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Open Journal of Science
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THE EFFECT OF TRANSLATION ON THE
APPROXIMATED FIRST ORDER POLARIZATION TENSOR
OF SPHERE AND CUBE
Suzarina Ahmed Sukri1, Taufiq Khairi Ahmad Khairuddin2*, Yeak Su Hoe3
1,3 Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi
Malaysia, Johor Bahru, Johor, 81310 Malaysia
2UTM Center for Industrial and Applied Mathematics (UTM-CIAM), Universiti
Teknologi Malaysia, Johor Bahru, Johor, 81310 Malaysia
*Corresponding Author email: taufiq@utm.my
ABSTRACT
Throughout this paper, the translation effect on the first order polarization tensor
approximation for different type of objects will be highlighted. Numerical
integration involving quadratic element as well as linear element for polarization
tensor approximation will be presented. Here, we used different positions of an
object of fixed size and conductivity when computing the first order polarization
tensor. From the numerical results of computed first order polarization tensor, the
convergence for every translation is observed. Moreover, discretization of the
geometric objects into triangular meshes was done by using meshing software
called NETGEN mesh generator while for the numerical computation, MATLAB
software was used. We found that the translation has no effect on the approximated
first order PT for sphere and cube after we have computed the first order PT for
both geometries with a few center of masses. The numerical results of approximated
first order polarization tensor is plotted by comparing the numerical results with
analytical solution provided.
Article History
Submission: July 25, 2020
Revised: August 30, 2020
Accepted: September 22, 2020
Keywords:
Numerical Integration;
Triangular Elements;
Electric Field
Abbreviations: NIL
10.31580/ojst.v3i3.1672
INTRODUCTION
In the applications of electric or electromagnetic such as medical imaging and metal
detection, some properties of an electrical conducting object will be described for
classification of object purposes. These properties include the material (or the
conductivity) of the object, size, shape and also the orientation of the object. The object
could be a tumor with larger value of conductivity than normal tissues in medical
imaging (1) or a fish hook avoided by electrosensing fish (2-6) whereas, in metal
detection, the object could be a buried landmine or a gun carried by a person when
walking under metal detector (7-13).
There are probably many approaches exist that can be used to describe these objects
and their properties. One way to characterize these objects is to implement the
Polarization Tensor (PT) terminology, as can be seen in the previous listed references.
This approach seems to offer lower computational cost as it does not require for
example, complete image of the object to describe the object.
Research Article
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275
In applied mathematics, Ammari and Kang (14), they have shown that the leading
term of asymptotic expansion that represent the disturbance in electrical field because
of the existence of object with conductivity is called Generalized Polarization Tensor.
Given the conductivity and geometry of the object, by solving system of integral
equation, the associated GPT in the matrix form or in some literatures known as PT
for object with specific conductivity can be determined. In other words, given GPT
due to the presence of a known or unknown object, the geometry and conductivity of
the related object can be described or predicted (15-17).
PT for an object with some conductivity when presented in the electrical field does not
give any information about the specific location of the object. Other methods must be
integrated with PT to locate the object when reconstructing the image of the object
(14), for example. Similarly, a person can carry a gun at any parts of the body before
entering a metal detector, and when the gun is detected, the body must be searched
manually by a security officer.
Therefore, the translating effect of the object on the computation of its first order PT
will be investigated throughout this study. Theoretically, as will be stated later on, the
PT of an object with specific conductivity are independent of the position of the object.
Therefore, our numerical results on computing the first order PT of an object at
different locations must be the same or at least, have only very small differences
between them. This will justify whether our numerical method produces a good
approximation to the PT. Besides, for some cases, computing the PT of an object at
different locations will also help to verify that the resulting PT is correct.
The paper will is organized as follows. First, the background of mathematical term of
the PT related to the disturbance in electric field because of the existence of object with
specific conductivity will be reviewed in the next section. From the mathematical
background, we will explain the translation of the PT and methods used to generate
the numerical results of each translation. Then, numerical results showing the effect
of translation on the approximation of the PT will be presented in graphical form.
Lastly, discussion and some conclusions about the numerical results obtained are
presented.
FIRST ORDER POLARIZATION TENSOR FORMULATION
Polarization tensor as considered here actually initiates from a transmission problem
of the inverse conductivity equation which has being reviewed by a numbers of
literatures. Ammari and Kang (14) consider a Lipschitz bounded domain
B
in
3
where the origin
O
is inside the boundary domain
B
and
k
will represent the
conductivity of domain
B.
The condition of the conductivity
k
must satisfy the
condition where,
01k +
. By assuming harmonic function, which denotes as
H
is
in
3
and the solution of the problem in (1) is
u,
then
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276
3
2
div(1 ( 1) ( )grad( )) 0 in
( ) ( ) (| | ) as | |
k B u
u x H x O x x
−
+ − =
− = →
(1)
where
represents the characteristic function of
B
. There are numerous industrial
application that employ the mathematical formulation in (1); for example in medical
imaging where we can see from Electrical Impedance Tomography (EIT) system,
material sciences as well as in detection of landmine (1, 14, 18).
Far-field expansion that represent PT has been introduced by Ammari and Kang (14)
where it yields to
||
| |,| | 1
( 1)
( )( ) ( ) ( , ) (0) as | |
!!
i
ij
x ij
ij
u H x x M k B H x
ij
+
=
−
− = → +
(2)
for
,ij
that denotes the multi indices, while
is the fundamental solution of the
Laplacian.
( , )
ij
M k B
is the generalized polarization tensor (GPT) or can be simply called
as PT. Since GPT can show the conductivity distribution of an object, therefore,
physicist usually assigned GPT as the dipole in electromagnetic applications. In this
case, (2) represent the perturbation on the electrical field
u
because of the presence of
a conducting object
B
.
Furthermore, according to Ammari and Kang, the GPT in (2) can also be defined by
system of integral equations over the boundary of
B
which is
( , ) ( ) ( )
j
ij i
B
M k B y y d y
=
(3)
where
()
iy
is formulated as
( ) ( )
1
*
( ) ( )
i
i B x
y I K x y
−
= −
(4)
for
,x y B
,
I
is the identity while
x
is the outward normal vector of
x
to the boundary
B
. In this case, the definition of
is
( 1) / 2( 1)kk
= + −
which contain the conductivity,
k.
Singular integral operator,
*
B
K
associated with Cauchy principal value
P
.
V
. is given
and yield to
*
3
()
1
( ) p.v. ( ) ( )
4
x
B
B
xy
x y d y
xy
−
=−
. (5)
In order to directly compute the PT of
B
for conductivity
k
, where
k
must satisfy
01k +
, we can simply determine it by substituting the conductivity and
B
to
equation (3), (4) and (5).
In this work, we restrict our investigation on the first order PT, when the multi indices
of GPT is
| | | | 1ij==
. Thus, by construction, the first order PT is represented by matrix
of 3 by 3. If
B
is a sphere of volume
||B
, then from the analytical formula that has been
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277
derived and introduced by Ammari and Kang (14), its first order polarization tensor
will yield to
3 (2 ) 0 0
( , ) ( 1) | | 0 3 (2 ) 0 .
0 0 3 (2 )
k
M k B k B k
k
+
= − +
+
(6)
In the next section, we will revise previous theoretical results which suggest that the
first order PT is independent of the position of
B
. After that, we explain our numerical
method for the first order PT approximation for sphere and cube geometries of
constant density at specified conductivity but with different center of mass.
METHODOLOGY
In the previous section, it is required that the origin must be in
B
but not the center of
mass for
B
. So, the center of mass for
B
can be anywhere when its first order PT is
computed. The following proposition is considered from the study of Ammari and
Kang (14), which explains the first order PT is independent of the position of object
B
.
Proposition 1
Let
( , )M k B
be the first order polarization tensor for an object
B
at conductivity
k
,
where
01k +
and
z
be a translation vector. If
( , )M k B+z
is the first order PT for
B
after
B
is translated by
z
then
( , ) ( , )M k B M k B+=z
.
The above proposition specifically tells us that if the center of mass of
B
is changed,
then its first order PT will remain the same. If we regard the first order PT as a physical
parameter of
B
such as surface area, volume or density, the first order PT is
independent of position of
B
. This information is useful to us as it can help us to verify
whether the first order PT of
B
that is numerically computed is correct or not; thus, we
can compare the first order PT for
B
, computed at two different center of mass and
ensure that they are the same.
In general, the first order PT can only be computed by numerical method, except when
B
is a sphere. In this case, finite element approach will be employed to approximate
the first order PT for sphere and cube based on equations (3), (4) and (5). Here, to
numerically compute first order PT for sphere and cube geometries, each object will
be initially created in the software Netgen (19) at a specified center of mass. After that,
mesh of the object consisting of a set of triangular elements will be automatically
created by the software. Since the integrals involved as given in (3) and (5) are surface
integrals, the triangular elements generated are actually the surface elements of the
object. The required nodes for each triangle given by Netgen will be exported to
Matlab for computation of the PT by finite element method. For comparison in this
study, quadratic and linear element which involve six and three nodes in each triangle
will be used (20).
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278
After generating the triangular elements for both objects, their first order PT will be
computed at a specified conductivity
k
. The procedures when using linear elements
are based on Khairuddin and Lionheart (21) and we refer to Reddy (20) when
developing numerical algorithm with quadratic elements to approximate the first
order PT. Then, we will increase the size of the mesh with Netgen and repeat the
processes until satisfactory results are obtained. Usually, the size of mesh must be
increased until the numerical computations of the first order PT converge to one value.
The convergence first order PT will then be used as the convergence approximation
of the first order PT. However, at the moment, approximating the first order PT is time
consuming because of the large size for the mesh used. So, our results here might only
be preliminary results.
For sphere, the error in approximating the first order PT can be computed by
comparing the approximated first order PT with (6). Since the analytical solution of
the first order PT for cube geometries has yet to be found, we cannot compute its error.
However, we can still further investigate what happens to the first order PT
approximation for both cube and sphere when the center of mass for both objects are
changed. In order to achieve this, the center of mass for both sphere and cube are
redefined in Netgen before the new meshes are created for computing the new first
order PT but still at the same conductivity. Mesh for sphere and cube at a few center
of masses will be considered in this study when approximating the first order PT for
both objects.
NUMERICAL RESULTS AND DISCUSSION
For the purpose of first order PT approximation for sphere of radius 0.01, at a few
center of masses, four coordinates are chosen and defined as the center in Netgen. The
four chosen centers of mass for the sphere are
(0,0,0)
,
(0.01,0,0)
,
(0,0.01,0)
and
(0,0,0.01)
. For each sphere with different center of mass, the meshes are then generated in
Netgen and surprisingly, the size for the mesh for each sphere can still be the same.
As the positions of each sphere are different from each other, the nodes for the
triangular elements will be different although the total elements are the same. For the
computation purposes, the sizes of the mesh for each sphere used are
620
,
2480
and
9920
.
Using the generated meshes, the first order PT for the sphere are numerically
approximated at conductivity
1.5k=
. After that, the error for each computation will
be computed. If
ˆ
M
is the approximated first order PT for sphere and
M
is the first
order PT for sphere given by (6), then the error,
e
is given by
2
2
ˆ
|| ||
|| ||
MM
eM
−
=
. (7)
where for a
33
matrix
A
,
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279
( )
33 2
2
11
|| || ij
ij
Aa
==
=
(8)
and
ij
a
are the elements of the matrix
A
. Fig. 1 then shows the error against total
surface elements of the approximated first order PT for sphere of radius
0.01
at
conductivity,
1.5k=
when the center of mass for the sphere are respectively
(0,0,0)
,
(0.01,0,0)
,
(0,0.01,0)
and
(0,0,0.01)
. As we can see in all graphs of Fig. 1, the error
decreases as the total surface elements increase in approximating the first order PT for
each sphere and the error when using quadratic elements for the sphere is smaller
than the linear elements for each mesh, as expected.
Now, in order to study the translation effect on the approximated first order PT,
graphs in Fig. 2 are plotted. These graphs show the average of the elements,
a
for the
approximated first order PT,
ˆ
M
against the total surface elements for each mesh used.
In this case,
a
is computed by the formula
( )
33
11
1ˆ
9ij
ij
am
==
=
(9)
where
ˆij
m
are the elements of the matrix
ˆ
M
. Based on the graphs, we have noticed
that
a
increases when
M
is computed either by linear or quadratic elements increased.
Both average approach to the straight line which represents the average elements of
the first order PT for the sphere computed based on the analytical formula (6). It is
expected that, for each mesh,
a
computed based on
M
that is approximated by
quadratic elements (Quad) are closer to the straight line than
a
computed based on
M
that is approximated by linear elements (Lin). Moreover, it is also observed that, for
each mesh, the average elements of
M
are all the same when
M
are approximated by
both linear or quadratic elements although the center of mass for the sphere are
(0,0,0)
,
(0.01,0,0)
,
(0,0.01,0)
and
(0,0,0.01)
.
(a) (b)
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280
(c) (d)
Fig. 1. The error,
e
against the total surface elements in approximating the first order PT for sphere of radius 0.01
at conductivity
1.5k=
where the center of the mass for the sphere are respectively (a)
(0,0,0)
, (b)
(0.01,0,0)
,
(c)
(0,0.01,0)
and (d)
(0,0,0.01)
Finally, Fig. 3 shows the average of the elements for the approximated first order PT
for cube against the total surface elements for each mesh used. Here, the total surface
elements or the size of the mesh for the cube generated automatically in Netgen are
192, 768, and 3072. In addition, the first order PT for the cube of size
0.04 0.04 0.04
, in
this case, are approximated based on (3), (4) and (5) also at conductivity
1.5k=
.
However, only quadratic elements are used during the numerical approximation.
Here, it is found that, for each mesh, the average elements of first order PT
approximation for cube resulted in similar number of mesh although the center of
mass for the cube are
(0,0,0)
,
( 0.02,0,0)−
and
(0.02,0.02,0.02)
.
Fig. 2. The average elements,
a
of the approximated first order PT by both linear and quadratic elements against
the total surface elements for the mesh. The first order PT for sphere of radius 0.01 at conductivity
1.5k=
are
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281
approximated when the center of the mass for the sphere are respectively
(0,0,0)
,
(0.01,0,0)
,
(0,0.01,0)
and
(0,0,0.01)
. The straight line represents the average elements of the first order PT for sphere computed based on
the analytical formula (6).
Fig. 3. The average elements,
a
of the approximated first order PT by quadratic elements for cube (size
0.04 0.04 0.04
and conductivity
1.5k=
) against the total surface elements for the mesh, where, the center of
the mass for the cube are respectively
(0,0,0)
,
( 0.02,0,0)−
and
(0.02,0.02,0.02)
.
CONCLUSION
In this study, we can observe that as the first order PT for sphere is evaluated either
by linear or quadratic elements, the error of the computation is smaller if quadratic
elements is used. The results are also true when the center of mass of the sphere are
changed, as presented in Fig. 1. Moreover, the average elements of PT when it is
approximated for both sphere or cube geometries are similar when the center of mass
for both objects are relocated at a different place. This suggests that the approximated
PT are also similar for both objects independent of their center of mass. Therefore,
these results agree with the previous proposed theory stated in Proposition 1.
Acknowledgements
The authors would like to acknowledge the Ministry of Education (MOE) for the MyBrainSc scholarship and
Research Management Center (RMC), Universiti Teknologi Malaysia (UTM) for the financial funding through
research grant with vote number 5F251.
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