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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

thierry.gilles@rma.ac.be , marc.piette@rma.ac.be

3 Université catholique de Louvain, Laboratoire TELEPlace du Levant 2, 1348 Louvain-la-Neuve, Belgium

craeye@uclouvain.be

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

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

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

INTRODUCTION

EXACT THEORY

source

m source

E j H M

H j E J

Harmonic waves ( ejt )

EXACT THEORY

EFIE – MFIE

source

m source

E j H M

H j E J

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

EXACT THEORY

EFIE – MFIE

EFIE :

MFIE :

From Differentialto

Integral equations

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 ˆ. . 0i i i j j jn E n E

ˆi i i

1ˆ ˆ' ' 'i i

i i i i i iS SiG n E dS G n H dS

2( ) ' ' ' ' ' ( )

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

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

1ˆ ˆ'sn E n Hj

ˆ ˆ ˆ( ) ( ) ( ) ( ) ' ' ' ( ) ' '

ˆˆ ( ) ' '( ) ' ' 'ˆ ( )( ) ( )

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

jZn r M G dSn r k G J J G dSM rn r E r

EXACT THEORY

Normal and tangentiel component

nEFIE :

tEFIE :

Re (M) Im (M)

0 1,000 0,500

Re (J ) Im (J )

0 2,386 1,193

EXACT THEORY

Mie and MoM solutions D = 0/3

Flat PEC sheet

ˆ ˆan 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

METHOD OF MOMENTS

( ) ( )

i j jj

J r J f r

( ) ( )

i m mm

M r M f r

RWG function

METHOD OF MOMENTS

Discretization

3rd order expansion

Linear expansion

(Full) linear divergence conforming basis function

Transverse gradient

METHOD OF MOMENTS

On the choice of basis functions

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 J V

MZ Z V

,( , ) ( ) ( ) dS

i ii i

EJ n Ji e i i je jS

Z Z N r N D f r ,( ) ( ) ( )dS

E n ince ie

V N r E r

,( , ) ( ) ( ) dS

i ii i

HM t Mh i mh m

Z T r D f r

,( , ) ( ) ( ) dS

i ii i

HJ t Ji h i i jh jS

Z Z T r T K f r

,( , ) ( ) ( ) dS

i ii i

EM n Me i me m

Z N r K f r

,( ) ( ) ( )dS

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

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

OOOOOOOOOOVI9-10(l-r)

OOOOOOOOOOV07-08

OM4OOOOOOOOOIV05-06

OM3OOOOOOOOOIII03-04

OM2OOOOOOOOOII2

OM1OOOOOOOOO

J13RRRRRRVII

J12RRRRRRV

J11RRRRRRIII

J10RRRRRR

J10RRRRRRVII10u

J10RRRRRRV07-08

J9RRRRRRIII03-04

J9RRRRRR

J8LLLLLLVII

J7LLLLLLIV

J6LLLLLLIII

J5LLLLLL

J4LLLLLLVII10u

J3LLLLLLIV05-06

J2LLLLLLIII03-04

J1LLLLLL

VJMJZm

IIIIIIVIIVIVIVIIIIIIType

03-042110u

03-0421n

OOOOOOOOOOVI

OOOOOOOOOOV

OOOOOOOOOOIV

OOOOOOOOOOIII

OOOOOOOOOOII

OOOOOOOOOO

OOOOOOOOOOVI9-10(l-r)

OOOOOOOOOOV07-08

OM4OOOOOOOOOIV05-06

OM3OOOOOOOOOIII03-04

OM2OOOOOOOOOII2

OM1OOOOOOOOO

J13RRRRRRVII

J12RRRRRRV

J11RRRRRRIII

J10RRRRRR

J10RRRRRRVII10u

J10RRRRRRV07-08

J9RRRRRRIII03-04

J9RRRRRR

J8LLLLLLVII

J7LLLLLLIV

J6LLLLLLIII

J5LLLLLL

J4LLLLLLVII10u

J3LLLLLLIV05-06

J2LLLLLLIII03-04

J1LLLLLL

VJMJZm

03-042110u

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

LROJ9LROROLOLROROLOLROROLOIII

ROJ9ROROOROROOROROOII

LOJ8LOOLOLOOLOLOOLO

LROJ7LRRLLRRLLRRLVII10u

OJ6OOOOOOOOOVI9-10(l-r)

ROJ5ROROOROROOROROOV07-08

LOJ4LOOLOLOOLOLOOLOIV05-06

LROJ3LROROLOLROROLOLROROLOIII03-04

ROJ2ROROOROROOROROOII2

LOJ1LOOLOLOOLOLOOLO

VIMJZm

03-042110u

03-0421n

LROJ9LROROLOLROROLOLROROLOIII

ROJ9ROROOROROOROROOII

LOJ8LOOLOLOOLOLOOLO

LROJ7LRRLLRRLLRRLVII10u

OJ6OOOOOOOOOVI9-10(l-r)

ROJ5ROROOROROOROROOV07-08

LOJ4LOOLOLOOLOLOOLOIV05-06

LROJ3LROROLOLROROLOLROROLOIII03-04

ROJ2ROROOROROOROROOII2

LOJ1LOOLOLOOLOLOOLO

VIMJZm

03-042110u

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

RMA-UCL FEKO™

MIXED MATERIALS

MoM example

MIXED MATERIALS

MoM example

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

E E En

M M Mn

2-edges

3-edges

10 115

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

.EJ EM E

HJ HM H

JZ Z V

MZ Z V

Infinite when R = 0

Basis RWG

2( ) ( ) ( ') ' ( ') ' ( ) 'm n

EJmn m n sS S

jZZ T r k G R f r f r G R dS dS

ˆ ( ) ( )( ) ( ') ' ( ) '

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

(1,0,0)

T’ 'p (0,0,0)

'p p ( , ,0)r x y

(0,1,0) p

1( ) ( ') '

RC T r r p dS dS

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

-0,578

-0,576

-0,574

3 7 12 16 19 25 42 61 73

1mx1mx1m (0 / 30) PEC pyramid

10 MHz

ACCURACY

Polynomial quadratures and regular triangles

424242

T = 1 = Z

I < 0 I > 0

1( ) ( ') '

RC T r r p dS dS

ACCURACY

Polynomial quadratures and stretched triangles

incH 300 MHz

Exact ~ 16 nodes integration 7 nodes integration

Polynomial quadratures and stretched triangles

ACCURACY

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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