Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
Thierry Gilles 1, Marc Piette 2, Christophe Craeye3
1,2 Ecole Royale Militaire, Laboratoire d’Electromagnétisme Appliqué (LEMA)Avenue de la Renaissance 30, 1000 Bruxelles, Belgium
[email protected] , [email protected]
3 Université catholique de Louvain, Laboratoire TELEPlace du Levant 2, 1348 Louvain-la-Neuve, Belgium
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
1. INTRODUCTION
2. EXACT THEORY
3. METHOD OF MOMENTS
4. MIXED MATERIALS
5. E-MFIE for EMBEDDED SHEETS
6. ACCURACY
7. CONCLUSIONS
OUTLINE
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
INTRODUCTION
Far Field
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
INTRODUCTION
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
INTRODUCTION
45m
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
INTRODUCTION
Near Field..differences..
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
INTRODUCTION
Mobile phone dual band antenna
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
INTRODUCTION
Benchmarking of software tools for small planar antenna analysisGuy A. E. Vandenbosch(1), A. Vasylchenko(1,2), and W. De Raedt(2)
Conference iWAT 2010
Impedance..differences..
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
INTRODUCTION
H
E
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
INTRODUCTION
Total Ez Field
Reaction to an EM excitation
by a set of LinearHomogeneousIsotropic
Volumes and/or Sheets
with bounding closed surface Sj
RESOLUTION METHOD : Maxwell EFIEi / MFIEi MoM Ji,Mi
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
INTRODUCTION
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
EXACT THEORY
0 0 0
0 0 0
0 0
0 , 0
.
.
source
source
source
m source
E j H M
H j E J
E
H
Harmonic waves ( ejt )
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
EXACT THEORY
EFIE – MFIE
0 0 0
0 0 0
0 0
0 , 0
.
.
source
source
source
m source
E j H M
H j E J
E
H
S1V
V0
0n̂
0 0 0 0 0 0 0 0 0 0 0ˆ ˆ ˆ( ) ( ) ' . ' 'inc
SE r E r j G n H n E G n E G dS
0 0 0 0 0 0 0 0 0 0 0ˆ ˆ ˆ( ) ( ) ' . ' 'incS
H r H r j G n E n H G n H G dS
S2
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
EXACT THEORY
EFIE – MFIE
EFIE :
MFIE :
From Differentialto
Integral equations
r
S1b
S2a
S3b
S4b
S4a
V0
1 ˆ ˆ ˆ( ) ( ) ( ) ' ' '2 a b
inca b S S
E r E r E r j G n H n E G n E G dS
1 ˆ ˆ ˆ( ) ( ) ( ) ' ' '2 a b
inca b S S
H r H r H r j G n E n H G n H G dS
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
EXACT THEORY
EFIE – MFIE .. for embedded sheets
EFIE :
MFIE :
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
EXACT THEORY
Boundary conditions
ˆ ˆ 0i i j jn E n E
ˆ jn
ˆin
iEjE
ˆ ˆ. . 0i i i j j jn E n E
ˆ
ˆi i i
i i i
J n H
M n E
1ˆ ˆ' ' 'i i
i i i i i iS SiG n E dS G n H dS
j
2( ) ' ' ' ' ' ( )
i i
inc ii i i i s i i i i iS Si TOTAL
scattered
jZE r k G J J G dS M G dS E r
k
ˆ ˆ ˆ( ) ( ) ' . ' 'i
inci i i i i i i i i i i iS
E r E r j G n H n E G n E G dS
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
EXACT THEORY
Introducing the current densities
2( ) ( ) ' ' ' ' 'i i
inc ii i i i i s i i i iS Si
jZE r E r k G J J G dS M G dS
k
1ˆ ˆ'sn E n Hj
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
2
2
ˆ ˆ ˆ( ) ( ) ( ) ( ) ' ' ' ( ) ' '
ˆˆ ( ) ' '( ) ' ' 'ˆ ( )( ) ( )
i i
ii
inc i ii i s i i i i i s i i i i iS Si i
iinc i i ii i i i s i iii i SSi
jZ jZn r E r J r n r k G J J G dS n r M G dS
k k
jZn r M G dSn r k G J J G dSM rn r E r
k
EXACT THEORY
Normal and tangentiel component
nEFIE :
tEFIE :
Re (M) Im (M)
0 1,000 0,500
r = 4
r = 1
Re (J ) Im (J )
0 2,386 1,193
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
EXACT THEORY
Mie and MoM solutions D = 0/3
Flat PEC sheet
Sa
Sb
ˆ ˆan n
bJ ˆ ˆ2 inca a a b a bn H n H H J J
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
EXACT THEORY
(Flat) PEC sheets
Physical Optics Approximation
=> Exact solution
aJ
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
METHOD OF MOMENTS
e
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
1
( ) ( )
Ji
i i
i
NJ
i j jj
J r J f r
1
( ) ( )
Mi
i i
i
NM
i m mm
M r M f r
p+ p-
T+ T-
1
2
e
r r
RWG function
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
METHOD OF MOMENTS
Discretization
Exact
3rd order expansion
Linear expansion
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
p+ p-
1
2
T T
(Full) linear divergence conforming basis function
Transverse gradient
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
METHOD OF MOMENTS
On the choice of basis functions
p+ p-
T+ T-
1
2
e
r r
RWG function
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
Curl conforming basis functions ?
Electric current density Magnetic current density
Cube with side 0/4 and r = 2
METHOD OF MOMENTS
On the choice of basis functions
( ) ( )dS ( ) ( ) ( ) dS
J Mi i
i i i i i i
i ii i
N Ninc J M
e i e i j i i j m i mj mS S
N r E r N r Z J N D f r M K f r
( ) ( )dS ( ) ( ) ( ) dS
J Mi i
i i i i i i
i ii i
N Ninc J M
h i i h i j i i j m i mj mS S
T r Z H r T r Z J T K f r M D f r
ˆ( ) ( ) ( )i ie e iN r w r n r
ˆ( ) ( ) ( )i ih i hT r n r w r
Arbitrary normal function for the nEFIE or nMFIE
Arbitrary tangential function for the tEFIE or tMFIE
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
METHOD OF MOMENTS
Testing
, , ,
, , ,
, , ,
, , ,
EJ n EM n E ni i iHJ n HM n H ni i i iEJ t EM t E tii i iHJ t HM t H ti i i
Z Z V
Z Z J V
MZ Z V
Z Z V
,( , ) ( ) ( ) dS
i ii i
i
EJ n Ji e i i je jS
Z Z N r N D f r ,( ) ( ) ( )dS
ii
i
E n ince ie
S
V N r E r
,( , ) ( ) ( ) dS
i ii i
i
HM t Mh i mh m
S
Z T r D f r
,( , ) ( ) ( ) dS
i ii i
i
HJ t Ji h i i jh jS
Z Z T r T K f r
,( , ) ( ) ( ) dS
i ii i
i
EM n Me i me m
S
Z N r K f r
,( ) ( ) ( )dS
ii
i
H t inci h ihS
V Z T r H r
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
METHOD OF MOMENTS
Local ZI=V system of equations
tMFIE for a PEC body
--- Complex vector ---
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
METHOD OF MOMENTS
Current densities from I
, , ,0,(1,1) 0,(1, ) 0,(1) 0,(1)
, , ,0,( )0,( ,1) 0,( , ) 0,( )
...
... ... ... ... ...
...
HJ t HJ t H tN
HJ t HJ t H tNN N N N
Z Z VJ
JZ Z V
tMFIE for a PEC body
N=2799
r27
9
0 0,(2) 0,(2) 0,(7) 0,(7) 0,(9) 0,(9)( ) ( ) ( ) ( )J r J f r J f r J f r
Re [J /| H inc| ] Im [J /| H inc| ]
tEFIEG-f CN=14374
tMFIE-nxf CN=98
0 3,88 1,94
Side = 0/4
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
METHOD OF MOMENTS
PEC rounded cube
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
MIXED MATERIALS
Consequence :16 J + 4 M = 20 unknowns2x6+2x6+2x15 = 54 equations
OOOOOOOOOOVI
OOOOOOOOOOV
OOOOOOOOOOIV
OOOOOOOOOOIII
OOOOOOOOOOII
OOOOOOOOOO
H
I
OOOOOOOOOOVI9-10(l-r)
OOOOOOOOOOV07-08
OM4OOOOOOOOOIV05-06
OM3OOOOOOOOOIII03-04
OM2OOOOOOOOOII2
OM1OOOOOOOOO
E
I1
J13RRRRRRVII
J12RRRRRRV
J11RRRRRRIII
J10RRRRRR
H
II
J10RRRRRRVII10u
J10RRRRRRV07-08
J9RRRRRRIII03-04
J9RRRRRR
E
II2
J8LLLLLLVII
J7LLLLLLIV
J6LLLLLLIII
J5LLLLLL
H
I
J4LLLLLLVII10u
J3LLLLLLIV05-06
J2LLLLLLIII03-04
J1LLLLLL
E
I1
VJMJZm
IIIIIIVIIVIVIVIIIIIIType
03-042110u
07-08
05-06
03-0421n
OOOOOOOOOOVI
OOOOOOOOOOV
OOOOOOOOOOIV
OOOOOOOOOOIII
OOOOOOOOOOII
OOOOOOOOOO
H
I
OOOOOOOOOOVI9-10(l-r)
OOOOOOOOOOV07-08
OM4OOOOOOOOOIV05-06
OM3OOOOOOOOOIII03-04
OM2OOOOOOOOOII2
OM1OOOOOOOOO
E
I1
J13RRRRRRVII
J12RRRRRRV
J11RRRRRRIII
J10RRRRRR
H
II
J10RRRRRRVII10u
J10RRRRRRV07-08
J9RRRRRRIII03-04
J9RRRRRR
E
II2
J8LLLLLLVII
J7LLLLLLIV
J6LLLLLLIII
J5LLLLLL
H
I
J4LLLLLLVII10u
J3LLLLLLIV05-06
J2LLLLLLIII03-04
J1LLLLLL
E
I1
VJMJZm
IIIIIIVIIVIVIVIIIIIIType
03-042110u
07-08
05-06
03-0421n
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
N L N L N P N P N R N R N O N O
ML Z EJ Z EM J V E
ML Z HJ Z HM 0 M V H
MP Z EJ Z EM J V E
MP Z HJ Z HM * M = V H
MR Z EJ Z EM J V E
MR Z HJ Z HM M V H
MO 0 Z EJ Z EM J V E
MO Z HJ Z HM M V H
Z I = V
Boundary conditions for RWG
MIXED MATERIALS
B.C.
M4
M3
M2
M1
J13
J12
J11
J10
J10
J10
LROJ9LROROLOLROROLOLROROLOIII
ROJ9ROROOROROOROROOII
LOJ8LOOLOLOOLOLOOLO
H
I
LROJ7LRRLLRRLLRRLVII10u
OJ6OOOOOOOOOVI9-10(l-r)
ROJ5ROROOROROOROROOV07-08
LOJ4LOOLOLOOLOLOOLOIV05-06
LROJ3LROROLOLROROLOLROROLOIII03-04
ROJ2ROROOROROOROROOII2
LOJ1LOOLOLOOLOLOOLO
E
I1
VIMJZm
IIIIIIVIIVIVIVIIIIIIType
03-042110u
07-08
05-06
03-0421n
M4
M3
M2
M1
J13
J12
J11
J10
J10
J10
LROJ9LROROLOLROROLOLROROLOIII
ROJ9ROROOROROOROROOII
LOJ8LOOLOLOOLOLOOLO
H
I
LROJ7LRRLLRRLLRRLVII10u
OJ6OOOOOOOOOVI9-10(l-r)
ROJ5ROROOROROOROROOV07-08
LOJ4LOOLOLOOLOLOOLOIV05-06
LROJ3LROROLOLROROLOLROROLOIII03-04
ROJ2ROROOROROOROROOII2
LOJ1LOOLOLOOLOLOOLO
E
I1
VIMJZm
IIIIIIVIIVIVIVIIIIIIType
03-042110u
07-08
05-06
03-0421n
Obtain a square Z matrix !
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
Reduction
MIXED MATERIALS
Common schemes :
Non PEC edges (1,2,3,4)
PMCHWT (addition)Müller (weighted addition)
PEC edges
EFIE only (drop MFIE)MFIE only (drop EFIE)CFIE = f [EFIE,MFIE]
!! Problem with embedded sheets
PEC
r= 4
0,30
0,10
iE
iH
RMA-UCL FEKO™
PEC
r= 4
0,30
0,10
iE
iH
RMA-UCL FEKO™
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
MIXED MATERIALS
MoM example
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
MIXED MATERIALS
MoM example
0,150
r = 4
PEC
iE
iH
RMA-UCL FEKO™
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
E-MFIE for EMBEDDED SHEETS
1,1 1, 1,12
7,1 7, 7,12
8,1 8, 8,12
9,1 9, 9,12
10,1 10, 10,12
11,1 11, 11,12
12,1 12, 12,12
... ...
... ... ... ... ...
... ...
... ...
... ...
... ...
... ...
... ...
M M Mn
M M Mn
M M Mn
E E En
E E En
M M Mn
M M Mn
z z z
z z z
z z z
z z z
z z z
z z z
z z z
1
1 7
8
9
1012
11
12
...
...
...
M
M
M
n E
E
M
M
v
J v
vJ
v
vJ
v
v
2-edges
3-edges
64
3
21
10 115
7
8
9
12
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
E-MFIE for EMBEDDED SHEETS
E-MFIE
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
E-MFIE for EMBEDDED SHEETS
Bowtie with solderline
Gap length = 0/20
Gap length = 0 , PEC solder line
Gap length = 0 , air solder line
0 6,523,26
Im[J]/|Hinc|
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
ACCURACY
Hm
N
nn
HMmn
N
nn
HJmn
Em
N
nn
EMmn
N
nn
EJmn
VMZJZ
VMZJZ
11
11
..
..
.EJ EM E
HJ HM H
JZ Z V
MZ Z V
tEFIE
tMFIE
Infinite when R = 0
mT
nf
Test
Basis RWG
Sm Sn
R
2( ) ( ) ( ') ' ( ') ' ( ) 'm n
EJmn m n sS S
jZZ T r k G R f r f r G R dS dS
k
ˆ ( ) ( )( ) ( ') ' ( ) '
2m n
HJ m nmn m nS S
n r f rZ T r f r G R dS dS
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
ACCURACY
Numerical integration issue
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
ACCURACY
Polynomial quadrature
singular edge
# nodes fI = -0,03505 n fI = 0,01074
Numerical Rel error Numerical Rel error 1 -0,01693 -0,52 -0,00564 -1,53 3 -0,02545 -0,27 +0,00384 -0,63 4 -0,02613 -0,25 +0,00384 -0,64 6 -0,02969 -0,15 +0,00685 -0,36 7 -0,03102 -0,12 +0,00775 -0,28 12 -0,03217 -0,082 +0,00868 -0,19 16 -0,03517 +0,0033 +0,01074 +0,0007 19 -0,03341 -0,047 +0,00954 -0,11 25 -0,03420 -0,024 +0,01026 -0,044 42 -0,03510 +0,0012 +0,01088 +0,0136 61 -0,03450 -0,016 +0,01034 -0,037 73 -0,03481 -0,007 +0,01056 -0,017
X
Z
Y
(1,0,0)
T
T’ 'p (0,0,0)
'p p ( , ,0)r x y
(0,1,0) p
3'
1( ) ( ') '
4T T
RC T r r p dS dS
R
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
ACCURACY
Polynomial quadratures and regular triangles
0,0845
0,0850
0,0855
0,0860
0,0865
0,0870
0,0875
0,0880
0,0885
3 7 12 16 19 25 42 61 73
-0,586
-0,584
-0,582
-0,58
-0,578
-0,576
-0,574
Z12
Z33
2,17
2,18
2,19
2,20
2,21
2,22
2,23
3 7 12 16 19 25 42 61 73
3,056
3,057
3,058
3,059
3,060
3,061
3,062
3,063
3,064
J3 J5
1mx1mx1m (0 / 30) PEC pyramid
incE
incH
10 MHz
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
ACCURACY
Polynomial quadratures and regular triangles
10-8
10-6
10-4
10-2
100
102
104
106
10-4
10-2
100
102
104
1
3 4
67
12
1619
25
42
61
73
S
Rel
ativ
e er
ror
in %
424242
42
16
73
7
T = 1 = Z
I < 0 I > 0
3'
1( ) ( ') '
4T T
RC T r r p dS dS
R
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
ACCURACY
Polynomial quadratures and stretched triangles
Z
S T
incE
incH 300 MHz
Exact ~ 16 nodes integration 7 nodes integration
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
Polynomial quadratures and stretched triangles
ACCURACY
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
CONCLUSIONS
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics
CONCLUSIONS
1. VECTOR FORMS OF EFIE / MFIE for VOLUMES and SHEETS Generalisation of the physical optics formula
2. CHOICE OF BASIS and TESTING Functions LIMITATIONS of RWG Curl Conforming functions
3. GENERAL CANVAS for MIXED MATERIALS
4. E-MFIE for EMBEDDED SHEETS
5. ACCURACY IMPORTANCE OF A CORRECT TREATMENT OF SINGULARITIES Reference solutions EFIE / MFIE discrepancy only close to edges (with RWG)
Advances in the Formulations and Accuracy of theMethod of Moments Applied to Electromagnetics