2d Shape Optimization !

8/12/2019 2d Shape Optimization !

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118 S .Y . Han

o r s t re s s d e s ig n e r r e q u i r e d a n d t o i m p r o v e t h e c o n v e r g e n c e r a t e . F i n a l l y , a n a u t o m a t i c sh a p e

o p t i m i z a t i o n s y s t e m w a s b u il t b y t he i m p r o v e d g r o w t h - s t r a in m e t h o d w i t h c o m m e r c i a l s o ft -

w a r e u s i n g t h e f in i te e l e m e n t m e t h o d f o r t h e s h a p e o p t i m i z a t i o n o f g e n e r a l t w o d i m e n s i o n a l

s t ruc tures .

2 T h e g r o w t h - s t r a i n m e t h o d

T h e g r o w t h - s t r a in m e t h o d o p t i m i z es a s h a p e b y g e n e r a t i n g th e b u l k s t r a i n t o m a k e d i s t r i b u te d

p a r a m e t e r s u n i f o r m . T h e o p t i m i z a t i o n c o n s i s t s o f a t w o - s t e p i t e r a t i o n . T h e f i r s t s t e p is a

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

T h e s e c o n d s t e p i s a g r o w t h a n a l y s i s t o c a l c u l a t e t h e g r o w t h d i sp l a c e m e n t o r t h e sh a p e m o d i -

f i c a ti o n , b a se d o n t h e g e n e r a t i o n l a w o f th e b u l k s t r a i n u n d e r sh a p e c o n s t r a i n t c o n d i t io n s .

T h e a n a l y s i s a t e a c h s te p i s p e r f o r m e d b y t h e f i n i te e l e m e n t m e t h o d .V o n M i se s s t r e ss , th e sh e a r s t r a i n e n e r g y d e n s i t y a n d t h e m a x i m u m p r i n c i p a l s t r e s s f o r t h e

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

m a x i m u m s t if f n e s s c a n b e u se d a s t h e d i s t r i b u t e d p a r a m e t e r s . I n t h i s s t u d y , y o n M i se s s t r e s s i s

s e le c te d a s th e d i s t r i b u te d p a r a m e t e r s i n ce t h e p r o b l e m s o f m a k i n g s t r e n g t h u n i f o r m a r e

t r e a t e d .

I f th e d i s t r ib u t e d p a r a m e t e r i s d e f in e d a s q u a n t i t y p e r u n i t v o l u m e o r a r e a s u c h a s y o n

M i se s s t r es s , th e d i s t r i b u t e d p a r a m e t e r g e n e r a l l y h a s t h e p r o p e r t y o f d e c r e a s i n g w i t h t h e

i n c r e a se in v o l u m e i n a l o c a l i n f i n it e s i m a l v o l u m e . T h e r e f o r e , t h e b u l k s t r a i n i s g e n e r a t e d a s a

f u n c t io n o f th e d i s t r ib u t e d p a r a m e t e r d u e t o t h e g r o w t h l a w o f E q . ( 1) t o m o d i f y t h e s h a p e s o

t h a t t h e d i s t r ib u t e d p a r a m e t e r s a r e u n i f o r m . T h e c o n t r a c t i n g b u l k s t r a i n i s g e n e r a t e d w h e r e

i s l e ss t h a n 9 , w h i l e t h e e x p a n d i n g b u l k s t r a i n is g e n e r a t e d w h e r e a i s g r e a t e r t h a n i n a l le l e m e n t s ,

~ - - f f - - f f ~ i j h , 1 )O

w h e r e c B is t h e b u l k s t r a in , ~ i s t h e d i s t r i b u t e d p a r a m e t e r ( y o n M i se s s t re s s ), ( ~ i s t h e b a s i c

v a l u e ( f o r e x a m p l e , t h e a v e r a g e s t r es s o r t h e m a x i m u m s tr e ss ) o f t h e p a r a m e t e r , ~ j i s t he

K r o n e c k e r d e l t a , a n d h i s t h e g r o w t h r a t e w h i ch a d j u s t s t h e m a g n i t u d e o f t he g r o w t h d e f o r -

m a t i o n a n d i s a n a r b i t r a r y c o e f f i c i e n t t o b e h < < 1 .

A ssu m i n g t h a t t h e m a t e r i a l i s a t h e r m a l l y i so t r o p i c o n e , t h e t h e r m a l s t r a i n g e n e r a t e d i s a

b u l k s t r a i n w i t h o u t s h e a r in g c o m p o n e n t s . H e n c e , t h e t h e r m a l s t r a i n c a n b e d e f i n e d w i t h t h e

s i m i l a r E q . ( 2) t o t h a t o f t h e g r o w t h a n a l y s i s ,

= 2 )

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

u s i n g v o l u m e c o n t r o l , a n d t h e o t h e r i s t o r e d u c e v o l u m e u s i n g s t re s s c o n tr o l . V o l u m e c o n t r o l

m a k e s t h e d i s tr i b u t e d s t re s s u n i fo r m , a n d t h e n t h e s t r e n g th i s m a x i m i z e d u n d e r a v o l u m e c o n -

s t r a i n t . S t re s s c o n t r o l m a k e s t h e d i s t r i b u t e d s t r es s t h e o b j e c t iv e s tr e s s , a n d t h e n t h e v o l u m e i s

m i n i m i z e d u n d e r s t r e s s c o n s t r a i n t s .

W h e n a d e s i g n e r e s t a b l i sh e s t h e o b j e c t i v e v o l u m e , E q . ( 3 ) m o d i f i e s t h e g r o w t h l a w o f

E q . ( 1 ) b y a p p l y i n g t h e P I D c o n t r o l t h e o r y . W h e n a d e s i g n e r e s t a b l i sh e s t h e o b j e c t i v e s t r e s s ,

E q . ( 5) is a p p l ie d . T h e r e f o r e , v o l u m e a n d s t r es s c a n b e c o n t r o l l e d e f f e c t iv e l y b y E q s . ( 3 ) - ( 5 ) ,



Shape optimization for general two-dimensional structures 119

respectively:

s B ~ ) O - , , -1 ) _ ~ , , - 1 )ij ~ - n - 1 )

n - 1 V k ) _ _ ~ o b j

P ~ o b j k : l

V n - l) _ V n - 2 ) ~

t - / ~ D V o b j j ,

a )

O-eVe

e

O-(mk)ax_ ~ - i ) ~-2)

n - __ O-obj + K D - - - - ,K~ O m a x - - O - m a x

h = l O - ~ O - o b j

5 )

O-ff ~) - Gobjn ) o ( n - 1 ) - - O-~ 6 i jh + K p - -

C i J ~ - - O-obj O-obj

where (n) is the number of the n-th iteration, K p , K r , K D are proport iona l constants, ve is the

volume of each element, O-~ is the representat ive stress of each element, V~bj is the objective

volume, V (~) mea ns the tota l vo lume of the n-th iteration, O-obj is the objective stress, and O-(~)is the maximum stress of the n-th iteration.

The term with K p corresponds to the proportional action of PID control and generates

the bulk strain according to the deviation f rom the objective volume or stress. The term with

K 1 corresponds to the integral action, and eliminates the residual deviation. And the term

with K 9 corresponds to the derivative action and accelerates the response to a disturbance.

But in this paper, K D is set to be zero because it was found that it causes oscillations during

the optimiza tion process [9].

3 Improvem ent of the growth strain method

The growth strain method has successfully been applied for shape optimization of structures

without a hole inside or with only one deformable free surface. But a valid optimal shape can-

not be obtained for the structures with two or more deformable free surfaces by the growth-

strain method as it is.

In the mathematical programming method the design variables of a shape are selected by

boundary parameterization. This causes a changing shape at each iteration compared to the

initial shape. Likewise, in the case that only one free surface of a structure exists, a changing

shape in the growth-strain method is similar to the initial shape since the bulk strain occurs

on the free surface only. But in case that two or more free surfaces of a structure exist, thebulk strain occurs by the growth law of Eq. (1) between the two or more free surfaces. This

causes an optimized shape far from the initial shape.

Therefore, in order to maintain the initial shape it is necessary that an additional con-

straint should be given, and by doing so a problem with two or more free surfaces is replaced

by that with only one free surface. In this paper, the additional constraint was given by the

type of a line conne cted with the points where the bulk strains are zero.

For example, consider a two-dimensional bracket with a hole. In case that both the inter-

nal hole and the outer surface are deformable, the optimized shape by the growth-strain

method is obtained as shown in Fig. 1. It is a shape far from the initial shape. It is because the

shape has been changed with the growth law by thermal deformation between the hole and

the outer surface as mentioned before.



120 S.Y. Han

Fig 1 Fig 2

Fig. 1. Shape optimized by the growth-strain meth od for a bracket with a hole

Fig. 2. A bracket with an additional bou ndar y condition aro und a hole

In orde r to apply the growth-s t ra in method to the problems which occur for s t ruc tures wi th

two or m ore de f ormable f ree surfaces, an add i t iona l cons t ra in t be tween the two free surfaces

should be given, as show n in Fig. 2. The c onstr aint is given by a l ine conn ected w ith the points

having zero stra ins, that is , where the bulk stra in has not occurred. By doing so, the two

deform able f ree surfaces a re de formed by the grow th law based on the g iven l ine cons t ra in t s .

Then, th is problem has been replaced by a prob lem wi th only one f ree surface .

As a result , the final optimized shape can be obtained as a reasonable shape, similar to the

sha pe op t imi z e d by t he ma t he ma t i c a l p rog ra m mi ng m e t hod .

4 Shape optimization system

A schem atic flowchart o f the developed shap e optim izat ion sy stem is show n in Fig. 3. The

sys tem cons ist s of commerc ia l sof tware us ing the f in i te e lement me th od I -DEA S) [10] and the

deve loped shape opt imiza t ion a lgor i thm us ing the growth-s t ra in method.

After modeling the ini t ia l shape with the fini te e lement method, a stress analysis is per-

formed. The da ta obta ined f rom the s t ress ana lys i s a re t ransformed to new da ta f i l e s by a

pos t -process ing ta sk . U sing the values of the d i s tr ibuted pa r amete rs von Mises s t re ss) andthe generated volumetric stra ins for each element in the post-processing task, new data fi les

a re genera ted in orde r to g ive the condi t ions of t empera ture and geo met ry for growth ana ly-

s is . I t is pe rforme d by a deve loped exte rna l progr am wi th C-language .

Next , the grow th ana lys i s i s pe rform ed by the rmal d e form at ion ana lys i s of the comm erc ia l

sof tware . The bou ndar y condi t ions of the growth ana lys i s can be es tabl i shed independent ly of

those of the s t ress ana lys i s. The bou ndary condi t ions o f the grow th ana lys is a re the condi t ions

of tempera ture and geomet ry . F ina lly , the shape is modi f ied by a deve loped program . An

opt imal shape i s obta ined by th i s procedure i t e rat ively . As the above proced ure i s pe rfo rmed

iterat ively, some of the fini te e lements become severely distorted. Then, mesh refinement is

pe rfo rmed be fore accompl i shing the next s t ress ana lys is by examining the shape of the f in ite

elements.



Shape optimization for general two-dimensional structures 121

I n i t i a l S h a p e

I i iSt res s An a ly s is i iii ll

0

[~ 1 : :Genera l Pu rpose9i:i~i,~!,i.i FEM Cod e

: Deve lopedprog ram

F i g 3 F l o w c h a r t o f a s h a p e o p t i m i z a t i o n s y s t e m

App lication examples

5 1 A torque arm

A shape optimization was accomplished by volume control for a two-dimensional torque arm

as shown in Fig. 4. The objective volume was established at 70 of the initial volume of the

torque arm. The values of h and Kp, Kr, KD were 0.05 and 0.5, 0.1, 0.0, respectively. These

D i m e n s i o n s i n c m

t = 0 3

E = 2 0 7 4 1 0 6 N / c m 2

=8 .0 .104 N/cm 2

p = 0.0081 Kg/cm 3

. 4 2 0

5066 N

2 . 5

4 27F i g 4 I n i t ia l d e s ig n c o n d i t i o n s o f a

t o r q u e a r m

3.0

2.5

2.O

1.5

1 .0 ~

0.5

0.0

- - I - - V o [ u m e

- - ~ - - S t r e s s

I te ra t ion Number

F i g 5 History of iterations of a torque

arm by volume control



122 S.Y. Han

Fig. 6. Optimized shape for a torque

arm by volume control

Fig. 7. Initial shape of a bracket

3 , 0

2 . 5

2 .0

1.5

1 .0

0 . 5

0 . 0

t = V o l u m e~ S t r e s s

~ ~ O ~ O ~ l ~ o - - o - - e _ O _ O _ o _ g l _ d 0 _ e _ o _ i _ 0

P i5 10 15 20 25

teration Numbe r

Fig. 8. History of iterations for a bracket by stress control

Fig. 9. Optimized shape of a bracket by stress contro l

values were obt aine d from a prev ious study [9]. The change s of the volume (area) to the initial

volume and the yon Mises stress to the maximum yon Mises stress are shown in Fig. 5, and

the optimized shape is shown in Fig. 6. Maximum yon Mises stress of the optimized shape

was obta ined as almos t the same value as the initial max imu m von Mises stress, and the fina l

volum e was reduced to 70 of the initial volu me as it was established.

5.2 A bracket

A shape optimization was accomplished by stress control for a two-dimension al bracket as

shown in Fig. 7. The objective von Mises stress was established at the initial maximum von

Mises stress of the bracket. The values of h and Kp B2z Ko were 0.05, and 0.5, 0.1, 0.0,



S h a p e o p t i m i z a t i o n f o r g e n e r a l t w o - d i m e n s i o n a l s t r u c tu r e s

E:2 Gpa

~ = 0 . 3

24

r2 kN

1 2 3

F i g . 1 0 . I n i t i a l b o u n d a r y c o n d i t i o n o f

M i c h e l l ty p e b e a m

F i g. 11 . O p t i m a l t o p o l o g y o f M i c h e l

t yp e b e a m b y E S O m e t h o d

. 12

F i g . 1 2 . I n i t i a l sh a p e f o r sh a p e o p t i m i -

z a t i o n

9 '-'i~se.:N ..... / .: ;~ :' x ':<.+;>~ , . . . . . . . . . : . . . . . . . . : . Z , ~ . - - ~ . ~ . . . .

i i :i i ; i: :5..L i:; i.L . i:i :.:: ] . i i :~: ;: ::l~i~ iiiiYi; i i: ]i:::~iiiiXi~:};i: ;~ ii~ii} ~ii( iii~: ii?.{ii)~i~i~{iiii:ii~ [':; i:i~i?ilili:.: Ii i

F i g . 1 3 . O p t i m i z e d s h a p e o f M i c h e l l

t y p e b e a m

r e s p e c t iv e l y . T h e c h a n g e s o f t h e v o l u m e ( a r e a ) t o th e in i ti a l v o l u m e a n d t h e y o n M i s e s s t r e s s

t o th e m a x i m u m v o n M i s e s s t re s s a re s h o w n i n F ig . 8 , a n d t h e o p t im i z e d s h a p e i s s h o w n i n

F i g . 9 . T h e m a x i m u m v o n M i s es s t r es s o f t h e o p t i m i z e d s h a p e w a s m a i n t a i n e d a s t h e s a m e

v a l u e a s t h e in i ti a l m a x i m u m v o n M i s e s s t re s s a n d t h e f in a l v o l u m e w a s r e d u c e d t o a b o u t

6 0 % o f t h e in i ti a l v o l u m e .

5 . 3 M i c h e l l t y p e b e a m

R e c e n t l y , t h e s t u d y o f t o p o l o g y o p t i m i z a t i o n [5 ], [6 ] h a s b e c o m e v e r y a c t iv e . T h e o b j e c t i v e o f

t o p o l o g y o p t i m i z a t i o n is t o o b t a i n a n o p t i m a l t o p o l o g y s a t i sf y in g t h e d e s ig n c o n d i t i o n s a t t h e

i ni ti al d e s i g n s ta g e . T h e o p t i m a l t o p o l o g y i s b a s e d o n t h e c o n c e p t o f g r a d u a l l y r e m o v i n g

r e d u n d a n t e l e m e n t s o f th e l o w s t r e ss e s p a r t o f t h e m a t e r ia l f r o m a s tr u c t u re . B u t t h e o b t a i n e d



124 S .Y . Han

t o p o l o g y c a n n o t b e a p p l i e d a s it is . T h e r e f o r e , a f t e r d e t e r m i n i n g a n i n it ia l s h a p e b a s e d o n t h e

o p t i m i z e d t o p o l o g y , s h a p e o p t i m i z a t i o n o f t h e d e t e r m i n e d i n i ti a l s h a p e s h o u l d b e p e r f o r m e d

f o r a p p l ic a b l e a n d m o r e p r e c i se o p t i m a l s h a p e .

F i g u r e 1 0 s h o w s t h e in i ti a l t o p o l o g y a n d c o n s t r a i n t s o f a M i c h e l l t y p e b e a m f o r t o p o l o g y

o p t i m i z at i o n . T h e t o p o l o g y o p t i m i z a t i o n w a s a c c o m p l i sh e d b y E S O ( E v o l u t i o n a r y S t r u c t u r a l

O p t i m i z a t i o n ) [ 6] . T h e o p t i m i z e d t o p o l o g y w a s o b t a i n e d a s s h o w n i n F ig . 1 1. T h e d e t e r m i n e d

i n i t i a l s h a p e f o r s h a p e o p t i m i z a t i o n i s s h o w n i n F i g . 1 2 .

S h a p e o p t i m i z a t i o n w a s a c c o m p l i s h e d b y v o l u m e c o n t r o l f o r a M i c h e ll t y p e b e a m a s

s h o w n i n F i g . 1 2. T h e o b j e c t i v e v o l u m e w a s e s t a b li s h e d a t 8 0 o f t h e in i ti a l v o l u m e o f t h e

M i c h e l l t y p e b e a m . T h e o p t i m i z e d s h a p e w a s o b t a i n e d a s s h o w n i n F i g . 1 3 . F r o m t h e r e s u l t s ,

i t w a s f o u n d t h a t t h e f i n al v o l u m e w a s r e d u c e d t o 8 0 o f t h e i n it ia l v o l u m e , a n d t h e m a x i -

m u m v o n M i s es s t r es s o f 2 .1 M P a w a s r e d u c e d t o 0 . 8 7 M P a . I t w a s ve r if i ed t h a t t h e i m p r o v e -

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

was s ucces s fu l .

Conc lu s i on s

I n t hi s p a p e r , a n i m p r o v e m e n t o f t h e g r o w t h - s t r a i n m e t h o d w a s s u g g e s te d b y g i v in g a d d i -

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

i m p r o v e m e n t o f t h e g r o w t h - s t ra i n m e t h o d f o r a s tr u c tu r e w i t h t w o o r m o r e d e f o r m a b l e f r ee

s u r f ac e s w a s s u c ce s sf u l. T h e r e f o r e , a s h a p e o p t i m i z a t i o n s y s t e m w a s b u i l t f o r g e n e r a l t w o -

d i m e n s i o n a l s t r u c tu r e s . T h i s s y s t e m a d o p t e d t h e P I D c o n t r o l f o r d e s i g n e r s t o c o n t r o l t h e

d e s i r e d v o l u m e o r s t r e s s e f f e c t i v e l y . I t i s e x p e c t e d t h a t t h i s s h a p e o p t i m i z a t i o n s y s t e m c a n b e

e x p a n d e d t o g e n e r a l t h r e e - d i m e n s i o n a l s t r u c t u re s w i t h t w o o r m o r e d e f o r m a b l e f r e e s u r fa c e s .

T h e s h a p e o p t i m i z a t i o n b y t h e i m p r o v e d g r o w t h - s t ra i n m e t h o d c o u l d b e m u c h m o r e e f fe c ti ve

t h a n t h e m a t h e m a t i c a l p r o g r a m m i n g m e t h o d f o r c o m p l i c a t e d t h r e e- d i m e n s i o n a l s t ru c tu r e s.

Ackn ow l e d g emen t

The au thor w ithes to acknowledge the f inancia l supp ort o f Ha nya ng Univers i ty , Kore a , made in the pro-

gram y ear of 1999.

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Shape opt im iza t ion for genera l two-dimen s ional s t ruc tures 125

[7] Azegami, H.: Shape op t imiza t ion o f solid s t ruc tures us ing the growth-s t ra in m ethod. SA E 921063

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Autho r s address : Prof . S. Y. Han, School of Mechanica l Engineering, Ha nya ng Univers i ty , 17 Haen g-

dang -Do ng, Sungdo ng-K u, Seoul 133-791, Ko rea (E-mail: syhan@ email .hany ang.ac .kr)

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2d Shape Optimization !