Sensitivity Analysis of Discrete Structural Systems

NASA Technical Memorandum (YASA-TH-86333) SENSIT1V:IY DISCRETE STBUCTPBAL S Y S I E B S : (UASA) 39 HC A03/8P A01

86 333

NeS-lElE6

ANALYSIS OE P SUBVBY

C S C L 20K

Onclas G3/39 SENSITIVITY ANALYSiS FOR DISCRETE STRUCTURAL SYSTEMS

HOWARD M. ADELMAN AND RAPHAEL T. HAFTKA

DECEMBER 1984

Nat~onalAeronautics and Space Adrn~n~stratlon hngley Rmbarch Center Hampton,Vlrgmia 23665

-A

13023

SURVEY

SENSI T I V I rY ANALY 513 FOR 01SCHETE STRUCTURAL SYSTEMS

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A SURVEY

In t roduct ion The f i e l d o f s e n s i t i v i t y analysis i s emergiog as a f r u i t f u l area o f engineering research.

The reason f o r t h i s i n t e r e s t i s t h e r e c o g n i t i o n o f t h e

v a r i e t y o f uses f o r s e n s i t i v i t y d e r i v a t i v e s .

I n i t s e a r l y stages, s e n s i t i v i t y

analysis found i t s predominant use i n assessing t h e e f f e c t o f varying parame t e r s i n mathematical models of c o n t r o l systems:

see, f o r example, the t e x t s

o f Tomovic (1963); Brayton and Spence (1980); Frank (1978); and Radawvic (1'4bb) f o r discussions o f the e a r l y development o f s e n s i t i v i t y theor-y.

I n t e r e s t i n optimal c o n t r o l i n t h e e a r l y 1960's (see, f o r example, Kelley, 1Y62), and automated s t r u c t u r a l o p t i m i z a t i o n (see, f o r example, Schmi t, 1981) led t o the use of gradient-based mathematical p r o g r a m i n g methods i n which a e r i v a t i ves were used t o f i n d search d i r e c t i o n s toward o p t i m m solutions. More r e c e n t l y , there has been s t r o n g i n t e r e s t i n promoting systematic s t r u c t u r a l o p t i m i z a t i o n as a u s e f u l t o o l f o r t h e p r a c t i c i n g s t r u c t u r a l design engineer on l a r g e problems--a process s t i l l underway.

E a r l y a t t e n p t s t o use

formal o p t i m i z a t i o n f o r l a r g e s t r u c t u r a l systems r e s u l t e d i n excessively lony and rxpensi ve computer runs.

Examination o f t h e o p t i m i z a t i o n procedures

i n d i c a t e d t h a t the predominant c o n t r i b u t o r t o t h e cost and time was the c a l c u l a t i o n of d e r i v a t i v e s .

As a consequence, t h e r e has been an emergence o f

i n t e r e s t In s e n s i t i v i t y a n a l y s i s ecnphasizing e f f i c i e n t computational proceddres.

I n a d d i t i o n , researchers have developed and applied s e n s i t i v i t y

analysis f o r dpproximate analysis, a n a l y t i c a l model improvement, and assessment of design trends-so

t h a t s t r u c t u r a l s e n s i t i v i t y analysis has become

more than a u t i l i t y f o r optimization, but i s a v e r s a t i l e design t o o l i n i t s own r i g h t

.

Most recently, researchers i n d i s c i p l ines such as physiology

(Leonard, 1974), thermodynamics ( I r w i n and 0' Brien, 1982), physi c s l chemi stry (Hwang, e t a1

Dwyer e t al.,

., l978),

and aerodynamics (Dwyer and Peterson, 1980;

1976; B r i s t w and Hawk, 1983), have been using s e n s i t i v i t y t o assess the e f f e c t s of

method01oqy

parameter var i a t i o n s i n t h e i r a n a l y t i c a l models, and t o create designs which are i n s e n s i t i v e t o parameter v a r i a t i o n (Schy and Giesy, 1981; 1983). This paper i s a survey of methods applicable t o t h e c a l c u l a t i o n o f s t r u c t u r a l s e n s i t i v i t y derivatives f o r f i n i t e element modeled structures. Except f o r c i t i n g several general references, the paper does not deal w i t h continuous ( d i s t r i b u t e d parameter) models.

The survey p r i n c i p a l l y discusses

1it e r a t u r e pub1ished during the past two decades and t h e paper concentrates on four main topics:

derivatives o f s t a t i c response (displacements and

stresses) , e i genval ues and e i genvectors, t r a n s i e n t response, and d e r i v a t i ves o f optimum s t r u c t u r a l designs w i t h respect t o prob lem parameters.

The bulk o f

t h e survey deals w i t h derivatives o f the aforement ioned responses w i t h respect t o gage-type variables such as rod cross-sectional areas, beam cross-sect iona 1 dimensions, anc p l a t e thicknesses.

Additionally, some works are reviewed i n

which the derivatives are calculated w i t h respect t o variables which def ine t h e shape o f s t r u c t u r a l elements.

Methods f o r c a l c u l a t i n g s t r u c t u r a1 sensi-

t i v i t y derivatives are summarized i n Table 1. S e n s i t i v i t y o f S t a t i c Response General Equat ions This section o f the paper focuses on the c a l c u l a t i o n of derivatives o f s t a t i c s t r u c t u r a l response (displacements and stresses) computed from f i n i t e element models.

The governing equation f o r displacement i s

K

where

i s the symmetric s t i f f n e s s matrix o f order

nxn

U i s the vector o f displacement F

i s the vector o f applied forces

Both K and F are, i n general, functions o f design variables,

v.

A

a c o n s t r a i n t ) w i 11 be respresented as

t y p i c a l function of displacement (e.g.,

F i n i t e Difference Method A straightforward method o f c a l c u l a t i n g d e r i v a t i v e s o f

f i n i t e difference approximati on.

g

i s t o use a

For example

A serious shortcoming o f the f i n i t e d i f f e r e n c e method i s the uncertainty i n

the choice o f a perturbation step s i z e t r u n c a t i o n e r r o r s may be excessive.

h.

If the step s i z e i s too large,

These can be thought o f as e r r o r s due t o

r e t e n t i o n of only the lowest-order terms o f a Taylor series representation of a perturbed function. occur.

I f the step s i z e i s too small, cond,ition e r r o r s may

Condition e r r o r s are due t o inaccuracies i n the c a l c u l a t i o n o f t h e

displacements and round-off e r r o r s i n the f i n i t e d i f f e r e n c e calculation. G i 11, e t al. (1980, 1983) developed an algorithm t o determine t h e optimum

f i n i t e difference step size; i.e., condition errors.

one which balances the t r u n c a t i o n and

The a1 g o r i thm i s based on approximatin,

e r r o r as a l i n e a r function o f step s i z e l i n e a r function of l / h .

the t r u n c a t i o n

h and t h e condition e r r o r as a

This technique has been tested on functions which

could be d i f f e r e n t i a t e d a n a l y t i c a l l y f o r check purposes and was found t o be very e f f e c t i v e .

Other work on f i n d i n g optimum step sizes was done by Stewart

(1967); Kelley and Lefton (1980); and Haftka and Malkus (1981).

A recent

paper by Haftka (1984) describes s technique f o r reducing c o n d i t i o n e r r o r s i n f i n i t e difference d e r i v a t i v e s of response q u a n t i t i e s obtained by i t e r a t i v e methods. Analytical Methods Analytical c a l c u l a t i o n s o f d e r i v a t i ves o f displacements and functions thereof have been described by Arora and Haug (1976, 1979); and Haug and Arora (1971j

.

I n these references, three methods are described:

the di rect o r

design space method ( a t t r i b u t e d t o Fox, 1965), the a d j o i n t v a r i a b l e o r s t a t e space method, and the v i r t u a l load method ( a t t r i b u t e d t o Barnett and Hermann, 1968).

The v i r t u a l load method i s a special case o f the d i r e c t method.

Both

t h e d i r e c t and a d j o i n t methods begin w i t h the d i f f e r e n t i a t i o n of equations (1) and (2).

D i r e c t Method.

and s u b s t i t u t e

dU/dv

The d i r e c t method i s t o solve equation (4) f o r i n t o equation (5).

dU/dv

Equation (4) needs t o be solved

once f o r each design variable ( v ) so t h a t t h e d i r e c t method i s c o s t l y when t h e number o f design variables i s large, Adjoint Method.

The a d j o i n t variable o r s t a t e space method has been

extensively used i n optimal c o n t r o l theory; see, f o r example, Kel l e y (1962).

The method s t a r t s by d e f i n i n g a vector o f a d j o i n t variables which s a t i s f i e s the equation

where

ag/aU

i s sometimes r e f e r r e d t o as the dummy load vector.*

Then using

equations (4), (5). and (6)

The a d j o i n t variable method requires t h e s o l u t i o n of equation (6) once f o r each function

g.

Therefore,

if the number o f functions i s smaller than

t h e number o f design variables, the a d j o i n t variable method i s more e f f i c i e n t and conversely if the number of design variables i s smaller, t h e d i r e c t approach i s more e f f i c i e n t .

Roth the d i r e c t and a d j o i n t methods involve fewer

computations than the f i n i t e difference approach which requires repeated f a c t o r i z a t i o n o f the s t i f f n e s s matrix, whereas the d i r e c t and a d j o i n t methods require a s i n g l e f a c t o r i z a t i o n w i t h several right-hand sides. Chon (1984) developed a variant o f the a d j o i n t method v i a s t r a i n energy d i s t r i b u t i o n and implemented i t i n a p r o p r i e t a r y version o f NASTRAN.

Hsieh

and Arora (1983); and Gurdal and Haftka (1984) extended t h e a d j o i n t method f o r boundary conditions which require special ized treatment w h i l e Haug and Choi (1984) suggest a generalization o f the a d j o i n t method t h a t eliminates many o f the problems associated w i t h mu1t i - p o i n t boundary conditions.

Adaptation o f

*Note t h a t i f g i s a p a r t i c u l a r displacement component, then ag/alJ corresponds t o a force o f u n i t magnitude i n the d i r e c t i o n o f the conponent.

t h e a d j o i n t variable method t o substructured f i n i t e element models i s described by Arora and Govil (1977). Calculation of

aK/av.

An important computational task i n the a d j o i n t

and d i r e c t methods i s the c a l c u l a t i o n o f

aK/av.

I f t h e s t r u c t u r a l model

contains only elements whose s t i f f n e s s matrix i s proportional t o rods where where

v

v

v

(such as

i s the cross-sectional area, o r membranes and shear panels

i s the thickness),

i s a constant matrix.

aK/av

But for elements

having bending s t i f f n e s s such as beams and plates, the s t i f f n e s s matrix i s a nonl inear function o f the cross-sectional dimensions and t h e s t i f f n e s s m a t r i x derivatives are not e a s i l y evaluated (see Giles and Rogers, 1982). p r e f e r r e d approach i s t o compute

aK/av

Hence, t h e

by f i n i t e differences as i n Prasad

and Emerson (1982); Camarda and Adelman (1984); and Wal l e r s t e i n (1984). Derivatives w i t h Respect t o Shape Design Variables Shape design variables t y p i c a l l y c o n t r o l t h e shape o f the boundary o f the structure--for example, variables c o n t r o l l i n g the shape o f a hole (and thereby the stress concentration f a c t o r a t the hole boundary).

The c a l c u l a t i o n o f

d e r i v a t i v e s w i t h respect t o shape design variables i s i n t h e e a r l y stages of development and there are unresolved issues.

Differentiating the f i n i t e

element equations t o obtain equation (4) has two disadvantages.

F i r s t , even

small changes of the boundary can change t h e e n t i r e f i n i t e element mesh and therefore, the c a l c u l a t i o n o f

aK/av

can be q u i t e costly.

Second, changes i n

shape can lead t o the d i s t o r t i o n o f the f i n i t e elements and reduced accuracy.

Thus, the d e r i v a t i v e s obtained from equation (4) have a spurious

component which r e f l e c t s the changing accuracy o f the s o l u t i o n when the mesh i s d i s t o r t e d (Botkin, 1982;fjennett

and Botkin, 1983)

.

Because o f the above, there has been substantial wwk i n obtaining s e n s i t i v i t y d e r i v a t i v e s by d i f f e r e r ~ tai t i n g the continuum equations before d i scretizing.

Derivations based on t h e concept o f material d e r i v a t i v e s have

been proposed by Chun and Haug (1978, 1979, 1983); Rousselet and Haug (1981, 1983); Rousselet (1983b); Zolesio (1981); Choi and Haug (1983); Dems and Mrdz (1984a); Braibant and Fleury (1984); Yoo, Haug, and Choi (1984); Choi (1984) ; and Yang and Chef (1984).

However, computational experience using

equation (4) does not i n d i c a t e t h a t mesh-di s t o r t i o n - d e r i v a t i ve e r r o r s are s i g n i f i c a n t (possibly due t o t h e use of elements which are not s e n s i t i v e t o distortion).

The material -deri v a t i ve approach seems t o s u f f e r from numerical

d i f f icu! t i e s associated w i t h the evaluation o f boundary integral s (see Yang

and Choi, 1984).

While some of these computational d i f f i c u l t i e s may be

e l i n d nated by replacing boundary by volume i n t e g r a l s (Choi and Haug, 1984). a t t h e present there i s no c l e a r i n d i c a t i o n as t o which method i s preferable. Calculation o f Second Derivatives Second d e r i v a t i ves o f displacement and c o n s t r a i n t functions are used f o r approximate analysis (e.g.,

Noor ana Lowder, 1975). and f o r t h e c a l c u l a t i o n o f

d e r i v a t i v e s o f optimal s o l u t i o n s (see subsequent section on t h i s t o p i c ) . d e r i v a t i v e s may be obtained by d i f f e r e n t i a t i n g equations (4) and ( 5 ) , f o r example,

Such

However, f o r

m design variables there are m(m + 1)/2

second derivatives,

and equations (8) need t o be solved f o r t h a t many right-hand sides.

It i s

possible t o proceed w i t h an extension o f t h e a d j o i n t variable method proposed by Haug (1981b); and Oem and Mrdz (1984b).

However, a mom e f f i c i e n t

approach proposed by Haftka (1982) i s t o use equation (6) t o obtain

This approach requires t h e s o l u t i o n of equation (4) f o r a l l the f i r s t derivatives and equation (6) f o r a l l vectors o f a d j o i n t variables. Second derivatives were also derived by Van B e l l e (l982), using f l e x i b i l i t y rather than s t i f f n e s s matrices.

F i n a l l y , Jawed and Morris (1984)

described a procedure f o r approximating higher order d e r i v a t i v e s from the f i r s t d e r i v a t i v e information, which i s equivalent t o introducing intermediate variables. Stress Deri vat ives The stresses i n an element may be obtained from the displacements using

where

u

i s a vector of element stresses

T I s an element temperature S

and G are stress-di spl acement and stress-tenperature matrices,

respectively. Derivatives o f stresses may be obta ined by d i f ferent i a t i n g equat i o n (10)

For f i n i t e elements such as rods, membranes, and shear panels, independent o f

v

v

and G

arc

and stress d e r i v a t i ves are obtained by simply s u b s t i t u t i n g

dU/dv i n t o equation (11). functions of

S

For bending-type elements,

S and G may be

and the complete expression must be used; see Camrda and

Adel man (1984). Nonlinear Analysis When geometric or material nonl i n e a r i t i e s are important, equation (1) i s no longer v a l i d and the displacement

U

i s calculated from a system o f t h e

form

where F

i s a vector o f nonlinear functions.

d i f f e r e n t i a t i n g equation (12) w i t h respect t o

where the Jacobian 3 ness matrix).

is

t h i s i s the d i r e c t method.

dU/dv

v

( o f t e n referred t o as the tangential s t i f f -

The d e r i v a t i v e o f any constraint

s o l v i n g equation (13) f o r

vector

aF/aU

Derivatives are obtained by

g may be calculated by

and then s u b s t i t u t i n g i n t o equation (5)--

A l t e r n a t i v e l y one can solve f o r t h e a d j o i n t

a from

and calculate

dgldv

from equation ( 7 )

using Rv

from equation (13).

Appl icat ions Appl i c a t i o n s o f d l spl acement sensi t iv i t y d e r i vat ives f o r formal optimi

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zation are described, f o r example, i n Nguyen and Arora (1982); Arora (1980); Prasad and Haftka (1980) ; and Schmi t and Farshi (1974).

Use of d i s p l a c e w l

and stress derivatives t o construct expl i c i t constraint approximations ir described, f o r example, by Schmit and Farshi (1974); Storaasli and and Noor and Louder (1975). S ~ b i e s r c z a n ~(1974); ki

A basic example o f such an

approximation i s

where ~ ( v ) i s the displacement vector f o r the design variable the vector corresponding t o the new design variable

v* = v +

AV

v,

.

U(v*)

is

Numerous

examples o f application o f stress derivatives i n formal optimization are c i t e d i n the survey by Schmit (1981).

Less well known i s t h e use of s e n s i t i v i t y

derivatives of stresses t o effect design changes without formal optimization. A good example o f t h i s i s reported by Musgrove, e t al. (1983).

The most

common appl i c a t i o n o f sensi t i v i t y calculations i n nonlinear s t a t i c response are of derivatives o f

U w i t h respect t o a load parameter.

Such d e r i v a t i v e s

are useful i n incremental s o l u t i o n procedures of equation (12) o r fur reduced basis s o l u t i o n of t h i s equation (see, f o r example, Noor and Peters (1980). F i n a l l y , readers interested i n the t o p i c o f s t a t i c response s e n s i t i v i t y o f d i s t r i b u t e d parameter systems are referred t o Haug and Komkov (1977); Haug and Rousselet (1980a); Haug (1981a); and Rousselet (1983a); as well as t h e t e x t o f Haug, Kordtov, and Choi (1984).

L

S e n s i t i v i t y o f E i genvalues and E i genvectors The general problem i s t o compute d e r i v a t i v e s o f eigenvalt -s and eigenvectors w i t h respect t o design variables o r system parameters.

For

reference purposes, the most general case considered i s the f o l lowing e i genvalue problem:

where

A

i s an e l genvalue (general l y complex).

nxn matrices

A

and B

of design variables

v.

And

real

respect ively

.

The general l y nonsymnetric

are assumed t o be e x p l i c i t functions of a set

X and Y

are r i g h t and l e f t eigenvectors,

The f ir s t r e s u l t on e i genval ue d e r i v a t i ves was pub1ished by

Jacobi (1846) who developea the r e s u l t f o r the special case o f symmetric A,and B =

1

f

1

W i t t r i c k (1962) applied Jacobi's f o r m l a for the case of a symmetric matrix t o the d e r i vat ives o f buckl ing e i genvalues and presented r e s u l t s f o r the change i n buckl i n g loads o f plates w i t h respect t o aspect r a t i o and thickness. Lancaster (1964) developed a rigorous treatment of eigenvalue d e r i v a t i v - i and, i n p a r t i c u l a r , showed t h a t f o r m l t i p l e eigenvalues, the d e r i v a t i v e s themselves are solutions o f an eigenvalue problem.

The issue of m u l t i p l e eigen-

values was also investigated by Simpson (1976); and Haug and Rousselet (1980b). who showed t h a t while simple eigenvalues are d l f f e r e n t i a b l e (Frechet) , nu1t i p l e eigenvalues are only d i r x t i o n a l l y (Gateaux) differentiable.

Two methods developed f o r sensi t i v i t y analysis o f e l e c t r o n i c networks are notable f o r the1 r non-re1 lance on e l genvectors I n t h e e i genvalue d e r i vat1 ve f o r m 1 as.

Rosenbrock (1965) and Morgan (1966) developed fornulas f o r e l gen-

value derivatives i n terms of the matrix A and i t s elgenvalues.

According t o

Morgan's own assertion however, the computational e f f o r t i s not nuch less than i f eigenvectors were required and examination of the d e t a i l s of t h e i r methods

indicates t h a t the calculations are eolllvalent t o those required f o r computing e i genvectors. Other contributions from the e l e c t r o n i c s d i s c i p l i n e include the use o f the a d j o i n t network theory.

An a d j o i n t network o r s t r u c t u r e i s one w i t h t h e

same geometry and nodal connections as the actual configuration, but t h e elements of the a d j o i n t system may be l i n e a r even though t h e actual elements are Vanhonacker (1980) has used the theory of a d j o i n t structures t o

nonlinear.

derive f o r m l a s f o r derivatives o f eigenvalues and e i genvectors o f structures. Fox and Kapoor (1968) and Fox (1971) considered the special case o f symmetric

A

general cases.

and 6

matrices, hut developed techniques applicable t o more

For eigenvalues, t h e i r f o r m l a i s

wherein i t i s assumed t h a t the eigenvectors are normalized such t h a t

For eigenvector derivatives, two methods are presented by Fox and Kapoor.

The

f i r s t i s t o d i f f ~ r e n t i a t eequation (16). g i v i n g a set o f simultaneous equat i o n s f o r the eigenvalue and eigenvector derivatives.

A complication here i s

t h a t the equations f o r the eigenvector derivatives are s i n g u l a r and the set i s

solvable m l y a f t e r a1gebraic manipulation which destroys the banded nature o f equations, a p o i n t which arises l a t e r i n connection w i t h another method.

The

second method f o r eigenvector derivatives, developed by Fox and Kapoor, i s t o expand the d e r i v a t i v e as a series of eigenvectors. Thus, f o r the

Tne c o e f f i c i e n t s

aik

eigenvector

are obtained by s u b s t i t u t i n g equation (22) i n t o equa-

t i o n s r e s u l t i n g from d i f f e r e n t i a t i n g equatiun (16). necessary t o ase a l l

i-th

I n principle, i t i s

n modes i n the expansion o f equation (22).

However, as

w i t h the modal method generally, i t should be possible t o obtain reasonakle r e s u l t s w i t h fewer than

n eigenvectors.

o f equation (22) i s c l e a r l y c a l l e d f o r .

Study o f t h e convergence properties Fox and Kapoor's second method was

specialized by H i r a i and Kashiwaki (1977) f o r the case o f design variables c o n t r o l l i n g only a small p a r t o f the structure.

Rogers (1970) and Stewart

(1972) d e r i ved sensi t i v i t y formulas f o r e i genvalues and e i genvectors o f the general problem (eqs. (16) and (17)).

For eigenvalues, the equation i s

Rogers expressed the derivatives as an expansion i n terms of the eigenvectors

The c o e f f i c i e n t s

aik

and

bi

are computed by s u b s t i t u t i n g equations (24)

i n t o an expression obtained by d i f f e r e n t i a t i n g the eigenvalue problem and combining i t w i t h appropriate orthogonal it y conditions.

P l a u t and Husseyin

(1973), as well as R u d i s i l l (1974); and Doughty (1982), developed t h e same r e s u l t s as Rogers and, i n addition, developed a fornula f o r second d e r i v a t i v e s Form1as f o r the second d e r i vat ives o f e i genvectors are

o f e i genval ulrs.

presented by Taylor and Kane (1975).

A

and B

Garg (1973) investigated the case where

were complex and produced f o r m 1 as f o r t h e eigenvalue and eigen-

vector d e r i v a t i ves.

His e i genvector d e r i v a t i v e procedures are analogous t o

those o f Fox and Kapoor.

Rudisi 11 and Chu (1975) developed the same

eigenvalue d e r i v a t i v e f o r , w l a s as Rogers.

Additionally, f o r eigenvector

d e r i v a t i v e s they extended Fox and Kapoor's f i r s t formulation t o the case here A

and B are nonsymmetric.

equations f o r the derivatives:

They suggest two ways t o solve t h e

an i t e r a t i v e method which converges t o t h e

d e r i vat ives of the lowest e i genval ue and corresponding e i genvector; and an algebraic method which i s an extension of Fox and Kapoor's f i r s t method. Andrew (1978 and 1979) provided some proofs and refinements o f R u d i s i l l ' s and Chu's a1g o r i thm.

Brandon (1984) showed t h a t second d e r i v a t i v e s o f eigenvalues

may be calculated by using t h e f i r s t d e r i v a t i v e s of the eigenvectors. An a l t e r n a t e method f o r c a l c u l a t i o n o f eigenvector d e r i v a t i v e s i s due t o D i f f e r e n t i a t i n g the e i genval ue problem o f equation (16) gives

Nelson (1976).

(25) The matrix A

-

a0

i s s i n g u l a r since

a

i s an e i genvalue.

Nelson i s t o represent the eigervector d e r i v a t i v e as

The method o f

where

V

i s the s o l u t i o n o f a reduced version o f equation (25) obtained by

deleting the k t h row and column from

A

- XB

t r a r i l y ) , and s e t t i n g the k t h conponent of c

(where

V

k

i s chosen a r b i -

equal t o u n i t y

. The n u l t i p l i e r

i s evaluated by s u b s t i t u t i n g equation (26) i n t o an equation obtained by

d i f f e r e n t i a t i n g equation (21).

This method has c e r t a i n advantages over

previous e i genvector d e r i v a t i v e techniques:

i t requi res only t9e e i genvalue

and eigenvector f o r t h e mode being d i f f e r e n t i a t e d , and t h e equation f o r

V

retains the banded character o f c o e f f i c i e n t matrix ( u n l i k e the a1gebrai c methods o f Fox and Kapoor, Plaut and Huseyin, and Rudisi 11).

Cardani and

Mantegazza (1979) extended Nelson's method t o transcendental f l u t t e r e i genvalue problems.

F l u t t e r e i genvalue d e r i v a t i v e s were a1so derived by

R u d i s i l l and Bhatia (19723, Rao (1972), Seyranian (1982), and by Pedersen and Seyrani an (1983)

.

Deri v a t i ves o f nonl inear buck1 ing e i genvalues were obtained

by Kamat and Ruangsi 1i a n s i ngha (1984).

Final l y , f o r the s e n s i t i v i t y analysis

o f e i yenvectors o f d i s t r i b u t e d parameter systems papers by Farshad (1974), Haug and Rousselet (1980b) and the t e x t by Haug, Komkov, and Choi (1984) should be o f i n t e r e s t t o readers. Sens it i v i t y of Transient Response Genera 1 The discussion o f s e n s i t i v i t y analysis o f t r a n s i e n t s t r u c t u r a l response i s usually based on t h e equations o f motion which are w r i t t e n as a system o f second order d i f ferent ia1 equations.

However, t h i s form obscures the

s i m i l a r i t y o f s t r u c t u r a l s e n s i t i v i t y analysis t o s e n s i t i v i t y analysis i n other f i e l d s where f i r s t order d i f f e r e n t i a compact than a f i r s t order f o r r m l a t ion.

qudtions are employed and i s also less For these reasons the discussion w i 11

be based on a system o f f i r s t order ord i n a r y d i f f e r e n t i a l equations o f the form

where U i s t h e response,

F

i s a vector of functions,

t

i s time,

v

is a

t y p i c a l design parameter, and a dot denotes d i f f e r e n t i a t i o n w i t h respect t o time.

AO

I n many structural applications the left-hand side o f equations (27) i s

where A

i s a matrix, and the methods discussed b e l w are also applicable

t o t h a t more general form (see, f o r example, Haftka and Kamat, 1984). Direct Method The d i r e c t method o f obtaining s e n s i t i v i t y derivatives i s based on d i f f e r e n t i a t i n g equations (27) t o obtain

where the Jacobian J

is

aF/aU.

Note that equations (28) i s a system o f

1inear d i f f e r e n t i a l equations, even i f the o r i g i n a l system, equations (27) i s nonlinear.

Often, derivatives o f the e n t i r e vector

U are not required.

Instead i t i s required t o obtain the derivatives o f a function o f

U o f the

form

where t f obtains

i s a f i n a l time f o r the response calculation.

dg/dv

as

The d i r e c t approach

where

dU/dv

i s calculated i n equations (28).

Green's Function Method Equations (28) have t o be solved once f o r each design variable, and are c o s t l y when the number o f design variables i s large. design variables i s l a r g e r than the dimensionality o f

When the number o f

U, then t h e Green's

function approach (see Hwang, Dougherty, Rabitz, and Rabitz, 1978) i s more e f f i c i e n t than the d i r e c t approach.

An a p p l i c a t i o n o f t h i s approach i s

s e n s i t i v i t y analysis o f t r a n s i e n t s t r u c t u r a l response when the response i s conputed using reduction techniques such as modal analysis (e.g., and Kamat, 1984; Young and Shoup, 1982). dU/dv,

see Haftka

The s e n s i t i v i t y derivative,

i s w r i t t e n as

where the Green's function

K

s a t i s f i e s ( r e c a l l t h a t the dot denotes

d/dt)

T':e e f f i c i e n c y o f the Green's function approach i s p a r t l y governed by the method used t o integrate equations (32).

A l a r g e amount of work on the

e f f i c i e n t i.qlementation o f the Green's function approach has been performed by Rab i t z and co-workers (Demi r l a p and Rabi tz, 1981; Dougherty, Hwang, and

Rdbitz, 1979; Dougherty and Rabitz, 1979, 1980; Eslava, Eno, and Rabitz, 1980;

Kramer and Calo, 1981; Kramer, Calo, Rabitz, and Kee, 1982; Rabitz, 1981). Their approach i s implemented i n a general purpose computer code c a l l e d AIM (Kramer, Calo, Rabitz, and Kee, 1982).

The Green's function method i s also

known as the variational method (see, Oogru and Seinfeld, 1981). Adjoint Variable Method Further improvements i n e f f i c i e n c y may be possible i f less information i s needed.

I f instead o f the derivatives o f t h e e n t i r e vector

a few functionals (e.g., method i s c a l l e d for. a d j o i n t vector

A

U, only those of

eq. (29)) are required, then an a d j o i n t variable The a d j o i n t variable approach solves f i r s t f o r the

from the d i f f e r e n t i a l equation

It i s shown by Haftka and Kamat (1984) t h a t

(34)

Equat ion (33) i s a set o f 1inear d i f f e r e n t i a l equat ions which i s integrdted backwards from

tf

t o zero.

As i n the steady s t a t e case, t h e a d j o i n t

variable approach i s preferred over the d i r e c t approach when t h e number o f functionals i s less than the number o f design variables.

The a d j o i n t variable

approach has been applied t o a v a r i e t y of problems i n c l u d i n g dynamics (Ray, Pister, and Polak, 1978; Haug, Wehage, and Barman, l 9 8 l ) , atmospheric d i f f u s i o n (Hal 1, Cacuci , and Schlesinger, 19821, nuclear processes (Oblow, 1976), and heat transfer i n structures (Haftka, 1981).

F i n i t e D i fference Method For s e n s i t i v i t y analysis o f s t a t i c response, the f i n i t e d i f f e r e n c e approach i s almost always i n f e r i o r t o a n a l y t i c a l methods.

For the c a l c u l a t i o n

o f d e r i v a t i v e s of t r a n s i e n t response t h i s i s not always t h e case.

When

e x p l i c i t methods are used f o r i n t e g r a t i n g the d i f f e r e n t i a l equations, t h e 1i n e a r i t y of the s e n s i t i v i t y equations does not c o n s t i t u t e a computational advantage.

Therefore, f o r the case of e x p l i c i t i n t e g r a t i o n the f i n i t e

d i f f e r e n c e approach i s o f t e n computationally superior t o t h e d i r e c t method (see Haftka, 1981; and Haftka and Ma1kus, 1981).

When imp1i c i t i n t e g r a t i o n

techniques are used, the f i n i t e d i f f e r e n c e approach i s less a t t r a c t i v e computational l y , but remains easier t o implement than t h e d i r e c t approach. FAST Method A1 1 t h e approaches discussed above provide l o c a l sensi t i v i t y information.

-

The Fourier Amp1 i t u d e S e n s i t i v i t y Test (FAST) method (see review by

Cukier, Levine, and Shuler, 1978) provides global s e n s i t i v i t i e s . t y p i c a l l y used t o assess s e n s i t i v i t i e s t o parameter uncertainties.

FAST i s This i s

done by systematical ly sampl ing s o l u t i o n s obtained by varying the parameters which have a range o f uncertainty. v i , i = l,...,m,

I f there are

m parameters

the sampling i s performed i n an m-dimensional space.

FAST

converts t h i s m-dimensional space t o a one-dimensional space in terms o f a variable

s by using the transformation i

where

w

= ai

,i=

t

1,.

bi s i n wis

.,

are a set o f incomnensurate frequencies and a i , bi

are constants which depend on the range of v a r i a t i o n

vi

.

The solutions f o r a

l a r g e number o f s-values are sampled and a Fourier transform o f t h e response

i n terms of

wi

s

i s obtained.

The c o e f f i c i e n t o f the transform associated w i t h

i s a d i r e c t measure o f the s e n s i t i v i t y o f the s o l u t i o n t o

While FAST

Via

i s more e f f i c i e n t than a Monte Carlo sampling o f t h e parameter space, i t i s s u b s t a n t i a l l y more expensive than l o c a l s e n s i t i v i t y methods when

m i s large.

While i n t h e l i t e r a t u r e reviewed herein FAST has been used only f o r c a l c u l a t i o n of s e n s i t i v i t i e s of t r a n s i e n t response, the method i s equally appl i c a b l e t o steady-state o r e i genproblem s e n s i t i v i t y calculations. method has been applied extensively i n physical chemistry (e.g.,

The

Koda, McRde

and Seinfeld, 1979; Tilden and Seinfeld, 1982), and a corrputer inplementation i s described by McRae, Tilden, and Seinfeld (1982). Other Forms o f Transient Response Equations A specialized form o f t r a n s i e n t s t r u c t u r a l response i s the response t o

harmonic excitation.

The s e n s i t i v i t y analysis o f t h a t response i s very

simi l a t o the s e n s i t i v i t y analysis o f s t a t i c response--(see,

f o r example,

Wang, K i t i s , Pilkey, and Palazzolo, 1982 and 1983, and Yoshimrra, 1984). The system o f equations (27) i s t y p i c a l l y obtained by d i s c r e t i z a t i o n o f the s p a t i a l variat!on (e.g., analysis i s performed.

by f i n i t e elements) before the s e n s i t i v i t y

I n some applications (see, f o r example, t h e discussion

o f s t a t i c shape s e n s i t i v i t y ) i t may be advantageous t o perform t h e s e n s i t i v i t y analysis before d i s c r e t i zing.

Koda, Dogru, and Seinfeld (1979); Owyer and

Peterson (1980); and Koda and Sei n f e l d (l982), f o r example, discuss appl i c a t i o n s o f s e n s i t i v i t y techniques t o p a r t i a l d i f fereclt ia1 equations, w h i l e Gibson and Clark (1977) and Cacuci (1981) present s e n s i t i v i t y analysis i n the general s e t t i n g o f functional analysis.

Second Deri vat ives Part o f the motivation f o r second d e r i v a t i v e s i s t h a t they estimate nonlinear s e n s i t i v i t y e f f e c t s i n c l u d i n g i n t e r a c t i o n between variables. d e r i vatives may be calculated d i r e c t l y .

Second

For example, d i f f e r e n t i a t i n g

equati ons (28)

m design parameters r e s u l t i n

Unfortunately equation (36).

such as equation

m(m + 1)/2

systems such as

If second derivatives are needed only f o r a functional

g

(B), then the c a l c u l a t i o n can '. g r e a t l y s i m p l i f i e d .

In

fact,

Thus, the s o l u t i o n f o r a l l the second d e r i v a t i v e s requires only f i r s t derivatives of

U

plus the a d j o i n t variable vector.

This e f f i c i e n t approach t o

second order s e n s i t i v i t y calculations i s not y e t i n use. describes somewhat less e f f i c i e n t d i r e c t

d.,J

The l i t e r a t u r e

a d j o i n t techniques (e.g.,

Coffee

and Heimerl, 1983; Haug and Ehle, 1982) o r f i n i t e d i f f e r e n c e techniques (e.g.,

Behrens, 1979). S e n s i t i v i t y Derivatives o f Optimal Solutions As the use of optimization techniques has expanded, there has been an increasing i n t e r e s t i n the s e n s i t i v i t y o f optimal solutions t o p r o b l i m parameters. following:

A t y p i c a l s i t u a t i o n where such d e r i v a t i v e s are needed i s the Suppose the minimum weight design o f an a i r c r a f t wing i s obtained

by varying the sizes of the s t r u c t u r a l components while the geometry o f the

wing, the loading and the s t r u c t u r a l materials were f i x e d during t h e optimization process.

Now suppose the minimum weight design i s s t i l 1 t o o

heavy and the designer needs t o know which o f t h e f i x e d parameters i s a good candidate f o r change.

It would be useful t o have t h e s e n s i t i v i t y o f t h e

minimum weight design t o changes i n such parameters. The information required f o r obtaining t h e s e n s i t i v i t y o f an o b j e c t i v e function such as minimum weight w i t h respect t o problem parameters i s composed o f a d i r e c t e f f e c t on the objective function p l u s an i n d i r e c t e f f e c t through the change i n the constraints.

For example, the optimization problem may

be posed as Minimize

f(v)

such t h a t

where

f(v)

gj(v)

represent constraints.

and l e t

i s an objective function, Let

p be a problem parameter.

v*,

v

i s a vector o f design variables and f*

be t h e s o l u t i o n t o t h e problem

Then i t i s shown (see, f o r example,

Barthelemy and Sobieski , 1983b) t h a t

df*

where

A

x

af

(v*)

-

are the Lagrange n u l t i p l i e r s associated w i t h the constraints.

The

Lagrange m u l t i p l i e r s thus hare the r o l e o f the 'priceu o f the constraints, i n that gj.

A

i s the change i n the o b j e c t i v e f u n c t i o n due t o a u n i t change i n

Because most optimization a l g o r i t h m y i e l d the Lagrange m u l t i p l i e r s o r

estimates thereof as a by-product o f the solution, the s e n s i t i v i t y of the o b j e c t i v e function t o problem parameters i s easy t o obtain. The s e n s i t i v i t y o f the o p t i m m set o f design variables t o problem parameters i s more complicated,

w i t h respect

Lagrange m u l t i p l i e r s are not

s u f f i c i e n t and additional c a l c u l a t i o n s are requi red. McCormick (1968);

v*

E a r l y work by F i acco and

Armacost and Fiacco (1974); Fiacco (1976, 1980); Bigelow

and Shapiro (1974) and Robinson (1974) concentrated on t h e mathematical aspects (see also t e x t by Fiacco, 1983). (1980a.b) ; Sobieszczanski -Sobieski

More recent papers by YcKeown

, B a r t h e l e w , and

R i l e y (1982); and

Vanderpl aats and Yoshi da (1984) discuss applications t o the optimal design o f dynamic systems and t o structures.

The c a l c u l a t i o n o f the d e r i v a t i v e s of

v*

requi res second derivatives o f the o b j e c t i v e f u n c t i o n and constraints w i t h respect t o the design variables, and thus poses a need f o r e f f i c i e n t computat i o n a l techniques t o obtain these derivatives. As w i t h other s e n s i t i v i t y derivatives, d e r i v a t i v e s o f optimal s o l u t i o n may be used t o extrapolate solutions f o r problem parameter changes.

Unfortu-

nately, the s e n s i t i v i t y derivatives do not take i n t o account changes i n the a c t i v e constraint set brought about by the change o f parameters (see Barthelemy and Sobieski, 1983a).

Consider, f o r example, a c o n s t r a i n t which i s

almost but not q u i t e c r i t i c a l f o r the optimum design.

The Lagrange m u l t i p l i e r

associated w i t h the constraint mist be zero and therefore as indicated i n equation ( 3 9 )

, such a constraint does not c o n t r i b u t e t o the s e n s i t i v i t y o f t h e

o b j e c t i v e function.

However, a small change i n t h e value o f

p can make t h e

constraint c r i t i c a l and completely change the value o f the derivative.

This

problem makes the use o f optimal s o l u t i o n s e n s i t i v i t y d e r i v a t i v e s more r i s k y than some other derivatives.

Sobieszczansk i-Sobieski, Barthelemy, and R i l e y

(1982) suggested using d e r i v a t i ves o f the Lagrange mu1t i p 1 i e r s and t h e optimrm

s o l u t i o n vector

v*

t o a n t i c i p a t e changes i n the a c t i v e set.

However, t h e

effectiveness of t h i s approach i s s t i l l i n doubt w i t h p o s i t l v e r e s u l t s obtained by Schmit and Chang (1984) and negative r e s u l t s by B a r t h e l e w ant Sobi eski (1983a). Concluding Remarks This a r t i c l e surveys methods f o r c a l c u l a t i n g s e n s i t i v i t y d e r i v a t i ves for discrete s t r u c t u r a l systems and p r i m a r i l y covers l i t e r a t u r e published during t h e past two decades.

Methods are described f o r c a l c u l a t i n g d e r i v a t i v e s o f

s t a t i c displacements and stresses, e i genvalues and e i genvectors, t r a n s i e n t s t r u c t u r a l response, and deri v a t i ves o f optimum s t r u c t u r a l designs w i t h respect t o problem parameters. i n Table 1.

Methods and selected references are s u m r i zed

The survey i s focused on publications addressed t o s t r u c t u r a l

analysts, but a1so includes a number o f methods developed i n nonstructural f i e l d s such as controls and physical chemistry which are d l r e c t l y applicable t o s t r u c t u r a l formulations.

Most notable among the nonstructural -based

methods are the adjoi n t variable technique from control theory, and the Green's function and FAST methods from physical chemistry. For s t a t i c displacements and stresses, methods are well established f o r derivatives w i t h respect t o simple s i z i n g variables.

F i n i t e difference and

a n a l y t i c a l methods ( d i r e c t and a d j o i n t variable) are available and there are clear gui del ines g i v i ng c l asses o f problems where t h e v a r i ous methods are preferred.

F i n i t e differences have long been disparaged as a method as

compared t o the more elegant analyt ical approaches-and

indeed the t h e o r e t i c a l

e f f o r t (as measured by operation counts, f o r example) o f f i n i t e differences does greatly exceed t h a t of the a n a l y t i c a l approaches except f o r very small numbers of design variables.

"owever,

f i n i t e d l f ferences have a major

advantage-i t i s extremely simple t o formulate and implement.

This factor,

together w i t h the increased speed o f recent and expected computers, may explain i t s popularity i n many applications. Methods f o r derivatives w i t h respect t o shape design variables are less well established and consequently there are no c l e a r choices o f preferred techniques.

One approach i s t o d i f f e r e n t i a t e a set of discretized equations

from a f i n i t e element model w i t h respect t o the shape design variables.

This

method has the advantage o f v e r s a t i l i t y bbt t h e disadvantage t h a t when the shape changes, the f i n i t e element mesh may be d i s t o r t e d leading

inaccuracies.

i

wnerical

An a1t e r n a t i ve approach i s t o d i f f e r e n t i a t e the cwr. I iNUUm

equations (before d i s c r e t i z a t i o n ) using a material derivative.

This approach

avoids the mesh d i s t o r t i o n problem and i s p o t e n t i a l l y more e f f i c i e n t but i s more complex t o implement. With regard t o d e r i vat1 ves o f s t r u c t u r a l e i genvalue problems, we1 1establ ished formulas are aval l a b l e f o r both r e a l and complex eigenvalues. Deri vat ives o f e i genvectors my be obtained by several methods i n c l u d i n g expanding the d e r i vat ives as a series of e i genvectors, an a1gebraic approach based on sirml taneous equations f o r e i genvalue and eigenvector d e r i vat1 ves, and a s i m p l i f i e d but rigorous a n a l y t i c a l approach developed by Nelson.

The

method o f Nelson i s most appealing as i t combines mathematical r i g o r w i t h computational s',npl i c i t y .

The modal expansion method also merits considera-

t i o n but requires a study o f the convergence properties o f the technique. Derivatives o f t r a n s i e n t s t r u c t u r a l response may be obtained using f i n i t e differences, d i r e c t and adjoi n t variable a n a l y t i c a l methods, a Green's funct i o n technique and the Fourier amplitude t e s t

- FAST (the l a t t e r two methods

developed by physical chemistry researchers).

As i n t h e s t a t i c case, there

are established guide1 ines f o r deciding when t o choose among the various methods.

Unlike t h e s t a t i c case, the f i n i t e d i f f e r e n c e method may be

c o n p e t l t l v e on the basis o f computational e f f l c l e n c y .

For example, I f an

expl i c l t numerical I n t e g r a t i o n algorithm i s used f o r the nominal solution, a f i n l t e d l fference c a l c u l a t l o n o f t h e d e r l v a t l v e m y be more e f f l c l e n t than an

a n a l y t l c a l method. Methods f o r d e r l vatlves of optlnum deslgns w l t h respect t o problem parameters are revlewed. l i t e r a t u r e was not large.

Recause t h i s I s a relatively new topic, t h e body of The d e r i v a t l v e o f the o b j e c t l v e functlon can be

easi ly obtained by a reasonably simple fornula.

The derlvatlves o f t h e

optinum design variables are somewhat more d l f f i c u l t t o obtain.

A compllca-

t i o n which arises i n using these d e r i vat1 ves t o extrapolate an opclnum design i s t h a t one m s t keep track o f canstralnts which change from nor:crltical t o c r i t i c a l as a r e s u l t o f small parameter changes.

Flnally, a signiflcant

by-product o f the i n t e r e s t i n derivatives of optinum designs I s the m t l v a t l o n i t has provided f o r research i n improved methods f o r second d e r l vatlves of

response quant it i e s

.

References Andrew, A. L. (1978): Convergence o f an I t e r a t i v e Method f o r Derivatives o f Eigensystems. Journal o f Computational Physics, Vol, 26, pp. 107-112.

.

Andrew, A. L. (1979): I t e r a t i v e Computation o f Derivatives o f Eigenvalues and Eigenvectors. Journal o f Inst. Math. ApplJcs., Vol 24, pp. 209-218, Armacost, R. L.; and Fiacco, A. V. (1974): Computational Experience i n S e n s i t i v i t y Analysis f o r Nonlinear Programni ng. Mathematical Programni ng, Vol. 6, pp. 301-326. Arora, J. S.; and Govil, A, K. (1977): Design S e n s i t i v i t y Analysis w i t h Substructuri ng. J. Engineering Mechanics Division, ASCE, Vol 103, No. EM4, pp. 537-548.

.

Arora, J. S.; and Haug, E. J. (1976) : E f f i c i e n t Optimal Design of Structures by General', ed Steepest Descent Programing. I n t e r n a t i o n a l Journal Num. Meth. i n Engineering, Vol. 10, pp. 747-766. Arora, J. S.; and Haug, E. 3. (1979): Methods of Design S e n s i t i v i t y Analysis i n S t r u c t ~ :a1 Optimization. A I M J., Vol. 17, No. 9, pp. 970-974. Arora, 3 . S. (1980) : Analysis o f Optimal it y C r i t e r i a and Gradient Projection Methods f o r Optimal Structural Design. Comp. Meth. Mech. and Engineering, Vol. 23, pp. 185-213. Barnett, R. L.; NASA CH-1038.

and Hermann, P. C. (1968):

High Per:ormante

Structures.

Barthelemy, 3. F.; and Sobieski, J. (1983a): Extrapolation o f Optimal Solutions Based on i e n s i t i v i t y Derivatives. A I M J., Vol. 21, No. 5, pp 797-799. Barthelemy, J. F.; and Sobieski. J. ( 1 9 8 3 ~ ) : Optimum S e n s i t i v i t y Derivatives o f Objective Functions i n Nonlinear Programing. A I M J., Vol. 21, No. 6, pp. 913-915. Behrens, J. C. (1979): An Exemplified Semi-Analytical Approach t o the Transient S e n s i t i v i t y o f Nonlinear Systems. Applied Mathematical Modeling, Vol. 3, pp. 105-115. Bennett, 3. A.; and Botkin, M. E. (1983): Shape Optimization o f TwoDimensional Structures w i t h Geometric Problem Description and Adaptive Mesh Refinement. A I A A Paper 53-0941 presented a t the AIAAIASMEIASCEIAHS 24th Structures, Structural Dynamics and Material Conference, Lake Tahoe, CA. Bigelow, 3. H.; and Shapiro, N. 1 . (1974): I m p l i c i t Function Theorems f o r Mathematical Programmi ng and f o r Systems o f Inequal it i es. Mathematical Programing, Vol. 6, No. 2, pp. 141-156. Botkin, M. E. (1982): Shape Optimization o f P l a t e and Shell Structures. Journal, Vol. 20, No. 2, pp. 268-273.

AIAA

Braibant, V.; and Fleury, C. (1984): Shape Optimal Design: A Performing CAD Oriented Formulation. AIAAIASMEIASCEIAHS 25th Structures, Structural Dynamics and Materials Conference, Paper No. AIAA-84-0857, Palm Springs, C a l i f o r n i a . Brandon, 3. A: (1984) : Deri v a t i on and S i gni f icance o f Second-Order Modal Design S e n s i t i v i t i e s . AIAA J., Vol. 22, No. 5, pp. 723-724. Brayton, R. K.; and Spence, R. E l s e v i e r S c i e n t i f i c Pub. Co.

(1980):

S e n s i t i v i t y and Optimization.

Bristow, D. R.; and Hawk, J. D. (1983): Subsonic Panel Method f o r Designing Wing Surfaces from Pressure D i s t r i b u t i o n s . NASA CR-3713.

.

Cacuci, D. G. (1981): S e n s i t i v i t y Theory f o r Nonlinear Systems. I NonLinear Functional Analysis Approach. II. Extensions t o Additional Classes o f Response. J. Math. Phys , Vol 22, No. 12, pp. 2794-2802 and 2803-2812.

.

.

Camarda, C. J.; and Adelman, H. M. (1984): I m p l e w n t a t i o n o f S t a t i c and Dynamic Structure S e n s i t i v i t y D e r i v a t i v e Calculations i n t h e F i nite-ElementBased Engineering Analysis Language System (EAL). NASA TM-85743.

.;

and Mantsgazza, P. (1979) : Calculation o f Ei genvalue and Ei genCardani , C vector Deri v a t i ves f o r 41 gebraic F l u t t e r and Divergence Ei genproblems. A I A A J., Vol. 17, No. 4, pp. 408-412. Choi, K. K., (1984): Shape Design S e n s i t i v i t y Analysis o f Displacement and Stress Constraints. J. Struct. Mech. ( i n press). Choi, K. K.; and Haug, E. J. (1983): Shape Design S e n s i t i v i t y Analysis o f E l a s t i c Structures. J. Struct. Mech., Vol. 11, pp. 231-269. Choi, K. K.; and Haug, E. 3. (1984): Shape Optimization o f E l a s t i c Structures. Paper presented a t the 21st annual meeting o f the Society o f Engi neeri ng Science, Bl acksburg, V i r g i n i a. Chon, C. T. (1984): Design S e n s i t i v i t y Analysis v i a S t r a i n Energy D i s t r i bution. A I A A J., Vol. 22, No. 4, pp. 559-561.

.

Chun, Y. W.; and Haug, E. J. (1978): Two-Dimensional Shape Optimal Design. I n t e r n a t i o n a l Journal f o r Numerical Methods i n Engineering, Vol 13, pp. 311-336. Chun, Y. W.; and Haug, E. J. (1979): Shape Optimal Design o f an E l a s t i c Body of Revolution. Paper No. 3526, ASCE Annual Meeting, Boston. Chun, Y. W.; Revolution.

and Haug, E. 3. (1983): Shape Optimization o f a S o l i d o f Journal of Engineering Mechanics, Vol. 109, No. 1, pp. 30-46.

Coffee, T. P.; and Heimerl, J. M. (1983): S e n s i t i v i t y Analysis f o r Premixed, Laminar, Steady State Flames. Combustion and Flame, Vol. 50, pp. 323-340. Cukier, R. I; Levine, H. B.; and Shuler, K. E. (1976): Nonlinear S e n s i t i v i t y Analysis o f Mu1t i parameter Model Systems. Journal of Computational Physics, Vol. 26, pp. 1-42.

Demirlap, M.; and Rabitz, H. (1981): Chemical K i n e t i c Functional S e n s i t i v i t y Analysis: Derived S e n s i t i v i t i e s and General Applications. 3. Chemical Physics, Vol. 75, No. 4, pp. 1810-1819. Dems, K.; and Mrbz, 1. (1984a): Variational Approach by Means o f A d j o i n t Systems t o Structural Optimization and S e n s i t i v i t y Analysis I 1 : Structure Shape Variation. I n t . Journal Sol i d s and Structures ( i n press).

-

Dents, K.; and Mrbz, Z. (1984b): V a r i a t i o n a l Approach t o F i r s t - and SecondOrder S e n s i t i v i t y Analysis o f E l a s t i c Structures. I n t e r n a t i o n a l Journal f o r Numeri cal Methods i n Engineering ( i n Press). Dogru, A. H.; and Seinfeld, J. H. (1981): Comparison o f S e n s i t i v i t y C o e f f i c i e n t Calculation Methods i n Automatic History Matching. Society o f Petroleum Engineers Journal, pp. 551-557. Dougherty, E. P.; Hwang, J. T.; and Rabitz, H. (1979): Further Developments and Applications o f the Green's Function Method o f S e n s i t i v i t y Analysis i n Chemical Kinetics. Journal of Chemical Physics, Vol. 71, No. 4, pp. 2794-1808. Dougherty, E. P.; and Rabitz, H. (1979): A Computational Algorithm f o r the Green's Function Method o f S e n s i t i v i t y Analysis i n Chemi ;a1 Kinetics. I n t e r n a t i o n a l Journal o f Chemical Kinetics, Vol 11, pp. 1237-1248.

.

Dougherty, E. P.; and Rabitz, H. (1980): Computational K i n e t i c s and S e n s i t i v i t y Analysis o f Hydrogen-Oxygen Combustion. Journal o f Chemical Physics, Vol. 72, No. 12, pp. 6571-6586. Doughty, S. (1982) : E i genvalue Deri v a t i ves o f Damped Torsional Vibrations. ASME Journal o f Mechanical Design, Vol. 104, Apri 1, pp. 463-465. Dwyer, H. A.; and Peterson, T. (1980): A Study o f Turbulent Flow w i t h Sensit i v i t y Analysis. A I M Paper 80-1397 presented a t the AIAA 13th F l u i d & Plasma Dynamics Conf , Snow-Mass, Colorado.

.

Dwyer, H. A,; Peterson, T.; and Brewer, J. (1976): S e n s i t i v i t y Analysis t o Boundary Layer Flow. Proc. 5th I n t ' l . Conf. on Numerical Methods i n F;uid Dynamics. Eslava, L. A.; Eno, L.; and Rabitz, H. A. (1980): Further Devslopments and Applications o f S e n s i t i v i t y Analysis t o C o l l i s i o n a l Energy Transfer. J. Chem. Physics, Vol. 73, No. 10, pp. 4998-5012. Farshad, M. (1974): Mechanics. A I A A J.,

Variations o f Eigenvalues and Eigenfunctions i n Continuum Vol. 12, No. 4, pp. 560-561.

Fiacco, A. V. (1976): S e n s i t i v i t y Analysis f o r Nonl inear Programi ng Using Penalty Methods. Mathematical Programing, Vol. 10, pp. 287-311. Fiacco, A. V. (1980) : honlinear Programming S e n s i t i v i t y Analysis Results Using Strong Second Order Assumptions. I n Numerical Optimization o f Dynir.nics Dixon and G. P. Szego, Editors), North Holland, Systems. (L. C. pp. 327-348. a!.

Fiacco, A. V. (1983): I n t r o d u c t i o n t o S e n s i t i v i t y and S t a b i l i t y i n Nonlinear Programni ng. Academic Press. Fiacco, A. V.; and McCormick, G. P. (1968): Nonlinear Proyamning: Sequential Unconstrained Minimization Techniques. W i ley. Fox, H. L. (1965): Constraint Surface Normals f o r S t r u c t u r a l Synt hesi s Technic ies. AIAA J., Vol. 3, No. 8, pp. 1517-1518. Fox, R. L. (1971) : Optimization Methods f o r Engineering Design. Wesley Pub. Co., Inc., pp. 242-249.

Addison-

Fox, R. L.; and Kapoor, M. P. (1968): Rates o f Change o f Ei genval ues and Eigenvectors. A I M J., Vol. 6, No. 12, pp. 2426-2429. Frank, P. M. (1978):

I n t r o d u c t i u n t o S e n s i t i v i t y Theory.

Academic Press.

Garg, S. (1973): Derivatives o f Eigensolutions f o r a General Matrix. A I A A J., Vol. 11, No. 8, pp. 1191-1194. Gibson, J. S.; and Ciark, L. G . (1977): S e n s i t i v i t y Analysis f o r a Class o f Evolution Equatims. Journal o f Mathematical Analysis and Applications, Vol. 58, pp. 22-31. G i les, G. L.; and Rogers, J. L., Jr. (1982): Implementation o f S t r u c t u r a l Response S e n s i t i v i t y Cal cul stions i n a Large-Scale F i n i te-Element Analysis Systm. A I A A Paper 82-0714, presented a t the AIAA/ASME/ASCE/AHS 23rd Structures, Structural Dynamics and Materi a1s Conference, N e w Orleans, LA.

G i l l , P. E.; Murray, W.; Saunders, M. A * ; and Wright, M. H. (1980): Computing the Finite-Difference Approximations t o Derivatives f o r Numerical Optimization. U.S. Army Research O f f ice, Report No. DAAG26-79-C-0110. G i l l , P. E,; Murray, W.; Saunders, M. A.; and Wright, M. H. (1983): Computing Forward Difference I n t e r v a l s f o r Numerical Optimization. S I A M J. Sci. Stat. Cornput., Vol. 4, pp. 310-321.

Gurdal, 2,; and Haftka, R. T. (1984): S e n s i t i v i t y Derivatives f o r S t a t i c Test Loading Boundary Conditions. A I A A Journal ( i n press). Haftka, R. T. (1981): Techniques f o r Thermal S e n s i t i v i t y Analysis. Numerical Math. i n Engineering, Vol. 17, pp. 71-80.

I n t . J.,

Haftka, R. T. (1982) : Second-Order S e n s i t i v i t y Derivatives i n S t r u c t u r a l Analysis. AIAA J., Vol. 20, pp. 1765-1766. Haftka, R. T. (1984): S e n s i t i v i t y Calculations f o r I t e r a t i v e l y Solved Problems, Paper presented a t the 21st meeting o f the Society o f Engineering Science, Blacksburg, V i r g i n i a , t o be pub1ished i n t h e I n t e r n a t i o n a l Journal f o r Numerical Methods i n Engineering. Haftka, R. T.; and Malkus, D. S. (1982): Calculation o f S e n s i t i v i t y Derivatives i n Thermal Problems by F i n i t e Di fferences. I n t e r n a t i o n a l Journal f o r Numerical Methods i n Engineering, Vol 17, pp. 1811-1821.

.

Haftka, R. T.; and Kamat, M. P. (1984): Martinus Ni j h o f f , The Netherlands.

Elements o f S t r u c t u r a l Optimization.

H a l l , M. C. G.; Cacuci, D. G.; and Schlesinger, M. E. (1982): S e n s i t i v i t y Analysis of a Radiative-Convective Model by the Adjoint Method. Journal o f t h e Atmospheric Sciences , 901 39, pp. 2038-2050.

.;

.

and Ehle, P. E. (1982): Second-Order Design S e n s i t i v i t y Analys i s Haug, E. J o f Mechanical System Dynamics. I n t e r n a t i o n a l Journal f o r Numerical Methods i n Engineering, ~ o l18, bp. 1699-1717.

.

Haug, E. 3 . (1981a): A U n i f i e d Theory o f Optimization o f Structures w i t h Displacement and Compliance Constraints. Jo Struct. Mech., Vol. 9, No. 4, pp. 415-437. Haug, E. 3 . ( l 9 8 l b ) : Second-Order Design S e n s i t i v i t y Analysis o f Structura Systems. A I A A J., Vol. 19, pp. 1087-1088. H u g , E. 3.; and Arora, J. S. (1978): Design S e n s i t i v i t ~Analysis o f E l a s t Meihani cal Systems. Computer Methods i n Applied Mechanics and Engineering, Vol. 15, pp. 35-62.

Haug, E. J. ; and Choi , K. K. (1984) : S t r u c t u r a l Design S e n s i t i v i t y Analysis With Generalized Global S t i f f n e s s and Mass Matrices. AIAA Journal, Vol. 22, No. 9, pp. 1299-1303. tiaug, E. 3 . ; and Rousselet, B. (1980a): Design S e n s i t i v i t y Analysis i n Structural Mechanics - I. S t a t i c Response Variations. 3. Struct. Mech., Vol. 8, No. 1, pp. 17-41. Haug, E. 3 . ; and Rosselet, B. (1980b): Design S e n s i t i v i t y Analysis i n Structural Mechanics. II, Eigenvalue Variations. Journal of Structural Mechanics, Vol. 8, No. 2, pp. 161-186. Hzug, E. J.; and Komkov, V. (1977): S e n s i t i v i t y Analysis i n D i s t r i b u t e d Parameter Mechanical System Optimi z a t i on. Journal o f Optimization Theory and Applications, Vol. 23, No. 3, pp. 445-464. Haug, E. J.; Komkov, V.; and Choi, K. K. (1984): o f Structural Systems. Academic Press. Haug, E. 3 . ; Wehage, R.; and Barman, N. C. (1981): Analysis o f Planar Mechanism and Machine Dynamics. Design, Vol 103, pp. 560-570.

.

Design S e n s i t i v i t y Analysis Design S e n s i t i v i t y ASME Journal of Mecnani c a l

H i r a i , I.; and Kashiwaki, M. (1977): Derivatives o f Eigenvectors o f Local l y Modified Structures. I n t e r n a t i o n a l Journal f o r Numerical Methods i n Engineering, Vol. 11, pp. 1769-1773. Hsieh, C. C.; and Arora, J. S. (1983): S t r u c t u r a l Design S e n s i t i v i t y Analysis w i t h General Boundary Conditions, Uni versi t y o f Iowa Technical Report CAD-SS-83.5.

i I

I

Hwang, J. T.; Dougherty, E. P.; Rabitz, S.; and Rabitz, H. (1978): The Green's Function Method o f S e n s i t i v i t y Analysis i n Chemical Kinetics. J. Chem. Physics, Val. 69, pp. 5180-5191. Irwin, C. L.; and OIBrien, T. J. (1982): S e n s i t i v i t y Analysis o f Thermodynamic Calcul ~ t i o n s U.S. Dept. of Energy Report DOEIMETCl82-53.

.

i;

1

Jacobi, C. G. J. (1846): Uber e i n l e i c h t e s Verfahren d i e i n der Theorie der Saecul arstoerungen vorkonnenden G l eichungen numerisch aufzuloesen, Zei t s c h r i ft f u r Reine unde Angewandte Mathematik, Vo1. 30, 1846, pp. 51-95. Also a v a i l a b l e as NASA TT F-13,666, June 1971

. J . (1984) :

.

Jawed, A. H ; and Morris, A. Approximate Higher-Order S e n s i t i v i t i e s i n Structural Design. Engineering Optimization, Vol 7, No. 2, pp. 121-142.

.

Kamat, M. P.; and Ruangsiliasingha, P. (1984): Design S e n s i t i v i t y Derivatives i n Nonl inear Response. A I M Paper 84-0973, presented a t the AIAA/ASME/ASCE/AHS 25th Structures, Structural Dynamics and Materials Conference, Palm Springs, CA, May 14-16. Kel ley, H. J. (1962): Method of Gradients. I n Optimization Techniques w i t h Appl ications t o Aerospace Systems (George L e i tmann , e d i t o r ) , Academic Press. Kel ley, H. J. ; and Lefton, L. (1980): Perturbation - Magnitude Control f o r Difference Quotient Estimation of Derivatives. Optimal Control Applications and Methods, Vol 1 , pp. 89-92.

.

Koda, M.; Dogru, A. H.; and Seinfeld, J. H. (1979): S e n s i t i v i t y Analysis o f P a r t i a l D i f f e r e n t i a l Equations w i t h Application t o Reaction and Diffusion Processes. Journal o f Computational Physics, Vol 30, pp. 259-282.

.

Koda, M. ; McRae, G. J.; and Seinfeld, J. H. (1979): Automatic S e n s i t i v i t y Analysis o f Kinetic Mechanisms. International Journal o f Chemical Kinetics, Vol. 11, pp. 424-444. Koda, M. ; and Seinfeld, J. H. (1982) : S e n s i t i v i t y Anakysis o f D i s t r i b u t e d Parameter Systems. IEEE Transactions on Automatic Control, Vol. AC-27, No. 4, August, pp. 951 -955. Kramer, M. A. ; and Calo, J . M. (1981 ) : An Improved Computational Method for Sensi t i v i t y Analysis: Green's Function Method w i t h "AIM. " Appl Math. Model ing, Vol. 5, pp. 432-441.

.

Kramer, M. A. ; Calo, J. M. ; Rabitz, H. ; and Kee, R. J. (1982) : AIM: The Analytical l y Integrated Magnus Method f o r Linear and Second-Order S e n s i t i v i t y Coefficients. Sandi a National Laboratories Report SAND-82-8231

.

Lancaster, P. (1964): On Eigenvalues o f Matrices Dependent on a Parameter. Numerische Mathematik, Vol. 6, No. 5, pp. 377-387. I

.I

;

i ! I

Leonard 3 . 1. (1974): The Application o f S e n s i t i v i t y Analysis t o Models o f LargeScal e Physiological Systems. NASA CR-160228.

McKeown, J. J. (l98Oa) : Parametric S e n s i t i v i t y Analysis o f Nonlinear Theory and A1 g o r i thms Programing Problems. I n Nonlinear Qptimi z a t i on (L. C. W. Dixon, E. S. Spedicato, and G. P. Szego, e d i t o r s ) , Bi r k h a k e r , Boston.

-

McKeown, J. J. (1980b): An Approach t o S e n s i t i v i t y Analysis. I n Numerical Optimization o f Dynamic Systems, North Holland (L. C. W. Dixon and G. P. Szego, e d i t o r s ) , pp. 349-362. McRae, G. J.; Tilden, J. W.; and Seinfeld, J. H. (1982): Global S e n s i t i v i t y Analysis A computational Implementation o f t h e Fourier Amp1 itude S e n s i t i v i t y Test (FAST). Computers and Chemical Engineering, Vol 6, No. 1, pp. 15-25.

-

.

Morgan, B. S. (1966): Computational Procedure f o r the S e n s i t i v i t y o f an E i genvector. E l e c t r o n i c Letters, pp. 197-198. Musgrove, M. D.; Reed, J. M.; and Hauser, C. C. (1983): Optimization Using S e n s i t i v i t y A m l y s i s . J. o f Spacecraft and Rockets, Vol. 20, No. 1, pp. 3-4. Nelson, R. B. (1976): Simp1i f i e d Calculation o f Ei genvector Derivatives. A I A A J., Vol. 14, No. 9, pp. 1201-1205. Nguyen, D. T.; and Arora, 3 . S. (1982): F a i l - s a f e Optimal Design o f Complex Structures w i t h Substructures. ASME Journal o f Mechanical Design, Vol. 104, pp. 861-868. Noor, A. K.; and Lowder, E. (1975): Structural Reanalysis Via a Mixed Method. Computers and Structures, Yo1 5, No. 1, pp. 9-12.

.

Noor, A. K.; and Peters, J. M. (1980): Reduced Basis Technique f o r Nonlinear Analysis of Structures. AIAA J., Vol. 18, No. 4, pp. 455-462. Oblow, E. M. (1976) : S e n s i t i v i t y Theory From a D i f f e r e n t i a l Viewpoint. Sci. Eng., Vol. 59, pp. 187-189.

Nucl

.

Pedersen, P.; and Seyranian, A. P. (1983): S e n s i t i v i t y Analysis f o r Problems of Dynamic Stabi 1ity. I n t e r n a t i o n a l Journal o f Sol i d s and Structures, Vol. 19, No. 4, pp. 315-335. Plaut, R. H.; and Husseyin, K. (1973): Derivatlves of Eigenvalues and Eigenvectors i n Non-Self-Adjoint Systems. AIAA J., Vol. 11, No. 2, pp. 250-251. Prasad, €3.; and Emerson, J. F. (1982): A General C a p a b i l i t y o f Design Sensit i v i t y f o r Finite-Element Systems. A I M Paper 82-0680, presented a t t h e AIAA/ASME/ASCE/AHS 23rd Structures, S t r u c t u r a l Dynamics and M a t e r i a l s Conference, New Orleans, Louisiana.

-

Prasad, B.; and Haftka, R. T. (1980): Organization of PARS A S t r u c t u r a l Resizing System. Advances i n Computer Technology, Vol. 111, ASME Pub. NO. 80-52584. Rabi tz, H. (1981): Chemical S e n s i t i v i t y Analysis Theory w i t h Applications t o Molecular Dynamics and Kinetics. Computers and Chemistry, Vol. 5, No. 4, pp. 167-180.

Radanovic, L. e d i t o r (1966) : Sensi t iv i t y Methods i n Control Theory. Pergamon Press Ltd. Rao, S. S. (1972): Rates o f Change o f F l u t t e r Mach Number and F l u t t e r Frequency. AIAA J., Vol. 10, No. 12, pp. 1526-1528. Ray, D.; Pister, K. S.; and Polak, E. (1978): S e n s i t i v i t y Analysis f o r Hysteretic Dynamic Systems: Theory and Applications. Computer Methods i n Applied Mechanics and Engi neeri nq, Vol 14, pp. 179-208.

.

Robinson, S. M. (1974): Perturbed Kuhn-Tucker Points and Rates of Convergence f o r a Class o f Nonlinear Programming A1 g o r i thms. Mathematical Programni ng, Vol. 7, No. 1, pp. 1-16. Rogers, L. C. (1970): Derivatives o f Eigenvalues and Eigenvectors. Vol. 8, No. 5, pp. 943-944.

A I A A J.,

Rosenbrock, H. H. (1965): S e n s i t i v i t y o f an Eigenvalue t o Changes i n t h e Matrix. Electronics Letters, Vol. 1, No, 10, pp. 278-279. Rousselet, B. (1983a): Note on the Design D i f f e r e n t i a b i l i t y o f the S t a t i c Response o f E l a s t i c Structures. 3. Struct. Mech., Vol. 10, No. 3, pp. 353-358. Rousselet, B. (l983b): Shape Design S e n s i t i v i t y of a Membrane. Journal o f Optimization Theory and Applications, Vol. 40, No. 4, pp. 595-622. Rousselet, B.; and Haug, E. J. (1981): Desigrt S e n s i t i v i t y Analysis o f Shape Variation. Optimization o f D i s t r i b u t e d Parameter Structures (E. J. Haug and 3 . Cea, Ed.), S i j t h o f f Noordhoff, Netherlands, pp. 1397-1442. Rousselet, B.; and Haug, E. J. (1983): Design S e n s i t i v i t y Analysis i n Struct u r a l Mechanics. 111. E f f e c t s o f Shape Variation. 3. Struct. Mech., Vol. 10, NO. 3, pp. 273-310. R u d i s i l l , C. S. (1974): Derivatives o f Eigenvalues and Eigenvectors o f a General Matrix. A I A A J., Vol. 12, No. 5, pp. 721-722. R u d i s i l l , C. S.; and Bhatia, K. G, (1972): Second Derivatives of t h e F l u t t e r Velocity and the Optimization o f A i r c r a f t Structures. A I M J., Vol. 10, pp. 1569-1572. R u d i s i l l , C. S.; and Chu, Y. (1975): Numerical Methods f o r Evaluating t h e Derivatives o f Eigenvalues and Eigenvectors. AIAA J., Vol. 13, No. 6, pp. 834-837.

-

Schmit, L. A., Jr. (1981): S t r ~ r c t u r a lSynthesis I t s Genesis and Develoynent, A I A A J., Vol. 19, No. 10, pp. 1249-1263. Schmit, L. A., Jr.; and Chang, K. J. (1984): Optimm Design S e n s i t i v i t y Based on Approximation Concepts and Dual Methods. I n t e r n a t i o n a l Journal f o r Numerical Methods i n Engineering, Vol. 20, pp. 39-75.

a

.

Schmit, L. A., Jr.; and Farshi, B. (1974): Some Approximation Concept f o r Structural Synthesis. A I A A J., Vol. 12, No. 5, pp. 692-699. Schy, A. A.; and Giesy, D. P. (1981): M u l t i o b j e c t i v e I n s e n s i t i v e Design o f A i rplane Cont r o ? Systems With Uncertain Parameters. Paper presented a t A I A A Gu i dance and Control Conference, A1 buquerque, NM. Schy, A. A,; and Giesy, D. P. (1983): Tradeoff Studies i n M u l t i o b j e c t i v e I n s e n s i t i v e Design of A i rplane Control Systems. Paper presented a t A I A A Guidance and Control Conference, Gat 1inburg, TN. Seyranian, A. P. (1982): S e n s i t i v i t y Analysis and Optimization o f Aeroelastic S t a b i l i t y . I n t e r n a t i o n a l Journal o f Solids and Structures, Vol. 18, No. 9, pp. 791-807. Simpson, A. (1976): On the Rates o f Change o f Sets o f Equal Ei genvalues. Journal o f Sound and Vibration, Vol. 44, No. 1, pp. 83-102. Sobieszczanski-Sobieski , J.; Barthelemy , J. F.; and Riley, K. M. (1982): S e n s i t i v i t y o f Optimum Solutions t o Problem Parameters. A I M J., Vol. 20, pp. 1291-1299.

.

Stewart, G. W. (1967): A Modification o f Davidon's Minimization Method t o Accept Difference Approximations o f Derivatives. ACM J., Vol 14, pp. 72-83. Stewart, G. W. (1972): On the S e n s i t i v i t y o f the Eigenvalue Problem Ax = ABX. S I A M Journal o f Numerical Analysis, Vol. 9, No. 4, December, pp. 669-686. Storaasli, 0.; and Sobieszczanski, J. (1974): On the Accuracy of t h e Taylor Approximation f o r Structure Resizing. A I A A J., Val. 12, No. 2, pp. 231-233. Taylor, D. L.; and Kane, T. R. (1975): Multiparameter Quadratic Eigen Problems. J. Applied Mech., June 1975, pp. 478-483. Tilden, J. W.; and Seinfeld, J. H. (1982): S e n s i t i v i t y Analysis o f a Mathemati cal Model f o r Photochemical A i r Pol l u t i o n . Atmospheric Environment, Vol. 16, No. 6, pp. 1357-1364. Tomovic, R. (1963):

S e n s i t i v i t y Analysis o f Dynamic Systems.

McGraw-Hi 11.

Van Be1le, H. (1982): H: gher Order S e n s i t i v i t i e s i n S t r u c t u r a l Systems. A I A A J., Vol. 20, No. 2, pp. 286-288.. Vanderplaats, G. N.; and Yoshida, H. (1984): E f f i c i e n t Calculation o f Optimum Design S e n s i t i v i t y . A I A A Paper No. 84-0855, presented a t t h e AIAAIASMEIASCEIAHS 25th Structures, S t r u c t u r a l Dynamics, and M a t e r i a l s Conference, Palm Springs, California, May 14-16. Vanhonacker, P. (1980) : D i f f e r e n t i a l and Difference S e n s i t i v i t i e s o f Natural Frequencies and Mode Shapes o f Mechanical Systems. AIAA J., Vol. 18, No. 12, pp. 1511-1514.

Wal l e r s t e i n, D. V. (1984) : Desi gn Enhancement Tools i n MSCINASTRAN. NASA CP-2327, pp. 505-526. Wang, B. P.; K i t i s , L.; Pilkey, W. 0.; and Palazzolo, A. B. (1982): He1icopter Vibration Reduction by Local Structural Modi f icat1ons. Journal o f t h e Ameri can He1icopter Soci e t y , pp. 43-47. Wang, B. P.; Pilkey, W. Dm; and Palazzolo, A. 0. (1983): Reanalysis Modal Synthesis and Dynamic Design. Chapter 8, State-of-the-Art Surveys on F i n i t e Element Technology (A. K. Noor and W. D m Pilkey, editors), American Society o f C i v i 1 Engineers

.

.

W i t t r i c k , W. H. (1962): Rates o f Change o f Eigenvalues, w i t h Reference t o Buck1ing and Vibration Problems. 3. Royal Aeronaut! cal Society, Vol 66, pp. 590-591. Yang, R. J.; and Choi, K. K. (1984): Accuracy o f F i n i t e Element Based Shape Design S e n s i t i v i t y Analysis. J. Struct. Mech. ( t o be pub1ished). Yoo, Y. M.; Haug, E. 3.; and Choi, K. K. (1984): Shape Optimal Design o f an Engine Connecting Rod. J . Mech. Design ( I n Press). Yoshinura, Y. (1984) : Design S e n s i t i v i t y Analysis o f Frequency Response i n Machi ne Structures. ASME Journal o f Mechani sms, Transmissions, and Automation i n Design, Vol. 106, March, pp. 119-125.

.

Young, S. 0.; and Shoup, T. E. (1982): The S e n s i t i v i t y Analysis of Cam Mechanism Dynamics. ASME J. Mech. Design, Vol 104, pp. 476-481. Zolesi o, 3. P. (1981): The Material Derivative (or Speed) Method f o r Shape Optimization. Optimization o f D i s t r i b u t e d Parameter Structures (E. 3. Haug and 3. Cea, Ed.), S i j t h o f f Noordhoff, Netherlands, pp. 1152-1194.

2

Direct Adjoint variable Mixed

Direct Adjoint variable ~ & n ' s function FAST

Direct Modal expansion Direct Modal expansion

Annacost 6 Fiacco (l974), kl(eorm (l98O), Sobieski, e t a1 (1982) Barthelerny 6 Sobieski (1983a), Schlnit 8 Chang (l984), Fiacco (1983)

Coffee and Heimerl (1983) Haug and Ehle (1982) Haftka (1982)

Haftka (1981 ) Ray, e t a1 (1981 ) Kramer, e t a1 (1982), Hwang, e t a1 (1978) Cukier, e t a1 (1978)

Fox and Kapoor (1971), Rogers (IWO), Nelson (1976) Fox and Kapoor (1971), Rogers (IWO), Stewart (1972) Taylor and Kane (1975) Taylor and Kane (1975)

Plaut 8 Husseyin (1973). R u d i s i l l (l974), Doughty (1972) , Brandon (1 984)

Jacobi (1846). Rogers (1970), Stewart (1972), Plaut 6 Husseyin (1973)

Fox 6 Kapoor (1968), Fox (1971), W i t t r i c k (1962), Vanhonacker (1980) Lancast e r (1964). Simpson (1976), Haug and Roussel e t (1980)

Jawed and Morris (1984), ,Van Be11e (1982) Haug (1981). Dems and Hroz (1984b) Haftka (1982 )

Rousselet 6 Haug (1983), Rousselet (1983), Dens 6 h ' z (1984b)

Botkin (1981). Bennett and Botkin (1983)

Fox (l965), Haug 6 Arora (1978). Arora 6 M u g (1979) Barnett Herman (l968), Kelley (l962), Haug 6 Arora (1978)

Sel ected references

methods f o r s t r u c t u r a l s e n s i t i v i t y derivatives

* F i n i t e difference methods are generally applicable. See, for example, 6111 (1980, l983), Stewart (l967), kll e y and Lefton (1980), Haftka and Ma1kus (1981 )

Optimum designs Objective function Design variables

Second derivatives

Transient displacement F i r s t derivatives

Second derivatives

Eigenvectors F i r s t derivatives

Direct

Direct

Nonsymnetric matrices

Second derivatives

Direct Direct

E igenval ues Symmetric matrices D i s t i n c t e i genval ues Mu1t ip l e eigenval ues

Direct Adjoint variable M i xed

Second der ivat ives

Direct Adjoint variable

Method

Suranary o f analytical

Differentiate disc r e t e equations Material d e r i v a t i v e

.

WT shape variables

S t a t i c displacenrent NRT s i z i n g variables

Type o f d e r i v a t i v e

Table 1