The artifical heat source method in numerical modelling of non-linear conduction problems

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S u m m a r y . In th e p a p e r a c e rta in a lg o rith m w h ic h c a n b e c a lle d th e a r t if ic ia l h e a t s o u rc e m e th o d is p re s e n te d . P ro p o s e d a p p ro a c h is u s e fu l in th e c a s e o f n o n - lin e a r an d n o n - ste a d y h e a t c o n d u c tio n p ro b le m s .

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S tre s z c z e n ie . W p r a c y p rz e d s ta w io n o p e w ie n a lg o r y tm , k tó ry n a z w a n o m e to d a s z tu c z n e g o ź ró d ła c ie p ła . M e to d a m o ż e b y ć w y k o rz y s ta n a d o n u m e ry c z n e g o m o d e lo w a n ia n ie lin io w y c h i n ie s ta c jo n a rn y c h z a g a d n ie ń p rz e w o d n ic tw a c ie p ln e g o .

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c ( T ) p ( T ) d T ( * ' *)■ = d i v [ A ( r ) g r a d T ( X , r ) ]

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B. Mochnacki, E.Majchrzak

X e l \ : T ( X , t ) = 7 \ ( X , t )

X e r 2 : - A n g r a d T X X , i ) = q n ( X , t )

X e r 3 : - A .n - g r a d r ( X , i ) = a [ T ( X , r ) - T „ ]

t = 0 : 7 ( X , 0 ) = T0 ( X )

w h e r e c , p , X a re th e th e rm o p h y s ic a l p a ra m e te rs (s p e c ific h e a t, m a ss d e n s ity , th e rm a l c o n d u c t iv it y ), T , X , t d e n o te a te m p e ra tu re , s p a tia l c o - o rd in a te s an d tim e , n g r a d f is a n o rm a l d e r iv a tiv e a t b o u n d a ry p o in t X , q n is a g iv e n h e a t flu x , a , T „ a re th e h e a t tra n s fe r c o e ffic ie n t a n d a m b ie n t te m p e ra tu re , T t, T 0 a re th e b o u n d a ry an d in it ia l te m p e ra tu re s .

T h e b a s ic m a th e m a tic a l m o d e l c a n b e r e b u ilt b y th e in tro d u c tio n o f so - c a lle d K ir c h h o f f s te m p e ra tu re , it m e a n s

T

U ( T ) = f X ( t i ) d p (3 ) r,

w h e re T r is a n a r b itr a r y a ss u m e d re fe re n c e le v e l.

T h e K ir c h h o f f s tra n s fo rm a tio n lin e a riz e s th e rig h t- h a n d s id e o f e n e rg y e q u a tio n (1 ), n a m e ly

d iv [ X ( r ) g T a d r ( X , f)]

T h e le ft- h a n d s id e o f e q u a tio n (1 ) c a n b e tra n s fo rm e d b y in tro d u c in g to th e c o n s id e ra tio n s the p h y s ic a l e n th a lp y re la te d to a n u n it o f v o lu m e

T

H ( T ) = J c ( p ) p ( p ) d p (5 )

r,

B e c a u s e H a n d U a re th e fu n c tio n s o f te m p e ra tu re a n d th e re a re m o n o to n e o n e s s o , it is p o s s ib le to c o n s tru c t th e fu n c tio n / / = $ (£ / ) — c o m p . F ig u r e s 2 , 3 , 4 . A d d itio n a lly

d H ( X , t ) _ d H ( U ) 3 t / (X , t ) = , V ) d U ( X , t ) = T ( t / ) 8 t / ( X , t ) (6 )

d t d U d t d t d t

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T h e b o u n d a ry a n d in it ia l c o n d itio n s s h o u ld b e a ls o tra n s fo rm e d in a d e q u a te w a y [1 ]

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T],

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w h e re a m= a / \ m. I t s h o u ld b e p o in te d th a t th is n o n - lin e a rity o f c o n d itio n (1 0 ) d o e s n o t c a u se th e e s s e n tia l d if f ic u lt ie s in n u m e ric a l re a liz a tio n .

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F ig . 3 . E n th a lp y fu n c tio n H = H ( T ) R y s . 3 . F u n k c ja e n ta lp ii

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F ig . 4 . F u n c t io n H = H ( U ) F ig . 5 . F u n e tio n 'k = $ ' R y s . 4 . F u n k c ja H = H ( U ) R y s . 5 . F u n k c ja 4 ' = $ '

2 . T H E A R T I F I C I A L H E A T S O U R C E M E T H O D ( A H S M )

C o n s id e r n o w , a fu n c tio n ' i ' ( U ) w h ic h is c o n v e n tio n a lly e x p re s s e d as a su m o f tw o c o m p o n e n ts , it m e a n s a c o n s ta n t p a rt 4 '0 a n d a c e r ta in in c re m e n t A 'k

W ( U ) = 7 0 + A > P (I/ ) (1 1 )

T h e e n e rg y e q u a tio n (7 ) c a n b e w r itte n in th e fo rm

y o d U ( - * ’ = d iv [g r a d l/ ( Y , i ) ] - A Y 3 t / (^ ’ l ) (1 2 )

o r

T 0 W ( 5 = d iv [g r a d l/ (X , 0 ] + « v ( X . 0 <13>

o t

w h e re q v ( X , t ) is a s o u rc e fu n c tio n (a c a p a c ity o f in te rn a l h e a t s o u rc e s ). T h e e s s e n tia l fe a tu re o f e q u a tio n (1 3 ) c o n s is ts in a fa c t, th a t le a v in g o u t th e la s t te rm o n e o b ta in s th e lin e a r fo rm o f e n e rg y e q u a tio n . T a k in g in to a c c o u n t th e p o s s ib ilitie s o f b o u n d a ry e le m e n t m eth o d a p p lic a tio n in th e ra n g e o f n o n - ste a d y p ro b le m s m o d e llin g , it is th e v e r y c o n v e n ie n t fo rm o f b a s ic d iffe r e n t ia l e q u a tio n (a n o n - lin e a rity a p p e a rs o n ly in th e c o m p o n e n t d e te rm in in g the in te rn a l h e a t s o u rc e s , a n d th e fu n c tio n d e s c rib in g s o - c a lle d fu n d a m e n ta l s o lu tio n fo r c o n s id e re d p ro b le m is w e ll k n o w n ). T h e c a lc u la tio n o f a s o u rc e fu n c tio n re q u ir e s , o f c o u rs e , th e in tr o d u c tio n o f a c e r ta in ite r a tiv e p ro c e d u re (th e d e ta ils c o n n e c te d w ith n u m e r ic a l asp ects o f p ro p o s e d a lg o r ith m w ill b e p re s e n te d in th e fu rth e r p a rt o f th e p a p e r ), b u t it s h o u ld be p o in te d th a t i f | A 'i [ < th e n th e a d e q u a te ite r a tiv e a lg o r ith m is c o n v e rg e n t. In th is p ap e r

th e A H S M w ill s u p p le m e n t a v a r ia n t o f b o u n d a ry e le m e n t m e th o d c a lle d th e B E M u s in g d is c re tis a tio n in tim e [2 , 3 , 4 , 5 ].

3 . B O U N D A R Y E L E M E N T M E T H O D U S IN G D IS C R E T IS A T IO N I N T IM E

T h e I D p ro b le m w ill b e c o n s id e re d , it m e an s th e e n e rg y e q u a tio n in th e fo rm

v 0 d U ( * ’ 0 = -- U (x ;~ + o

dr dx2

(1 4 )

w ith a ss u m e d b o u n d a ry c o n d itio n s f o r x = x lt x = x 2 an d in it ia l c o n d itio n fo r r = 0 . It s h o u ld be p o in te d th a t a g e n e ra liz a tio n o f p re s e n te d a lg o rith m o n 2 D o r 3 D p ro b le m is v e r y s im p le [5 ].

A t fir s t th e G r e e n ’ s fu n c tio n o f th e fo rm

ex p

2 y J a A t (

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is in tro d u c e d ( x is a s p a tia l c o - o rd in a te w h e re a s £ is a p o in t w h e re a c o n c e n tra te d h e a t s o u rc e is a p p lie d ).

T h e fu n c tio n (1 5 ) f u lf ills th e e q u a tio n

d x a A t

(1 6 )

w h e re A (£ , x ) is a D ir a c ’s fu n c tio n .

T h e id e a o f d is c u s s e d v a r ia n t o f th e B E M c o n s is ts in a s u b s titu tio n o f a tim e d e r iv a tiv e b y its fir s t o r d e r a p p ro x im a tio n a n d th e n th e e q u a tio n (1 4 ) is tra n s fo rm e d to th e fo rm

d 2 U ( x , t + A t ) 1 d x 2

4 - l / ( x , r + A f ) = - q y ( x , t ) - - 4 - U ( x . 0 ( 1 7 >

a A t a A t

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U ' U . x ) A x

246

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T h e la s t in te g r a l e q u a tio n c o rre s p o n d s to so - c a lle d W e ig h te d R e s id u a l M e th o d c r ite r io n . In te g r a tio n b y p a rts th e le ft- h a n d s id e o f e q u a tio n (1 8 ) an d u s in g e q u a tio n (1 5 ) le a d s to

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A b o v e s y s te m o f e q u a tio n s a llo w s to fin d th e ’m is s in g ’ b o u n d a ry K ir c h h o f f s te m p e ra tu re s o r h e a t flu x e s , a n d n e x t a p p ly in g th e e q u a tio n (1 9 ) o n e c a n d e te rm in e a s e a rc h e d fu n c tio n U (x , t + A t ) a t th e in te rn a l p o in ts o f d o m a in D . T h e h e a t flu x c o n tin u ity c o n d itio n (1 0 ) c a n b e a ls o ta k e n in to a c c o u n t.

T h e it e r a t iv e p ro c e s s o f s o u rc e fu n c tio n d e te rm in a tio n is th e fo llo w in g . 1 . T r a n s itio n fro m f °= 0 to t ‘ = i°+ A f :

— it is a ss u m e d th a t q v ( x t, r ') = 0 , a t th e sam e tim e x , d e n o te s a c e n tra l p o in t o f in te rn a l c e lls d is tin g u is h e d in c o n s id e re d d o m a in ,

— fo r th is a s s u m p tio n th e K ir c h h o f f s te m p e ra tu re fie ld fo r w h o le d o m a in is c a lc u la te d ,

— th e lo c a l c o o lin g ra te s [ t / ( x „ t l ) - U ( x „ f °) ] / A f a re e s tim a te d ,

— a lo c a l v a lu e s o f s o u rc e fu n c tio n q v ( x „ t ‘) a re c o rre c te d ,

— th e it e r a t iv e p ro c e s s is sto p p e d i f re q u ire d a c c u ra c y is o b ta in e d . 2 . T r a n s itio n fro m t f ~' to t f , / = 2 , 3 , . . . , F :

— it is a ss u m e d th a t q v (x f, t f ) is e q u a l to th e la s t v a lu e o f q v fo u n d d u rin g th e p re v io u s ite r a tiv e p ro c e s s (a t c o n s id e re d p o in t),

— fo r th is a s s u m p tio n th e K ir c h h o f f s te m p e ra tu re fie ld fo r w h o le d o m a in is c a lc u la te d ,

— th e lo c a l c o o lin g ra te s [ i/ ( x (, t f ) - U ( x „ t f ~ ' ) ] / A t a re e s tim a te d ,

— a lo c a l v a lu e s o f s o u rc e fu n c tio n q v (x f, t / ) a re c o rre c te d ,

— th e ite r a tiv e p ro c e s s is sto p p e d i f re q u ire d a c c u ra c y is o b ta in e d .

T h e te s t c o m p u ta tio n s s h o w (a n d it c a n b e p ro b a b ly p ro v e d in a n a ly t ic w a y ) th a t th e ite r a tiv e p ro c e s s is c o n v e rg e n t i f | A ' i \ < ♦<>.

4 . E X A M P L E O F N U M E R I C A L S IM U L A T IO N

T h e s te e l p la te w ith th ic k n e s s L = 0 .1 [m ] h a s b e e n c o n s id e re d . T h e rm o p h y s ic a l p a ra m e te rs o f th e m a te r ia l (C = 0 .0 8 , S i= 0 .0 8 , M n = 0 .3 1 , S = 0 .0 5 , P = 0 .0 2 9 , C r = 0 .0 4 5 , N i= 0 .0 7 , M o = 0 .0 2 ) h a v e b e e n assu m e d o n th e b a s is o f e x p e rim e n ta l d a ta q u o te d in [6 ], T h e p ro b le m is s tro n g ly n o n - lin e a r b e c a u s e , fo r e x a m p le , s p e c ific h e a t (re la te d to a n u n it o f v o lu m e ) ch a n g e s fro m 3 .5 1 0 6 to 8 .7 - 1 0 6[J/ m 5K ] , w h e re a s th e th e rm a l c o n d u c tiv ity fro m 3 0 to 5 0 [W / m K ], U s in g th e n u m e ric a l in te g ra tio n m e th o d s th e fu n c tio n s H ( T ) a n d U ( T ) h a v e b e e n fo u n d - F ig u r e s 2 a n d 3 , n e x t fu n c tio n H = H ( U ) h a s b e e n c o n s tru c te d (F ig u r e 4 ) a n d f in a lly its d e r iv a tiv e (F ig u r e 5 ). T h e fo llo w in g b o u n d a ry - in itia l c o n d itio n s h a v e b e e n a s s u m e d : x , = 0 :

<7„(0, i) = 0 , x t = L : - d U l d x = a m ( U - U „ ) , a t th e sa m e tim e a = 3 0 0 , i / „ = 0 (c o m p , e q u a tio n (1 1 )), fo r f = 0 : U 0= 2 7 5 1 0 (t h is v a lu e c o rre s p o n d s to r = 5 5 3 °C ).

In F ig u r e 6 th e c o o lin g c u rv e s a t s e le c te d p o in ts o f in te r io r D a re s h o w n , in p a r tic u la r : 1 : x = 0 .0 9 5 , 2 : x = 0 .0 6 5 , 3 : x = 0 .0 3 5 , 4 : x = 0 .0 5 [m ]. T h e f u ll lin e s illu s tr a te th e n u m e ric a l s o lu tio n o b ta in e d o n th e b a s is o f re p e a te d ly v e r ifie d F D M a lg o r ith m fo r n o n - lin e a r e q u a tio n s , w h e re a s th e s y m b o ls s h o w th e n u m e ric a l s o lu tio n fo u n d b y m e a n s o f a r t if ic ia l h e a t s o u rc e m e th o d . T h e m a x im u m d iffe re n c e b e tw e e n p re s e n te d n u m e ric a l s o lu tio n s d o e s n o t e x c e e d 0 . 5 % . I t se e m s th a t p ro p o s e d m e th o d c a n b e v e r y u s e fu l in th e c a s e o f th e B E M a p p lic a tio n f o r n u m e ric a l c o m p u ta tio n s o f n o n - ste a d y an d n o n - lin e a r h e a t c o n d u c tio n p ro b le m s .

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F ig . 6 . C o o lin g c u r v e s a t s e le c te d p o in ts o f d o m a in D R y s . 6 . K r z y w e s ty g n ię c ia w w y b ra n y c h p u n k ta c h o b s z a ru D

A C K N O W L E D G E M E N T

T h is re s e a rc h w o r k h a s b e e n su p p o rte d b y K B N (G r a n t N o 3 P 4 0 4 0 3 8 0 7 ).

R E F E R E N C E S

[1 ] M o c h n a c k i B . , S u c h y J . : M o d e llin g and S im u la tio n o f C a stin g S o lid ific a tio n . P W N , W a rs a w 1993.

[2 ] C u ra n D .A .S ., C ro ss M . and L e w is B .A . : S o lu tio n o f p a ra b o lic d iffe re n tia l eq u ation s b y the b o u n d ary elem en t m ethod u sin g d isc re tis a tio n in tim e. ,,A p p l. M a th . M o d e llin g ” , N o 4, 1980, 3 9 8 - 4 0 0 .

[3 ] B re b b ia C .A ., T e lle s J . C . F . and W ro b e l L .C .: B o u n d a ry E le m e n t T e ch n iq u es.

S p r in g e r - V e r la g , B e r lin 1984.

[4 ] S ic h e rt W .: B e re ch n u n g v o n in statio n a ren th erm ish en P ro b le m e n m itte ls d e r R andelem ent- m eth o d e, E rla n g e n : 1989.

[5 ] M a jc h rz a k E . and W ite k H .: A n a ly s is o f C o m p le x C astin g S o lid ific a tio n U s in g C o m b ined B E M . „ S o lid ific a t io n o f M e ta ls an d A llo y s ” , V o l. 18 (1 9 9 3 ), 1 1 3 - 1 2 0 .

[6 ] T e p lo fiz ic e s k ije s v o js tv a ve sc e s tv, S p ra v o c n ik , N au k a, M o s k v a 1960.

R e ce n ze n t: p ro f. d r hab. in ż . W .N o w a c k i

W p ły n ę ło d o R e d a k c ji w g ru d n iu 1994 r.