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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

[email protected]

mailto:[email protected]

mailto:[email protected]


1. INTRODUCTION

2. EXACT THEORY

3. METHOD OF MOMENTS

4. MIXED MATERIALS

5. E-MFIE for EMBEDDED SHEETS

6. ACCURACY

7. CONCLUSIONS

OUTLINE


INTRODUCTION

Far Field


INTRODUCTION


INTRODUCTION

45m


INTRODUCTION

Near Field..differences..


INTRODUCTION

Mobile phone dual band antenna


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..


INTRODUCTION

H

E

http://physics-animations.com/Physics/English/ellf_tmp.htm


INTRODUCTION

Total Ez Field

http://www.met.reading.ac.uk/clouds/maxwell/circle20.html

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


INTRODUCTION


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 )


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


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


EXACT THEORY

EFIE – MFIE .. for embedded sheets

EFIE :

MFIE :


EXACT THEORY

Boundary conditions

ˆ ˆ 0i i j jn E n E

ˆ jn

în

iEjE

ˆ ˆ. . 0i i i j j jn E n E

ˆ

î 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


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


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


EXACT THEORY

Mie and MoM solutions D = 0/3

Flat PEC sheet

Sa

Sb

ˆ ân n

bJ ˆ ˆ2 inca a a b a bn H n H H J J


EXACT THEORY

(Flat) PEC sheets

Physical Optics Approximation

=> Exact solution

aJ


METHOD OF MOMENTS

e


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


METHOD OF MOMENTS

Discretization

Exact

3rd order expansion

Linear expansion


p+ p-

1

2

T T

(Full) linear divergence conforming basis function

Transverse gradient


METHOD OF MOMENTS

On the choice of basis functions

p+ p-

T+ T-

1

2

e

r r

RWG function


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


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


METHOD OF MOMENTS

Local ZI=V system of equations

tMFIE for a PEC body

--- Complex vector ---


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


METHOD OF MOMENTS

PEC rounded cube


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


03-042110u

07-08

05-06

03-0421n


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


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


03-042110u

07-08

05-06

03-0421n

Obtain a square Z matrix !


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™


MIXED MATERIALS

MoM example


MIXED MATERIALS

MoM example

0,150

r = 4

PEC

iE

iH

RMA-UCL FEKO™


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



E-MFIE



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|


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


ACCURACY

Numerical integration issue


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


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


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


ACCURACY

Polynomial quadratures and stretched triangles

Z

S T

incE

incH 300 MHz

Exact ~ 16 nodes integration 7 nodes integration


Polynomial quadratures and stretched triangles

ACCURACY


CONCLUSIONS


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)


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