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8/18/2019 Finite-Element Formulation
1/7
8/18/2019 Finite-Element Formulation
2/7
900
IEEE TRANSACTIO NS ON MICROWAV E THEORY AND TECHNIQUES, VOL.
MTI
-3 3, NO.
la,
OCT08ER
Finite-Element Formulation in Terms of the
Electric-Field Vector for Electromagnetic
Waveguide Problems
MASANORI KOSHIBA, SENIOR t-tEMBER , IEEE , KAZUYA HA YATA,
AND
MICHIO SUZUKI, SEN I
OR
MEMBER, IEEE
A
ftruct
- A veclo r fin ite-element method for the analysis of anisotropic
\\'1lvegui
des wilh off-diagonal elements in Ihe permeability tensor is for
mulated in terms of all th ree COmponents of Ihe electric field. In this
approach, spurious, nonphyslcal so lutions do
rIOl
appear anywhere above
the
a
ir-line. The appli cation of this finite-element method to waveguides
with an abrupt discontinuity In the permittivjty is discussed. In particular,
we discu
ss bow
to use the bou
nd
ary co nditions of the electric field al the
[nt
.,.
rface
e t w ~ n
two
medi
a
with
different pennlttivilies. To
show
the
validity and usefulness of this formulation, exa
mpl
es are
co
mputed for
dielectric- loaded waveguides
Ilnd
fe
rn i
e-l
oa
ded waveguides.
I. INTRODUCTION
T
HE VECTOR · finite-element method is widely used
either in an axial-component E, - Hz) formulation
[1)- (4) or in a three-component (either th e electric field E
or the magne tic field H ) fo rmu lation
(5
), [6), which enables
one to compute accurately the mode spectrum of an el
ec
tromagnetic waveg
ui
de with arbitrary cross section: The
most serious diffi
cu
hy in u
si ng
the vector
fi
nite-element
analys is is the appearance of spurious, nonphysi
ca
l
so lu
tions [1)-[6]. Hano
[7
has presented a three-component
finite-element formulation with rectangular elements. In
his formulation, spurious so
lu
tions, except for zero e
ig
en
values, do not appear, but a diagonal permittivi ty tensor
and a d iagonal permeability ten sor are assumed. Rece ntly,
an improved finite-element method with triangular
ments has been fo
rm
ulated for the anal
ysis
of anisotropic
di e
le
ctric waveguides in terms of all thr
ee
components of H
[8) - [l1J. In dielectr
ic wavegu
ides, the permeability is al
ways assumed to
be that of free space. Therefore, each
ever,
t
IS difficult to apply this H-field formulation
tQ:
waveguides containing anisotropic media such as
f e r r i
because the tenso r permeability may vary from material to :;
materia
l.
In such cases, it
is
advantageous
to
solve for
j
rather than for H . .
In this paper, an imp roved finite-element method with.
triangular elements is formulated for the analysis of n i s ~
tropic waveguides with a nondiagonal permeabili
ty
t e n s o r
using all three components of E. II) ferrite-loaded wave-
guides, the permittivity is assu med to be constant in each-
•
material,
but
may vary
fr
om material
to
material.
At
abrupt discontinuity in the permittivi ty,
th
ere is an abrupt
change in
E.
In
thi
s work, the application
of
the
E-field
fo rmulation to wavegu ides with abrupt discontinuities
in
the perrrtittivity is discussed in det
aiL
In particula
r,
we
discuss how to use the boundary conditions of
E
at
th
e
interface between two media wit h different permittivities.
In this improved E-rield
fo
rmulation, no spurious solu
ti
onf'
appear anywhe
re
above the air-line. The appearance
of
spurious solutions is
limi
ted to the region / ko< 1 and
these solutions are equivale
nt to t h ~
TM modes of
hollow
waveguides. To show the validity and usefuln
ess
of this
formulation, ex.amples are computed for dielectric-loaqed
waveguides and ferrite-loaded waveguides.
II . FUNCTIONAL FORMULATION
We
co
nsider an anisotropic waveguide with a
tensa
permeability and a scalar pe·rmittivity. With a time depen
8/18/2019 Finite-Element Formulation
3/7
el Ill. : ELECTROMAGNETIC WAVEGUIDE PROBLEMS
-
,
2( 0 11 0
(4)
functional
[12], (13)
for (3) is known to be
- k
J L ( , E . E d n (5)
where
n
represents the cross section of the waveguide and
asterisk denotes complex conjugation. In the finite-el
e
analysis using (5), spurious solutions appear scattered
the propagation diagram (51-112] , [141. 115]. These
SP
' ' '; '
). s solutions belong to
tw
o distinct cat
ego
ri
es [11)
,
The first one (SI) can be characterized as follows:
V
x E =O
'\7'(,£ + 0
o r k
~
6)
,· 1,, second group (S2) can be characterized as follows:
vxE*'O 'C'
rE-.l-O
for k5 > O.
(7)
In order to eliminate the spurious solutions SI and S2
we
propose the following functional according to the H-field
fo rmulation (8]- (11) :
(8)
For the functional (8), the first variation
aft
is given by
~ t - o E
. [ X
([ X
X E) - ( , , •,E) -
k <
,E] dO
-
I r ~ E [ n
x
r X
E)
- n(,,·.,E)]dr
(9)
where f represents the contour of the region 0 ,
n
is the
outward unit normal vector to
f ,
and the term
n
X
([
p,
,1-
1
v X E) correspond s to the tangent
ia
l components
of the magne tic
fi
eld H on
f .
The stationary requirement
901
Fi
g.
l . Interface with an abrupt discontinuity in the permittivity.
obey the following equations:
€,E - v1/;
( \7
2
+
k5)'
-= 0
in
region n
(12a)
(12b)
' _ 0 on perfect electric conductor (12c)
af an -= 0 on perfect magnetic conductor
(12d)
where , is the scalar field. The electric field E of (12)
satisfies the stationary requirement
8F
- 0, but the diver
gence of
€,E
is not ze ro. Therefore, in the finite-element
analys
is
using (8), spurious solut
io
ns
S
3'
which are not
included in (5), do appear. The solutions S3 are equivale
nt
to the TM modes of hollow waveguides (replace' in
(12b)- (12d) with E
z
) and the appearance is limited to the
region f3
/
ko < 1. They do not appear anywhere above the
ai r
-line.
III.
FINITE-ELEMENT DISCRET IZATION AN D
BOUNDARY CONDITIONS
Dividing the cross section n of the waveguide into a
number of second-order triangular elements
as
shown in
Fig. 1 and using the finite-elemenLmethod on (8), we can
write the functional fo r the whole reg ion 0 in the form
t -
Lt
.
(Il)
•
t. -
(E [AJ.(E
).
14
)
[AJ . -
[SJ. +[UJ k5[TJ . 15)
wh
ere
{ E } is
the electric-field vector
co
rr
es
ponding to the
nodal points within each element, thy matrices
8/18/2019 Finite-Element Formulation
4/7
902
IEEE TRANSACTIONS ON MTCROW.WE THEORY AND TECHNIQUES, VOL, MIT-33, NO. 10, OCTOBER
I f the functional for € is used in its origirial fo rm (14),
we
should modify the functional
for
e
2
in order
to
satisfy
the boundary
conditions
of the electric field Eon f , For
e
2
•
the functional (14) can be rewritten as
A x x ]
lA,,)'
[AI,
[A.xll
[Ax'xli
IA ,L
[Az'xL
F , ~ E ) i l A I , E ) ,
(16a)
(EL
lA,,),
IA,,],
lA,,],
lA",],
lA",),
lA,,],
(E, ),
(E,) ,
(E,),
(E,,),
(E
),
(E,,),
[A.i:zL
lA,, ],
[A,,] ,
[A
",],
lA",],
[A
u'
],
IA,,),
[A u' )'
[A
x
'>;'
] 2
lA ,, ],
[A
,
),
' [A ,,
),
(16b)
lA,,),
IA
",
],
IA
",),
I
A
, ],
lA",,],
lA ,,),
Using (17), (16) can be transformed as follows:
[Ax:'L
lA ,],
[A ,
),
[A
,,
),
IA
, ] ,
[A ,,),
F , ~
(E)i XI,(E),
(E),
(E,\,
(E,
),
(E,
)2
(E,,),
(E
),
(E,,),
[AX •
J2
lA
,,],
[Au]2 [Ad],
I
A,,]
,
[A , ),
IA,,],
IA,,], IA,,], IAp),
IA ,],
I
A
,),
h ~
[A,J,
lA,,),
[A,,),
[Au'],
I
A",],
[A ,]'
[A
,,'],
lA,,),
[A ,],
[AN]'
IA,
,,,],
[AN]'
IA",),
IA ,I,
I
A
,I,
lA ,
, ,
IA""I,
IA
I,
[A
J,
lA ,),
[A ,],
[A
i',' ],
IA ,,),
[ A
where
the
components
of
the
{E
j
}
2
vector are the values
of
the eiectric field Ej l
=
x, y, z) at the nodal points within
,
the element e
z
except P, the components of the {Ei'h
vector are the values of
E;
t
the nOda l points
on
f
, ,
where
[Ax'x'b = q';x[A
X
X']2+ qxAxy([ A
x
,y
·12+
[A
y;x' ]2)
+q';y[Ay·y,12
[A X'Y'] 2
=
qxxqxy
(A x'x']2+
qxxqyy [
AX'y']2
19b)
16c)
19c)
(20a)
8/18/2019 Finite-Element Formulation
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KOSHIBA el al.: ELECTROMAGNETIC W
AV
EGUIDE PROBLEMS
•
•
-
b , . ; t"QI2
( 1.5
Fi
g.
2. Dispersion characteristics of a h
alf-fi 1ed
dielectric waveguide.
By using the original functional (14) fo r e
1
and 'the mod
ified functional (19) for e
2
• the boundary conditions of the
electric field E at the interface with an abrupt disco
nt
inu
ity in the permittivit
y
are sat
is
fied.
IV. NUMERICAL RE
SU LT S
A. Dielectric- Loaded Waveguides
Fi
rs
t, let us consider a rectangular waveguide half-
fil
led
wfth
a
dielectric
of
permittivity (\ (relati
ve
permittivity
f
,\
-
( 1 / (
0 ) ' .
We subdivide one half of the cross section into second
order triangular elements
as
shown in
th
e insert in Fig. 2,
where
(
1 - 1.5, the
pl
ane of
sy
mm etry
is
assumed
to
a
perfect magnetic conductor, 36 elements (N£) are used,
and the number of the nodal points N
p
is 91. Computed
res
ults (solid lines) for
th
e LSM
n
a
nd L S E m ~
modes agree
well with the exact results [16J. Spurious solutions S, and
S2
which are included in (5), do not appear. Spurious
solutions S3 (dashed lin es) corresponding to the sol utions
of (12) appear on
ly
in the region Il/ko <
1.
The solutions
903
•
,
,
•
,
•
.
•
,
,
,
,
,
,
100
_0
15
-0
F
ig. 3.
p-depcndence for the spurious
s o l u t i o n s ~ .
becomes F. For
(2
1),
(12)
is reduced to
( p 2 V 1 k ~ ) 1 f _ in
regi
on 0 (22a)
If - a on perfect electric conductor
(22b)
on per
fect
magnet
ic co
nductor.
(22c)
The appearance of the solutions of (22) is limited to the
region Pl ko i p and the cutoff frequencies of these
so lutions
va
ry in proportion to the value of
p.
Fi
g.
3 shows the p-dependen
ce for
the
so
lutions S3 in
the same waveg
uid
e
as
sh
ow
n in Fig.
2.
So
li
d and dashed
lines in Fig. 3 correspo
nd to
the TMIl and TM
I2
modes in
Fig.
2, re
spectively. When
p
= 2, the
so
lutions
S3
appear in
th
e
re
gion /3 l k(j<
0.5
and the cutoff
fre
quencies of
th
e
solutions corresponding to the TMA) and TM Il
mod
es in
Fig. 2 become
koQ
- 2fi IT and 2v 5
IT,
respective l
y.
Wh en
p
=
0.5, the so lutions S3 appear in the region Plk
o
< 2 and
the cuto
ff
freq
uen
cies of the
sol ut io
ns corresponding to the
TM
ll and TM
12
modes in
Fi
g.
2 become
k o
Q
0.5fi IT
and
0.51S IT,
respectively. The p-dependence is very sm
all
for the physical solutions. For larger valu es of
p,
how ever,
the degree of accura
cy
for
th
e physical
so
lutions becomes
poorer. For smaller val ues of p. on the other ha
nd
, more
8/18/2019 Finite-Element Formulation
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IEEE TRANSACTIONS ON
MI
CIlOWAVE THEOIlY AN D TECHNIQUES,
VOL
. MIT-B , NO.
10
,
8/18/2019 Finite-Element Formulation
7/7
[6]
[ ]
[8]
[9]
[IO[
[ll]
[ 2J
[13J
[14]
[15]
[16J
[17]
[18J
t'r al,:
ELECTROMAGNETIC WAVEGUIDE PROIJLEMS
M. Ikeuch
i, H.
Sawami, and H.
Nih
Analysis of open-type
dielectric waveguides by the finite·element iterative method, IEEE
Trans, Microwave Til/wry Tech" vol.
MTI-29, pp. 234 - 239 , Ma
r.
I n l
N. Mabaya, P. E. Lagasse, and P. Vandenbukke, Finite element
analysis
of
optical waveguides,
IEEE Trans. Microwave Theory
Tech., vol.
MIT-29, pp. 600- 605, June 1981.
K. Oyamada and T. Okoshi, Two-dimensional finite-element
calculation
of
propagation characteristics of axially nonsymmetrical
op
tical fibers,
Radio Sci., vol.
17, pp. 109- 116,
Jan.-Feb
. 1982.
A. Konrad, High-order triangular finite elements for electromag
netic waves in anisotropic media,
IEEE Trans. Miuowaue Theory
Tech.,
vol. MIT-25, pp.
353
- 360. May 1977.
B. M. A Rahman and J. B. Davies, Finite-element analysis of
optical and microwave waveguide problems,
IEEE Tram. Micro
wave Theory Tech., vol. MIT-n
pp.
20
-
28,
Jan. 1984.
M. Hano, Finite-element analysis of dielectric-loaded waveguides,
I
Trans.
Microwave
Theory
Tech ..
vol.
MIT-32, pp. 1275-1279,
.Oct. 1984.
M. Koshiba, K. Hayata, and M. Suzuki, Vectorial finite-element
formulation without spurious modes for dieleclric waveguides,
Trans. lnsl. Eleclron. Commull .
Eng.
Japall, vol.
E67,
pp.I91
- 196,
Apr. 1984.
M.
Koshiba,
K.
Hayata, and
M.
Suwki, Vectorial finite-element
method without spurious solutions for dielectric waveguide prob
lems, Elec/ron. Lell.,
vol. 20,
pp. 409 - 410, May 1984.
B.
M. A Rahman and
J.
B. Davies, Penalty function improvement
of waveguide solution by finite elements,
IEEE Truns. Mi{'rQwuve
TheaI) Te,·h.,
voL
MIT
-
n
pp. 922-
928,
J\ug. 1984,
M.
Koshiba, K. Hayata, and
M.
Suzuki, Improved finite-clement
formulation in terms of the magnetic-field vector for dielectric
waveguides, IEEE Tram. Microwave Theory
Tech.
, vol. MTT-33,
pp. 227- 233, Mar. 1985.
A
Konrad, Vector variational formulation of electromagnetic
fields in anisotropic media, IEEE Tral/-$. Microwave Theory Tech.,
vol. MIT;24,
pp. 553-559, Sept. 1976.
A.
D. Berk, Variational principle for
el
ectromagnetic resonators
and waveguides, IR Tran$. Amellnas Propaga/.,
vol.
AP-4, pp.
104-111, Apr. 1956.
J.
B.
Davi
es
,
F. A.
Fernandez, and G. Y. Philippou, Finile element
analysis of aU modes in cavities with circular symmetry, I
Trall$. Microwave Theory Tech
..
voL
MTT-30, pp. 1975- 1980, Nov.
1982.
M.
Hara
. T. Wada, T. Fukasawa, and
F.
Kikuchi,
A
three dimen
sional analysis of
RF
electromagnetic fields by the finite element
method,
IEEE Trall . MagI ., vol.
MAG·19, pp. 2417-2420, Nov.
1983.
N. Marcuvitz,
Wuveguide Halldbook.
New York: McGraw-Hill,
1951.
Z.
J. Cscndcs and P. Silvester, Numerical solution of dielectric
loaded waveguides:
II
- Modal approximation technique
,
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Trans. Micro
wave
Theory Tech., vol.
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905
Masanori Koshiba
(S
M'84)
was
born in Sapporo,
Japan, on November 23,. 1948. He received the
B.S.,
M.S.
, and Ph.D. degrees in electronic en
gineering from Hokkaido University, Sapporo,
Japan, in 1971, 1973, and 1976, respectively.
In 1976 , he joined the Department
of El
ec
tronic Engineering, Kitami Institute
of
Technol
ogy, Kitami. Japan. Since 1979,
he
has been an
Assistant Professor
of
Electronic Engineering at
Hokkaido University. He has been engaged in
research on surface acoustic waves, dielectric
optical waveguides, and applications
of
finite-element and boundary-ele
ment methods to field problem's.
Dr
. Koshiba is a member of the Institute of Electronics and Communi
cation Engineers of Japan, the Institute
of
Television Engineers of Japan,
the Institute
of
Electrical Engin
ee rs of
Japan, the Japan Society for
Simulation Technology, and Japan Society for Computational Methods
in
Engineering
KazllJ a
Hayata was born in Kushiro, Japan,
on
December 1,1959. He rece
ived
the
B.S.
and M
,S.
degrees in electronic engineering from Hokkaido
University, Sapporo, Japan, in 1982 and 1984,
respectively.
Since 1984, he has been a Research Assistant
of Electronic Engineering at Hokkaido Univer
sit
y. He
has
been
engaged in research on dielec
lric optical waveguides and surface acoustic
waves.
Mr . Hayata
is
a member of the Institute
of
Electronics and Communication
n g i n ~ r s
of Japan.
Miehio Suzuki (SM'57) was born in Sapporo,
Japan, on November 14,
1923.
He received the
B.S.
and Ph.D. degrees in electrical e n g i n ~ r i n g
from Hokkaido University, Sapporo, Japan, in
1946 and 1960, respectively.
From 1948 to 1962, he was an Assistant Pro
. f
es
sor of Electrical Engineering at Hokkaido
University. Sin
ce
1962,
he
has ~ n a Professor
of Electronic Engineering at Hokkaido Univer
sity. From 1956 to 1957, he
was
a Research
Associate at the Microwave Research Institute
of
the Polytechnic Institute of Brooklyn, Brooklyn, NY.
Dr. Suzuki
is
a
m m ~ r
of the Institute
of El
ectronics and Communi
cation Engineers of Japan, the Institute of Electrical Engin
ee
rs of Japan,
the Inst itute 'of Television Engineers of Japan, the Japan Society of
Information and Communication R
ese
arch, and the Japan Society for
Simulation Technology.