Approximation methods for combined thermal/structural design

,

NASA TP

1428 c.1

1

NASA Technical Paper 1428

Approximation Methods for Combined Thermal/Structural Design

Raphael T. Haftka and Charles P. Shore

JUNE 1979

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NASA

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I

NASA Technical Paper 1428

Approximation Methods for Combined Thermal/Structural Design

Raphael T. Haftka Illinois Iizstitzrte of Techuology Chicago, Illiizois Charles P. Shore Larzgley Research Ceizter Hanzpton, Virginia

NASA

National Aeronautics and Space Administration

Scientific and Technical Information Branch 1979

SUMMARY

?

Two approximation c o n c e p t s f o r combined t h e r m a l / s t r u c t u r a l d e s i g n are evaluated. The f i r s t c o n c e p t is an approximate t h e r m a l a n a l y s i s based on the f i r s t d e r i v a t i v e s of s t r u c t u r a l temperatures w i t h respect to d e s i g n v a r i a b l e s . Two commonly used f i r s t - o r d e r Taylor series expansions are examined. The d i r e c t and reciprocal e x p a n s i o n s are shown to be special members of a g e n e r a l f a m i l y of approximations, and it is shown t h a t f o r some c o n d i t i o n s o t h e r memb e r s of t h a t f a m i l y of approximations are more accurate. S e v e r a l examples are used t o compare t h e a c c u r a c y of t h e d i f f e r e n t expansions.

The second approximation c o n c e p t is t h e u s e of c r i t i c a l t i m e p o i n t s f o r combined t h e r m a l and stress a n a l y s e s of s t r u c t u r e s w i t h t r a n s i e n t l o a d i n g conditions. I t is shown t h a t s i g n i f i c a n t t i m e s a v i n g s may be r e a l i z e d by i d e n t i f y i n g c r i t i c a l t i m e p o i n t s and performing t h e stress a n a l y s i s f o r t h o s e p o i n t s only. T h i s approach is u s e d to d e s i g n an i n s u l a t e d p a n e l which is exposed t o t r a n s i e n t heating conditions. INTRODUCTION One of t h e major o b s t a c l e s t o t h e widespread u s e of automated s t r u c t u r a l d e s i g n (member s i z i n g ) p r o c e d u r e s is t h e need f o r l a r g e c o m p u t a t i o n a l resources t o perform repeated a n a l y s e s of a s t r u c t u r e throughout t h e d e s i g n process. To a l l e v i a t e t h i s problem, it is nuw common practice d u r i n g major p o r t i o n s of t h e d e s i g n p r o c e s s to u s e approximations f o r response f u n c t i o n s such as d i s p l a c e (See r e f s . 1 to 6 . ) Such ments, stresses, b u c k l i n g loads, or f l u t t e r speeds. approximations are o f t e n based on t h e f i r s t d e r i v a t i v e s of t h e response funct i o n s w i t h respect t o t h e d e s i g n v a r i a b l e s . One obvious t e c h n i q u e i n t h e u s e of t h e s e approximations is t o employ a f i r s t - o r d e r Taylor series ( i . e . , a l i n ear approximation) , h e r e i n c a l l e d t h e d i r e c t exp'ansion. An a l t e r n a t e approach is to approximate t h e d e s i r e d f u n c t i o n by a f i r s t - o r d e r Taylor s e r i e s i n t h e reciprocals of t h e d e s i g n v a r i a b l e s . Such a t e c h n i q u e , h e r e i n c a l l e d t h e reciprocal expansion, is e x a c t f o r stresses and d i s p l a c e m e n t s of s t a t i c a l l y d e t e r m i n a t e trusses and has been shown to be more accurate t h a n t h e d i r e c t expansion f o r o t h e r s t r u c t u r e s ( r e f . 5). Another u s e f u l approximation concept is t h a t of c o n s t r a i n t d e l e t i o n (e.g., r e f . 1 ) . C o n s t r a i n t s which are f a r from being c r i t i c a l are ignored d u r i n g p a r t s of t h i s d e s i g n process. P e r i o d i c updates are made of t h e number For c o n s t r a i n t s which are f u n c t i o n s of a of c o n s t r a i n t s t h a t are r e t a i n e d . parameter such as t i m e (parametric c o n s t r a i n t s ) , t h e c o n c e p t of c o n s t r a i n t d e l e t i o n can be extended ( r e f . 7 ) . When t h e dependence on t h e parameter is s u f f i c i e n t l y smooth, c r i t i c a l v a l u e s of t h a t parameter ( c r i t i c a l p o i n t s ) may be i d e n t i f i e d . The c o n s t r a i n t s f o r o t h e r v a l u e s of t h e parameter may be i g n o r e d ; however, t h e v a l u e s of t h e c r i t i c a l p o i n t s are updated p e r i o d i c a l l y as the d e s i g n changes.

T h i s paper is concerned w i t h a p p l i c a t i o n of t h e two c o n c e p t s , r e s p o n s e approximation and c r i t i c a l p o i n t s , to combined t h e r m a l / s t r u c t u r a l design. Both d i r e c t and reciprocal expansions of t h e response f u n c t i o n s are i n v e s t i g a t e d f o r accuracy i n p r e d i c t i n g t e m p e r a t u r e f i e l d s f o r s t e a d y - s t a t e and t r a n s i e n t t h e r m a l problems. The t i m e s a v i n g s a s s o c i a t e d w i t h t h e use of t h e s e expansions are also e v a l u a t e d w i t h t h e a i d of t w o examples; one of a s t e a d y - s t a t e response and t h e I n t h e second example, use is also made of o t h e r of t r a n s i e n t t h e r m a l response. t h e second approximation c o n c e p t (i.e., c r i t i c a l t i m e p o i n t s ) , and an i n v e s t i g a t i o n i s made of t h e r e s u l t i n g a c c u r a c y and t i m e s a v i n g s .

SYMBOLS

V a l u e s are g i v e n i n both S I and U.S. C u s t o m a r y U n i t s . and c a l c u l a t i o n s were made i n U.S. Customary U n i t s .

bar element areas, rmn2 ( i n 2 )

a1 r a 2 r a 3

Young's modulus, Pa ( p s i )

E f

(VI

fD (v) fR

(v)

response f u n c t i o n d i r e c t approximation t o f u n c t i o n

reciprocal approximation t o f u n c t i o n

g

constraint function

gC

cr it i c a l c o n s t r a i n t f u n c t i o n

k

thermal c o n d u c t i v i t y , W/m-

m (VI

mass o f structure, k g (lb)

"C

number o f c o n s t r a i n t s

"V

number of d e s i g n v a r i a b l e s

n,

number of t i m e p o i n t s

N,

I

f (V)

a p p l i e d l o a d s , kN/m

Ny I N q

6

h e a t rate, W/m2

t

t h i c k n e s s , mm ( i n . )

f

(Btu- i n/sec- f t 2- 9)

(lbf/in.)

(Btu/in2-sec)

tl ,t2,t3,t4

p l a t e element t h i c k n e s s e s , rmn ( i n . )

to,t41j,tgo

lamina t h i c k n e s s e s , mm ( i n . )

2

(v)

The measurements

T

temperature, OC (OF)

Ta

allowable temperature, OC ( O F )

Teq

transient, surface equilibrium temperature,

Vi

ith design variable

Vmi

offset value of ith design variable

V

vector of design variables

a

coefficient of thermal expansion, per OC (per OF)

P

mass density, kg/m3 (lb/in3)

‘I

time, sec

TC

critical time, sec

OC (?E’)

Subscripts: max

maximum

0

initial value APPROXIMATION CONCEPTS First-Order Taylor Series Expansions of Response Functions

The design problem considered herein is to find a vector of design varinv) that minimizes the mass ables V (with components ~ j ;j = 1, 2, m(V) of a structure and/or its thermal protection system subject to a set of nc constraints

. . .,

. where T is time. The constraint functions gi represent design limits on response functions such as displacements, stresses, or temperatures. Many algorithms for resizing the structure require evaluation of the derivatives of the response functions with respect to the design variables. These derivatives may be used to obtain approximations for the response functions which can replace the full analysis during parts of the design process. One approximation for a response function f(V), which is based on the value of the function and its derivatives at a design point Vo, is a first-order Taylor series expansion

3

where Voj are the components of Vo. T h i s appr ximation is herein called the direct expansion. An alternate expression (e.g., refs. 1 t o 3 , 5 , and 6), herein called the reciprocal expansion, is a first-order Taylor s e r i e s i n the reciprocals of the design variables. I t can be written as

t

The reciprocal expansion was originally proposed for approximating stresses and displacements i n structural analyses. For s t a t i c a l l y determinate trusses, stresses and displacements are indeed linear functions of the reciprocals of the cross-sectional areas of the members. For s t a t i c a l l y indeterminate structures, the reciprocal expansion is not exact, b u t studies (ref. 5) have shown it to be superior t o the d i r e c t expansion for a number of problems. The reciprocal expansion also has the added attraction t h a t it i s usually more conservative than the d i r e c t expansion i n estimating response functions such as stresses, displacements, or temperatures ( r e f . 2 ) . If f (V) is the temperature a t a point and V is a vector of insulation thicknesses, it may be expected t h a t af(V)/avj < 0. Since a l l design variables are positive, it i s easy to check from equations ( 2 ) and ( 3 ) that f R ( v ) > fD(v). The reciprocal expansion predicts an i n f i n i t e value for f ( V ) when any one of the variables v j goes t o zero. I t is, therefore, especially suitable for approximating functions, such as stresses and displacements i n a s t a t i c a l l y determinate truss, which go t o i n f i n i t y when the member areas go t o zero. Approximation of stresses i n redundant structures (or temperatures i n situations where multiple heat paths e x i s t ) is s l i g h t l y different i n that several variables m u s t vanish simultaneously for f ( V ) t o become i n f i n i t e . 7

Some understanding of the relative performance of the d i r e c t and reciprocal expansions for a redundant structure may be gained by an examination of the following simple example:

f(V) =

11 lOV1

+

v2

(4)

This example i l l u s t r a t e s a redundant situation where one of the design variables has a much larger effect than the other. Assume that f(V) is approximated 4

4

The d i r e c t and r e c i p by u s i n g i t s v a l u e and d e r i v a t i v e s a t vol = v02 = 1.0. rocal expansions f o r s e v e r a l o t h e r v a l u e s of vi and v2 are c o n t a i n e d i n t a b l e 1. Two c o n c l u s i o n s may be drawn from t a b l e 1 . F i r s t , t h e r e c i p r o c a l expansion is better f o r approximating f ( V ) for changes i n t h e dominant d e s i g n v a r i a b l e VI b u t n o t as good f o r approximations w i t h changes i n t h e o t h e r d e s i g n v a r i a b l e v2. Second, t h e reciprocal expansion is much b e t t e r w i t h changes i n t h e d e s i g n v a r i a b l e s which r e i n f o r c e each o t h e r (vi = v2 = 2) t h a n w i t h changes t h a t c a n c e l e a c h o t h e r (vi = 0.5, v2 = 6). G e n e r a l i z e d Approximation The f a c t t h a t t h e f u n c t i o n f ( V ) becomes v e r y l a r g e o n l y when s e v e r a l v a r i a b l e s go t o z e r o s i m u l t a n e o u s l y c a n be accounted f o r by t h e f o l l o w i n g more g e n e r a l form of t h e approximation:

where V m i is t h e i t h component of a v e c t o r of c o n s t a n t s chosen to account for t h e f a c t t h a t as one v a r i a b l e goes to zero t h e f u n c t i o n remains f i n i t e because o t h e r v a r i a b l e s may have nonzero values. The d i r e c t and r e c i p r o c a l e x p a n s i o n s a r e s p e c i a l c o n d i t i o n s of e q u a t i o n (5) w i t h t h e d i r e c t expansion corresponding to vmi + and t h e reciprocal expansion c o r r e s p o n d i n g t o ‘ V m i = 0. Because t h e approximation becomes unbounded when V i = V m i , V m i may be chosen as t h e v a l u e of v i which d r i v e s t h e e x a c t f u n c t i o n f ( V ) t o i n f i n i t y i f a l l other v a r i a b l e s v i remain c o n s t a n t (i.e., v j = voj where j # i ) . -Q)

For t h e f u n c t i o n o f e q u a t i o n ( 4 ) w i t h vOl = v02 = 1 , such a c o n s i d e r a t i o n and vm2 = -1 0. The l a s t column i n t a b l e 1 l e a d s t o t h e c h o i c e of vml = -0.1 shows t h a t t h e g e n e r a l approximation f o r t h i s choice is g e n e r a l l y superior t o t h e other t w o , i n t h e range c o n s i d e r e d . I n g e n e r a l , i t may n o t be possible to p i c k optimum v a l u e s of v m i , b u t small n e g a t i v e v a l u e s of V m i may be used t o a v o i d l a r g e errors i n t h e r e c i p r o c a l expansion when one v a r i a b l e g o e s t o zero. C r i t i c a l Time P o i n t s

The u s e of r e s p o n s e approximations a p p l i e s t o d e s i g n s under e i t h e r s t e a d y s t a t e or t r a n s i e n t t h e r m a l c o n s t r a i n t s ; however, a second approximation c o n c e p t may also be u s e f u l i n t r a n s i e n t problems. For t r a n s i e n t t h e r m a l loads, t h e t h e r m a l / s t r u c t u r a l d e s i g n problem is c h a r a c t e r i z e d by c o n s t r a i n t f u n c t i o n s (see eq. ( 1 ) ) which must be e n f o r c e d for an e n t i r e t i m e i n t e r v a l . Numerically, t h i s means t h a t a s e t of c l o s e l y spaced time p o i n t s , where j = 1 , 2, nT, m u s t be chosen such t h a t serious c o n s t r a i n t v i o l a i o n s a t i n t e r m e d i a t e t i m e p o i n t s are u n l i k e l y . Equation ( 1 ) may t h e n be replaced by

‘2

. . .,

5

L

i = 1, 2,

. . .,

j = 1 , 2,

. . ., nT

nc

I n most practical a p p l i c a t i o n s , t h e p r o d u c t nc+ w i l l be a v e r y l a r g e number and c a r r y i n g o u t t h e d e s i g n process w i t h t h i s f u l l complement of c o n s t r a i n t s may become p r o h i b i t i v e l y expensive.

*

I n s t e a d of monitoring a c o n s t r a i n t a t a s e t of + t i m e p o i n t s , it may be monitored o n l y a t i t s most c r i t i c a l p o i n t s (ref. 7 ) . T h i s concept is e x p l a i n e d by f i g u r e 1 which s c h e m a t i c a l l y shows t h e v a r i a t i o n of a c o n s t r a i n t The times of t h e Local minima of t h e c o n s t r a i n t f u n c t i o n f u n c t i o n with t i m e . (A, B, and C i n f i g . 1 ) are h e r e i n called c r i t i c a l t i m e p o i n t s . Instead of monitoring t h e c o n s t r a i n t f u n c t i o n a t a l l t i m e s , it is monitored o n l y a t t h o s e t i m e s . However, t h e c r i t i c a l t i m e p o i n t s d r i f t as t h e d e s i g n changes, and t h e c o n s t r a i n t must be c a l c u l a t e d a t many t i m e p o i n t s to a c c u r a t e l y locate the c r i t i c a l t i m e p o i n t s . The u s e f u l n e s s of t h e c r i t i c a l - t i m e - p o i n t s c o n c e p t d e r i v e s from t h e f a c t t h a t , f o r p o r t i o n s o f t h e d e s i g n process, t h e d r i f t o f t h e c r i t i c a l t i m e p o i n t s may be n e g l e c t e d . For t h e s e p o r t i o n s of t h e d e s i g n process, t h e c o n s t r a i n t f u n c t i o n h a s t o be e v a l u a t e d f o r o n l y a r e l a t i v e l y small number of t i m e p o i n t s . To prove t h a t t h e d r i f t i n t h e l o c a t i o n of t h e c r i t i c a l p o i n t s may be n e g l e c t e d to a first-order approximation, it is assumed a t f i r s t t h a t a cons t r a i n t f u n c t i o n g(V,-c) h a s a s i n g l e local minimum i n t h e t i m e r e g i o n of i n t e r e s t . Equation ( 1 ) f o r t h e c o n s t r a i n t gi(V,.r) is replaced by

where T~ is t h e t i m e for which g i is most c r i t i c a l ( t h e c r i t i c a l t i m e p o i n t ) . The partial d e r i v a t i v e o f g c i w i t h respect t o a d e s i g n v a r i a b l e is

vj

. For i n t e r i o r c r i t i c a l p o i n t s (A and B i n f i g . l ) , 3gifa.r = 0, and for boundary minimum p o i n t s ( p o i n t c ) , aTc/avj = o i f agi/aT # 0. T h e r e f o r e ,

6

' t

Equation ( 9 ) shows t h a t t h e d r i f t i n t h e c r i t i c a l t i m e p o i n t may be n e g l e c t e d to a f i r s t - o r d e r approximation. D r i f t may be n e g l e c t e d n o t because it is small b u t because it does n o t a f f e c t t h e f i r s t d e r i v a t i v e s of t h e c o n s t r a i n t gci. In practice, t h e c r i t i c a l time p o i n t s are c a l c u l a t e d p e r i o d i c a l l y and are f r o z e n between u p d a t i n g s . However, t h e c o n c e p t remains u s e f u l , even i f t h e c r i t i c a l p o i n t s r e q u i r e f r e q u e n t u p d a t i n g , because of t h e s a v i n g s i n t h e d e r i v a t i v e calculation. When t h e c o n s t r a i n t f u n c t i o n has more t h a n one local minimum as i n f i g u r e 1 , each c r i t i c a l p o i n t must be a s s i g n e d a separate c o n s t r a i n t gci; o t h e r w i s e , e q u a t i o n ( 7 ) may y i e l d a d i s c o n t i n u o u s d e r i v a t i v e of g c i when t h e g l o b a l minimum is switched from one l o c a l minimum t o a n o t h e r .

I .

EVALUATION OF APPROXIMATION TECHNIQUES

To judge t h e u s e f u l n e s s o f an approximation c o n c e p t , t h e q u e s t i o n of what magnitude of change i n d e s i g n parameters r e s u l t s i n u n a c c e p t a b l e errors m u s t be answered. Although it is d e s i r a b l e t o o b t a i n a g e n e r a l answer to t h i s quest i o n , t h i s d i d n o t appear p o s s i b l e i n t h e p r e s e n t s t u d i e s . Instead, the present w o r k u s e s examples of b o t h s t e a d y - s t a t e and t r a n s i e n t t h e r m a l / s t r u c t u r a l d e s i g n r e s u l t s to assess t h e u s e f u l n e s s of t h e t w o approximation c o n c e p t s d i s c u s s e d i n t h e preceding section. The s t e a d y - s t a t e example is a t i t a n i u m p a n e l w i t h i n t e g r a l aluminum b a r s , and t h e t r a n s i e n t one is an i n s u l a t e d p a n e l .

Approximate Thermal A n a l y s i s of a Titanium P a n e l The accuracy o f b o t h t h e d i r e c t and reciprocal expansions f o r s t e a d y - s t a t e t h e r m a l a n a l y s i s w a s i n v e s t i g a t e d i n r e f e r e n c e 8 f o r a c o n f i g u r a t i o n which cons i s t e d o f a t i t a n i u m p a n e l w i t h aluminum b a r s as shown i n f i g u r e 2. T h i s conf i g u r a t i o n is r e p r e s e n t a t i v e o f a c l a s s of s t r u c t u r e s i n which one m a t e r i a l s a t i s f i e s s t r e n g t h r e q u i r e m e n t s and t h e o t h e r acts as an e f f i c i e n t conductor t o t r a n s f e r i n c i d e n t h e a t t o a h e a t s i n k . The s i n k is r e p r e s e n t e d by t h e p a n e l edge maintained a t T = -18O C ( O o F ) . The i n c i d e n t h e a t l o a d , material propert i e s , and remaining boundary c o n d i t i o n s are also shown i n f i g u r e 2. The accur a c y of t h e d i r e c t and reciprocal e x p a n s i o n s f o r p r e d i c t i n g t h e maximum tempera t u r e is shown i n f i g u r e 3 f o r changes i n t h e t h i c k n e s s of one-quarter of t h e p l a t e and i n t h e area of one aluminum b a r ( b o t h v a r i a b l e s are changed proportionately). I n i t i a l l y t h e e i g h t t r i a n g u l a r p l a t e elements were 12.7 rmn (0.5 i n . ) t h i c k and t h e f o u r bar e l e m e n t s had areas of 645.2 mm2 (1.0 i n 2 ) . The r e s u l t s shown i n f i g u r e 3 i n d i c a t e t h a t t h e reciprocal expansion is more a c c u r a t e t h a n t h e d i r e c t expansion. The p o r t i o n s of t h e p l a t e and t h e b a r s t h a t were r e s i z e d c o n s t i t u t e t h e major p a t h of h e a t conduction to t h e h e a t s i n k . When t h e i r t h i c k n e s s and area are reduced to z e r o , t h e temperature becomes unbounded so t h a t t h e reciprocal expansion may be expected to be b e t t e r t h a n t h e d i r e c t one. The b i a s toward t h e reciprocal expansion can be c a r r i e d t o an extreme by uniformly i n c r e a s i n g or d e c r e a s i n g t h e t h i c k n e s s and areas of a l l elements. When t h i s is done, t h e error i n t h e reciprocal expansion becomes minute. A d i f f e r e n t s i t u a t i o n o c c u r s when some o f t h e d e s i g n v a r i a b l e s are 7

i n c r e a s e d and o t h e r s are d e c r e a s e d (a common Occurrence i n o p t i m i z a t i o n probl e m s ) . Performance of t h e e x p a n s i o n s f o r such mixed changes i n d e s i g n variables is shown i n f i g u r e 4. The p e r c e n t a g e change i n maximum plate temperature is shown as a f u n c t i o n o f t h e p e r c e n t a g e of t h e t o t a l change i n d e s i g n v a r i a b l e s from t h e o r i g i n a l d e s i g n to t h e f i n a l d e s i g n . This is i n d i c a t e d i n t h e f i g u r e The reciprocal i n s e t . For t h e s e c o n d i t i o n s , t h e d i r e c t expansion is b e t t e r . expansion does n o t perform as w e l l because of t h e redundancy of t h e h e a t p a t h s . T h i s redundancy creates a s i t u a t i o n similar to t h a t of t h e simple example of e q u a t i o n ( 4 ) p r e v i o u s l y noted. r

Approximate Thermal A n a l y s i s of an I n s u l a t e d P a n e l The q u a l i t y o f t h e approximate t h e r m a l a n a l y s i s was i n v e s t i g a t e d f o r t r a n s i e n t h e a t i n g of t h e i n s u l a t e d p a n e l shown i n f i g u r e 5. A t r a n s i e n t s u r f a c e e q u i l i b r i u m t e m p e r a t u r e Teq, typical of a r e e n t r y h e a t i n g t r a j e c t o r y , is applied to t h e o u t e r s u r f a c e of t h e i n s u l a t i o n l a y e r . The i n s u l a t i o n protects a balanced, symmetric, graphite/epoxy composite p a n e l w i t h Oo, +45O, and 90° plies. The m a t e r i a l properties of t h e composite p a n e l and i n s u l a t i o n are g i v e n i n t a b l e 2. The t h e r m a l and stress a n a l y s e s of t h e p a n e l were carried o u t u s i n g a modified v e r s i o n of t h e computer program employed to o b t a i n t h e r e s u l t s of r e f e r e n c e 8. The dependence of t h e l a m i n a t e temperature on i n s u l a t i o n t h i c k n e s s is shown i n f i g u r e 6 f o r one t i m e p o i n t . Also shown are t h e d i r e c t and reciprocal expans i o n s f o r t h a t t e m p e r a t u r e based on d e r i v a t i v e s c a l c u l a t e d for a 1 .52-mm (0.060-in.) l a m i n a t e t h i c k n e s s and 102-mm (4.0-in.) i n s u l a t i o n t h i c k n e s s . The reciprocal expansion is r e a s o n a b l y good over t h e e n t i r e range of i n s u l a t i o n The d i r e c t expansion t h i c k n e s s e s , which is e q u a l to t h e base v a l u e A50 p e r c e n t . is c o n s i d e r a b l y less a c c u r a t e . The b e t t e r performance of t h e reciprocal expans i o n is u n d e r s t a n d a b l e because t h e t e m p e r a t u r e becomes v e r y high when t h e insul a t i o n t h i c k n e s s goes t o zero. The dependence o f t h e l a m i n a t e temperature on l a m i n a t e t h i c k n e s s is shown i n f i g u r e 7 . The temperature is much less s e n s i t i v e t o t h e l a m i n a t e t h i c k n e s s t h a n t o t h e i n s u l a t i o n t h i c k n e s s so t h a t t h e errors f o r both approximations are smaller. For t h i s set of c a l c u l a t i o n s however, t h e d i r e c t expansion is b e t t e r t h a n t h e reciprocal one. The r e a s o n is t h a t t h e t e m p e r a t u r e is n o t expected to become v e r y h i g h when t h e l a m i n a t e t h i c k n e s s goes to z e r o because of t h e p r e s e n c e of t h e i n s u l a t i o n ( t h e same phenomenon w a s noted i n t h e example of eq. ( 4 ) f o r changes i n v 2 ) . It is a l s o shown i n f i g u r e 7 t h a t an even b e t t e r approximation can b e o b t a i n e d w i t h Vm = -4.06 mm (-0.16 i n . ) .

For t h e c a l c u l a t i o n s i l l u s t r a t e d i n f i g u r e 7 , t h e d e s i g n v a r i a b l e w a s a s s i g n e d to t h e l a m i n a t e t h i c k n e s s . O f t e n , d e s i g n v a r i a b l e s are a s s i g n e d to t h e t h i c k n e s s of each p l y . When one p l y t h i c k n e s s is changed w h i l e o t h e r s a r e k e p t c o n s t a n t , t h e reciprocal expansion w i l l b e even less a c c u r a t e . This i n a c c u r a c y was demonstrated by r e p e a t i n g t h e c a l c u l a t i o n s p r e s e n t e d i n f i g u r e 7 with i n d i v i d u a l d e s i g n v a r i a b l e s a s s i g n e d t o t h e Oo, t45O, and 90° p l i e s . The i n i t i a l l a m i n a t e t h i c k n e s s , 1.52 mm (0.060 i n . ) , c o n s i s t e d of 0.51 mm (0.020 i n . ) f o r each p l y d i r e c t i o n . Then, t h e 90°- and +45O-ply t h i c k n e s s e s were k e p t c o n s t a n t and o n l y t h e Oo-ply t h i c k n e s s was i n c r e a s e d . The reciprocal

8

t

I

I

expansion f o r t h e s e c o n d i t i o n s is shown i n f i g u r e 7 by t h e c i r c u l a r symbols. The d i r e c t expansion is n o t a f f e c t e d by a s s i g n i n g d e s i g n v a r i a b l e s i n t h i s manner. The curve w i t h t h e c i r c u l a r symbols i n f i g u r e 7 shows t h e danger o f choosing V m i = 0 (reciprocal expansion) f o r a h i g h l y redundant system such as a composite l a m i n a t e . The c h o i c e of V m i i s based on e s t i m a t i o n s of t h e combined e f f e c t of a l l o t h e r d e s i g n v a r i a b l e s when v i goes t o zero. Based on t h e i n s u l a t e d p a n e l example, t h e f o l l o w i n g recommendations can be made:

t

1 . I f t h e r e s p o n s e f u n c t i o n is n o t expected to become v e r y l a r g e when

goes to z e r o , t h e d i r e c t expansion is appropriate and be a l a r g e n e g a t i v e number.

Vmi

Vi

should b e chosen t o

2. I f t h e response f u n c t i o n may be expected to become large b u t f i n i t e when V i = 0 , choose V m i to be a small n e g a t i v e f r a c t i o n of a typical v a l u e of V i . T h i s is a s a f e t y measure to guard a g a i n s t e x c e s s i v e errors i n t h e approximation when V i changes by an o r d e r of magnitude.

U s e of C r i t i c a l T i m e P o i n t s

The approximation c o n c e p t o f f o l l o w i n g o n l y c r i t i c a l t i m e p o i n t s w a s also checked f o r t h e i n s u l a t e d p a n e l . Shown i n f i g u r e 8 are typical changes i n temperature h i s t o r i e s as t h e d e s i g n changes. I n d i c a t e d i n t h e f i g u r e is an a p p r e c i a b l e d r i f t i n t h e c r i t i c a l t i m e f o r t h e maximum t e m p e r a t u r e c o n s t r a i n t . The same phenomenon is i l l u s t r a t e d i n f i g u r e 9 by showing t h e d i f f e r e n c e between f o l l o w i n g t h e maximum t e m p e r a t u r e , t h a t t a k e s i n t o account t h e d r i f t of t h e c r i t i c a l t i m e p o i n t , and f o l l o w i n g t h e temperature a t t h e t i m e p o i n t which w a s maximum f o r t h e base d e s i g n ( i n s u l a t i o n t h i c k n e s s = 101.6 mm ( 4 . 0 i n . ) , l a m i n a t e t h i c k n e s s = 1 . 5 2 mm (0.06 i n . ) ) . The t w o c u r v e s i n f i g u r e 9 have t h e same d e r i v a t i v e s a t t h e b a s e d e s i g n p o i n t a s p r e d i c t e d by e q u a t i o n ( 9 ) . However, t h e d i f f e r e n c e between t h e two c u r v e s becomes s u b s t a n t i a l due to t h e d r i f t of t h e t i m e p o i n t of maximum temperature. There are i n d i c a t i o n s i n f i g u r e s 8 and 9 t h a t t h e c r i t i c a l t i m e p o i n t s r e q u i r e updating more o f t e n t h a n t h e approximate (Compare f i g s . 6 , 8 , and 9 . ) thermal a n a l y s e s . APPLICATION TO OPT1M I ZATION O p t i m i z a t i o n o f a Titanium P l a t e The t i t a n i u m p l a t e w i t h aluminum b a r s shown i n f i g u r e 2 w a s designed f o r minimum mass under a c o n s t r a i n t on t h e maximum temperature. A r e s i z i n g a l g o r i t h m based on o p t i m a l i t y c r i t e r i a ( r e f . 9 ) was used f o r o p t i m i z a t i o n . D e r i v a t i v e s of t h e t e m p e r a t u r e s w i t h respect to d e s i g n v a r i a b l e s were calculated analytically.

Mass i t e r a t i o n h i s t o r i e s f o r t h i s d e s i g n problem are shown i n f i g u r e 1 0 . During t h e f i r s t few i t e r a t i o n s , a l l d e s i g n v a r i a b l e s were reduced u n t i l cons t r a i n t v i o l a t i o n s o c c u r r e d ; t h u s , a reciprocal expansion may be e x p e c t e d to b e more a c c u r a t e t h a n a d i r e c t expansion. Succeeding d e s i g n i t e r a t i o n s t e n d to 9

r e d i s t r i b u t e material among t h e s t r u c t u r a l e l e m e n t s t o s a t i s f y t h e d e s i g n cons t r a i n t s ; t h e r e f o r e , a d i r e c t expansion may be expected t o be more a c c u r a t e f o r t h a t phase of t h e o p t i m i z a t i o n process. Accordingly, t h r e e separate calc u l a t i o n s were made u s i n g approximate t h e r m a l a n a l y s i s : i n the f i r s t , a reciprocal expansion w a s used i n both phases of t h e o p t i m i z a t i o n process; i n t h e second, a d i r e c t expansion was used i n b o t h p h a s e s ; and i n t h e t h i r d , a reciprocal expansion w a s used i n the f i r s t phase and a direct expansion was u s e d i n t h e second phase. For t h e i n i t i a l d e s i g n , t h e p l a t e t h i c k n e s s w a s set a t 76 m ( 3 .O i n . ) and t h e aluminum bars had areas of 9290 nun2 (1 4 . 4 i n 2 ) each. D e r i v a t i v e s were calculated f o r t h e i n i t i a l d e s i g n , and t h e t e m p e r a t u r e w a s approximated up to i t e r a t i o n 1 0 . A t t h a t p o i n t , a new f u l l a n a l y s i s was u s e d to g e n e r a t e a new approximation which w a s used f o r t h e rest of t h e d e s i g n process. The d e c i s i o n to g e n e r a t e a new approximation a t i t e r a t i o n 1 0 was based on f u l l a n a l y s i s r e s u l t s where changes i n t h e d e s i g n v a r i a b l e s became mixed r a t h e r than uniform. Although t h e mass changed v e r y l i t t l e after a b o u t t h e 4 0 t h i t e r a t i o n , t h e o p t i m a l i t y c r i t e r i a r e q u i r e d an a d d i t i o n a l 40 iterat i o n s t o a c c u r a t e l y s a t i s f y t h e temperature c o n s t r a i n t s . The f i n a l d e s i g n s and maximum temperature errors (which were o b t a i n e d from a f u l l a n a l y s i s o f t h a t f i n a l d e s i g n ) are g i v e n i n t a b l e 3 . I t can be seen t h a t t h e d i r e c t expansion l a c k s s u f f i c i e n t accuracy t o g u i d e t h e optim i z a t i o n process c o m p l e t e l y to a converged s o l u t i o n . The reciprocal expanThis results i n a s i o n also b e g i n s to d i v e r g e a f t e r a b o u t 30 i t e r a t i o n s . d e s i g n s u b s t a n t i a l l y d i f f e r e n t from t h e f u l l a n a l y s i s w i t h 20 p e r c e n t greater mass and a maximum t e m p e r a t u r e error of 49.9O C (89.8O F ) . However, t h e u s e of t h e reciprocal expansion i n t h e f i r s t phase o f t h e o p t i m i z a t i o n and t h e d i r e c t expansion i n t h e second phase y i e l d s a f i n a l design v e r y close to t h e f u l l a n a l y s i s d e s i g n w i t h o n l y a 1 . 1 p e r c e n t mass d i f f e r e n c e and a maximum A d d i t i o n a l l y , t h e approximate t h e r m a l temperature error o f 1 4 . go C (26.8O F) a n a l y s i s reduces the t o t a l o p t i m i z a t i o n t i m e by a b o u t 40 p e r c e n t . The optim i z a t i o n algorithm was m d i f i e d to a u t o m a t i c a l l y select t h e expansion c o n c e p t based on whether t h e changes i n d e s i g n v a r i a b l e s a l l have t h e same s i g n or d i f f e r e n t s i g n s . The results o b t a i n e d by such a modified algorithm were essent i a l l y t h e same as b e f o r e .

.

O p t i m i z a t i o n o f an I n s u l a t e d P a n e l The i n s u l a t e d pane1 shown i n f i g u r e 5 was designed for minimum mass under temperature and stress c o n s t r d n t s . The stress c o n s t r a i n t w a s based on t h e Tsai-Wu f a i l u r e c r i t e r i o n (ref. IO). The applied loads and t h e material s t r e n g t h allowables are g i v e n i n t a b l e 2. The o p t i m i z e r f o r t h e programs f o r a n a l y s i s and r e s i z i n g of s t r u c t u r e s 11 , was used for o p t i m i z a t i o n . PARS is a c o l l e c t i o n of programs developed for t h e minimum mass d e s i g n of s t r u c t u r e s s u b j e c t to stress, d i s p l a c e m e n t , and f l u t t e r c o n s t r a i n t s . The PARS o p t i m i z e r employs t h e s e q u e n t i a l u n c o n s t r a i n e d m i n i m i z a t i o n t e c h n i q u e (SUMT), described i n r e f e r e n c e 1 2 , w i t h a quadratic extended i n t e r i o r p e n a l t y f u n c t i o n , described i n r e f e r e n c e 13. Newton's method w i t h approximate second d e r i v a t i v e s ( r e f . 1 4 ) is used to g e n e r a t e d i r e c t i o n s for one-dimensional s e a r c h e s for each uncon(PARS) , described i n r e f e r e n c e

10

f'

s t r a i n e d minimization. The program approximates t h e c o n s t r a i n t f u n c t i o n s duri n g one-dimensional s e a r c h e s u s i n g v a l u e s of V m i s p e c i f i e d by t h e user. I n r e s i z i n g t h e p a n e l , new temperatures were approximated based on t h e v a l u e s and d e r i v a t i v e s of t h e i n i t i a l design temperatures w i t h o u t any updating. That is, a f u l l t h e r m a l a n a l y s i s w a s performed o n l y t w i c e ; once f o r t h e i n i t i a l d e s i g n and once f o r t h e f i n a l design. Exact c a l c u l a t i o n s of t h e stresses from t h e approximate temperatures were per formed once f o r each one-dimensional s e a r c h . During each s e a r c h , t h e stresses were approximated u s i n g t h e same v a l u e s of vmi used f o r approximating t h e temperatures. f

S t a r t i n g w i t h t w o i n i t i a l d e s i g n s , s t u d i e s were c a r r i e d o u t to t e s t t h e a c c u r a c y and t i m e s a v i n g s associated w i t h t h e two approximation concepts. The f i r s t i n i t i a l d e s i g n had a 101.6-mm (4.0-in.) i n s u l a t i o n t h i c k n e s s and p l y t h i c k n e s s o f t o = t g o = 0.76 mm (0.03 i n . ) and t 4 5 = 1.52 mm (0.06 i n . , combined t h i c k n e s s o f +4S0 and -45O p l i e s ) . T h i s is f a i r l y close t o t h e optimum i n s u l a t i o n t h i c k n e s s , b u t f a r from t h e optimum l a m i n a t e t h i c k n e s s . The second i n i t i a l d e s i g n had a 203-mm (8.0-in.) i n s u l a t i o n t h i c k n e s s and p l y t h i c k n e s s of t o = t 4 5 = t g 0 = 0.508 nun (0.02 i n . ) , which is close t o t h e o p t i m u m l a m i n a t e t h i c k n e s s b u t f a r from t h e optimum i n s u l a t i o n t h i c k n e s s .

.-

Performance u s i n g response approximations The r e s u l t s of t h e optimizat i o n s t a r t i n g w i t h t h e f i r s t i n i t i a l d e s i g n are summarized i n t h e f i r s t t h r e e r o w s of t a b l e 4. For t h i s d e s i g n , both approximations compare f a v o r a b l y w i t h t h e f u l l a n a l y s i s . T h i s may be e x p e c t e d , because t h e f i n a l d e s i g n d i f f e r s from t h e i n i t i a l d e s i g n p r i m a r i l y i n t h e l a m i n a t e t h i c k n e s s which h a s a w e a k e f f e c t on t h e temperature. (See f i g . 7 . ) The f i n a l d e s i g n t h i c k n e s s e s determined u s i n g t h e reciprocal expansion are s u b s t a n t i a l l y d i f f e r e n t from t h o s e o b t a i n e d from t h e f u l l a n a l y s i s . However, t h e mass is o n l y 3.6 p e r c e n t g r e a t e r t h a n t h a t from t h e f u l l a n a l y s i s d e s i g n . The maximum t e m p e r a t u r e error f o r t h e r e c i p r o c a l expansion f i n a l d e s i g n is o n l y 3 . 6 7 O C ( 6 . 6 O F ) , which is smaller t h a n t h e error f o r t h e d i r e c t expansion. This does n o t i n d i c a t e t h a t t h e reciprocal expansion is superior t o t h e d i r e c t expansion; r a t h e r , it i n d i c a t e s t h a t t h e d i f f e r e n c e between t h e i n i t i a l and f i n a l d e s i g n t h i c k n e s s e s from t h e reciprocal expansion is less t h a n t h a t from t h e d i r e c t expansion.

*

C o n s i d e r a b l e t i m e s a v i n g s are achieved by u s i n g t h e approximate t h e r m a l a n a l y s i s . The approximate thermal a n a l y s i s requires less t h a n 5 p e r c e n t of t h e t i m e r e q u i r e d f o r a f u l l a n a l y s i s . However, t h e t o t a l o p t i m i z a t i o n t i m e can o n l y be reduced to 30 p e r c e n t of t h e time r e q u i r e d f o r f u l l a n a l y s e s because of t h e t i m e needed f o r t h e i n i t i a l f u l l a n a l y s i s and f o r t h e o p e r a t i o n o f t h e PARS o p t i m i z e r . The f i r s t t h r e e r o w s o f t a b l e 5 summarize t h e r e s u l t s o b t a i n e d w i t h t h e second i n i t i a l design. T h i s time t h e r e w a s a l a r g e change i n i n s u l a t i o n t h i c k n e s s , and as a r e s u l t , b o t h approximations performed r a t h e r p o o r l y . The d i r e c t expansion y i e l d e d an i n f e a s i b l e d e s i g n w i t h a maximum temperature error o f 292O C (525O F ) . The reciprocal expansion produced a much b e t t e r d e s i g n which is v e r y close to t h e t r u e optimum w i t h a maximum t e m p e r a t u r e d i f f e r e n c e o f Based on t h e t e m p e r a t u r e error , t h e f i n a l d e s i g n is, i n 66.7O C (1 20° F ) f a c t , much closer to t h e t r u e optimum t h a n may be expected. T h i s is because

.

11

t h e error peaks f o r n o n c r i t i c a l t i m e p o i n t s . For t h e c r i t i c a l t i m e p o i n t s , t h e temperature error is an o r d e r of magnitude smaller. The r e s u l t s i n t a b l e s 4 and 5 for t h e reciprocal expansion were o b t a i n e d w i t h V m i = 0 for t h e p l y t h i c k n e s s d e s i g n v a r i a b l e s . When t h e same calcul a t i o n s were r e p e a t e d f o r V m i = -1.02 mm (-0.04 i n . ) , t h e r e was l i t t l e change. However, t h e changes i n p l y t h i c k n e s s from t h e i n i t i a l d e s i g n t o t h e f i n a l d e s i g n were moderate. When p l y t h i c k n e s s e s are changed d r a s t i c a l l y d u r i n g t h e d e s i g n process, it becomes n e c e s s a r y t o compensate f o r t h e t h i c k n e s s of o t h e r p l i e s v i a Vmi. An example is an i n i t i a l d e s i g n w i t h i n s u l a t i o n t h i c k n e s s of 122 mm (4.8 in.) and p l y t h i c k n e s s of t o = t g 0 = 0.127 mm (0.005 i n . ) and With V m i = -1.02 mm (-0.04 i n . ) , t h e f i n a l mass t 4 5 = 1.27 mm (0.05 i n . ) . per u n i t area w a s 20.67 kg/m2 (4.233 l b / f t 2 ) , or 3.6 p e r c e n t h e a v i e r t h a n t h e t r u e optimum, w i t h a maximum temperature error o f 7.4O C ( 1 3.3O F ) With V m i = 0 mm, t h e f i n a l mass per u n i t area was 21.35 kg/m2 (4.372 l b / f t 2 ) , or 7.1 p e r c e n t h e a v i e r t h a n t h e t r u e optimum, w i t h a maximum temperature error o f 16.4O C (29.6O F ) .

.

Based on t h e s e s t u d i e s , it appears t h a t it may be f e a s i b l e to employ an approximate thermal a n a l y s i s u s i n g t h e r ec ipr oca1 expansion w i t h o u t updating t h e approximation. I f t h e temperature error for t h e f i n a l d e s i g n is l a r g e enough to p u t t h e o p t i m a l i t y of t h a t d e s i g n i n t o doubt, t h e o p t i m i z a t i o n may be r e s t a r t e d from t h a t f i n a l d e s i g n . This w a s done f o r t h e f i n a l d e s i g n g i v e n i n table 5 f o r t h e reciprocal expansion. The r e s t a r t e d o p t i m i z a t i o n r e q u i r e d 74.0 seconds of c e n t r a l p r o c e s s i n g u n i t (CPU) t i m e t o converge to t h e e x a c t optimum i n a few i t e r a t i o n s .

.-

The performance of t h e critical-time Performance u s i n g c r i t i c a l times approach was e v a l u a t e d next. Because of t h e d r i f t i n t h e c r i t i c a l t i m e p o i n t d i s c u s s e d i n " U s e o f C r i t i c a l T i m e P o i n t s " and shown i n f i g u r e s 8 and 9, c r i t ical t i m e s were updated a t t h e beginning of each one-dimensional s e a r c h . For each one-dimensional s e a r c h , stresses were c a l c u l a t e d o n l y a t t w o t i m e p o i n t s : t h e time of maximum stress r a t i o and t h a t o f maximum temperature. The results of t h e o p t i m i z a t i o n u s i n g t h i s c o n c e p t are summarized i n t h e l a s t two rows of t a b l e s 4 and 5. I t can be seen t h a t a d d i t i o n a l s a v i n g s of a b o u t 50 p e r c e n t i n computation t i m e are r e a l i z e d w i t h o u t any s u b s t a n t i a l change i n t h e d e t a i l s of t h e f i n a l design.

CONCLUDING REMARKS

Two approximation c o n c e p t s were e v a l u a t e d f o r u s e i n t h e d e s i g n of thermal/ s t r u c t u r a l systems for minimum mass. The f i r s t c o n c e p t w a s t h e u s e o f response approximations, and t h e second concept w a s t h e u s e o f c r i t i c a l t i m e p o i n t s . The e v a l u a t i o n s were c a r r i e d o u t by t h e use of s t e a d y - s t a t e and t r a n s i e n t example problems. The f o l l o w i n g c o n c l u s i o n s are t h e r e f o r e v a l i d o n l y t o t h e e x t e n t t h a t t h e y a p p l y t o t h o s e examples. The f i r s t concept which was e v a l u a t e d made u s e of approximate thermal analy s i s . Two commonly used approximations, t h e d i r e c t and reciprocal expansions, were shown to be special members of a more g e n e r a l f a m i l y o f approximations c h a r a c t e r i z e d by a set o f parameters V m i which c o n t r o l s t h e p o i n t where t h e 12

.

i

approximations become unbounded. The f o l l a w i n g c o n c l u s i o n s were reached regardi n g t h e performance of t h e approximations. 1 . The reciprocal expansion is more a c c u r a t e than t h e d i r e c t expansion when t h e temperature becomes v e r y h i g h as a s i n g l e d e s i g n v a r i a b l e goes t o zero.

2. I f t h e t e m p e r a t u r e does n o t become v e r y h i g h when a s i n g l e d e s i g n v a r i a b l e goes t o z e r o b u t as s e v e r a l d e s i g n v a r i a b l e s g o to z e r o s i m u l t a n e o u s l y ( r e d u n d a n t system) , t h e reciprocal expansion p r o v i d e s a b e t t e r approximation The d i r e c t when t h e s e d e s i g n v a r i a b l e s e i t h e r i n c r e a s e or d e c r e a s e t o g e t h e r . expansion is b e t t e r when some d e s i g n v a r i a b l e s are i n c r e a s e d as o t h e r s d e c r e a s e .

3. The reciprocal expansion is a c c u r a t e enough f o r approximating tempera t u r e s w i t h o u t needing u p d a t i n g by a f u l l a n a l y s i s f o r changes of up to 50 perc e n t i n t h e v a l u e s of d e s i g n v a r i a b l e s d u r i n g t h e d e s i g n process. 4 . For a redundant system such as a composite l a m i n a t e , t h e g e n e r a l i z e d approximation w i t h a value of t h e parameter which compensates f o r the redundancy is found t o be b e t t e r t h a n e i t h e r t h e d i r e c t or reciprocal expansions.

The second c o n c e p t t h a t w a s e v a l u a t e d , t h e u s e of c r i t i c a l t i m e p o i n t s , w a s found t o w o r k w e l l i f t h e c r i t i c a l t i m e p o i n t s were updated p e r i o d i c a l l y . For t r a n s i e n t example problems, t h e combined u s e of t h e t w o approximation c o n c e p t s r e s u l t e d i n an o r d e r o f magnitude r e d u c t i o n i n computation t i m e . Langley Research Center N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o n Hampton, VA 23665 A p r i l 1 0 , 1979

13

REFERENCES 1 . Schmit, L. A., Jr. ; and F a r s h i , B.: Some Approximation Concepts f o r S t r u c t u r a l S y n t h e s i s . AIAA J., v o l . 1 2 , no. 5, May 1974, pp. 692-699. 2. S t a r n e s , James H., J r . ; and H a f t k a , Raphael T.: P r e l i m i n a r y Design of Composite Wings for Buckling, S t r e n g t h and Displacement C o n s t r a i n t s . A C o l l e c t i o n of T e c h n i c a l P a p e r s AIAA/ASME 1 9 t h S t r u c t u r e s , S t r u c t u r a l ( A v a i l a b l e as Dynamics and Materials Conference, A p r . 1978, pp. 1-13. AIAA Paper No. 78-466.)

-

3. A u s t i n , Fred: A Rapid O p t i m i z a t i o n Procedure for S t r u c t u r e s S u b j e c t e d to M u l t i p l e C o n s t r a i n t s . AIAA/ASME 1 8 t h S t r u c t u r e s , S t r u c t u r a l Dynamics & Materials Conference, Mar. 1977, pp. 71-79. ( A v a i l a b l e as AIAA Paper No. 77-374.) 4. S t o r a a s l i , O l a f 0.; and S o b i e s z c z a n s k i , Jaroslaw: On t h e Accuracy of t h e Taylor Approximation for S t r u c t u r e Resizing. AIAA J., v o l . 1 2 , no. 2 , Feb. 1974, pp. 231-233.

5. Noor , Ahmed K. ; and Lowder, Harold E. : S t r u c t u r a l R e a n a l y s i s V i a a Mixed Method. Comput. & S t r u c t . , v o l . 5, no. 1 , A p r . 1975, pp. 9-12. 6. Schmit, Lucien A., J r . ; and Miura, Hirokazu: Approximation Concepts f o r E f f i c i e n t S t r u c t u r a l S y n t h e s i s . NASA CR-2552, 1976. 7. Haftka, Raphael T. : P a r a m e t r i c C o n s t r a i n t s With A p p l i c a t i o n t o Optimizat i o n for F l u t t e r Using a Continuous F l u t t e r C o n s t r a i n t . AIAA J., v o l . 13, no. 4, A p r . 1975, pp. 471-475. Develop8. Adelman, Howard M.; Sawyer, P a t r i c i a L.; and Shore, C h a r l e s P.: ment of Methodology f o r O p t i m u m Design o f S t r u c t u r e s a t E l e v a t e d Tempera t u r e s . A C o l l e c t i o n o f T e c h n i c a l P a p e r s - AIAA/ASME 1 9 t h S t r u c t u r e s , S t r u c t u r a l Dynamics and Materials Conference, A p r 1978 , pp. 23-36. ( A v a i l a b l e as AIAA Paper N o . 78-468.)

.

9. R a o , G. Venkateswara; Shore, C h a r l e s P.; and Narayanaswami, R.: An O p t i m a l i t y C r i t e r i o n for S i z i n g Members of Heated S t r u c t u r e s With Temperature C o n s t r a i n t s . NASA ?N P 8 5 2 5 , 1977. 1 0 . T s a i , Stephen W. ; and Wu, Edward M. : A G e n e r a l Theory of S t r e n g t h f o r A n i s o t r o p i c Materials. J. Compos. Mater. , vol. 5, J a n . 1971 , pp. 58-80.

T.; and P r a s a d , B.: Programs f o r A n a l y s i s and R e s i z i n g of Complex S t r u c t u r e s . Trends i n Computerized S t r u c t u r a l A n a l y s i s and S y n t h e s i s , Ahmed K. Noor and Harvey G. McComb, J r . , eds., Pergamon P r e s s ( A l s o a v a i l a b l e i n Comput. & S t r u c t . , v o l . 10, Ltd., c.1978, pp. 323-330. no. 1 / 2 , A p r . 1979, pp. 323-330.)

1 1 . H a f t k a , R.

12. Fiacco, Anthony V. ; and M c C o r m i c k , G a r t h P. :

Nonlinear Programming: S e q u e n t i a l Unconstrained Minimization Techniques. John Wiley & Sons, I n c . , c.1968.

14

13. H a f t k a , Raphael T.; and S t a r n e s , James H., Jr.: A p p l i c a t i o n s of a Q u a d r a t i c Extended I n t e r i o r P e n a l t y F u n c t i o n f o r S t r u c t u r a l Optimization. AIAA J., vol. 14, no. 6, J u n e 1976, pp. 718-724. 14. Haftka, Raphael T.:

Automated Procedure for Design of Wing S t r u c t u r e s To S a t i s f y S t r e n g t h and F l u t t e r Requirements. NASA TN D-7264, 1973.

15

TABLE 1

.- APPROXIMATIONS FOR

f (V) = 1 1/ (1ovl

VALUES OF FUNCTION AND DERIVATIVES AT

V1

.5 2.0 1 .o 1 .o 2.0 .5

v2

1 .o 1 .o 1 .o .2 5.0 2.0 6.0

1 .ooo 1 .a33 .524 1.078 .733 .500 1 .oo

1

.ooo

1.455 .091 1.073 .636 1 ~.

16

Vol

BASED ON

V2)

= v02 = 1

Generalized Direct . Reciprocal approximation approximation approximation vml = -0.1, vm2 = -10 (eq. (2)) (eq. (3)) (eq. (5))

-1 .o

+

.ooo .ooo

.- - -

1 .ooo 1 .909 .545 1.364 .927 .500 1.833 . ..

1 .ooo 1 .a33

.

....

.524 1.078 .733 .441 1.521

..

.

.

~

TABLE 2.-

MATERIAL PROPERTIES AND LOADING FOR INSULATED PANEL

( a ) Mechanical p r o p e r t i e s f o r g r a p h i t e epoxy

Value a t Pr ope r t y

Symbol

Room temperature

Poisson's r a t i o , major

~ 1 2

Shear modulus

G12

C o e f f i c i e n t s of thermal expans ion

a1

a2

-

Units

I

----------------GPa psi per OC per 9

per

Oc

per 9'

I

0.30

'

5.10 0.74 x

0.23

lo6

-0.234 x -0.13 x 34.0 18.9

x x

177O C (350O F)

10-6

6'01

0.551 0.08 x lo6 -0.126 -0.07

x

78.7 43.7

x

x 10'6

10-6

x

10-6

Continued

TABLE 2.-

( a ) Concluded

-

Value a t Property

Units

Symbol

Room temperature

177O C (350° F)

I

Material allowables S t r e n g t h component i n f i b e r d i r e c t i o n , t e n s i l e and compressive

GPa

X t

ps i XC

S t r e n g t h canponent i n d i r e c t i o n normal t o f i b e r s , t e n s i l e and canpressive

I1

'I

yt

yc

MPa

psi MPa

psi MPa

psi

1 1 .ll 0.161 x

lo6

1.03 0.150 x

lo6

-9 72 -0.141 x

lo6

-8 48 -0.123 x

lo6

35.8 5.19 x 103

21 .o 3.04 103

-1 70 -24.7 x 103

-1 19 -17.3 103

i j

I

Shear s t r e n g t h

s

i

MPa

psi

-

57.9 8.40 x 103

25.9 3.76 x 103

Loads e

Applied loads

NX

kN/m

l b f / in. NY

kN/m

lbf/in.

-700 -4000 2 28 1300

I

I

1

L

NXY

kN/m

lbf/in.

22.4 128

I

TABLE 2.-

Concluded

(b) Thermal p r o p e r t i e s

1 Symbol

Proper t y

Mass d e n s i t y

Material Units Graphite/Epoxy

1

Insulation

P ~

~~~~

Thermal c o n d u c t i v i t y Btu-in/sec-f t 2-0F Heat c apac it y B t U/ 1b- OF

0.23 962

I

1200 0.291

N 0

TABLE 3.-

FINAL DESIGNS FOR TITANIUM PLATE W I T H ALUMINUM BARS

Maximum F i n a l mass

t e m p ra ture error

Design v a r i a b l e s

Procedure

analysis

I

11.79

,5.35

I

I i ,

I

0.836 i n . .272 .185 -103 1.398 i n 2

a2 a3

21.23 mm 6.91 4.70 2.62 901.9 "2 2491 .O 314.8

t1 t2 t3 t4 a1 a2 a3

28.85 mm 8.70 4.66 ' 2.76 , 1036.8 mm2 2507.7 342.6

1.136 i n . .342 .183 .lo9 1.607 i n 2 3.887 .531

149.9

ti t2 t3 t4 a1 a2 a3

20.84 mm 6.91 4.63 2.55 917.4 mm2 2485.8 302.6

0.821 i n . .272 .182 .lo1 1.422 i n 2 3.853 .469

14.9

t2 t3

I

t4

a1

7, 7

Reciprocal e x p a n s i o n 16.41

'

-

I

1I 1

, i

I

Combined reciprocal and d i r e c t expansions

I

5.29

O

lo

11.66

1

optimization, sec

'

1.99

I)

I

167

I I

I

1

14.14

C P U ~ for t o t a l

U.S. Customary Units

Units

1 Full

C P U ~per iteration,

1

'1

1

1

89.8

I

i

~

1.12

100

1.12

100

I

I

I

I

I

- -

!

26.8

' I

TABLE 4.- FINAL DESIGNS OF INSULATED COMPOSITE PANEL

[Initial design: t l = 101.6 mm, t o = 0.76 mn, t45 = 1.52 mm, and t g o = 0.76 mm]

I

Procedure

F u l l analysis

19.94 (4.084) 1

Direct expansion

Design variables, mm ( i n . )

mass,

1 1 121.4 (4.781)

1

19.98 (4.093)

~

1

121.7 (4.792) 113.5 (4.468)

1

1

0.848 (0.0334) 0.859 (0.0338) 0.724 (0.0285)

1 1

1

error, OC (OF) 0.175 (0.0069) 0.173 (0.0068) 1.384 (0.0545)

1

1

0.541 (0.0213)

(16.5)

0.688 (0.0277)

3.67 (6.6)

0.49

63.0

9.50 (17.1)

0.23

28.9

0.23

28.7

Direct expansion plus c r i t i c a l time points

19.97 (4.090)

121.7 (4.790)

0.853 (0.0336)

0.163 (0.0064)

0.528 (0.0208)

20.68 (4.235)

113.6 (4.472)

0.673 (0.0265)

1.422 (0.056)

0.681 (0.0268)

~

sec

0 (0i

20.70 (4.239)

~~

1 1 .oo

To t a l CPU,

0.546 (0.0215)

Rec ipr oca1 expansion

~

~

analysis, sec

I

230.0 59.1

0*49

~~

~~

Reciprocal expansion plus c r i t i c a l time points

ICentral processing u n i t time on CDC Cyber 173; FTN compiler,

3.72 (6-7)

OPT = 2.

TABLE 5.-

FINAL DESIGNS OF INSULATED COMPOSITE PANEL t l = 203.2 mm

[ I n i t i a l design:

Procedure

F u l l analysis

1

Direct expansion

mass. kg/m2 (lb/ft2)

I

and

to

= t 4 5 = t g 0 = 0.51 m]

Design v a r i a b l e s , mm ( i n . )

error , ti

t0

t45

t 90

19.93 (4.081)

121.5 (4.783)

0.848 (0.0334)

0.163 (0.0064)

0.549 (0.0216)

13.91

80.6

0.864 (0.0340)

0.102 (0.0040)

0.508 (0.0200)

R e c ipr oc a1 expa n s i o n

20.73

127.6

0.833 (0.0328)

0.124 (0.0049)

0.551 (0.0217)

Direct expans ion plus critical points

14.49

79.9

0.777 (0.0306)

0.480 (0.0189)

0.663 (0.0261)

R e c i p r o c a l expansion plus critical points

20.67

0 (0)

0.49

(525)

I I

j

I

53.6

I

66.7 (120)

0.49

58.6

275 2(495)

0.23

21.5

0.23

21.4

I

127.4

0.836 (0.0329)

0.114 (0.0045)

0.538 '(0.0212)

l C e n t r a l p r o c e s s i n g u n i t time on CDC Cyber 173; F?N compiler, I n f e as i b l e d e s i g n

.

1

68.3 (123)

OPT = 2.

'

E

Tine, T

F i g u r e 1.-

\

T

max

T y p i c a l v a r i a t i o n of c o n s t r a i n t f u n c t i o n w i t h t i m e .

23

Plate dimensions305 mm x 305 mm (12 in. x 12 in.)

f i

Insulated edges bars

Titanium plate

-+r

Q =

3300 W/m2 (0.002 Btu/in2-sec)

I

1

-18OC (O°F)

(a) Configuration and loads.

I Mater ia1

Property

Aluminum

Titanium

-

103

69.6 10.1

-~

15

lo6

23.8

10.4

13.2

5.8

2770

4437

0.100

0.160

2 34

lo6 _

~-

-

X

9.49

0.451

0.018 -

.~

177

260

350

500 -

(b) Material properties. Figure 2.- Titanium plate with aluminum bars. 24

Design v a r i b l e s

T = -18OC (OOF)

Reciprocal expansion

-- - D i r e c t

_--

expansion

Full analysis

40

80

P e r c e n t change i n d e s i g n v a r i a b l e s

F i g u r e 3 . - Approximations of maximum’ t e m p e r a t u r e i n t i t a n i u m p l a t e w i t h aluminum bars (from r e f . 8 ) .

25

"r ---- - -

4

0 Percent change i n

Tmax

/ Full analysis Direct expansion Reciprocal expansion

/

/

/

/

/'

/

I

-2 -4 -6 -8

-10 -12

1

I

I

1

I

1

0

20

40

60

80

100

P e r c e n t a g e o f t o t a l change i n d e s i g n v a r i a b l e s from o r i g i n a l t o f i n a l d e s i g n

Figure 4 . - Approximation performance f o r mixed changes i n d e s i g n v a r i a b l e s for titanium p l a t e with aluminum bars. 26

t45 t90

0

500

1000

1500

2000

Time, sec

Figure 5.- Mathematical model and loadings for insulated composite panel

27

3

Full analysis

- --

Direct expansion

L

Temperature ratio, T/Ta

1 -

50

75

100

125

150

Insulation thickness, t, mm

F i g u r e 6.-

Q u a l i t y of approximations i n e s t i m a t i n g t e m p e r a t u r e dependence on i n s u l a t i o n t h i c k n e s s ; T = 2000 sec.

1.4 .-

1.0

Full a n a l y s i s

0.8

---- --

-

Temperature ratio,

D i r e c t expansion Reciprocal expansion

n

Approximation w i t h vm = -4.06 mm (-0.16 i n . )

T/Ta

0

Reciprocal expansion w i t h i n d i v i d u a l p l y design v a r i a b l e

0.2

1.5

I

1

1

1

2.0

2.5

3.0

3.5

Laminate t h i c k n e s s , t , mm

Figure 7.- Q u a l i t y of approximation i n e s t i m a t i n g temperature dependence on l a m i n a t e t h i c k n e s s ; T = 3200 sec, t l = 101.6 m (4.0 i n . ) .

W 0

tl,

----

---

Temperature ratio, 0.5' T/Ta

Initial design

"

to + t 4 5

203

1.52

Intermediate desigR 118

3.26

Final design

121

+

2.86

I

0

1600

3200

4000

6400

Time, T , sec

F i g u r e 8.- Laminate temperature h i s t o r y f o r d i f f e r e n t i n s u l a t e d composite-panel d e s i g n .

t,O,

"

Laminate t h i c k n e s s = 1 . 5 2 nun (0.06

75

50

100

125

ill,)

150

I n s u l a t i o n t h i c k n e s s , t , mm

.-

Figure 9 V a r i a t i o n w i t h i n s u l a t i o n t h i c k n e s s of m a x i m u m l a m i n a t e t e m p e r a t u r e and l a m i n a t e t e m p e r a t u r e a t T = 3200 sec for i n s u l a t e d composite p a n e l .

31

60 D i r e c t expans i o n 50

R e c i p r o c a l expansion

1

Combined d i r e c t and r e c i p r o c a l expansion

40

-

Full a n a l y s i s Mass,

kg

l2O

loo

80

M lb ass,

30

20

10

0

1

20

I

I

I

40

60

80

-

60

-

40

-

20

100

Iter a t i o n

F i g u r e 10.-

32

Performance of Taylor series approximations i n o p t i m i z a t i o n of t i t a n i u m p l a t e w i t h aluminum b a r s .

-_

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I

.

1. Report No.

NASA TP-1428 _ _ -

. -..--

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I--

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T1

3. Recipient's Catalog NO.

2. Government Accession No. - -.

5. Report Date

4. Title and Subtitle

APPADXIMATION METHODS FOR COMBINED THERMAL/STRUCTUFUU DESIGN . .

.

.

~

_- ._

~

I

7. Author(s)

Raphael T. Haftka and Charles P. Shore -

.A

1

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9. Performing Organization Name and Address

NASA Langley Research Center Hampton, VA 23665

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--

~

-

June 1979

6. Performing Organization Code ..

.-

E. Performing Organization Report No.

L-12674 10. Work Unit No.

505-02-53-01 11. Contract or Grant No.

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.

13. Type of Report and Period Covered

~

2. Sponsoring Agency Name and Address

National Aeronautics and Space Administration Washington, DC 20546 ~~

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5. Supplementary Notes

Raphael T. Haftka: Illinois Institute of Technology, Chicago, Illinois. Charles P. Shore: Langley Research Center. __

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~-

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6 Abstract

approximation concepts for combined thermal/structural design are evaluated. The first concept is an approximate thermal analysis based on the first derivatives of structural temperatures with respect to design variables. Two commonly used first-order Taylor series expansions are examined. The direct and reciprocal expansions are shown to be special members of a general family of approximations, and it is shown that for some conditions other members of that family of approximations are m r e accurate. Several examples are used to compare the accuracy of the different expansions. Two

The second approximation concept is the use of critical time points for combined thermal and stress analyses of structures with transient loading conditions. It is shown that significant time savings may be realized by identifying critical time points and performing the stress analysis for those points only. This approach is used to design an insulated panel which is exposed to transient heating conditions.

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7. Key Words (Suggested by Author(s))

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9. Security Classif. (of this report1

20. Security Classif. (of this.page)

Unclassified .

Unclassified .. .

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-

I _.

- .~

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Unclassified

Optimization Approximation methods Thermal structures

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- - ._ .-

18. Distribution Statement

- Unlimited Subject Category 39 .. . .. ~

21. No f;i .

...

Pages .

22. Rice'

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$4.50 .

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