RECENT DEVELOPMENTS IN FIELD AND FORCE COMPUTATION

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RECENT DEVELOPMENTS IN FIELD AND FORCE COMPUTATION

J. Simkin

To cite this version:

J. Simkin. RECENT DEVELOPMENTS IN FIELD AND FORCE COMPUTATION. Journal de

Physique Colloques, 1984, 45 (C1), pp.C1-851-C1-860. �10.1051/jphyscol:19841174�. �jpa-00223649�

JOURNAL _DE PHYSIQUE

Colloque Cl, suppl6ment a u n o I, Tome 45, janvier 1984 page C1-851

RECENT DEVELOPMENTS I N F I E L D AND FORCE COMPUTATION

J. Simkin

Rutherford Appleton Laboratory, Ghilton Didcot, Oxon OX11 OQX, U . K .

RQsum6 - Les progrBs r s c e n t s e n m a t i s r e d e m6thodes d e c a l c u l d e s champs Blectromagn6tiques s o n t p a s s d s e n revue, en s e r d f 6 r a n t e n p a r t i c u l i e r aux t r a v a u x p r d s e n t d s B l a Conf6rence COMPUMAG q u i s ' e s t t e n u e 1 GGnes en j u i n 1983. Les p r 6 c i s i o n s o b t e n u e s 1 l ' a i d e d ' d q u a t i o n s i n t d g r a l e s e t aux

r i v 6 e s p a r t i e l l e s s o n t compardes. Le c a l c u l d e s f o r c e s e s t examin6.

A b s t r a c t - Recent developments i n methods f o r e l e c t r o m a g n e t i c f i e l d compu- t a t i o n a r e reviewed, i n p a r t i c u l a r work r e p o r t e d a t the COMPUMAG Conference h e l d i n Genoa i n June 1983. The accuracy of p a r t i a l d i f f e r e n t i a l and i n t e g r a l e q u a t i o n s o l u t i o n s i s compared and t h e e v a l u a t i o n of f o r c e s i s examined.

1 - INTRODUCTION

The computation of e l e c t r o m a g n e t i c f i e l d s i s important f o r many s c i e n t i f i c and i n d u s t r i a l a p p l i c a t i o n s . Most of t h e s e a p p l i c a t i o n s r e s u l t i n computational problems t h a t a r e unique to t h i s d i s c i p l i n e . One of t h e major problems i s t h a t r e s u l t s must be extremely a c c u r a t e i f they a r e t o be of any p r a c t i c a l u s e . T h i s requirement would be i m p o s s i b l e t o s a t i s f y i f t h e m a t e r i a l s used i n the construc- t i o n of e l e c t r o m a g n e t i c d e v i c e s were l e s s easy t o model. Commonly used m a t e r i a l s a r e e a s i l y measured and c h a r a c t e r i s e d , and t h e i r p r o p e r t i e s a r e r e a s o n a b l y c o n s t a n t . However, t h e r e a r e e x c e p t i o n s , f o r example permanent magnet m a t e r i a l s s u c h a s Alnico. There a r e many o t h e r s p e c i f i c problems, f o r example t h e unbounded n a t u r e of t h e f i e l d , t h e g e o m e t r i c a l complexity of t h e d e v i c e s and s k i n e f f e c t s i n eddy c u r r e n t s o l u t i o n s .

T h i s paper reviews t h e p r o g r e s s i n e l e c t r o m a g n e t i c f i e l d computation, paying p a r t i c u l a r a t t e n t i o n t o work r e p o r t e d a t t h e June 1983 COMPUMAG conference.

C u r r e n t r e s e a r c h i s dominated by t h e a p p l i c a t i o n of f i n i t e elements to t h e s o l u t i o n of p a r t i a l d i f f e r e n t i a l e q u a t i o n s , t h e reasons f o r f a v o u r i n g t h i s approach a r e examined. The c a l c u l a t i o n of f o r c e s produced by e l e c t r o m a g n e t i c f i e l d s i s o f t e n s u b j e c t t o l a r g e e r r o r s , t h e t e c h n i q u e s used f o r f o r c e c a l c u l a t i o n s a r e compared and the cause of the e r r o r s i s i d e n t i f i e d .

2

- ACCURACY

I n many a p p l i c a t i o n s of e l e c t r o m a g n e t i c d e v i c e s t h e f i e l d s must be computed t o an a c c u r a c y of t h e o r d e r of 1 p a r t i n 1000 o r b e t t e r . T h i s can only be achieved when t h e m a t e r i a l s involved a r e s t a b l e and t h e i r p r o p e r t i e s can be e a s i l y measured.

Even t h e n , g u a r a n t e e i n g t h e r e s u l t s t o t h i s accuracy i s extremely expensive and f o r t r u l y t h r e e dimensional f i e l d s may be i m p o s s i b l e . Problems t h a t i n v o l v e l i n e a r magnetic m a t e r i a l s can be s o l v e d t o v e r y high p r e c i s i o n i n two dimensions u s i n g boundary i n t e g r a l m e t h o d s / l / . A s i m p l e problem has been solved w i t h c o n s t a n t p e r m e a b i l i t y i n o r d e r t o show r a t e s of convergence of t h e s o l u t i o n f o r d i f f e r e n t methods. Although r e l a t i v e l y simple the problem shown i n F i g u r e 1 does provide a n o n - t r i v i a l t e s t . One q u a r t e r of a 'H' frame magnet with a s l o p i n g p o l e was s o l v e d

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19841174

Cl-852 JOURNAL DE PHYSIQUE

using a s c a l a r p o t e n t i a l boundary i n t e g r a l method and a v e c t o r p o t e n t i a l , p a r t i a l d i f f e r e n t i a l e q u a t i o n , f i n i t e element method. The problem was assumed i n f i n i t e l y long out of the p l a n e , with an i r o n r e l a t i v e p e r m e a b i l i t y of 1000. A v e r y l a r g e (500 degrees of freedom) boundary i n t e g r a l s o l u t i o n was used t o o b t a i n t h e ' c o r r e c t ' r e s u l t . F i g u r e 2 shows the f l u x d i s t r i b u t i o n i n the problem and from t h i s f i g u r e t h e boundary c o n d i t i o n s should be obvious. F i g u r e s 3a and 3b show t h e convergence of the i n t e g r a t e d f l u x a c r o s s the a i r gap f o r the two s o l u t i o n methods and a l s o show t h e time required f o r the s o l u t i o n s . This problem was s e l e c t e d because the r a p i d l y varying f i e l d gives a more r e a l i s t i c t e s t of t h e f i n i t e element method, accuracy i s much e a s i e r t o o b t a i n i f t h e f i e l d i s c o n s t a n t . I n both methods q u a d r a t i c v a r i a t i o n of t h e p o t e n t i a l was used i n each d i s c r e t e element.

The p a r t i a l d i f f e r e n t i a l equation s o l u t i o n employed smoothing on t h e f i e l d s o l u t i o n , d i r e c t d i f f e r e n t i a t i o n of the element shape f u n c t i o n s g i v e s much l a r g e r e r r o r s .

F i g u r e 1. Cross s e c t i o n of an H frame magnet with a s l o p i n g pole. Only one q u a r t e r of t h e magnet is d i s p l a y e d .

F i g u r e 2. Flux d i s t r i b u t i o n i n t h e H frame magnet.

The r e s u l t s c l e a r l y show t h a t f o r t h i s problem t h e boundary i n t e g r a l method is by f a r t h e b e s t . However, t h i s cannot be used f o r non-linear problems except by i n t r o d u c i n g a r e a d i s c r e t i s a t i o n , whereas rhe p a r t i a l d i f f e r e n t i a l e q u a t i o n method can be used f o r non-linear s o l u t i o n s and w i l l g i v e s i m i l a r p r e c i s i o n . The f i g u r e s do not t e l l t h e whole s t o r y , the i n c r e a s e i n accuracy from 0.08% t o 0.02% with p a r t i a l d i f f e r e n t i a l equatioh s o l u t i o n was obtained by only i n c r e a s i n g t h e

d i s c r e t i s a t i o n w i t h i n t h e a p e r t u r e of the magnet. This s t r o n g c o r r e l a t i o n between

accuracy and $rely l o c a l s u b d i v i s i o n i s one of t h e s i r e n g t h s . o f t h e method, i t i s

not a property of the i n t e g r a l method except i n a r e a s c l o s e to the d i s c r e t i s e d

Error

0 100 200 300 0 2000 4000 6000 8000 Degrees of Freedom Degrees of Freed om

CPU seconds CPU seconds

Fig. 3 A Fig. 38

10

Error in / B dx as a function of Discretisation

0 Y

F i g u r e 3. R e s u l t s f o r t h e H frame magnet.

a ) from a boundary i n t e g r a l s o l u t i o n

b) f r a n a p a r t i a l d i f f e r e n t i a l e q u a t i o n s o l u t i o n 2.1 Cost Comparison

The r e s u l t s p l o t t e d i n F i g u r e 3 were o b t a i n e d by programs t h a t g i v e n e a r l y optimal performance f o r t h e t y p e s of method they use. Table 2 shows how t h e o p e r a t i o n count v a r i e s a s a f u n c t i o n of accuracy f o r boundary i n t e g r a l , volume i n t e g r a l and p a r t i a l d i f f e r e n t i a l e q u a t i o n methods i n 2D s o l u t i o n s . Accuracy i s assumed t o depend on t h e element s i d e l e n g t h r a i s e d t o a power t h a t i s independent of the method. Table 3 shows t h e same comparison f o r 3 D s o l u t i o n s .

Table

A comparison o f ' o p e r a t i o n counts f o r d i f f e r e n t s o l u t i o n methods i n 2D problems

Boundary Volume P a r t i a l

i n t e g r a l i n t e g r a l D i f f e r e n t i a l

a c c u r a c y

na

d i s c r e t i s a t i o n

n

m a t r i x e v a l u a t i o n

n 2

e q u a t i o n s o l u t i o n a n 3

f i e l d r e c o v e r y n

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Table 3

A comparison of o p e r a t i o n counts f o r d i f f e r e n t s o l u t i o n methods i n 3D problems

Boundary Volume P a r t i a l

i n t e g r a l i n t e g r a l D i f f e r e n t i a l

a c c u r a c y

n

d i s c r e t i s a t i o n a n 2 m a t r i x e v a l u a t i o n a n 4 e q u a t i o n s o l u t i o n a n 6 f i e l d recovery a n2

The most obvious c o n c l u s i o n from t h e s e t a b l e s i s t h a t i n t h e l i m i t of v e r y l a r g e d i s c r e t i s a t i o n s all methods a r e dominated by e q u a t i o n s o l u t i o n . The next

c o n c l u s i o n is t h a t f o r s o l u t i o n s i n t h r e e dimensions p a r t i a l d i f f e r e n t i a l methods must become more e f f e c t i v e than t h e o t h e r approaches; a g a i n , i n t h e l i m i t of l a r g e d i s c r e t i s a t i o n s when t h e c o n s t a n t s i n t h e p r o p o r t i o n a l i t y r e l a t i o n s h i p s become i n s i g n i f i c a n t . It is only i n two dimensions t h a t boundary i n t e g r a l methods a r e p a r t i c u l a r l y a t t r a c t i v e , F i g u r e 3 d e m o n s t r a t e s t h i s q u i t e e f f e c t i v e l y . I n s o l v i n g p r a c t i c a l problems on e x i s t i n g computing hardware t h e c o n s t a n t s i n the propor- t i o n a l i t y r e l a t i o n s h i p s become i m p o r t a n t . Experience with e x i s t i n g programs i n t h r e e dimensions shows t h a t once t h e number of unknowns r i s e s above 1000 f o r i n t e g r a l methods, t h e p a r t i a l d i f f e r e n t i a l methods a r e b e t t e r . Accuracies of t h e o r d e r of .2% can be achieved w i t h t h i s l e v e l of d i s c r e t i s a t i o n .

2.2 L i n e a r Algebra

I n d e r i v i n g t h e o p e r a t i o n c o u n t s f o r T a b l e s 2 and 3 i t was assumed t h a t d i r e c t e q u a t i o n s o l u t i o n methods were used f o r i n t e g r a l methods (Gaussian e l i m i n a t i o n ) and t h a t the most e f f e c t i v e methods a v a i l a b l e were used f o r t h e p a r t i a l d i f f e r e n t i a l methods. I n t h e l a t t e r , t h e r e have been s i g n i f i c a n t developments i n t h e l a s t 5

y e a r s and i t is o n l y a s a r e s u l t of t h e s e t h a t the p a r t i a l d i f f e r e n t i a l methods have been s o s u c c e s s f u l . The m a j o r i t y of new, l a r g e , f i n i t e element programs now u s e p r e c o n d i t i o n e d con j u g a t e g r a d i e n t methods/2,3/. The computer s t o r a g e r e q u i r e d by t h e s e methods i n c r e a s e s l i n e a r l y w i t h t h e number of unknowns and i s independent of o r d e r i n g of t h e unknowns. S i m i l a r l y t h e s o l u t i o n times a r e almost independent of t h e o r d e r i n g of t h e unknowns, t h i s i s t o be c o n t r a s t e d with o t h e r s p a r s e m a t r i x methods where t h e bandwidth o r p r o f i l e of t h e m a t r i x was v e r y important e s p e c i a l l y f o r problems i n t h r e e dimensions. Table 4 g i v e s an i n d i c a t i o n of t h e s o l u t i o n t i m e s r e q u i r e d u s i n g an incomplete cholesky c o n j u g a t e g r a d i e n t method f o r a f i n i t e element s o l u t i o n o f L a p l a c e ' s e q u a t i o n i n t h r e e dimensions.

Table 4

S o l u t i o n times f o r incomplete cholesky c o n j u g a t e g r a d i e n t methods.

An I B M 3081D was used f o r t h e s e c a l c u l a t i o n s .

Number of S o l u t i o n

unknowns time ( s e c o n d s )

The R u t h e r f o r d Appleton Laboratory implementations of t h e s e methods do not use any d i s k backing s t o r e , t h e s o l u t i o n of a 25000 unknown s e t of e q u a t i o n s f o r t h e L a p l a c i a n problem r e q u i r e s 5MB of v i r t u a l s t o r a g e .

2.3 3D S o l u t i o n s

I n o r d e r t o show the convergence of s o l u t i o n s t o t h r e e dimensional problems an

axisymmetric model w i t h t h e c r o s s - s e c t i o n

2D a x i s y m e t r i c model and a f u l l 3 D model i n which one o c t a n t of t h e geometry was d i s c r e t i s e d . F i g u r e 4 shows a computer generated d i s p l a y of the model with hidden l i n e s removed. F i g u r e 5 shows t h e convergence of t h e i n t e g r a t e d f i e l d a s a f u n c t i o n of d i s c r e t i s a t i o n . Moving from F i g u r e 3 t o F i g u r e 5 using Tables 2 and 3 i t would seem reasonable t o expect t o r e q u i r e 600000 degrees of freedom t o g i v e 0.2% accuracy. Figure 5 shows t h a t only 24000 unknowns were r e q u i r e d , t h i s was achieved by using high o r d e r small elements i n t h e r e g i o n of i n t e r e s t and low o r d e r elements i n t h e o u t e r a r e a s . These r e s u l t s were obtained using the TOSCA

F i g u r e 4. A 3 D model of an axisymmetric H frame magnet.

E r r o r %

I I I

0 10,000 20.000

Number of Degrees of Freedom

F i g u r e 5. Convergence of t h e i n t e g r a t e d f l u x c r o s s i n g the gap of

t h e 3 D H frame magnet.

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3 - FAR FIELD BOUNDARY

I n t e g r a l methods can be formulated such t h a t t h e f i e l d decays c o r r e c t l y t o i n f i n i t y . I n s o l v i n g p a r t i a l d i f f e r e n t i a l e q u a t i o n s using f i n i t e element

procedures t h e f i e l d i s i n g e n e r a l a r t i f i c i a l l y terminated a t t h e boundary of t h e mesh w i t h e i t h e r a D i r i c h l e t o r Neumann boundary c o n d i t i o n . This can r e s u l t i n l a r g e e r r o r s i f t h e f a l s e t e r m i n a t i o n of t h e f i e l d i s s t r o n g l y coupled to t h e r e g i o n s of i n t e r e s t .

S e v e r a l methods of overcoming t h i s l i m i t a t i o n have been developed. A boundary i n t e g r a l method141 o r boundary G a l e r k i n method151 can be a p p l i e d t o t h e f a l s e boundary o r a b a l l o o n i n g technique161 could be used. The b a l l o o n i n g t e c h n i q u e uses r e c u r s i v e g e n e r a t i o n of s i m i l a r r i n g s of elements around t h e problem s p a c e and t h e d e g r e e s of freedom s o c r e a t e d a r e condensed out of t h e system of e q u a t i o n s . T h i s t e c h n i q u e and t h e boundary i n t e g r a l method, both produce c o u p l i n g between a l l d e g r e e s of freedom a s s o c i a t e d w i t h t h e boundary and hence c r e a t e a dense block i n t h e f i n a l system m a t r i x . Another a l t e r n a t i v e i s t o use an element i n t h e e x t e r i o r which e x t e n d s from t h e boundary of the mesh to i n f i n i t y l 7 1 , t h e s e a r e u s u a l l y c a l l e d ' i n f i n i t e e l e m e n t s ' . Various decay f u n c t i o n s may be s e l e c t e d i n o r d e r t o aproximate t h e v a r i a t i o n from the boundary to i n f i n i t y , t h e most commonly used f u n c t i o n s a r e e i t h e r r e c i p r o c a l , e x p o n e n t i a l o r orthogonal polynomial/7,8/. The advantage of t h i s approach over t h e o t h e r s i s t h a t t h e system m a t r i x does not c o n t a i n a d e n s e l y populated block connecting t h e nodes on t h e boundary. I t s d i s a d v a n t a g e is t h a t t h e decay f u n c t i o n must be c l o s e l y r e l a t e d t o t h e a c t u a l decay i f l a r g e e r r o r s a r e not t o be produced.

4 - EDDY CURRENTS

There have been e x c i t i n g developments i n the c a l c u l a t i o n of eddy c u r r e n t s . I n two s p a c e dimensions boundary i n t e g r a l methods have been extended t o i n c l u d e formu- l a t i o n s capable of modelling eddy c u r r e n t s / 9 / . I n t h r e e dimensions v e c t o r p o t e n t i a l o r e l e c t r i c f i e l d s o l u t i o n s have been d i r e c t l y coupled t o s c a l a r p o t e n t i a l s f o r t h e a r t e r i o r s p a c e l l O / . Uniqueness has been demonstrated f o r t h e v e c t o r p o t e n t i a l s / l l / , w i t h a d e f i n i t e i m p l i e d g a u g e / l 2 / , p r o v i d i n g t h i s

f o r m u l a t i o n i s o n l y used i n s i d e conducting r e g i o n s . Such methods minimise the number of unknowns r e q u i r e d a t each d i s c r e t i s a t i o n p o i n t and a l s o avoid t h e c a n c a l l a t i o n problems i n h e r e n t i n o t h e r t e c h n i q u e s / l 3 / . The e l e c t r i c f i e l d and v e c t o r p o t e n t i a l approaches a r e v e r y s i m i l a r / l 4 / , i t is n o t y e t c l e a r which of them w i l l be b e s t f o r t r a n s i e n t s o l u t i o n s .

The most s u c c e s s f u l methods t h a t a r e now a v a i l a b l e a r e based on coupled network f o r m u l a t i o n s / l 5 / . These methods have been g r e a t l y extended and now use networks based on t e t r a h e d r a , t h i s makes t h e modelling of r e a l d e v i c e s much more s t r a i g h t - forward. Two o t h e r developments a r e p a r t i c u l a r l y i n t e r e s t i n g ; eddy c u r r e n t s o l u t i o n s using coupled f i n i t e elements and boundary e l e m e n t s / l 6 / , where s p e c i a l f i n i t e elements f o r v e c t o r f i e l d s a r e used; f i n i t e d i f f e r e n c e s o l u t i o n s , which have been developed and r e f i n e d t o g i v e v e r y good r e s u l t s / l 7 / .

5 - ^FORCES

The f o r c e s a s s o c i a t e d w i t h e l e c t r o m a g n e t i c f i e l d s a r e i n many c a s e s t h e primary motive f o r t h e d e s i g n of t h e d e v i c e o r they a r e a c o n s t r a i n t on t h e d e s i g n . T h i s

i s u n f o r t u n a t e because

is p a r t i c u l a r l y d i f f i c u l t t o r e l i a b l y compute t h e f o r c e s . I n a l l methods any e r r o r i n t h e f i e l d s is enhanced i n computing t h e f o r c e s . The methods of computing f o r c e s a r e ,

v i r t u a l w o r k / l 8 / ;

body f o r c e i n t e g r a t i o n / l 9 / ; Maxwell s t r e s s i n t e g r a t i o n l l 8 1 ; change i n s t o r e d energy.

I t has been shown t h a t v i r t u a l work c a l c u l a t i o n s become, i n t h e l i m i t of s m a l l

e l e m e n t s , e q u i v a l e n t t o Maxwell s t r e s s i n t e g r a t i o n .

Body f o r c e i n t e g r a t i o n i s s u b j e c t t o l a r g e e r r o r s because t h e e x t e r n a l f o r c e s on components may be many times s m a l l e r than t h e i n t e r n a l f o r c e s t h a t a r e balanced by s t r e s s e s i n t h e component. I n a s i m i l a r way a s u r f a c e i n t e g r a t i o n of t h e Maxwell s t r e s s t e n s o r of t e n r e s u l t s i n n e a r l y b a l a n c i n g p o s i t i v e and n e g a t i v e c o n t r i - b u t i o n s . V i r t u a l work h a s t h e same problem and c a l c u l a t i o n s based on a c t u a l d i s p l a c e m e n t s of o b j e c t s a r e v e r y i n c o n v e n i e n t .

F i g u r e 6: The f l u x d i s t r i b u t i o n f o r a p a i r of i n f i n i t e l y long p a r a l l e l conductors c a r r y i n g e q u a l and o p p o s i t e c u r r e n t s , with an i r o n b a r s l i g h t l y o f f s e t from t h e c e n t r e .

The e r r o r s t h a t w i l l occur a r e very dependent on t h e problem being s o l v e d , v e r y a c c u r a t e answers a r e e a s y t o o b t a i n i f t h e r e is no c a n c e l l a t i o n i n t h e i n t e g r a l s , f o r example, computing t h e f o r c e between a c u r r e n t c a r r y i n g c o i l and an i n f i n i t e h a l f plane. F i g u r e 6 shows a problem t h a t i s p a r t i c u l a r l y d i f f i c u l t to s o l v e . This i s a two dimensional s e c t i o n through a p a i r of conductors c a r r y i n g e q u a l and o p p o s i t e c u r r e n t s , s l i g h t l y d i s p l a c e d from the c e n t r e is an i r o n r a i l , t h e f i g u r e a l s o shows t h e f l u x d i s t r i b u t i o n . The problem i s t o compute t h e f o r c e on t h e i r o n r a i l . T h i s can be solved v e r y e a s i l y by computing t h e change i n s t o r e d energy o r by computing t h e r e a c t i o n on t h e c o i l s u s i n g t h e body f o r c e . It i s p a r t i c u l a r l y d i f f i c u l t to s o l v e by Maxwell s t r e s s i n t e g r a t i o n . The r e s u l t a n t f o r c e on the r a i l i s l e s s t h a n 10% of t h e i n t e g r a t e d modulus of t h e s t r e s s t e n s o r . For t h e r a i l t h e r e is a weak s i n g u l a r i t y i n t h e f i e l d near the c o r n e r s ; i f a p a r t i a l

d i f f e r e n t i a l f i n i t e element method is used t h i s s i n g u l a r i t y i s only approximated

c r u d e l y and t h e answers on t h e l e f t and r i g h t hand c o r n e r s a r e very dependent on

t h e l o c a l d i s c r e t i s a t i o n . F i g u r e 7 shows t h e v a r i a t i o n of t h e modulus of t h e f i e l d

around a h a l f c i r c l e c e n t r e d on t h e r a i l , with a r a d i u s such t h a t t h e c i r c l e passes

c l o s e t o , b u t does not t o u c h , t h e s u r f a c e of t h e r a i l . There is a s l i g h t l e f t t o

r i g h t asymmetry due to t h e displacement of the r a i l , b u t t h e d i s c o n t i n u i t y on the

l e f t i s p u r e l y due t o t h e l e f t t o r i g h t asymmetry of t h e d i s c r e t i s a t i o n . The

e f f e c t s of t h e c o r n e r damp out q u i c k l y a s the r a d i u s of t h e p a t h i s i n c r e a s e d such

t h a t p o i n t s on t h e path do not l i e w i t h i n the bard of elements touching t h e s u r f a c e

of t h e r a i l . T h i s e f f e c t has a tremendous i n f l u e n c e on t h e computed f o r c e s , F i g u r e

8 shows t h e f o r c e computed by Maxwell s t r e s s i n t e g r a t i o n a s a f u n c t i o n of t h e

r a d i u s of t h e c i r c l e used f o r t h e i n t e g r a t i o n . The expected r e s u l t was obtained

u s i n g a v e r y l a r g e number of unknowns, f r a n t h e body f o r c e on t h e conductors. The

f i g u r e shows r e s u l t s f o r two l e v e l s of d i s c r e t i s a t i o n , i f v e r y s m a l l elements a r e

used c l o s e t o t h e s u r f a c e of t h e b a r t h e poor modelling of t h e s i n g u l a r i t y does n o t

a f f e c t the computed f o r c e s .

JOURNAL DE PHYSIQUE

F i g u r e 8: F o r c e on t h e i r o n b a t a s a f u n c t i o n of r a d i u s of t h e p a t h of i n t e g r a t i o n .

F i g u r e 7: EIodulus of t h e m a g n e t i c f i e l d p l o t t e d on a s e m i c i r c l e c e n t r e d on t h e i r o n b a r .

6 - ^GENERAL 0.6-

0.4-

0.2 -

0

The major headache a s s o c i a t e d w i t h f i n i t e element a n a l y s i s is i n t h e p r e p a r a t i o n of d a t a . E l e c t r o m a g n e t i c f i e l d s a r e p a r t i c u l a r l y hard t o model because t h e e x t e r n a l s p a c e must a l s o be d i s c r e t i s e d . There a r e major developments. i n t h i s a r e a t h a t s h o u l d be a v a i l a b l e t o t h e d e s i g n e r i n t h e n e a r f u t u r e , t h e s e i n c l u d e a u t o m a t i c mesh g e n e r a t i o n on a g e o m e t r i c model, and a d a p t i v e mesh g e n e r a t i o n t o s a t i s f y some

X

Force from Maxwell Stress

X

rn

0

J x B result

I I

0 1 2 3

Radius of Integration P a t h

s p e c i f i e d a c c u r a c y c r i t e r i a . Both developments a r e i m p o r t a n t p a r t i c u l a r l y i f t h e y c a n be used i n o r d e r t o g i v e r e l i a b l e r e s u l t s w i t h o u t a h i g h l y e x p e r i e n c e d and a b l e f i n i t e element e x p e r t . Also l i n k e d w i t h t h e a d a p t i v e mesh g e n e r a t i o n is t h e i d e a of u s i n g d u a l i t y p r i n c i p l e s t o o b t a i n upper and lower bounds on s o l u t i o n

v a r i a b l e s / 2 0 / , t h i s must be i m p o r t a n t i n f u t u r e s o f t w a r e developments. There a r e a l r e a d y some programs t h a t u s e a d a p t i v e mesh g e n e r a t i o n i n t h e s o l u t i o n of e l e c t r o m a g n e t i c f i e l d s / 2 1 / .

7 - CONCLUSION

The review paper on f i e l d computation/22/ t h a t was p r e s e n t e d i n 1972 a t MT-4, i n c l u d e d t a b l e s showing programs f o r f i e l d c o m p u t a t i o n , a t t h e 1983 Compumag c o n f e r e n c e a review of eddy c u r r e n t programs/23/ produced a l i s t a t l e a s t t h r e e t i m e s l o n g e r . Methods f o r e l e c t r o m a g n e t i c f i e l d c o m p u t a t i o n a r e s t i l l b e i n g a c t i v e l y developed and t h e a c c u r a c y of t h e r e s u l t s i s o n l y l i m i t e d by t h e computer r e s o u r c e s a v a i l a b l e . I n p r a c t i c e t h i s means t h a t r e l i a b l e r e s u l t s c a n o n l y b e o b t a i n e d by e x p e r i e n c e d p e r s o n n e l . The t r e n d t o a d a p t i v e mesh g e n e r a t i o n s h o u l d improve p r o d u c t i v i t y and r e d u c e t h e d e g r e e of e x p e r t i s e t h a t i s r e q u i r e d , however, i n c r e a s e d computing power w i l l be needed f o r v e r y h i g h a c c u r a c y . Todays s e r i a l computers f a v o u r t h e use of p a r t i a l d i f f e r e n t i a l e q u a t i o n methods, p a r a l l e l computers on t h e o t h e r h a n d , would r e v i v e t h e i n t e r e s t i n i n t e g r a l e q u a t i o n s . It must b e s a i d t h o u g h , t h a t i n t e g r a l s o l u t i o n s a r e more prone t o d i s a s t r o u s e r r o r s .

REFERENCES

1. Armstrong, A G A M , C o l l i e , C J , Simkin ,

, Trowbridge, C

T , "The s o l u t i o n of 3-D m a g n e t o s t a t i c problems u s i n g s c a l a r p o t e n t i a l s " , R u t h e r f o r d Appleton L a b o r a t o r y Report RL-78-088, 1978.

2. Kershaw,

S , "The i n c o m p l e t e c h o l e s k y - c o n j u g a t e g r a d i e n t method f o r t h e i t e r a t i v e s o l u t i o n of systems of l i n e a r e q u a t i o n s " , J Comp Phys 6 ^43-65,

1978.

3 . Duff , ^I S , ( E d i t o r ) , " C o n j u g a t e g r a d i e n t methods and s i m i l a r t e c h n i q u e s " , AERE H a r w e l l R-9636, 1979.

4 . Friedman, M J , " A new f i n i t e element-boundary i n t e g r a l p r o c e d u r e f o r t h e s o l u t i o n of t h e m a g n e t o s t a t i c f i e l d problem", t o a p p e a r IEEE Trans Mag, November 1983 (COMPUMAG C o n f e r e n c e ) .

5. Mayergoyz, I D, C h a r i , M V K , Konrad, A, "Boundary G a l e r k i n ' s method f o r t h r e e d i m e n s i o n a l f i n i t e element e l e c t r o m a g n e t i c f i e l d computation", t o a p p e a r IEEE T r a n s Mag, November 1983 (COMPUMAG Conference).

6. S i l v e s t e r , P P , Lowther, D A , C a r p e n t e r , C 3, Wyatt, E A , " E x t e r i o r f i n i t e e l e m e n t s f o r 2-Dimensional f i e l d problems w i t h open b o u n d a r i e s " , P r o c IEE, 1 2 4 , 1267-1270, 1977.

-

7. Z i e n k i e w i c z , 0 C , Ernson, C , B e t t e s s , ^{P ,} "A n o v e l boundary i n f i n i t e e l e m e n t " , IJNME , vo1.19 , 393-404, 1983.

8 . P i s s a n e t s k y , S , " S o l u t i o n of t h r e e d i m e n s i o n a l , a n i s o t r o p i c , n o n - l i n e a r problems of m a g n e t o s t a t i c s u s i n g two s c a l a r p o t e n t i a l s , f i n i t e and i n f i n i t e m u l t i p o l a r e l e m e n t s and a u t o m a t i c mesh g e n e r a t i o n " , IEEE T r a n s Mag, v o l . Mag-18, No.2, March 1982.

9 . Rucker, W , R i c h t e r , K R, " C a l c u l a t i o n of two d i m e n s i o n a l eddy c u r r e n t problems

by t h e boundary element method", t o a p p e a r IEEE Trans Mag, November 1983

(COMPUMAG Conference).

C1-860 JOURNAL DE PHYSIQUE

P i l l s b u r y ,

D J r , " A t h r e e dimensional eddy c u r r e n t formulation using two p o t e n t i a l s : The magnetic v e c t o r p o t e n t i a l and t o t a l magnetic s c a l a r p o t e n t i a l " , t o appear IEEE Trans Mag, November 1983 (COMPUMAG Conference).

P o l a k , S, Wachters, A , van W e l i j , J , "A note on p o t e n t i a l s f o r eddy c u r r e n t problems", t o appear IEEE Trans Mag, November 1983 (COMPUMAG Conference).

Emson, C R I , Simkin, J , "An optimal method f o r 3-D eddy c u r r e n t s " , t o appear -1EEE Trans Mag, November 1983 (COMPUMAG Conference).

Biddlecombe, C S, Heighway, E A, Simkin, J , Trowbridge, C W , "Methods f o r eddy c u r r e n t computation i n t h r e e dimensions" , IEEE Trans Mag, v o l . Mag-18 , No .2 ,

March 1982.

Rodger, D , Eastham, J F , "A f o r m u l a t i o n f o r low frequency eddy c u r r e n t s o l u t i o n s " , t o appear IEE5 Trans Mag, November 1983 (COMPUMAG Conference).

Davidson, J A M , B a l c h i n , M J , "Three dimensional f i e l d c a l c u l a t i o n by network methods using s c a l a r magnetic p o t e n t i a l s and loop e l e c t r i c c u r r e n t s " , t o appear IEEE Trans Mag, November 1983 (COMPUMAG Conference).

B o s s a v i t , A , V e r i t e , J C , "The TRIFOU code: s o l v i n g t h e 3-D eddy c u r r e n t problem by using H a s a s t a t e v a r i a b l e " , t o appear I E E E Trans Mag, November 1983 (COMPUMAG Conference).

Knoblauch, A, M u l l e r , W , " F i n i t e d i f f e r e n c e s o l u t i o n of 3-dimensional eddy c u r r e n t d i s t r i b u t i o n s " , t o appear IEEE Trans Mag, November 1983 (COMPUMAG Conference).

Coulomb, J C , "A methodology f o r the d e t e r m i n a t i o n of g l o b a l electromechanical p r o p e r t i e s from a f i n i t e element a n a l y s i s and i t s a p p l i c a t i o n t o t h e

e v a l u a t i o n of magnetic f o r c e s , torques and s t i f f n e s s " , t o appear IEEE Trans Mag, November 1983 (COMPUMAG Conference).

J a c o b s , A, M u l l e r , W , "Numerical s o l u t i o n of f o r c e s and t o r q u e s " , to appear IEEE Trans Mag November 1983 (COMPUMAG Conference).

Penman, J , L e e s , P , F r a s e r , J R, Smith, J R, "Complementary energy methods i n t h e computation of e l e c t r o s t a t i c f i e l d s " , t o appear IEEE Trans Mag, November 1983 (COMPUMAG Conference) .

G i r d i n i o , P , Molfino, P , M o l i n a r i , G , P u g l i s i , L , V i v i a n i , A, " F i n i t e

d i f f e r e n c e and f i n i t e element g r i d o p t i m i s a t i o n by t h e g r i d i t e r a t i o n method", t o appear IEEE Trans Mag, November 1983 (COMPUMAG Conference).

Trowbridge, C W , "Progress i n magnet d e s i g n by computer", P r o c 4 t h I n t e r n Conf on Magnet Technology, Brookhaven L a b o r a t o r y , 1972.

L a r i , R J , "Comparison of eddy c u r r e n t programs", t o appear IEEE Trans Mag,

November 1983 (COMPUMAG Conference).