neural network and response surface methodology for rocket engine ...

1 Payload.

Apay

(lbs)(All

NEURAL

variables

are normalized

NETWORK

AND

ROCKET Rajkumar

RESPONSE

ENGINE

Vaidyanathan

_:',

Nilav

respective

COMPONENT

tIaftka

baseline

SURFACE

}'apila '_''. Wet

Raphael 'l>Department

bv their

values).

METHODOLOGY

OPTIMIZATION

Shyy :__i: p. Kevin

_'= and Norman

Tucker

Fitz-Cof

Mar,shall

Space

Flight

Center,

ABSTRACT

of

-oal

response

of this ,.,,ork

surface

neural

networks

Huntsville,

1.1

cn,:ine_

components..\

rest

points,

'.hree clement.

from

76

employed

to select

response

hack-pr(_pagation layered ilig(qlIhnls,

is

Various

issues

_ie

addressed,

_p|lmlZall_H1 !+1c

alH{}l-ilhm

choices, n.',lS

the te

nl

considered. linear

RBNN

_,ptimization

lhe

i]etna[

promise strategy

This the

of

possibility

tbr engineering

is

,,pt_mum

design

!he

'.,Hllaccs

_t design

been

,t O_i_, cIli,rt tn

in

,+L1..tdr;.ItlC

Jcpendent

i_cated

,NN

more

using

c,,n>tstcnc; as

problems.

RSM

have

lor rocket

,in?.' _,pec_fic

an

data

is

a concern,

and

RSM

enables

for

of

The associated

of with

reeions

design

data.

are normalized

bv thmr respective

baseline

space.

be

both

effectively

any number

ot

injectors

generality depth

of

the

breadth

_'i.e., details

and

it on

values_. AIAA-2000-4880

of

the

global

filters

noise

the

shape of the actual substantial errors et al. 4 have

of and

allows

levels

in representing

Shyy

to

of the

through

can

space

used tied

dimensionality

Depending

and the introduce

engine

is not

{_btained

and

is effective design

of the design

approach

The

Surface

rocket

et al. -_ have

at varying

the

polynomial employed suttee, the RSM can

Response for

types

This

information

RSM

polynomial

to combine

_i.e.. ,,,cope of design _ariables) the desi,an variables _.

characteristics

The

methods

combinations.

consideration

desion

used.

data

different

the

polynomials

['he

the designer

variables

propellant

and

experimental

which cubic

Tucker

or >_urce.

and

and

before

For example,

type

or RSM

space.

method.

design.

and

objective

linear regression. The ltle surface can then be

used

rejector

data

n()t

numerical

search

been

design.

an

used. Graduate Student Assistant, Student Member AIAA : Graduate Student Assistant, Student Member AIAA • I)_olessor and Dept. Chair., A_,sociate Fclh),,_ .\I.V\ .-\cmspace Engineer. Member AI:XA _][Aerospace Engineer. Member AIAA :# Distinguished Professor. Fellow AIAA _* Associate Professor. Member AIAA 1 INTRODUCTION

a gradient

methodoh)gies

to rocket

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in

is bx (ff

for

constraints.

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models

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is represented

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have

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to

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based

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die

to

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as

used

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{RSM)

been

data

conditions,

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simulations.

c_mponents,

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icDlesclltcd

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representing experimental

space

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by, development

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propuisum

optimum

;Inc

Methodoh)gy

not

design

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techniques

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evaluation

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than

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designed

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performance

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decreasing

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using

performance Fhe

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and complex

systems

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preliminary study,

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et al 3 used

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analysis. 2.4.1

Radial

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on

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the size

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of data

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(Figure

be designed

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radial-basis

layer

of neurons, can

(RBNN)

networks

output

number but

Networks

neural

neurons to train

for training.

AIAA-2000-4880

The transfer radbas,

which

function

is shown

and minimum

outputs

the function

of

is given

for

in Figure 1 and

radial

2b.

basis

Radbas

neuron

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with does,

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Figures

maximum

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output

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a neuron

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also

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error

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In BPNN

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This

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in Matlab.

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so its to start

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incorporated

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betx_een

t l and

"apparent"

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levels

a

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teaction.

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bx K,

the

RPM,

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in r/and

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maximize

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bias

in turn

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it is

tot

worst.

causes

more

accurate.

rmalizcd

76

number

37

except that BPNN is clearly based on the two cases, it seems

most

design

D - 300.440,(_

RPM

is

i_ptimal

response

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+5.__SD, _,_''

- 0.00362k

in.jector, Overall.

have hidden

and 60 neurons

supersonic

lbr the to RBNN.

constraints. \vith

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but pertbrms

8 for the 4

in the

D_-114.407C

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me reduced

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respectively

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that _oh'crh evaluated.

S

cklbic polynomial

with

W.

uses

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t[t_e

designed

t? and

The accuracy of the various available in Table 6A and

data

ba_,ed L_n the _btaincd

,elected

0 _,uggcsts

terms.

(19)

be due to overfitting,

trend

,_t these

45

polynommb,

eqtmtion

l'be

presence

W was

of

cubic

that are products ol three because ,_1 the number of

the

Dk, RPM

DA ,,,,,k , -- O.O162 DC,kr

networks

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was

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+ 921.053

-- 0.692

for

BPNN

fable generated.

of levels

also

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initially

the number

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RPM

respectively, in a single hidden chosen are listed in "Fable 7.

V.,_:,_,.

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_erms ,_ere included. Cubic terms different ,,ariables were included

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5A.

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The

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nf the weights

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D -- 23.359C¢,

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-- 162.604

a

process.

with

values

neuron

through

training

dependent

of one

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evaluated

D + 10.744Q,

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but

the linearity evaluations.

A,,,," -- 0.00856Q

be due

Otherwise

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The

tbr to the it

is

stage

we are dealing

AIAA-2000-4880

onlywiththesinglestageof theturbine,ftencethereis no splitonthestage reaction. 4

5

The

SUMMARYANDCONCLUSIONS In

the

constructed employed and

adding combined

helps in

NN

are

first

test

data

are

then

of

polynomial

the

BPNN

and

Ihe

is

or,, and

accuracy

and

various

improvement

between

evaluate

neurons

model

best

evaluation

RS

data.

performance

into

The

Marshall

the

training

the

insight

terms.

stud.,,,,

the

to assess

to offer

data

present

using

selected

on

NN,

netv_orks

a

for

optimum

gradient-based surfaces networks.

Based

c,n the

polynomutls

and

obtained.

as

the

both

present all

',kith

perform

the','

have

stud>',

both

NN

data

sizes.

the

NN

than

seems

more

to be

cases.

networks,

even

s(s[ver/L

[cndt()

,wer

fact

a large

l;curt>lls

respond

space.

Thus,

bas_s neurons that

to

larger

can

in

b:t"

neurons

can

>pace,

while

ICl_.ltl\C{\

a radial for

MIILI[J

Based

i/axe

output.,,

tadia[

the

for

network

ellen

i_l

more

takes

network

shown

technique

_s not

method

t)f making

designing

,'.

less

ume the

K,.

Rocket

Eneine

ASEE ] 3-15.

Joint 199g

Shy.v,

W.,

10.

procedure

fast

such

out cases.

of efficient.

of the be

aimed modest

and should

data

size

sizes to

the

are criucat

be add'ressed

to

pertbrm a The

number for pracucat

of

12.

better work is

Response Product John

Wiley

R. G., httroduction

to

Lecture

Publishing W.

Tucker,

and

Notes

Company,

Sloan,

J.

G,,

"An

Methodology

34 th AIAA

P.

Snr#lce Engine

and

K.

June

and

Nettr¢ll

InJector / ASEE

/ ASME

For / SAE

Vaidyanathan,

Network

_'0-24.

for

35 th AIAA

Propulsion 1999,

R.,

techniques

Opttmtzatton.'" Joint

/

July

Conference

/ and

AIAA-99-2455,

Los

CA. N.,

Shy5,,

V¢.. Fitz-Cm
tcnt

and turbine

basis

this is the

that

t)_e

Mxver t_rder

flexibilitx.

configurati_m>.

comparable back-propagation si,_,moid neurons in the hidden

--

better

neural

reached

Myers,

Integrated

response

trained

ha_c

more

l\_r modest

in teeter

P, adial

!_.

3.

a standard. the

,a¢

statistical measure needs the best terms to include.

,_fl,,'erb

the

using _wer

4.

polynomials

,\mong

5.

out

algorithm

results

polynomials

In

4.

the

t li,.z,her order

ct_mparably 3.

is carried

conclusions.

appropriate determine 2.

by

represented

following 1.

design

optimizanon

has

Center.

REFERENCE:

the

RBNN.

Thus the test data are adopted to help select appropriate RSM and NN models. Once an RSM or NN model is constructed, a search

study

Flight

Optimization Using & Sons, 1995. 2.

xarying

m

present

Space

Surface

the test

with

constants

1. and

based

[:_r the

spread

6

polynomials

by removing

or.

of

varying

ACKNOWLEDGMENT

Propulsion (2000),

liquid NAS3F.

and

For

A 36 a'

Conference Huntsville,

AIAA-2000-4880 Neural

13. Haftka, R. T., Gurdal, Z. and Kamat, M, P., Elements of Structural Optimization, 3 'd edition, Kluwer Academic Publishers, 1990. 14. SAS Institute Inc. (1995). JMP version 3. Cary, NC. 15. Dernuth, It. and Beale M., Matlab Neural Network Toolbox, The Math Works Inc, 1992. 16. Stepniewski, S. W., Greenman, R. M., Jorgensen. C. C. and Roth, K. R., "Designm.? Compact Feedlbrward O/F 4.6.8 4.6,8

[

4,6,8 1: Range of design

Table

Model # [

Coefficient

= 0

V/Vo _

Training

!i

.

Sets,"

4.5.6.7.8

Terms Removed

l

Terms

Included

: V,/Vo]

I

Quadratic

and less

!

.......... r l 'd'_)" ! J_":q',:_l ......... , _'./_'::) _',A"o) _ I

_,, (%)

0"(%)

0.218 0.0857 0.0799

0.280 0.212 0.214 0.214 0.213 0.212 0.212

0.0799 0.0859 J 0.0936

t_'/_:°f2l

, l..,,,,,/ L,,,,,..,)"_,(V/V,,):(L,,,,,,:,):

7

[ :VYVoj:L,,,,,,,. _'/Vo_ _ :l.,,,,,u) _, :I, ,A,,) _L,,,,,,U . Vcq/o(L,.,,,,,f,)' 0.0988 'Fable 21a): Different cubic pob'nomiats for ERE. (Dependent variables: _i/_:,, and L,,,,,,,, t5 training points, 10 test pointsl (Errors are given in percentages of the mean value of the responses). ('oefficient

Model

: I)

I

ferms !,?.cmoxcd

Terms

Included

_, l";)

[

_l 1

_

# of lxp,'ers

Scheme RBNN (Soh'crbe) RBNN (Soh'erb)

, ,

') ] _ 4 i used to design

"

, :_'A'o)-fO/t:)-

5.445 5.584 5.584 5.584 5.584

i

3.490 2.234 2.094 2.094 2.234

3.909 5,584

_ i

2.094 2,094

lot Q. i Dependent variables: O/F and V/_.:,, 9 training (,f lhe mean value of the responses).

# of neurons in I # of neurons !' the hidden lax er ! the_

_ RBNN {Soh'e,be) ," i 1_ RBNN(Soh'erb_ 2 14 BPNN 2 8 "Fable 3: Neural Network architectures spread constant }

! !

( t"/_o_. I OIF)' ( I'A,'ol'. : O/F) J, r _iA:o _: _ _,'/_,'-). _C)/t":'. t _'/Vo)-

Table 2(b): Dit'fl:rent cubic polynomials points) (Errors are g,ven in percentages Scheme

and less

, t,, _.o: :_,'/_'or, !O/F)' '

6 7

0"(%)

1-

()uadratic

/V/V:n'.: 0/f:)' __)/l,b'

] i

O/FI2 3 4 5

38 th

clement.

i

5

Data

L,.,,_,, in. 4,5.6,7,8 4.5.6.7,8

8 for the shear coaxial injector

r _'./_"_)/:L.........

4

4

Snzall

i

i

V'./Y_/

3

with

V/V, 4 6

I considered

variables

Models

AIAA Aerospace Sciences Meeting and Exhibit, January l0-13, 2000. 17. Coleman, T., Branch. M. A. and Grace, A., Optimization Toolbox for Use with Matlab, Version 2, The Math Works Inc, 1999.

in

: .'7 I _ 1 i' I 1 / I the model for

cr tiw ERE I';:f ) 0.207 0.133

points.

Error qoal aimed for during

4 test

training

e i O.O {so= 3.25) I).0 {sc = 1.20} ,: 0.001 {so= 1.05} 0.001 {sc= 1.05} I 0.01 I 0.01 shear coaxial injector element. {sc =

cr tbr Q (%) 1.396 1.536

BPNN

0.i80

0.832

Partial Cubic RS

0.213

2.234

Quadratic RS 0,280 Table 4: RMS err_}r i_, predicting the values ot the objecIive function bv various schemes coaxial in ector element (Errors are L,iven in percentages of the mean value of the responses).

3.490 for the shear

AIAA-2000-4880 V/V,,

Scheme

4

O/F

L,.,,,,_, in.

RBNN

(Soh'erbe)

i

8.0

7.0

98.60

(0.00)

0.588

(0.00)

RBNN

(So/verb)

8.0

7.0

98.60

(0.00)

0.588

(0.00)

8.0

6.9

98.64

(0.14)

0.578

(1.70)

8.0

7.0

98.61

(0.01)

0.594

(1.02)

8.0

7.0

98.67

(0.07)

0.591

(0.51)

Model

8.0

7,0

Model

8.0

6.9

8.0

7.0

99.20

BPNN Partial

Cubic

Quadratic

6

RBNN

RS RS

(Soh'erbe)

RBNN

(Soh'erbl

BPNN Partial Cubic

[ ]

Quadratic Model RBNN RBNN

_

Cubic

Quadratic J Fable

5

Solution,,,

and RSM

schemes

are given

in parenthesis

lot

for the _hear

0.588

(0.00)

0.512

(0.00)

99.20

(0.00)

0.512

(0.00)

7.0 7.0

99.18 99.15

(0.02) (0.05)

0.513 0.500

(0.20) (2.34)

l _

8.0 8.0

7.0 7.0

99.17 (0.03) 99.20

0.531 (3.71) 0.512

j t

80

7.0

99.40

(0.00)

0.493

(0.00)

8.0

7.0

99.40

(0.00)

0.493

(0.00)

7.0 7.0 70

99.41 99.42 09.67

(0.01) (0.02 } (0.27)

0.500

(1.42)

, ti\cd

'.atuc',

coaxi;.t[

8.0 8.0 S.O

i 1

,',.0

!

I

,ff _,,,"_ and

mlector

for cizch prediclion

TypeofRS

98.50

7.0

Model

Optimal

0.588

8.0

RS RS

98.60

8.0 8.0

BPNN Partial

Q, Btu/inZ-sec

!

i

(Soh'erbe (SolverS)

%

I RS RS

ERE,

7.0 given

elemem.

T

0.499 ( 1.22) 0.47 ! (4.46)

99.40

_('onstraints:

t

0.493

and l ....... ,, obtained

ran,_,e _t 0/1: 4 _< O/F

_< 8.4

\_ith

NN

tlktltll_

_'II I }ptlllltH1?

i.:1'".I

,:2 :h_c

..----O--,

RS

i +::_ J4

%,

I 24 --

+

-El

--

RI/NN ",,4 .cibca

-7

--

'_I

--

RBNN naK¢lb¢)

t:

ca -

-

-X o -

"

.!,:FINN

"

-)

1.20

ISolver_l

-

\

130

-

-

-K-

-

-A-

--

_1: No

+

'_

t Solve

BPNN

t'ase

#

constraints:

0

2

I 1: No

2: ('oll',,ll_llllls

on

AN `) and

Case

# 2 2; (_onstraints

and

\rllldl_

I constraints: \N2

tit ('OIl_ffalllt_

[!IfCCt

_

......

on

(d)

or1 ( )l_tilllum

[$1allc

I!_tCCl

\ _LLI k It, ,hl

-

Ol! ()ptill/tlm

OI ( OIP, ltalllt>,

_'Jtag¢

]4CaCIIOI1

50

_

........ -------'4_t

ublc

RS

'

•

7

&-----t_

t tl)

? _i_

7-

rb

BPNN

"¢Iltd_

(el

±

"RBNN

erb)

I OO I

fl

:25

RS

I 60

..............

I 41)

Vane

I

os 07

:

of Constraints

150erbe

'RBNN

_

t 71)

7

I,LBNN I SoN

o -
,N

- -X-

-

5

-

'-X-

- -RBNN , Sohcrb+

E

] (16_

'

--

-_"

--

BI'NN --

-d_" --

F;F'NN

H _tl

I'c

NOCO_IMIdHIthl ,n \N2

(,,nMr,tzn[_

_ .rod

\ [l',_lu

_','.!

at/d

\ [ll;]_J

_o_

I_

I:igLIlC 14: t!li'o-'t due to prc:,cn,.'c l)iameter, D (in.L (b) Optimum C'_ tin.),

n_wtnallzcd

by

{e) Optimum their

Blade

respective

,.\',:h.t[ (ihurd.

t_a,_clillC

\ ;lilACS

(

_ln.) and

,pCC[I',

C

{";t>Clllle

iack

t,t 11

+lbs). (ci

\ttJllegJ.

Ct)Ib,

ll;.lil]t_,