I

. y . d

DEPARTIqENT OF MECHANICAL ENGINEERING AND MECHANICS

OLD I>OFJIINION UNIVERSITY SCHOOL OF ENGINEERING !

NORFOLK, V I R G I N I A 23508 ~

I

OPTIMAL CURE CYCLE DESIGN FOR AUTOCLAVE PROCESSING OF THICK COMPOSITES LAMINATES: A FEASIBILITY STUDY

c f /$/J;.", i:"d +#PPM -7-

.*d'/-C 2 "rJ"--& t,

/- *I" BY

Jean W . HOU, P r i n c i p a l I n v e s t i g a t o r

Progress Report For t h e p e r i o d January 15 t o October 15, 1985

Prepared f o r Nat iona l Aeronaut ics and Space A d m i n i s t r a t i o n Langley Research Center Hampton, VA 23665

Under Research Grant NAG-1-561 Robert M. Baucom, Technical Mon i to r MD-Polymeri c Mater i a1 s Branch

(NASA-CR-181261) CPTXEAL CUEE CYCLE DESIGN N ~ J - ~ W I F C B AUTCCLAVE E6CCBSSIblG CE TEICK C O B P C S I T E S L A M I N A T E S : A EEASIBXLIIY S51UCb Proqress Feport, 15 Jan. - 15 Cct, 1915 (Gld Unclas

0 4 1 c n A n i o n U n i v , ) 42 p A v a i l : h31S HC G3/24 0093187

November 19 8 5

I

T I !

I

https://ntrs.nasa.gov/search.jsp?R=19870018301 2018-05-07T11:52:25+00:00Z

DEPARTMENT OF MECHANICAL ENGINEERING AND MECHANICS SCHOOL OF ENGINEERING '

OLD DOMINION UNIVERSITY NORFOLK, V I R G I N I A 23508

OPTIMAL CURE CYCLE DESIGN FOR AUTOCLAVE PROCESSING OF THICK COMPOSITES LAMINATES: A FEASIBILITY STUDY

BY

Jean W. Hou, P r i n c i p a l I n v e s t i g a t o r

Progress Report For t h e p e r i o d January 15 t o October 15, 1985

Prepared f o r Nat iona l Aeronaut ics and Space Admin i s t ra t i on Langley Research Center Hampton, VA 23665

Under Research Grant NAG-1-561 Robert M. Baucom, Technical Moni tor MD-Polymeric Mater i a1 s Branch

Submitted by t h e Old Dominion U n i v e r s i t y Research Foundation P. 0. Box 6369 Nor fo lk , V i r g i n i a 23508

Nov ember 19 8 5

TABLE OF CONTENTS

INTRODUCTION ...................................................... CONCLUSIONS ....................................................... REFERENCES...................................... .................. APPENDIX A: NUMERICAL STUDIES ON THE DESIGN SENSIT IV ITY

CALCULATION OF TRANSIENT PROBLEMS WITH DISCONTINUOUS DERIVATIVES. . . . . . . . . . . . . . . . . . . . . . . . . . . .

LIST OF TABLES

Table

1 Thermal Design S e n s i t i v i t y Analysis f o r Compression Molding ....................................................

A1 Comparisons o f Design S e n s i t i v i t y Calculat ions of Example 1 a t b = 400 .................................................

A2 Perturbat ion o f Cost Funct ion .............................. (a) Exact I n teg ra t i on based on t h e given c r i t i c a l t ime €. . (b) Design S e n s i t i v i t y by Using DE Program ................ (c ) Design S e n s i t i v i t y by Using Simpson's Rule ............ Effect o f f Misca lcu la t ion on Design S e n s i t i v i t y Ca lcu la t ion ................................................ (a) DE Program. ........ ............... .......... .......... (b) Simpson's Rule... .. .. .. . . . . . ..... . . .. . . . .. .. . . .. .. . .. .

A3

A4 Design S e n s i t i v i t y Analysis o f Example 2 a t T=475"K and 6 T = 1°K. .............................................

Ad jo in t va r iab le method with jump condi t ion. . . . . . . . . . . Ad jo in t va r iab le method with l o g i c a l funct ion. . . . . . . . .

LIST OF FIGURES

F igure

1 Temperature P r o f i l e a t t he Center o f Laminate S o l i d L ine -- Data i n Loos' Report D iscre te Points -- Calculated Data. .. . . . . . . . . . . . . . . . . . . . . . .

2 Temperature P r o f i l e s f o r 10-mm Thick Sheet... .............. 3 Cure P r o f i l e s f o r 1O-mm Thick Sheet...... ..................

ii

Page

1

6

6

14

Page

7

31

32 32 33 34

35 35 35

36 36 36

Page

8

9

10

TABLE OF CONTENTS - continued

LIST OF FIGURES - continued Figure

4 Profile of the Change of Temperature Uniformitry by R i s i n g 1" K Mold Temperature .................................

5 Profiles of Thermal Design Derivatives of Temperature w i t h Resepect t o Mold Temperature ............................

6 Profiles of Thermal Design Derivatives of Percent Cure w i t h Respect t o rvlold Temperfature .......................

A 1 State Variable and Adjo in t Variable for Example 1 ............ A2 State Variable and Adjoint Variable for the

C u r i n g Process.. .............................................

i i i

Page

11

1 2

13

37

38

I INTRODUCTION

The research progress i n the p ro jec t "Optimal Cure Cycle Design f o r

Autoclave Processing of Thick Composite Laminates: A F e a s i b i l i t y Study,"

dur ing the per iod o f January t o October, may be repor ted separately i n th ree

aspects; namely:

1. design s e n s i t i v i t y ana lys is f o r the chemical-k inet ic reac t i on dur ing prepreg processing, wh i le the temperature i s considered

as a design var iable,

2. f i n i t e element ana lys i s for the thermal system o f heat conduction coupled w i t h the chemical-k inet ic reac t i on dur ing

prepreg processing,

3. design s e n s i t i v i t y ana lys is for the thermal system o f heat conduction coupled w i t h the chemical-k inet ic reac t i on dur ing prepreg processing, wh i le the temperature o f cure cycle i s

considered as a design var iable.

The chemical-k inet ic reac t i on o f Hercules 3501 dur ing autoclave

processing has been modelled and expressed i n terms o f an equation of degree

of cure:

0 (K1 + K2 a) (1 - a) (B - a)

a = K3 (1 - a) a < 0.3 a > 0.3 (1)

where B i s constant, and K1, K2 and K3 are func t ions o f temperature. Note

t h a t the r a t e o f cure presents d i scon t inu i t y a t a = 0.3. The design

s e n s i t i v i t y c a l c u l a t i o n o f t r a n s i e n t problems w i t h discontinuous d e r i v a t i v e i s

t h e o r e t i c a l l y d i f f i c u l t . I n add i t ion , i t i s numerical ly d i f f i c u l t t o

p rec i se l y monitor the c r i t i c a l time a t which the d i s c o n t i n u i t y occurs. The

research r e s u l t s i n t h i s regard have been documented i n Appendix A. I n a

2

sensi t ivi ty numerically. Two approaches were developed. One maintains

jump condition. The other uses a logic function t o smoothly approximate

d i scon ti n u i t y

Never the1 ess , that the a

numeri ca 1 ly d

position.

0

summary, one f i r s t studied the effect of the accuracy of the c r i t i ca l time

* evaluation on the accuracy of the thermal d e s i g n sensi t ivi ty analysis. I t

showed that the design sensit ivity calculated by the adjoint variable

technique is n o t sensitive to the accuracy of the c r i t i ca l time evaluation.

Next, the adjoint variable technique was employed to f i n d the thermal design

the

the

ts.

the l a t t e r one i s to be used for further study. The reason i s

i s a function of time as well as position; therefore, i t is

f f i c u l t to keep track of the a discontinuity a t every spatial 0

w i t h i n a small1 region. Both approaches showed good resu

The second stage of research i s devoted to the computer code development

for the simulation of a heat conduction model coupled w i t h a chemical-kinetic

model d u r i n g prepreg processing. The equation for the chemical-kinetic model

i s already given i n equation 1. In addition, the heat condction equation i s

given as

where p is the mass density, c i s the coefficient of heat capacity, k i s the

heat conduction coefficient, and HR i s the heat generated by cured resin.

The f in i t e element discretization i s introduced to convert the i n i t i a l -

boundary value equations (1) and ( 2 ) i n t o a s e t of f i rs t order different ia l

equations. T h i s s e t of equations are then solved simultaneously by a

numerical integration code called DE. To preserve analysis accuracy, the

temperature and the degree of cure are subjected to the same numerical error

control d u r i n g the numerical integration.

3

Two numerical example have been performed. One simulates the autoclave

processing o f 192 p l y prepregs w i t h 32% Hercules r e s i n content. The r e s i n

f low can be neglected i n t h i s example, because the r e s i n content i s low. To

focus on the heat conduction and the chemical-k inet ic models, the measured

temperature on the surfaces o f the composite laminate are used as boundary

temperature, instead o f the temperature o f cure cycle. Although the heat

f l ux , induced by the heat convection o f autoclave a i r temperature, i s

neglected, the numerical r e s u l t i s i n an exce l l en t agreement w i t h Loo's data,

as shown i n f i g u r e 1. The long CDC computer time (17,056 CPU seconds) i s

requ i red to t o analyze a complete cyc le of autoclave processing. This may be

because the autoclave processing needs a long r e a l t i m e (275 minutes) t o

perform. I n addi t ion, the numerical i n teg ra t i on requ i res small time step s ize

becuase the system equations are q u i t e " s t i f f " .

The second example i s taken from the r e s u l t s o f compression model l ing o f

composite laminates [l]. The research was done i n General Motor Research

Center. The degree o f cure o f r e s i n i n term o f temperature i s given as

where m and n are constants, and K1 and K2 a re funct ions o f temperature. Note

t h a t ne i the r a d i s c o n t i n u i t y nor r e s i n flow are needed to be considered i n

t h i s example. The developed computer code can be used to solve equations 2

and 3 wi thou t d i f f i c u l t i e s . The resu l t s ca lcu la ted are c lose to those

published, as shown i n f i gu res 2 and 3.

0

I n the t h i r d stage, one concentrates on the de r i va t i on o f the thermal

design s e n s i t i v i t i e s o f temperature uni formi ty r e l a t e d t o the change o f

surface temperature. Note t h a t the temperature on the surface o f prepreg i s

c a l l e d the surface temperature which i s cont ro lab le and i s considered as a

4

design var iab le. Moreover, the goal o f the opt imal design i s t o achieve a

uni form temperature d i s t r i b u t i o n across t h e th ickness o f the laminates. The

performance index o f the temperature uni formi ty . (b. may be def ined as the

l e a s t square o f the dev ia t ion between the pointwise temperature and the

average temperature as:

where T i s the temperature d i s t r i b u t i o n and h i s the th ickness o f the

laminate.

Two methods have been developed t o analyze the thernal design

s e n s i t i v i t y . One i s the a d j o i n t var iab le method. The other i s the approach

o f d i r e c t d i f f e r e n t i a t i o n .

Using the a d j o i n t method, the design d e r i v a t i v e o f the temperature

un i fo rmi ty i s der ived as

where f i s the r i g h t s ide o f equation 3 and the a d j o i n t va r iab le

solved by the fo l l ow ing equations:

A and u are

n

w i t h homogeneous boundary condi t ions and terminal cond i t ions defined a t t =

tf. The above equations are coupled l i nea r equations which can be solved by

the same computer code developed f o r the thermal ana lys is discussed

5

prev ious ly .

Using the d i r e c t d i f f e ren t i a t i on , equation 4 can be taken for de r i va t i ves

w i t h design var iable, Tc, d i r e c t l y as

*

[j: T T' dz - j! T dz ,," T' dz/h] d t

where T ' : dT/dTc

and 3 as can be obtained by taking the de r i va t i ves o f equations 2

and

w i t h homogeneous boundary and i n i t i a l condi t ions. Again, the f i n the above

equat ions denotes the r i g h t side o f equation 3 and a' da/dTc. The

pe r tu rba t i on of the performance index, A+, due t o the change of the design

var iab le, ATc, can be approximated by the design der iva t ive , d4/dTc, i.e.,

AJ, ATc (7)

where the ac tua l change A4 can be obtained by the f i n i t e d i f f e rence scheme,

1 .e.,

The combination o f the preceding two equations provides a good mean t o check

the accuracy o f the thermal design s e n s i t i v i t y . As l i s t e d i n Table 1, the

ac tua l changes are ca lcu la ted f o r various pe r tu rba t i on o f design var iab le,

ATc, based on equation 8; and the l a s t two columns ind ica ted the change of

J, p red ic ted by methods presented i n equations 5 and 6. Using the

compression molding o f a po lyester [11 as an example, i t i s noted t h a t the

6

d i r e c t d i f f e r e n t i a t i o n method i s super ior to the a d j o i n t method i n t h i s study.

The d i r e c t d i f f e r e n t i a t i o n method provides a b e t t e r eva lua t ion o f the ac tua l

change o f the performance index. Besides, the approach o f d i r e c t

I d i f f e r e n t i a t i o n provides the time h i s t o r i e s o f design de r i va t i ves d+/dTc,

dT/dTc and da/dTc, as shown i n f igures 4 t o 6, respec t ive ly . It i s o f

g r e a t i n t e r e s t t o observe t h a t the change o f the con t ro i temperature has I

I s i g n i f i c a n t e f f e c t on the temperature un i fo rmi ty on ly when the opera t iona l I

~

t ime of processing i s over 100 seconds. The f i gu res dT/dTc and da/dTc

I conf i rm t h i s observation.

Conclusions

Two goals l i s t e d i n the proposal have been achieved, namely, the thermal

I ana lys is and the ca l cu la t i on o f thermal s e n s i t i v i t y . A f i n i t e element program 1

f o r the thermal ana lys is and design der iva t ives c a l c u l a t i o n f o r temperature I I

I d i s t r i b u t i o n and the degree o f cure has been developed and v e r i f i e d . It i s

found t h a t the d i r e c t d i f f e r e n t i a t i o n i s the best approach f o r the thermal I I

design s e n s i t i v i t y analys is . In addi t ion, the approach of the d i r e c t

d i f f e r e n t i a t i o n provides time h i s t o r i e s o f design de r i va t i ves which a r e o f

g rea t value t o the cure cyc le designers. The approach o f d i r e c t d i f f e r e n t i a -

t i o n i s t o be used f o r f u r t h e r study, i.e., the optimal cyc le design. I'

Ref e re n ce

1. Barone, M. E. and Caulk, 0. A . , "The E f f e c t o f Deformation and Thermoset Cure on Heat Conduction i n a Chopped-Fiber Reinforced Polyester During Compression Molding," I n t . J. Heat Mass Transfer, Vol. 22, pp. 1021-1032, 1979.

7

Table 1 Thermal Design Sensitivity Analysis for Compression Molding

Mold Temp. Cost Func t i on Actual Change $ 1 AT: J, I AT;

423°K

422°K

418°K

413°K

408 " K

403°K

393°K

428°K

433°K

21019.648

20501.695

18157.462

16137.194

14772.308

13479.446

11083.596

23238.107

25042.554

----- 517.95

2862.186

4882.45

6247.34

7540.20

9936.05

2218.46

4022.90

----- 341.97

1720.32

3445.93

5160.96

6881.28

10321.92

1720.32

3440.64

--e--

501.051

2505.257

5010.514

7515.771

10021.028

15031.542

2505.251

5010.514

~ ______ ~~

* Calculated by the adjoint variable method **

Calculated by the direct differentiation

8

$.I, I 0

aa I

v) 0 0

n

-

I c

(a cd

.- -w

n I

u u

c> I I

I! \.

0 Lc) N

0 0 N

0 0 0 Lc) -

0 0 r9

0 0 c\1

0 0 N

0 Lc) - 0 0 -

0 0

0 0

9

200

I 6 0

I 2 0

80

-Data in R e f . I --Calculated Da ta

200

40 =

24 I 1 I 8 . I I I I

60

20

80

40 24

DISTANCE FROM MIDPLANECmm)

Figure 2. Temperature Profiles f o r IO-mm Thick Sheet.

10

W QL 3 V I- z W V QL W a.

I O 0

80

60

40

20

0

1 -- .---

- Data - - Calcu

in Ref . I ated Data

2.. 5

I O 0

8 0

60

40

2 0

0 5.0

DISTANCE FROM MIDPLANE (mm) Figure 3 . Cure Profiles for 10-mm Thick Sheet.

11

I

8 b r9 nl

e

I 1

8 8 0 cu

d- e -

I 1 -

0 00 6) * d d

0 nl - 0 0

0 00

-

,o (D

0 ,*

,o c\I

0

L 0 , o

(L

0

aJ m E rrJ c . V a J

L aJJ

33NVNUOdU3d 3 H l 3 0 33NVH3 lN33U3d

,* u \ t-

W > , > U W t3

u

- a -

I

t QL W I: I-

a

DISTANCE FROM MIDPLANE(mm1

Figure 5. Profiles of Thermal Design Derivatives o f Temperature w i t h Respect t o Mold Temperature.

13

o. 0. 0.

18 17 I O

0.05 4

1

:

see

6 sec

I

“e IO

-0.05

2.5 5.0

DISTANCE FROM MIDPLANECrnm)

Figure 6. Profiles of T.herma1 Design Derivatives o f Percent Cure w i t h Respect t o Mold Temperature.

APPENDIX A

NUMERICAL STUDIES ON THE DESIGN SENSITIVITY CALCULATION OF TRANSIENT PROBLEMS WITH DISCONTINUOUS DERIVATIVES

15

SUMMARY

The aim o f t h i s study i s t o f i n d a r e l i a b l e numerical a lgor i thm t o

ca l cu la te design s e n s i t i t i v t y o f t rans ien t problem w i t h d iscont inous

der iva t ives . The f i n i t e d i f ference method and the ad j o i n t va r iab le technique

us ing both Simpson's r u l e and DE program, a p red ica tor - cor rec tor algor i thm,

are invest igated. I t i s shown t h a t the design s e n s i t i v i t y ca lcu la ted by the

f i n i t e d i f f e rence method i s q u i t e sens i t i ve t o the numerical e r ro rs .

Nevertheless, the design sensi t i v i t y ca lcu lated by the a d j o i n t va r iab le

technque i s r e l a t i v e l y s tab le aga ins t various numerical i n teg ra t i ons and time

steps. It i s concluded t h a t the a d j o i n t va r iab le technique i n conjunct ion

w i t h the DE program w i t h appropr ia te t runcat ion e r r o r con t ro l provides very

sa ti s fac tory numerical resu l ts.

I. INTRODUCTION

The d e r i v a t i v e of the thermal response w i t h respect t o the design

va r iab le i s usua l ly c a l l e d thermal design d e r i v a t i v e o r s e n s i t i v i t y . The

in format ion o f design d e r i v a t i v e i s n o t on ly very usefu l for the t rade-o f f

design b u t i t i s a l so requ i red f o r the i t e r a t i v e design opt imizat ion.

The c a l c u l a t i o n o f design der iva t ives i n thermal problems has a t t r a c t e d

research i n t e r e s t i n such areas as design of space s t ruc tu re sub jec t t o

temperature cons t ra in t s [l], and chemical process con t ro l C21.

For the problems o f i n te res t , the s tate equat ion i s usua l ly expressed as

w i t h i n i t i a l cond i t i on

z (b, 01 = zo

16

where a do t denotes d i f f e r e n t i a t i o n w i t h time, t, b i s the design var iab les

and zo i s prescr ibed i n i t i a l condi t ion. The s ta te funct ion z(b, t ) i s a

func t ion o f time and design var iable. Although the design va r iab le b i s

genera l l y a func t ion o f time, i n the fo l lowing discussion, i t i s l i s t e d as a

constant parameter f o r s i m p l i c i t y .

Four d i f f e r e n t methods [l]; the Green's function, f i n i t e di f ference,

d i r e c t d i f f e r e n t i a t i o n s a n d a d j o i n t var iab le technique, are c u r r e n t l y used i n

the c a l c u l a t i o n of thermal design s e n s i t i v i t y , i.e., dz/db. Haftka [l, 31

t) used t o

e f f i c i e n c y

con ti nuous

ind i ca ted t h a t the numerical i n t e g r a t i o n scheme (expl

so lve i q . 1 i s an impor tant f ac to r i n detemining the

and accuracy of thermal design s e n s i t i v i t y f o r the

d e r i v a t i v e 2.

c i t or i m p l i c

computational

problem w i t h

On the o ther hand i t i s n o t uncommon i n engineering app l i ca t i ons

t h a t o r the func t ion f (b , z, t ) def ined i n Eq. 1 e x h i b i t s d i s c o n t i n u i t i e s .

The c r i t i c a l time a t which the d i scon t inu i t y happens i s usua l ly monitored by

the state va r iab le z. Engineering examples can be found i n the mul t i -s tage

con t ro l problem [41, con t ro l o f chemical k i n e t i c s [51 , and the mechanical

system w i t h i n t e r m i t t e n t motion C6, 71. The i n t e r m i t t e n t motion i s

character ized by the occurence o f near ly discont inuous force and v e l o c i t y

caused by impuls ive force, impact, mass capture, and mgss re lease. The I I

I opt imal design problems o f mechanisms w i th i n t e r m i t t e n t motion have been I

discussed by Huang, Huag and Andrews 161. Their method i s based on the

i d e n t i f i c a t i o n o f c r i t i c a l times a t which d i s c o n t i n u i t i e s i n forces o r

v e l o c i t i e s occur. Jump condi t ions o f those d i s c o n t i n u i t i e s are employed i n an

a d j o i n t va r iab le approach f o r the ca l cu la t i on o f design sensi t i v i t y

c o e f f i c i e n t s . Ehle and Haug [7] introduced the " l o g i c a l funct ion" t o smoothly

approximate d i s c o n t i n u i t i e s . An example i n t h e i r work shows t h a t time step and

I

17

the s ize o f t r a n s i e n t zone used for d i s c o n t i n u i t y approximation have

s i g n i f i c a n t e f f e c t on the accuracy of design s e n s i t i v i t y ca lcu la t ion .

It i s the ob jec t i ve of t h i s study t o i nves t i ga te the numerical accuracy

o f design sensf t i v i t y ca lcu la t ion , focused on the numerical i n t e g r a t i o n

schemes and the approximation o f o f c r i t i c a l time.

I I. Design Sensi t i v i t y Analysis

A f unc t i ona l i s g iven as

7

(4 = g ( z ( t ) , b, t) d t ( 3 ) 0

where z i s the terminal time which i s assumed t o be independent o f the design

va r iab le b. The s ta te var iab le z ( t ) i s governed by the s ta te equations as

fo l lows:

when z ( t ) < C

when z ( t ) 2 c ( 4 )

w i t h i n i t i a l cond i t i on

The constant c o f jump cond i t ion i n Eq. 4 i s assumed again t o be independent

on design var iab le. It i s obvious tha t the s ta te va r iab le z ( t ) and the

c r i t i c a l t ime 5 a t which z (E) = c depend on the design var iab le b because z ( t )

i s the s o l u t i o n of Eq. 4. I n o ther words, z and 5 can be def ined as z ( t , b )

and ?(b), respec t ive ly . The problem o f i n t e r e s t i s t o determine the design

s e n s i t i v i t y o f func t iona l (1,. The va r ia t i on o f func t iona l (1, w i t h respec t t o

the design va r iab le i s def ined as

I I

I

= +' 6b

where 6b i s the per tu rba t ion o f design va r iab le and +' i s def ined as

d+/db. The v a r i a t i o n o f s ta te function, 62, can be def ined by a s i m i l a r

fashion.

According t o the d e f i n i t i o n o f va r ia t i on and L e i b n i t z ' s ru le , the

v a r i a t i o n o f func t iona l i s der ived as

Note that , because o f the dependence of b, 6z = (az/ab)bb and

6't = (aT/ab)6b. The l a s t term o f above equat ion can be dropped prov ided

t h a t the c o n t i n u i t y assumption o f g a t i i s maintained. The var ia t ions ,

can be determined by using the e q u a l i t y o f Eq. 4 and the jump

cond i t i on z(i) = c. However, by carefu l se lec t i on o f the a d j o i n t equation,

i t i s n o t necessary t o ob ta in e x p l i c i t expressions f o r 6z and 6z i n order t o

ob ta in the v a r i a t i o n 641.

6z and 65

I n accordance w i t h the e q u a l i t y o f Eq. 4, i t i s ev ident t h a t

for an a r b i t r a r y func t ion A ( t ) and the design va r iab le b. The v a r i a t i o n of

the preceding func t iona l y i e l d s :

19

0 = Io t 6h(;-fl) d t + I,+ z 6h (;-f2) d t t

Note that the summation o f f i r s t two terms should be zero because the

equality o f Eq. 8 i s also true for an arbitrary 6h. Furthermore, i t i s

understood that the c r i t i ca l time 't depends on design variable implicitly

through the relation Thus, employing the Leibnitz's rule, the

variation 65 appears i n the derivation. However, these two boundary terms may

be dropped out because - f 2 are equal t o zero when the time t

approaches t o z- and ?, respectively, according t o Eq. 4. After

simplification and integration by parts, the preceding equation can be

rewritten as

z ( f , b) = C.

- f l and

Note that the operators "6" and " * I ' are exchangable provided t h a t z i s a

continuous function o f b and t in the time domains 0 t < f' and

t < t z. The i n i t i a l condition z(b,O) = zo i s assumed to be independent

on the design variable. Consequently, 6z = 0 a t t = 0. As to the total

-+

20

v a r i a t i o n of the jump cond i t i on z(b,?) = c,

o r 6z = -bf f o r t approaches to e i t h e r t o r ?+. The boundary terms i n

Eq. 10 can then be rearranged as

i t i s der ived as i6 ' t t 6z = 0 --

+ ( h f 2 ) 3 6 i + h6z( I 'i+ 7 = [-(hfl) I 't-

Adding Eqs. 7 and 10 up, one has the va r ia t i on o f the func t i ona l JI as

Since h s t i l l r e t a i n s i t s a rb i t ra r iness , the only unknowns i n the l a s t

fo rmula t ion are 6z and 65. One may now speci fy the va r iab le h i n such a way

t h a t a l l o f terms associated w i t h Sz and S't are dropped. Def in ing the

f o l l o w i n g ad jo in ' t equations:

afl + ag 5 = - 3 r E ,

5 = - E - a z , af2 + ag

w i t h terminal condi t ions:

O < t < ' t

' t < t < z

(12)

(13)

h2 (7) = 0 and

(14) I

F i na l l y , the combination of Eqs. 11-15 provides a simple formula f o r 6+,

I f the design var iab le i s a parameter instead of a funct ion, the general

expression of Eq. 16 can be s imp l i f i ed to y i e l d the design s e n s i t i v i t y ,

Example 1 The state equation i s given as

0 < z < 1440 1440 < z

w i t h i n i t i a l cond i t ion z(b,O) = 0.

Given the func t iona l J, as

G1 = ,," z 2 d t ,

the d e r i v a t i v e dJ,/db i s simply obtained as

- dJ,l ,: 2hl b t 2 d t + ,I A,$ d t ,

t db=

where the a d j o i n t var iab les hl and h2 are determined by the a d j o i n t equations

hl = - 2z, {Az = - 22,

22

w i t h terminal condi t ions,

I f the design va r iab le and the t o t a l t ime i n t e r v a l are assigned as b =

400 and z = 2, i t fo l lows t h a t the c r i t i c a l time f i s exac t l y equal

t o i = 0.3 f o r jump cond i t i on z (5 ) = 1440 and the design s e n s i t i v i t y i s

obtained as dJl/db = 5598.3. The state var iab le and a d j o i n t var iab le can be

solved a n a l y t i c a l l y . They are p l o t t e d i n Fig. 1. The jumps o f and h are

ind icated.

Example 2 A model o f chemical k i n e t i c s i s inves t iga ted here. The r e l a t i o n

between the degree o f cure a and the react ion o r cure r a t e dur ing the cu r ing

process o f Graphi te/Epoxy composite i s determined exper imenta l ly as [51,

0 fl(as T, t ) = (K1+Kza) (1-a) (B -a ) , 0 < a < 0.3

a = ‘f2(a, T, t ) = K3(1-a) S 0.3 < a

w i t h i n i t i a l cond i t i on a(0) = 0 and the fo l low ing d e f i n i t i o n s

(17)

K1 = AA1 Exp (-AE1/RT)

K2 = AA2 Exp (-AE2/RT)

K3 = AA3 Exp (-AE3/RT)

where MI, AAzs AA3, AE1, AEz, A E 3 , R and B are mater ia l constants C53,and T

i s O K temperature. The problem i s t o analyze the s e n s i t i v i t y o f a fgnct ional

of a w i t h respect t o the temperature, i.e., dJl/dT where (1, i s def ined as

23

2 G2 = a d t

The a d j o i n t equation i s der ived as

2 k , = 2a t hl (K1 + KIB - K2B 't 2K Ba t K a - 2Kla - K 2 a ), 2 2

o c t c t t c t c z k2 = 2a - K + ~ ,

(18)

w i t h terminal condi t ions,

-h2(E) K3 hl (a = -

Kla2 - KIB + K2aZ + B K,a2 - 2KZa5 , a t t = t .

I n add i t ion , the v a r i a t i o n o f func t iona l J12, d+2/dT, i s given as

a f 1 af 2 -

d+2 t - - I ( - A, TI d t + I" ( - A -) d t 2 aT - t

dT 0 (19)

I t i s no t easy t o solve Eqs. 17 and 18 a n a l y t i c a l l y . Instead, they are

And dG2/dT i n Eq. 19 i s evaluated by solved numer ica l ly i n the next section.

a numerical i n t e g r a t i o n method.

i

As mentioned e a r l i e r , the discontinuous d e r i v a t i v e can be replaced by a

l o g i c a l f unc t i on L(z ,E ) which smoothly approximates a Heaviside step func t i on

~

1 " I ,

24

w i t h i n a given reg ion O<Z<E for a small number E . The l o g i c a l func-

t i o n L(z,E) i s def ined as C71:

2n+l (20)

1 1z12n+1 + z L(Z,E) = - 2

( Z p + l + 1 [ ( Z ' E J 2n+l - ( P E ) 2n+l 3 2

where n i s an i n tege r selected so as t o ensure the c o n t i n u i t y o f the

d e r i v a t i v e up t o order d, i.e., 2n+l >d . The n i s taken as 1 i n t h i s

study. Note t h a t the values of l o g i c a l funct ion L(z,E) are 0, l/2 and 1 f o r

z=O,~/2 and E. , respec t ive ly . The d i f f e ren t i a l equation, equat ion "4, can be

combined i n t o a s ing le funct ion by us ing the l o g i c a l f unc t i on L(z,E) as

Since L i s a smooth funct ion, there i s no d i s c o n t i n u i t y i n the o f the

preceding equation. Thus, the formulat ion of the design s e n s i t i v i t y ana lys i s

can be s i m p l i f i e d a g rea t deal.

The design d e r i v a t i v e o f func t ion +2 , def ined i n the example 2, can be

e a s i l y obtained by using the standard a d j o i n t va r iab le technique:

a f a f d% 1 2 - = I"[ - - (1-L)h - - LA] d t dT 0 aT aT

(21)

~ where the a d j o i n t va r iab le h s a t i f i e s the fo l l ow ing a d j o i n t equation:

3 A + 2a=o af2 aL L + f 2 E aL + + aa ( 1 4 ) - fl 2f 1

w i t h terminal cond i t i on

h ( z ) = 0 . With the d e f i n i t i o n o f L(a-0.3,~) Eq, 20, the de r i va t i ve aL/2a i s n o t

25

d i f f i c u l t to calculate.

A 1 though using a logical function t o smooth the derivative d i scon t inu i ty ,

one may avoid the need to identify the cr i t ical time of jump condition, y e t

the selection o f e , the domain where t h e logical function i s defined,

introduces a new difficulty. The numerical resul ts l is ted i n Table 4.b are

obtained by i n t e g r a t i n g Eqs. 20 - 22. To o b t a i n these results, the report

time step required i n DE program is given as AT=0.05 .

I

I 11. Numerical Considerations

The c a l c u l a t i o n o f the design s e n s i t i v i t y o f thermal system w i t h

discont inous de r i va t i ves encounters i n numerical d i f f i c u l t i e s as expected.

Numerical e r r o r s a r i s e n o t only i n so lv ing the s t a t e equation b u t a l so i n

i d e n t i f y i n g the c r i t i c a l t i m e of jump condi t ion.

There are two numerical i n t e g r a t i o n schemes employed here t o solve the

s ta te and a d j o i n t equations discussed i n the preceding sect ion. One i s the

Simpson's method, the o ther i s DE program C81 using the Adam fami ly o f

formulas. .,

Since the t runcat ion e r r o r o f Simpson's method i s p ropor t iona l t o the

f o u r t h order de r i va t i ve of unknown function, i t provides exact i n t e g r a t i o n f o r

so l v ing equat ion i n the f i r s t example.

The DE program i s one o f predic tor -corrector i n t e g r a t i o n a lgor i thm us ing

Adams fami ly o f formulas. The t runcat ion e r r o r i s con t ro l l ed by vary ing both

the step s i ze and the order o f the method. The t runcat ion e r r o r a t time step

tn+l i s requ i red to s a t i s f y

I t runc I < ABSERR + RELERR 1zn1 I

where the zn i s the s o l u t i o n o f d i f f e r e n t i a l equat ion a t tn and the values

ABSERR and RELERR are suppl ied by the user. The DE program i s q u i t e easy t o

be used and has c a p a b i l i t y to manage moderate s t i f f equation which happens

commonly i n the problem o f chemical k ine t ics .

The c r i t i c a l time t a t which the jump cond i t i on occurs i s determined by

se lec t i ng the time g r i d p o i n t c loses t t o the cond i t ion z(?) = c f o r a given

constant c and s ta te var iab le z. Thus, the accuracy of t s t rong ly depends on

the step size.

-

\

I

27

The cost function and design derivative obtained by the adjoint variable

The comparisons between technique of the f i r s t example are examined f i r s t .

different numerical integrations used to solve equat

and bounds of errors are l i s ted i n Table 1. I t is

provide satisfactory results. Since b is taken as

ons, as well as time step

ndicated that a l l o f them

400 i n this calculation,

t i s exactly 0.3. Therefore, there is no approximation error a t a l l on the

c r i t i ca l time i . Note that when the DE program i s used for solving the

s ta te and adjoint equations, the cost function 447) = ,," a2 d t i s solved by an 2 additional differential equation (t = a

-

. ..

To investigate possible sources of errors i n the numerical calculations

of thermal design derivatives, the change of c o s t functional w i t h respect t o

different perturbation size of design variables are l i s ted i n Table 2. The

design derivatives, $', i n Table 2 are obtained by u s i n g the f i n i t e

difference method and adjoint variable techniques w i t h different numerical

integration algorithms. I t shows that the f i n i t e difference method, the

adjoint variable techniques w i t h Simpson's rule and w i t h DE program

introducing error bounds less than 10E-4 exhibit quite a deviation a g a i n s t the

exact calculation. The major source o f error m i g h t be the miscalculation

of 't i n the analysis.

! As an example, when b is reduced t o 399.9 (0.025% change) the difference

I between cost functions evaluated exactly, 5293504.6, and evaluated by the

Simpson's rule, 6194664.7, soars up t o 901160 ( 1 7 % ) . T h i s b i g discrepency

resul ts from the numerical prediction of cr i t ical time ? which should be 0.299

analytically instead of 0.3 numerically. Although the error of f i s small, i t

happens i n a neighborhood of steep which causes significant error of z.

Moreover, t h i s error i s squared and accumulated through t = 0.3 to t = 2.0.

I

28

It has been shown i n Table 2 t h a t the accuracy o f the design s e n s i t i v i t y

ca lcu la ted by the a d j o i n t var iab le technique i s l ess sens i t i ve t o d i f f e r e n t

numerical i n t e g r a t i o n schemes than the accuracy of the cos t f unc t i ona l

eva lua t ion does. To see the e f f e c t of the misca lcu la t ion o f 't on the accuracy

o f the design s e n s i t i v i t y ca lcu la ted by the a d j o i n t var iab le technique,

var ious i ' s , which i s supposed to be 0.3 a n a l y t i c a l l y , are used f o r the design

s e n s i t i v i t y ca lcu la t ion . The r e s u l t s are l i s t e d i n Table 3. It i s c l e a r l y

shown t h a t the a d j o i n t va r iab le approach i s q u i t e i nsens i t i ve , compared to the

f i n i t e d i f f e rence method, t o the numerical e r r o r s a r i s i n g i n the ana lys is and

i n the est imat ion of 't a t the jump conditions.

..

I n example 2, ne i the r s t a t e var iab le z nor c r i t i c a l time 5 has a n a l y t i c a l

so lu t ion . The design s e n s i t i v i t i e s l i s t e d i n Table 4.a are ca lcu la ted by the

f i n i t e d i f fe rence method and the a d j o i n t va r iab le technique w i t h DE program.

It i s again shown t h a t the a d j o i n t var iab le technique provides a more s tab le

s o l u t i o n than the f i n i t e d i f f e rence method does, aga ins t numerical e r ro rs .

The jumps o f state va r iab le and a d j o i n t var iab le for example 2 a re i nd i ca ted

c l e a r l y i n Fig. 2.

Furthermore, the numerical r e s u l t s l i s t e d i n Table 4.b are obtained by

us ing the l o g i c funct ion approximation and by i n t e g r a t i n g Eqs. 20-22. The

accuracy o f the approach i s a l so q u i t e sa t is fac to ry .

I V . Conclusions and Remarks

The c a l c u l a t i o n o f design s e n s i t i v i t y i s discussed f o r the thermal

t r a n s i e n t problem w i t h discont inuous der iva t ive . The numerical d i f f i c u l t i e s

depend on the approximation e r r o r o f i n teg ra t i on and the evaluat ion o f the

c r i t i c a l t ime o f jump condi t ion.

Because o f the s i m p l i c i t y , i t i s a very common p rac t i ce i n the opt imal

29

design community t h a t the f i n i t e d i f fe rence approach i s used as a standard

reference t o check the accuracy of design s e n s i t i v i t y ca lcu la t ions . However,

i t i s revealed i n t h i s i nves t i ga t i on that the f i n i t e d i f f e rence method may

prov ide very un re l i ab le in format ion o f design s e n s i t i v i t y . On the other hand,

the ad j o i n t var iab le technique using numerical i n t e g r a t i o n a lgor i thm w i t h

var ied step s ize and e r r o r con t ro l performs s a t i s f a c t o r i l y i n the design

s e n s i t i v i t y analysis. F ina l l y , i t i s also suggested i n t h i s i n v e s t i g a t i o n

t h a t the design s e n s i t i v i t y analys is can be used as an accuracy i n d i c a t o r f o r

analyz ing the t r a n s i e n t problems w i t h discontinuous der iva t ives . .I

Acknowledgment

Research sponsored by NASA Langley Research Center, NASA Grant NAG-1-561.

References of Appendix

1. R. T. Haftka, "Technique fo r Thermal S e n s i t i v i t y Analysis," I n t . J. Num. Meth. Engng., 17, 71-80 (1981)

2. L. t. Biegler , "Solut ion of Dynamic Opt imizat ion Problems by Successive Quadrat ic Programming and Orthogonal Collocation," Design Research Center, Carnegie-Mellon Un ive rs i t y , P i t t sbu rgh (1983)

3. R. T. Haftka and D. S. Malkus, "Calcu lat ion of S e n s i t i v i t y Der i va t i ves i n Thermal Problems by F i n i t e Differences," I n t . J. Num. Meth. Engng. 17, 1811-1821 (1981)

4. A. E. Bryson and Y. C. Ho, Applied Optimal Control, Ginn and Co., Massachusetts , 1969

5. A. C. Loos and G. S. Springer," Curing o f Epoxy Ma t r i x CompositesIn J. .* Composite Mater ia ls , 17, 135-169 (1983)

6. R. C. Huang, E. J. Haug, and J. G. Andrews, " S e n s i t i v i t y Analysis and Optimal Design o f a Mechanical System w i t h I n t e r m i t t e n t Motion," ASME J. Mech. Design, 100 (31, 492-499, (1978)

7. P. E. Ehle and E. J. Haug, "A Logical Function Method f o r Dynamic and Design S e n s i t i v i t y Analysis o f Mechanical System w i t h I n t e r m i t t e n t Motion," ASME J. Mech. Design, 104(1), 90-100 (1982)

8. L. Shampine and M. Gordon, Computer So lu t i on o f Ordinary D i f f e r e n t i a l , Equations W. H. Freeman and Co., San hancisco, ( 1 9 / n

31

E CT, .C

Y- O

v) E 0 v)

L a E 0 V

.r

n c, V (d X W

c, E aJ .C

QI d 0 d QI N In

m h 0 d m (v In

m h 0 -9. QI (v m

co h u) d m N In

N h 0 d Q, N m

3

(u

10 m 0 ro

. W m 0 W

h

;n m 0 rD

W d 0 ro

h

4 d 0 W

-3

32

Table A2. Per turbat ion o f Cost Funct ion

(a) Exact I n teg ra t i on based on the given c r i t i c a l t ime

0.1

1

4

10

20

40

-0.1

-1

-4

-10

- 20

-40

5294641.

5299760.

5316879.7

5351383.7

5409692.

5530494.8

5293504.6

5283205.2

5265617.6

5230743.3

5168474.0

5047690.4

568.2

5687.2

22807.

57311.

115619.2

236422.1

-568.1

- 10867.5

-28455.0

-63329.4

-125598.6

-246382.3

604.1

6041.7

24166.8

60417.

120834.

241668.

-604.1

-6041.7

-24166.8

-60417.

-120834.

-241668.

0.3

0.3

0.3

0.3

0.3

0.29

0.3

0.31

0.31

0.31

0.32

0.33

* by the f i n i t e d i f f e rence method

by us ing the a d j o i n t va r iab le technique

I

I

I **

m 0 .

L n

m 0 W

.

. m 0 W

W h W d m N v)

m 0 .

W

d 0 W

. 4 m 4 N

m W h W m N v)

4 . 0

m 0 .

h

v) m d W

.

. W m 0 W

ul 0 4 0 0 m v)

m 0 .

'9 d 0 W

h W d W

. v) W 0 rl 0 m v)

4

m 0

rl

m

d N

. z

. 0 d 4 d cu

m 4 N co 4 m v)

(3

0

d Q) 4 d N

W v) h 0 d

. d m m v) m m v)

d

. h v) m 0 W

. v) m m 0 W

. W 0 d d Ln m v)

m 0 .

0 W d 0 W

. 0 W 0 m W

. co v) W h v) m v)

E:

(3

0 .

. v) 4 h 0 N 4

4 m W 0 N rl

d 0 h d 4 d v)

m 0

0 N cn 0 N 4

. h m W (u (u 4

Ln 4 N h rl

b,

0 (u

Q! N

0

N

4 m d 4 d N

.

e m cn 4 4 d (u

W W N v) m v) v)

m cu 0 . . 0 d W 4 d N

. W 4 (u W v) (3

W ul h O v) W Ln

0 d

m 0

v)

m 0 W I

d 0 W I

. cn w d m m (u v)

m 0 .

10

d 0 W I

.

00 0 h

W W (v v) ul cu v)

l-l . 0 I

d m 0 .

h

Ln m 0 rD I

. b m 0 W I

. rD m 0 aJ co N v)

rl m 0

W d 0 W I

. AI m m v) I

. W d rD a3 aJ N v)

d I

4 m 0

rl

m

d N I

2

rl v) rl d N I

N (v ul m W N ln

4 m 0

. d W rl d N I

. N ul d d N I

. W W 0 0 h N Ln

d I

rl m 0

W

h v) m 0 W I

. m ul m 0 W I

. e h W m m N v)

4 m 0

0 W d 0 W I

. d v) W 0 W I

. d cy h m m N v)

$: I

N m 0

W

v) l-l h 0 N 4 I

. h W W 0 cy 4 I

W W 4 m h 4 v?

N m 0 .

0 (v m 0 N 4 I

. 0 QI W 4 4 I

. W W W Ln h 4 v)

0 N

I

m (3

0 .

cu 4 (3 e- 4 d (u I

.

. (v Ln (v (v d (u I

l-l cy W 4 u) 0 u)

m m 0

. 0 d a0 4 d (u I

. Qo (u m N d (u I

. 0 In N cu v) 0 v)

0 d I

al c

J w v) - c 0 v) n E .C

h c, w c,

.r

-r

.- v) c al v)

h

V Y

4 0

0 II

c,

9

a

4 0 0

11

c, a

c,

n ro 3

3 a

3

c,

n rg

3

3 a

3

n 10

0 0 m 0 .

W

m 0 W

.

m m m co d

.

m co co co co m N v)

m . W

m 0 ID

.

0 d d co d

(v

d co 03 a3 m N v)

.

d . 0

0 0 m 0 .

(v

W m 0 W

.

v) m m 00 d

9 d co m (v d m v)

m

W m 0 W

. h 0 m co d

9 4 co m (v d m v)

4

co m (v

0

v)

d (v

z

4

m co m OD 0 4

4

(v m d (v 0 d v)

1

. d d d d (v

v)

v) (v (v v) m 4

W

co m (v m co d rn

.

d

v) m (v

0

v)

(v W m 0 W

.

v)

m 0 m co m 4

.

v)

co v) m (v m d v)

.

m

0 W m 0 W

h

00

co m d

2

m 4 W (v (v m h m

E:

0 m (v

0

. v) cu h 0 (v 4

d

(v m m m h 4

.

d 4 co Q, h W d v)

.

m (v

0 (v h 0 N 4

co v) m v) 0 m E:

m co 0 u d (v m W

.

0 (v

(v co cu 0

0 In d 4 d (v

0 v) m W d Q) cu

.

9 d a0 rD 00 h v) In

co (v . 0 d d 4 d cu

9 4 h v) v) h

2

4

d d W m W m W

0 d

0 0 m 0

W

m 0 W I

W

0 h Q, m I-

.

W

m d 0 d h m v)

.

1

W m 0 W I

v)

d m v) 0 0 m

.

I-

d W W d m 4 W

4

0 I

4 0 m 0

(v

W m 0 W I

v)

cu W 0 W m

v)

4 d 4 0 m m v)

m . W m 0 W I

h

0 (v W m d a3

co m m W m d 4 W

4 1

(v 0 m 0

v) d 4 d (v I

W

co h d co v)

W

h (v v) (v v) m v)

1

d d d d (v I

d 4 h m 4 co W

.

W

d d 0 W h m v)

.

d I

v) 0 m 0

v)

cu W m 0 W I

h

0 Ln v) m 4

h

m m v) h 0 m v)

.

4 m . 0 W m 0 W I

h

h W m W v) m

.

el

0 d 0 4 v) W v)

52 I

2 m 0 .

v) (v h 0 (v d I

0 W CI) d v) h I

Q,

(v 4 rn co d (v v)

4 m

. 0 (v h 0 (v d I

m W m v) m m W

.

d . 0 h W m m m v)

0 (v I

(v (v m 0

0 v) d 4 d (v I

d

02 d (v m (v (v I

? 0 0 a2 d 10 0 la

(v m

0 d d

d (v I

v) W v) W 4 d

43 . 3 W 0 4 h v)

0 d I

35

E m L m 0 L

CL.

W 3

0

m Y

-I n 3

3 3

3

n N 3

-.(

3

3 3

3

n N 3

c,

4 0 0 0

I1

2

4

3

t;

0

II

c P

03 In 0 W

In (v

W m 0 W

m l-4

z 0 W

Q) 4

N r- (v W

m m W m 0 W

(u m m m a3 In

*

36

Y 0 Ln h * II I- c, 4

(v

aJ n E

- 4 X W

y. 0 v) *I- v)

h c 4 r U h c, w - .-

* 4 aJ

c ro

3 -

* ++ a

I- a

p: 0 p: p: W

h h rT) m m 4

0 I

00 d 0 m d

0 I

4

Ln 0 0 0

00 I

W

4

w 4 m m m 4

0 I

co * 0 m d 4

0 I

In 0 0

a I

W

4

s 00 m m

0 t

d

(v W c3 In Ln 4

0 I

Ln 0 0 0

d I

W

4

m 4 00 d m 4

0 I

4 4 d 4 4 4

0 I

ln 0 0

d

I

W

4

* 5

* 3 a

* *Id

E 0 p: p: W

(v 0 (v v) 4

0 I

4 (v m d

0 I

4

h 4 0 0 0 0

co I

W

4

4 Ln 4

0 I

4 (v m d

0 I

4

m 0 0 0 0

00 I

W

4

QI c3 4 Ln 4

0 I

m 4 4 4

0 I

m 0 0 0 0 0

d

1

W

4

Y

d h * 0 w Y

In h *

0

0

E L Y-

U aJ 0, E 4 r V

v)

aJ L J c, 4 L aJ p. E aJ c,

al r I-

.r

* * *

16

0

CI

- x Y

n

0.5- 0

o - a - Y-

e

0

..

F

0 0.5 I .o g u r e A l . S t a t e V a r i a b l e and A d j o i n t V a r i a b

2;O Time (Min.)

e f o r Example 1.

38

n a U U t

I .6

0.8

0 I I I 1 0 4 8 12 16 Time ( M i d

1.0 -

0.5 -

0 I I I 1

0 4 8 12 16 Time (Min.)

20 -

IO-

0-/

-10 I I I t 16 Time ( M i d 0 4 8 12

Figure A2. State Variable and Ad jo in t Variable for the C u r i n g Process.