Download kseee_paper2 - university of nairobi staff profiles

A Hybrid Finite Element/Moment Method for Solving Electromagnetic
Radiation Problem of Arbitrarily-Shaped Apertures in a Thick
Conducting Screen
ABUNGU N ODERO1, DOMINIC B O KONDITI1
ALFRED V OTIENO2
1Jomo
Kenyatta University of Agriculture and Technology
2University of Nairobi
Abstract – In this paper a hybrid numerical technique is presented for characterization of the transmission
properties of a three-dimensional slot in a thick conducting plane. The slot is of arbitrary shape and is excited
by an electromagnetic source far away from its plane. The analysis of the problem is based on the
"generalized network formulation" for aperture problems. The problem is solved using the method of
moments(MOM) and the finite element method(FEM) in a hybrid format. The finite element method is
applicable to inhomogeneously filled slots of arbitrary shape while the method of moments is used for solving
the electromagnetic fields in unbounded regions of the slot. The cavity region has been subdivided into
tetrahedral elements resulting in triangular elements on the surfaces of the apertures. Validation results for
rectangular slots are presented. Close agreement between our data and published results is observed.
Thereafter, new data has been generated for cross-shaped, H-shaped and circular apertures.
Index Terms – Finite–Element Method, Method of Moments, Perfect Electric Conductor(PEC)
I. INTRODUCTION
T
HE problem of electromagnetic transmission
through
apertures
has
been
extensively
investigated[1-6]. Many specific applications, such as
apertures in a conducting screen, waveguide-fed
apertures,
cavity-fed
apertures,
waveguide-towaveguide coupling, waveguide-to-cavity coupling, and
cavity-to-cavity coupling have been investigated in the
literature.
Both intentional and inadvertent apertures of various
shapes are encountered in the many applications
afforded by the advances in technology and increased
scale of utilization of microwave and millimeter-wave
bands for communications, radar, industrial and
domestic applications. Examples of undesirable
coupling are leakage from microwave ovens and printed
circuit boards, electromagnetic penetration into vehicles,
and electromagnetic pulse interaction with shielded
electronic equipment. These lead to problems of
Electromagnetic
Compatibility
(EMC)
and
Electromagnetic Interference (EMI) which should be
minimized or eliminated. Desirable coupling exist in
slotted antenna arrays, microstrip-patch antennas,
directional couplers, cavity resonators, etc.
For the purposes of either minimizing or enhancing
electromagnetic coupling through apertures, it is
desirable to quantify the electromagnetic penetration
through apertures. The past seventeen years have
witnessed an increasing reliance on computational
methods for the characterization of electromagnetic
problems. Although traditional integral-equation
methods continue to be used for many applications, one
can safely state that in recent times the greatest progress
in computational electromagnetics has been in the
development and application of hybrid techniques, such
as
FEM/MOM
and
(finite-difference
timedomain)FDTM/MOM .
The FDTM requires a fine subdivision of the
computational domain for good resolution and so is
computationally intensive just like the MOM. The
method of moments is an integral equation method
which handles unbounded problems very effectively but
becomes computationally intensive when complex
inhomogeneities
are
present.
In
contrast,
inhomogeneities are easily handled by the finite element
method, which requires less computer time and storage
because of its sparse and banded matrix. The matrix
filling time is also negligible when simple basis
functions are used [7]. However, it is most suitable for
boundary value problems. Since the investigation to be
carried out encompasses unbounded problems with
complex inhomogeneities, a procedure combining the
finite element method and the method of moments
would be effective.
It is the objective of this paper to develop and
demonstrate the validity and accuracy of a new hybrid
FEM/MOM method. Apertures having various shapes,
such as rectangular, circular, cross and H have been
studied.
II.
GENERAL FORMULATION OF THE PROBLEM
Consider the 3–dimensional ( aperture–cavity–
aperture ) structure illustrated in Fig.1.
Using an approach similar to that presented in [8],
one can replace the tangential electric fields in the
aperture planes with equivalent magnetic currents M 1 in


the z  0 aperture plane and M 2 in the z   d
aperture plane. In the interior region, the equivalent
magnetic currents  M 1 and M 2 in the z  0 and the
z   d  aperture
planes,
respectively.
Perfect
conductors can be placed in the z  0 S1  and the
z   d S 2  planes, as illustrated in Fig. 2, without
altering the fields. The problem geometry is then
decomposed into 3 regions as illustrated in Fig. 3.
y
cavity wall
Region A ( z > 0 )
upper aperture
o
S1
M1(x)
x
x
-M1(x)
Region C
lower aperture
d
M2(x)
-M2(x)
(a)
Ji, Mi
Region A
Region B ( z < -d )
z
y
o
x
S2
Fig. 2: Equivalent currents introduced in the aperture
planes.
z
 A, A
Ji, Mi
Region C
S1
M1(x)
x
conductor
Region B
conductor
z =0
S1
(b)
-M1(x)
Fig. 1. Problem geometry: (a) Top view (b) Cross-sectional view
The specific configuration consists of a cavity in a
thick conducting plane having an aperture at the top
surface and another one at the bottom surface. The free
space region above the plane ( z > 0 ) is denoted as
region A and that below the plane ( z < -d ) as region B.
Also, the volume occupied by the cavity ( -d < z < 0 )
will be referred to as region C. It is assumed that the
cavity is filled with an inhomogeneous material having
a relative permittivity  c r  and relative permeability
M2(x)
 C , C
S2
z = -d
x
-M2(x)
S2
 c r  .
 B , B
-z
Fig. 3: Problem geometry decomposed into
three regions; A, B, and C.
The total magnetic field in regions A and B can be
expressed as a superposition of the short-circuited

magnetic field H sc ( i.e., the incident magnetic field

plus the specular magnetic field H scat , produced by the
equivalent surface currents. The tangential components
of the total magnetic field in the aperture planes are then
expressed as



(1)
nˆ  H tot  nˆ  H sc  nˆ  H scat
In the interior region C, the closed cavity is
perturbed by the equivalent currents M 1 and M 2 , as
illustrated by Fig. 3. The total tangential magnetic field
that is produced on S 1 is expressed as the superposition
of the fields produced by each current




(2)
nˆ  H1tot  M1  H1tot  M 2
 



with a similar expression on S 2 . To ensure the
uniqueness of the solution, the tangential magnetic field
must be continuous across the aperture planes. To this
end, continuity of the total tangential magnetic fields
across the aperture planes is enforced. Therefore, on S1







(3)
nˆ  H1sc  nˆ  H1tot  M 1  H1tot  M 2  H1scat M 1
At this point, the equivalent currents are still
unknown. Their approximate solution is derived using
the method of moments. To this end, the equivalent
magnetic currents are expanded into a series of known
basis functions weighed by unknown coefficients.
 

M 1 x, y, z  0 



N
V
1n M 1n

 
P
V

2n M 2n
  Y  V
  Y  V

 H 1sc   Y11C
  sc    C
 H 2   Y21
(4)
x, y 
(5)
n 1
 
 Y A

 0
2n 
C
12
C
22
1n
0
YB
 V1n 
 
 V2 n 
 
(8)
The advantage of using this function is that the
admittance matrices of the interior and the exterior
regions can be constructed independently. The aperture
admittance matrices Y A  and Y B  are computed by
solving the problem of the equivalent magnetic currents
radiating into a half-space. The aperture admittance
matrix Y C  is computed by solving the interior cavity
problem for each equivalent magnetic current basis
function to compute in the aperture planes, and then
taking the inner products with the appropriate testing
functions. The following section describes the solution
of this interior problem using the FEM.
III.
x, y 
n 1

M 2 x, y, z  d  
V2 n can be
the unknown coefficients V1n and
obtained. The first two inner products of equations (6)
and (7) are referred to as the aperture admittance
matrices for region C [10]. These matrices relate the
tangential magnetic fields in the aperture to the
equivalent currents, which are equivalent to the
tangential electric fields. The last inner products
appearing in equations (6) and (7) are referred to as the
interior aperture admittance matrix for regions A and B
respectively. In terms of the aperture admittance
matrix, the coupled equations (6) and (7) are expressed
as
COMPUTING THE INTERIOR
ADMITTANCE MATRIX BY THE
FEM
A.
SOLUTION OF THE INTERIOR
PROBLEM
The approximate equivalent currents are substituted
into equation (3). Two sets of vector testing functions


are also introduced : W1m x, z  0 and W2m x, z  d 
Within the cavity region, the magnetic fields must
satisfy the vector Helmholtz equation
in the S1 and S 2 planes, respectively. Taking the inner
 
advantage of the linearity of the operators results in
where H is the total magnetic field. The cavity region
is defined as a closed space confined by PEC walls,

product of equation (3) with W1m x, z  0 and taking
 
W1m , H1sct  
N
V
1n
  V
 

W1m , H1tot
t M 1n 
n1

N
V
1n
P
2n
 
 

W1m , H1tot
t M 2n
n1
(6)
 
 

W1m , H1scat
M 1n
t
n1
where, a similar expression is derived on S 2


W2 m , H 2sct 
P
V
2n
n 1

P
V
2n
  



W2 m , H 2tott M 2 n 
N
n 1
 

 t 
W2 m , H 2sca
M 2n
t
1
r


  H  k 2 r H  0

which will be referred to as the domain V . A
functional described by the inner product of the vector

wave equation and the vectors testing function, H , is
expressed as
   H


FH 
 



V1n W2 m , H 2tott M 1n
V
(7)
n 1
where the subscript t denotes the tangential fields. By
introducing N  P vector testing functions, equations
(6) and (7) provide N  P linear equations from which
(9)
*
. 

 
  H  k 2  r H * .H dV
r

1
(10)
is defined. The solution of the boundary-value problem
is then found variationally, by solving for the magnetic
field at a stationary point of the functional via the first
variation
 

F H  0

Wej being the Whitney basis
with the spacial vector
(11)
function defined by
The functional has a stationary point ( which occurs at
an extremum ) and equations (9) and (10) can be used to
derive a unique solution for the magnetic field.
For an arbitrary V the solution of the above variational
expression cannot be found analytically and is derived
via the FEM. To this end, the domain V is discretized
into a finite number of subdomains, or element domains,
Ve ( Fig. 4). It is assumed that each element domain Ve
has a finite volume and that the material profile within
this volume is constant. Within each element domain

the vector magnetic field is expressed as H e . If there
are N e element domains within V , the functional can
be expressed in terms of the approximate magnetic
fields as
   

FH 
Ne
e 1






  

 *
1

  H e  k 02  r H e* .H e dVe 
 H e . 
Ve 
r



   

FH 
Ne
Ve
e 1

*  
 *
1
2
 H e .    H e  k 0  r H e .H e dVe

r



Wij  i  j   j i
and
as
  
 1 r  rb . Ai
i r   
4
3Ve


(13)
position vector of the barycenter of the tetrahedron
defined as
   
 r0  r1  r2  r3 
rb 
4

in which ri is the position vector of node i .
Using equations (17) in (16) leads to:

(14)
Also

k  
1 *
H e .nˆ    H e  j 0 H e* .M
r
(15)
0
implying contribution from only regions of implied

magnetic current sources, i.e, at the apertures only. M
= surface magnetic current source.
So that
   


F H 
Ne
e 1
Nb

b 1
S


 *

*   
1

2
   H e .   H e  k0 r H e .H e dVe 
Ve   r
 

k0  * 
H b .M n dS
  j 

(16)
0
The approximate vector field in each element domain is
represented as
m


H   ejWej
j 1

       


Ne
e 1
*
b
Nb

m
m
*
ej
j 1
ei
i 1
Ve



  
1

  Wej .   Wei  k 2Wej .Wei  dVe 



 r





 
  j   W .M dS 
b 1
 *

*   

1
2
V    H e .   H e  k 0  r H e .H e dVe  N ji
e


 r
 
F H  



1 *

e 1 

ˆ

H
.
n



H
dS
e
e
e
 S e  
 and
 r



 
(20)
k0
0
S
b

(21)
n
Letting
and the divergence theorem, (12 ) can be written as

(19)

where Ai is the inwardly directed vectorial area of the
tetrahedron face opposite to node i , Ve the element


volume, and r the position vector. Also, rb is the

F He 


   1
 

 1
1
H e* .    H e  . H e*     H e      H e .   H e*
r


 r
  r



is the barycentric function of node i expressed
(12)
Applying the vector identity
Ne
i
(18)
(17)





  
1
     Wej .   Wei  k 2Wej .Wei  dVe
Ve 
 r

Pnb  j
(22)
k0
0
 
W
 b .M n dS
S
(23)
equation (21 ) becomes
Ne
m

m
 Nb
F H e    ej*  ei N ji    b* Pnb
e 1  j 1
i 1
 b 1
 
(24)
By enforcing the continuity of the magnetic field, a
global number scheme can be introduced.
The
following symmetric and highly sparse matrix results
N eb   e   0 

(25)
N bb   b   Pnb 
where :  b are coefficients weighting the edge
elements lying in the aperture plane S1 and S 2 .  e are
 N ee
N
 be
coefficients weighting the remaining edge elements in
V.
B.
EVALUATION OF N ji
 Pib  j
k0
0

 W
S

pq .M i1dS
,for
the
i th common non-
Consider the tetrahedron in Fig ( ) :
boundary edge whose end nodes are p and q .
0
l
l
01
l
02
D0
03
3
l
 li


 2 A   i1 , r in Ti
i

 
 l
M i1 (r )   i   i1 , r in Ti 
 2 Ai
0 ,
otherwise


l
13
23
A0
1
2
l
12

W pq   p q  q  p
  
 1 r  rb . Ap
 p r   
4
3Ve
Fig( 4): Tetrahedron simplex 3-dimensional
finite element
 K12  M12  N12
 K 2 K1  M 2 M1  N 2 N1

D   KK3KK1MM 3MM1  NN3NN1
 K4 K1  M 4M1 N 4N1
 51 5 1 5 1
 K 6 K1  M 6M1  N6 N1
 R12 Q12  P12
 R2 R1 Q2Q1  P2 P1

K    RR3RR1QQ3QQ1 PP3PP1
 R4 R1 Q4Q1 P4P1
 5 1 5 1 51
 R6 R1 Q6Q1  P6 P1
K1K 2  M1M 2  N1N 2
K 22  M 22  N 22
K 3 K 2  M 3M 2  N 3 N 2
K4K2  M 4M 2  N4 N2
K1K 3  M1M 3  N1N 3
K 2 K3  M 2 M 3  N 2 N3
K 32  M 32  N 32
K 4 K3  M 4 M 3  N 4 N3
K5 K 2  M 5M 2  N5 N 2
K6 K 2  M 6M 2  N6 N 2
K5 K3  M 5M 3  N5 N3
K 6 K3  M 6 M 3  N 6 N3
R1R2  Q1Q2  P1P2
R2 2  Q2 2  P2 2
R3 R2  Q3Q2  P3 P2
R4 R2  Q4Q2  P4 P2
R5 R2  Q5Q2  P5 P2
R6 R2  Q6Q2  P6 P2
R1R3  Q1Q3  P1P3
R2 R3  Q2Q3  P2 P3
R32  Q32  P32
R4 R3  Q4Q3  P4 P3
R5 R3  Q5Q3  P5 P3
R6 R3  Q6Q3  P6 P3
N ji are the elements of matrix N   D  K 
C.
Pnb  j
EVALUATION OF
k0
0

S
 
Wb .M n dS
Pnb
K1K 4  M1M 4  N1N 4
K2K4  M 2M 4  N2 N4
K1K 5  M1M 5  N1N 5
K 2 K5  M 2 M 5  N 2 N5
K 3 K 4  M 3M 4  N 3 N 4
K 42  M 42  N 42
K5 K 4  M 5M 4  N5 N 4
K 3 K 5  M 3M 5  N 3 N 5
K 4 K5  M 4 M 5  N 4 N5
K52  M 52  N52
K 6 K5  M 6 M 5  N 6 N5
K6 K 4  M 6M 4  N6 N 4
R1R4  Q1Q4  P1P4
R2 R4  Q2Q4  P2 P4
R1R5  Q1Q5  P1P5
R2 R5  Q2Q5  P2 P5
R3 R4  Q3Q4  P3 P4
R4 2  Q4 2  P4 2
R5 R4  Q5Q4  P5 P4
R3 R5  Q3Q5  P3 P5
R4 R5  Q4Q5  P4 P5
R52  Q52  P52
R6 R4  Q6Q4  P6 P4
R6 R5  Q6Q5  P6 P5


K 3 K 6  M 3M 6  N 3 N 6 

K 4 K6  M 4M 6  N 4 N6

K5 K 6  M 5M 6  N5 N 6

K62  M 62  N62 
K1K 6  M1M 6  N1N 6
K 2 K6  M 2M 6  N 2 N6


R3 R6  Q3Q6  P3 P6 

R4 R6  Q4Q6  P4 P6

R5 R6  Q5Q6  P5 P6

R6 2  Q6 2  P6 2 
R1R6  Q1Q6  P1P6
R2 R6  Q2Q6  P2 P6
 
 

[Y B ]  [ Tq , H 2scat
M p  ] PxP
t
The discretization of the MOM computational domain is
based on triangular patch modeling scheme as proposed
by Rao et al [12] and as implemented by Konditi and
Sinha [ paper u carried ].
where

rb is the position vector of the tetrahedron
barycenter.
k l
Pib  j 0 i 
 0 2 Ai
A.


 W pq . i1 dS
S
Eliminating surface integration over each triangle by
approximating

W pq by its value at the centroid of each
EVALUATION OF MATRIX
ELEMENTS
As explained in [paper u carr ], following Galerkin

procedure ( Tm  M m ), a typical matrix element for the
rth region is given by
r
Ymn
 2  M m , H t r (M n ) 


= 2
Pib  j
k0 
  
c
 Ai W pq ri
 0  


li
 ic1   Ai W pq ri c 

2
A


 i

 .
 
li

 ic1  

2
A

 i
 

 .
  
  

lk 
 j i 0 W pq ri c  . ic1  W pq ri c  . ic1
2 0
D.



 Tm
triangle :
 
COMPUTING Y C
 
The computation of the nth column of Y C in ( ) and ( )
is performed by perturbing the cavity with an equivalent
current basis function M n . Then, with the use of ( ) or (
), the interior tangential magnetic fields in the aperture
planes S1 and S2 are computed. The first N row
 
C
elements of the nth column of Y
are then computed
by taking the inner product of the aperture field on S1

successively with the N testing functions Tm . The next
P row elements of the nth column are computed by
taking the inner product of the aperture field on S 2 with

with the P testing functions Tq . Similarly, the
= 2
 M
IV.
COMPUTING THE EXTERIOR
ADMITTANCE MATRICES BY THE MOM
 
 

[Y A ]  [Tm , H1scat
M n  ]NxN
t

Tm
m


M m  H t r ( M n ) ds 


 H t r ( M n ) ds
Tm
where the notation
 (
) ds
has been introduced
Tm
for compactness.
In terms of the electric vector potential F (r )
and the magnetic scalar potential  (r ) , the magnetic
field H r ( M n ) can be written as
H r (M n )   j Fn (r )  n (r )
where Fn (r ) =  r
 G (r / r ' )  M
n ( r ' ) ds '
T
n (r ) 
  Fn (r )
 j  r  r
In Eqn. (14), G ( r / r ' ) denotes the dyadic Green's
function of the half space.
Substitution of Eqn. (13) in Eqn. (12) and use of twodimensional divergence theorem leads to
r
Ymn
  2 j
 M
Tm
 
remaining P columns of Y C can be computed by
perturbing the cavity with equivalent current basis
functions M p .
M m  H t r ( M n ) ds 
m  Fn
ds  2
 m
m n
ds
Tm
(2.19)
where
mm
 Mm

 j
(2.20)
Eqn. (16) contains quadruple integrals; a double integral
over the field triangles Tm and a double integral over
the source triangles Tn involved in the computation of
Fn (r ) and n (r ) . In order to reduce the numerical
computations, the integrals over Tm can be

 c
 c

I mi   lm  H t i (rmc  )  m  H t i (rmc  )  m
2
2


approximated by the values of integrals at the centroids
of the triangles. This procedure yields
r
Ymn


 mc
 mc 
c
c

 j  Fn (rm )  2  Fn (rm )  2 
  2l m 



c
c

 n (rm )  n (rm )
where
Fn (rmc )   r
 G (r
Tn
c
| r ' )  M n (r ' ) ds'
1
j  r
(21)
Following the procedure as outlined in [ paper u
carried], transmission coefficient and transmission
cross-section are evaluated for the aperture problem.
V.

(19)
n (rmc ) 



 (18)



   G (r
c

| r ' )  M n (r ' ) ds'
Tn
(20)
In (18),  mc  are the local position vectors to the
centroids of Tm and rmc  = ( rm1  rm2   rm3 ) / 3 are the
position vectors of centroids of Tm with respect to the
global coordinate system.
Similarly, as explained in[ paper u carried],
using the centroid approximation in Eqn. (2.10), an
element of excitation vector can be written as





RESULTS AND DISCUSSION
REFERENCES
1.
2.
3.
4.
5.
6.
7.
Collin, R.E.: Field Theory of Guided Waves, New York:
McGraw-Hill, 1960
Harrington, R.F.: Time-harmonic Electromagnetic Fields,
New York : McGraw-Hill, 1961
N. Amitay. V. Galindo. and C. P. Wu, Theory and
Analysis of Phased Array Antennas. New York: WileyInterscience, 1972
W. F. Croswell. "Antennas and wave propagation,"
Electronics Engineers' Handbook, D. G. Fink and A. A.
McKenzie, Eds. New York: McGraw-Hill, 1975, pp. 1826.
R.F. Harrington, J.R. Mautz, and D.T. Auckland:
"Electromagnetic coupling through apertures," Dept.
Elec. Comput. Eng., Syracuse Univ., Syracuse NY, Rep.
TR-81-4, Aug. 1981.
C. M. Butler. Y. Rahmat-Samii. and R. Mittra.
"Electromagnetic penetration through apertures in
conducting surfaces," IEEE Trans. Electromagn. . vol.
EMC-20. pp. 82-93, Feb. 1978.
Xingchao Yuan, “Three-dimensional electromagnetic
scattering from inhomogeneous objects by the hybrid
moment and the finite element method”, IEEE Trans
Microwave Theory Tech., Vol. MTT-38,
pp. 1053-
1058, 1990.
Harrington, R.F. and Mautz, J.R. : "A generalized
network formulation for aperture problems," IEEE Trans.
Antennas Propagat., Vol. AP-24, pp. 870-873, 1976
9. Harrington, R.F. and Mautz, J.R. : "A generalized
network formulation for aperture problems," IEEE Trans.
Antennas Propagat., Vol. AP-24, pp. 870-873, 1976
10. S. K. Jeng, "Scattering from a cavity-backed slit in a
ground plane TE case," IEEE Tram. Antennas Propagat.,
vol. 38, pp. 1523-1529, 9, Oct. 1990.
11.
8.