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EngineeringAnalysi~withBoundary Elements12 (1993) 293-303 1994 Ehevier Science Limi tedPrintd in Great Britain. All rights roservedE L S E V I E R 0 9 5 5 - 7 9 / 9 4 / $ 7 . 0 0

Boundary e lement analys i s for ax i symmetr ic heatconduct ion and thermal s tress in s teady stateShu Yao Long, Xing Cheng Kuai , Jun Chen

Department of Engineering Mechanics, Hum an University, 410082, People's Republic of C hina&

C. A. BrebbiaComputational Mechanics Insti tute, Ashurst Lodge, Ashurst, Southampton, UK, S04 2AA

(Received 12 March 1992; revised version received 3 November 1993; accepted 4 November 1993)

In this paper, axisymmetric heat conduction and thermal stress problems withthree types of boundary conditions are analysed by the boundary elementmethod. The temperature and thermal stress fields for the piston of a diesel engineare calculated using triangular finite elements and constant boundary elements,respectively, and the two results agree. However, BEM needs fewer data, lesscomputer time (about one-sixth that of FEM) and storage volume. Theadvantages of BEM are sufficientlydemonstrated.Key words:Boundary elements method, finite elements method, heat conduction,elliptic integral, heat stress.

1 INTROD UCTION BEM needs fewer data, less computer time (about one-sixth that of FEM) and storage volume. Advantages

Heat conduction and thermal stress problems is one of of BEM are sufficiently demonstrated.the most importa nt subjects in engineering and technol- Routines for the above-mentioned problem haveogy, for example, for analyses of pressure vessels of an also been developed. The numerical example includedatomic reactor, the cylinder and piston o f a diesel demonstrates the computat ional accuracy and reliabilityengine and so on. Several papers have so far been pub- of the program.lisbed for heat conduction and stress problems by theboundary element method. 6-13 However, there are fewinvestigations for axisymmetric heat conduction and 2 GOVERNING DIFFEREN TIAL EQUATIONSthermal stress problems. AND BOUNDARY CONDITI ONS FOR HEAT

In this paper, axisymmetric heat conduction including COND UCTI ONthree types of boundary conditions and thermal stressproblems are analysed by the boundary element Let us consider an isotropic axisymmetric body whichmethod. In order to obtain better evaluation accuracy, occupies a domain f~ surrounded by the boundary F.the complete elliptic integrals are approximated by poly- The variable, i.e. temperature T(r , z ) of the heat conduc-nomial expressions; the singular integrals are humeri- tion problem under consideration in f~ should satisfy thecally calculated using the cubic non-linear polynomial following governing differential equation: ~3transformation at singular points. Therefore, the uni-fied standard Gaussian integral scheme can be used; k V 2 T + p r Q = 0 in f~ (1)discontinuous and partly discontinuous elements are where k denotes the material conductivity constant; andemployed to deal with boundary normal discontinuity, p is the material density. Throughout this study theAt the end of the paper, temperature and thermal stress material constants are assumed to be constant in spacefields for the piston of a diesel engine are computed and time; Q = Q(r , z ) is the intensity of the internalusing FEM and BEM, and results obtained from the heat source in ~, the domain under consideration; andabove two methods are in good agreement. However, V 2 the Laplace operator. No coupling of stress and

2 9 3

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2 9 4 S . Y . Long , X . C . Kuai , J . Che n , C . A . Brebb ia

I ie m p e r a t u r e i s a s s u m e d , a n d o n l y t h e s t a t i o n a r y c a s e i s = q(x) T* (~ , x )d~ o(x) r ( x ) d ~ ( x )s tudied . ~+~2T h e b o u n d a r y c o n d i t io n s f o r th e h e a t c o n d u c t i o n e q n I Ii1 ) a r e s u c h t h a t : + ~ T f T " ( ( , x ) d ~ o ( x ) r ( x ) d f ' ( x )T = ~P on F t (2) ~3

k O T I p Q ( x ) [ 2 ~ T , ( ~ , x l r ( x l d a ( x ) (6 /q = - ~ n = o n F 2 (3 ) + h k J0q = h ( T - T f ) o n F 3 ( 4 ) w h e r e ~ i s t h e g e n e r a t i n g b o u n d a r y c o n t o u r w h i c h i s t h e

i n t e r s e c t i o n o f F w i t h t h e r + - z s e m i p l a n e ; a n d ~ i s t h ew h e r e F = I~ l q - 1~2 d - 1 ~3 ; T f d e n o t e s t h e a m b i e n t t er n - d o m a i n s u r r o u n d e d b y t h e b o u n d a r y c o n t o u r ~ .p e r a t u r e ; h i s t h e h e a t t r a n s f e r c o e f fi c ie n t . 2P i s t h e p r e - W r i t in g t h e t h r e e - d i m e n s i o n a l f u n d a m e n t a l s o l u t i o ns c r i b e d t e m p e r a t u r e ; q t h e o u t - f l u x , O T /O n = V T . n ; n i n c y l i n d r ic a l c o o r d i n a t e s :t h e o u t w a r d u n i t n o r m a l ; a n d # a p r e s c r i b e d i n - fl u xs c a la r . T h e t h i r d t y p e o f b o u n d a r y c o n d i t i o n a p p e a r s T * ( ~ ,x ) - 1w h e n h e a t i s l o s t o r g a i n e d f r o m t h e s u r r o u n d i n g m e d i a 4 7rr(~ , x )b y c o n v e c t i o n . A s c a n b e s e e n f r o m e q n s ( 3 ) a n d ( 4 ), w h e n 1~ = 0 o r h = 0 t h e h e a t - i n s u l a ti n g b o u n d a r y c o n d i t i o n i s =ob ta i ne d , a nd w he n h - -~ o in eq n (4) T = Tf . d r 2 ~) + r 2 x) - 2r (~) r (x) cos [~o ~) - ~ (x) ] + [z (~) - z (x) ] 2

E q u a t i o n s ( 1 ) - ( 4 ) u n i q u e l y d e f i n e t h e p r o b l e m . ( 7 )t h e a x i s y m m e t r i c f u n d a m e n t a l s o l u t i o n c a n b e c a l c u -l a t e d e x p l i c i tl y i n t e r m s o f t h e c o m p l e t e e l l i p ti c in t e g r a l

3 F O R M U L A T I O N O F T H E I N T E G R A L o f t h e f i r s t k in d K ( m ) a s :E Q U A T I O N F O R T H E H E A T C O N D U C T I O N I 2* ( ~ , x ) = T * ( ~ , x l d ~ ( x )T h e b o u n d a r y i n te g r al e q u a t i o n f o r h e a t c o n d u c t i o n c a nb e f o r m u l a t e d u s i n g t h e w e i g h t e d r e s i d u a l t e c h n i q u e : ~ K ( m )

- ( 8 )J J r h T(x) 7r (a+ b) ' /2( ~ ) T ( ~ ) + T ( x ) q * ( ~ , x ) dr(x) + ~r 3 i n w h i c h :

P

r * ( ~ , x ) d r ( x ) = J q ( x ) T * ( ~ ,x ) d r (x ) rn = 2 b / ( a + b )P~ +1"2J r h T f T * ( ~ , x ) d I ~ ( x ) a = r 2 ( ~ ) + r 2 ( x ) + [ z( ~ ) - z ( x ) ] 2+ 3~

b = 2 r ( ~ ) r ( x ) ( 9 ) I n ~ ~ ( x ) T * ( ( , x ) d rY ( x) ( 5 ) T h e r a n g e o f v a r i a t i o n o f t h e p a r a m e t e r m i s 0 ~< ~ ~< 1 .U n l i k e t h e t w o - a n d t h r e e - d i m e n s i o n a l c a s e s , t h e a x i -i n w h i c h I" = I'~ + r ~ + r~ ; a n d c i s t h e v a l u e t h a t c a n b e s y m m e t r i c f u n d a m e n t a l s o l u t i o n c a n n o t b e w r i t te n a sd e t e r m i n e d o n l y f r o m t h e g e o m e t r ic a l p r o p e r t i e s o f th e s i m p l y a fu n c t i o n o f th e d i s t a n c e b e t w e e n t w o p o i n t s :b o u n d a r y . F o r a s m o o t h b o u n d a r y p o i n t c = 0. 5 an d i t a l s o d e p e n d s o n t h e d i s ta n c e o f t h e p o i n t s t o t h e a x isf o r a n in t e r n a l p o i n t w e c a n p u t c = 1. ( d e n o t e s t h e o f r e v o l u t i o n .

s o u r c e p o i n t ; a n d x t h e f i n d p o i n t i n t h e d o m a i n f~ o r T h e n o r m a l d e r i v a ti v e o f t h e f u n d a m e n t a l s o l u t i o no n t h e b o u n d a r y r . I n th e a x i s y m m e t r ic c a s e , f o r a a l o n g t h e b o u n d a r y c o n t o u r f ' is g i v e n b y :c o n v e n i e n t r e p r e s e n ta t io n o f b o u n d a r y a n d i n t er n a lvalueSfollowing.aylindriCalowingcoordinate systemindependenceWille u s e d i n 1 { 2 _ ~(x ) [( r~ ( ( ) _ r 2 ( x )t h e t o t h e o f a ll O * ( ( ' x ) - ~ r ( a + b ) ~ / ~b o u n d a r y a n d i n t e r n a l v a l u e s i n t h e a x i s y m m e t r i c c a s eo n t h e c o o r d i n a t e ~o, e q n ( 5 ) c a n b e i n t e g r a t e d a l o n g + ( ~ ( ( ) - ~ ( x ) ) ~ ) E ( ~ ) / ( a - b ) - K ( ~ ) ] n ~ ( x )t h e r i n g d i r e c t io n o f ~o w i t h r e s p e c t t o t h e f u n d a m e n t a ls o l u t i o n : - t ~ ( ( ) - ~ ( x ) ~~ _ b . e ( , , , / n . ( ~ ) ~ ( 1 0 )

I 2( ( ) T( ( ) + T(x ) q" (~ , x ) d~o x) r ( x ) d ~ ( x ) w h e r e E ( ~ ) i s t h e c o m p l e t e e l li p ti c i n t e g ra l o f t h e s e c o n dk i n d .I r h r ( x ) l ~ S u b s t i t u t i n g e q n s (8 ) a n d ( 1 0 ) i n t o e q n ( 6 ) y i e ld s

+ ~ J 0 T * ( ~ , x ) d g ( x ) r ( x ) d ~ ( x ) t h e f o l l o w i n g b o u n d a r y i n t e g r a l e q u a t i o n f o r th e

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B E a n a l y si s o r a x i s ym m et r ic t em p er a t u r e a n d s tr e s s i e l d s 295a x i s y m m e t d c h e a t c o n d u c t i o n : . F.e~

c(~)T(~) + I~ T ( x ) q * ( ~ , x ) r ( x ) d ~ ( x ) + I ' ~ - ~x T ( x ) ~ ' * ( C x ) r ( x ) d ~ ( x ) = [ q ( x )~ ' *( C x )JP +[ '2

I r h r f r ' ( ( , x ) r ( x ) d ~ ( x )r ( x ) df ' (x ) + -~ ~+ I f i pQ k(x) ~ . . (~, x) r (x) d ~( x) ( 11 )

t i e r

4 B O U N D A R Y I N T E G R A L E Q U A T I O N O F T H ED I S PL A C E M E N T A N D S T R E S S F O R T H EA X I S Y M M E T R I C C A S EW e know f r om R e f . 1 , s t a r t i ng w i th t he t h r e e -d i m e n s i o n a l f u n d a m e n t a l s o l u t io n , t h a t o n e c a n o b t a i nthe i n t e g r a l e qua t ion ho ld ing a t a n a r b i t r a r y po in t ~ i nf L The n , t a k ing the l im i t ing p r oc e s s i n wh ic h po in t ~a pp r oa c he s t he bou nda r y po in t a nd e va lua t ing s ingu la r F ig. 1 . P ro f il e o f t he p is ton .in t e g r a ls , we c a n a r r ive a t t he f o l low ing bou nda r yin t e g r a l e qu a t ion f o r t he a x i sym m e t r i c al body : f+ 2~r j h fi/~ ~, x ) p j ( x ) r ( x ) d ~ ( x )

c ~ j( ~ )u j( ~) + 2 7 r p q ( ~ , x ) u j ( x ) r ( x ) d r ( x )+ 27r ~ ( ~ , x ) b j ( x ) d ~ ( x ) ( i , j = r ,z )

= 27r L ~,;.(~,x ) p : ( x ) r ( x ) d r ( x ) (12)l 1

101 103 105/ 102 / 104 / 106~ S l I ~ o 1 . 1 r ~ 7 1 ~ / 5 ~ 5 0 , ~ 4 2 4 1 ~. . . . ~ . ~ . ~ - - ~ r ~ r ~ ; . . . . . . . . . . . . ~ 7 . ~ . ~ . . ~ - - -7 t - I

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I / 1 - 3 7 7 5 I ~8 6 # . . . # 1 ( b ) ~ - ~ 3 1 .3 2 ~6 1 - ~ 7 ~ 067 - ~ 7~ .5 301 .287 - 93 31.3 51.5~ - 1 ~ 3 1 ~ 5 2.5101 31.3 2 7102 - 107 31.3 37.7

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: ,H = 3 5 7" C / _ ~ ~ ' ~ H ,,, 360C " - IL = 45"C L = 43"C - - I/ ~

(a) (b)F i g . 3 .

wh e r e the f a c to r 2~ r r ( x ) a p pe a r s i n the in t e g r a ls be c a use the r m a l l oa d ing due to r o t a t ion a b ou t a f ixe d a x i s o rin t e g r a t ion wi th r e spe c t t o ~ wa s a l r e a dy a c c om p l i she d c e n t r i f uga l l oa d in the a x i sym m e t r i c c a se, dom a in in t e -a nd , c onse qu e n t ly , ~ a n d ~ c o r r e sp ond to a two- g r a l s ha ve to be c om pu te d , bu t t he se c a n be su i t a b lyd im e ns iona l r e g ion on ly . P i j a nd u ij c o r r e s p o n d t o t h e t r a n s f o r m e d i n to a b o u n d a r y i n te g r al b y t h e G a l e r k i nf u n d a m e n t a l s o lu t io n s o f t h e a x i s y m m e t r ic p r o b l e m , v e c t o r c o r r e s p o n d i n g t o t h e a x i s y m m e t r ic f u n d a m e n t a lNo t i c e tha t t he a bov e e qua t ion i s a l so va l id f o r i n t e r io r so lu t ion . 1'~2po int (x E f~) i f c~ j(~) = ~ i j. W hen the te rm involving the bod y in tegra l in eqn

I t i s wo r th m e n t ion ing tha t , i n c on t r a s t w i th two- a nd ( 12 ) is su i t a b ly t r a ns f o r m e d in a b oun da r y in t e g ra l ,t h r e e - d im e ns io na l c a ses , c om pu ta t ion o f t he c oe f fi c ie n t t he bou nda r y in t e g r a l e qua t ion invo lv ing the r m a l a ndc ij (~ ) t oge the r w i th the a s soc ia t e d p r inc ipa l va lue in t e - c e n t r i f uga l l oa ds be c om e s :g r a l c a n n o t b e c a r r i e d o u t b y r i g i d - b o d y m o v e m e n t s i n gt h e d i r e c ti o n o f r . H o w e v e r , c o m p u t a t i o n o f t h e c o e f - c ~ 7 ( ~ ) u j ( ~ ) 2 ~ | ~ ( ~ , x ) p j ( x ) r (x ) d ~ ( x )f i ci e n t c~ . ( ~) c a n be m a de by the c o r r e spon d ing a na ly t i c J~e x p r e s s i o n . 1,3 p- 2 ~ ~ P ~ / ~ , x ) ~ ( x ) r ( ~ ) d r ( x )W e c a n o b t a i n t h e d i s p la c e m e n t f u n d a m e n t a l s o lu - ~t ions u~j f r om R e f s 1 , 4 , 10 - 12 f o r t he a x i sym m e t r i c c a se. ~ g

The e xpr e s s ions f o r t he c o r r e spon d ing s t re s ses c a n be - J~ P y T d ~ ( x ) - J ~ Q ~ T ,~ n ~ d~ ' (x)o b t a i n e d f r o m t h e s t r e s s - d i s p l a c e m e n t r e l a t i o n s a n dHo oke ' s La w in c y l ind r i c a l c oo r d ina te sys te m s . f P ~ d~ ' (x ) (13)I f t he r e i s t he bod y f o r c e c a use d by a s t e a dy s t a t e Jl'-

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L :,, - 1 2 0 M P a 4 5 . 6 L f fi - 1 2 3 M P a 4 5 . 6

. . . .

( a ) ( b )

Fig. 4 .

i n w h i c h e x p r e s s i o n s o f p / T , Q /~ a n d p /c a r e g i v e n i n i n w h i c h S i j k , D I j k , E i ~ . , i ~ and V~.~c a n b e o b t a i n e d .R e f . 1 . T h e y w i l l n o t b e l i s te d i n d i v i d u a l l y o w i n g t o t h eO n c e al l v a l u e s o f b o u n d a r y d i s p l a c e m e n t a n d tr a c - l i m i t a ti o n o f s p a c e.t i o n s a r e k n o w n , i n t e r n a l d i s p l a c e m e n t s a n d s t r e s s e s

c a n b e c o m p u t e d b y u s i n g e q n ( 1 3 ) w i t h c i j = ~ i j a n dt h e f o l l o w i n g i n t e g r a l e q u a t i o n :O ' i j ( ~ ) : 2 ~ r [ ~ S i j k ( ~ , x ) p k ( x ) r ( x ) d f ' ( x) 5 I M P L E M E N T A T I O N O F N U M E R I C A L

dx C O M P U T A T I O N- 2 ~ r I t " D i jk ( ~ ' X ) U k ( X ) r ( x ) d ~ ( x ) F o r t h e n u m e r i c a l s o l u t i o n o f b o u n d a r y i n t e g r a l e q u a -

t i o n s ( 6 ) a n d ( 1 3 ) , i t i s a s s u m e d t h a t t h e b o u n d a r y ~ 'Jt"[E i ~ T d ~ , ( x ) _ J t" [F i ~ T , n k d ~ ( x ) w h i c h s u r r o u n d s t h e d o m a i n ~ i s d i s c r e t i z e d i n t o t h eN b o u n d a r y e l e m e n t . E q u a t i o n s ( 1 1 ) a n d ( 1 3 ) a r e n o wa p p r o x i m a t e d b y t h e f o l l o w i n g d is c r e ti z cd e q u a t i o n s ( it

+ f V ~ d ~ ' ( x ) ( 1 4 ) i s a s s u m e d t h a t t h e i n t e n s i t y o f t h e h e a t s o u r c e i n e q nJr" ( 1 1 ) i s , f o r t h e m o m e n t , i g n o r e d ) :

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29 8 S. Y . Lon g, X. C. Kuai , J . C hen, C. A. Brebbia

' ~ H - 8 . 3 M P a / ~- 8 .7 MP a I1"1 L - -268 MP a r " lL - , -273 MP a I ' r ' l H -44 .2 H-111, - ~ . . ! ~

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j=l y=lN+ ~N ipjh~ T(x)~ '*(~,x)r (x)d~3(x) - j = l IP; QTT'knkd ~(x)

j = l

N j ~ jv I~, + ~ P~ d~(x) (16)= ~ q ( x ) t * (~ , x ) r (x ) d ~ l + 2 ( x ) j = lj = l ~ u n kn o w ns o n t he b o ~ d a ~ c l i e n t s , i . . t he~ ~ ~ t em ~r a tu r e s T o r t tu x es q i n eq n ( 1 5 ), an d th e d i sp l a~-+ ~ TrY* ( ( , x ) r (x ) d ~ (x ) (15) mer i ts u~ or t rac t ions p~ in eqn (16) , a re approx im ated by~ t h e ir n o d a l v a l ~ a n d s h a ~ f u n ct io n s , t h a t is :~ ~~ ~ r , r (n~ = ~ T ~ , q (n ) = ~ q ~ (17)~ o(~ )" ~(~ l = ~ ~ ( ~ , ~ l ~ ( ~ l ~ ( ~ l a ~ ( ~ l ~ ~

]~1 ~ ~.,(~) ~ : , ~,(~) ~ : (18)N- 2 ~ I :~(~,xl.:(xl~(x)d~ (x) ~=~ ~

:= ~ r : ( i = r , z )

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B E analysis fo r axisymme tric temperature and stress fields 29 9

" i- 134 MPa 75. ~ H - 143 MPa 75.9L = 7.3 MPa L - 8.5 MPa25 81 ~) 42.5 25.! 25.842 .5 ~ 2.~_.~. 42 .5

(~) (b)F i g . 6 .

i n w h i c h m , n d en o t e t h e n o d a l n u m b er s o f e l emen ts ; an d w h e re [ c] i s a d i ag o n a l m a t r i x w h i ch co n t a i n s h e a tr / i s t h e l o ca l co o rd i n a t e an d v a r i es b e t w een -1 an d + 1 co n d u c t i o n an d t r an s f e r co e ff ic ien ts ; {T f} i s a n o d a lo v e r t h e e l emen t . T h e g l o b al co o rd i n a t e s (r , z ) a r e a ls o co l u m n v ec t o r o f t h e am b i en t t emp era t u r e ; {B} is a l soex p re s sed b y i n t e rp o l a t io n fu n c t io n s : a co l u m n v ec t o r p ro d u c ed b y b o d y fo r ce s ; an d [ / /] , [(~ ]~, q and [HI , [G] den o te the in f luence ma t r i ces in the usualr ( r /) = E ~ o ~ r~ , z (~ 7 ) = ~ - ~ ~ z . r ( 1 9 ) m a n n e r .

-r=l - r= l By app ly ing the p rescr ibed bo un da ry cond i t ions o fthe p rob lem , eqns (21) and (22) can be reo rder ed inT h e b o u n d a r y e l emen t s h av e t o b e t r an s fo rm ed f ro m t h e s u ch a w ay t h a t t h e f in a l s y s tems a r e o b ta i n ed :g lobal coord ina te sys tem in to the loca l one as ~ arefun ct ion s of the local co ord ina te ~7, i .e . : [ .~]{~} = {P } (23)d ~ " = I J I d o (20) [A]{x} = {F } (24)

He re IJ I i s the va lue o f the Jacob ian de term inan t , wh ere [~] , [A] are fu l ly pop ula ted mat r i ces ; and {~}, {x}W h en a l l t h e n o d es a r e t ak en i n t o co n s i d e ra t io n , eq n s a r e v ec t o r s co n t a i n i n g a ll t h e b o u n d a ry u n k n o w n s .(15 ) an d (16 ) p ro d u c e t h e r e s p ec t iv e s y s tem o f eq u a t i o n s N o t i ce t h a t f o r t h e h ea t co n d u c t i o n p ro b l em w i t hw h i ch can b e r ep re s en t ed i n ma t r i x fo rm a s: b o u n d a ry co n d i ti o n s o f t h e D i r i ch le t t y p e , t h e( [ ~ ] + [ c ] [ d ] ) { T } = [ d ] ( { Q } + [ c] { Tf } ) (2 1) u n k n o w n s a r e t e m p e r a t u r es o n t h e b o u n d a r y ; c o n -v e r s e l y , f o r N eu man n b o u n d a ry co n d i t i o n s , t h ey a r e[H]{u} = [G]{p} + {B} (22) f luxes . In a Ne wto n (o r Ro bin) p rob lem , there i s a

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30 0 S. Y. Long, X. C. Kuai, J . Chen, C. A. Brebbia

. ., : 7H - 6 2 . 8 MP a H , , 5 7 . 9 MP a " 71

o .

( a ) ( b )F i g . ~.

l i nea r co m b i n a t i o n o f b o t h t em p e r a t u r e an d f lu x, w h i l e ( i) T w o k i n d s o f t h e co m p l e t e e l l ip t ic i n teg r a lsi n a m i x ed p r o b l em t h e r e a r e s o m e o f th e t em p e r a t u r e K( m ) , E ( m ) a r e a p p r o x i m a t e d b y p o ly n o m i a lan d o t h e r s o f t h e f lu x. I n a p u r e N e u m an n p r o b l em i t ex p r e ss i o n s. T h e i r p o l y n o m i a l ex p r e s si o n s a r ei s n eces s a r y t o ad d o n e co n d i t i o n b ecau s e t h e s o l u t i o n g i v en i n R e f . 5 .i s unde termine d . Us ual ly th i s i s don e by f ix ing the va lue ( ii ) W hen source po in ts loca te w i th in the in tegra tedof the po ten t ia l in a po i n t and in th i s sense i t can be sa id e lement , there ar e s ingu lar in tegra l s in the ca lcu-tha t the p rob lem has been t r ans fo rm ed in to a mixed la t ion o f in f luence mat r ices [ /~] , [ d ] , [HI , [G]t y p e o n e . an d [ B ]. I n o r d e r t o p r o v e i n t eg r a ti o n accu r acyE q u a t i o n s ( 2 3) an d ( 2 4) can b e s o l v ed b y em p l o y i n g an d t h a t t h e u n i fi ed G au s s i an i n teg r a l s ch em et h e s t an d a r d G au s s i an e l i m i n a t io n m e t h o d . O n ce a l l co u l d b e em p l o y e d , t h e cu b i c p o l y n o m i a l n o n -t h e v a l u e s o n t h e b o u n d a r y a r e k n o w n , o n e co u l d l in ea r t r an s f o r m a t i o n i s ap p l i ed a t a s i n g u la rca l cu l a t e t h e v a l u e s a t an y i n t e r io r p o i n t s u s i n g eq n p o i n t . T h i s t r an s f o r m a t i o n can au t o m a t i ca l l y b e(11) w i th cij(~) = ~iy and eqn (14). conc en t r a te d m ore Ga uss ia n po in ts in the v icin-

The ca lcu la t ion s o f mat r ice s [ / / ] , [ d ] , [HI , [G] and i ty o f the s ingu lar po in t ; therefo re , cons ide rab ly{ B } co n t a i n t w o k i n d s o f t h e co m p l e t e e l li p ti c i n teg r a ls h i g h accu r acy can b e o b t a i n ed . 2an d s i n g u la r i n teg r a ls . I n o r d e r t o o b t a i n b e t t e r c a l cu la - ( ii ) D i s co n t i n u o u s o r p a r t l y d is co n t i n u o u s e l em en t st i o n a l a ccu r acy , t h e f o l lo w i n g i m p o r t an t m eas u r e s a r e a r e em p l o y ed t o d ea l w i t h t h e b o u n d a r y n o r m a lado p ted : d i scon t inu i ty . Fo r th i s r eason on ly , there i s a

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B E a n a l ys is f o r a x i s y m m e t r i c t e m p e r a tu r e a n d s t re s s f i e ld s 301

. . 8 o 3 . . 9 . 1 M P .-252 MPa H L = -2 ~ MPaL = ~ - ~ H

. ~ 7.9 7 9

(a) (b)] ~ i g . $ .

smooth boundary at the nodes; thus, c = 1/2 (eqn the cooling water is in the major proportion, and the(11)) and ci j= 6i j (eqn (13)). The analytic quantity of heat transferred from the petticoat partcalcula tion o f coefficients c (~) or cij(~) is and the wrist pin pedestal is limited (abou t 15% of theavoided. ~4 total quantity of heat absorbed by the piston). There-

fore, error due to ignoring heat transfer of the wrist6 EXA MPL E pin pedestal is quite small.

Temperatures Tf of the media and their heat transferTemperature and stress fields for the piston of a diesel coefficients h in the outer surroundings of the piston areengine are analysed as a practical example in engineer- usually known, and their values are determined byexperimental or semi-experimental formulae in a designing in the present paper. Strictly speaking, the geo- of the diesel engine. The power of a single cylinder ismetric shape of the piston and its distributions of assumed to be 200 HP and the rotations per minute astemperature and stress are not completely axisymmetri-cal, but results obtained in the paper show tha t consid- 350. The piston, whose maximum diameter is 350ramering it as an approximate ly axisymmetric problem here and tota l length is 640 mm, is assumed as a non-liquidhas less influence on the distribut ions of the temperature cooling one and is made o f aluminium casting; otherand stress of the piston. For a non-liquid cooling physical parameters of the piston are given as follows:piston, heat transfer mainly depends on the piston rings, Heat conduction coefficient k = 155 kcal m -2 h -~ C -~especially on the first piston ring via which the quantity Linear thermal expansion a = 2.3 x 10-5C -~of heat transfer red from the cylinder of a diesel engine to Strength trb = 200-250 MPa

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302 S. Y. Long, X. C. Kuai, J. Chen, C. A. Brebbia

~ . 5 .

~. , ' r 1

L = 5 . 9 M P a 5 7 ,7 1 } [ , ~ ] 2 8 . 2 L = 4 . 8 M P a " ~ 2 8 . 2/ / / 2

ia ) ( b )F i g . 9 .The pro f i l e o f the p is ton i s show ~ in F ig . 1 . A F E M p lo t ted in F ig . 3 . Com pared to the FF .M so lu t ions , theanalys is o f the heat conduct ion prob lem was car r ied agreements, in genera],excellent.

out by Kong and Wang, 13 employing 279 linear triangu- Isostresses of the maximal, the minimal principal andlar elements of which there were 172 interna l elements the Mises equivalent stresses produced by variat ionaland 196 nodes (Fig. 2(a)), and there were no boundary tempera tures are plotted in Figs 4-6 , respectively. Thecondit ions of the Dirichlet and Neumann types. Accord- agreements are also, in general, excellent, and differ-ing to the temperatures and heat transfer coefficients of ences between FEM and BEM cannot be distinguished.the outer media, the boundary condition of the con- We also plot the isostress of the maximal, the minimalvective heat transfer type is divided into eight groups; and the Mises equivalent stresses in Figs 7-9 under com-the parameters of each group are given in Fig. 2(b). binat ion of thermal loading and 100 pressures of fuelFor comparison purposes the BEM discretization gas. Notice that stresses computed here are producedemploys the same division as FEM, i.e. 107 cons tant at the moment of the working stroke, i.e. the maximalboundary elements (Fig. 2(b)), and temperatures and pressure in a combustion chamber. We can see fromstresses in internal points corresponding to FEM are the above figures tha t the thermal stress is much highercomputed. A FEM analysis for the stresses was not than that produced by internal pressure, i.e. the stresscarried out by Kong & Wang. ~3 We compute stress produced by internal pressure has less influence on thefields of the piston by using FEM and BEM, employing total stress of the piston and it improves the distribu-the same division as the heat analysis and taking the tion and the level of total stress of the piston.initial reference temperature as 20C. Fro m the above figures we can see that results

Isotherms of the piston of the marine diesel engine are obtained from FEM and BEM are in good agreement.

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B E a n a l ys i s f o r a x i s y m m e t r i c t e m p e r a t u r e a n d s t re s s f i e l d s 30 37 C O N C L U S I O N S 4 . B re bb ia , C . A . e d. Progress in Boundary Element Methods,Vols 1 & 2, Pentech Press, L ond on, 1980.5. Abramaw itz , M. & Slegum, I . A. Ha n d b o o k o f Ma th ema -T h e b o u n d a r y e l e m e n t m e t h o d h a s b e e n a p p l i e d t o tical Functions, Dover Publications, New York, 1965.a n a l y s e s o f t h e a x i s y m m e t r i c t e m p e r a t u r e a n d s t re s s 6 . T a n a k a , M . et al. Boundary element method appl ied tof i e lds . A typica l prac t i ca l pro ble m in engin eer ing was 2-D the rmoe last ic problems in s t eady and non-s teadyc o m p u t e d b y t h e c o m p u t e r p r o g r a m d e v e l o p e d i n t h is s ta te s, Engg Anal. , 1985, 2, 51 -7.s tudy . A ser i es of i so th e rm s an d i sos t res ses of a p i s to n 7 . Boll eus , L . & Tul lbe rg , O . Boundary e lement method

applied to two-dimensional heat conduction in non-a re g iven for prac t i ca l use . homogeneous media , Engg Anal. , 1985, 3, 44-50 .A l th ou gh the pres ent inves t iga t io n i s re s t r i c t ed to 8 . Anza , J . J . & Ala rcon , E . The e ffec t of boundaryco ns t an t bo un da ry e l em ent s , we can a l so co m pu te these condi tions in the numerica l so lu tion of 3-D the rmoe last icp r o b l e m s b y l i n e a r o r h i g h e r - o r d e r e l e m e n t s w i t h o u t a n y p ro b le m s, Engg Anal. , 1986, 3, 18-24.d i f f i cu l ty ; these resu l t s a re no t pres ente d he re ow ing to 9 . S ladek , V . & S ladek, J . Boun dary in tegra l equa t ionthe l im i ta t io n of space an d the go od resu l t s ob ta in ed method in two-dimensiona l the rmoe last ic ity, Engg Anal. ,1984, 3, 135-48.on ly by co ns tan t bo un da ry e l em ent . 10. Mayr , M. , Drexle r, W. & Kuhn, G . A semiana ly ti ca lboundary integral approach for axisymmetric elasticbodies with arbi t rary boundary condi t ions , Int. J. SolidsStructures, 1980, 10, 863-71.R E F E R E N C E S 11. Kerman idis , T. A numerical solut ion for axial ly symmetricelasticity problems, Int. J. Solids Structures, 1975, 11,1. Brebbia, C. A., Telles, J . C. F. & Wrobel, L. C. Boundary 493-500.Element Techniques, Springer-Verlag, Berlin, 1984. 12. Cruse, T. A., Snow, D. W . & Wilson, R. B. Numerical2. Tells, J . C. F. A self adap tive coo rdina te tran sform ation solutions in axisymmetric elasticity, Comput. Structures,for efficient num erical evalu ation of general bo un dar y 1977, 7, 445-51.element integrals, Int. J. Numer. Meth. Engg, 1987, 24, 13. Kon g, X. Q. & Wan g, Zh. P . Fini te e lement meth od959-73. applied to hea t transfer, Science Publishing Hou se, Beijing,3. Hartm ann, F . Com puting the C-matrix in non-sm ooth 1981 ( in Chinese).boundary poin t s , i n New Developments in Boundary 14. Brebbia, C. A. & Lo ng, S. Y. B oun dary element analysis ofElement Methods, ed. C. A. Brebbia, Butterw orths, plates using Reissuer's theor y, in Proc . 9 th I nt . BE M Co n f ,Lo nd on , 1980. Springer-Verlag, 1987, 3-8 .