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Dipole simulation method for two-dimensional potential problems in exterior regions and
periodic regions
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2014 J. Phys.: Conf. Ser. 490 012054
(http://iopscience.iop.org/1742-6596/490/1/012054)
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2nd International Conference on Mathematical Modeling in Physical Sciences 2013
IOP Publishing
Journal of Physics: Conference Series 490 (2014) 012054
doi:10.1088/1742-6596/490/1/012054
Dipole simulation method for two-dimensional
potential problems in exterior regions and periodi
regions
Hidenori Ogata
Department of Communiation Engineering and Informatis,
Graduate Shool of Informatis and Engineering,
The University of Eletro-Communiations,
1-5-1 Chofugaoka, Chofu, Tokyo 182-8585 JAPAN.
E-mail: ogataim.ue.a.jp
Abstrat.
In this paper, we propose a dipole simulation method for two-dimensional potential
problems. The proposed method is a modiation of the harge simulation method (the method
of fundamental solutions), whih is a numerial solver for potential problems, and it gives an
approximate solution as a superposition of the potentials due to eletri dipoles positioned
outside the problem region instead of point harges as in the harge simulation method. We
nd that the proposed method ahieves high auray under some onditions from a numerial
example. Besides, it is advantageous to the harge simulation method in approximating omplex
analyti funtions. We also show the dipole simulation method for two-dimensional potential
problems with one-dimensional periodiity.
1. Introdution
Consider the two dimensional potential problems, that is, Dirihlet problems of the Laplae
equation
8
>
in <4u = 0
(1)
u=f
on ; R2 (exterior region)
>
:u is bounded in where f is a given funtion on . The harge simulation method (the method of fundamental
solution method), whih is well-known numerial solver for potential problems [3℄, gives an
approximate solution of (1) in the form1
( ) uN (z ) = Q0
u z
N
1 X
2 j =1 Qj log jz
j
j ;
(2)
where j 2 C ; j = 1; 2; : : : ;P
N; are given harge points and harges Qj 2 R ; j = 0; 1; 2; : : : ; N are
subjet to the onstraint Nj=1 Qj = 0 and determined so that uN satises the olloation
1
Hereafter we identify R2 with C .
Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution
of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Published under licence by IOP Publishing Ltd
1
2nd International Conference on Mathematical Modeling in Physical Sciences 2013
IOP Publishing
Journal of Physics: Conference Series 490 (2014) 012054
doi:10.1088/1742-6596/490/1/012054
ondition that uN (zi ) = f (zi ); i = 1; 2; : : : ; N for given olloation points zi 2 ; i =
1; 2; : : : ; N: In terms of physis, uN is a superposition of the potentials due to point harges
outside the problem region. We remark that the approximate solution uN satises the Laplae
equation 4uN = 0 exatly in . The harge simulation method is used in various elds of
siene and engineering suh as numerial onformal mappings [1℄, wave problems [4, 5℄ and so
on.
In this paper, we propose the dipole simulation method for two-dimensional potential problems
in exterior regions2 . In the proposed method, the solution is approximated by a superposition
of the dipole potentials.
2. Dipole simulation method
The approximate solution of the dipole simulation method for the potential problem (1) is given
by
N
X
nj
1
u(z ) uN (z ) = C0 +
(3)
2 pj Re z j
j =1
where j j 2 C n ; j = 1; 2; : : : ; N; are given dipole points, nj 2 C ; j = 1; 2; : : : ; N; are given
dipole diretions satisfying jnj jP
= 1 and C0 ; pj 2 R; j = 1; 2; : : : ; N are dipole magnitudes whih
are subjet to the onstraint Nj=1 pj = 0 and determined by the olloation ondition that
uN (zi ) = f (zi ); i = 1; 2; : : : ; N for given olloation points zi 2 ; i = 1; 2; : : : ; N: uN satises
the Laplae equation 4uN = 0 exatly in . In terms of physis, uN is a superposition of the
potentials due to the dipoles outside the problem region.
The dipole simulation method has the following advantage to the harge simulation method
in approximating omplex analyti funtions suh as numerial onformal mapping. Let f (z )
be a omplex funtion analyti in a region ( C ). Regarding that Re f (z ) is harmoni in ,
the harge simulation method approximates f (z ) by
( ) Q0
f z
N
1 X
2 j =1 Qj log(z
0
)
j ;
Q0 ; Qj
2 R;
N
X
j =1
1
Qj
= 0 A;
j
2 C n ;
j
= 1; : : : ; N:
In atual omputations, the omplex analyti funtion log(z j ) is omputed as the prinipal
value, that is, the branh suh that < arg(z j ) , and there appears a disontinuity of
2i on the half innite line f j + t j 1 < t 0 g : This disontinuity makes the omputation
diÆult and we need some tehniques to avoid the disontinuity. On the other hand, the dipole
simulation method approximates f (z ) by the rational funtion
N
X
pj nj
1
f (z ) C0 +
2 j =1 z j ;
C0 ; pj
2 R;
nj
2 C ( jnj j = 1 ) ;
j
2 C n ;
j
= 1; : : : ; N:
This approximate funtion is free of the disontinuity due to the prinipal values of the omplex
logarithmi funtions and we ompute it easily ompared with the approximation by the harge
simulation method.
Numerial Example We show a numerial example of the dipole simulation method. The
omputations were arried out on a HP Compaq 8000 Elite SFF PC personal omputer with a
3.00GHz Intel Core2 Duo CPU using programs oded in C++ with double preision working.
The example is the problem (1) of the boundary data f = x=(x2 + y2 ) in the exterior of an ellipse
2
0
0
-2
-2
log10(error)
log10(error)
2nd International Conference on Mathematical Modeling in Physical Sciences 2013
IOP Publishing
Journal of Physics: Conference Series 490 (2014) 012054
doi:10.1088/1742-6596/490/1/012054
-4
-6
-8
-4
-6
-8
q=0.9
q=0.8
q=0.7
q=0.6
-10
q=0.9
q=0.8
q=0.7
q=0.6
-10
-12
-12
0
20
40
60
80
100
0
N
method.
40
60
80
100
N
dipole simulation method
Figure 1.
20
harge simulation method
The error estimates of the dipole simulation method and the harge simulation
= (x; y) 2 R2 x2 =A2 + y2 =B 2 > 1 ( A = 1; B = 0:5 ): We omputed an approximate
solution of the problem by the dipole simulation method and the harge simulation method
with the dipole points (the harge points) j , the dipole diretion nj and the olloation points
zj respetively given by
j
= J (q!j 1 );
zj
= J (!j 1 );
nj
j 1 0 (q! j 1 )
= ! jJ 0 (Jq!
j 1 )j ;
j
= 1; 2; : : : ; N;
(4)
= exp(2i=N ), q is an assignment parameter suh that 1 < q <p 1, =
(A + ) ( B ) and J () is the Joukowski transform J ( ) = (=2)( + 1= ); = A2 B 2 :
We omputed the error estimates N maxz2 juN (z ) u(z )j; where the maximums are
estimated on 256 points distributed uniformly on the boundary , and show the results in
Figure 1. From the gures, we nd that the magnitudes of the errors of the dipole simulation
method is approximately the same as the magnitudes of the errors of the harge simulation
method, and the errors of the dipole simulation method deay exponentially as N inreases.
where
p
!
B = A
3. Problems in periodi regions
We show a modiation of the dipole simulation method for two-dimensional potential problems
with one-dimensional spatial periodiity. We onsider the problem of two-dimensional eletri
eld with ondutors in one-dimensional periodi array parallel to the real axis, and assume that
the eletri eld is uniform at innity in the diretion perpendiular to the ondutor array. We
write the eletri potential as = E (Im z ) + u; where E is the magnitude of the uniform
eletri eld at innity. Then, u satises the boundary value problem
8
>
<4u = 0
in u = E (Im z )
on Dn ( n 2 Z )
>
:u onstant as Im z ! 1;
2
(5)
The dipole simulation method was originally proposed in [2℄ for the two-dimensional potential problems in disk
regions and our work is an extension of his work for problems in general regions.
3
2nd International Conference on Mathematical Modeling in Physical Sciences 2013
IOP Publishing
Journal of Physics: Conference Series 490 (2014) 012054
doi:10.1088/1742-6596/490/1/012054
where D0Sis a simply-onneted
interior region, Dn = D0 + na( n 2 Z; a > 0 ) and
= C n n2Z Dn : Considering that u inludes a periodi funtion, we approximate u by
N
n
h
X
1
pj Re nj ot
(z
u(z ) uN (z ) = C0 +
2a j =1
a
N
X
1
= C0 + 2 pj Re
j =1
nj
z
j
+ nj
j
)
io
X
1
z
n6=0
1
+
na
na
j
(6)
;
where 1 ; 2 ; : : : ; N 2 D0 are the given dipole points, n1 ; n2 ; : : : ; nN 2 C are the given dipole
diretion suh that jnj j = 1, j = 1; 2; : : : ;P
N and C0 ; p1 ; p2 ; : : : ; pN 2 R are determined by the
olloation ondition and the onstraint Nj=1 pj = 0: The gure (a) in Figure 2 shows the
eletri eld with ylindrial ondutors Dn = f z 2 C j jz naj < g ; n 2 Z ( a = ) in a
periodi array omputed by the dipole simulation method with the dipole points j , the dipole
diretions nj and the olloation points zj respetively given by j = q!j 1 ; nj = !j 1 ; zj =
! j 1 ; j = 1; 2; : : : ; N; where q is an assignment parameter suh that 0 < q < 1. We also
omputed the error estimates N = maxz2D0 juN (z ) u(z )j; whih are shown in the gure (b)
of Figure 2. From the gure (b), the error N deays exponentially as N inreases.
0
4
-2
-4
log10(error)
Im z
2
0
-2
-6
-8
-10
q=0.9
q=0.7
q=0.5
q=0.3
q=0.1
-12
-14
-4
-16
-4
-2
0
Re z
2
4
0
20
40
60
80
100
N
(a)
(b)
(a) The eletri eld with ylindrial ondutors in a periodi array and (b) the error
estimates of the dipole simulation method.
Figure 2.
Aknowledgments
This study is supported by JSPS KAKENHI Grant number 22540116.
Referenes
[1℄
[2℄
[3℄
[4℄
[5℄
Amano K 1994 J. Comput. Appl. Math. 53 353
Katsurada M 1989 J. Fa. Si. Univ. Tokyo Set. IA, Math. 36 135
Murashima S 1983 Charge simulation Method and Its Appliations (Tokyo: Morikita) (in Japanese)
Sanhez-Sesma F R 1981 Arh. Meh. 33 167
Sanhez-Sesma F R and Rosenblueth E 1979 Earthq. Eng. Strut. Dyn. 7 441
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