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Sukri et al. | Malaysian Journal of Fundamental and Applied Sciences, Vol. 17 (2021) 343-353
RESEARCH ARTICLE
Study of a Different Order of Gaussian
Quadrature using Linear Element
Interpolation in Evaluation of First Order
Polarization Tensor
Suzarina Ahmed Sukria,b,*, Yeak Su Hoea,b, Taufiq Khairi Ahmad
Khairuddinb
a Department of Mathematical Sciences, Faculty of Sciences, Universiti Teknologi
Malaysia, 81310 UTM Johor Bahru, Johor Malaysia; b UTM Center for Industrial and
Applied Mathematics (UTM-CIAM), Universiti Teknologi Malaysia, 81310 UTM Johor
Bahru, Johor Malaysia
Abstract The mathematical terminology of first order polarization tensor (PT) has been widely used
in engineering applications especially in electric and magnetic areas due to its capability to define an
object with conductivity. In this case, a suitable method must be employed in the evaluation of first
order PT to make sure that the tensor obtained is higher in its accuracy. Our aim in this paper is to
provide a simple yet effective implementation method for computing first order PT which is based on
Gaussian quadrature numerical integration involving linear interpolation method. This study provides a
comparison between two different orders of Gaussian quadrature, which are order one and order three.
The numerical approximation of first order PT when computed using higher order of Gaussian
quadrature gives higher accuracy and convergence compared to the Gaussian quadrature which has
the lower order. In this study, the validation of the results obtained by using our proposed method is
provided by comparing it with the analytical solution derived from the previous researcher. We illustrate
the behavior of a tensor of a sphere and ellipsoid with graphical representation.
Keywords: Polarization tensor, Boundary integral equation, Gaussian quadrature, Numerical
integration.
Introduction
Polarization tensor (PT) is an object-specific property that depends on the shape, size, orientation, and
material [1]. In some studies, PT is simply called a tensor. PT is first described as a virtual mass of a
moving solid in a fluid for which the solid has zero conductivity [2]. Later, this term of virtual mass has
been generalized by researchers as in [3] where the researcher combined the infinite numbers of PT
and named this generalization as Generalized Polarization Tensor (GPT) [4]. From the generalization,
the first term of GPT is known as first order PT. The application of GPT can be seen where some of the
researchers used GPT formulation in medical applications, particularly in electrical imaging. The
simulation of cloaking devices based on PT terminology is performed by using a mathematical
formulation of PT, for example in [5,6]. Other than that, the application of PT, especially in landmine
detection is very common among researchers, see [1,7–11] and also electrosensing fish [12–17].
*For correspondence:
suzarina3@graduate.utm.my
Received: 12 Nov 2020
Accepted: 15 August 2021
© Copyright Sukri. This
article is distributed under
the terms of the Creative
Commons Attribution
License, which permits
unrestricted use and
redistribution provided that
the original author and
source are credited.
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Sukri et al. | Malaysian Journal of Fundamental and Applied Sciences, Vol. 17 (2021) 343-353
Several studies [18,19] have suggested different types of methods to compute PT such as a semi-
algebraic method, finite element method (FEM) as well as boundary element method (BEM). The semi-
algebraic method has been applied to a two-dimensional case of PT, where, the research has shown
that objects with higher conductivity would produce lower convergence of PT compared to objects with
lower conductivity [20]. A new technique that is based on the programming scheme according to BEM
which is BEM++ has been employed in [19] where they computed various kinds of geometry with different
conductivity. Sukri et al. [21] implement the quadratic element approach to compute first order PT.
In this study, we provide two different orders of Gaussian quadrature for computing first order PT, namely
the one-point Gaussian quadrature and three-point Gaussian quadrature involving linear finite element
shape functions. The comparison of the numerical approximation of first order PT between two different
orders of Gaussian quadrature will be investigated based on their convergence rates. Three-point
Gaussian quadrature based on linear element integration will provide higher convergence rates
compared to the one-point Gaussian quadrature. It is obvious since the increased number of points
inside an interval can increase the accuracy of numerical results of a problem, especially a problem
involving integral. Since we apply these approaches to a problem involving surface integral of first order
PT, we observe the behavior of the numerical solutions for both orders as we change the parameters of
PT such as conductivity and the size of meshes used. The validation of the numerical results is then
compared with the analytical solution provided by Ammari and Kang [3]. We will observe the behavior of
the numerical results for first order PT by implementing the linear element numerical integration with a
different number of Gaussian quadrature points. The numerical results obtained throughout this study is
very important in ensuring that
In the next section, we are going to review the formulation of the first order PT together with its analytical
solution, the method proposed, and the formula used to analyze the convergence of the numerical
results. The numerical results of first order PT for two types of geometry are illustrated in the graphical
presentation where different values of conductivity and the number of mesh are used. The last section
will summarize the study where several recommendations are stated.
Methodology
Formulation of first order PT
In this section, we will review the concept of first order PT that has been studied by researchers as in
[6,22–24]. First order PT which has been notated as
!"#
, is an integral that originates from a transmission
problem which has been discussed in [3,25,26]. The solution for the integral of the first PT can be
expressed as a matrix system with size
$%$
as
!&
'
!(( !() !(*
!)( !)) !)*
!*( !*) !**
+
,
(1)
In some cases, the matrix solution of first order PT in equation (1) is used to represent the
electromagnetic behavior for an object, for example, see [1]. The integral representation of first order PT
can be written as
!"#
-
./0
1
&
2
3#4"
-
3
1
56 78
-
3
1
/39:0
(2)
where
;
and
<
are multi-indices which equal to
=/>
and
$
. The conductivity of object,
0
is denoted as
.
where
.
must satisfy
?@.A=@BC
.
3#
is the element of object domain
0
while
4"
-
3
1 is the linear
system of the equation that can be represented as
4"
-
3
1
&
-
DEFGH
I
1
J(
-
KLMNO"
1-
3
1
,
(3)
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From equation (3),
P
is defined as
P&-.B=1Q->.F>1
while outward normal vector of element
O
in
object domain
0
is denoted as
KL
.
RH
I
is the singular integral operator which has been defined as Cauchy
Principal Value (
S,K,
) and is written as
THI4"
-
O
1
&(
UVS,K,
2W
L!JX"/Y#!
Z
[
L!JX"
[
$4"
\
3#
]
^H 78
\
3#
]
/
(4)
where
O"F3#
is the distance between element
O
and element
3
. Hence, our main concern throughout
this study is mainly in equations (2), (3), and (4). Hereafter, we are going to review the analytical solution
that has been provided by Ammari and Kang [3] for specific geometry which is sphere and ellipsoid.
The analytical solution of first order PT
The analytical solution is derived by Ammari and Kang [3] where the researchers firstly considered the
geometry
0
to become ellipsoid. The Cartesian coordinate system of the ellipsoid is represented by
_`
a`B
b`
c`Bd`
e`&=
, where
f/g
and
h
are the semi principle axes such that
?@hig@f
. Then, the first order
PT with conductivity
.
is given as
!
-
./0
1
&
-
.F=
1j
0
j'
=Q7(? ?
? =Q7)?
? ? =Q7*
+
/
(5)
where
j0j
is the volume of an ellipsoid while
7(/7)
and
7*
are defined as
7(&
k
=Fgh
f>
l
=
m>
n
m>F=B
o
gf
p
>
n
m>F=B
o
hf
p
>q7m
BC
=
r
B.
k
gh
f>
l
=
m>
n
m>F=B
o
gf
p
>
n
m>F=B
o
hf
p
>q7m
BC
=
r
/
7)&
s
t
u
=Fgh
f>
v
=
w
m>F=B
o
gf
p
>
x
$>
n
m>F=B
o
hf
p
>q7m
BC
=
y
z
{
B.
s
t
u
gh
f>
v
=
w
m>F=B
o
gf
p
>
x
$>
n
m>F=B
o
hf
p
>q7m
BC
=
y
z
{
/
7*&
s
t
u
=Fgh
f>
v
=
n
m>F=B
o
gf
p
>q
w
m>F=B
o
hf
p
>
x
$>7m
BC
=
y
z
{
B.
s
t
u
gh
f>
v
=
n
m>F=B
o
gf
p
>q
w
m>F=B
o
hf
p
>
x
$>7m
BC
=
y
z
{
,
(6)
From the Cartesian coordinates of the ellipsoid, by setting
f&g&h
, the first order PT for ellipsoid then
will become the first order PT for sphere which is given by
!
-
./0
1
&
-
.F=
1j
0
j|
$Q->B.1 ? ?
? $Q->B.1 ?
? ? $Q->B.1
}
,
(7)
We will evaluate the numerical approximation using our proposed method and compare it with the
analytical solution computed by using the equation in (5) and (7). For ellipsoid geometry, we can also
validate the numerical approximation of first order PT obtained by using theorem proven by Yunos et al.
[23]. Here, we will review the theorem that has been proven
Theorem 1 Let
!-./01
be the first order PT for ellipsoid
~)Qf)B•)Qg)B€)Qh)&=
at any conductivity
.
, where
?@.A=@BC
.
a.
f&g
, if and only if
!((&!))
.
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b.
f&h
, if and only if
!((&!**
.
c.
g&h
, if and only if
!))&!**
.
We will observe whether the numerical results of first order PT that are obtained for an ellipsoid satisfies
the theorem provided by the previous researcher in [23].
Linear interpolation method using one-point and three-point Gaussian
quadrature
Throughout this section, we will review the linear interpolation method by using one point and three-
points Gaussian quadrature. First and foremost, the desired geometry is being triangularized by a free
software which is Netgen Mesh Generator that is developed by Joachim Schöberl [27]. This software
requires us to define the geometry in
,•‚ƒ
file. Then, by clicking the
•‚„‚…fm‚
button in the interface of
the software, it will automatically generate the triangular meshes of the geometry. There are a total of
five different sizes of meshes options that have been provided which are
†‚…•qhƒf…‡‚/hƒf…‡‚/ˆƒ7‚…fm‚/‰;„‚/
and
†‚…•q‰;„‚
. Unfortunately, we cannot define the number of
meshes for every geometry because the software will generate the number of meshes automatically
based on the defined geometry.
Figure 1. The representation of triangular mesh containing point
S(/S)
and
S*
changing from local
coordinates to global coordinates.
Figure 1 illustrates one triangular mesh in local coordinates containing three points (linear) in each vertex
of the mesh where
S(&
W
~(/•(/€(
Z
/
q
S)&
W
~)/•)/€)
Z
/
q
S*&
W
~*/•*/€*
Z
,
(8)
From the local coordinate of
~/•
and
€
as in equation (8), they will be transformed into global coordinates
of
Š
and
‹
under transformation formula as
~
-
Š/‹
1
&Œ(~(B•)~)B•*~*/
q
&
Ž
••~•/
*
••(
•
-
Š/‹
1
&•(•(B•)•)B•*•*/
q
&
Ž
••••/
*
••(
€
-
Š/‹
1
&•(€(B•)€)B•*€*/
q
&
Ž
••€•/
*
••(
(9)
where Ž
••
*
••(
is the shape function and defined as
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Sukri et al. | Malaysian Journal of Fundamental and Applied Sciences, Vol. 17 (2021) 343-353
•(&Š/
q
•)&‹/
q
•*&=FŠF‹,q
(10)
From equation (4), the Cauchy principal integral can be represented as a
THI4"
-
O
1
&(
UVq
Ž
‘#
W
L!JX"/Y#!
Z
[
L!JX"
[
$4"
\
3#
]
’
#•( M“
-
Š/‹
1
M”
-
Š/‹
1
/
(11)
where
‘#
is the area of the triangular mesh (for one-point Gaussian quadrature) but for three-points
Gaussian quadrature,
‘#
is the weighting points of the element
3#
.
“-Š/‹1
is the Jacobian matrix while
”
-
Š/‹
1 is the surface projection for object domain
0
.
“
-
Š/‹
1 can be expressed as
“
-
Š/‹
1
&
'
~•~–
•••–
€•€–
+
/
(12)
such that
~•&:~Q:Š
,
q••&:•Q:Š
,
q€•&:€Q:Š
,
~–&:~Q:‹
•–&:•Q:‹
and
€–&:€Q:‹q
obtained by
finding the partial derivative from equation (9). The surface projection,
”-Š/‹1
for the triangular meshes
is obtained by using the standard equation of a triangular plane which is
—~B˜•B…€&7/
(13)
where
—/˜
and
…
are the vector orthogonal to the plane with
~
,q
•
and
€
coordinates of the triangular plane.
From equation (13),
7
is the distance of the plane from its origin. By rearranging equation (13), it will
yield to
—Q7q~B˜Q7q•B…Q7q€&=,
(14)
Then, by letting
—Q7q&™/
˜Q7q&0
and
…Q7q&š
, equation (14) will eventually become
™~B0•Bš€&=,
(15)
Since we have a linear triangular mesh (meaning to have three-point coordinates), therefore, the
equation in (15) is expressed as
™~(B0•(Bš€(&=/
q
™~)B0•)Bš€)&=/
q
™~*B0•*Bš€*&=,
(16)
We then transform equation (16) into a matrix system and then evaluate the values of
™/0
and
š
by
using the simple inverse matrix method. By obtaining the values of
™/0
and
š
, the surface projection of
the plane which contains three points coordinates is equivalent to
”
-
Š/‹
1
&
›
œ%•H%•ž%
ž,
(17)
From equation (11), the distance between the element,
O"F3#
for one-point Gaussian quadrature can
be obtained by finding the distance between the barycenter of each element. The distance between
elements for the three-point Gaussian quadrature is obtained by finding the Gaussian points in each
triangular element. Figure 2(a) describes the distance of element of barycenter of each element while
Figure 2(b) shows the distance between Gaussian points of element
O
and
3
.
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Sukri et al. | Malaysian Journal of Fundamental and Applied Sciences, Vol. 17 (2021) 343-353
(a)
(b)
Figure 2. Representations of the distance of elements. (a) Barycenter of element
O
and
3
. (b) Gaussian
points of element
O
and
3
.
Let the distance of element for one-point Gaussian quadrature be
O"F3#
, while the distance between
elements for three-point Gaussian quadrature be
O"IF3#I,
Next, for the case of one-point Gaussian
quadrature, the outward normal vector for element
O
,
KL
, can be computed by finding the cross product
for each triangular meshes. The normal vector for one-point Gaussian quadrature is
KL&
W
•)(M€*(F•*(M€)( ~)(M€*(F~*(M€)( ~)(M€*(F~*(M€)(
Z
,
(18)
The outward unit normal vector for one-point Gaussian quadrature is defined as
KL&
W
~/•/€
Z
ŸW
~/•/€
ZŸ
)/
q
&
W
~/•/€
Z
›
~)B•)B€),qqqqqqqqqqqqqqqqqqqqqqq
q (19)
The unit normal vector in equation (19) is positive in its orientation and is also guaranteed to be an
outward unit normal vector. For the normal vector involving three-point Gaussian quadrature, since the
direction of the vector in surface projection in (17) is similar to the direction in (19), the normal vector can
be expressed as
KLI&
W
œ/H/ž
Z
›
œ%•H%•ž%/qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
(20)
where
™/0/
and
š
come from the values computed as in equation (16). Then for one-point Gaussian
quadrature, equation (11) will eventually become matrix with size
•%•
, where
•
is the number of mesh
and is expressed as
THI4"
-
~
1
&(
UV
'
‘(T
-
~(/•(
1
‘’T
-
~(/•’
1
¡ ¢ ¡
‘(T
-
~’/•(
1
‘’T
-
~’/•’
1+
/
(21)
where
T
\
~"/•#
]
&
W
L£JX¤/Y¥£
Z
[
L£JX¤
[
¦M“
\
Š"/‹#
]
M”
\
Š"/‹#
] while for three-point Gaussian quadrature, the singular
integral operator will eventually lead to a matrix system of
$•%$•
which is expressed as
THI4"
-
~
1
&(
UV
|
‘(§TI-~(¨/•(§1 ‘’§T-~(¨/•’§1
¡ ¢ ¡
‘(§TI-~’
¨/•(§1 ‘’§T-~’
¨/•’§1
}
/
(22)
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Sukri et al. | Malaysian Journal of Fundamental and Applied Sciences, Vol. 17 (2021) 343-353
where
THI
o
~"¨/•#§
p
&
W
L£©JX¤ª/Y¥£©
Z
«
L£©JX¤ª
«
¦M“¨
\
Š"/‹#
]
MӬ
\
Š"/‹#
] with
¬/-&=/>q
and
$
(similar to the number of
Gaussian points). Next, after we obtained the singular integral operator for both one-point and three-
point Gaussian quadrature, we substitute equation (21) and (22) to the linear system in (3). For one-
point Gaussian quadrature, the solution of the linear system,
4"-3#1
will have
•%$
matrix size, while for
three-point Gaussian quadrature,
4"I-3#1
the matrix size is
$•%$
. The linear system is then solved as
mentioned earlier using the matrix inverse method. Finally, the last step is to obtain the first order PT by
solving the integral in equation (2). By using a similar approach in handling singular integral operator in
(4), the integral for one-point Gaussian quadrature is expressed as
!
-
./0
1
&
®
‘¯
’
¯•( 3¯#M“
\
Š"/‹#
]
M”
\
Š"/‹#
]
M4"
\
3#
]
/
q
&
°
4"
\
3#
]±
²‘¯M“
\
Š"/‹#
]
M”
\
Š"/‹#
]
M3¯#
q
(23)
where °
4"I
\
3#
]±
²
has size
$%•
.
3¯#/‘m/“
\
Š"/‹#
] and
”
\
Š"/‹#
]
q
has a similar matrix size which is
•%$
. For
three-point Gaussian quadrature, equation (4) yields to
!I
-
./0
1
&
®
‘¯¨
’
¯•( 3¯#M“
\
Š"/‹#
]
M”I
\
Š"/‹#
]
M4"I
\
3#
]
/
q
&
°
4"I
\
3#
]±
²‘¯¨M
°
Ҭ
\
Š"/‹#
]±
²M”¨
\
Š"/‹#
]
M3¯#
q
(24)
where °
4"I
\
3#
]±
²
has size
$%$•
.
/3¯#/‘¯¨/“¨
\
Š"/‹#
] and
Ӭ
\
Š"/‹#
]
q
has a similar matrix size which is
$•%$
. Finally, the first order PT in equation (23) and (24) will eventually have the same matrix size of
$%$,
Convergence of the numerical approximation of first order PT
The main objectives of this research are to verify and validate whether the proposed approach is suitable
to compute first order PT. Therefore, for convergence purposes, we will estimate the error of first order
PT using the relative error formula which is
Relative Error,
³&
Ÿ
´&J´
Ÿ
Ÿ
´
Ÿ (25)
where
!œ
is the approximated first order PT computed using our proposed approaches while
!
is the
analytical solution evaluated using the formula in equation (5) and (7). In order to study the convergence
of the numerical results, a different number of meshes were generated for two types of geometry,
particularly sphere and ellipsoid.
Results and discussion
Numerical approximation of sphere and ellipsoid
In this section, we will present numerical results for the computation of first order PT which focuses on
its accuracy and convergence. We implemented our algorithm in MATLAB while for the discretization
purpose, we used a user-friendly software which is a Netgen Mesh Generator as mentioned earlier. For
a sphere with radius
…&?,?=
, the software generates a different number of meshes, which are
µµ/>$?/¶>?/>µ·?/
and
¸¸>?
. We observed that as finer meshes are used to compute first order PT, the
numerical results showed higher in its accuracy compared to coarse meshes. Figure 3 depicted the first
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Sukri et al. | Malaysian Journal of Fundamental and Applied Sciences, Vol. 17 (2021) 343-353
order PT for a sphere with conductivity,q
.&=,¹
computed for different points of Gaussian quadrature. It
is observed that, by using three-point Gaussian quadrature, the numerical results tend to converge to
the exact solution of first order PT. The relative error for both quadrature points tends to become 0.
However, the numerical results for three-point Gaussian quadrature shows low relative error compared
to one-point Gaussian quadrature. We can conclude that; we must use finer meshes together with a
higher number of Gauss points to ensure that the numerical results obtained from the first order PT are
higher in its accuracy.
(a) (b)
c
Figure 3. (a) The main diagonal
!((/!))
and
!**
of approximated first order PT of sphere for
…&
?,?=/.&=,¹
and
•&µµ/>$?/¶>?/>µ·?
to
¸¸>?
. (b) The relative error for approximated first order PT for
sphere using one-point and three-point Gaussian quadrature for
…&?,?=/.&=,¹
and
•&
µµ/>$?/¶>?/>µ·?
to
¸¸>?
.
We have already validated that the numerical results for three-point Gaussian quadrature is better in its
convergence and has higher accuracy than one-point Gaussian quadrature for computing first order PT
of the sphere. The main diagonal of first order PT for a sphere is similar to each other while for the non-
diagonal, for both one and three-point tend to 0. Next, we will study the effect of conductivity,q
.
on first
order PT and how the behavior of the numerical results computed using our proposed method.
We computed the first order PT for at different conductivity and compared the results of its main diagonal
with the analytical solution in (6).
From Figure 4(a), increase of
.
would produce less accurate results of the main diagonal compared to
the lower value of conductivity. As conductivity used is less than 1 or approaching 1, the numerical results
depicted are higher in its accuracy compared to the numerical results as conductivity used is greater
than 1. This may be due to the singularity of the operator
-PºFTHI1
as
.
qis greater than 1. As we
calculated its relative error, the computed first order PT for three-point Gaussian quadrature portrayed
less error compared to the one-point Gaussian quadrature as in Figure 4(b). This is the result of one-
point Gaussian quadrature where we only consider the barycenter of each mesh and also the area of
each triangular meshes in its computation. However, for three-point Gaussian quadrature, we
implemented the concept of
Š
and
‹
which produced a better approximation of the first order PT. As
conductivity increases, the relative error shows a stagnant graph for both numerical results which is at
?,?¹
for three-point Gaussian quadrature and approximately
?,?·
for one-point Gaussian quadrature.
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Sukri et al. | Malaysian Journal of Fundamental and Applied Sciences, Vol. 17 (2021) 343-353
(a)
(b)
Figure 4. (a) The main diagonal of approximated first order PT of a sphere where
•&¶>?
and
increasing conductivity,
.&=,¹
until
=?»
. (b) The relative error computed for first order PT computed
using one-point and three-point Gaussian quadrature.
(a) (b)
c
Figure 5. (a) The first main diagonal of approximated first order PT,
q!((
for an ellipsoid where
•&
=¼?/µµµ/¶¶·/=¹¹¶
and
µ$$>
triangles at fixed conductivity,
.&¹??
. (b) The second and third main
diagonal of approximated first order PT,
q!))
and
!**
for an ellipsoid where
•&=¼?/µµµ/¶¶·/=¹¹¶
and
µ$$>
triangles at fixed conductivity,
.&¹??
.
For the first order PT of an ellipsoid with semi principal axes
f&>/g&=/q
and
qh&=
, we can observe that
the first main diagonal differs from the second and third main diagonal. This satisfies the theorem made
by Yunos et al. [22], where, from Theorem 1c, it stated that if semi principal axes of for
•
and
€
are equal
to each other,
g&h
, then the values of second and third main diagonal would probably be similar to
each other. From Figure 5, the main diagonal for first order PT of ellipsoid showed higher convergence
as we computed it using three-point Gaussian quadrature compared to one-point Gaussian quadrature.
The numerical results for both Gaussian points are far from the exact solution since we used less number
of surface elements which is only up to
µ$$>
meshes. Possibly, if we use more elements (finer meshes)
the numerical results would converge to the analytical solution.
352
Sukri et al. | Malaysian Journal of Fundamental and Applied Sciences, Vol. 17 (2021) 343-353
(a) (b)
c
Figure 6. (a) The first main diagonal of approximated first order PT,
q!((
for an ellipsoid where
•&µµµ
triangles at different values of conductivity,
.&?,>¹/?,¹/=,¹/=?/=?*/=?U/=?»
and
=?½
. (b) The second
and third main diagonal of approximated first order PT,
q!))
and
!**
for an ellipsoid where
•&µµµ
triangles at different values of conductivity,
.&?,>¹/?,¹/=,¹/=?/=?*/=?U/=?»
and
=?½
.
Figure 6 illustrates the behavior of first order PT as the conductivity increased. Similar patterns can be
noticed in both sphere and ellipsoid geometry as we increase the conductivity. At some point which is at
conductivity
/.&=,¹q
and
q•&µµµ
, the numerical solution for both points of Gaussian quadrature
depicted similar results as its analytical solution. However, after the conductivity increased, the results
for one point slowly diverge from its exact solution. The next section provides the conclusion of this study.
Conclusions
In this paper, we have introduced a simple and easily implemented method to approximate first order
PT. To verify that our proposed method can be used to compute PT, we compute first order PT for
objects such as the sphere and ellipsoid where their analytical solutions exist. Not only that, we validate
the computed results by referring to the theorem that has been proven by previous research.
From the numerical results obtained, we can observe that, as the number of surface elements increase,
the accuracy for first order PT for both objects will also increase. Hence, finer meshes must be adopted
in the computation of first order PT so that the results can converge to the exact solution faster. As for
conductivity, it is shown that higher values of conductivity have resulted in a lower convergence of first
order PT. The establishment of the study of the numerical methods to compute PT is very important
since PT is useful for many engineering problems especially in magnetic and electrical fields.
Conflicts of interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Funding statement
The authors would like to express gratitude to the Ministry of Higher Education (MOHE) for the
sponsorship through MyBrainSc scholarship and also Research Management Center (RMC), Universiti
Teknologi Malaysia (UTM) for the financial support through research grants with vote number 5F251.
353
Sukri et al. | Malaysian Journal of Fundamental and Applied Sciences, Vol. 17 (2021) 343-353
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