A general purpose two-dimensional mesh generator

A general purpose two-dimensional mesh generator

Citation for published version (APA):

Beukering, van, L. H. T. M., Schoofs, A. J. G., & Sluiter, M. L. C. (1979). A general purpose two-dimensional

mesh generator. In R. A. Adey (Ed.), Engineering software : proceedings of the 1st international conference,

held at Southampton University, September 1979 Pentech Press.

Document status and date:

Published: 01/01/1979

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A GENERAL PUR}OSE TI{O-DÏMENSIONAL MESH GENERATOR S c h o o f s A . J . G . , v a n B e u k e r i n g L . H . T h . M ' , S l u i t e r t ' 1 ' L ' C ' Department of Mechanical Engineering

S i n d h o v e n U n i v e r s i t y o f T e c h n o l o g y , T h e N e t h e r l a n d s '

INTRODUCTION

Cornposing and checking input for finite element programs is

.r"ty t"tórlt intensivel this is particular true for the

divi-s i o n o f r h e a r e a t o b e divi-s t u d i e d i n t o e l e r n e n t divi-s ' I n t h e p a s t

many programs, called mesh generators, have been developed in

o r d e r È o a u t o m a t e t h i s j o b . A s u r v e y o f t h e s e p r o g r a m s i s g i

-v e n b y B u e l l W . R . a n d B u s h B . E . ( 1 9 7 3 ) ' T h i s p a p e r d e a l s w i t h

. r""íl generaror for two-dimensional areas; the principal

charactáristics of this nesh generator, named TRIQUAMESH' are:

l . a u s e r o r i e n t e d i n p u t l a n g u a g e w i t h d e b u g g i n g a i d i s p r o

-vided; the user will only have to supply siurple cornposable

in-p u t d a t a .

i. Uottr single and rnultiple coherent two-dimensional areas

with a

"orp1.* geometry àan be divided

into triangular and/or

q u a d r i l a t e r a l e l e m e n t s . 3 . e a s y s p e c i f i c a t i o n o f t h e m a g n i t u d e o f t h e e l e m e n t s ' 4 . s u U s t r u c t u r i n g f a c i l i t i e s h a v e b e e n i n c o r p o r a t e d ' 5 . t h e s h a p e o f i h e g e n e r " t e d e l e m e n t s i s o p t i m i s e d ' 6 . t h e t " . | t g . r , " . . t o r h a s s o n e p o s s i b i l i t i e s t o r e d u c e t h e b a n d w i d t h o f t h e a s s e m b l e d s t r u c t u r a l m a t r i c e s ' 7 . t h e o u t p u t o f t h e m e s h g e n e r a t o r c a n b e u s e d d i r e c t l y a s a p a r t o f t h à i n p u t f o r t h r e á f i n i t e e l e m e n t p r o g r a m s , i n c l u d i n g

ASKA and }4ARC.

A f t e r i n t r o d u c t i o n o f s o m e b a s i c c o n c e p t s , t h e m e t h o d u s e d i n

TRIQUAMESH will be dealr \íith. Afterwards the use and

possibi-l i t i e s o f T R I Q U A M E S H w i l l b e i l l u s t r a t e d b y m e a n s o f s o m e e x

-a m p l e s . ,

BASIC CONCEPTS

T h e a r e a t o b e d i v i d e d G c a n b e d i v i d e d i n t o n s s u b a r e a s G , r

. . . . G i n o r d e r t o s p e c i f y e l e m e n t a n d m a t e r i a l p r o p e r t i à s

and tonflefine the subslructuring of the area G. It is dernanded

t! t h a t s u b a r e a s a r e s i m p l e c o h e r e n t ; n - f o l d c o h e r e n t a r e a s c a n b e m a d e s i m p l e c o h e r e n t b y r n a k i n g a t l e a s t n - l c u Ë s . T h e c o n t o u r ^ c ^ - ^ - ^ ^ r : ^ ^ _{s u b a r e a G . h a s C a s c o n t o u r a n d t h e o v e r a l l} U L é l l d l E d U I n ! , _{r *} c c o n t o u r o f a l l s u b a r e a s i s C ( f i g i l . r . j

G

_{w w}

(83"

F i g u r e l . A n a r b i t r a r y t w o - d i m e n s i o n a l a r e a ( a ) a n d a d i v i s i o n t h e r e o f i n s i m p l e c o h e r e n t s u b a r e a s ( b ) . I n G a n u m b e r o f s o - c a l l e d b a s i s p o i n t s i s f i x e d , b y m e a n s o f n u m b e r s a n d c o o r d i n a t e s . T h e s e p o i n t s w i ! 1 s u p p l y a b a s i s f o r t h e g e o m e t r i c a l d e s c r i p t i o n o f c o n t o u r C a n d f o r t h e d e t e r m i -n a t i o -n o f o t h e r u s e r w i s h e s , f o r e x a m p l e t h e d e s i r e d e l e m e -n t s i z e . W i t h t Í ^ 7 o o r m o r e b a s i s p o i n t s o r i e n t e d e l e m e n t a r y c u r v e s c a n b e d e f i n e d , f o r e x a m p l e s t r a i g h t l i n e s , a r c s e t c . ( f i g . 2 a ) .

A contour part (ident.ified by a number) is a non-branched

coup-l i n g o f e coup-l e m e n t a r y c u r v e s a n d h a s a n o r i e n È a t i o n . T h e g e o m e t r i c d e s c r i p t i o n o f t h e c o n E o u r p a r t c o n s i s t s o f t h e c o u p l i n g o f d e s c r i p t i o n s o f t h e c o n s t i t u e n t e l e m e n t a r y c u r v e s ( f i g . 2 b ) . ( b ) ( a )

l

i i ,

l

i l i ,'y't.,

{-.<--i

a . e 2 . (b) (c) 3 , 2 1 , - 3 ( a ) F i g u r e 2 . D e s c r i p t i o n o f ( b ) a n d s u b c o n t o u r s ( c ) . e l e m e n t a r y c u r v e s ( a ) , a c o n t o u r p a r t S u b c o n t o u r C c o n s i s t s o f t h e c o u p l i n g i n a c l o s e d c u r v e o f o n e s o r m o r e c o n t ó u r p a r t s . F o r r e a s o n s o f u n i v o c a l i t y , C ^ i s d e s -c r i b e d b y d e n ó t i n g t h e n u r n b e r s o f t h e a n t i - -c l o -c k w i s e - s e q u e n -c e o f j o i n i n g c ó n t o u r p a r t s . I f a c o n t o u r p a r t i s m e t i n a d i r e c -t i o n o p p o s i -t e Ë o i -t s o r i e n -t a -t i o n , t h e n u m b e r o f t h i s c o n t o u r p a r t i s d e n o t e d n e g a t i v e l y ( t i g . _{2 c ) . A s u b s t r u c t u r e} i s d e f i n e d b y o n e o r m o r e s u b a r e a s . A 1 l s u b s t r u c L u r e s t o g e t h e r f o r m o n e s t r u c Ë u r e : t h e a r e a t o b e d i v i d e d G . T h e c o n c e p t s m e n t i o n e d a r e h i e r a r c h i c a l l y o r d e r e d . T h e b a s i s p o i n t s d e f i n e e l e m e n t a r y c u r v e s . T h e s e d e f i n e c o n t o u r p a r t s e t c . : 3 R L 4 4 CM 6,5

l o l d c o h e r e n t a r e a s c a n b e ; t n - l c u t . s . T h e c o n t o u r l s c o n t o u r a n d t h e o v e r a l l

) .

( b ) a r e a ( a ) a n d a d i v i s i o n

) .

s i s f i x e d , b y m e a n s o f w i | 1 s u p p l y a b a s i s f o r C and for the determi-1 e t h e d e s i r e d e l e m e n t iented elementary curves i n e s , a r c s e t c . ( f i g . 2 a ) . ) is a non-branched coup-r i e n t a t i o n . T h e g e o m e t r i c t s o f t h e c o u p l i n g o f t a r y c u ï v e s ( f i g . 2 b ) .

,'--\'-'-..

/

\

\

i----3 Í c, fs c, )

' _{\ l} - /

\__-,-í

2 - 7 C 1 : 3 , 2 C 2 : 1 , - 3 (c) r v e s ( a ) , a c o n t o u r p a r t i n a c l o s e d c u r v e o f o n e u n i v o c a l i t y , C _ i s d e s -a n t i - c l o c k w i s e o s e q u e n c e p a r t i s m e t i n a d i r e c

-number of this contour

A s u b s t r u c t u r e i s d e f i n e d

u r e s t o g e t h e r f o r m o n e he concepts mentioned are t s d e f i n e e l e m e n t a r y

Basispoints + elementary curves + contourparts -+

s u b a r e a s + s u b s t r u c t u r e s + s f r u c È u r e .

D u r i n g t h e d i v i s i o n o f a r e a G i n t o e l e m e n t s , u s e i s m a d e o f t h e

r o u g h ; e s s f u n c t i o n g 2 ( x , y ) . I t i s p o s t u l a t e d t h a t i n G , 9 2 í s

d i r á c t l y p r o p o r t i o n a l t o t h e d e s i r e d m a g n i t u d e o f t h e e l e m e n t

sides; the length .0 of any element side between the points (xl '

y , ) a n d ( x r , ! r ) t h e r e f o r e s h a l l h a v e t o b e t t t h e b e s t f i t o o ê s i b l e " :

-v = \ t g 2 ( x r , y , ) + g 2 ( x r , -v 2 ) ]. R r

The proportional constant RÏ has the

w i l l b e c a l l e d t h e s t a n d a r d e l e m e n t

f i n i t i o n o f t h e r o u g h n e s s c o n c e p t .

DESCRIPTION OF THE },IETHOD

( l ) dirnension of length and s i d e . E q u a t i o n I i s t h e d e

-Globaly TRIQUMESH has been developed as follows:

l . C h e c k i n g a n d m a n i p u l a t i o n o f i n p u t d a t a . *

2 . G e n e r a t i o n o f n o d a l p o i n t s o n t h e t o t a l c o n t o u r C '

3 , D i v i s i o n o f s u b a r e a s i n t o e l e m e n t s .

4 . P o s t - p r o c e s s i n g , s u c h a s : o p t i m i z a t i o n o f e l e m e n t s h a p e t

bandwidth reduction, transformation to elements with more nodal

F o i n t s a n d o u t p u t .

T h e a s p e c t s m e n t i o n e d a b o v e w i l l b e d e s c r i b e d s u b s e q u e n t l y '

Checking and manipulation of input dgta

e h a s b e e n d e v e l o P e d s o t h a t t h e

i n p u t c a n b e i n t e r p r e t e d s i m p l y a n d e l a b o r a t e t e s t s f o r e r r o r s

i n t h e i n p u t d a t a a r e p o s s i b l e . T h e p r o g r a m e x p e c t s t h e i n p u t

t o b e d e l i v e r e d b y $ e a n s o f p u n c h c a r d s o r b y u e a n s o f a f i l e

t o b e f o u n d o n a d i s k - u n i t . F r o m t h i s i n p u t , a r r a y s a r e d e t e r

-mined which r^ri11 serve as input parameters for the next steps. D u r i n g p r o c e s s i n g , Ë h e i n p u t i s a l s o c h e c k e d f o r s y n t a x a n d s e -m a n t i c e r r o r s . P o s s i b l e e r r o r -m e s s a g e s a r e f o r i n s t a n c e :

I T R T A X 3 2 , 5 )

* >>>> LEFT PARENTITESIS ENECTED * >>>> UNKNOI^ÍN SUBAREA @ v a l u e s f o r t h e r o u g h n e s s f u n c t i o n g 2 ( x , y ) f o r e a c h o f t h e b a s i s p o i n t s i n a r e a G . D o i n g s o , u s i n g a c h o s e n r o u g h n e s s b e h a v i o u r a l o n g t h e c o n t o u r ' t h e r o u g h n e s s o n t h e t o t a l c o n t o u r C ^ i s f i x e d . S c a r t i n g f r o r n t h i s , a n d t o -g e t h e r r , r i t h t h e u s e r -g i v e n s t a n d a r d e l e m e n t s i d e R I , t h e n o d a l p o i n t s t o b e g e n e r a t e d o n C ^ w i l l b e d e t e r m i n e d . B e c a u s e o f t h e a s s e m b l y o f a s u b c o n t o u r o u t o f c o n t o u r p a r t s ' w h i c h i n t u r n c o n s i s t o f e l e m e n t a r y c u r v e s , i t L / i 1 1 o n l y b e n e c e s s a r y t o e x -p l a i n t h e g e n e r a t i o n o f n o d a l -p o i n t s o n a n e l e m e n t a r y c u r v e . Consider an elernentary curve K with length I and curvilinear c o o r d i n a t e s , 0 s s < l ; f o r r e a s o n s o f s i r n p L i c i t y a n o n e s s e n

6

t i a l s i m p l i f i c a t i o n t o a s t r a i g h t _{l i n e i s m a d e . 0 n c u r v e K a}

n u m b e r o f b a s i s p o i n t s a r e d e n o t e d , numbered locally with j,

j = 0 , 1 , . . . r 8 , _{w h i c h d i v i d e L . h e c u r v e i n m p i e c e s , á r r a , i y}

means

-of the user given values gk; r a piecewise harrnonic rough_

n e s s f u n c t i o n i s d e f i n e d . ( f i g . 3 J : 8 k i * B k . i * , g k . - g k . * , s - s .

g k ( s ) = - #

* - t - *

. c o s

; ; i - l

- j + l - j

? r ; i = 0..m (2)

i Q X e

I

t

-S = 0

o k

i -

i

? s = S Z S : s 3 f u n c t i o n o n a n e l e n e n t a r y c u r v e . - s l F i g u r e 3 . H a r m o n i c r o u g h n e s s S t a r t i n g _{f o r n t h i s f u n c t i o n g k ( s ) n o d a l p o i n t s a r e g e n e r a t e d} o n c u r v e K , t h i s í s d o n e i . n t w o s t e p s : f i r s t t h e d e t e r m i n a t i o n

o f t h e n u m b e r o f n o d a l p o i n t s a n d subsequently the computation

o f t h e c o r r e c t l o c a t i o n . _{S u p p o s e t h a t n - l n o d a l p o i n t s w i l l}

h a v e t o b e g e n e r a t e d ( a n d t h e r e f o r e n element siáes) on the

e l e m e n t a r y c u r v e ( f i g . 4 ) . l I /n ê\

rffi_

f--'

I

s=0 F i g u r e 4 . N o d a l p o i n r s a n d S : S i - l S:Si S = l e l e m e n t s i d e s o n t h e c u r v e F r o m e q u a t i o n I f o r e l e m e n t s i d e i t h e f o l l o w i n g can p n b e d e -r i v e d : t i - t i - l t i * l - t i B k i * B k i _ _{t B k l * 1} - B k ' \ r T h i s c a n b e r n e t "as good as b y c o m p u t i n g n a s f o l l o w s :

- - r ) u

_{r r - ; ? . Í - - - T o s} r ( r b g K ( s , = l r 2 r . . . r n - l ) ( 3 ) p o s s i b l e " f o r a l l e l e m e n t s i d e s , ( 4 )

i

\

, --sk(s)

) i s m a d e . 0 n c u r v e K a r m b e r e d l o c a l l y w i t h j ,

i n m p i e c e s , a n d , b y piecewise harmonic

rough-l : _{Í ; i = 0 . . m ( 2 )} + l " j gk (s)

-\\

@

i

J s= s3 on an elementary curve. 1 p o i n t s a r e g e n e r a t e d f i r s t t h e d e t e r m i n a t i o n

equently the computation n - l n o d a l p o i n r s w i l l e l e m e n t s i d e s ) o n t h e I ^ l . Q / , i o n-l n S : 1 " e s o n t h e c u r v e P Q . f o l l o w i n g c a n b e d e -- l _{) ( 3 )} f o r a l l e l e m e n t s i d e s , ( 4 ) 7 A f t e r w h i c h n i s r o u n d e d o f f t o a n i n t e g e r i n a s u i t a b l e w a y . A s s o o n a s t h e c u r v i l i n e a r c o o r d i n a t e s s r . . . s - _ , a r e k n o w n , o n e c a n e a s i l y d e t e r m i n e t h e c o o r d i n a t e s ' i n t t i e ' o v e r a l l t v r o -d i m e n s i o n a l s y s t e m . l G e n e r a t i n g o f e l e m e n t s i n a s u b a r e a j À c o n t o u r i s d e f i n e d b y s e q u e n t i a l l y c o n n e c t e d n o d a l p o i n t s o n t h i s c o n t o u r . T h e c o n n e c t i o n i s m a d e b y s t r a i g h t l i n e s ( t h e e l e m e n t s i d e s ) . E v e r y s u b a r e a w i l l h a v e t o b e d i v i d e d e i t h e r i n t r i a n g l e s o r q u a d r i l a t e r a l s , d e p e n d i n g u p o n t h e u s e r g i v e n e l e m e n t t y p e . T h e n o d a l p o i n t s o n c o n t o u r C - o f s u b a r e a G ^ a r e n u m b e r e d l o c a l l y I . . . n c p ( f i g . 5 ) . " et(r 7 <=rrq) F i g u r e 5 . L o c a l n u m b e r i n g o f c o n t o u r p o i n t s o n C - . J

Subarea G is concave whenever one of the angles enclosed by t h e c o n t o i r o : > r . L e t i a n d j , w h e r e i * 3 l b e n o d a l p o i n t s

o n C s u c h t h à t t h e i n t e r c o n n e c t i n e l i n e b e t w e e n i a n d i l i e s

c o m p l e t e l y w i t h i n G ^ . W h i l s t d i v i d í n g G ^ i n t o e l e m e n t s - s u c h

lines are frequentl! used, and an instaitanuous check will

h a v e t o b e m a d e t o s e e w h e t h e r t h i s l i n e i s a c t u a l l y w i t h i n G ^ ; f o r i n s t a n c e t h e c o n n e c t i o n b e t w e e n p o i n t s B a n d l 7 í n

f i g . 5 i s n o t a c c e p t a b l e . T h e s e c h e c k s f o r c o n c a v e a r e a s a r e

quite cornplicated and therefor a concave subarea is split into

Ë\^ro or more convex partial areas, after which thèse areas

a r e d i v i d e d i n t o e l e m e n c s . S P l i t t i n g a c o n c a v e s u b a r e a i n t o c o n v e x p a r t i a l a r e a s A n o d a l p o i n t i s c a l l e d c o n c à v e i f s - . > n , ( f i g . 5 ) . T h e s p l i t t . i n g i s d o n e b y t h e f o l l o w i n g s t e p s : r l . T a k e a c o n c a v e n o d a l p o i n t o n t h e c o n t o u r ; c a l l t h i s p o i n t P . I f n o s u c h o o i n t e x i s t s . t h e a r e a w i l l b e c o n v e x . 2 . D e t e r m i n e t i t e ; a c c u m u l a t i o n V , o f n o d a l p o i n t s o n t h e c o n t o u r w h i c h a r e v i s i b l e f r o m p . '

3 . D e t e r m i n e o u t o f V r E h a t n o d a l p o i n t Q in such a r,ray that,

b a s e d o n g i v e n c r i r e r i a , P Q i s t h e b e s t s p l i t t i n g l i n e .

4 . D e t e r m i n e the accumulation V. of nodal points on the contour

w h i c h a r e v i s i b l e f r o m 0 . L

5 . - D e f i n e o n PQ a roughness funetion, based on the roughness v a l u e s o f t h e n o d a l p o i n t s in V, and Vr.

b-ó

6 . G e n e r a t e , u s i n g t h a t r o u g h n e s s f u n c t i o n , n o d a l p o i n t s o n p O . 7 . D e f i n e t w o n e r r a r e a s s e p a r a t e d b y l í n e p Q . 8 . C o n t i n u e w i t h s t e p I f o r b o t . h a r e a s . E x p l a n a t i o n _{S t e p l . P o i n t P i s c h o s e n t o b e t h e m o s t c o n c a v e} p o i n t o n . t h e c o n t o u r o r t o b e t h e r n i d d l e p o i n t of a series of, a l m o s t e g u a l l y c o n c a v e p o i n t s . .steps 2 and 4. The determination

o f v i s i b l e p o i n t s i s i l l u s t r a t e d ! í i t h a n e x a m p l e ( f i g . 6 a ) . C o n -s i d e r a c o n t i n u o u -s c o n t o u r ; B " i -s d e f i n e d a -s b e i n g t h e a n g l e b e t w e e n l i n e P - K a n d r h e t a n g Ë n t o f t h e c o n t o u r i n f ; n i g . O t s h o w s 8 U a s f u n c t i o n o f c u r v i l i n e a r c o o r d i n a t e s . ( a ) ( b ) F i g u r e 6 . D e t e r m i n a t i o n o f f r o m p v i s i b l e p o i n t s .

9r.

ï

K 3 K L P + S P o i n t s o f i n t e r e s t i n t h e d e t e r m i n a t i o n o f t h o s e c o n c a v e p o i n t s L o f t h e c o n t o u r w h e r e v i s i b l e p o i n t s a r e B, is a local ex-is a local maximr:m; i s a l o c a l m i n i m u m ; t r e m u m ; n o t v i s i b l e w i l l b e : l . P o i n t s K w i t h s , , > s , a n d 8 . , < f o r e x a m p l e p o i n t s \ e t w à e n r c , à n d 2 . P o i n t s K w i t h s , , < s , a n d ' 8 , , > f o r e x a m p l e p o i n t s \ e t w Ë e n x . à n d

Ê

:L K ) . p -I K , . 4 . f B L . f B L

\_:_/

F i g u r e 7 . S e a r c h f o r a s u i t a b l e s p l i t t i n g - l i n e f r o m p o i n t p . S t e p 3 . C o n s i d e r f i g . b o u r i n g p o i n t s o f P . a r e a i n s e c t o r s I , I 1 / : r n e p o ] - n E s P T h e l i n e s P P ' a n d a n d I I I . A t f i r s t . _ + l a n d r P a r e t h e n e i g h -P -P d i v i d e t h e v i s i b l e E . h e m o s t s u i t a b l e p o i n t

. c t i o n , n o d a l p o i n t s o n P 0 . r l n e r q .

.n to be the most concave d l e p o i n t o f a s e r i e s o f 2 a n d 4 . T h e d e t e r m i n a t i o n . a n e x a m p l e ( f i g . 6 a ) . C o n -i n e d a s b e -i n g t h e a n g l e h e c o n t o u r i n P ; F i g . 6 b o o r d i n a t e s . K1 Kz K, ( b ) i b l e p o i n Ë s . o n o f v i s i b l e p o i n t s a r e

where L is a local

ex-L i f B - i s a l o c a l m a x i m u m l L i f B - i s a l o c a 1 m i n i n u m ; L

\

I t i n g - 1 i n e f r o m p o i n t P . ' a n $ , P ' a r e t h e n e i g h -n d P P d i v i d e t h e v i s i b l e s t t h e m o s t s u i t a b l e p o i n t * i - j i s a c c e p t a b l e l . t h e a n g l e s \ 1 r \ 2 a n ( l q . , a r e E n e c u t t l à g J l i n e t h e n K . P + S Q i n s e c t o r ï i s s e a r c h e d f o r ; i f s u c h a s u i t a b l e p o i n t i n t h a t s e c t o r e x i s t s , t h e c o n c a v e a n g l e i n P c a n b e e l i m i n a t e d .

However, if line PQ does not meet some minimum demands, the

s e a r c h i n g - a r e a i s e x t e n d e d ! í i t h s e c t o r s I I a n d I I I . P o s s i b l e s p l i t t ' i n g 1 í n e s a r e s e l e c t e d w i t h r e s p e c t t o t h r e e c r i t e r i a : l . t h e a n g l e s ' ( , . . . \ t , ^ w i 1 1 h a v e t o f i t a s g o o d a s p o s s i b l e m u l t i p l e s o f 6 0 " ' o r 9 Ó - ( r e s p e c t i v e l y f o r t r i a n g l e s a n d q u a d r i l a t e r a l s ) ; t h i s r e s u l t s i n a d i f f e r e n c e À y ; 2 . t h e l e n g t h o f t h e s p l i t t i n g l i n e , w h i c h s h o u l d b e a s s h o r t a s p o s -s i b l e , y i e l d s a d i f f e r e n c e A g ; 3 . t h e n u m b e r o f n o d a l p o i n t s o n a s p l i t t i n g l i n e i s r o u n d e d o f f , w h i c h g i v e s a À n . F r o m a l l p o s s i b l e s p l i t t i n g l i n e s t h a t l i n e i s c h o s e n f o r w h i c h A = C , . a - * C r . A o + C . . 4 . - r e a c h e s a m i n i m u m . T h e w e i g h t i n g f a c c o r c l , ë , " " á c l h a v È b 8 e n e m p i r i a l l y d e t e r m i n e d .

Step 5. A twó-dimensional roughness polynomial 92 is determined

b y m e a n s o f a w e i g h t e d r e s i d u a l m e E h o d ; l e t g k , b e t h e r o u g h

-n e s s i -n c o -n t o u r p o i -n t k , s o i t ' s w e i g h t i n g f a c t à r w i l l b e l / g 1 \ , .

The result can be reduced to an one-diurensional polynornial gl^

o n t h e s p l i t t i n g l i n e .

Dividing a convex area into triangles The contour of a convex

p a r t i a l a r e a i s d e t e r n i n e d b y n n o d a l p o i n t s . F o r a n y o f t h e s e

p o i n t s , 1 o c a 1 1 y n u m b e r e d i , i = 1 . . . n , E h e c o o r d i n a t e s , t h e c o n

-tour-enclosed angle o- and the roughness gk- are known, An

a n g l e o , . i s c a l 1 e d s h á r p ^ w h e n e v e r o , - . 8 0 - r ' b e c a u s e c u t t i n g

a n d s p l i t t i n e o f c . = 8 0 v i e l d s e q t a l d i f f e r e n c e s w i t h r e s

-p e c t i o 6 0 " ; s o o - t - 6 0 o = 6 0 " - o r / 2 . t t " a c t u a l d i v i d i n g o f a c o n v e x a r e a c a n ' b e d e s c r i b e d a s f o l l o w s :

l . I f n = 3 , o n e t r i a n g l e i s f o r m e d a n d t h e a r e a i s e x h a u s t e d .

2. Lf a sharp angle is found, subsequent layers are cut from t h e a r e a a s l o n g a s t h i s i s i n a c c o r d a n c e w i t h g i v e n c r i t e r i a . T h e s e l a y e r s a r e d i v i d e d i n t o t r i a n g l e s .

3 . S p l i t t h e r e m a i n i n g a r e a i n t o t r , í o p a r t i a l a r e a s .

4. Define tr^/o contours for those partial areas and continue for

b o t h a r e a s w i t h s t e p l . E x p l a n a t i o n _{S t e p 2 . A c u t t i n g l i n e i + j}

-r t i t s a t i s f i e s t w o c o n d i t i o n s ( f i g . 8 ) : Y 3 a n d y O s h o u l d b e > 4 0 - , a n d 2 . i f g ; + . i r o u g h n e s s v a l u e s i n t h e e n d p o i n t s o f t h é r o u g h n e s s g o o n t h a t l i n e s h o u l d s a t i s f y :

1

z n i n { g k i + j , B t i _ j } . g t . T h i s c r i t e r i o n _{r e d u c e s o o s s i b l e} f i g . _{8 b , w h i c h c a u s e s b à d s h a p e d}

o n a n ê c c e p t e d cutting line are r o u g h n e s s of a1l contour points

t i a l a r e a . . m a x { g k . . . , s k , . } _- ( 6 ) 1 + 1 - " 1 - 1 c o n f i g u r a t i o n s a s s h o w n i n e l e m e n t s . T h e n o d a l p o i n t s g e n e r a t e d b a s e d u p o n t h e

par-l 0

I

I

(b) (o) F i g u r e 8 . C u t t i n g l a y e r s s t a r t i n g a t s h a r p a n g l e I . F i g . 9 i l l u s t r a t e s t h e d i v i d i n g o f a l a y e r i n t o e l e m e n t s ; i n -s i d e a l a y e r n o n o d a l p o i n t -s w i l l b e c r e a t e d . ro) (D (c) F i g u r e 9 . F o r r n i n g t r i a n g l e s w i t h i n a l a y e r . I f L M < K N ( f i g . 9 a ) t h e n L K M w i l l b e t h e n e x r r r i a n g l e t o b e f o r m e d ; o t h e r w i s e K M N w i l l b e c r e a t e d . F i g . 9 b s h o w s a p o s s i . -b l e e n d s i t u a t i o n a n d f i g . 9 c a s i n i l a r s i t u a t i o n f o r t h e f i r s t l a y e r ; t h e t r i a n g u l a t i o n o f s u c h r e m a i n i n g a r e a s i s t r i v i a l . S t e p 3 . T h e a s p e c t s w i t h r e s p e c t t o w h i c h p o s s i b l e s p l i t t i n g l i n e s a r e s e l e c t e d ( i . e . a n g l e s , l e n g t h a n d n u m b e r o f n o d a l p o i n t s t o b e g e n e r a t e d ) a r e t h e s a m e a s f o r t h e s p l i t t i n g o f L V r r 9 4 V g é t g d S .

The method is

nain-l y a n a nain-l o g o u s t o t h e m e t h o d f o r t r i a n g nain-l e s . A n a n g l e o , i s n o w s a i d t o b e s h a r p i f ' o - . < 1 2 0 " . T h e a c t u a l d i v i s i o n oÈ " " o . r . r u * a r e a i n t o q u a d r i l a t e r à l s _{i s s h o w n i n t h e f , o l l o w i n g s t e p s :} l . I f n = 4 ; ' o n e q u a d r i l a t e r a l _{i s f o r m e d , a n d t h e a r e a i s e x} -h a u s te d .

2 . Í f . n = 6 t 2 , 3 , 4 o r 5 q u a d r i i " a t e r a l s will be formed, and t h e a r e a i s e x h a u s t e d .

3 . F i n d a c o m b i n a t i o n o f t w o c o n t o u r p o i n t s i and j for which b o t h a . à n d q . a r e " s h a r p , t a n d t h e b e t w e e n i a n d j s i t u a Ë e d a n g l e s * c U = 1 8 0 - , w h i c h m e a n s t h a t i a n d j are coinected with

--ff--- -í--l--.

h

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t t I I I t t a t . t ) I

I ' l

I

j

I

I t ' I l

-(b) s h a r p a n g l e i . l a y e r i n t o e l e m e n t s ; c r e a t e d . ( c l l a y e r . e t h e n e x t t r i a n g l e t o b e d . F i g . 9 b s h o w s a p o s s i -l a r s i t u a t i o n f o r t h e f i r s t a i n i n g a r e a s i s t r i v i a l . w h i c h p o s s i b l e s p l i t t i n g gth and number of nodal

a s f o r t h e s p l i t t i n g o f

erals The method is

main-l e s . A n a n g main-l e o - i . s n o w

t u a l d i v i s i o n o f a c o n v e x

t h e f o l l o w i n g s t e p s :

ned, and the area is

ex-a l s w i l l b e f o r m e d , a n d

r o i n t s i a n d j f o r w h i c h

t w e e n i a n d j s i t u a t e d

:.nd j are connected with

Ln-a s t r Ln-a i g h t l i n e . C u t o f f l Ln-a y e r s , Ln-a s l o n g Ln-a s t h i s i s p o s s i b l e w i t h r e s p e c t t o g i v e n c r i t e r i a , a n d d i v i d e t h e s e l a y e r s i n t o q u a d r i l a t e r a l s . 4 . S p l i t t h e r e m a i n i n g a r e a i n t o t w o p a r t i a l a r e a s . 5 . C o n t i n u e w i t h s t e p l . E x p l a n a t i o n S t e p 2 i s n e c e s s a r y b e c a u s e o f t h e g e o r n e t r i c a l l y a n d t o p o l o g i c a l l y d i f f i c u l t e l e r n e n t f o r m . T y p i c a l g e o m e t r i e s w h i c h w i l l c a u s e 2 , 3 , 4 o : 5 q u a d r i l a t e r a l s a r e s h o r " m b e l o w ( f i e . I 0 ) .

r u f r m

F i g u r e 1 0 . D i v i d i n g a h e x a g o n i n t o 2 , 3 , 4 o r 5 q u a d r i l a t e r a l s .

Step 3. Apart from the demands for the angles with the contour

and the minimum and maximtm roughness on the cutting-line as in

t h e c a s e o f t r i a n g l e s , i t w i l l b e a d e m a n d t h a t t h e n u m b e r o f

nodal points on the cutting-line is equal to the number on the

p r e v i o u s c u t t i n g - l i n e ( o r t h e o r i g i n a l l i n e b e t w e e n i a n d j ) .

I n Ë h a c c a s e , t h e d i v i s i o n o f t h e l a y e r i s t r i v i a l .

P o s t - p r o c e s s i n g

thape improvement of the elements It has been found that the

f internal nodal point k can

be improved in an efficient way by giving point k new

coordi-nates as the average of the coordicoordi-nates of his nk neighbouring

p o i n t s ( i . e . _{n o d a l p o i n t s w h i c h a r e c o n n e c t e d t o k w i t . h a n e l e} -n e -n t s i d e ) : . n k I K _{n k .} - . ' - k i 1 = l ._I n k r r . = - l - I r r - K _{n K ,} - K r T h e i t e r a t i o n p r o c e s s i s s t o p p e d a s s o o n a s n o n e o f

na1 nodal point changes significanrly.

( 7 ) the

inter-Bandwidth reduction The nethod r^rhich TRIOUAI"ÍESH uses for the

-ctrviding of an aiea G into elements causes that the

coefficient-o a t r i c e s o f s u b s t r u c t u r e s d e f i n e d i n G , w i l l h a v e a b a d o r n o

band structure. Because ilany finite element programs only work

e f f i c i e n t l y _{b y m e a n s o f a n a c c e p t a b l e b a n d s t r u c t u r e , s o m e p o s}

-sibilities have béen provided in order to reduce band-width by

renumbering the nodal points in a suitable way. The user can,

f o r e a c h s u b s t r u c t u r e , d e f i n e m o r e r e n u m b e r i n g s ; f i n a l l y t h a t

o n e i s c h o s e n w h i c h g i v e s t h e b e s t b a n d s t r u c t u r e . T h e a v a i l

-a b l e r e n u m b e r i n g m e t h o d s w i l l b e d e s c r i b e d s u b s e q u e n t l y . CMK-Benunbering This rnethod of renumbering was developed by

-U u t h i l l E . a n d M c k e e J . ( 1 9 6 9 ) a n d s t a r t s f r o m t h e t o p o l o g y o f

and

:'

È-l*

liri1,1,

ill

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I r

1

l"

i'l',ii;

l

ill

i,

.

I

ir d e g r e e = l 0 d e g r e e = F i g u r e I l . D e g r e e o f a n o d a l point CMK-remrmbering proceeds as follows:

I . C h o o s e a s t a r Ë i n g p o i n t k, and give this number l. The ac_ c u m u l a t i o n v o f r e n u m b e r e d p ó i n c s now consists of the nodal p o i n t k , : V = { k , } .

2' Nunbêr sequerltialry _{the not yet renumbered surround.ing points}

o f e a c h n o d a l p o i n t i n V , accoràing to an increasing degree,

with their new numbers

3 . D e f i n e V = { n o d a l p o i n t s to which new numbers were given in_{e t . a n ? ]}

_ a . i f V i s e m p t y t h e p r o c e s s stops otherwise repeat step 2. Because of the fact that the cMKlrenurobering ,r.", ,ro g"lrn.tri-c a 1 d a t a , t h i s m e t h o d w i l l be espeg"lrn.tri-cially u"áfrrl in thË g"lrn.tri-case of

s u b s t r u c t u r e s w i t h a n i r r e g u l a , g e o * e t . y .

Lir",l, Di"t""""- .

"irre These methods are based

o n Ë h e g e o m @ o o r d i n a t e s _{o f n o d a l p o i n t s ) ,}

and are realised _{by rneans of a, user-given, ,r.r,u C"rt.ria.,}

.oor_ o r n a t e - s y s t e m x , y . I n order to achieve this two basispoints p a n d Q w i l l h a v e t o b e s p e c i f i e d . The origin of the

" . r ' r y r a " *

w i l l _{b e e ; y i s o , " . r r r r " á a l o n g the line pQ and x along a line}

o r t h o g o n a l t o p Q . I n t h i s system a nodal point k can Ëe char_

Ë ï : Ï ; i . i n

t w o w a v s :

t h e p a i r ( i u , vu) à"a.r'" p.ir-lio,-iul.

t 2

E h e e l e m e n t d i v i s i o n o f t h e s u b s t r u c t u r e . L e t ê r , ê o r . . . r e

( n > l ) b e t h e elements in'hich conrain nodal poiJr; k?'rià'J"* r

1 3 d a f n o i n t " ï , k , . . . k _ d e s c r i b e r h e e l e r n e n t s e , . . . e s o t h a t n o d a l p o i n t t . h a s d Ë g r e e m a n d k , . . . k m . r " , r , r i r o . , * a i . , g p o i n t s o f k . ( f i g . l l ) . I d e g r e e = 7 1 . F i g u r e 1 2 . C h a r a c t e r i s a t i o n o f a n o d a l p o i n t i n t n e ( i , y ) _ s l s r e . . r .

u r e . L e t ê r , € ? r . . . r e n n o d a l p o i i r t k i T h e m ' ^ + I e e l e m e n t s e , . . . e s o r - ^ - ^ I ^ , , - * ^ , , D ' ' I . . . k m a r e s u r r o u n o r n g l 3 B y m e a n s o f t h e n o d a l p o i n t s k , w i g h c o o r d i n a t e s < = t l i , J - l l ( ; , , 6 , ) a n d k , w i t h c o o r d i n a t à s - ( x 2 , Y 2 ) o r ( r r , Q 2 ) , t h r e e r e i r u n b ê r i n g m e E h o d s c a n b e s p e c r t : - e c l : k , w ! 1 1 f o l l o w k r - í f t r l ( " r , - * z ) u t(ir = ir) ,r (r, , _{l r r ,} " L i n e " - r e n u m b e r i n g z . ( í t , i z ) , ( ( ; l = i r ) n { 0 , t l t l l " n i s t a n c e " - r e n u m b e r i n g 3 . ( O l ' 6 ) ' ( ( Ó l = o r ) n { i , ' r 2 ) ) " A n g 1 e " - r e n u m b e r i n g F i g . l 3 s h o w s s o m e ty p i c a l e x a m p l e s o f e l e m e n t d i v i s i o n s w h i c h " " á h i n t u r n a r e t y p i c a l f o r t h e d i f f e r e n t r e n u m b e r i n g m e t h o d s . -t a ^ - - ^ ^ - a u E Ë l s g - , e t h i s n u m b e r l . T h e a c -c o n s i s t s o f t h e n o d a l .numbered surrounding points

o a n i n c r e a s i n g d e g r e e , new numbers were given in . h e r w i s e r e p e a t s t e p 2 . .mbering uses no geometri-. 1 1 y u s e f u l i n t h e c a s e o f

g T h e s e d e t h o d s a r e b a s e d

i o o r d i n a t e s o f n o d a l P o i n t s )

' g i v e n , n e w C a r t e s i a n c o o r -: v e t h i s t w o b a s i s P o i n t s P

origin of the nev/ system

.ne PQ and x along a line rdal point k can

be_chaf-y 1 ) a n d t h e p a i r ( r u ' ó 1 ) .

F i g u r e 1 3 . T y p i c a l e l e m e n t d i v i s i o n s f o r ( a ) L i n e - , ( b ) D i s

-tance- and (c) Angle-renumbering.

G e n e r a t i o n o f n i d p o i n t s T h e t r i a n g l e s a n d ( o r ) q u a d r i l a t e r a l s

f f i b e e n d i v i d e d , c a n b e s e e n a s e l e m e n t s

v i t h t h r e e o r f o u r n o d a l p o i n t s . H o w e v e r , i n m a n y c a s e s o n e

w i s h e s t o u s e e l e m e n t s w i t h m o r e n o d a l p o i n t s . E x a m p l e s o f

these are the ASKA-elements TRIM6 and QUAMB; these have, apart f r o m t h e n o d a l p o i n t s o n t h e c o r n e r s , a l s o n o d a l p o i n t s o n t h e

e d g e s o f t h e e l e m e n t s ( s o c a l l e d n i d p o i n t s ) . I n o r d e r n o t t o

lose the band structure which was obtained by renumbering the

nodal point.s, for each new midpoint a number will have to be

chosen which does not exceed the maximum or minimum nrmrber of

t h e c o r n e r n o d a l p o i n t s o f t h a t e l e m e n t . T h i s w i l l b e a c h i e v e d

by giving a roidpoint between k, and k, the exact or nearest

" f r e e ' r v a l u e o f ( k , + k " ) / 2 . r L i s e v i d e n t t h a t a s p a c i n g w i l l

h a v e t o b e m a d e i n ' t h e 6 r i g i n a l b a s e o f c o r n e r p o i n t s ; t h i s i s

done by roultiplying the numbers of the corner points by 4

( w h i c h i s a q u i t e a r b i t r a r y v a l u e ) . A l l l i n e s d e f i n e d b y t w o

corner points are gathered and for each of these lines a

nid-point is generated by the trethod mentioned above. Afterwards a

n e w t o p o l o g y i s c o m p o s e d b y t h e o l d t o p o l o g y , t h e s e l i n e s a n d t h e n e w l y c r e a t e d m i d p o i n t s ; a t l a s t t h e s p a c e s l e f t i n t h e

nunbering will be eliminated.

u s i n g @ e _{u s e r c a n o b t a i n a n o u t p u t o f t h e r e}

-s u l t -s o f T R I Q U A M E S H o n a l i n e p r i n t e r , c a r d p u n c h e r , d i s c - p a c k

o r o n a p l o t t e r . T h e c h a r a c t e r i s a t i o n o f t h e g e n e r a t e d d i v i s i o n

in elements, is presented in such a r,ray on punch cards and (or) d i s e - p a c k , t h a t i t will match the demands of various finite e l e m e n t p r o g r a m s . At this moment it is possible to create outpuË r 1 p o i n t i n t h e ( x r Y ) - s Y s t e " . , .

--l +

w h i c h c a n s e r v e a s i n p u t f o r t . h e s y s t e m s A S K A , FEMSys and MARC: e x t e n s i o n f o r o t h e r s y s t e m s i s e a s i l y p o s s i b l e .

THE USE OF TRIQUAMESH

T R T Q U A M E S H ' h a s _{b e e n p r o g r a n r n e d i n B E A ( B u r r o u g h s E x r e n d e d A l g o l )}

a n d h a s , b e e n i m p l e m e n t e d o n t h e B u r r o u g h s 8 7 7 0 0 cornputer of the u n i v e r s i È y o f T e c h n o l o g y E i n d h o v e n . T h e p r o g r a m can be used both i n b a t c h a n d i n t h e 8 7 7 0 0 t i m e s h a r i n g systeÍ!. In the latter

c a s e a n e f f i c i e n t _{u s e c a n b e m a d e o f a c R T t e r m i n a l ( T E K T R O N T X} 4 0 1 4 ) i n o r d e r t o o b t a i n g r a p h i e a l ( i n t e r m e d i a r e ) _{o u t p u t . T h e} u s e o f T R I Q U A M E S H w i l l b e i l l u s t r a r e d b y a n e x a m p l e . I ^ I h i l s t c o m p o s i n g t h e i n p u t - d a t a t h e u s e r m a k e s a s k e t c h o f the area to b e d i v i d e d a n d s p e c i f i e s t h e r e i n b a s i s p o i n t s , c o n t o u r p o i n t s , s u b -3 r e a s e t s . ( f i g . I 4 a ) .

/ ' ^ \

\ 4 / F i g u r e 1 4 . I n p u t s k e t c h a n d h o o k . ( b ) g e n e r a t e d c o n t o u r p o i n t s f o r a c r a n e -T h e i n p u t f o r t h i s p r o b l e m i s : 1 $ { i * ï t ' { [ i t j l ' " t . f o u K [ : ' 1 . _{I 0 . $ I r A $ ï S i F i l l : N ï' $} J . : ; 0 J . l 0 r i l 0 0 i _{t i - . 4 . 3 y 1 . i 1 ï i i J l . i " : i ,} " i r y ? 0 i 4 l O r U i : t l O ï ; l * 4 O r { r i d ; ; - 4 r } , , 1 O i , r í - i ; , O y l . i ) i Í l i L 0 y O i 1 4 0 ? i , 7 Í y 4 O i L O i r ' , r 1 , , ! r i ) í _{l . 1 i . i l ; J y J . i i i i i L : : r ; 4 i r l . ? . J i} 1 ï . j 0 1 3 Í O y 1 7 0 i : 1 4 i 0 , . t : j ( : ) í _{t l j i i r y : 1 . * ï O i L s i : ï O y _ . , i O i} 1 ó0 l;{:0t{ ï'lltJ[ir:.[ [.[;[:ti 1 . 7 t i 1 . ( L l i : l . . : t . _ 4 i : l i l : j l . r t i . L / i : . : l . i : j r _ : r \ : 1 . " 1 r J . . i . , l . i ï 0 , ï ( I _{{ : 1 4 L , l y l i l f i l - ; : : i : l r : l { r r l : i l i ! . . . , í r f{ 1 . . . ; : . U i i ll r ?} l'ii) _{ftl-. L0 t. Ii l. l. y L",l i.li'l l.+ r.l.;} :,100 ÍÍ.St.iFi:ilÍ.t IÍl(.lti :.i 1. ti :L i.ï p :1. y -.1:l s -.:1. )

1s ASI<A, FEMSYS and MARC; ) o s s i b 1 e . i u r r o u g h s E x t e n d e d A l g o l ) ; h s 8 7 7 0 0 c o m p u t e r o f t h e r p r o g r a m c a n b e u s e d b o t h s y s t e Í r r . I n t h e l a t t e r CRT Eerminal (TEKTRONIX : e r m e d i a t e ) o u t P u t . T h e )y an example. Iíhilst ; a s k e t c h o f t h e a r e a t o > o i n t s , c o n t o u r P o i n t s , s u b -( b ) c o n t o u r p o i n t s f o r a c r a n e -+ . , : v L l O i 4 i 0 r 0 i ' ] 0 v l l . 0 i l l i 1 0 v 0 i : / y l . \ . , \ / r . L i ; { ó y l i : J y v .1. i(i í 1,5 i ,IO v "',ï0 i ,-:i"i .1.'1 r :i..3 i ,í fil... .1 Ui'1 $ r I :;l::j0 ir [iirrli.r :l i{f; ; l . i $ i r t I l ' l Ï : L i : ;

i;140 $ lir-Jtii:iï f':U{::ï

'.'liirO /:riiilrr:t .Lí:0 J. f fi t f4.1i 1I :.i:;ï(: Sft Ë:NLlllIlËtt :,:$l{i fri:t{.lF :t l-[:. :lïo ( 1. fi"lli l. ) ( .-ï00 ${lu1'F tJï ;:l;10 I Cfjf:'l-tÏN'f I :l::r0 $cïl)['' L! [i [::

)

I Ci''lF, .7 ) ( 1 1...t l.lE T 0 F ' R ï N ï ' p Í - : O L : É t F . t t v 7 ! v & ) T C l t , í \ R I r . ' F ' t . . 0 I ( A:1) 4'F O N { l i . . x o l a n a t i o n $ B A S I S P O I N T S : c o n t a i n s t h e b a s i s p o i n t s a n d r h e i r c o o r d i n a t e s .

$COtlfOUnpfSCES: defines how each of the contour parts is

com-p o s e d .

$suncoNtoun: defines subcontours by means of contour parts'

$susstnucruRE: defines which subarea belong to which

substruc-ture and the element type of each subarea'

$G36DING: contains infármation about the desired magnitudê of

t h e e l e m e n t s .

$RENID,ÍBER: control data for bandwidth-reduction'

$ O I J T P U T : d e f i n e s t h e w a y o u t p u t w i l l h a v e t o b e p r e s e n t e ' l ' F i g . l 4 b s h o w s t h e g e n e r a t e d c o n t o u r p o i n t s a n d f i g ' l 5 a t h e d i v i s i o n i n t o e l e m e n t s . ( a ) ( b ) ( c ) F i g u r e 1 5 . E l e r n e n t d i v i s i o n s f o r a c r a n e - h o o k . T h e d i v i s i o n s a c c o r d i . n g t o f i g . l 5 b , c a n d d a r e c r e a t e d b y

making some slight alterations in the input above

F i g . l 5 b : t h e c ó n p l e t e d i v i s i o n l e s s c o a r s e b y a f a c t o r 2 ' ( R I = 7 . 5 a t l i n e 2 3 0 ) . F i g . l 5 c : L o c a l l y r e f i n e m e n t i n t h e b a s i s p o i n t s 1 0 , l l , ! 1 2 , 1 3 a n d 1 6 b y a f a c t o r 2 ( a d d c V 0 . 4 ( 1 0 , l l , 1 2 , 1 3 , 1 6 ) a f t e r l i n e 2 3 0 ) . ( d ) I i g . t 5 d : a s f i g . l 5 c v r i t h q u a d r i l a t e r a l s . ( Q U A M 4 i n s t e a d o f T R I M 3 a r t i n e 2 6 0 ) .

-;-t 6 T h e n e x t e x a m p l e d e a l s t ^ t i t h a p i s t o n o f a d i e s e l e n g i n e which i s s i m p l i f i e d _{t o a n a x i - s y r n n e t r i c c o n s E r u c t i o n ( f i g . l 6 a ) . T h e} m o u l d e d a l u m i n i u m p i s t o n i s o i l - c o o l e d b y m e a n s o f a m o u l d e d -i n s t e e l t u b e w h -i c h s e r v e s a s a c o o l a n t s p -i r a l ; i n a r a d i a l p l a n e o n e , c a n s e e 4 c r o s s - s e c t i o n s o f t h e t u b e : t h e s u b a r e a s 2 , 3 , 4 _{a n d 5 . A - t t h e l o c a t i o n o f t h e u p p e r t w o p i s t o n r i n g s ,} a c o n s i d e r a b l y h i g h e r w e a r r e s i s t a n c e m a t e r i a l i s u s e d : a r n o u l d e d - i n c a s i i r o n r i n g , t h e c r o s s - s e c t i o n o f w h i c h i s e i v e n b y s u b a r e a 7 . ( a ) ( b ) ( a ) a n d e l e m e n t d i v i s i o n ( b ) F i g u r e 1 6 . G e n e r a t e d c o n t o u r p o i n t s I n o r d e r t o b e a b l e t o m a k e a n o p t i m a l d e s i g n o f t h e f o r m of t h e c a s t i r o n r i n g , _{i t i s p u t i n t o a s e p e r a t e s u b s t r u c t u r e t o} -g e t h e r w i t h i t s s u r r o u n d i n -g s ( s u b a r e a 6 ) . F i -g . l 6 b shows the m e s h g e n e r a t e d f o r t h e p i s t o n .

DISCUSSION AND FUTURAL DE\i'ELOPMENTS

The meshgenerator TRIQUMESH has proved to be quite an useful

t o o l ; t h e m o s t r e v e a l i n g a d v a n t a g e s o f w h i c h a r e :

- user orientation; o n e t e n d s t o w o r k w i t h i t q u i c k l y .

- freedom in choice of geom.etry and number of subareas. - simple control of element size.

- possibility o f q u a d r i l a t e r a l e l e m e n t s .

Irtrhilst using the program, a number of useful extensions have

b e c o m e c l e a r w h i c h s h a l l b e m a d e i n t h e n e a r f u t u r e ; t h i s w i l l

b e d o n e i n t h e f r a m e w o r k o f a p r e - p r o e e s s i n g s y s t e m , w i t h

which one can generate the complete input (mesh, kinematic and

d y n a m i c b o u n d a r y c o n d i t i o n s e t c . ) f o r t w o - d i m e n s i o n a l p r o b l e m s

L i t e r a t u r e

fi;IffiR. and Bush B.E.

o f a d i e s e l e n g i n e w h i c h s t r u c t i o n ( f i e . l 6 a ) . T h e d by means of a moulded-n t s p i r a l ; i n a r a d i a l t h e t u b e : t h e s u b a r e a s u p p e r t w o p i s t o n r i n g s , m a t e r i a l i s u s e d : a s e c t i o n o f w h i c h i s g i v e n ( b )

) and element division (b)

I d e s i g n o f t h e f o r m o f s e p e r a t e s u b s t r u c t u r e t o -6 ) . F i e . l -6 b s h o w s t h e ' e d t o be quite an useful , f w h i c h a r e : : w i t h i t q u i c k l y . L u m b e r o f s u b a r e a s . r t s . u s e f u l e x t e n s i o n s h a v e : h e n e a r f u t u r e ; t h i s w i l l : o c e s s i n g s y s t e m , w i t h

-nput (mesh, kinematic and

: t w o - d i m e n s i o n a l p r o b l e m s .

r G e n e r a t i o n - A s u r v e y , E n g í n e e r i n g f o r I n d u s t r y '

t 7 p p . 3 3 2 - 3 3 8 .

il,rrro,rgtrr 87000/86000 ALC,OL REFERENCE MANUAL (1977); Pub1. no.

5 O O / 6 3 9 ; B u r r o u g h s C o r P o r a t i o n . C u t h i l l E . a n d M c K e e J . ( 1 9 6 9 ) ; R e d u c i n g t h e B a n d w i d t h o f s p a r s e s y n m e t r i c m a t r i c e s ; p r o c . 2 4 t h N a t i o n a l c o n f ' , A s s o c ' C à m p u t . i { a c h . , A C M P u b l i c a t i o n P - 6 9 , ) 1 2 2 A v e ' o f t h e A m e r i c a n , N e w Y o r k . N . Y . p e t e r s , F . J . ( 1 9 7 6 ) ; F E M S Y S , e e n s y s t e e m v o o r o p d e e i n d i g e e l e

-rDentenmethode gebaseerde berekeningen, Deel I. Inleiding

ter-r n i n o t o g i e e n g á b ter-r u i k e ter-r s h a n d l e i d i n g . T e c h n i s c h e H o g e s c h o o l E i n d

-hoven, onderafdeling der líiskunde.

S c h o o f s , A . J . G . , v a n B e u k e r i n g , L . H . T h . M ' , S l u i t e r , M ' L ' C ' ( 1 9 7 8 ) ;

TRIQUAMESH. Gebruikershandleiding; THE-rapport WE 78-01,

Tech-nische Hogeschool Eindhoven, Afdeling der l^Ierktuigbouwkunde.

:.

: i - - .