d imens xonal scalar potential problems - CiteSeerX

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 D IMENS XONAL S C A L A R POTENTIAL P R O B L E M S G. M e u n i e r , J . L . Coulomb L a b o r a t o i r * e d'Electrotechnique de G r e n a b l e , France

Abstract

The 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 method 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 and cffic:ient method f o r the s o l u t j o n o f t w o

dimensional [ 11 and axisymmetric P I e l e c t r o m a g n e t i c f j e l d problems. T h e method allows f o r a n y r e g i o n in the p r o b l e m to b e represented b y e i t h e r f i n i t e elements or boundary ej.emetltsl 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 or c e r t a i n c l a s s e s of 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 method 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 capability 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 nonlinear materials. The method 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 scalar potential problems. An e x a m p l e is p r e s e n t e d here for a t h r e e dimensional p r o b l e m . The results w e r e t h e n compared t o a c l o s e d f o r m solution.

FOR

THREE

S . J . Safan RPI T r o y , N e w York

Boundary e l e m e n t s : A p p l y i n g Green's theorem to Laplace's e q u a t i o n w e o b t a i n a n expressioa for t h e potential i n terms oi' 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 .

y = 1 for 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 the r e g i o n y = t h e f r a c t j o n o f the - 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 straight l i n e )

Where

This expression is evaluated directly by the p o i n t matching m e l - h o d . Gauss q u a d r a t u r e is uscd t o perfarm the integration. 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 form the s y s t e m m a t r i x is as follows

Three Dimensional Formulation '

SOLUTIONS

t h e following formulation t h e unknown 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 equation, in b o t h the 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 regions. This s e c t i o n g i v e s t h e b a s i s o f the formulation f o r t h e finite elements a n d t h e b o u n d a r y elements which w a s uscd 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 sect i o n .

In

F i n i t e E l e m e n t Region: I n t h e f i n i t e element region the unk-nown p o t e n t i a l satisfies Laplace's equation. In G a J u r k i n f o r m t h i s b econ~es

where w is a w e i g h t i n g function a n d t h e new . variable 3 $ / 2 n appears only on t h e boundary o f the problem. E x p a n d i n g $j i n terms of * I

polynomials and choosing f o r an element,

+

s(e)@

w

zai

we

obtain

q = 0

Where

where t h e unknown 8 e x i s t s a t each node p o i n t and t h e unknown 3g/aa e x i s L s at each n o d e on the b o u n d a r y . Example problem: In order t u v e r i f y t h e formulation, an example was taken from elec-1,rost;z.t.-iC S . The e x a m p l c chosen was a c o n d u c t i n g s p h e r e having an a p p l i e d potential. The 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

space. The r e g i o n in between 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 elerne~t~s. In t h i s case the f i n i t e elements were t e t r a h e d r a a n d were g e n e r a t e d by t h e Delauny method [4], t h e boundary 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 the t e t r a h e d r a which were in common w i t h the e x t e r i o r c u b e . See F i g u r e 1. solution. The system matrix has t h e form shown b e l o w in f i g u r e 2 . It should b e p o i n t e d out that while t h e f i n i t e element equations 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 general 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 coupled t o each o t h e r . Another p o i n t influencing t h e c h o i c e of a s o l u t i o n technique i s t h a t the b o u n d a r y element equations will g e n e r a l l y b e nonsymmetric. The method chosen in this c a s e gradient was t h e preconditioned biconjugate method 151.

Matrix

hi aai

+--aai a a i

-t--

a c ~ i a& -) dxdydz a2 az

MI18-9464/86/0900-1040$01.00 01986 IEEE

I

t

1 I

SPM$E

t

I 1

-

-r

I -rrrLrrrrrc" 1

I 1 1 1

FULL

I

FIGURE 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 SPACE

Discussion of results: The problem was solved 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 the finite element region is shown in figure 3 The corresponding electric 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 large bandwith of t h e s y s t e m m a t r i x an o p t i o n for 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 of t h e problem which was solved n e x t . Figure 6 shows the corresponding equipotential p l o t which is s m o o t h e r t h a n t h a t of f i g u r e 3 due t o the smaller s i z e of t h e elements. Figure 7 shows t h e potential as a function of radius which agrees well with t h e analytic solution

Conclusions: The H y b r i d method h a s b e e q F I G U R E 4 . E ~ B C T R I C FIELD VECTORS successfully extended t o t h r e e dimensional TO FIGURE 3 problems. A solution t o a full three1 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 relatively small number o f u n k n o w n s .

FIGURE 3. E Q T 3 I P O T E N T I A t PLOT I N THE ELFNlrlNT R E G ION

CORRESPONDING

FINITE

FIGITRE 5 , SPHERE WTTTT S Y M M E T R Y CONBITTOM

References: 1.SJ . S a l o n , J.M.Schneider, " A Finite Element Boundary I n t e g r a l Formulation 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 Blement Boundary Element Farmulatian 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 Element Boundary Element Method in Blectromagnetics", IEEE Trans.,Vol.NAG-21,#5,Sept*l985,pp.18291834 4 3 . du Terrail, Modelisation Geometrique et Topologique en 3 Dimensions Pour l Y A p p l i c a t i o n de la Method des Elements Finis en Electromagnetism s PhD Thesis INPG G r e n o b l e

1986 5.D.A.H.Jacobs,"Generalizations Conjugate Gradient Method for Symmetric and Systems", C E G B , RD/L/M70/80

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

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