Numerical Airfoil Optimization

40 Pages·2001·0.97 MB·English

https://ntrs.nasa.gov/search.jsp?R=19790010744 2019-04-03T21:57:11+00:00Z
NASA Technical Paper 1370
NumericalA irfoilO ptimization
Garret N. Vanderplaats
MARCH 1979
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TECH LIBRARY KAFB, NM
NASA Technical Paper 1370
ApproximationC oncepts for
Numerical Airfoil Optimization
Garret N. Vanderplaats
Ames ResearchC enter
M of fett Field, Califorrzia
National Aeronautics
and Space Administration
Scientific and Technical
Information Office
1979

NOTATION
A airfoei ln closed area dividebd y c2
a participation coefficient
i
C chleon rgdt h
dragc oefficient
cD
lift coefficient
cL
pitching-momentc oefficient
cM
pressurec oefficient
cP
-
F(X) designo bjectivef unctiont ob em inimizedo rm aximized
-
f arbitraryf unction of X
(X)
G~ constraint function
[HI Hessionm atrixc ontainings econdp artiadl erivatives
n number od fe sigvn a riables
'5
searcdh i rection in optimization
tt/hc i ckness-to-chorrad t io
-
X vectocr o ntainintgh d e e sigvn a riables
lower boundo nd esignv ariable i
xi
U
upper boundo nd esignv ariable i
xi
-
Y array of airfoilc oordinates
Yi arrayo cf oordinatesd efining a shapef u nction
c1 airfoai ln gloae ft tack
a* move parameteiorn p timization
-
V gradoiep netr ator
A differeonpc eer ator
iii
Subscripts:
i variable number
RS lowsefu ra r c e
max maximum
min minimum
us uspupr efra ce
Superscripts:
0 nodmeisn iagln
4 iteration number
k design number
iv
APPROXIMATIONC ONCEPTSF OR NUMERICAL AIRFOIL OPTIMIZATION
Garret N. Vanderplaats
Ames ResearchC enter
SUMMARY
An efficienta lgorithmf ora irfoilo ptimization is presented. The algo-
rithmu tilizesa pproximationc onceptst or educet he numbero fa erodynamic
analysesr equiredt or eacht he optimum design.E xamples are presenteda nd
comparedw ithp reviousr esults.O ptimizatione fficiencyi mprovementso fm ore
than a factoro f 2 are demonstrated.I luchg reateri mprovementsi ne fficiency
are demonstrated when analysisd atao btainedi np reviousd esigns are utilized.
Them ethod is a generalo ptimizationp rocedurea nd is notl imitedt ot his
application. The method is intendedf ora pplicationt o a wider angeo fe ngi-
neeringd esignp roblems.
INTRODUCTION
Numericalo ptimizationt echniquesh aveb een shown top rovide a versatile
toolf ora irfoild esign. The usuala pproachh asb eent oc ouplee xistinga ero-
dynamica nalysisc odesw itha no ptimizationc odet oa chievet hed esignc apa-
bility. Thep rimarye fforth asb eend irectedt owarda pplicationo ft hese
techniquest o a widev arietyo fd esignp roblemsw hileu singi ncreasingly
sophisticateda ndt ime-consuminga erodynamica nalysisp rograms.
The costo ft hisa utomatedd esign,w hereby a veryt ime-consuminga nalysis
program is executedr epetitively( perhaps several hundredt imes), is neces-
sarilya ni mportantc onsideration when judging the practicality of these
techniques.P erhapst hes implestm eanso fe stimatingc ost is by the numbero f
times thea erodynamica nalysisp rogram is executedd uring a designs tudy.
That is, for a givena erodynamicsp rogram,t hec osto fo ptimization is a
directf unctiono ft he number of times the program is executedo nt he com-
puter.T herefore,a nyi mprovementi no ptimizatione fficiency is directly
measurable in design cost savings.
Very little efforth asb eend irectedt owardi mprovingt hee fficiencyo f
thea utomatedd esignp rocess as appliedt oa erodynamicd esign. The principal
improvement tod ateh asb eeni nt hem ethodo fd efiningt hea irfoil.I nr efer-
ences 1 and 2, polynomials were usedt od efinet hea irfoils hape,w itht he
coefficientso ft hep olynomialb eingt hed esignv ariables.I nr eferences 3
and 4, and ins ubsequentw ork,t hesep olynomials were replacedb ym oreg eneral
analyticalo rn umericallyd efineds hapef unctions. The result was ane ffi-
ciencyi mprovemento fm oret han a factoro f 2, togetherw ithi mproveda irfoil
definition( ref. 3). However, efficiencyi mprovements are still needed if
numerical airfoil optimization is to become an economically feasible design
approach when using sophisticated aerodynamic analysis codes.
Thep urposeh ere is top resent a techniquet hati mprovest hed esigne ffi-
ciencyb ya notherf actoro f 2 orm ore.T heb asica pproach is tod evelop
approximationst ot hed esignp roblemu sing a minimala mounto fi nformation.
Thea pproximatingf unctions are used in the optimization and the resulting
design is analyzedp recisely.T hisa nalysisi nformation is addedt ot he
availabled ataa ndt hep rocess is repeatedu ntilc onvergencet ot he optimum
is achieved.( Thei deao fu singa pproximationc onceptsi na erodynamico pti-
mizationo riginatedf romt heo bserveds uccesso f similar techniquesu sedb y
Schmita ndM iura( ref. 5) int hef ieldo fs tructuralo ptimization.)
To provide a backgroundf ort hem ethod,t heb asicc onceptso fn umerical
optimization are firstp resented.T his is followedb y a descriptiono ft he
presentm ethodo fc ouplinga na erodynamica nalysisp rogramt oa no ptimization
program fora utomatedd esign. The concepto fo ptimizationb ys equential
approximations is thenp resented,f ollowedb y a morep recisem athematical
formulationo ft hem ethoda nd a summary oft hed esigna lgorithm.E xamples
demonstratet hee fficiencya ndr eliabilityo ft hem ethoda ndf inally, some
oft hei mplicationso ft hism ethodf orf utured evelopment are discussed.
OPTIMIZATION CONCEPTS
Assume the airfoil is definedb yt her elationship
where is a vectocr ontainingt h eu ppear ndl o wecr oordinateso tf h e air-
foil and T i are shapef unctionst hat may themselvesd efinea irfoils. The
. . .,
coefficients al, a2 an are referredt o as participationc oefficients.
Now assume it is desiredt of indt hea irfoilt hatm inimizest hed ragc oeffi-
cient CD withc onstraintso nl iftc oefficient CL, thickness-to-chordr atio,
t/c, etc., at a specified Mach numbera nd angleo fa ttack. The participation
coefficients al - an are thed esignv ariables,a nd w i l l bec hangedd uring
theo ptimizationp rocess. The n-dimensionals paces pannedb yt hed esign
variables is calledt hed esigns pace.
The optimizationp roblemc anb es tatedm athematically as:
Minimize
CD
subjectt o
2
(t/c) 2 (t/c)min
. . .,
where CD, CL, and t/c are functionso f al, a2, an. Thisc anb e
generalizedt ob e:
Minimize
subjectt o
J
x U
Xi G i = l , n (7)
i
x
where is a vectorc ontaining thed esignv ariables, ai, i = 1,n.T here are
a totaol f m constraints. The liftc onstraint of equation (3) is writteni n
thef ormo fe quation (6) as
1 -- cL G O
'bin
Similarly,f rome quation (4)
The parameters Xi' and X iU ofe quation (7) are referredt oa ss ide con-
straintst hat l i m i t ther egiono fs earchf ort heo ptimum.A lthoughs ide
constraintsc ouldb ei ncludedi ne quation ( 6 ) , they are usuallyt reateds epa-
.
rately for convenience and efficiency
Theo ptimizationp roblemo fe quations (5)-(7) is quiteg eneral.I f it
is desiredt om aximize CL with a constrainto n CD, -CL is minimized.
Also,t hec onstraint set ofe quation (6) is notl imitedt oc onstraints at the
designf lightc ondition.W itht hisf ormulation,t hea irfoilc anb ed esigned
at a given flight condition with constraints at other flight conditions so
long as thea ppropriatei nformation is calculatedd uringt hea erodynamic
analysis. If thei nequalityc onditionso fe quations (6) and (7) are satis-
fied,t hed esign is saidt ob ef easible.I fa nyo ft hesec onditions are
violated,t hed esign is calledi nfeasible.
The optimizationp rocesst ypicallyp roceedsi na n iterative fashion as:
p
xq = xq-1 + a*
(10)
An initial design, go, mustb ep rovidedw hich may or may not defi-n e a feasible
design. The superscript q is thei t eration numbeVr .e ctor Sq is the
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search- direction and a* is a scaler determiningt he move distance in direc-
tion Sq. The notation a is usedf orc onsistencyw ithm athematicapl ro-
gramming literature ands houldn otb ec onfusedw itht hea irfoila ngleo f
attack.
Ifg radientm ethods are used,t heo ptimizationp rocessc onsistso f two
sq
steps. The first is determinationo f a move direction that w i l l improve
thed esignw ithoutv iolatinga nyc onstraints;t hes econd is calculation of
a* sucht hatt heo bjective is reduced as much as possiblei nt hisd irection.
This may be understood by considering a two-variabled esignp roblemw here
CD is minimizeds ubjectt oc onstraintso n CL and t/c. A hypothetical
problem is shown in figure 1 whichs howsc ontourso fc onstantd raga nds hows
the CL and t/c constraintb oundaries. Assume ani nitiald esign is given at
point A, withn o active orv iolatedc onstraints.U singg radientm ethods,t he
x
processb eginsb yp erturbinge achc omponento f tod etermine its effecto n
theo bjective.T hat is, theg radiento f CD is calculated by finited iffer-
enceu sing a singlef orwards tep,a ndt heg radientv ector is constructed as
-
VF = ?C =
D
It is obvioust hatt heg reatesti mprovemento ft heo bjectivef unction is
achieved b-y mo-v ing in the negative gradient, or st-e epest descent direction,
so that S = -VCD. Knowing thes earchd irection, S1, the scalar a* that
w i l l minimize CD int hisd irectionm ustb ef ound.T his is a one-variable
minimizationp roblemS. everal somewhat arbitrar_yv alueso f a are defined
+
andt hea irfoil is analyzed at eachp oint, = Xo as1. A polynomial is
usuallyf itt ot hesep ointsa nd a morep recise a = a* is calculated at point
B inf igure 1, endingt hef irsto ptimizationi teration. The secondi teration
beginsb ya gainp erturbingt hed esignv ariablest oo btaint heg radiento ft he
-
objective. Using the conjugate direction algorithm of Fletcher and Reeves
(ref. 6) a new searchd irection, S2, is foundw hich w i l l againr educet he
objective. A search is performed int hisd irection,l eading to point C,
completingi teration two. At C, thel iftc onstraint is active (Gj = 0) and
a direction is foundt hat w i l l reduce CD withoutv iolatingt hisc onstraint.
The gradiento fb otht heo bjectivea nd active constraint are calculateda nd
s3,
a new searchd irection, is foundu singZ outendijk'sm ethod of feasible
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directions( refs. 7,8). The process is repeatedu ntil a design at E is
obtainedw heren od irectionc anb ef oundt hat w i l l reducet heo bjectivew ith-
outv iolatingt hec onstraintsa ndt hisd esign is called optimum.L ogic is
includedi nt hea lgorithm so that if an initial design is defined at point F,
the constraint violations are overcome to yield a final design at point E.
The optimizationp rocedured escribeda bove is essentially that used in
the CONMIN program( ref. 9). In a typicald esign,a bout 10 iterations are
requiredt oa chievea n optimum design.F ore achi teration, n aerodynamic
analyses are usedt oc alculatet her equiredg radienti nformationb yf inite
difference. To determine a* requiresa na verage of threea nalyses so that
+ +
a total of1 0(n 3) = 10n 30 aerodynamica nalyses are requiredf oro pti-
mizationf or a singlef lightc ondition.A lthoughq uitee fficientf roma n
optimizationv iewpoint,t hat many executions of a sophisticateda erodynamics
programc anb ev erye xpensive.T herefore, it is desirablet or educet he number
ofr equireda nalysesa s much as possible.T hisi mprovementi no ptimization
efficiency is thes ubjecth ere. The technique w i l l bed evelopedb yf irst
reviewingt hea pproachc urrentlyu sedf ora erodynamico ptimization.
Previous Method ofA erodynamicO ptimization
At thep resent time, most airfoilo ptimization is performedb yc oupling
thea erodynamicsp rogramt ot heo ptimizationp rogram as shown in figure 2 .
Each time theo ptimizationp rogramd efines a new design, either for finite
differenceg radientc omputationso rf ord etermining a*, thea erodynamics
program is calledf or a completea nalysis.F ort hee xampleo ff igure 1, a
set ofa nalyses is performed as indicatedi nf igure 3. Duringo ptimization,
at iteration j , very little informationf romp reviousi terations is used.
- s2
At point B inf igure 1, thev ector S1 is usedt oc alculate so that,i f
no constraints are active,p riori nformation is usedH. owever, ifo neo r
morec onstraints are active orv iolated( theu suals ituation), no prior
information is used.
It canb ea rguedi ntuitivelyt hat all calculatedi nformations hould
beo fv aluei ng uidingt heo ptimizationp rocess.F urthermore,i n a design
study,n umerouso ptimizations are usuallyp erformed.F ore xample,o ne
optimization may bed one tom inimize CD withc onstraintso n CL and, later,
anothero ptimizationd onet om inimize CM withc onstraintso n CL and CD.
It may bee xpectedt hat,b ecause many airfoils were analyzedd uringt hef irst
optimization, a secondo ptimization at the same flightc onditions hould
utilizet hisa vailablei nformation. Onwe ay to do this wouldb et oa pproxi-
mate ther equiredf unctionsu singa vailablei nformation.T his would provide
explicitf unctionsw hichc ould now beo ptimizedi ndependento ft he time
consuminga erodynamica nalysisp rogramA. erodynamica nalysis is still used
to improvet hea pproximation,l eadingt o a precises olution.
Theg eneralp rocedure is outlinedi nt hef ollowings ection.
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