applications of numerical field modeling to electromagnetic methods of nondestructive testing

IEEE TRANSACTIONS ON MAGNETICS, VOL. MAG-19, NO. 6, NOVEMBER 1983

APPLICATIONS OF NUMERICAL FIELD MODELING TO ELECTROMAGNETIC METHODS OF NONDESTRUCTIVE. TESTING

William Lord

Abs t rac t - E l e c t r o m a g n e t i c f i e l d i n t e r a c t i o n s w i t h meta ls can be used to de te rmine bo th mater ia l p roper - ties and t he p re sence o f de fec t s . Such techniques a r e u s e d w i d e l y i n t h e n o n d e s t r u c t i v e t e s t i n g o f c r i t i c a l components f o r a e r o s p a c e , t r a n s p o r t a t i o n , e n e r g y a n d m e t a l s i n d u s t r i e s w h e r e r e l i a b i l i t y , s a f e t y and p roduc t qua l i t y cons ide ra t ions a r e impor t an t . Th i s pape r desc r ibes t he common p h y s i c a l b a s i s f o r act ive dc, res idual and ac methods of specimen magne- t i z a t i o n a n d shows how f in i t e e l emen t ana lys i s t ech - n i q u e s o r i g i n a l l y d e v e l o p e d f o r t h e s t u d y o f f i e l d s i n e l e c t r i c a l m a c h i n e r y c a n b e e x t e n d e d t o p r e d i c t e l e c t r o m a g n e t i c NDT t r a n s d u c e r s i g n a l s f o r a l l t h r e e forms of specimen exci ta t ion.

INTRODUCTION

The s c i e n c e o f n o n d e s t r u c t i v e t e s t i n g (NDT) i s i n t e r d i s c i p l i n a r y i n n a t u r e , c o v e r i n g t h e c l a s s i c a l a r eas o f r ad iog raphy , u l t r a son ic s , e l ec t romagne t i c s a n d v i s u a l i n s p e c t i o n . H i s t o r i c a l l y , p r o g r e s s i n NDT h a s c l o s e l y p a r a l l e l e d d e v e l o p m e n t s i n t h e s c i e n c e s i n t h a t a l l p h y s i c a l phenomena are p o t e n t i a l c a n d i d a t e s fo r t he nondes t ruc t ive measu remen t o f ma te r i a l p rope r - ties o r t h e d e t e c t i o n o f d e f e c t s . I n t h e c a s e o f e l e c t r o m a g n e t i c NDT methods, the use of eddy current phenomena f o r m e t a l s c l a s s i f i c a t i o n [ l ] i s o f t h e same v i n t a g e as Maxwell ' s Treat ise . I t is o n l y r e c e n t l y h o w e v e r , t h a t w i d e s p r e a d , s u s t a i n e d e f f o r t h a s gone into the development and model ing o f NDT techniques . Th i s r e su rgence o f i n t e re s t has been b rough t abou t by t h e u s e o f m e t a l s t r u c t u r e s i n t h e most demanding of environments and the concomitant costs , both economic a n d s o c i a l , o f c a t a s t r o p h i c f a i l u r e s , p a r t i c u l a r l y i n t he ae rospace , t r anspor t a t ion and power i n d u s t r i e s .

T r a d i t i o n a l e l e c t r o m a g n e t i c NDT methods cover a wide range of f requencies f rom dc to microwave. How- e v e r , t h e common i n d u s t r i a l t e c h n i q u e s a r e l i m i t e d t o act ive dc, res idual and eddy current forms of exci ta- t i o n ; a l l low frequency phenomena for which d i sp lace- ment c u r r e n t e f f e c t s c a n b e n e g l e c t e d .

Ana ly t i ca l app roaches t o t he mode l ing o f e l ec t ro - m a g n e t i c f i e l d / d e f e c t i n t e r a c t i o n s h a v e l a r g e l y b e e n unsuccess fu l , due t o bo th t he awkward boundar ies a s s o c i a t e d w i t h r e a l i s t i c d e f e c t s h a p e s a n d t h e l a c k o f g e n e r a l i t y t h a t e n s u e s when making t h e n e c e s s a r y a s s u m p t i o n s n e e d e d t o o b t a i n t r a c t a b l e a n a l y t i c a l s o l u - t i o n s [ Z ] . Viable models are needed i n o r d e r t o u n d e r - s t a n d t h e ways i n which f i e l d s and d e f e c t s i n t e r a c t t o produce measurable ind ica t ions , to he lp in the des ign o f d e t e c t i o n p r o b e s f o r d i v e r s e a p p l i c a t i o n s , t o simu- l a t e t h o s e t e s t i n g e n v i r o n m e n t s w h i c h a r e d i f f i c u l t a n d / o r e x p e n s i v e t o r e p l i c a t e i n t h e l a b o r a t o r y a n d , perhaps most important ly in many o f t h e c r i t i c a l t e s t i n g s i t u a t i o n s f a c i n g i n d u s t r i e s t o d a y , t o p r o v i d e t r a i n i n g d a t a f o r a u t o m a t e d d e f e c t c h a r a c t e r i z a t i o n schemes . Success fu l app l i ca t ion o f f i n i t e d i f f e rence and f i n i t e e l emen t ana lys i s t echn iques t o f i e ld p rob - lems in both dc and ac machinery has paved the way f o r similar, para l le l deve lopments in the model ing of e lec t romagnet ic NDT phenomena [ 3 ] .

This work has been supported by both the Army Research O f f i c e a n d t h e E l e c t r i c Power R e s e a r c h I n s t i t u t e .

W. Lord is wi th the E lec t r ica l Engineer ing Depar tment , Co lo rado S t a t e Un ive r s i ty , Fo r t Co l l in s , CO 80523.

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The purpose o f th i s paper is t o s e r v e b o t h as a rev iew of the more common methods f o r n o n d e s t r u c t i v e l y t e s t i n g metals using electromagnetic phenomena, and t o show how numerical methods have been applied as a model ing tool .

ACTIVE EXCITATION

Those nondes t ruc t ive t e s t ing t echn iques cha rac t e r - i z e d a s r e q u i r i n g a cont inuous dc source o f exc i ta t ion are c l a s s i f i e d a s ' a c t i v e ' methods [ 4 ] , and inc lude magne t i c pa r t i c l e t e s t ing [5 ] , magne tography [ 6 ] , per turba t ion techniques [7] , po ten t ia l d rop methods [8] and va r i ab le r e luc t ance p robes [ 9 ] . The major differ- e n c e i n t h e s e t e c h n i q u e s , a s a p p l i e d t o d e f e c t d e t e c - t i o n , r e l a t e s to t h e way in wh ich t he de fec t l eakage f i e l d s a r e d e t e c t e d . All of t h e methods a r e u s e d t o d e t e c t d e f e c t s i n f e r r o m a g n e t i c m a t e r i a l s ( a l t h o u g h po ten t i a l d rop and cu r ren t pe r tu rba t ion me thods can be used with any conducting specimen) and Fig. 1 shows how a d c e x c i t a t i o n c u r r e n t i n a s t e e l b i l l e t sets up a n a c t i v e l e a k a g e f i e l d a r o u n d a de fec t .

f

I

Fig. 1. A c t i v e e x c i t a t i o n o f a ferromagnetic specimen.

Each par t of the specimen under test o p e r a t e s a t a d i f f e r e n t p o i n t a l o n g t h e m a t e r i a S s i n i t i a l m a g n e t i z a - t i o n c u r v e a n d t h e a s s o c i a t e d m a g n e t o s t a t i c f i e l d s c a n b e modeled by the non l inea r Po i s son equa t ion

v X(vV x 2) = 3 (1)

where v=S /v , t he non l inea r r e luc t iv i ty ob ta ined f rom t h e m a t e r i a l B / H c h a r a c t e r i s t i c . I f a Ha l l e l emen t o r o t h e r f l u x d e n s i t y s e n s i t i v e t r a n s d u c e r is scanned over t h e s u r f a c e o f a r e c t a n g u l a r s l o t a 3 dimensional " l eakage f i e ld p ro f i l e " wou ld be ob ta ined similar t o t h a t shown i n F i g . 2. Such p r o f i l e s a r e a func t ion of a number o f va r i ab le s i nc lud ing t he shape o f t he de fec t ; i n d e e d , t h e p r o b l e m o f i n t e r e s t t o t h e NDT community is whe the r t he shape o f t he de fec t can be p red ic t ed f rom t h e d e f e c t s l e a k a g e f i e l d p r o f i l e i n s u c h a way t h a t t he t e s t ing p rocedure can be au tomated fo r a s p e c i f i c a p p l i c a t i o n .

Ana ly t i ca l ly , t he mode l ing o f l eakage f i e ld phenomena h a s l a r g e l y b e e n l i m i t e d t o e q u i v a l e n t d i p o l e approaches where "magnetic charge" i s assumed t o r e s i d e on t he walls o f d e f e c t s [ l o ] . Equat ions der ived for s i m p l e r e c t a n g u l a r s l o t s a r e i n f a c t i d e n t i c a l i n f o r m t o t h o s e d e r i v e d b y K a r l q u i s t f o r p r e d i c t i n g t h e l e a k - a g e f i e l d o f a magne t i c t ape head [ l l ] and a r e no t r e a d i l y e x t e n d i b l e t o a r b i t r a r i l y s h a p e d d e f e c t s i n m a t e r i a l s h a v i n g n o n l i n e a r B / H c h a r a c t e r i s t i c s . A key parameter i n t h e d i p o l e e q u a t i o n s i s t h e f i c t i t i o u s sur face magnet ic charge dens i ty , which can on ly be e s t i m a t e d f o r a g i v e n s l o t a n d e x c i t a t i o n c u r r e n t a f t e r i t s l e a k a g e f i e l d p r o f i l e h a s b e e n m e a s u r e d

0018-9464/83/1100-2437$01.00 O 1983 IEEE

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Fig . 2. 3-D l e a k a g e f i e l d p r o f i l e f o r a r e c t a n g u l a r s l o t i n a s t e e l b a r .

exper imenta l ly .

Ac t ive l eakage f i e ld p ro f i l e s have a l so been s t u d i e d u s i n g a conventional 2-dimensional and axisYmet- r i c m a g n e t o s t a t i c f i n i t e e l e m e n t c o d e t a k i n g i n t o a c c o u n t t h e n o n l i n e a r m a g n e t i z a t i o n c h a r a c t e r i s t i c s of t h e t e s t s p e c i m e n [ 4 ] . A v a r i e t y o f d e f e c t s h a p e s h a v e been s tudied, and Fig. 3 shows t y p i c a l f i n i t e e l e m e n t p r e d i c t i o n s o f t h e p r o f i l e s f o r t h r e e d i f f e r e n t l y s h a p e d s l o t s .

'- 0.4 r 0.4

- 0.4

Fig. 3. F i n i t e e l e m e n t p r e d i c t i o n s o f l e a l c a g e f i e l d p r o f i l e s f o r r e c t a n g u l a r a n d o b l i q u e s l o t s i n a steel b a r . (€3 B and B a r e t he no rma l ,

t a n g e n t i a l a n d t o t a l f i e l d s r e s p e c t i v e l y . , ) N' T

This work has l ed t o t he deve lopmen t o f a d e f e c t c h a r a c t e r i z a t i o n a l g o r i t h m c a p a b l e o f p r e d i c t i n g t h e equ iva len t w id th , dep th and i nc l ina t ion o f su r f ace d e f e c t s i n f e r r o m a g n e t i c test specimens [13]. More r ecen t ly [14 ] , t he magne tos t a t i c f i n i t e e l emen t code has been ex tended t o t h ree d imens ions , and t he fu l l 3-D l e a k a g e f i e l d p r o f i l e p r e d i c t e d f o r a r e c t a n g u l a r s l o t i n a s t e e l b a r b y f i r s t e s t i m a t i n g t h e 3-D c u r r e n t d i s t r i b u t i o n a r o u n d t h e s l o t [ 1 5 ] .

Under a c t i v e e x c i t a t i o n c o n d i t i o n s t h e n , a d i r e c t c u r r e n t i s app l i ed t o t he f e r romagne t i c spec imen s e t t i n g up l eakage f i e lds a round su r f ace and subsu r face f laws which can be de tec ted us ing any magnet ic f lux s e n s i t i v e t r a n s d u c e r s u c h as a Hal l e lement , moving co i l , magne t i c t ape head , e t c . [ 3 ] . The development o f l e a k a g e f i e l d p r o f i l e s f o r a v a r i e t y o f s u r f a c e a n d subsu r face f l aw shapes is o f impor t ance t o t he manu- f a c t u r e r s a n d u s e r s o f s t e e l p r o d u c t s . A p a r t i c u l a r example o f d i r ec t exc i t a t ion and its f i n i t e e l e m e n t m o d e l i n g f a m i l i a r t o t h e a u t h o r , r e l a t e s t o t h e u s e o f a v a r i a b l e r e l u c t a n c e (VR) p r o b e f o r t h e d e t e c t i o n o f magne t i t e bu i ldup i n t he c r ev ice gaps o f p re s su r i zed water r e a c t o r s t e a m g e n e r a t o r s .

F igure 4 shows d e t a i l s o f s u c h a g e n e r a t o r t o g e t h e r w i t h t h e f i n i t e e l e m e n t mesh s t r u c t u r e f o r m o d e l i n g t h e m a g n e t o s t a t i c f i e l d s i n t h e c r e v i c e gap region between tube and support p la te [16,17].

I+srrau io TURBlNE

b

Fig . 4 . PWR s t eam gene ra to r t ube t e s t ing ,

a ) S t eam gene ra to r de t a i l , b ) Eddy c u r r e n t p r o b e i n t h e c r e v i c e

c ) F in i t e e l emen t mesh f o r v a r i a b l e gap region,

r e l u c t a n c e p r o b e s t u d i e s .

The VR p robe cons i s t s o f a mild steel bobb in ca r ry ing a d c w i n d i n g ( a c t i v e e x c i t a t i o n ) . F l u x d e n s i t y l e v e l s a round t he pe r iphe ry o f t he bobb in can be r e l a t ed t o c r e v i c e gap c l e a r a n c e a n d a r e m o n i t o r e d i n t h e f i e l d p r o t o t y p e [la] by e i g h t Hall p l a t e s . F i g . 5 shows

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t y p i c a l f l u x p l o t s f o r t h e VR p r o b e a s i t passes through a s u p p o r t p l a t e . Use of t h e f i n i t e e l e m e n t code a l lowed des ign s tud ie s t o be made o n d i f f e r e n t b o b b i n s i z e s as w e l l as the deve lopment o f the inver - s i o n a l g o r i t h m f o r p r e d i c t i n g c r e v i c e gap c l e a r a n c e a t e igh t po in t s a round t he 'pe r iphe ry o f t he p robe .

Fig. 5. VR p r o b e f l u x d i s t r i b u t i o n s i n t h e c r e v i c e gap reg ion of a s team genera tor .

By i n c r e a s i n g t h e mesh d e t a i l i n t h e c r e v i c e gap region p r o b e s e n s i t i v i t y s t u d i e s c a n b e made f o r t h e d i f f e r e n t m a g n e t i t e c o n f i g u r a t i o n s l i k e l y t o b e e n c o u n t e r e d i n chemica l f lush ing procedures .

RESIDUAL LEAKAGE FIELDS

R e s i d u a l l e a k a g e f i e l d s e x i s t a r o u n d d e f e c t s i n f e r r o m a g n e t i c s p e c i m e n s a f t e r t h e d c a c t i v e e x c i t a t i o n c u r r e n t is removed. The m a t e r i a l w o r k i n g p o i n t s ( a t o f ) o n t h e i n i t i a l m a g n e t i z a t i o n c u r v e now re lax towards the s econd quadran t o f t he B/H c h a r a c t e r i s t i c ( p o i n t s a ' t o f ' ) a s shown i n F i g . 6. As i n t h e c a s e o f

Fig. 6. B/H p lane behavior o f a fe r romagnet ic Faterial a f t e r r e m o v a l o f a c t l v e d c e x c i t a t i o n .

Fig. 7. 3-D r e s i d u a l l e a k a g e f i e l d p r o f i l e f o r a s l o t i n a steel b a r .

where x and y a r e C a r t e s i a n c o o r d i a n t e s , b is o n e h a l f t h e s l o t w i d t h , h i s t h e s l o t d e p t h i n t h e y d i r e c t i o n and u is the exper imenta l ly de te rmined magnet ic charge d e n s i t y .

The c l a s s i c a l m a g n e t o s t a t i c f i n i t e e l e m e n t c o d e o r i g i n a l l y d e v e l o p e d f o r t h e s t u d y o f f i e l d s i n d c machinery [19] can be used to model the exc i ta t ion o f the specimen from 0 through f i n F i g . 6. However, a f t e r r e m o v a l o f t h e e x c i t a t i o n c u r r e n t e a c h w o r k i n g p o i n t of t h e m a t e r i a l r e l a x e s t o p o i n t s s u c h a s a' t o f ' i n t h e s e c o n d q u a d r a n t .

I n t h e m a c h i n e s l i t e r a t u r e s u c h phenomena (asso- c i a t ed w i th t he des ign o f pe rmanen t magne t s t ruc tu res ) have been modeled by e q u i v a l e n t c u r r e n t e f f e c t s . Such a n a p p r o a c h i n t h e NDT context would be phenomenclogi- c a l l y i n c o r r e c t , a n d c o n s e q u e n t l y a t t e m p t s h a v e b e e n made to fo rce each work ing po in t i n t he ma te r i a l a long i t s a p p r o p r i a t e h y s t e r e s i s c u r v e ( s u c h as dd ' ) , reducing t h e e x c i t a t i o n c u r r e n t g r a d u a l l y t o z e r o , a n d s t o p p i n g t h e r e l a x a t i o n p r o c e s s when t h e f l u x d e n s i t y a t some p o i n t i n t h e b a r s u c h as A i n F i g . 8 approaches i t s expec ted va lue [20-221.

a c t i v e e x c i t a t i o n t h e r e s u l t a n t l e a k a g e f i e l d s , a l t hough cons ide rab ly weake r t han ac t ive l eakage f i e lds can be measured by f l u x d e n s i t y s e n s i t i v e t r a n s d u c e r s a s b e f o r e . F i g . 7 shows a t y p i c a l r e s i d u a l l e a k a g e f i e l d p r o f i l e o b t a i n e d f o r t h e same s l o t t h a t . p r o d u c e d t h e a c t i v e l e a k a g e f i e l d p r o f i l e o f F i g . 2. A s can be seen f rom F igs . 2 .and 7 t h e r e a r e s i g n i f i c a n t d i f f e r - ences be tween ac t ive and r e s idua l l eakage f i e lds a round the same s l o t which unfor tuna te ly are no t t aken in to accoun t i n t he d ipo le mode l ing q f [ l o ] , where i t is assumed tha t t he equa t ion desc r ib*g bo th phenomena is of the form:

~ ~ l n [ ( x + b ) ~ + ( y + h ) ~ ] [(x-b) 2 2 +y ]

[(x+b)*+y2] [ ( ~ - b ) ~ + ( y + h ) ~ ] H = (2)

Fig. 8. F i n i t e e l e m e n t p r e d i c t i o n o f t h e r e s i d u a l l eakage f i e ld a round a r e c t a n g u l a r s l o t .

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Fig. 8 shows a t y p i c a l r e s i d u a l f l u x p l o t o b t a i n e d by fo l lowing th i s " forced re laxa t ion" p rocedure which would correspond to a s o l u t i o n o f t h e e q u a t i o n

Vx(vVXx) = 0 ( 3 )

By noting the normal f lux densi ty components a t each p o i n t on t h e s u r f a c e o f t h e b a r , r e s i d u a l l e a k a g e f i e l d p r o f i l e s similar to t hose o f F ig . 7 would be ob ta ined [ = I .

EDDY CURRENT NDT

With a l t e r n a t i n g c u r r e n t e x c i t a t i o n , b o t h f e r r o - magnet ic and nonfer romagnet ic mater ia l s can be t es ted n o n d e s t r u c t i v e l y . The presence o f d e f e c t s o r o t h e r inhomogeneities cause induced eddy current paths to c h a n g e , r e s u l t i n g i n m e a s u r a b l e v a r i a t i o n s i n t h e tes t

d e v e l o p e d f o r t h e study of eddy c u r r e n t s i n e lectr ical co i l impedance . A f i n i t e e l e m e n t code similar t o t h a t

machinery [23] has been used successfu l ly for p red ic t - i n g d i f f e r e n t i a l a n d a b s o l u t e eddy cur ren t p robe impedance p l ane t r a j ec to r i e s fo r bo th two dimensional and axisymmetric testing geometries [24-261. A s eddy c u r r e n t NDT p r o b e e x c i t a t i o n l e v e l s are r e l a t i v e l y l o w , a l i n e a r i n i t i a l permeabi l i ty value has been found adequate for model ing those t es t ing s i tua t ions where fe r romagnet ic materials are p r e s e n t .

In terms o f t h e B/H behavior o f fe r romagnet ic materials, a c e x c i t a t i o n c a u s e s e a c h w o r k i n g p o i n t t o move around a c o m p l e t e h y s t e r e s i s l o o p c l o s e t o t h e o r i g i n o f t h e B/H p l a n e f o r low e x c i t a t i o n l e v e l s as shown in F ig . 9.

Fig. 9 . AC e x c i t a t i o n o f a ferromagnet ic test specimen.

The f i e l d s a s s o c i a t e d w i t h s u c h phenomena s a t i s f y (4)

where t he exc i t a t ion angu la r f r equency , ma te r i a l con- d u c t i v i t y a n d s o u r c e c u r r e n t d e n s i t y are given by w, u and J respect ively. Both axisymmetr ic and 2-D eddy cus-rent code has been developed and confirmed by s tudying geometr ies such as a c o i l i n a i r and a con- ductor above a conduc t ing s l ab fo r wh ich known ana ly t - i ca l r e s u l t s e x i s t [25] . In add i t ion , t he ax i symmet r i c code has been appl ied to the model ing of eddy current NDT phenomena i n steam gene ra to r t ube i n spec t ion as i l l u s t r a t e d i n F i g . 4 c ) [ 2 4 , 2 7 ] .

A t y p i c a l f i n i t e e l e m e n t mesh f o r s t u d y i n g d i f f e r - en t i a l p robe impedance p l ane t r a j ec to r i e s is shown i n Fig. 10. From t h e r e s u l t i n g m a g n e t i c v e c t o r p o t e n t i a l va lues co i l impedance , eddy cu r ren t dens i t i e s , f l ux p l o t s , e t c . c a n b e c a l c u l a t e d . Numerous s t u d i e s h a v e

Fig. 10. F i n i t e e l e m e n t d i s c r e t i z a t i o n f o r t h e suppor t p l a t e r eg ion o f a PWR steam genera tor .

been made o f b o t h a b s o l u t e a n d d i f f e r e n t i a l p r o b e i m p e d a n c e p l a n e t r a j e c t o r i e s f o r a v a r i e t y o f a x i - symmetr ic defect shapes and the resul ts confirmed experimental ly . Such numerical s tudies can be used f o r p r o b e d e s i g n , s i m u l a t i n g f i e l d c o n d i t i o n s d i f f i - c u l t t o r e p l i c a t e e x p e r i m e n t a l l y , a n d g e n e r a t i n g t r a in ing da t a fo r au tomated s igna l p rocess ing s chemes . I n add i t ion t he f l ux p lo t s gene ra t ed can p rov ide u se - f u l i n s i g h t i n t o t h e p h y s i c s o f p r o b e f i e l d / d e f e c t i n t e r a c t i o n s . In t h i s r e g a r d , a shor t an imated f i lm sequence has been produced [27] which shows how t h e d i f f e r e n t i a l p r o b e f i e l d s i n t e r a c t w i t h a carbon steel s u p p o r t p l a t e t o p r o d u c e t h e c h a r a c t e r i s t i c L i s s a j o u s - pa t te rned impedance p lane t ra jec tory as i l l u s t r a t e d i n F i g . 11.

Movement o f t he p robe t h rough t he t ube i s simu- l a t e d by c a r r y i n g o u t a comple te f in i te e lement s tudy a t d i s c r e t e p o i n t s as t h e c o i l i n t h e mesh of F ig . 10 is moved p a s t t h e s u p p o r t p l a t e . To produce the com- p l e t e impedance p l a n e ' t r a j e c t o r y shown p a r t i a l l y comple t ed i n F ig . l l b ) r equ i r ed s even ty pos i t i ons pe r l obe . Th i s p rocedure t he re fo re neg lec t s any e f f e c t s o f ve loc i ty i nduced vo l t ages i n t he t ube walls and s u p p o r t p l a t e . A t t he p robe speeds u sed i n ac tua l s t e a m g e n e r a t o r t e s t i n g t h i s e r r o r a p p e a r s t o b e n e g l i g i b l e .

Using a magne t i c vec to r po ten t i a l fo rmula t ion and hexahedra l i sopa rame t r i c e l emen t s , Ida [14 ] has recent ly ex tended the 2-D f in i t e e l emen t code t o a f u l l 3-dimensional treatment. The code has been ver i - f i e d by s t u d y i n g eddy c u r r e n t d e n s i t i e s i n a s l a b benea th a current carrying conductor , and by predict ing t h e d i f f e r e n t i a l eddy c u r r e n t c o i l impedance plane t r a j e c t o r y f o r a square through-wall hole in a steam gene ra to r t ube .

COMPUTATIONAL CONSIDEPATIONS AND DISCUSSION

Most of the e l ec t romagne t i c NDT problems discussed i n t h i s p a p e r h a v e b e e n r u n on a VAX 780 computer o r a Cyber 720 system. The complexi t ies of extending t h e f i n i t e e l e m e n t c o d e t o t h r e e d i m e n s i o n s c a n b e imagined by examining Table 1 where the computer core requirements for producing Fig. l lb) f rom axisymmetr ic

2441

a

b

Fig. 11. Fin i te e lement p red ic t ion of eddy cur ren t p r o b e f l u x p a t t e r n s .

a ) Di f fe ren t ia l p robe pass ing th rough a s u p p o r t p l a t e .

b ) One frame of animation sequence showing par t ia l ly comple ted impedance p lane . t r a j e c t o r y .

code a r e summarized and compared to the requirements f o r a f u l l 3-D a n a l y s i s . C o n s i d e r a t i o n s l i k e t h i s h a v e l e d t o a carefu l ana lys i s o f computer requi rements and the development of minimum s t o r a g e m a t r i x i n v e r s i o n algori thms based on Gauss e l i m i n a t i o n a n d f r o n t a l t echniques [28] . Such codes .a re needed in the 3-D modeling of e l e c t r o m a g n e t i c f i e l d / d e f e c t i n t e r a c t i o n s i n o r d e r t o p r e d i c t l e a k a g e f i e l d p r o f i l e s and imped- a n c e p l a n e t r a j e c t o r i e s f o r r e a l i s t i c d e . f e c t s h a p e s . The research group a t Colorado S ta te Univers i ty has b e e n f o r t u n a t e i n h a v i n g t h e ' T e k t r o n i x g r a p h i c s s y s t e m o f F ig . 12 ava i lab le for the deve lopment o f the 3-D code. I n the nea r fu tu re t he g raph ic s sys t em w i l l b e connec ted t o a Cyber 205 supercomputer which, although

4954 DIGIIAL IABLEl HOST COMPUTER

4863 DIGITAL PLOllER

4805 MASS STORAGE MODULE

Fig. 1 2 . Graphics system used in development o f 3-D f in i t e e l emen t code .

r e q u i r i n g v e c t o r i z a t i o n o f a l l c o d e , w i l l speed t he 3-D runs cons iderably . A s w i t h a l l n u m e r i c a l s t u d i e s g r e a t c a r e must b e t a k e n i n i n t e r p r e t i n g r e s u l t s , a n d i t i s t h e a u t h o r ' s e x p e r i e n c e t h a t c a r e f u l e x p e r i m e n t a l work i s r e q u i r e d a t e v e r y s t a g e o f a lgor i thm deve lop- ment i n o r d e r t o v a l i d a t e t h e c o d e , p a r t i c u l a r l y f o r those t es t ing geometr ies where no a n a l y t i c a l s o l u t i o n s a re ava i lab le for compar ison . Al though the e lec t romag- n e t i c f i e l d s a s s o c i a t e d w i t h NDT t e s t i n g t e c h n i q u e s a r e w e l l b e h a v e d ( v e r y l o c a l i z e d a n d ' t i g h t ' ) when com- p a r e d t o f i e l d s i n e l e c t r i c a l m a c h i n e r y a n d f u s i o n magne t s fo r example , ca re mus t s t i l l ' be t aken in choos ing an appropr ia te mesh s t r u c t u r e i n r e l a t i o n t o bo th de fec t s i ze and sk in dep th . In add i t ion , t he a c c u r a t e s p e c i f i c a t i o n o f m a t e r i a l c o n d u c t i v i t y . a n d p e r m e a b i l i t y p r o p e r t i e s as code input da ta is important. The model ing of res idua l l eakage f ie lds is p a r t i c u l a r l y s e n s i t i v e t o t h e B/H da ta p rov ided fo r fo rced r e l axa - t i o n .

2-D Animation frame problem

Mesh - 120 x 25 t r i a n g l e s 6000 elements 3146 v a r i a b l e s Semibandwidth - 28 Matr ix - 3146 x 28 88,088 words of memory

3-D Animation frame problem

Mesh - 120 x 25 x 25 hexahedra 75,000 elements 245388 v a r i a b l e s Semibandwidth - 4209 Matr ix - 245388 x 4209

1.03 x 10 words of memory 9

Table 1. Comparison of 2-D and 3-D code core requi rements .

The purpose o f th i s paper has been to g ive an over - v iew of the appl ica t ion of numer ica l methods to e lec- t romagnet ic NDT problems. Space l imi ta t ions p rec lude t h e f u l l d i s c u s s i o n t h a t t h e s u b j e c t m a t t e r w a r r a n t s , a n d t h e i n t e r e s t e d r e a d e r i s encouraged to pursue the s u b j e c t m a t t e r f u r t h e r b y s t a r t i n g w i t h some o f t he r e f e r e n c e m a t e r i a l i n t h e n e x t s e c t i o n . C o n s i d e r a b l e work remains to be done in apply ing computer based mode l ing t echn iques bo th t o spec i f i c t e s t ing geomet r i e s and t o a r e a s s u c h as pulsed eddy current and microwave techniques .

2.442

ACKNOWLEDGMENTS

During the past decade i t has been my p r i v i l e g e t o work w i t h some e x c e l l e n t g r a d u a t e s t u d e n t s on t h e appl icacion of numerical methods to eLectromagnet ic NDT f i e l d p r o b l e m s . W i t h o u t t h e c o n s i d e r a b l e e f f o r t s of J. H . Hwang, R. Palanisamy, S. R. Sa t i sh and N. Ida, none of the work descr ibed in this paper would have heen poss ib le . The au thor i s a l s o d e e p l y i n d e b t e d t o D r . J. Hur t o f the Army Research Off ice and Drs. G. Dau and J. Quinn of the E lec t r ic Power R e s e a r c h I n s t i t u t e f o r t h e i r s u s t a i n e d s u p p o r t a n d encouragement.

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