Hybrid finite element boundary element solutions for three dimensional scalar potential problems

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Hybrid finite element boundary element solutions for three dimensional scalar potential problems

Gérard Meunier, Jean-Louis Coulomb, Shape Salon, Laurent Krähenbühl

To cite this version:

Gérard Meunier, Jean-Louis Coulomb, Shape Salon, Laurent Krähenbühl. Hybrid finite element boundary element solutions for three dimensional scalar potential problems. IEEE Transactions on Magnetics, Institute of Electrical and Electronics Engineers, 1986, 22 (5), pp.1040- 1042.

�10.1109/TMAG.1986.1064625�. �hal-00359878�

IEEE TRANSACTIONS

ON

MAGNETICS, VOL. MAG-22, NO. 5 , SEPTEMBER 1986

HYRRTU

F I N I T E ELEMENT B O U N D A R Y ELEMENT S O L U T I O N S FOR

THREE D

IMENS XONAL S C A L A R

P O T E N T I A L

P R O B L E M S

G. M e u n i e r , J . L . Coulomb S . J . S a f a n L a b o r a t o i r * e d ' E l e c t r o t e c h n i q u e R P I

d e G r e n a b l e , F r a n c e T r o y , N e w York

A b s t r a c t

T h e H y b r i d f i n i t e e l e m e n t - b o u n d a r y e l e m c n t m e t h o d h a s b e e n s h o w n t o b e an e f f s c f l i v e a n d c f f i c : i e n t m e t h o d f o r t h e s o l u t j o n o f t w o d i m e n s i o n a l [ 1

1

^{a n d} a x i s y m m e t r i c

P I

e l e c t r o m a g n e t i c f j e l d p r o b l e m s .

T h e

method a l l o w s f o r a n y r e g i o n i n t h e p r o b l e m t o b e r e p r e s e n t e d b y e i t h e r f i n i t e elements o r b o u n d a r y e j . e m e t l t s l Thus t h e u s e r can s o l v e o p e n b o u n d a r y p r o b l e m s ^{o r} c e r t a i n c l a s s e s ^{o f} e x t e r i o r p ~ * o b l a r s [ 3 ] b y u s i n g t h e b o u n d a r y e l e m e n t m e t h o d _{a n d} s t i l l r e t a i n t h e n o n l i n e a r c a p a b i l i t y o f t h e f i n i t e e l e m e n t m e t h o d f o r r e g i o n s w i t h n o n l i n e a r m a t e r i a l s .

T h e

m e t h o d

is now e x t e n d e d t o t h r e e d i m c n s i o n a 3 s c a l a r p o t e n t i a l p r o b l e m s . An e x a m p l e i s p r e s e n t e d h e r e f o r a t h r e e d i m e n s i o n a l p r o b l e m .

T h e

results w e r e t h e n c o m p a r e d ^{t o} a c l o s e d f o r m s o l u t i o n .

Three Dimensional

F o r m u l a t i o n

' In _{t h e} f o l l o w i n g f o r m u l a t i o n t h e u n k n o w n is t h e scalar p o t e n t i a l ,

w h i c h

is t h e s o l u t i o n o f Laplace's e q u a t i o n , in b o t h t h e f i n i t e c-:Iement a n d b o u n d a r y e l e m e n t r e g i o n s .

T h i s

s e c t i o n g i v e s t h e b a s i s o f t h e f o r m u l a t i o n f o r t h e

f i n i t e

e l e m e n t s a n d t h e b o u n d a r y e l e m e n t s w h i c h w a s u s c d on t h e e x a m p l e i n t h e f o l l o w i n g

s e c t i o n .

F i n i t e E l e m e n t R e g i o n : I n _{t h e} f i n i t e e l e m e n t region t h e unk-nown p o t e n t i a l s a t i s f i e s Laplace's e q u a t i o n .

In

G a J u r k i n f o r m

t h i s

b e c o n ~ e s

w h e r e

w

is a w e i g h t i n g f u n c t i o n a n d t h e

.

n e w ^*I- v a r i a b l e 3 $ / 2 n a p p e a r s o n l y o n t h e b o u n d a r y o f t h e p r o b l e m . E x p a n d i n g $j i n t e r m s o f p o l y n o m i a l s and c h o o s i n g w

z a i

we o b t a i n f o r a n element,

s ( e ) @

+

^{q = 0}

W h e r e

hi aai

+--

aai

a a i

^-t-- _a2 ^{a c ~ i -)}

a&

_az ^dxdydz

B o u n d a r y e l e m e n t s : A p p l y i n g Green's theorem t o Laplace's e q u a t i o n w e o b t a i n a n e x p r e s s i o a f o r t h e p o t e n t i a l i n t e r m s o i ' t h e p o t , e n t i a l a n d its n o r m a l d e r i v a t i v e o n t h e b o u n d a r y .

W h e r e y = 1 ^{f o r} a p o * i n t i n s i d e t h e r e g i o n y =

0

f o r a p o i n t o u t s i d e t h e r e g i o n y = t h e f r a c t j o n o f t h e - i n t e r n a l a n g l e made ^{b y} t h e s u r f a c e a t t h e f i e l d p o i n t . ( e . g . 0 . 5 i f t h e p o i n t is on a s t r a i g h t l i n e )

T h i s e x p r e s s i o n i s e v a l u a t e d d i r e c t l y b y t h e p o i n t

m a t c h i n g

m e l - h o d . G a u s s q u a d r a t u r e is u s c d t o p e r f a r m t h e i n t e g r a t i o n .

A s s c m l > Z y o f t,hi.: m a t r i x : I n s y m b o l i c f o r m t h e s y s t e m m a t r i x is a s f o l l o w s

w h e r e t h e u n k n o w n 8 e x i s t s a t e a c h n o d e p o i n t a n d t h e u n k n o w n 3g/aa e x i s L s at e a c h n o d e ^on

t h e

b o u n d a r y .

E x a m p l e p r o b l e m : I n o r d e r t u v e r i f y t h e f o r m u l a t i o n , a n e x a m p l e w a s t a k e n f r o m elec-1,rost;z.t.-i ^{C S .} T h e e x a m p l c c h o s e n was a c o n d u c t i n g s p h e r e h a v i n g an a p p l i e d p o t e n t i a l .

T h e

s p h e r e is i m b e d e d i n u n i f o r m h o m o g e n e o u s s p a c e . The r e g i o n

i n

b e t w e e n t h e s p h e r e a n d a n a r b i t r a r i l y c h o s e n c u b e was r e p r e s e n t e d b y f i n i t e e l e r n e ~ t ~ s .

I n _{t h i s}

c a s e

t h e

f i n i t e e l e m e n t s w e r e t e t r a h e d r a a n d were g e n e r a t e d b y t h e D e l a u n y m e t h o d [ 4 ] , t h e b o u n d a r y e l e m e n t s _werue t h e t r i a n g u l a r faces o f t h e t e t r a h e d r a w h i c h w e r e i n c o m m o n w i t h t h e e x t e r i o r c u b e . S e e F i g u r e

1.

M a t r i x ^solution.

T h e

s y s t e m m a t r i x

h a s

t h e f o r m s h o w n b e l o w i n f i g u r e 2 . It s h o u l d b e p o i n t e d o u t that

w h i l e

_{t h e} f i n i t e element e q u a t i o n s a r e s p a r s e

,

^{t h e} b o u n d a r y element, e q u a t i o n s a r e i n g e n e r a l f u l l y p o p u l a t e d , i + e . a l l u n k n o w n s on

t h e

b o u n d a r y a r e c o u p l e d t o e a c h o t h e r . A n o t h e r p o i n t i n f l u e n c i n g t h e c h o i c e o f a s o l u t i o n t e c h n i q u e i s t h a t t h e b o u n d a r y ^element e q u a t i o n s

w i l l

g e n e r a l l y b e n o n s y m m e t r i c .

The

method c h o s e n in t h i s c a s e w a s t h e p r e c o n d i t i o n e d b i c o n j u g a t e g r a d i e n t m e t h o d 151.

MI1 8-9464/86/0900-1040$01.00

0

1986 IEEE

I

t

1

SPM$E I t

I

1 I

- - r

- r r r L r r r r r c " 1

I 1 1

1

FULL

I

F I G U R E 2 .

F O R M

O F T H E

S Y S T E M M A T R T X

F I G U R E

1 .

C Q N D U C T J M G

S P H E R E

I N F R E E S P A C E

Discussion of results: The

problem was s o l v e d u s i n g b o t h

f i r s t

a n d s e c o n d o r d e r elements.

An equipotential plot

in t h e finite

element

r e g i o n is shown i n f i g u r e 3 The

c o r r e s p o n d i n g e l e c t r i c f i e l d vectors are

shown i n figure

4 . Due t o

t h e

l a r g e number

o f

unknowns

and t h e l a r g e b a n d w i t h

o f

t h e

s y s t e m

m a t r i x

an o p t i o n f o r

inclusion

o f s y m m e t r y boundary c o n d i t i o n s was a d d e d .

Figure 5 shows

t h e 1 / 8 t h section

o f

t h e p r o b l e m w h i c h was

s o l v e d

n e x t . F i g u r e

6

shows the

c o r r e s p o n d i n g e q u i p o t e n t i a l p l o t

w h i c h is

s m o o t h e r

t h a n

t h a t

o f f i g u r e

3

due

t o

the

smaller

s i z e

^of

t h e

e l e m e n t s .

F i g u r e

₇

s h o w s t h e

p o t e n t i a l as a

function

o f radius

which

a g r e e s

well with

_{t h e} a n a l y t i c

solution

C o n c l u s i o n s :

T h e H y b r i d m e t h o d h a s

b e e q

F I G U R E

4 . E ~ B C T R I C

F I E L D

VECTORS

CORRESPONDING

successfully extended t o

t h r e e d i m e n s i o n a l TO F I G U R E

3 p r o b l e m s .

A

s o l u t i o n t o a

full t h r e e 1

d i m e n s i o n a l

problem w a s o b t a i n e d

u s i n g

a r e l a t i v e l y

small _number

o f

u n k n o w n s .

F I G U R E 3 . E Q T 3 I P O T E N T I A t PLOT

I N

_{T H E} F I N I T E

ELFNlrlNT R E G

ION

FIGITRE 5 , _SPHERE WTTTT

S Y M M E T R Y C O N B I T T O M

F I G U R E

6 . E Q U I P O T E N T I A J J P L O T

OF F T G U R E 5

I 1 1 1 1 1 r 8 r

( 2

I r r r 1 1 r r~

II4 ^{I t t t t t l l '} 115 ^{r r t r T r l f f} 1'6 ^{1 1}

References:

1 . S J

. S a l o n , J.M.Schneider, " A

Finite

Element Boundary I n t e g r a l F o r m u l a t i o n o f Poisson's E q u a t i o n " , I E E E

Transactions,Vol.MAG-17.#6.

pp

2574-2576

Z.S.J.Salon, J.P.Peng,"A Hybrid

F i n i t e

B l e m e n t

Boundary E l e m e n t

F a r m u l a t i a n of

Poisson's

Equation

_For

Axisymmetric

Vector Potential

Problems",

J . A p p 1 .

Physics 53(11),November1982 PP

8420-8422

3. S.J.Salon,"The Hybrid

F i n i t e

E l e m e n t

Boundary E l e m e n t

Method i n

Blectromagnetics", I E E E

Trans.,Vol.NAG-21,#5,Sept*l985,pp.1829-

1834

4 3 .

d u Terrail, Modelisation

Geometrique et Topologique en

3

Dimensions

P o u r

l Y A p p l i c a t i o n

de l a Method

des

Elements

F i n i s en

Electromagnetism

s

P h D

Thesis

INPG

G r e n o b l e

1986

5.D.A.H.Jacobs,"Generalizations of t h e

Conjugate Gradient

Method

f o r S o l v i n g Non-

S y m m e t r i c and C o m p l e x

Systems", C E G B , RD/L/M70/80

F I G U R E

7 .

P O T R N T T A I

v s . R A D I U S

TN THE

F I N I T E E L E M E N T R E G I O N