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I nt e grate d s i mu l ati on of t he i nj e c t i on mold i ng pr oce s swi t h s t e r e oli t hography molds Ab s t r act F u nct i onal par t s a r e nee ded f or des i gn v e r i fi c a t i on t e s t i ng, fi e l d t r i a l s , c ust om e r e val uat i on, a nd product i o n pla n ning. B y e l i m i na t i ng m ult i ple s t e ps , t he c r e a t i on o f t he i nje c t i on m old dir e c t l y by a r a pid prototyping ( R P ) proce s s holds t he bes t prom i s e of r e duci ng t he t i m e a nd c ost n e e ded t o m old l ow- v olum e qua nt i t i e s of par t s . T he pote nti a l of t his i nte gr a t i o n of i nje c t i on m olding w i t h R P h a s bee n dem onst r a t e d m a ny t i m e s . What i s m i s s i ng i s t he f undam e nt a l under s t a nding of h ow t he m odi fi c a t i o ns t o t he m old m a t e r i a l a nd R P m a nufa c t uri ng proce s s i m pac t both t he m old des i gn a n d t he i nje c t i on m old i ng proce s s . I n a ddit i on, n um e r i c a l s i m ula t i o n t e c hniq ues h a ve nowbec om e hel pful t ools of m old des i gner s a nd proce s s e ngi ne e r s f or t r a dit i o nal i nje c t i on m olding. B ut a l l c u r r e nt s i m ula t i on pac kag e s f or c onventi o nal i nje c t i on m olding a r e no l onger a p p l i c a ble t o t his new t ype o f i nje c t i on m olds , m a i nly bec a use t he proper t y o f t he m old m a t e r i a l c h a nges gre a t l y. I n t his paper , a n i nte gr a t e d a pproac h t o a c c om pli s h a n um e r i c a l s i m ula t i on of i n j e c t i on m olding i nto r a pid- protot yped m olds i s e s t a bli s hed a nd a c orr e s ponding s i m ula t i on s yst e m i s devel oped. C om par i s ons w i t h e x per i m e nta l r e s ult s a r e e m ployed f or v e r i fi c a t i o n, w hic h s h ow t hat t he pre s e nt s c hem e i s w e l l s u i t e d t o handle R P f a bri c a t e d s t e r e oli t hogra phy ( S L ) m olds . K e yword sK K K I n j e c t i on m olding N u m e r i c a l s i m ula t i on R a pid protot yping 1 I n t r odu c t i on I n i nje c t i on m old i ng, t he polym e r m e l t a t high t e m pe r a t ur e i s i nje c t e d i nto t he m old u nder high pre s s ur e 1. T hus, t he m old m a t e r i a l n e e ds t o have t h e r m a l a nd m e c hanic a l proper t i e s c a pa b l e o f w i t hst a nding t he t e m pe r a t ur e s a nd pre s s ur e s of t he m old i ng c y c l e . T he f ocus of m a ny s t udie s has bee n t o c r e a t e t he i nje c t i on m old dir e c t l y by a r a pid protot yping ( R P ) proce s s . B y e l i m i na t i ng m ult i ple s t e ps, t h i s m e t hod of t ooli ng h old s t he bes t prom i s e of r e duci ng t he t i m e a nd c ost nee ded t o c r e a t e l ow- v olum e quanti t i e s of par t s i n a product i o n m a t e r i a l . T he pote nti a l of i nte gr a t i ng i nje c t i on m olding w i t h R P t e c hnologie s ha s bee n dem onst r a t e d m a ny t i m e s . T he proper t i e s of R P m olds a r e v e r y dif f e r e nt f r om t hose of t r a dit i o nal m e t a l m olds . T he key dif f e r e nce s a r e t he proper t i e s of t her m a l c onduct i vit y a n d e l a s t i c m od ulus ( r i gid i t y). F or e xam ple , t he polym e r s u s e d i n R P - f a bri c a t e d s t e r e oli t hogra phy ( S L ) m olds h a ve a t her m a l c onduct i vit y t hat i s l e s s t han one t housa ndth t hat of a n a l um i num t ool. I n usi ng R P t e c hnolo gie s t o c r e a t e m olds , t he e nti r e m old des i gn a n d i nje c t i on-m old i ng proce s s par a m e t e r s nee d t o be m odi fi e d a nd opti m i z e d f r om t r a dit i o nal m e t hodolo gie s due t o t he c om ple t e l y dif f e r e nt t ool m a t e r i a l . H owe ver , t her e i s s t i l l no t a f undam e n t a l under s t a nding of howt he m odi fi c a t i ons t o t he m old t ooli ng m e t hod a nd m a t e r i a l i m pac t both t he m old des i gn a nd t he i nje c t i on m olding proce s s par a m e t e r s . O n e c a nnotobta i n r e a s onable r e s ult s b y s i m ply c hanging a f e w m a t e r i a l proper t i e s i n c urr e nt m odel s . A l s o, u s i ng t r a dit i o nal a pproac hes w hen m a king a c t ual par t s m a y be g e ner a t i ng s ub-opti m a l r e s ult s . S o t her e i s a dir e n e e d t o s t udy t he i nte r a c t i o n bet w e e n t he r a pid t ooli ng ( R T ) pro c e s s a nd m a t e r i a l a n d i nje c t i on m olding, s o a s t o e s t a bli s h t he m old des i gn c r i t e r i a a nd t e c hniq ues f or a n R T - ori e nte d i nje c t i on m olding proce s s . I n a ddit i on, c om pute r s i m ula t i on i s a n e f f e c t i ve a pproac h f or pre dic t i ng t he qual i t y of m olded par t s . C om m e r c i a l l y a vai l a ble s i m ula t i on pac kages of t he t r a dit i o nal i nje c t i on m olding proce s s have n ow bec om e r outi ne t ools o f t he m old des i gner a nd pro c e s s e n gine e r 2. U nfor t unat e l y , c urr e nt s i m ula t i o n progra m s f or c onventi o nal i nje c t i on m olding a r e no l onger a ppli c a ble t o R P m olds , bec a use of t he dra m a t i c a l l y dis s i m i l a r t ool m a t e r i a l . F or i nst a nce , i n usi ng t h e e xis t i ng s i m ula t i on s oft w a r e w i t h a l u m i num a n d S L m olds a n d c om par i ng w i t h e xper i m e nta l r e s ult s , t hought he s i m ula t i on val ues of par t dis t ort i o n a r e r e a s onable f or t he a l um i num m old, r e s ult s a r e u nac c e pta ble , w i t h t he e r r or e xce e ding 50%. T he dis t ort i o n dur i ng i nje c t i on m old i ng i s due t o s hri nk a ge a nd w a r page of t he pla s t i c par t , a s w e l l a s t he m old. F or ordinar i l y m olds , t he m a i n f a c t or i s t he s h r i nkage a n d w a r pa ge of t he pla s t i c par t , w hic h i s m odel e d a c c ura t e l y i n c ur r e nt s i m ula t i ons. B ut f or R P m old s , t he dis t or t i o n of t he m old h a s pote nti a l l y m ore i n fl uence , w hic h have bee n negle c t e d i n c urr e nt m odel s . F or i nst a nce , 3 use d a s i m p l e t hre e - s t e p s i m ula t i on proce s s t o c onsi der t he m old dis t or t i on, w hic h had t oo m uch devia t i on. I n t his paper , bas e d on t h e a bove a nal ysi s , a new s i m ula t i o n s yst e m f or R P m olds i s dev e l oped. T he propose d s y s t e m f ocus e s on pre dic t i ng par t dis t ort i o n, w hic h i s domi nat i ng def e c t i n R P - m olded par t s . T he dev e l oped s i m ula t i on c a n be a ppli e d a s a n e v a l ua t i o n t ool f or R P m old des i gn a n d proce s s opti m i z a t i on. O u r s i m ula t i o n s yst e m i s ver i fi e d b y a n e xper i m e nta l e x a m p l e . A l t hough m a ny m a t e r i a l s a r e a vai l a ble f or use i n R P t e c h no l ogie s , w e c once ntr a t e o n u s i ng s t e r e oli t ho gra phy ( S L ) , t he ori gina l R P t e c hnology, t o c r e a t e polym e r m olds. T he S L pro c e s s use s photopolym e r a n d l a s e r e ner gy t o buil d a par t l a yer b y l a yer . U s i ng S L t a kes a dvanta ge of both t he c om m e r c i a l domi na nce o f S L i n t he R P i ndust r y a nd t he s ubse quent e xper t i s e bas e t hat has bee n dev e l oped f or c r e a t i ng a c c ura t e , high-qua l i t y par t s . U n t i l r e c e nt l y, S L w a s pri m a r i l y u s e d t o c r e a t e physi c a l m odel s f or vis ual i nspec t i on a nd f or m - fi t s t udie s w i t h ver y l i m i t e d f unc t i o nal a ppli c a t i o ns. H owe ver , t he new e r gener a t i on s t e r e oli t ho gr a phic photopolym e r s h a ve i m pr oved dim e ns i onal , m e c hanic a l a nd t her m a l proper t i e s m a king i t poss i ble t o u s e t hem f or a c t ual f unct i o nal m olds. 2 I n t e grate d s i mu l ati on of t he mold i n g p r oce s s 2 . 1 Me t h o d o l o g y I n order t o s i m ula t e t he use of a n S L m old i n t he i nje c t i on m olding proce s s , a n i t e r a t i ve m e t hod i s propose d. D i f f e r e nt s oft w a r e m odule s h a ve bee n dev e l oped a nd use d t o a c c om pli s h t his t a s k. T he m a i n a s s um pt i on i s t hat t e m pe r a t ur e a n d l oad bound a r y c ondit i o ns c a u s e s i gnifi c a nt dis t ort i o ns i n t he S L m old . T he s i m ula t i o n s t e ps a r e a s f oll o w s : 1 T he par t g e om e t r y i s m odel e d a s a s oli d m odel , w hic h i s t r a ns l a t e d t o a fi l e r e a dable by t he fl ow a nal ysi s pac kag e . 2 S i m ula t e t he m old- fi l l i n g proce s s of t he m e l t i nto a pho t opolym e r m old, w hic h w i l l out put t he r e s ult i ng t e m pe r a t ur e a n d pre s s ur e pro fi l e s . 3 S t r uct ura l a nal ysi s i s t hen per f orm e d on t he photopolym e r m old m odel usi ng t he t her m a l a nd l oad boundar y c ondit i o ns obta i ned f r om t he pre v i o us s t e p, w hic h c a l c ula t e s t he dis t or t i o n t hat t he m old under go dur i ng t he i nje c t i on proce s s . 4 I f t h e dis t ort i o n of t he m old c onver ges , m ove t o t he next s t e p. O t h e r w i s e , t he dis t or t e d m old c a vit y i s t hen m odel e d ( c hanges i n t he dim e ns i ons of t he c a vit y a f t e r dis t ort i o n), a nd r e t ur ns t o t he s e c ond s t e p t o s i m ula t e t he m e l t i nje c t i on i nto t he dis t ort e d m old. 5 T he s hri nk a ge a n d w a r pa ge s i m ula t i on of t he i nje c t i on m olded par t i s t hen a ppli e d, w hic h c a l c ula t e s t he fi na l dis t or t i o ns of t he m olded par t . I n a boves i m ula t i on fl ow, t h e r e a r e t hre e bas i c s i m ula t i o n m od ule s . 2 . 2 F i l l i n g s i m u l a t i o n o f t h e m e l t 2 . 2 . 1 Ma t h e m a t i c a l m o d e l i n g I n order t o s i m ula t e t he use of a n S L m old i n t he i nje c t i on m olding proce s s , a n i t e r a t i ve m e t hod i s propose d. D i f f e r e nt s oft w a r e m odule s h a ve bee n devel oped a nd use d t o a c c om pli s h t his t a s k. T he m a i n a s s um pt i on i s t hat t e m pe r a t ur e a n d l oad boundar y c ondit i ons c a use s i gnif i c a nt dis t or t i o ns i n t he S L m old. T he s i m ula t i o n s t e ps a r e a s f oll o w s : 1. T he par t geom e t r y i s m odel e d a s a s oli d m odel , w hic h i s t r a ns l a t e d t o a f i l e r e a dable by t he f l o w a nal ysi s pac kag e . 2. S i m ula t e t he m old- f i l l i ng proce s s of t he m e l t i nto a photopolym e r m old, w hic h w i l l out put t he r e s ult i ng t e m pe r a t ur e a n d pre s s ur e profi l e s . 3. S t r uct ura l a n a l ysi s i s t hen per f orm e d o n t he phot opolym e r m old m odel u s i ng t he t her m a l a n d l oad boundar y c ondit i ons obta i ne d f r om t he pre vious s t e p, w hic h c a l c ula t e s t he dis t or t i on t hat t he m old under go dur i ng t h e i nje c t i on proce s s . 4. I f t he dis t ort i o n of t he m old c onve r ges , m ove t o t he n e xt s t e p. O t her w i s e , t he dis t ort e d m old c a vit y i s t h e n m odel e d ( c hanges i n t he dim e ns i ons of t he c a v i t y a f t e r dis t ort i o n), a nd r e t ur ns t o t he s e c ond s t e p t o s i m ula t e t he m e l t i nje c t i on i nto t he dis t ort e d m old. 5. T he s hri nk a ge a nd w a r page s i m ula t i on o f t he i nje c t i on m olded par t i s t hen a ppli e d, w hic h c a l c ula t e s t he f i nal dis t ort i o ns o f t he m old e d par t . I n a boves i m ula t i on f l ow, t her e a r e t hre e bas i c s i m ula t i o n m odule s . 2 . 2 F i l l i n g s i m u l a t i o n o f t h e m e l t 2 . 2 . 1 Ma t h e m a t i c a l m o d e l i n g C om pute r s i m ula t i on t e c hniques h a ve had s ucc e s s i n pre dic t i ng f i l l i n g behavior i n e x t r e m e l y c om pli c a t e d g e om e t r i e s . H owe ver , m ost of t he c urr e nt n um e r i c a l i m ple m e nt a t i o n i s bas e d o n a hybri d f i nit e - e l e m e nt/ f i nit e - dif f e r e nc e s olut i on w i t h t he m i ddle pla ne m odel . T he a ppli c a t i o n proce s s o f s i m ula t i on pac kages bas e d o n t his m odel i s i l l us t r a t e d i n F i g. 2-1 . H owe ver , u nli k e t he s urf a c e / s oli d m odel i n m old- des i gn C A D s yst e m s , t he s o- c a l l e d m i ddle - pla ne ( a s s h own i n F i g. 2- 1 b) i s a n i m a gina r y a r bit r a r y pla nar geom e t r y a t t he m i ddle of t he c a vit y i n t he g a p- w i s e dir e c t i on, w hic h s hould bri ng a boutgre a t i nconvenie nce i n a ppli c a t i ons. F or e xam ple , s urf a c e m odel s a r e c om m only u s e d i n c urr e nt R P s yst e m s ( gener a l l y S T L f i l e f orm a t ) , s o s e c ondar y m odel i ng i s u navoid a ble w hen u s i ng s i m ula t i on pac kages bec a use t he m odel s i n t he R P a nd s i m ula t i on s yst e m s a r e dif f e r e nt . C ons i d e r i ng t hes e def e c t s , t he s urf a c e m odel of t he c a v i t y i s i ntr oduce d a s dat um pla nes i n t he s i m ula t i o n, i nst e a d of t he m i d dle - pla ne. A c c or ding t o t he pre vious i nves t i gat i ons 4 6, f i l l i nggover ning e qua t i ons f or t he f l ow a nd t e m pe r a t ur e f i e l d c a n be w r i t t e n a s : w her e x , y a r e t he pla nar c oordinat e s i n t he m i ddle - pla ne, a n d z i s t he g a p- w i s e c oordinat e ; u , v ,w a r e t he v e l o c i t y c om ponent s i n t he x , y , z dir e c t i ons; u , v a r e t he a ver a ge w hole - gap t hic knes s e s ; a n d , , CP ( T ) , K( T ) r e pre s e nt vis c os i t y, densi t y, s pec i f i c h e a t a nd t her m a l c onduct i vit y of polym e r m e l t , r e s pec t i vel y. Fi g.2-1 a d. S c hem a t i c pr oce dure of t he s i m ul a t i on w i t h m i ddle - pl a ne m odel . a T he 3- D s urf a c e m odel b T he m i ddle - pl a ne m odel c T he m e s hed m i ddle - pl a ne m odel d T he dis pl a y of t he s i m ul a t i on r e s ult I n a ddit i o n, boundar y c ondit i o ns i n t he g a p- w i s e dir e c t i on c a n be def i ned a s : w her e T W i s t he c onst a nt w a l l t e m pe r a t ur e ( s hown i n F i g. 2a) . C om bining E qs . 1 4 w i t h E qs. 5 6, i t f oll o w s t hat t h e dis t r i buti ons of t h e u , v , T , P a t z c oordinat e s s hould be s ym m e t r i c a l , w i t h t he m i r r or a xis bei ng z = 0, a nd c onse quentl y t he u , v a ver a ged i n hal f - gap t hic knes s i s e qual t o t hat a v e r a ged i n w hole gap t hic knes s . B a s e d o n t his c har a c t e r i s t i c , w e c a n divide t he w hole c a v i t y i nto t w o e qual par t s i n t he g a p- w i s e dir e c t i on, a s des c r i b e d by P art I a n d P art I I i n F i g. 2b. A t t he s a m e t i m e , t r i a ngula r f i nit e e l e m e nt s a r e gener a t e d i n t he s urf a c e ( s ) of t he c a vit y ( a t z = 0 i n F i g. 2b), i nst e a d o f t he m i ddle - pla ne ( a t z = 0 i n F i g. 2a) . A c c or dingly, f i nit e - dif f e r e nce i ncr e m e nts i n t he gapwi s e dir e c t i on a r e e m ployed only i n t he i nsi de of t he s urf a c e ( s ) ( w a l l t o m i ddle / c e nte r - l i ne ) , w hic h, i n F i g. 2b, m e a ns f r om z = 0 t o z = b . T his i s s i ngle - s i ded i ns t e a d of t w o- s i ded w i t h r e s pec t t o t he m i ddle - pla ne ( i .e . f r om t he m i ddle - l i ne t o t w o w a l l s ) . I n a ddit i o n, t he c oordinat e s yst e m i s c h a nged f r om F i g. 2a t o F i g . 2b t o a l t e r t he f i nit e - e l e m e nt/ f i nit e - dif f e r e nce s c hem e , a s s hown i n F i g. 2b. Wi t h t he a bovea djust m e nt , gover ning e qua t i ons a r e s t i l l E qs . 1 4. H owe ver , t he ori gina l boundar y c ondit i o ns i n t he gapwi s e dir e c t i on a r e r e w r i t t e n a s : Me a nwhil e , a ddit i o nal boundar y c ondit i o ns m ust be e m ployed a t z = b i n order t o kee p t he f l ows a t t he j unct ur e of t h e t w o par t s a t t he s a m e s e c t i o n c oordinat e 7: w her e s u bsc r i pt s I , I I r e pre s e nt t he par a m e t e r s o f P art I a nd P art I I , r e s pec t i ve l y, a nd C m - I a n d C m - I I i ndic a t e t h e m oving f r e e m e l t - f r onts of t he s u r f a c e s of t he divided t w o par t s i n t h e f i l l i ng s t a ge. I t s h ould be note d t hat , u nli k e c ondit i o ns E qs . 7 a nd 8, e nsuri ng c ondit i o ns E qs . 9 a nd 10 a r e uphel d i n nume r i c a l i m p l e m e nta t i ons bec om e s m ore dif f i c ult due t o t he f oll o w i ng r e a s ons: 1. T he s urf a c e s a t t he s a m e s e c t i o n h a ve bee n m e s hed r e s pec t i vel y, w hic h l e a ds t o a dis t i nct i ve pat t e r n of f i nit e e l e m e nts a t t he s a m e s e c t i o n. T hus, a n i nte r pola t i on oper a t i o n s hould be e m ployed f or u , v , T , P duri ng t he c om par i s on bet w e e n t he t w o par t s a t t he j unct ur e . 2. B e c a us e t he t w o par t s h a ve r e s pec t i ve f l o w f i e l d s w i t h r e s pec t t o t he n odes a t point A a nd point C ( a s s hown i n F i g. 2b) a t t he s a m e s e c t i o n, i t i s poss i ble t o have e i t he r both f i l l e d or one f i l l e d ( a nd one e m pty) . T hes e t w o c a s e s s hould be h a ndle d s e par a t e l y, a ver a ging t he oper a t i o n f or t he f or m e r , w her e a s a s s i gning oper a t i o n f or t he l a t t e r . 3. I t f oll o w s t hat a s m a l l dif f e r e nce bet w e e n t he m e l t - f r onts i s per m i s s i ble . T hat a l l owa nce c a n be i m ple m e nt e d b y t i m e a l l owa nce c ontr ol or pre f e r a ble l oca t i on a l l owa nce c ont r ol of t he m e l t - f r ont n odes . 4. T he boundar i e s of t he f l ow f i e l d e xpand b y e a c h m e l t - f r ont a dvance m e nt , s o i t i s n e c e s s a r y t o c hec k t he c ondit i o n E q. 10 a f t e r e a c h c hange i n t he m e l t - f r ont. 5. I n v i e w of a bove- m e nti oned a n a l ys i s , t he physi c a l par a m e t e r s a t t he n odes of t he s a m e s e c t i o n s h ould be c om par e d a nd a djust e d, s o t he i nform a t i o n des c r i bing f i nit e e l e m e nts of t he s a m e s e c t i o n s hould be pre par e d bef ore s i m ula t i on, t hat i s , t he m a t c hing oper a t i on a m ong t he e l e m e nts s hould be pre f orm e d. Fi g. 2a,b. I l l us t r a t i ve of boundar y c ondit i ons i n t he gap- w i s e dir e c t i on a of t he m i ddle - pl a ne m odel b of t he s ur f a c e m odel 2 . 2 . 2 N u m e r i c a l i m p l e m e n t a t i o n P r e s s ure f i e l d. I n m odel i ng vis c os i t y , w hic h i s a f unct i o n o f s h e a r r a t e , t e m pe r a t ur e a nd pre s s ure o f m e l t , t he s h e a r - t hinning beh a vior c a n be w e l l r e pre s e nte d by a c r oss - t ype m odel s uch a s : w her e n c orr e s pondst o t he powe r - l a w i ndex, a n d * c h a r a c t e r i z e s t he s h e a r s t r e s s l e vel of t he t r a nsi t i o n r e gion bet w e e n t he N e w t onia n a n d powe r - l a w a s y m p t ot i c l i m i t s . I n t e r m s of a n A r r henius- t ype t e m per a t ur e s e nsi t i vit y a nd e xponenti a l pre s s ure dependence , 0 ( T , P ) c a n be r e pre s e nt e d w i t h r e a s onable a c c ura c y a s f oll o w s : E quat i ons 11 a n d 12 c onst i t ut e a f i ve- c onst a nt ( n , * , B , T b , ) r e pre s e nt a t i o n f or v i s c osi t y. T he s hea r r a t e f or v i s c osi t y c a l c ula t i on i s obta i ned b y: B a s e d on t he a bove, w e c a n i nfe r t he f oll o w i ng f i l l i ng pre s s ure e quat i o n f r om t he gover ning E qs. 1 4: w her e S i s c a l c ula t e d b y S = b 0 / ( b z ) 2 d z . A pplying t he G a l e r kin m e t hod, t he pre s s ur e f i nit e - e l e m e nt e quat i o n i s deduc e d a s : w her e l _ t r a ver s e s a l l e l e m e nts , i ncl uding n ode N , a nd w her e I a nd j