L _ ;=- _[_

NASA Technical Memorandum 102639

AVSCOM Technical Memorandum 90-B-006

Differential Equation Based Method For Accurate

Approximations in Optimization

Jocelyn I. Pritchard and Howard M. Adelman

(_ASA-TM-IO?639) OIFFERENTIAL E_UAT[nN

_ASEO MFTHOD FOP ACCURATE APPQOXIMATI_NS [N

OPTIMIZATION (NASA) I4 p CSCL OIC

April 1990

G3/05

N90-22558

National Aeronautics and

Space Administration

langley Reseerch Center

Hampton, Virginia 23665-5225

US ARMY_

AVIATION

SYSTEMS COMMAND

AVIATION R&T ACTIVITY
_L
DIFFERENTIAL EQUATION BASED METHOD FOR ACCURATE

APPROXIMATIONS IN OPTIMIZATION

Jocelyn I. Pdtchard*

U. S. Army Aerostructures Directorate

USAARTA-AVSCOM

Langley Research Center

Hampton, Virginia

and

Howard M. Adelman**

National Aeronautics and Space Administration

Langley Research Center

Hampton, Virginia

Abstract

This paper describes a method to efficiently and

accurately approximate the effect of design changes on

structural response. The key to this new method is to

interpret sensitivity equations as differential equations

that may be solved explicitly for closed form

approximations, hence, the method is denoted the

Differential Equation Based (DEB)method.

Approximations were developed for vibration

frequencies, mode shapes and static displacements.

The DEB approximation method was applied to a

cantilever beam and results compared with the

commonly-used linear Taylor series approximations and

exact solutions. The test calculations involved

pedurbing the height, width, cross-sectional area, tip

mass, and bending inertia of the beam. The DEB

method proved to be very accurate, and in most cases,

was more accurate than the linear Taylor series

approximation. The method is applicable to

simultaneous perturbation of several design variables.

Also, the approximations may be used to calculate other

system response quantities. For example, the

approximations for displacements are used to

approximate bending stresses.

t_m_J_Jatu_

K stiffness matrix

M mass matrix

s number of steps

v design variable

v vector of design vanables

_v change in design variable

" Research Engineer, interdisciplinary Research

Office, Member AHS

"* Deputy Head, Interdisciplinary Research Office

Member AIAA,ASME

o_2

d_

e

Superscript

T

Subscript

o

i

vibration eigenvalue (frequency)

vibration eigenvector (mode shape)

distance along move direction

designates transpose

designates nominal value

ith component of vector

Introduction

it is highly desirable in optimization to be able to

estimate the effect of design changes on system

behavior without having to perform expensive finiteelement analyses inside an optimization loop. This

need has fed to an increased interest in the

development of accurate and efficient approximation

techniques. Presently, many optimization procedures

use linear Taylor series approximations of the objective

function and constraints so that excessive calls to the

full analysis can be eliminated.1 Taylor series and other

approximations 2 are also used outside of optimization

for quick assessment of the effect of design changes.

Other techniques and demonstration of their benefits

are described in references 3-11. In this paper an

efficient method is described which significantly

extends the range of applicability of approximations

beyond the range of the linear Taylor series approach.

The key to this new approach is to recognize that the

formulas for the sensitivity derivatives of system

response quantities can be interpreted as differential

equations which may be solved to obtain closed form

approximations. Herein, these approximations will be

referred to as the Differential Equation Based (DEB)

approximations.

Basis of DEB Method

in this section of the paper the DEB method is

developed for approximating the effect of design
changes on response. First, the method will be

developed for the approximation of frequencies, then

mode shapes, and finally for static displacements.

Approximation of Freauencies

From reference 12, the equation for the derivative

of a non-repeated vibration eigenvalue, co2with respect

to a design variable, v is

dco2 dDT[dK co2dM ]e

-_-= Ld-;"- -_vJ

(1)

where K is the stiffness matrix, M is the mass matrix and

¢, is the eigenvector normalized such that ¢'TM4)= t.

Define two scalar quantities a and b as follows

a = 4DT dM ¢, and b = d)TdK 4D (2) d'-;"

then Equation (1) may be written as

din---J-: = b- a¢2 (3)

dv

Equation (3) may be interpreted as a first order

differential equation in co2 with variable coefficients. If a

and b do not vary excessively with v, they may be

evaluated at the nominal design and considered to be

constant. Then the general solution to equation (3),

provided a is nonzero is

co: = Ce_aV+ b (4)

a

Solving for C from the condition of co2= O)2owhen V=Vo

and substituting into Equation (4) gives an explicit

approximation for co' as

o.=Io+_ + ,,,

Aooroximation of Mode Sha_es _

=

From reference 13, the equation for the derivative

of the mode shape, d) with respect to a design variable

is

dd)

-- = Q + De (6) dv

where Q satisfies the equation

K-co2M]Q= dco2Me dKd=+co2 dMd,

and

Id_TdM d_

O = -OI"MQ- _•

(7)

(8)

Derivations of Q and D may be found in reference 13.

Equation (6) is a nonlinear first order vector differential

equation in _ with variable coefficients. In order to

solve this equation, Q and D are evaluated at the

nominal design and assumed constant. Equations (1)

and (5) may be used for explicit forms of de°2and co" in

dv

the calculations of Q and D. The solution to equation

(6) is

• = Ae0v - D Q (9)

where A is determined from the nominal condition that

= O o when v=v o. Solving for A and substituting into

equation (9) gives the explicit approximation for • as

l O(V-Vo) 1

O=(*o +_.Q)e -_,Q (10)

Equation (10) is a vector equation, however, it is

uncoupled in the sense that each component of 4>

varies independently with the design variable, v. The

difference between components is reflected in the

corresponding components of ¢b and Q.

Aooroximation of Disolacements

Assuming a linear elastic structure in which the

applied force does not vary with v, the equation for the

displacement derivative is

dU

--=su (11)

dv

(12)

where

B =_K-t d_._. K

dv

Equation (11) is a differential equation that can be

solved explicitly for an approximation of the

displacement, U. Similar to solving the differential

equations for frequencies and mode shapes, the matrix

2
B is evaluated at the nominal design and assumed

constant for moderate excursions from the nominal

design. Equation (11) is a vector differential equation

having the following solution

U = eBVA (13)

where eB is the matrix exponential defined as

B2 B3

eB = l + S +._l +.._(+... (14)

Solving for A in equation (13) from the condition that

U=Uo at V=Vogives

u = eB(V-V°)Uo (15)

Using equation (14) in equation (15) gives the

displacement approximation

B2(V-Vo) 2

U = Uo + B(v-vo)U o+ 2! ' Uo+.-. (16)

Rather than compute the matrix B and its powers and

then multiply by Uo, the BnUo terms were obtained

iteratively as follows

dK U

KBUo = --_" o

KB2U o__dK BU

- dv o

dK

KBnUo = --_-- Bn-Iuo

(17)

The Taylor series expansion of U is

U dU, , ] _U, vo'2+-..) = U o +--(V- V,.,/+----..-_ IV- dv " "" 2! dv" "

(18)

As pointed out in reference 14, when K is a linear

function of v, equation (18) can be written as

U = Uo + BUo(V- Vo)+ B2Ue(v- Vo)2+... (19)

Also pointed out in reference 14, equations (16) and

(19) differ only by the factorial in each term. Equation

(16) is less accurate than the Taylor series equation (19)

when they both converge. Equation (16) is based on

the power series for the exponential and will converge

for any value of AV, whereas the Taylor series may not

always converge. Since the terms in both equations

(16) and (19) are easily evaluated using equation (17),

the DEB method would not be the preferred

approximation unless the Taylor series did not

converge.

DEB Method Extended to Multiple Desian Variables

The approximations in equations 5,10,and 15 are

directly applicable only when a single design variable is

perturbed. To calculate the effect of simultaneous

changes in several design variables, the DEB method

is modified as follows. If vo is the vector of nominal

design variables and v is the vector of perturbed

design variables, then V-Vo is the direction of the

move from the nominal to the perturbed design. Any

design along the move direction may be associated with

a scalar0 between 0 and 1. Then

,,(e)=Vo+e(v-,o) (20)

In particular, v(1) = v and v(0) = vo. Now the sensitivity

equations (1), (6), and (1 t) may each be written with 0

as the independent variable. For example, equation (1)

is rewritten as

or

d°'= ,t, Fd .=dM -._-.- L.._.-tu -_-j_ (21a)

dco2

= be - aoco2 (21 b)

where

_ _.T dM _ d_TdK • (22)

ae - ,_, -'_-<> and be - d'-'e"

and
dM _oT_dv i ndM,

(23)

dK _dKdv _dK'v v _

where n is the number of design variables.

solution to equation (21b) is

0")2(0) = ( =2°- be)e-ae(0-g°)ag ,) *beae (24)

Now the

Using equation (20) _2(].)=_2(v) and by definition

0o = 0 and O= 1 at the perturbed design. Therefore,

the final form of equation (24) is

of-(v)= o_o2- e-a° +_'e (25)

Equation (25) corresponds to one step Iv-vo I. For a

step size of sty- %J

bele-' '+beao (26)

Extension of the method to multiple design variables

has also been carded out for • and g.

DEB Method Extended To Nonlinear Desian Variables

We will define a nonlinear design variable to be one

that contributes nonlinearly to either the mass or

stiffness matrix and a linear variable is one that

contributes linearly to both the mass and stiffness

matrices. An early attempt at applying the DEB method

in a straight forward manner to nonlinear design

variables proved to be highly inaccurate. In retrospect,

the reason for the inaccuracy was that the coefficients

which were assumed constant (e.g. a and b) were in fact

highly nonlinear functions of the design variable. The

solution to this difficulty is as follows. Define y to be a

vector of nonlinear design variables and v to be a

vector of linear variables. It is assumed that the linear

variables may be written as functions of the nonlinear

design variable such that vo = v(yo)and v = v(y). For

example, the bending inertia I and the cross-sectional

area A contribute linearly to M and K and they are both

functions of the nonlinear variable H. Then equation

(20) may be rewritten with v as a function of y and 0

,,(o,y)=,,o+e(,,-vo):v(yo)+o[v(y)-,,(yo)] (27)

The method for nonlinear design variables then

becomes very similar to the DEB method for multivle

linear variables (equation 26). Forthis case

dK" dKdvi ndK

=i=,l. vidO= Vi(y°)]OVi

(28)

Then ae and be are calculated from equation (22).

Unlike the method for linear variables, ao and be are

evaluated at each step along the move direction due to

the nonlinearity of the perturbed variable and in

equation (26) s is always equal to 1. Evaluating ae and

be involves a recalculation ofvi(y) for each perturbed

step of y and the multiplication of a matrix by a scalar.

The derivatives of the mass and stiffness matrices do

not have to be recalculated. They are evlluate_d o_nceat

the n0rninal design=and:used throu_outTh6 iange of

the move direction. Again, a similar extension has been

carried out for • and U.

Results

The approximations for frequencies, mode shapes,

and displacements were tested on a 193 inch

cantilever beam modeled with ten equal elements (see

fig.l). The beam has a box cross-section and a tip

mass. Properties of the beam are listed in table 1.

!

Fig.1 Test problem.

4
Table 1 Material properties and cross-sectional

dimensions of beam model

Element No.

E (psi)

P{Ib/in 3)

H {in.}

B {in.)

t (in.)

d (in.)

1

4.90E6

9.07

5.00

3.75

0.80

0.10

2-10

5.85E6

i

0.07

5.00

3.75

0.80

! o.10

The test cases involved perturbing the height H, width

B, cross-sectional area A, tip mass M, and bending

inertia I of the beam. All calculations were generated in

a general purpose finite-element program (EAL) 15.

Results were compared with exact solutions and the

linear Taylor series approximation.

Freouency Aooroximation Results

The DEB method shows excellent capability for

approximating frequencies of perturbed designs. For

example, figure 2 shows a graph of the fundamental

eigenvalue versus the height of the beam, H for values

perturbed from the nominal value of 5.0 inches. For as

much as a 50 percent increase in H, the new

approximation is within 2 percent of the exact solution

compared to 12 percent for the linear Taylor series

approximation. It is also evident from the figure that

beyond a 45 percent decrease in H, the linear Taylor

series method gives negative values for the

eigenvalues. This is an example ol the DEB method

applied to a nonlinear variable. Figure 3 shows a

--- Linear Taylor serle=

•_¢"_2500 ..... DEB method 7' I .... .o,°,,o°

=--_-500 k = l , , , ,

_= 0.0 2.0 3.0 4.0 5.0 6.0 7.0 810

Beam depth, H (In.)

Fig. 2 Comparison of DEB and Taylor series

approximations of frequency for perturbation of beam

depth, H ( B=3.75 in., M=0. Ibm. ).

comparison between the approximations for the

frequency clue to a change in the magnitude of the tip

mass. Again the new approximation is very accurate.

Only a four percent deviation from the exact solution is

seen for a 100 percent increase in the tip mass

compared to a ten percent deviation for the linear Taylor

series approximation. Figure 4 illustrates the application

215" I

195. p_ Linear Taylor series

_----_o _, ..... DEB method

¢ = 175"r'_ ----- Exact solution

 55.

95. - , , , , ",q,

0.0 4.0 8.0 12.0 16.0 20.0

Tip mass, M (Ibm)

Fig. 3 Comparison of DEB and Taylor series

approximations of frequency for perturbation of tip mass

( B=3.75 in., H=5.0 in. ).

of the DEB method to simultaneous changes in three

design variables: tip mass, bending inertia, and crosssectional area of the beam. Each step, s represents a

ten percent perturbation in the design variables from

their nominal values. In comparing the DEB

approximation with the exact solution there was a 36

percent error for 50 percent increase in the design

variables. The linear Taylor series approximation was in

error by nearly 100 percent. Figure 5 shows the DEB

and linear Taylor series approximations of the second

bending frequency compared to the exact solution for

perturbed values of the beam depth, H. Again, the new

method is very accurate. For as mUch as a fifty percent

increase in H, the DEB approximation is within 2 percent

of the exact solution compared to 12 percent for the

linear Taylor series. This result is very similar to the result

for the first bending mode with H as the design variable.

Similar trends occurred in perturbation studies of the

other variables for approximating frequencies of the first

and second bending modes.

5
1100 I- Taylor series

l Exponential

o_._. 900 N_.,.

c _ Exact

= soo-

"a=0 -\-..'-..

_

300 \..

100 -

-100 ..... I i l l

0 1 2 3 4 5

Number of steps, s

Fig. 4 Comparison of DEB and Taylor series

approximations of frequency for simultaneous

perturbation of tip mass M, bending inertia I, and crosssectional area A ( Nominal values: M=5.0 Ibm., 1=28863

in4, A=7.0 in2 ).

11.4

_,_, 10.4

•- "- 9.4 e

"o >(

_.--. 8.4 ///

.a_ 7.4 ,///

6.4 ..._, 4.4

o o Linear Taylor series 3.4

_ 24 / -----DEBmethod

1.4 .q --" -- Exact solution

" 0|/"/

-0 6 L__L_..L.L._ ,_ , 2L,

• 0.i 2.0 4.0 6.0 8.0 10.0

Beam depth, H (in.)

Fig. 5 Comparison of DEB and Taylor series

approximations of frequency for perturbation of beam

depth, H (B=3.75 in., M=0. Ibm. ).

Mode ShaDe ADoroximation Result._

Figure 6 shows the first bending mode shape for

the nominal design, and the linear Taylor series

approximation, the DEB approximation, and the exact

solution for a 50 percent increase in the beam width, B.

Figure 7 shows the second bending mode shape for

the nominal design and both approximations and the

exact shape for an 80 percent increase in the tip mass.

Although the linear Taylor series approximation is very

accurate, the DEB approximation is slightly closer to the

exact curve in both figures. However, the mode shapes

do not seem to be sensitive to design changes for this

test problem as indicated in figures 6 and 7.

/

i I

/

/ ,,2"

,,/,,,2"

,//_J

/'/_F

,/'/fP" _ Linear Taylor Series

/

,/"'7"I ..... DEB method

Exact solution

Nominal

II1|_|11

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

Normalized distance along beam

F_j. 6 Comparison of DEB and Taylor series

approximations of first bending mode shape for 50

percent increase in beam width, B ( Nominal values:

H=5.0 in., B=3.75 in. ).

Linear Taylor Series

..... DEB method

Exact solution

Nominal

| •

1 ! I I t I I I i I

0 .1 .2 .3 .4 .5 .6 .7 .9 .9 1.0

Normalized distance along beam

Fig. 7 Comparison of DEB and Taylor series

approximations of second bending mode shape for 80

percent increase of tip mass, M ( Nominal values:

B=3.75 in., H=5.0 in., M=5.0 Ibm. ).
In some cases the DEB method did not approximate the

mode shapes as accurately as the linear Taylor series

approach. Usually, these were cases in which the

change in the mode shape due to perturbations in the

design were small. Other test cases may provide a

more challenging problem for testing these

approximation methods for mode shapes and will be a

topic for future investigation.

Static Disolacement Aobroximation Results

The cantilever beam of figure 1 was also used to

test the method for approximating static displacements.

The design variables were the dimensions H and B. A

cubically distributed loading was used to represent the

applied force and the load values at each grid point of

the model are given in table 2.

Table 2 Values of applied force at the grid points of

the beam model

Grid point

1

2

3

w

4

5

6

Force (Ibf}

0.000

0.025

0.200

0.675

1.600

3.125

Grid point

7

8

9

10

11

I Force {Ibf}

5.400

8.575

12.800

18.225

25.000

Figure 8 shows a typical convergence plot for the DEB

method equation (16) for the maximum displacement.

This figure suggests that the number of terms needed

for a converged value is 3 or 4. Forthis same number of

terms and the same amount of computation, the Taylor

series approximation (equation 19) is known to be more

accurate and would be chosen over the DEB method

unless it did not converge for certain values of _v.

Results for the linear Taylor series will be included

because it is a frequently used method of

approximation.

Figures 9, 10, and 11 show results for perturbation

of the nonlinear design variable, H. In figure 9, the

maximum displacement for values of H from 3.5 inches

to 8.5 inches was plotted for the DEB, linear Taylor

series approximations, converged Taylor series and the

exact solutions. As H is decreased, the bending

stiffness of the beam is decreased causing the beam

displacement to become very large. At this end of the

range it is difficult for the approximation methods to

follow the curvature of the exact results. At the other

end, however, the DEB method appears to follow the

trend of the exact curve. The linear Taylor series

approximation gives negative values for the

displacement beyond 40 percent perturbation of H from

the nominal value.

A

co

E

o

ca

u

Q.

u}

om

qD

E

::3

E

x

tll

I

21.4 -

21.0 -

20.6 -

20.2 -

19.8 -

19.4 -

19.0 -

0

/

/

I

I

I

/

t

/

I

/

t

/

I t I I I

3 5 7 9 11

Number of terms

Fig. 8 Convergence plot of DEB method

39.

A

c 31.

c

E 23.

I1

o

_- 15.

E

= 7.

E

x

_ O.

I

"9.

0.0

\

\

y.,,C_ _ Linear Taylor series

DEB method

Exact and converged

Taylor series

I I I X I I

3.0 5.0 7.0 9.0 11.0

Beam depth, H (in.)

Fig. 9 Comparison of DEB and Taylor series

approximations of maximum displacement versus

perturbation of beam depth, H.

7
Figure 10 is a graph of the displacement of the beam for 8. -

a 30 percent decrease in the nonlinear variable H. The Linear Taylor series

figure compares the DEB and linear Taylor series

approximations to the exact solution. Both DEB method

approximations are very good with the DEB method 6, -

slightly more accurate. .-.. Exact and converged

Taylor series ./

_4. _./

Linear Taylor series -_Z ,, /

40. - .... DEB method I_ ':- J ...-

.... Exact and converged _ ,, ___:._....-

32. Taylor series ,/ =.u. -

. // "/

i,°- -, , , , , ,

o ,/ -- 0. .2 .4 .6 .8 1.0

_. 8. ,/y Normalized distance along beam

}'_ _ Fig. 11 Comparison of DEB and "rayior series

o, approximations of displacement for 50 percent increase

in beam depth, H.

Figures 12, 13, and 14 show results for linear Taylor

-8. J l I i I series and DEE] displacement approximations,

0. .2 .4 .6 .8 1.0 converged Taylor series and exact solutions for the

Normalized distance along beam linear variable B. In figure 12, the maximum

displacement for values of B from 1.875 inches to

5.625 inches was plotted for each approximation

Fig. 10 Comparison of DEB and Taylor sedes method and compared to the exact solution. Again, it

approximations of displacement for30 percent appears to be more difficult to approximate the

decrease in beam depth, H. displacement for small beam dimensions, but the DEB

method follows the exact curve, particularly for the

larger values of B. Figure t3 shows a plot of the

approximate displacements and the exact solution for a

A similar compadson is shown in figure 11. For a 50 50 percent decrease in B. Both approximations are

percent increase in H from the nominal value, the DEB good but the DEB curve is closer to the exact. Figure

method had an error of 47 percent for the maximum 14 shows the result for a 50 percent increase in the

displacement. However, the linear Taylor series design variable. It shows that the DEB approximation is

approximation gives a displacement in the opposite within 8 percent of the exact solution compared to the

direction. This is in accordance with the result we saw in linear Taylor series error of 21 percent.

figure 9.

8
29,

c 25.

t-

® 21

E "

o

_.17.

"D

E13.

E

x

_ 9.

0

40.

32.

A

C

E16.

o

¢U

8.

o_

Oi

-8.

\ _ Linear Taylor series

,\ ..... DEB method

\',\'_'_ Ex a ct and converged

"_ Taylor series

i, i I I I I

2.5 3.5 4.5 5.5 6.5

Beam width, B (in.)

Fig.12 Comparison of DEB and Taylor series

approximations of maximum displacement versus

perturbation of beam width, B.

Linear Taylor series

DEB method

.... Exact and converged

i Taylor series

/

/

I I I i !

.2 .4 .6 .8 1.0

Normalized distance along beam

Rg. 13 Comparison of DEB and Taylorsedes

approximations of displacement for 50 percent

decrease in beam width. B.

A

e

*m

E

G)

O

0=

i

¢3.

¢3

11.

So

o

5.

3.

=

0.

"1.

0

Linear Taylor series

DEB method

Exact and converged

Taylor series ,"

/,/

/ /

/,//

i I I I I

.2 .4 .6 .8 1.0

Normalized distance along beam

Fig. 14 Comparison of DEB and Taylor series

approximations of displacement for 50 percent increase

in beam width, B.

Additional Aooroximations

An additional application of the DEB approximation is

in estimating system response quantities that depend

on approximated quantities. Inserting the approximate

displacement field into the stress-displacement

equations to obtain approximate stress field is an

example of this. Figure 15 shows a graph of maximum

normal bending stress for perturbed values of B. The

stresses were obtained using the approximate values of

U from the DEB method and the linear Taylor series

approach and are compared to the converged Taylor

series and exact solutions. The approximate stress field

shown in figure 15 is very accurate. This was a

particularly good result in view of the fact the the

stresses are in effect proportional to the second

derivatives of the approximate displacements.
29.

25.

o 21.

_17.

_13.

9.

0

r

| -1-903.

.100 _, t .904. \ Linear Taylor series

-_ .092 - _ -I -905.

,\ DEB method _ --906.

\',\ Exact and converged

"_,,_ Taylor series _¢_ .084- _/_ a9 --907. _

- -908. <_

9o9

.068- _-bo/ao _ "910.

"911.

.060 J L [ I -912.

2.5 3.5 4.5 5.5 6.5 0 1 2 3 4 5

Beam width, B (in.) NumberOf steps, S

,,t

Fig. 15 Comparison of DEB and Taylor series

approximations of maximum stress versus perturbation

of beam width, B.

Validity of Assumptions Made in DEB

ADoroximation Method

Recall that the methods developed in this paper were

based on the assumption that certain variables were

assumed constant, for example a and b in equations (5)

and (26). In order to assess the validity of these

assumptions, the actual values of the coefficients ae

and bewere plotted for values of the tip mass, bending

inertia, and cross-sectional area perturbed

simultaneously. Figure 16 shows a graph of ae and b...._.e

ae

versus the number of 10 percent step increases in the

design variables. The lower curve, shows that be is

ae

essentially constant. The upper curve representing the

coefficient as shows a deviation of up to 19 percent

from the nomial value when the design variables are

changed by up to 50 percent. Assuming ae to be

constant was not detrimental to the approximation

results. As shown in figure 4, the DEB method was able

to obtain very good approximations for the full range of

the pertubed design variables.

Fig. 16 Actual values of coefficients assumed constant

in DEB frequency approximation for simultaneous

perturbation of tip mass, bending inertia, and crosssectional area

Figure 17 shows a graph of the coefficients that appear

in the mode shape approximation of equation (10).

The plot of D versus perturbed values of tip mass show

plus and minus 20 percent deviations in D based on up

to 100 percent perturbation in the tip mass. The

second plot shows the vector _ for the nominal tip

D

mass and for plus and minus 100 percent perturbations

from the nominal mass. These plots show that the

vector varied considerably, however, as shown in figure

6, this did not seem to have a large effect on the

approximate mode shapes obtained with the DEB

method.

10
References

20.

18.

a 16.

14.

12.

10.

0.0

10"3 ,o.==oo,'bm7

. / J'-(Nominal)

I /._ = ,o.oibm

II ,

,0 .2 .4 .6 .8 l.O

Normalized distance =,long beam

.32

.24

.15

o_

0.8

0.0

.... ' ' -0.6

4.0 8,0 12,0.14.0 16.0

Tip mass, M (ibm)

Fig. 17 Actual values of coefficients assumed constant

in DEB mode shape approximation for mass

perturbation

(_oncludino Remarks

This paper described a method that is based on

interpreting sensitivity equations of structural response

as differential equations that may be solved for closed

form approximations. The method was developed for

approximating vibration frequencies, mode shapes and

static displacements. The test cases used to evaluate

the approximations involved perturbations of the

height, width, cross-sectional area, tip mass, and

bending inertia of a cantilever box beam. The results

were compared to the linear Taylor series

approximations and exact solutions. The results

showed that the new method is very accurate and in

most cases more accurate than the linear Taylor series

approximations. The method is applicable to both linear

and nonlinear design variables and to simultaneous

perturbation of several design variables. Also, the

approximated response may be used to approximate

behavior derived from the response, for example,

approximating stresses from an approximate

displacement field. To date, the method has been

developed and demonstrated for frequencies, mode

shapes, static displacements, and stresses derived

from the displacement. In principle, the method is

applicable to approximating any quantity for which an

analytical sensitivity formula is available.

Acknowledgement

The authors wish to express their special thanks to

Professor Raphael T. Haftka of Virginia Polytechnic

Institute and State University for his valuable

discussions and suggestions during the course of this

research.

1Walsh, J. L., "Application of Mathemetical Optimization

Procedures to a Structural Model of a Large FiniteElement Wing," NASA TM-87597, January 1986.

2Storaasli, O. O. and Sobieszczanski, J., "On

theAccuracy of the Taylor Approximation for Structure

Resizing," A/AA Journal, Vol. 12, 1974, pp. 231-233.

3Haftka, R. T. and Shore, C. P., "Approximate Methods

for Combined Thermal Structural Design," NASA TP1428, 1979.

4Haftka, R. T., "Design for Temperature and Thermal

Buckling Constraints Employing a Noneigenvalue

Formulation," Journal of Spacecraft, Vol. 20, 1983, pp.

363-367.

5Salajegheh, E. and Vanderplaats, G. N., "An Efficient

Approximation Method for Structural Synthesis with

Reference to Space Structures," Space Structures

Journal, Vol. 2, 1986/1987, pp. 165-175.

6Vanderplaats, G. N. and Salajegheh, E., "New

Approximation Method for Stress Constraints in

Structural Synthesis," AIAA Journal, Vol. 27, No. 3,

March, 1989.

7Schmit, L. A. and Farshi, B., "Some Approximation

Concepts for Structural Synthesis," AIAA Journal, Vol.

12, No. 5, 1974, pp.692-699.

8Lust, R. V. and Schmit, L. A., "Alternative

Approximation Concepts for Space Frame Synthesis,"

AIAA Journal, Vol. 24, October 1986. pp.1676-1684.

9Statues, J. H., Jr. and Haftka, R. T., "Preliminary

Design of Composite Wings for Buckling Stress and

Displacement Constraints," Journal of Aircraft, Vol. 16,

1979, pp.564-570.

10prasad, B., "Potential Forms of Explicit Constraint

Approximations in Structural Optimization-Part

I:Analysis and Projections," Computer Methods for

Applied Mechanical Engineering, 1983, pp. 245-261.

111namura, T., "Eigenvalue Reanalysis by Improved

Perturbation," International Journal for Numerical

Methods in Engineering, Vol. 26, 1988, pp. 167-181.

12Fox, R. L. and Kapoor, M. P., "Rate of Change of

Eigenvalues and Eigenvectors," AIAA Journal, VoI. 6,

December 1968, pp. 2426-2429.

13Nelson, R. B., "Simplified Calculation of Eigenvector

Derivatives," AIAA Journal, Vol. 14, No. 10, 1976, pp.

1201-1205.

14Kirsch, U., Optimum Structural Design, McGraw Hill,

New York,1981, pp.67.

15Whetstone, W. D., "Engineering Analysis Language

Reference ManuaI-EISI-EAL System Level, 2091,"

Engineering Information Systems, Inc. EISI-EAL, July

1983.

11
I. Repo_ No.

NASA TM-102639

AVSCOM TM-90-B-O06

4. Title and Subtitle

Report Documentation Page

2. Government Accession No. 3. Recipient's Catalog No.

5. Report Date

Differential Equation Based Method For Accurate

Approximations in Optimization

7. Author(s)

Jocelyn I. Pritchard

Howard M. Adelman

9. Performing Organization Name and Address

NASA Langley Research Center, Hampton, VA 23665 and

Aerostructures Directorate, USAARTA-AVSCOM, Hampton

VA 23665

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Washington DC 20546 and U. S. Army Aviation Systems

Command, St. Louis, MO 63120-1798

Apri I 1990

6+ Performing Organization Code

8. Performing Organization Report No.

10. Work Unit No.

505-63-51-10

11, Contract or Grant No,

13. Type of Report and Period Covered

Technical Memorandum

14. Sponsoring /_gency Code

15. Supplementary Notes

Jocelyn I. Pritchard, Aerostructures Directorate, USAARTA-AVSCOM, Langley

Research Center, Hampton, VA

Howard M. Adelman, Langley Research Center, Hampton, VA. Presented at the 31st

AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference,

April 2-4, 1990, Long Beach, California _

16. Abstract +

This paper describes a method to efficiently and accurately approximate the effect

of design changes on structural response, The key to this new method is to interpret sensitivity equations as differential equations that may be solved explicitly

for closed form approximations, hence, the method is denoted the Differential

Equation Based (DEB) method. Approximations were developed for vibration frequencies, mode shapes and static displacements. The DEB approximation method was

applied to a cantilever beam and results compared with the commonly-used linear

Taylor series approximations and exact solutions. The test calculations involved

perturbing the height, width, cross-sectional area, tip mass, and bending inertia

of the beam. The DEB method proved to be very accurate, and in most cases, was

more accurate than the linear Taylor series approximation. The method is

applicable to simultaneous perturbation of several design variables. Also, the

approximations may be used to calculate other system response quantities. For

example, the approximations for displacements are used to approximate bending

stresses.

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

approximation

optimization

vibration analysis

vibration frequencies

mode shapes

18. Distribution Statement

Uncl assi fied-Unl imi ted

Subject Category 05

19. Security Classif. (of this report)

Uncl assi fi ed

20. Security Classif. (of this page)

Unclassified

121. No. of pages

12

22. Price

A03

ql

NASA FORM 1626 OCT 86