in z direction of
magnetic potential and current density vectors.
The sensor impedance ( Z ) is then computed
using the above impedance’s relationship:
a. Cartesian
Z
jN
2
J  S2
 2rAds
b. Transformed
(9)
s
N and J are total number of turns coil and current
density respectively; S: section of the inductor coil;
4 Applications and Results
4.1 2D Case
The application considered in this work, also studied
in [4], is a non-destructive analysis of a tube by a
differential transducer with axisymmetric external
defect. The tube parameters are inner radius 9.75
mm, 1.27 mm width, 40 mm high and electric
conductivity 1E+6 (S/m). The sensor excited by a
current of 5 mA at 100 kHz frequency, is
characterised by 7.75 mm inner radius, 0.75 mm
width and 2 mm high.
Axisymmetric
transformed tube
Sensor
Fig. 4: Vector potential distribution
In fig.5 is presented a comparison between
Lissajous curves (variation of imaginary part
with the real one of the impedance) obtained
with the proposed models, posed models, when
applying a current density J to the inductor coils,
and reference data [4].
-J +J
+J
1
1
0. 0.
8 8
plate
External defect
0. 0.
6 6
External defect
+J
-J
Im (Z)
0. 0.
4 4
Im(Z)
-J
c. Axisymmetric
0. 0.
2 2
0
- 0. 0.
2 2
- 0. 0.
4 4
0
Analytical Models
Given Results
Fig.7 Constructed cylinder
Like it is shown in Fig.8.1 and Fig.8.2 which
represent the top view and the lateral one of the
constructed cylinder, the last is free of deformation.
Fig. 5 Lissajous curves for an external
defect of 40%.
The comparison in terms of real part of the vector
potential obtained when using the proposed models
and those obtained with finite element analysis along
the inner radius of the tube show a good agreement
between the results (Fig. 6).
Fig.8.1 Top view
2
Transformations models
f.e.m
1.5
real part of the vector potential [ T.m]
Fig.8.2 Lateral view
x10-8
1
0.5
Applying the direct model to the constructed
cylinder leads to the original plan configuration, i.e.
a plate scanned with a differential sensor, as it is
represented in Fig.9.
0
-0.5
-1
-1.5
-2
0.02
0.025
0.03
0.035
0.04
z [m]
0.045
0.05
0.055
0.06
Fig.6 Comparison between real parts of vector
potential
4.2 3D Case
4.2.1 Geometrical Model
The upcoming application is considered in order to
extend the validity of the developed models to the
3D cases. In Fig.8 is shown the constructed cylinder
with the differential sensor. It is obtained using the
geometrical inverse model to transform the plate and
the differential sensor defined above.
Fig.9 Original configuration
This allows us to conclude the validity of the
proposed geometrical model.
4.2.2 Vectorial Model
In this case the inverse vectorial model is applied to
the results given in team workshop N°13 for the
magnetic flux density. The geometric characteristics
of the system are as follow.
z
Steel chanel
x
y
y
Plate
Coil
2
Cylindrical object
Given results for the plate
Magnetic induction [T]
1.8
1.6
a) Front view
b) Top view
1.4
1.2
Fig.10 Geometrical arrangement
1
The given results are depicted in Table 1.
Coordinates [mm]
N°
1
2
3
4
5
6
7
X
Y
-25 <= x <= 25
0 <= y <= 1.6
0.8
0.6
0
Induction in [T]
Z
0
10
20
30
40
50
60
1.333
1.329
1.286
1.225
1.129
0.985
0.655
The obtained results are given in Fig.11 below.
One can see that a good repartition of the magnetic
flux density surrounding a cylinder object.
0.01
0.02
0.03
High [m]
0.04
0.05
0.06
Fig.12 Predicted results for the cylindrical object
In a second step, the direct model is applied to the
obtained results when using the inverse vectorial
model. The planar repartition of the magnetic flux
density is depicted in Fig.13. This shows a good
repartition of the vectors.
0.07
0.06
0.05
Z
0.04
0.07
0.03
0.06
0.05
Z
0.02
0.04
0.01
0.03
0.02
0
5*10-4
0.01
0
0
0.01
-5
0.005
0.01
0.005
0
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
Y
X
0
-0.005
Y
-0.005
-0.01
X
Fig.11 Predicted repartition of the magnetic flux
density in the cylindrical object
The result of the magnetic flux density obtained
using the inverse vectorial model is represented in
Fig.12, where some difference could be note
between the initial and the transformed quantities.
Fig.13 Planar repartition of the magnetic flux
density
The bijectivity of the proposed models is checked
when using the inverse model, for the predicted
results in the cylindrical body.
We note that the initial values in the plate are the
same than the results obtained when the inverse
models are applied to the geometry and flux density
values, like it is represented in Fig.14. This offers
some interesting perspectives 3D numerical
solutions in NDT systems or Non Destructive
Evaluation NDE)
1.4
1.3
Magnetic Induction [T]
1.2
1.1
1
Analytical Models
[T]
Given results
0.9
0.8
0.7
0.6
0
0.01
0.02
0.03
High [m]
0.04
0.05
0.06
Fig.14 Bijectivity of the proposed models
5 Conclusion
Transformation functions which allow to predict the
electromagnetic phenomena behaviour in a
cylindrical object with external axisymmetric defect
by means of equivalent plate geometry, in nondestructive testing domain, are presented. The
validity of results is obtained by comparison with
reference data, in 2D case. Then the proposed
models were extended to 3D structure and applied
successfully to magnetic flux density quantities
(3D).
References
[1] A. Diche, H. Mohellebi, M. Féliachi and G.
Berthiau,
Development
of
Cartesianaxisymmetric Transformations Models for NonDestructive Testing Problems, ISEM’03,
Versailles, France, 12-14 Mai 2003, 144-145.
[2] S. Mziou, Application de transformations
mathématiques à la résolution des équations de
Maxwell pour le contrôle des tubes de
générateur de vapeur par une sonde ponctuelle,
Stage de DEA Université Paris-Centre Jussieu
Août 1999, LCME, CEA Saclay.
[3] F. Henrotte, B. Meys, H. Hedia, P. Dular and
W. Legros, Finite Element Modelling with
Transformation Techniques, IEEE Transaction
on Magnetics, vol.35, No.3, May 1999, pp
1434-1437.
[4] J. L. Thomas, Modélisation Simplifiée du
contrôle par Courants de Foucault de Tubes de
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