von KARMAN INSTITUTE

FOR F., UID DYNAMICS

GRANT AFOSR. 80-0177

9! Final Scientific Report1980 May 01-1981 April 30

0PTIMIZATION OF MULTI-ELEMENT AIRFOILS

K.P. ieas

mAyI 198

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4. Title (and Subtitle) . Type of Re. o__t_&.er.ioi Covered*-~FinaL cientific Xept L

OPTIMIZATION OF NMLTI-ELEIENT AIRFOILS,

'IT, TON OFMULTIELDIEPerformin Org.-R-eprt Taib-or(.....VKI-PR-198P-5 V7-8. Coat ox--G mL-Numbdr

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airfoils; wing flaps; lift devices; flight pptimization20. Abstract / " '- A review of techr iques- aiined at maximi,.ing Cg,-n -J %f or multi-element airfoils: Ishowed the need for Lnore exhaustive testing s§sib configurations of flap

dflection, slot geometry and airfoil angle of actack. For a typical 4-element

aiLfoil, the nun.,er of possible configurations can easily be in the billions. Anew technitite, based on evolution strategy, has been developed for the problem ofoptimizinoa multi-clement airfoil w1th respect to its envelope of riaximun/ I/C '

versus T, This Lechuique was applied to an airfoil having a slotted leadi --d

edge flap and a double-sloLted trailing edge flap using an existing computer

program fo: 2D subsoniL multi-elEiaent airfoils to provide values of aerodynnir cco.fficient.s. Althou,it limitations of the program precluded th? full determina-tion cf Lhie envelope o! (fiICAivs Ci the problem of madimizig y 1.; for-

-nIu, C costr L w,.o ! ed. The result of testing approxi'a)-eiy 400 con-"Igc..1t io;; ,ener:ted by the random mutatioa-naLural selet: "n proccdure ofevolut.ion sratey was an 8-fold increase in ith a 12 % increase i4~FOM@ 1473 ' i ''

- Cont.inued on Reverse

.9 'M

Continu d..

! over t1e initial geometry.f2 he conclusions reached from this work are 1)evolution strategy applied to the optimization of julti-element airfoils canyield substantial improvements in aerodynamic performance, 2) the number ofconfigurations required to find the optimum can be reduced from the totalnumber possible to a small fraction of the total with the same final result,3) the inherent simplicity and speed of the technique develoed lends itselfwell to further application in wind tunnels, 4) evolution thecry appears tobe the best choice for techniques aimed at the optimization of systems definedby a large number of degrees of freedom.

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This report has been reviewed by the EARD Information Office and isreleasable to the National Technical Information Service (NTIS). At

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This technical report has been reviewed and is approved for publication.

C:1 RN W. BAILEYMajor, USAFChief, Electronics

FOR THE COMMANDER

SGORDON L. HERMANN'-

Lt Colonel, USAFDeputy Commander

*14

J St

ACKNOWLEDGEMENTS

I

'1 The author extends his sincerest appreciation to*i the supervisor of this work, Professor Roland Stuff, who

first proposed the application of an evolution strategy -

based scheme to the optimization of complex aerodynamic

configurations. His enthusiasm and unselfish counsel was

largely responsible for the promising results of this work.Alsoto be recognized is the assistance of Professor John

*.1 Sandford with regards to the fundamental concepts of multi-

elcment airfoil optimization and to his review of the

Yeport manuscript, and to MBB Hamburger Flugzeugbau for

the contribution of a realistic aerodynamic design problem.

Finally, the author would like to acknowledgethe sponsorship of Detachment 1, Air Force Office of

Scientific Research (EOARD/AFSC), USAF.

"But it can't fly, Newtonz taws prove it 1"

Th. von Karman, early 1900's

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1 INTRODUCTION 1

1.1 Multi-Element Airfoil Optimization

It is no secret today that comoetition among civilian and "

military aircraft manufacturers in the international marketplace is

tough. This competition has demanded improved flight efficiency, and

4. as a result a strong emphasis has been placed on dcign optimization

with respect to take-off, cruise, and landing aerodynamic character-

istics. Design considerations for take-off and landing aerodynamics

are primarily directed towards the reduction of wing area, inj,rase in

maximum take-off weight, improved fuel efficiency, reduced field

length requirements mnd lowerl, arouui nnir q levn1 s. ThP rIntivR

importance of the nondimensional coefficients of lift and drag, 01

* f and Cd resnectivelv, for toke-off and landinp are shown in figure (I)! .

Over the entire regime of flight, the nerformance at take-off,

A when aircraft weigh+ and enrine noise are at a maximum for the flight,is the most critical facter in the design of lifting surfaces. To

achieve the maximum rate O1 angle of climb, thus reducing runway

renuirements and noise footorint, meximum Cl/Cd is desired for a

given C1 . The envelone of values of maximum Cl/Cd versus C1 for the

Boeing 727 wing, obtained from flight measurements, is given in figure

(2), and is more or leos- similar to the envclone obtained for all

other airfoils2 .

The complexity of multi-element airfoils, as seen in figure (2)

and in more detail in figure (3A), has been dictated by the need for

h miaooth, and ior high-speed transnort aircraft shock-free airflow

at cruise conditions, combined with the high-lift configurations re-

quired at take-off and lanaina , 3 . Although significant effort has

been devoted to the study of advanced high-lift devices such as

those employing blown f1rps, jet flans, boundary laver control or

augmnentor wings, it is reasonable to expect the continued use of

4 C.- mechanical devices for the next generation of civilian and military

aircraft. Indeed, as shown i-:. figure (4), values of Clmax for large,

high-subsonic aircraft nearly doubled between 1950 and 1970 as a

result of imnroved desig-,n of mechanical flao sl: tems4 . Once the

choice of high-lift system anO associated flan and airfoil oeome-

tries has been ,r:ade, the relative nositioning of the elements must

be determined such that the obtainable values of C1 and Cl/Cd are

sufficient to meet performance demands (or preflight promises!)

The ideal set of configurations would be those that would result in

the largest Cl/Cd-vs-Cl envelone. The number of all possible

2

configurations, however, is very large even for the typical oase o .

an airfoil having a double-slotted trailing edge fiap and a, stf, -leading edge flap, as depicted in figure O B) . ,' .." .,:¢" r "

1.2 Empirical MethodsThe most common means of determining the best airfoil-flapI"', ,,

configurations involves many hours of wind tunnel testing for a, rep.-

resentative set of variations of slot gap width and overlap,, flap

deflection and angle of attack. Optimum configurations are then

made by adding to the main airfoil the leading and trailing edge-

devices, whose individual slot geometries have been optimized withrespect to on!- one .djncent element. An expiple of thie empirical.

method, renroduced from the work of LjungstroM5, appears in figure

(5). Althoug-h this emniricel method can greatly improve tli aero-

A dynamic performance of a multi-element airfoil, there are two signif-

icant disadvantages that must be considered. First, the accuracy of

an empirical method is dependent on v large number of test config-

urations; however, for oven'the most well-funded design relatively

few different configarations are actually tested. For instance, for

a 4-element airfoil, 500 test cases is only 5.10-6 % of the numberof possible cases, if each of the 10 doegrees of freedom can be

varied across 10 increments. Increasin4 the number of test cases

comes at the exnetse of time and ilirect onerntin- cost of windtunnels. If wind tunnel time and cost trends increase at the rate de-

pictedl in figure (6), however, it is doubtful that empirical methods

will produce better results than at the nresent time, when a great

emnhasis is being nlaced on im-nroving high-lift device performance6 .

4JThe second disadvantage of empiricPl methods is that by combiningAelements ontimized with resnect to only adjacent elements, the

imi)ortant viscous interaction between r7ll elements is unaccountedfor. This interaction, however, is strong end tests have shown thatthe combination of individu .y-o-tioized co-,'onents raely results

in an ontimum overall configurntion5 .

,; 1.3 Self-Ontii, inr AirfoilsIt would be desireeble then to have a self-optimizing scheme,

where the syite-n seeks it3 ovin best confimszrrtion without the need for

testi". every nosiibilit-. Far fro-, ec'f,'ve, such sche-As hsvpbeen used with a good deal of succass in several different problems

where systems are described by several degrees of freedom and

relative 'factors of fitness' result for each variation of narame-

4 ters. Ono such anlication described by Levinsky7 attempted toF r. I

maximize Cl for a fixed maximum Cd and minimize Cd for a fixed

i minimum Clfor a flexible 2-dimensional airfoil mounted in a

I ' transonic intermittant wind tunnel. Results of this work, and

results of further investigation of a 3-dimensional flexible airfoil8

in a transonic continuous wind tunnel are given in figures (7) and

(8). In both cases, the hydraulically-actuated leading and trailing

edges were modified automatically by on-line computers programmed

with a gradient-strategy ontimization scheme. This is described

in section 2. The only human input was TGhe factor of fitness to be

maximized or minimized (Cl,Cd, or volume) and a constraint. Gradient

strategy has also been used in the work of Hicks9, where 2-dimen-

sional transonic airfoils were modified to improve C1, Cd or

maximize volume. Some of the more significant results of this work,

performed numerically as onnosed to the above mentioned wind tunnelwork, are shown in figure (9). Although these examples have resulted

in substantial imniovements in airfoil performance, the a&olication

of their ontimization methodolo"y was found to be more clunbersome

than the method described in this work. flxisting ontimization

methods are comnored below to a new scheme based on 'evolution

strategy'. This scheme hus been developed to determine the config-urations of a 4-elerent airfil giving maxirmn Cl/Cd for each value

of Cl, where only the r-eo-itry of e .cb e!.me-ot is kno,,:n inititlly.

i/

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2 THFOR"2.1 Simrle OTtimization Problem-Existing Solution Methods

Conside, the ontimization problem faced by an experimentalist

standing above a room in which there is a single geographical peak

* whose location with respect to a 2-dimensional coordinate system is

to be determined. In this problem, the 'system', this being the hu-

man, has two degrees of freedom, his location with respect to the

x-axis and the y-axis. At each x,y coordinate positioi, the value

is determined of the geographical 'fitness factor', or the distance

between the x-y rlane and the surface below. The most rudimentary

method to find the peak would be to make a number of random soundings

in order to obtain a rough impression of the surface. In order to

improve accuracy, the grid could be divided into a fine mesh and the

height is then recorded at each mesh point. Although this scheme

guarantees that all possible x,y cobinations have been checked,

the number of posnible combinations increases with the seuare of the

number of grid divisions and with a rower e-ual to the number of

derzrees of freedom for higher order systems. Figure (10) denicts

these 2 basic schemes along with two gradient-based strategies that

were develoned to converge to the -eak with a reduced arnount of

effort1 0 .

In the first of these two gradient-based strategies, knownm

as the Gauss-3eidol Strategy, the experimentalist, trying to find the

neak with the least s-rount of effort, simnly proceeds in the direc-

tion of the stee-nest positive graOient, determined at the starting

noint, until t':e gradient becomes zero or negative. From this

noint, he turns in the direction of a new, locally-stee-est gradient

.nd continues in this now direction. Thisi orocedure is continued

until the Peak is found, that is, r nosition hns been reached on the

x-y nlane where no nositivA wrrdients exist below. The second of

these technioues, known s the 'gonerv! r-redient strate-y', is sim-

ilar to the fir.t excent that at etch sto-) the mar.niturle of gradients1 in all directions is re-evpluatec. Although the second scheme may

result in the shorter total distance betw:,een the stertinT -oint and

the nceak with resnect to the first scheme, it renuires more work

-oer sten). Clearly, the use of eithe" of the pradient stretegies

renresents a substnti! i!nrove,!ient over the insnection of every

oossible co:nbination of the vtriable parameters. It is noted that

in order to reduce the a,-ount of effort involved in scheme B of

figure (10), one is teinnted to incres- the -rid s-acin-. The

danger of this, of course, is that there is a stton,,4-os6sib '

that the peak will be completely missed- this is precisely the

point mentioned previouxsly related to optimizing airfoils by

empirical methods. The danger of using gradient strategies in-

more complicated systems is that only a relative optimum may be

reached, as the smaller of two peaks for the simple problem just

described.

2.2 Simpole Optimization Probl em-Eivolut ion Stategy, SolutionA fifth, and relatively new optimization scheme that may

further reduce the effort reouired to improve a system described by

a numb cr o f dogr ̂-c ao f fr o o 4-. 1 b ac ad o n I ov olut ion --t rat e y.

'Relatively' new is stressed, because the basis of this strate.!y, is

derived from the theories of natural selection first postulated by the

10th century biologint, Charles Darwin. As in biologicvl systems,

which in their simnlest claspifications can be grouped according to

a nunber of naramiote-'s, siich as size, number of anpendaaes, means

of renroduction or mobility, en enginonrinr system survives if the

combrltion of each of' its definin.! Paraeers results in a 'beast'

that is surerior to all other combinations. Only reenntly has this

concept been anplied to realintic nhysicl-I problems$ thb majority

of the work to dvte being attributed to Professor Ingo flechenberg,

of the Technischen Universitaxt Berlin. In ord r to describe thin

scheme, wie considr'r agisin the tasic of the oe:nerimentri'ist. Tjoing

evoluition stratepy, rs deinoted in firu--o (11A), a number of test

points are sel~ected randomly over a region noer the starting noint.

The heipht is recorded at eRCh point, then tVie P. eririentv-list

moves to the hi .hest of thnse noints. 'ro-i there, the, same num~bsr

of rev. noints are rendonly chosen, and the norocedure continues as

before.

The distance fromi one Ihir,'h -point' to enanh now noint in a

succes!aive set is contr~lle6 by the nrevious distance, that is if a

large sten resiulted in a, lar~er heig'ht thvan a smaller sten0, all new

points wiould be deter*.ined using steps anroximately enual to the

large step. This im-nortent factor of' evolution strateiy allows

1ar',e stens to be t.akcen ear 1 v in the o-otimization process, and

decreasinp sten sise as the ontinu nopition is aponroached. The

inclusion of lar.7e mutation steps elso nrevents the convergence to

local maxi"na, a nroble- associated w,.ith gradient strategies. In the

cane of a locel maxim~a or small ba~rrier in the ontiinization process,

evolution strate~y will iiunn aher-d as the result of a large

mutationa length.-6

2.3 Evolution Strategy Examples

Several examples of evolution strategy an-lication are shown

in figures (11) and (12) As a simple verification test, figidre,

(11B) depicts a segmented plate having 5 degrees of freedom thai

is mounted in a wind tunnel. The objective is to reduce the 0vdrA-l

drag of the plate as measured by the momentum defect between the' -' "," "

leading and the trailing edges. From the initial configuration,-

several new configurations, or cases, were created by random mod-

ifications of the angle of each segment with resnect to an adjacent,

mgment., Afte. 1.40 csips, tbA s rgenter nI. pl.A hAC.nA n IRly plarar,

the expected result. The important result is that 140 cases reo-

resents only 4.06"j0-5% of the total possible, 345 million.

The second example is an attempt to modify a system whose op--

timum configuration is unknownl l . In this case, shown in figure

(12A), the objective was to maximize the efficiency of a two-phase

supersonic nozzle constructed from a number of discs. The optimal

configuration is radically different from conventional designs and

verifies the ability of evolution strategy to converge to unknown

solutions. It is also interesting to note that flow mixing regions

seen in the figure have been previously suggested for improving noz-

zle efficiency.

The third exm nle, tht of reducing the head loss for a 900

bend of flexible tubing, also has an unpredictable result. As

depicted in figure (12B) the final shane is a subtle change from

the initial, a standard circular arc, but the result of 300 muta-

tions is a halving of the head loss ll .

2.4 Selection of Ontimization Mlethod

The advantares of using evolution strategy over an exhaustive

evaluation of every noscoible configuration ere clear, excent perhaps

for the case of a system having only one derree of friiedom. The

advante-es of evolution strategy over a 'strictly determined

mathematical stratej', such as gradient methods, have been sum-

marized by Rechenberg'O:

1 When a large number of warameters are involved, theevolution strategy attains the desired result more ranidlythan the more fe,-iliar strictly determined search strategies,assuming the size of the search steps to be the same in bothcases. So far, this could only be nroved for the case of ann-dimensional hyperplane risin- in any arbitrary direction.A more general proof is being attempted.

2 '.Vhere-s the mathematical search strategies used so far

require very small stops (in the Sense of the truncation o!I' the ' ITaylor series after the first term),

the evolution method ca'6-

and should operate also with much larger steps Which exceed the '.

linear region in the neighborhood. Taking larger stepss ig-nifies(a) in many situations a more rapid advance towards the -desired aim, and(b) a shorter time to decide whether the step taken has been ',-:".successful or not. (The change in the value of a function will-generally be greater in the case of a large parameter :hangethan in the case of a small change.)

3 A so-called "steady signal" in the measurement of thevalue of a function is a mathematical fiction. Disturbances arealways present, which give rise to errors in measurement. Theeffectiveness of the strictly determined mathematical searchstrategies is markedly reduced by small errors of measurement.It is of the nature of the evolution method (as a stochasticsearch method) that small random errors of measurementcannot have a decisive influence on the develonment of theprocesr.

4 There are esses in which the more familiar mathematicalsearch strntegies must each deadlock. Such ceases are fre-nuently observed in nhysics, exanles are hysteresis phenomenaand locally limited extremal values, The evolution rethod cangenerally cone with such situations without difficulty.5 The algorithm of the evolution stratery is extraordin-arily simple. This imnlies that the effort reouired for anautoriintion of the search process is relatively low.

Since the work associated with this renort was strictly numer-

ical, as onnosed to experimentnl, the small error effect noted above

related to the use of mathematicsJ. '-trategies could not heve enysignificance; thus an arru-ent could stil. be retde in frvor of gra-'

dient methods, Rechnnberp, how:ver, hrs founOd that o cross-over

point exists at which seprch effort for innroved configurations

using evolution strategy is loes than the effort required for

gradient strategies 9 . This noint is for systems having 5 or more

degrees of freedom.

The s,-tem of interest in tbe vi:rk described 11ore i- a A-

ele pnt airfnil described by 10 parameters Ps shown in figure (13).

Three 'oivot noints describe the relative nosition of adjacent ele-

ments. The individual reometry of an element is given with resnectto a local coordinate system with the origin on its leading edge.

Pivot points 1 and 2 are fixeo with resoect to element 2 and are

vrriable with resnect to elements 1 and 3. Likewise, pivot noint

3 is fixed ,vith resnect to element 3 and variable with resnect to

element 4. The two deprees of freedom for each of the -ivot points

olus the reletivp deflection between adjacent elerents define 9

degrees of freedom. The 10th de7cree of freedom is the angle of

-8-

attack between the mnin airfoil anO the fre',tream vplocity vector.

As previously stated, the objective of this optimization is

to obtain the best envelorie of Cl/Cd-vs-Cl values. The fitness fac-

tor for this system is the magnitude of C1'/dd for each value of C1,

which is equivalent to minimizing Cd for fixed C1. This implies a

series of individucl ortirizations over all desired values of C .

The flexible nature of evolution strato allows a two-step proce-

dure, however. First Cl/C d is maxi-ized while maintaining C1

above the initial value; the limit of this mnaxiizntion is P point

on the envelone. The second ster is then to mnove P.loni this envel-

one with moeerate mutntinn len!rths, nelectin! those eonnfi,7urations

havi n the beqt combination of 01/(d Pm ('.. It in becruse of thisC

flexibility counled with the eboveme.tioned asnects of evolution

strateg-y th t this method wnr- chosen over grndient stratVgies P.s a

bvsis for multi-eleent ,irfoil o,t~miz-tion.

4

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3C-AlCULATION OF ARRODYNAIA'C COEFFTCIR'NTS

3.1 Description of Program Theory-and Us e,-

Aerodynamic Coefficients are cacltd b tg rad

detailed in reference 13. This prograrn is composed of .the- followintask areas:

A. Potential Flow Solution

B. Ordinary Boundary Layer Solution

CConfluent Boundary Layer Solution

D. Slot-Flow Analysis

B. Combined SolutionThe itiviscia, potentip&. flow solution inales use of the distributed'vortex concert with the vortex singularity con~nrising the f-unda-

*1mental solution t ) -the ILanlace equation. Airfoils, arbitrarilyarranged and comnosod of fron ono to four elements, have surfacesep'nroxirated by a dlosdi-olyron whose linear segmnents are representedby distributed singu).aritios. Airfoil contours are limited to smooth,regular shanes with shorn or -nointod trailinr edges. The only*restriction on the slot formed by two adj&,cent elements il that themagnitude of flan ovorlsr must be grepter then abotit l1/ of theairfoil chord.

The ordinary boundary layer solution is comprised of maths-

matical models for laminar, transition, end turbulent boundvry

layers in subsonic flow. The Iiinc~r boundary layer model is based

needs of the nrogn'am. Lanine r stall criterion develonmeO by the authorofthe pr ora relict ' the formation of short or long senexation

bubbles or bubble burst, a flow condition which causes the tcrintinat~.onthe further -orof riin execution. The transition model, evolving frontv

theinsabiitycritei-ia of Schlicting end Ulrich , establisheslimiting codtosfor dnfininm *t-he, DositioA of transition on theairfoil. Two septarate mcthemeptical models for ordinpry turbulentboundsry layer develornirnt are used. The first, an approximatem'odel develoned by Goradia, is used in the initial iterative calcu- [lations. The second and more accurate model, based on the methods

of INash, deter-ines the turbulent boundary layer in the final, vis-

couo solufion.

A significant feature o~' the nroprom i8 the inclusion of aconfluent baundEary l~yer model thet reflects ihe merging of an upper

surface boundEary layer with slot efflux. This model, developed fromthe exneri-rnental an"' -cn,-lvticel wor'k of Uroradia, accounts for the

.4

-10-highly complex viscous phen~omena as-ociated with slotted. airfoils.

-'Associated with-n the confluent boundary layer,, a slot-f-low: model

is defined for flow between slot regions.

-combine the inviscid solution with the boundary layer calculations'!The geometry of the 'equivalent airfoil', reflecting local boundaylayer thicknesses, is successively define oe a aibennb f

iterations until. oressure distributions have converged to a steady,c ondaitiont. As the work described in this renort was more of arqualitative nature than uantitative, only one iteration betweenvi-Scous and inviscid calculations is miade, in the interest ofreduced computation time. A com-orrison of -predicted and experimenta.pressure coefficients using the progrum for a two-element airfoiiappears In figure (15). A listing and descrintion of nrograminuis given in ainnendix (i) for the 4-eleiaent airfoil considered in

this work.

C4?4K4

: , - 1I1 -

S0PTIMIZATION PROCEDUREJ 4..l Geometry ModificationsIL

The choice of evolution strategy required the preliminary

development of the following:A. Scheme for Random Geometry iodifications

B. Mutation Iength Control

C. Criterion for Convergence to OptimumD. Initial Airfoil Geometry

Of fundamental importance to the effectiveriess of evolution strategy

is the combination of largo and small mutation lengths. If only

small., sizes are used. the number of imoroved cases will. be hig.h, but

* the rate of progress will be slow. On the other hand, if only large

sizes are used, the optimum mey be entirely skioned over. For this

reason, new cases were divided into 3 groups, one having all mutation

lengths of a base value, the second group having lengths of 505 of the

base value, and the third groun having lengths 150O of the base.,

The control. of the value of this base mutation length will bedescribed below. For each new geometry, then, 18 new configurations

are generated, divided into 3 groups of 6 cases each. The choice of

4 18 cases per set was somewhat arbitrary, but it was felt that this

would be a good trade-off between effort and resulting improvement.As the optimization nrocedure was carried out, it was found that 18cases was a safe choice, as the orogran used to determine C1 and 0 dhad certain limitations. Tn oxoeriment, a smaller number of eases

per set migbt be desireable.

o.4 All mutations were generated by a computer program, whoselisting appears in aonendix (ii) and is described by its logic

diagram in figure (16A). The random choice requioP by this Oro ranis made using a. random number generator that determines. for each

case! -ich of the 10 -wrameters is to be modified, and in the event ofa modification, whether the increment is positive or negative. As

seen in figuire (160), two coliumns of mutation lonoths are printed.The first gives the three lengths used to modify the x,y position

4 of pivot points. These values reoresent a percentea-e of the range

defined for each nivot noint, as shown in fi-nire (16B). The second

colu'nn ip derrees of muitation for flan deflection and angle of attack.

The Random number generator is a comouter system function that4

returns an even distribution of rep! values in the range 0.0-1.0.

t k £ descrintion of its algo&ithm is given in awnendix (iii).

-1A

12 -

4.2 Mutation Length Control -"

As described above, three modification lengths are xsed 'for

each new set of configuprations. The control of the base, or

middle length is critical, as rapid convergence to. an optimum requires

an expansion of mutation size far from the optimum and a dompression

close to it. In this optimization problem, the base length for a

new set was controlled by the best case from the previous set.

For instance, if case 1 , a member of the third mutation length group:,

is the best in its set, its mutation length will be used as the base

value for the next set. In this way, increasing lengths will be

favored. On the other hand, if case 3 is the most improved, smaller

lengths will be favored in the next set. Using this scheme, the

optimization straterzy self-adjusts to the distancn from convergence.

4.3 Convergence Criterion

Concerning the question as to when convergence has been

reached, a criterion is used that recognizes that mutation lengths

will automaticelly compress as the noint of convergence is apnroached.,

When the modification of an im-nroved configuration fails to increase

0l/Cd for a minimum C1 , the airfoil. has 'apnroximately converged'.For absolute convergence, an additional sot of configurations is gen-

erated, based on the best case of the set that failed to improveover the point of apnroxiirate convergence. If acraiuq there is no

improvement, convergence is said to exist.

The procedure proposed to determine the envelope of aero-

dynamic coefficients is comprised of the following two basic steps:

1. TLaximize 01,/Cd while maintaining Cl g'eater than Clmin ,the value obtained from the initial geometry

2. Nove across the envelone from the maxima point found from

ste- 1 to the maximum Cl limit, when flow separation occurs

Results from this -rocedure are presented in section 5.

4.4 Initial Airfoil Geometry

The initial airfoil configuration was an arbitrary but

realistic placement of elements so as to guarantee good flow con~i-

tions through -the slots. The value of the 10th parsmeter, angle of

attack, was determined from the polar shown in figure (14). t.n

angle of 0.50 was used as it resulted in the maxi.-um Cl/Cd. This

initial configuration and angle of attack thus set the value of

Slmin at 2.71. In all subsequent program calculations, mach numberand Reynold c number, refetrenced to.'an-airfoil chord of 350mm,

were 0.125 Ltnd 1.26.106, respectiv-ly. lement .rofiles were

-13 -

taken from reference 14, with aw ropriate moaification to meet

tho requirement of smooth contours.

.!

71

"A

Itt

-14-

5 RE SUMIS -

5.1 First Three Configuration Sets

The first sten of the optimization procedure, the maxim-

ization of C1/C d with the constraint of was started at

the initial geometry. Figure (17) shows the scatter of data.)

points for the first three sets (54 6ases) of airfoil configurations.

: ome points have been omitted for clarity. Circled data points are

'best cases for 'each set; these cases serve as the bases for subse-

, quent modifications. "' The automatic mutation length &ontrol reacts

well in these first sets' far from the point of convergence, with

an increase in bese lpnghb hetwqen sets nn. and two and a onntant,base length between sets two and three. As exnected, points scattere 4in the innediate vicinity of the initializing case were from the firs"

mutation length group, and points scattered further away were from the

second and third groups. As seen in figure (17), the ratio Cl/Cdapnears more sensitive to airfoil modification than C1. This effect

continued for all other sets, with most points locate,,' in a narrow

Cl band between 2.0 and 3.0 and a scatter of other points in a rel-atively low Cl/Cd - low Cl region. The tabulation of all configurations

(37 sets, 666 cases) is given in Appendix (iv) to-ether with initialand final configuration data.

5.2 Descrixtion of Complete Ontimization

• .Figure (18) showis the result of continued maximnization'of

Cl/Cd, plotting the best case for each of 37 sets. Up to set 11,

mutation lengths genorclly expanded or remained constant. Sets

12-14, however, resulted in a comprossion of lengths, and convergence

was thought to be imminent. Beyond set 14, lengths began to expand

again until the base value stabilized at approximately 2% and 10for pivot point location and deflection angles, respectively. This

effect was considered to be a 'small barrier' in the optimization

procesr, which the evolution strategy was able to step over with the

4 larger of the three mutation len,ths. A gradient strategy-based

technioue would have converged to such a barrier, or local maximum.

Beyond set 22, the accuracy of the aerodynamic coefficients

beca~ne questionable, resulting in erratic behavior of the optimiza-

tion nrocess. When the nrocess was continued without the Clmin

constrrint, lrge ch.nges in Cl/Cd and Cl occurred for relatively

k.-. - small -eometrv modifications. As an attempt to return to smaller.variations of coefi~_ci-ens, the nrocess was restarted at set 24

with a: forced reduction of step Sizes, as shown by the dotted lines ,

of figure (18). This reulted in sets 27 and 28 Which againyere

characterized by large variations of cefficientq for smali m di i

cations. This sequence of forced length Compression and optiniiza-

tion restart wb.s continued to the 37th, but by this point values-

of Cl/Cd were unrealistically high, even for the simplified two-

dimensional airfoil model without induced drag or body interference

effects. Indeed, during the process of optimization, numerous

configurations resulted in unsuccessful program termination or

extremely high values of Cl/Cd,and in several cases, negative 6d, '

The reasons for this inaccuracy are thought to be one or a com-

bination of the following:

1 Invalid geometry; i.e., overlap less than 1% of chord

2 Confluent boundary layer or slot flow model failures for• small slot cross-sections

j 3 ~Inadeauate verification of roambauhsfr4-elementsmall of ~program by authors for4-lenairfoil confiurations (note-the frenuency and severity of

inaccuracies were substantially les for the preliminary

optimization of a 2-element airfoil)

In the further presentation of results, sets 21-22 are taken to be theL aproximate limit of data accuracy, althouph all points are shown.

! t Cl/Cd and C1 are shown with rosnect to set number in figure

(19). The points again nre from the beet cases for each set. As

seen in this figure, the result of 22 sets of c onfigurations and_ 74

296 total cases, Cl/Cd has been increased from 4.28 to 36.6, a factor

of 8.55. At the same time, C1 has been increased from 2.31 to 2.59,

a result that evolved by fPvoring the configuration having the larger

Cl when two or more hrd a-orocimately the same Cl/Cd. Increasing

Cl, however, was not the priiary objective of the ontimization.

Because of the li-iit of data accuracy, the optimization process

nas terminate. at the 37th set, thus riakinr it impossible to find

the envelope of aerodynamic coefficients and thus carry out the

s,)cond sten of the nropose3 ontimization.

5.3 Initial-Finri onfi.irption Comnarison

P -ure (20) shows nolars generated for the initial configur-1 ation end the configurations rasulting from the process of Cl/Cd

Smaximization. The resulting draiatic improvement in aerodynamic

-aerformance is aoarent. During the process of ootimization, a num-

-, ber of configurctions produced values of Cl/Cd and Cl that when plot-

ted fell to the right of the polar for maximized Cl/Cd. A polar

-16-was generated from the point th.t was furthest to the right and. i

denoted as the 'optimized Cl configuration'. This c6nlfiguration, it

is interesting to note, was a member-of the optimum dl/! set,

number 22. The dotted lines have been added to these poiars to show

the real flow behavior for multi-element airfoils such as shjown, inI figure (2).

A comarison of initial and final configurations is given in

figure (21). A noticeable effec't of the. optimization is the

reduction of slot area and improved contour smoothness of the trail-

ing edge flans. Both of these effects have been shown experimentally

to improve aerodynamic performance. The same changes did not occur

for the leading edge flap. It was found thiat the multi-element

airfoil program was very sensitive to reduction of slot aree. of the

leading edge device when both trailing edge flaps were present. It

is suspected that the reasons for this sensitivity are the same as

I those mentioned above.

A comparison between the optimum Cl/Cd configaration found

using evolution strategy and that suggested by the eripirical reco-n mandations of reference 5 was a~ttemapted', but the slot geometries

Irequired ere not capable of bing modelled byt airfoil program,

Any change from these recommendations so as to suit program inputj requirements would have resulted in a meaningless comparison.

;! JA

V AI-.7

CONCT~USIoUS ,-

Although the inaccuracy of the program used to determine a4ero-

dynamic 6oefficidnts prebluded the full solution of the airfoil per-

formance envelope, the problem of maximizing Cl/Cd was easily handled

by the evolution strategy.-based optimization techniaue. A review of,

existing techniques showed that evolution strategy is the best choice

for systems described by 5 or more degrees of freedom, due to its

relative incomplexity, flexibility to handle a wide variety of prob-

lems, and to the knowledge that the converged solution is an absolute

optimum within the range of variation of these degrees of freedom.

The problem exnerienced with the use of the above-mentioned

program reinforces the opinion that as of yet numerical solution

methods are severely limited in their range of apnlication.to complex

engineering systems, such as for a multi-element airfoil with itolarge number of configurations. The value of numerical methods,however, lies in their ability to provide preliminary results to be

used as a starting point for further design refinement in thelaboratory. Of course, as long as the limits of numerical methods

are not exceeded, experimental optimization may be well predicted.

The next logical step related to the work described here is the

apnlication of evolution strategy to optimization in a wind tunnel.

A setun is envisioned, similar to those of references 7 and 8, where

modifications to the airfoil, analysis of aerodyhemic coefficients,

and optimization procedure are carried out by on-line computers.

The importance of using the wind tunnel is that all configurations of

an airfoil are capable of being tested, as onnosed to computer model-

imposed restrictions such as slot geometry and separation-free flow.

An important note is thr-t multi-element airfoils often are designed

to perform with senarated flow and negative overlap, two conditionsthat can not exist for the successful use of the program of refer-

ence 13.

The effect of increasing Cl/Cd through the modification of14 I slot geometries while at the same time maintaining adeouate C1 isbest seen for the design trade-off between take-off and cruise flight .

Increascd Cl/Cd at take-off allows increase6, rate or angle of climb,

decreased overall distance to reach a requireO screen height,

increased teke-off weight for the same runway requirement, or a

combination of the three. An increase in Cl/Cd at take-off also has

a stronr, influence on cruise flight efficiency througa the reduction

of the installed nower needed for take-off.

I- ia -

By using an evolution-based strategy, the converged 6ptimum

is an 'absolute' optimum within the range of Variations of the

degrees of freedom era particular system. The question of Whetherconvergence has been obtained or not is thus eliminated'i this W~as'

not the case for the design Of the Boeing 73.This is seen in

figure (4) by the 3 increments in 0lmax for its high-lift devices.

At each point, the design was probably considered to be aia optimm!

Other areas of aerodynamic design that should lend themselVes

well to evolution-strategy optimization arc. the wing-fuselage

junction, empennage, sunercritical airfoils, engine nacelles, and

fuselage rear up-sweep. These design areas are currently anproached

using procedures similar to those of flap design, the wind tunnel

"*! testing of a relatively small number of different configurations.

Each of these problems is described by many degrees of freedom, a

characteristic that has been shown to be well-suited to evolution

strategy. The technioue described in this work is by no means restr-icted to aerodynamic desirn, however. A wide range of annlications 7

can be imagined; the examples included in this roort are only a few

preliminary oases whose results show promise for this new technique to

be used as a powerful tool in the design process.

One can consider the development of evolution strategy as a

result of man learning from his observations of nature. Figure (22)-

depicts what might result if nature learns from the Perodynamlic

achievements of man.

#1

I"I

High Speed Aircraft," the Boeing Company, R en-6on, Yladhington,D6-.16168, June 1.965.

z 2. Steiner, J., et.a.., "Case 'Study in Aircraft Design: The j3 6eIrn t727,t" AIAA Professional Study Series, Sept 10,78.

3.Bruner, G., "Le Boeing 747,"1 DOC-AiR-E.sp-AcE., No.112, Sept. -196,8,YP9. 7.

4. Technical Staff, AEDO (USAF) and ARO, Inc., edited by

J.D. Whitfield? J.P. Hartin, and S.R. Pate, "Aerodlynamic

Testing-a Look at Future Requirements," AIAA No.78-765, 1978.

5. Ljun ,strbm, B.Tj.G., "Exp~erimental High Lift Optimization ofIA] Mulutiple Element Airfoils," Proceedings of the AGARD Conferenceon V/ST0TL Aerodynamics, AGARD-OP-143, Apr. 1974, pn. 13-1 - 13-16.

1 6,. Rottie, I.H., "Air Transportation for the 1980's: Aerodynamic

Design," from collected Daners of the 1975-76 Seminar Series

supnorted by the 1Minta N'artin Fund for Aeronautical. Research,University of 11,aryland, College Park, De-oartment of AcrosiaceEngineerinw, June 1976, w). 115-152.

7. Levinsky, E., Schanpwelle, R., and Po-Lntney, S., "Airfoil~ i Ontiinization Through the Adaptive Control of Camber and

Thickness, Phase 11: Transonic Wind Tunnel Test and ProgramiSumi ,ary," General Dlynamnics Corporation, Convair Divisioh,CASD-NSC-75-004, Sept. 1975.

8. Iievinsky, E,.S., 1'allko, R.L., "Semispan Wind Tunnel Tert of a

Comnnuter Controlled Self-Optimizing Flexible Technology Wing,"AIAA Daper 78-786, 1978.

9. Hicks, V% , F~urman, E. , and Variderplaats, G., "An Assessment ofAAirfoil Design by Numnerical Ontirnization," NASA TV~ X-30921

July 1974.

10. Rechenberg, I., "Cybernetic Solution Path of an Expnerimiental

Problem," Royal Aircraft TPstablishment Library Translation

* No. 1122, Aua. 1965.

AA

REFERENCES (cont'd) -2

11. Rechenberg,1. , "Evolutions strat egie , Optirnierung TeclhisoherSysteme nach Prinzi-pien der Biologischeni Evolution," No. 1"5

of Problemata Series, Friedrich Frommanin Verlag, Stuttgart-Bad Cannstatt, 1973.

12. Discussion between Professors I.Rechenberg aand R. Siu Cf',

Technischen Universitat Berlini, Feb. 1980.

13. Stevens,, W., Goradia, S., and Braden, J., "1Nathematicai Mdodelfor Two-Dimensional I~ulti-Component Airfoils in Viscous Flow,"TLockhond-Georgia. Company, ?Marietiua, Georgia, NASA CR-1843 and

Su~nlement to NASA CR-l843, July' 1971.

14. Hertha, K. and Scheerer, J., IlKonstrukt ion des ZICP-Panelhalbmodellr, esserschmitt-Bd5lkov-Blohm- Gmbh, Hamburger Flugzeugbau, ILFK 7512,v

A Ergebnisbericht Nr.19, Arrbeitspaket:E-I.4, April 1977.,

7 ~ 15. Schwefel, Hans-Paul, "Numerische QOtimierung von CompOuter-1, odd11en mittels der Evolutionsstrategie, volumne 26,

I Interdfisciplinary Systems Research (ISR 26), Birkh~Iuscr Verleg,Basel, 1977.

44

A

TAKEOFF

GROUND F WIUST CLIMBOU-JI QSRUN . WEIGHT ]WEIGH

WING I OADING I IFTt TAKEOFF L AG

CD

ROLLING FRICTION

At TITIIDF{I OW L ,kx

CLIMBOUT - , - HIGH L/FLARE-- _ , oc ,×,LAR 7 HGH L MAX

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TAKEOFF AND LANDING FAERODYNAMICS

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TAKEOFF

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0 727 TAKEOFF") 727 LANDING

CL 20

I C0 50-- HEEORETICAL DRAGt ' " \FLAPS UP\~J-A PPROXIMA TE2 "0 ENVELOPE

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40'o FLAPS"'

0.0 0.5 1.0 1.5 2.0 2.5 3.0L IF T COEFFICIENT- CL

B F

BOEI:"l NG-727 DRAG POLARS FIGURE 2

-- A - A

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BARL DEGREES OF FREEOM

A.LHIG-LIFET DEVI DETAILS3

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101-_______0 10 - 20

FOUR ELEMENT A IRFOIL OP TIMIZA -

TiON BY EMPIRICAL METHOD

• ... . .. i i A

106 100 YEARS 1000

SHUTTLEF 111 ,.

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TIME, HR F15

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10 WRIGHT FLYER DC.010 -I I I I I I I I . .01

1900 1920 1940 1960 1980 2000

TIME, YEAR

WHAT PRICE ACCURACY ?

.... 1010 CONFIGURATIONS

1 MEASUREMENT/30 SECONDS

.... TOTAL TIME 9.51.10 5 YEARS

... TOTAL COST $8.331010

I: IWIND TUNNEL TEST PROGRAM FIGURE 6TRENDS

i .

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AIRFOIL _

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6 :

IDEG START OF TEST2 END OF TEST

PROBLEM:.MIN. CD~A A A A A 'FOR CL=O. 5, M=0.85

CL

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OP TIM UM

CD START OF TESTEND OF TEST I

0 5 10 15 20 ITERATION NUMBER

V THREE- DIMENSIONAL FLEXIBLEA iRFOIL OPTIMIZA TRON FGR

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cI 0 -110* 1 -.4 11

0 10 20 30 40 50 60 70 80 90 100

- - IITIAL X/CPERCENT CHORDIIILAIRFOIL., CD=O.OO& 3, V='0.721I

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0 10 20 30 40 50 60 70 80 90 100

X/C, PERCENT CHORDINI TIA L A IRFOIL, CD= 0.0524, V=0.680

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TWO-DIMENSIONAL TRANSONICFI E9AIRFOIL OPTIMIZATION,. GRADIENT ST

MODEL FOR SIMPLE OPTIMIZATION PROBLEM

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EVOLUTION STRATEGY EXAMPLES IGUE1* (CONTINUED) -

N' b

AIRFOIL WITH 10 PARAMETERS FIGURE 13

5.0 C=.lCL~ 2 ~S

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100 A-bB-PRFILE Al-725-146 SbOTTED KRUEGER DOUBLE-SbOTTED TE FLAP OP'IlIZATION200 4 85 1300 1 23 23400 1.610) -Q.4200500 0.0000 0.0344 0.0689 0.1378 0.2067 0.2756 0.3415 0.413460O 0.5512 O.0b90 0.8261 o.qbb 1.1024 1.2402 1.3780 1.51577o0 1.ob35 1.7913 1.9291 2.0670 2.2047 2.3,125 2.4459800 0.4134 0.5994 Ob.683 0.7441 V.992 0.b543 0.8h19 0.8951900 0.9232 0.9232 0.9163 0,902o 0.8888 0.8o12 0.8406 0.8130100Q 0,7923 0.189 0.7303 0.6890 0.b407 0.5925 0.55121100 0.0000 0.0)344 0.0689 0.1378 0,2067 0.275t 0.3445 0.41341200 0.5512 0.6890 0.828 0.964 1.1024 1.2402 1.3780 1:51571300 1.0535 1.7913 1.9291 2.0670 2.2047 2,3425 2,44591400 0.4134 0,2070 0.1378 0.0689 0.0207 0.00(9 0.0000 0.02071500 0.09o5 0.24)70 0.2t94 0,3445 C,3258 0.4272 0.4b1b 0.49bi1500 0.523b (0.

b443 0.5512 0.b512 0.5512 0.5512 00b512

1760 2 37 37I80 -0.50O0 0.0000 12,0000 0.00001900 0.000 0.0344 0.1033 0.1722 0.2411 0.3100 0.3789 0.44782000 0.5107 0.58tb 0.bl4 0.8304 1.0480 1.3049 1.6150 1.989U2100 2.4145 2.9929 3.t571 4.35.13 5.39868 t4433 7 140b 7.735d2200 8.2181 80.122 H.9264 9.1901 9.0870 10.1004 10o003 11.21102300 II.559b l.l0 12.2252 12,b091 12.53942400 0.0000 0.16b4 0.2o32 0.3238 0.3707 0.4079 0.4396 0.46852500 0.4919 O.lb4 0.5374 0.5843 0.h339 0.0b48 0.7372 0.789b2000 0.8433 0.8929 0.9384 0.9701 0.9894 0.9156 0.9467 0.90072700 0.8654 0.8254 0.7896 0.7551 0.6862 0.6228 0.5314 0.4258200 0.3596 0.2921 0.2301 0.1791 0.17222900 0.0000 0.0344 0,1033 0.1722 0.2411 0.3100 0.3789 0.44783)1p0 0.517 0.5d5o .o~ol4 0.8364 1.0486 1.3U49 1.6150 1.989V3100 2.4445 2.4929 3.o571 4.3513 5.39b8 ).4433 7.1406 7.73583200 8.2181 U6122 8.9264 9,1951 9.6870 10.1004 10.6033 11,21103300 ll.bb9o 11.9110 12.2252 12.5091 12.53943400 0.0000 -0.2287 -0.3514 -0.4203 -0.4713 -0.5140 -0.5484 -0.57873500 -0.6063 -0.b311 -0.6559 -0,70bg -0.700b -0.8157 -0,d681 -0,9163o0o -0.9577 -0.9894 -1.0031 -0,9921 -0.939N -0 8312 -0.7427 -0.6476b

3700 -0.6o .4974 -0.4437 -0.3982 -0.3142 -0,246! -0.1o95 -0.08973800 -0.u551 0°0344 0.1378 0.1o54 0.172239v0 2 23 2340U0 -0.10)00 0.2200 2.06b9 -0.13784100 0.0000 0.013d 0.0413 0.089( U.1516 0.2136 0.275t 0.41344200 0.5h12 0. 890 0.H2b8 0.9646 1.1024 1.2402 1.3780 1.51574300 1.653b 1.7913 1.9291 2.0O09 2.2047 2.342 2.48034400 0.0000 0.09bt 0.137b 0.129 0.2343 0.2756 0.3031 3445451 0 0.3l20 0.385d 0.3927 I1.3g51 0.3'179 0.3583 0.3307 304o00 U.27,h 0.2343 0.1929 0.1516 0.1033 0.0482 0.00004700 0.0000 0.013" 0.0413 O.Ob6 ). IbIb 0:213o 0.27bo 0.41344800 0.5512 0.6890 0.8208 09646 1.1024 1,202 1,31h0 1,51574900 1.o53b 1.1913 1.9291 2.Out9 22047 2.34!! 2.4803bu0O 0.0000 -0.0344 -0.0ob9 -0.1033 -0.1240 -0.139 -0.1378 -0.1378b100 -0.1419 "(.146b -0. ,1 51 -0.1526 -0.14o1 -0.1419 -0.13095200 -0.1033 (0.0)5o -0.04b2 -0021 -0.0138 "0.0069 0.00005300 1 11 175400 -0.100u o.00005500 0.)()00( 0 .0202 0.0524 0.1047 0.1571 0.2094 0.3142 0.4189bbo0 0.52 6 0.n283 0.8378 1.0472 1.2567 1.3180 1.7913 2.2047b'lOO 2.75595800 0.0000 0.082 0.1171 0.1557 0.1846 0.201 0.2425 0.265911 5900 0,2797 0.2s60 0.2d6b 0.2673 0.2343 0,2130 0.140o 0.07996000 0.0000o1OO 0.0000 0.0262 0,0524 0.1047 0.1571 0.2094 0.3142 0.4189b200 0b23o 0o2b3 0.8318 1.0472 1.25o7 1.378) .17913 2.2047b300 2.75596400 0.0000 -0.0482 -0,020 -0.0758 -0.075H -0,0690 -0 0620 -0.0413b500 -0.0276 -0.02u7 -0.0069 0.0009 0.013u 0.0207 0.027b 0.0207

n"-600 O 000u.... 1 000 2

6800 1 1 2 1 -22.87006900 3 1 2 2 11.bo07000 4 1 3 2 15.loOO7100 1£ .,: 7200 0.00 IIT

S7300 SLOT DATA74uj 0.120:* l"7boo 1.1483 0.08337t)Ou 51U.70 1.20 0.710 1.0

8300 0 0

8400 0 0 07900 OI .... . ......

8600 [HE ENE

T\'

200 C.3ii 0 C ~ 5U( IR Al- % i,-1 2 Pu S :u~q iUUI 'IC lit, &hj I Iu h tL

400 C kj KD1500 C600 c CA1 P Az fli. ., I f I 1.,I. 6 A ~ O If Y- 1.0 1'AiitE RA11~ t-7 0 u 0. 1% 'iX I M U 11 1) .,I1'I' IL114 ibix iGrI ~(.;R~dS

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1400 C 'I A -U1CWLJ u., 4~)P1 Jvo k'Ulu.1V15001 C Y-;)f ~b J'J 0. N 1)L 111.0 i1i;'

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3300 L'c'r. III k'IU" #I P Ii

f 3kiov C 1 ) .wC. a:s I I$ 0u..'ltA ~ ~ L~~

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8100 C8200 C8300 C8400 iF(I,3.bO.1) PAR (bi IL2,L3) =IPAIR(1)+(MULi'~j.NcR(L I)#X I)

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8700 11AL3:EU.4) 'l(il,,j)IA()+MI,~lil(1*28800) 1F(L3.E0.5) PAIdl,lll2,LI3)=IPAR(5)*I-'ITj*C(lj1)*ZI)8900 1'(L3.EAJ~b) PA l,6 3)IA 6 W1p*NR(Ll+IC9000 iF(L3.E0.7) PAI<(I ,I12,b3)=1PAR(7)+(M1i~'L*IJCR(b1)

4 X3 )

9100 IF( L3 .FU. 8 PAR(L1 L2U)iAd)(MiiIJRL1*39200 I F( L3 .EO(.9) PA 1 21 ,3)= A ()+(UL*INR( h14

9300 IF(L3.E0. 10) PARUJl1,b2,L3)=PA(10)+(ULT (IJC(jl+t4iI C) )

9400 U (PAR(bj1 ,I210).Lr.0.0) GOT(J 5009500 GOTO 7509600 600 b4=64+19700 C ------- CHECK\ TI PREVE~NT 1)UP1L1CATT~f' OF 11ILI.JAbj GEOMETRY9800 1F(L4.EO.rwAR) GOTO 4009900 700 A(,,L2,i3)1PAtdt.3)10000 750 CONITINUOE10100 80o COi'TI1UK

F10200 900) CIT'f NIUoE10300u boo CONTINUE10400 WR1TI!. (6, 1100 ) mL)0NI.,1 2 e12 NINC, ICASK10500 1100 FUR 14AT ( ~/ / / , X , IuUMKTH Y MU D I FiCA1T1LUr 12,2/ ,7X Ill=

1 0)0(1) A I IrC Nc UM W ,3X, Iwr ,3,r h'(: x ; kLt ' I

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11200 1240 iTI:(,24)11IC(),UC(1N C

11300 er<iTE (t),IzOO) ((d'ARO) 1=1,10)

lino WRIT'E (6,1300) (uIAR(i)pJ=1,10)

11700 1300 FURMAT ( 1 l0 .(,E. 10 ( ~Ft. 2)

11900 14o0 FWi(MAT (I 114 CS12000 c0 1-100 [,1=1 ,NIwc

12200 bj3=(L1-1)fNCAS1IW+b

12300 WIc1T- 6150 Li13(i~i1L4~l-h0

12400 1500 FUIU4AT (/ f 1 2 1 Xi 1 ,Oh.)/J~ tA

12500 x IcL)= 1,6x* '.L / (I.U= ', o CM, )

~-A12600 1600 CUI~N~iU12700 17M0 C 1)HT14 U h12800 STOP ' ,UD)WICAT1IOi COMPLET' i

12900 NL

;44

AT,1,1"DIX (iii) RAITni, ",G7, ' , , ; ,- ,, r'-.' -'A,,M OR .. ..

D.4.9 RANDU Subroutine

The RANDU subroutine computes a pseudo-random number, as a singleprecision value uniformly distributed in the range:

0.0 .LE. value .LT. 1.0

Format:

CALL RANDU(il,i2,x)

Arguments: 4

il,i2INTEGER*2 variables or array elements that contain the seed forcomputing the random number.

xa real variable or array element where the computed random numberis stored.

Notes:

1 1. The values of il and 12 are updated during the computation to* contain the updated seed.

2. The algorithm for computing the random number value is as2 follows:

If .I=0, I2=0, set generator base

X(n+l) 2**16 + 3

otC~her wise

X(n+l) (2**16+3) X(n)mod 2**32

t -Store generator base X(n+l) in 11,12.

4_4w Result is X(n+l) scaled to a real value Y(n+l), for 0.0 .LE.

Y(n+l) .LT. 1.

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