[American Institute of Aeronautics and Astronautics 36th AIAA Aerospace Sciences Meeting and Exhibit...

Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

HSCT Configuration Design Using Response Surface Approximations ofSupersonic Euler Aerodynamics

Duane L. Knill*, Anthony A. Giunta^, Chuck A. Baker*, Bernard Grossman§,William H. Mason1, Raphael T. Haftka11, and Layne T. Watson**

Multidisciplinary Analysis and Design Center for Advanced VehiclesVirginia Polytechnic Institute and State University

Blacksburg, Virginia 24061-0203

AbstractA method has been developed to efficiently implement

supersonic aerodynamic predictions, from Euler solutionsinto a highly constrained, multidisciplinary design opti-mization of a High-Speed Civil Transport (HSCT) con-figuration. The method alleviates the large computationalburden associated with performing computational fluid dy-namics (CFD) analyses and eliminates the numerical noisepresent in the analyses through the use of response surface(RS) methodologies, variable-complexity modeling (VCM)techniques, and coarse grained parallel computing. In thisresearch, simple conceptual level aerodynamic models pro-vide the functional form of the drag polar. Response sur-face models are created for the intervening functions (dragpolar shape parameters) instead of for the drag itself. As ameans to reduce the errors in the RS models of Euler solu-tions, optimization results using linear theory RS modelsare used to select the allowable ranges of the design vari-ables. Then stepwise regression analysis, performed usinglinear theory aerodynamic results, provides information onthe relative importance of each term in the quadratic RSmodels. With this information, reduced term RS modelsrepresenting a correction to the linear theory RS modelpredictions are constructed using Euler solutions. Studiesinto five, ten, fifteen, and twenty variable HSCT designproblems show that accurate results can be obtained with

'Graduate Research Assistant, Dept. of Aerospace and OceanEngineering. Current Position: Postdoctoral Research Associate,Dept. of Aeronautics and Astronautics, University of Washington,Seattle, WA, Member AIAA.

t Postdoctoral Research Associate, National Research Coun-cil/NASA Langley Research Center, Hampton, VA, Member AIAA.

* Graduate Research Assistant, Dept. of Aerospace and Ocean En-gineering, Student Member AIAA.

§ Professor and Dept. Head of Aerospace and Ocean Engineering,Associate Fellow AIAA.

'Professor of Aerospace and Ocean Engineering, Associate FellowAIAA.

II Professor of Aerospace Engineering, Mechanics and EngineeringScience, University of Florida, Gainesville, FL, Fellow AIAA

"Professor of Computer Science and Mathematics

Copyright ©1998 by Duane L. Knill. Published by the AmericanInstitute of Aeronautics and Astronautics, Inc. with permission.

the reduced term models at a fraction of the cost of cre-ating the full term quadratic RS models. Specifically, asavings of 255 CPU hours out of 392 CPU hours requiredto create the full term RS model is obtained for the twentyvariable problem on a single 75 MHz IP21 processor of aSGI Power Challenge.

1. IntroductionWith advances in computational fluid dynamics (CFD)

code maturity, grid generation capabilities, and computerperformance, the application of CFD in the aircraft designprocess1 has received much attention. According to Nico-lai,2 about eighty percent of the aircraft life cycle cost isset at the conceptual design stage. Using more accurateaerodynamic predictions early in the design process, whenthe aircraft is initially defined, can result in less time andmoney spent in redesign and an overall improved product.However, the relatively large computational expense asso-ciated with CFD analyses can discourage its application inhigh dimensional design optimization procedures. A newprocedure must be developed to enable the efficient imple-mentation of aerodynamic predictions from CFD solutionsinto high dimensional, highly constrained multidisciplinarydesign optimization (MDO) procedures.

Previous work performed at Virginia Tech3"6 tackledthe computational expense problem by employing a variable-complexity modeling (VCM) technique to the MDO of aHigh-Speed Civil Transport (HSCT) configuration (Fig. 1).This technique utilized both conceptual level and prelimi-nary level models for predicting aerodynamic performanceand structural weights. The simple conceptual design levelmethods were used predominantly in the optimization dueto their low computational costs. More accurate and morecomputationally expensive methods (linear theory aerody-namics and structural optimization) were used to periodi-cally update the simpler models. In this paper, 'linear the-ory aerodynamics' refers to a combination of slender bodytheory wave drag results, supersonic panel results for thedrag due to lift, and a strip boundary layer correction forthe viscous drag. Details are given in Section 4. The VCM


Figure 1: Typical HSCT Configuration.

technique combines the accuracy of the higher level codeswith the computational efficiency of the simpler models.

This procedure was effective in reducing the computa-tional cost, but it was plagued by poor convergence andthe appearance of artificial local minima. These problemswere a result of high frequency, low amplitude variationsin the results from aerodynamic and structural analyses.This numerical noise is common with iterative proceduresor discrete representations of continuous geometric shapesor physical phenomena.7'8 Low amplitude, high frequencyoscillations have been found in wave drag predictions fromslender body theory7 and in panel level drag-due-to-liftresults.9 Numerical noise can also result from subopti-mizations which are performed within the complete opti-mization framework. Balabanov10 et al. show such noisefrom structural optimization and wing camber optimiza-tion procedures.

Response surface (RS) modeling offers a number of ben-efits for aerodynamic and structural design.7'10'12'13 First,the RS models smooth out numerical noise present in theanalyses, improving the performance of gradient based op-timizers and eliminating artificial local minima. Second,use of the RS modeling approach allows the analysis codesto be separated from the optimization routines. This elim-inates problems associated with integrating large produc-tion level grid generators, analysis codes, and post pro-

cessing utilities with the optimizer. It also allows analysesto be performed by experts in the specific discipline onparallel architecture machines.

Response surface (RS) modeling techniques14 have beenused for a number of years to solve complex, computa-tionally intensive engineering problems. In 1964, Powers15

used RS modeling techniques to compute minimum dragbodies of revolution. To use RS models for high dimen-sional problems, one must overcome problems associatedwith the curse of dimensionality, in that the accuracy ofRS models often degrades with increasing numbers of de-sign variables, and the cost of creating RS models increasesrapidly with the dimension of the problem. As the di-mension of the problem increases, the number of terms ina quadratic polynomial RS model increases quadratically.The number of design point evaluations required to ac-curately model the polynomial terms, and the associatedcomputational expense, grows even faster. Balabanov11

dealt with this by applying constant and linear correctionsto the lower fidelity RS models to create RS models fordetailed level structural optimization results. Kaufman16

et al. found that the accuracy of RS models for the wingweight in high dimensions can be increased by using con-ceptual level analyses to select an appropriate set of inter-vening variables.

Studies at Virginia Tech were performed using variable-complexity RS modeling techniques for both the aerody-namic design7 and structural design10'16 aspects of HSCTdesign. To this point, the aerodynamic design was per-formed using only panel level methods as the highest fi-delity aerodynamic model instead of CFD solutions. Thesame is true for HSCT MDO performed by other groupsat the Georgia Institute of Technology,17'18 the Universityof Notre Dame,19 and Stanford University.20 Computerprograms such as ACSYNT21 and FLOPS22 also performaircraft MDO with lower fidelity aerodynamic and struc-tural models.

Studies performed by Knill23 et al. demonstrate sig-nificant changes in the HSCT performance estimates andaircraft weight resulting from the use of more accurateEuler/Navier-Stokes aerodynamics in place of linear theorymethods. The HSCT is a highly leveraged design, and itsperformance is very sensitive to small changes in the drag.An underprediction in the cruise drag of only 2 counts re-sults in a 120 n.mi. overestimate of the range. The result-ing aircraft take-off gross weight (TOGW) is also sensitiveto changes in the drag. A 2 count increase in the dragover the entire mission results in an increase in the opti-mal TOGW of 56,000 Ib. Clearly, the best possible aero-dynamic predictions must be included in the design opti-mization. However, due to the large computational costs


associated with complex CFD analyses, novel approachesmust be devised to retain an efficient optimization proce-dure.

A method has been developed which utilizes informa-tion gained from lower fidelity aerodynamic methods tomore efficiently create RS approximations to the super-sonic Euler drag predictions. This method is tested us-ing an existing multidisciplinary HSCT design optimiza-tion procedure developed at Virginia Tech.3"6 Conceptuallevel aerodynamic models provide the functional form ofthe drag polar:

Table 1: Twenty-Nine Variables in HSCT Design.

CD(x) = CDo(x) (1)

Response surface models of the intervening functions, CDO(X)and K(x), are created. Details of the RS modeling tech-niques are given in Section 5. Response surface modelingtechniques are employed using linear theory aerodynamicsto determine which terms in the RS models play a signif-icant role in the evaluation of the drag. With this infor-mation, reduced term RS models representing a correctionto the linear theory RS model predictions are constructed.By eliminating unnecessary terms in this manner, the ac-curacy of the RS models is not compromised, and fewerCFD evaluations are required to evaluate the coefficientsof the polynomials. In addition, results from the inexpen-sive linear theory analyses are used to identify the likelyneighborhood of the optimal designs from.Euler analyses.This enables smaller bounds on each design variable, whichimproves the accuracy of the RS models. This method istested on simplified five, ten, fifteen, and twenty variableHSCT design problems.

2. HSCT Design TestbedThe design problem involves minimizing the TOGW

of a HSCT with a 5500 n.mi. range, designed to cruise atMach 2.4 and carry 250 passengers. The general HSCTconfiguration and mission is parameterized by 29 designvariables. The aircraft geometry (Table 1) is describedwith 26 design variables. Eight variables are used to de-scribe the cranked delta planform. The airfoil sections aredescribed using five design variables. The axi-symmetricfuselage is defined with eight design variables which pro-vide the fuselage radii at four axial restraint locations. Theshape of the body between these points is then determinedby considering it as a minimum wave drag body of a fixedvolume.4'24 The spanwise location of the nacelles is de-fined with two design variables. A single design variabledescribes the thrust of each engine. The horizontal andvertical tail areas are given by the final two geometric de-sign variables. The idealized mission profile is composedof take-off, subsonic climb, supersonic cruise/climb, and

DV1234567891011121314151617181920212223242526272829

Descriptionwing root chord, croot (ft)LE break, x (ft)LE break, y (ft)TE break, x (ft)TE break, y (ft)LE wing tip, x ( ft)wing tip chord, CUP (ft)wing semispan, 6/2 (ft)location airfoil max. thickness, (x/c)max_t

LE radius parameter, RLEt/c at wing root, (t/c)rogt

t/c at LE break, (t/c)hreak

t/c at wing tip, (t/c\ip

fuselage axial restraint #1, x/USl (ft)fuselage radius at axial restraint #1, r/USl (ft)fuselage axial restraint #2, XfUS2 (ft)fuselage radius at axial restraint #2, r/US2 (ft)fuselage axial restraint #3, XfUS3 (ft)fuselage radius at axial restraint #3, r/us^ (ft)fuselage axial restraint #4, z/US4 ( ft)fuselage radius at axial restraint #4, rfUS4 (ft)location of inboard nacelle, ynaceiie ( ft)location of outboard nacelle (ft)mission fuel weight, W/uei ( Ib)starting cruise altitude (ft)cruise climb rate (ft/min)vertical tail area (ft )horizontal tail area (ft*)thrust per engine ( Ib)

landing segments. Three design variables describe the mis-sion: the cruise climb rate, initial cruise altitude, and fuelweight.

The optimization uses up to 68 inequality constraintsdealing with the aircraft geometry and performance/aero-dynamics. These constraints are devised to ensure feasibleaircraft geometries and impose realistic performance andcontrol capabilities. Fuel volume and wing chord lengthlimits are examples of geometric constraints. Aerodynamicconstraints include, for example, landing angle-of-attacklimits; balanced field length requirements; and wing, tail,and engine scrape prevention criteria. Emergency con-ditions are used to enforce the landing constraints. Itis assumed that the aircraft lands on a runway 5000 ftabove sea level at 145 knots, carrying 50% of its initial fuelweight. Other aerodynamic constraints establish control-lability during adverse flight conditions. For example, the


no constraint violationsconstraints activeconstraints violatedrange constraintlanding CLam and tip scrapetip spike constraint

Optimum 2

Table 2: Active Design Variables in the SimplifiedOptimization Problems.

TOQW658000657000656000655000654000653000652000651000650000649000648000647000646000645000644000

Figure 2: Nonconvex Design Space in a Ten VariableProblem.

aircraft must be capable of trimmed flight with both en-gines on one side of the aircraft inoperable. These are com-plicated, nonlinear constraints that require aerodynamicforces and moments, stability and control derivatives, andcenter of gravity and inertia estimates.

An example of the complicated constraint boundariespresent in the design spaces associated with this HSCToptimization using the RS modeling approach is shown inFig. 2. This figure represents a plane in ten dimensionaldesign space, passing through three design points. Two ofthe design points represent local optima found by the op-timizer, and the third point is a suboptimal feasible point.The design points represented by the open circles are feasi-ble points, while those represented by the filled circles haveviolated some constraints. The plot clearly shows the non-convexity of the design space caused by the aerodynamicconstraints. If the optimizer drives the design near Opti-mum 1, it cannot cross the boundary created by the rangeconstraint to arrive at Optimum 2 which is 2000 Ib lighter.In addition to enabling the rapid creation of these typesof useful plots, using RS modeling techniques allows oneto investigate a number of different starting points in theoptimization to discover these local optima. The bulk ofthe computational effort is spent initially in the creation ofthe RS models, and performing a number of optimizationsusing the polynomial RS models is relatively inexpensive.

3. Simplified HSCT Design ProblemsCreating an accurate full term quadratic RS model in

29 dimensions requires at least 4000 Euler evaluations,

Design VariableNo. of Design Variables5 10 15 20

Planform VariablesCroat

Ctip6/2

SLE,ALE/ALEO

STB,ATE,

X X XX X X

X XX

X X XX X

straight TEstraight TE

XXXXXXXX

Airfoil Variables(x/c)max-tRLE(t/c)root

(t/CJbreak

(t/C\ip

X XX X

X X X(t/tfbreak = (*/C)root

(*/4i» = (*/cLot

XXXXX

Fuselage Variables

*;;;;"el^full

r/T*

X

X

X

X

X

X

X

XNacelle, Mission, and Empennage Variablesynacelle

^nacellewfuelStarting Cruise Alt.Cruise Climb RateVertical Tail AreaHorizontal Tail AreaEngine Thrust

X X

X X X

No horizontal tail

XXX

which is too time consuming to be viable in our HSCT de-sign process. A series of simplified optimization problemsof five, ten, fifteen, and twenty variables serve to evalu-ate our methods of including Euler analyses in multidisci-plinary HSCT design. Because of the reduced number ofevaluations required, full term quadratic RS models for theEuler aerodynamics are created for the five, ten, and fifteenvariable problems, providing a means to assess the perfor-mance of our methods. The design variables (Table 2) usedto define the geometry and mission are subsets of those


used in the full 29 variable design, although some appearin a different form. For example, instead of specifying thex and y locations of the leading edge (LE) and trailingedge (TE) break points, the simplified problems use moremeaningful variables specifying the wing sweep angles andthe inboard surface lengths. In the same way, the variablespecifying the x-location of the wing tip LE is replacedwith a variable stipulating the outboard LE sweep angle.The upper and lower limits of the active design variables(Table 3) remain the same throughout the series of opti-mization problems. These bounds on the design variablesare selected from optimization results performed using thelinear theory RS models. These optima provided clues tothe expected general location of the optimal designs fromEuler analysis and allow smaller ranges on the design vari-ables, resulting in more accurate RS model predictions.

The simplified designs are subject to a reduced set ofconstraints (Table 4). There are three basic reasons whycertain constraints are eliminated:

1. The design variable(s) on which the constraints de-pend are not active.

2. Explicit limits on the design variables prevent geo-metric constraints from becoming active.

3. Simplifications to the mission eliminate related con-straints.

Vertical tail sizing is not active for the simplified designs,therefore constraints number 47 and 48 do not appear.Unlike the 29 variable design, these configurations haveno horizontal tail. This eliminates constraints 49, 50, and51. The engine thrust also remains constant, removingconstraints 52, 67, and 68 from the active list.

The optimization problem is the minimization of theTOGW subject to a number of constraints related to boththe geometry and the mission. Side constraints limit thevalues of the design variables. The optimization problemcan be written as

Table 3: Design Variable Limits in the SimplifiedOptimization Problems.

min TOGW (x),

subject to: g(x) < 0,

(2)

<

where x is the m dimensional vector of design variables,and g(x) is the vector of nonlinear inequality constraints.The minimum and maximum values of the design variablesare given by xmin and xmax.

A simplified five variable wing design is considered first.The ten variable design problem is an extension of the

Design VariableNo.5

of Design Variables10 15 20

Planform VariablesC-root

Ctipb/2SLE,ALB,ALEOSTE,ATB/

150- 190 ft7 -13 ft

[74 ft][132 ft]

58-105

78ft- 135ft

67° - 76°[25°]

straightstraight

12°-TETE

-32°10-30/t-55° - 16°

Airfoil Variables(*/c)ma*-tRLE(t/cLat(t/^break(t/c)tiv

[40%][2.5]

1

38-2.1-

52%-4.1

5 - 2.7%(t/C\reak = (t/^root

(*/cU = (*/cLot1.5 - 2.7%1.5 - 2.7%

Fuselage Variables•Efus-\

rius,•E/MS2

r/«*2

2/US3r/«S3xfusi

[50 ft][5.2ft]

[5.7ft]

[5.9/i]

4.5 -6.0/t[100 ft]

4.5 -6.0ft[200ft]

4.5 - 6.0ft[250 ft]

[5.5ft] 4.5-6.0/tNacelle, Mission, and Empennage Variables

ynacelle

ÛnacelleWfuel

Starting Cruise Alt.Cruise Climb RateVertical Tail AreaHorizontal Tail AreaEngine Thrust

[20 ft] 10 -35 ft[6ft] 6 -18 ft

No Limits[65, 000 /i]

[WO ft/ min][548 ft*]

No horizontal tail[39, 000 lb]

[ • ] variable value when not active in the design

five variable problem, enabling more general planform andairfoil geometries, and allowing variation in the spanwiselocation of the nacelles. Fuselage shaping is enabled withthe fifteen variable design. Finally, the twenty variabledesign provides a complete description of the wing, airfoil,fuselage, and nacelles.


Table 4: Active Constraints in the SimplifiedOptimization Problems.

#123-2021222324-4142-4344-454647484950515253545556-585960616263646566

67-68

Constraint (Abbrev.)Range > 5, 500 n.mi.CL at landing speedSection Ce < 2Landing a < 12°Fuel volumeSpike preventionWing chord > 7.0 ftEngine scrapeEngine scrape (5° roll)Wing tip scrapeRudder deflectionBank angle < 5°Takeoff rotationTail deflectionWing TE < HT LEBalanced field lengthTE break scrape (5° roll)LE break < semispanTE break < semispan(t/c) > 1.5%*Efus-\ — bji'

^/USO "^/ttSl — *•*-' J "

Xfus3 - Z/usa - 10/*Xfus4 - Xfusz > 10 ft300ft-xfusd > 10 ftVnacelle > fuselage

Î/nacelle > 0Engine-out limit;(vertical tail design)Maximum thrust

Total Active Constraints

No. of Variables5XXX

XXX

X

X

42

10XXXXXXXXXX

XX

X

49

15XXXXXXXXXX

XX

X

49

20XXXXXXXXXX

XXX

X

50

3.1. Five Variable HSCT Design

The geometry for the five variable wing design (Fig. 3)is created with four design variables specifying the rootchord, Croat', tip chord, ctip; inboard leading-edge (LE)sweep angle, A.LEI ', and the thickness-to-chord ratio, t/c.The fifth design variable gives the fuel weight, Wfuei. Theallowable ranges of values for the variables are shown inTable 3. These ranges are chosen using results from pre-vious optimization studies in an attempt to bracket theoptimal designs within the bounds of the design variables.To uniquely describe the aircraft, a number of geometricparameters are specified. The fuselage and vertical tailshapes remain constant. The trailing edge (TE) for these

*'\4

F

1/ — leading-ea

^ }'•-*.

ge radius (fixed)

Xjsroot chordx2=tip chordx3=inboard LE sweep anglex4=thickness-to-chord ratioxs=fuel weight

sweep (fixed)

wing semispan (fixed)

Figure 3: Geometry for Five Variable Design Problem.

configurations is straight with no TE break. The lengthof the LE from the wing apex to the LE break is con-stant (SLE, — 132/i), as is the sweep angle of the outboardLE (A.LEO

= 25°) and the wing semispan (6/2 = 74ft).Two airfoil parameters are held fixed: the leading-edgeradius parameter (RLE — 2.5) and the chordwise locationof the maximum thickness ((x/c]max,t = 40%). The en-gine thrust, spanwise nacelle positions, relative position ofthe wing to the fuselage, and the cruise altitude are alsoinvariant. Table 4 details the 42 constraints that are activefor the five variable design.

3.2. Ten Variable HSCT Design

The wing planform for the ten variable design (Fig. 4)is created with five design variables specifying the rootchord, Croat, tip chord, CUP; semispan, 6/2; inboard LEsweep angle, ^.LE,', and outboard LE sweep angle, ALEO-The airfoil sections are described using three design vari-ables: the leading-edge radius parameter, RLE', locationof maximum thickness, (x/c)max_t\ and thickness-to-chord


f

!iju

-*-x

*i=rootchordx2=tip chord\3=wing semispan\4=inboard LE sweep anglexsôutboard LE sweep angleX6=Iocation of max. thickness\7=LE radius parameterxg=thickness-to-chord ratioX9=location of inboard nacellex!0=fuel weight

length of inboard LE(fixed)

A

ma

Figure 4: Geometry for Ten Variable Design Problem.

xj=root chord^ V ^ x2=tip chord

\ xa=wing semispaned) x4=length of inboard LE

X'

,

ed)

'— *•

csd) '•$

Sfcjs

— »•

ed)

j /

xs=inboard LE sweep angle

•* —— *lo '

iii

•* —— "u

'-S1&.1

•* ——— "121

1

r 1 x6=outboard LE sweep angle\4 xs x7=location of max. thickness\ x8=LE radius parameter\ X9=thickness-to-chord ratio\ xjo - Xj3=fuselage radii\ X]4=location of inboard nacelle\ xis=fuel weight

X4vV6

" >~~Tr~Tr~~~~— -.J -L-

-—— .» *""

/ ' " '

_i"2

1

ratio, t/c. The final two variables specify the inboardnacelle placement, ynacelle, and the mission fuel weight,Wfuel- The allowable ranges of values for these variablesare shown in Table 3. For these configurations, the fuselageand vertical tail shapes are fixed. The length of the lead-ing edge from the wing apex to the leading-edge break isconstant (SLE, — 132 f t ) , and the trailing edge is straight.The engine thrust, spanwise distance between nacelles, rel-ative position of the wing to the fuselage, and the cruisealtitude are also fixed. Table 4 details the 49 constraintsthat are active for the ten variable design.

3.3. Fifteen Variable HSCT Design

The wing planform for the fifteen variable design (Fig. 5)is created with six design variables specifying the rootchord, croot; tip chord, ctip; semispan, 6/2; inboard LElength, SLE, ', inboard LE sweep angle, ALB, ; and outboardLE sweep angle, A.LEO- The airfoil sections are describedusing three design variables: the leading-edge radius pa-rameter, RLE] location of maximum thickness, (x/c)max_t;

Figure 5: Geometry for Fifteen Variable DesignProblem.

and thickness-to-chord ratio, t/c. The fuselage radii, r/USi,are specified at four axial locations. The final variablesspecify the inboard nacelle placement, y nacelle, and fuelweight, Wfuei. The allowable ranges of values for thesevariables are shown in Table 3. For these configurations,the wing TE is straight. The vertical tail shape, enginethrust, distance between nacelles, relative position of thewing to the fuselage, and the cruise altitude are constant.Table 4 details the 49 constraints that are active for thefifteen variable design.

3.4. Twenty Variable HSCT Design

The wing planform for the twenty variable design (Fig. 6)is created with eight design variables specifying the rootchord, croot', tip chord, ctip; semispan, 6/2; inboard LElength, SLE,', inboard LE sweep angle, A.LE,', outboardLE sweep angle, A.LEO ', tne inboard TE length, STE: ', and


J J1 / — "10

r@—— —————— ^ 1 +(9 - - X]=root chord

x2=tip chordr ——— ————— - r ————— -y x3=wing semispan

\ x^=length of inboard LE. \ x5=in board LE sweep angle

V

X•

:

td)

1

— "i

1ijc

ed) -£5bOc

^<•:,

,

frf;

^

x6=outboard LE sweep angle--1 ——— i \7=Iength of inboard TE

•* —— "14 '

•* —— "is

"1

!1:j

%>s?31'!,

•* ———— "16

i! ] f

-^x,7-^^

\

xs=inboard TE sweep angleX9=location of max. thickness

x10=LE radius parameterxn ~ X]3=thickness-to-chord ratios

x]4 - x17=fuse)age radiix18=location of inboard nacelleXi9=nacelle separationx2o=fuel weight

X4

\

\ \\^X6

~^\• XTT" — ~r-i»^^^ "7 1 1 rr^^*'s^** LJ

^/ ^ Xjs ——— X!9 ^

/ •* ——————— »3 ——————— *"

A*2

|

Figure 6: Geometry for Twenty Variable DesignProblem.

the inboard TE sweep angle, A.TE,- The airfoil sectionsare described using five design variables: the leading-edgeradius parameter, RLE', location of maximum thickness,(z/c)max-t; and the thickness-to-chord ratios at the wingroot, (t/c)root, LE break, (t/c)brea^ and win§ tJP> (*/c)tip-The thickness-to-chord ratio is varied linearly between thesethree spanwise locations. The fuselage radii, rfUSi, arespecified at four axial locations. The final variables spec-ify the inboard nacelle placement, ynacelle, the separationbetween the inboard and outboard nacelles, &y nacelle, andfuel weight, W/uei. The allowable ranges of values for thesevariables are shown in Table 3. The vertical tail shape, en-gine thrust, relative position of the wing to the fuselage,and the cruise altitude are fixed. Table 4 details the 50constraints that are active for the twenty variable design.

4. HSCT Design Tools

4.1. Linear Theory Aerodynamics

What we term supersonic linear theory predictions areactually obtained from three codes, each computing a par-ticular component of the drag. The volumetric wave dragis computed using the Harris25 wave drag program. Dragdue to lift is calculated using a panel method by Carlson26

et al. with attainable leading edge thrust corrections.27

Viscous drag estimates are obtained using standard alge-braic estimates28 of the skin friction assuming turbulentflow. These viscous drag predictions are also added to theEuler solutions.

The camber for our HSCT designs is determined usinga linear theory code with empirical corrections to accountfor nonlinear effects named WINGDES.26'29 WINGDESattempts to find the camber distribution along the wingwhich minimizes the drag due to lift. Two runs of WING-DES per wing were required to get the proper camber dis-tribution. The second run serves to smooth the camberdistribution and provide the maximum leading edge suc-tion parameter at the design lift coefficient.

4.2. Computational Fluid Dynamics

Version 2.2 of the General Aerodynamic SimulationProgram30 (GASP) is used to obtain the Euler solutions.GASP is a fully conservative CFD code which solves theReynolds averaged Navier-Stokes equations and many ofits subsets. The code uses an upwind three dimensional fi-nite volume spatial discretization, and for our calculations,a third order upwind biased interpolation of the Roe fluxesis used in each of the marching planes.

The finite volume formulation of the Reynolds averagedNavier-Stokes equations is written in terms of the vectorof conserved variables, Q, the vector of primitive variables,q, the cell volume, V, and a residual vector, R(q), as

d ddq dt V + R(q) = 0.

The cell averaged quantity, <Q>, is defined as an integralover the volume of the cell

The cell averaged quantity for the primitive variables, <q>,is defined in the same manner. The residual vector is writ-ten as a function of the cell edge area averages of the in-viscid fluxes, F and viscous fluxes, Fv, the unit normalvectors, n, to the cell faces, and the areas, A.A, of the nf


Figure 7: HSCT Wing-Fuselage with CFD Grid Planes.

cell faces as

nf

The norm of this residual vector represents the conver-gence to the steady state solution. GASP iteratively solvesthe system of equations until a prescribed reduction in theresidual norm is reached. Due to the large savings in com-putational time, space marching has been performed forall of the CFD calculations. The accuracy of the CFDpredictions has been studied extensively by Knill23 et al.These studies confirmed that the Euler drag results are ac-curate within 1/2 counta in the range of flight conditionsconsidered using 405,000 grid cells.

4.3. Grid Generation

Grids suitable for space marching calculations on theHSCT wing-fuselage configurations are created using a gridgenerator originally developed by Barger.31 The code wasmodified32 to provide better resolution of the leading edgeand remain robust for large changes in the aircraft geome-try. The grid generator receives as input the aircraft con-figuration stored in the Craidon33 geometry format, ex-tends the wing to join the fuselage, performs filleting ofthe wing-fuselage intersection,34 and then creates a gridfor space marching calculations. Since our HSCT opti-mization code creates a Craidon description file from itsset of design variables, the conversion from a set of designvariables to a space marching CFD grid is straightforward.

The space marching planes are created along planes ofconstant x-value. Two of these computational planes for awing-fuselage configuration are shown in Fig. 7. A Mach

AC.D = 1.0xlO~4 corresponds to 1 count of drag

cone analysis is used to form the outer boundary to ensurethat all shocks are contained within the computational do-main. The grid generator allows for flexible stretching ofthe grid points around and normal to the aircraft to creategrids suitable for both Euler and Navier-Stokes calcula-tions. Measures are employed to reduce grid skewness atthe wing tip and wing-fuselage juncture. The grid gener-ator is automated and robust for large planform changes,essential qualities for application in design optimization.

4.4. Weight Estimation

All components of the TOGW are calculated using em-pirically based functions obtained from the Flight Opti-mization System22 (FLOPS) weight equations.

4.5. Optimization Routine

Optimization is performed using Design OptimizationTools35 (DOT). The constrained optimization is performedusing sequential quadratic programming (SQP) and cen-tral difference gradient approximations.

5. Response Surface ModelingTo use relatively expensive Euler solutions for the large

number of constraint evaluations required in our multidis-ciplinary optimization, RS models of the supersonic aero-dynamics are created. Response surface modeling tech-niques for aerodynamic and structural design improve theperformance of highly constrained gradient based optimiza-tions.7'10'12'13 The RS models smooth out numerical noisepresent in the analyses. This noise distorts gradient infor-mation and can lead to artificial local minima in the designspace. In addition, the analysis codes are separated fromthe optimization routines, eliminating the need to integratelarge, production level grid generators, analysis codes, andpost processing utilities with the optimizer. This also al-lows analyses to be performed by experts in the specificdiscipline on parallel architecture machines. Results fromparallel computing efforts are described in Appendix 7. Fi-nally, by replacing complex analysis codes with simple qua-dratic polynomials, one can readily obtain information ondesign trade-offs, sensitivities to certain variables, and in-sight into the highly constrained, nonconvex design spaces.These aspects are presented in Appendix 7.

5.1. Functional Form of the Response

Not all response functions can be modeled accuratelyby low order polynomials, but often accuracy can be im-proved by transformations of the function or its arguments.Analytic solutions obtained under simplifying assumptionsoften help reveal the needed transformations. The use of


such transformations was pioneered for approximations ofstructural response by Schmit and Vanderplaats and theircoworkers (see Ref. 36). They called the transformed vari-ables and functions 'intervening variables' and 'interveningfunctions,' respectively.

Our group has made use of intervening variables to im-prove the accuracy of the RS model for optimal structuralweight as a function of the HSCT configuration variables.16

Golovidov37 showed that approximating the drag coeffi-cient by a quadratic polynomial and calculating the rangefrom the drag coefficient improves the accuracy over directapproximation of the range. The drag coefficient becamethe intervening function for the range.

In the present work, we use the knowledge that foruncambered wings the drag coefficient is approximately asimple quadratic function of the lift coefficient, written as

We then fit the intervening functions, C£>0(x) and K(x.),by quadratic RS models. The use of this form has theadded advantage of removing the dependence of the dragcoefficient on the fuel weight, introduced only through thelift coefficient.

The HSCT wings have little camber since they are op-timized for cruise at Mach 2.4; therefore, this form of thedrag polar for uncambered wings is still fairly accurate.The error in fitting the Euler drag polars to this form isless than 0.5 count over 0.05 < CL < 0.12. These values ofCL cover the range of cruise lift coefficients found for ourHSCT designs.

5.2. Creating Quadratic RS Models

The procedure for creating the RS models for CDO(X)and K(x) is illustrated in Fig. 8. The first step is togenerate a pool of candidate designs using a full facto-rial, central-composite, or small-composite experimentaldesign. Details of the experimental design are presentedin Appendix 7. Then, geometric constraints are applied toeliminate infeasible designs. The D-optimality criterion isused to choose the desired number of points from the poolof feasible designs.

WINGDES is used to provide a camber distribution foreach of the D-optimal design points. The lift and drag arethen computed from both linear theory and Euler analysis.Since the Euler calculations are only for wing-fuselage con-figurations, wave drag predictions for the vertical tail andnacelles from linear theory analysis are added to the Eulerdrag data. The values of CDO(X) and K(x.) are computedusing data from two points on the drag polar. Response

Euler RS Modelsfor CDoand K

Linear TheoryRS Models for

Când*

Euler RS Modelsfor ACDoand AK

Figure 8: Flowchart for Creating RS Models.

surface models for Cx>0(x) and K(x} are found from theset of computational experiments using the statistical soft-ware package JMP.

A quadratic response surface model in m variables hasthe form

(4)

where y is the response, the Xj are the design variables,and CQ, a,-, and CJK are the polynomial coefficients. Thereare (m+l)(m+2)/2 coefficients for a quadratic responsesurface model in m variables. Giunta38 et al. found thatusing approximately twice as many points as the numberof coefficients is sufficient to accurately compute the valueof the coefficients for problems with five variables. How-ever, as the dimension of the problem increases, the ratioof the required number of points to the number of termsp/n in the response surface also increases. We use a sys-tematic increase from p/n = 2.0 in the five variable case top/n = 3.5 in the twenty variable case to provide adequateaccuracy. This translates to 60, 276, 720, and 1470 CFDevaluations per RS model for the 5, 10, 15, and 20 vari-able problems, respectively. To compute the coefficientsfor the quadratic RS models of both Cx>0(x) and K(x.)for a 30 variable problem would require 4184 CFD evalu-ations. On a single processor of the SGI Power Challenge

10


R6000, this would require over 46 days of computation!Clearly, a method must be developed which enables accu-rate response surface estimates with a reduced number ofrequired CFD analyses.

5.3. Reduced Term RS Models

Linear theory RS models can be readily created formoderately high dimensional spaces because the evalua-tions are so computationally inexpensive. Regression anal-ysis, performed to obtain the coefficients of the polynomialmodels, also provides a means of systematically removingterms14 from the RS models that have little or no impacton the response through a process called step-wise regres-sion analysis. Terms are eliminated using a measure ofthe significance level of the term called the p-value. Thisrepresents the probability that the coefficient of a partic-ular term is actually zero, not the value computed. Typ-ically, a p-value of 0.05 or less indicates that the term issignificant in predicting the variation in the response. Asthe prescribed value of the p-value is reduced, more termsare eliminated from the RS model. At some point in thestepwise regression process, the error in the RS model fitincreases noticeably, indicating that too many terms havebeen eliminated to satisfy the p-value limit. For this re-search, JMP39 is used to perform the stepwise regressionanalysis.

The root mean square (RMS) error estimate is used toindicate the error in the RS model fit. The RMS error iscalculated as

RMS error =

NZ3 (?/«~~ y»)

N (5)

where yt is the observed value of the response and & is thepredicted value of the response at the N sample points.The TV sample points are a randomly selected subset of thepoints used in the initial screening experimental design notincluding the design points used in the creation of the RSmodels.

The stepwise regression technique is applied to the RSmodels for the linear theory aerodynamics to create reducedterm RS models. Since the Mach 2.4 cruise regime is pre-dominantly linear, performing regression analysis on theRS models for the linear theory aerodynamics should givea polynomial with nearly the same terms as the reducedterm RS models for the Euler aerodynamics. Instead ofcreating full term quadratic RS models for the Euler re-sults, one needs only to create the reduced term modelsfound using linear theory analyses. The Euler RS model isevaluated using only the terms deemed necessary from the

RMS

Crui

se D

rag

Erro

r (co

unts)

O

—

K>

21

l/i

—

in

NJ

In

<j

; - - -A- - - L.T. RS Models

\; \

- \ - •• • -; \

\. . . . . . . A . . . . . . . .\

a— a B^D — — n n q ____ ,_ L-^ -A---a-M-" — i

Terms in Polynomial (From Stepwise Regression on L.T. Aero)

Figure 9: Stepwise Regression (5 Variable Problem).

linear theory regression analysis. Computational time istherefore not wasted evaluating coefficients which do nothave a significant effect on the response.

5.4. Incremental RS Models

We have found that the best means to implement thereduced term RS models with Euler analyses is by using anincremental form. (See Ref. 32 for details on other alter-natives.) This approach to implementing the reduced termmodels uses reduced term RS models for the difference be-tween the Euler and linear theory RS model predictions,ACL>0(X) and AJi'(x). To estimate the Euler values ofthe response, these correction RS models are added to thefull term quadratic linear theory RS models. The sum ofthe linear theory and correction RS models will be referredto as the incremental RS models (i.e., n term incremen-tal RS model = n term correction RS model + full termlinear theory RS model). Results from this procedure arepresented in Section 6.

6. Optimization Results

6.1. Five Variable HSCT Design

As discussed in Section 5, using knowledge of the func-tional form of the drag polar eliminates the fuel weightdependency of the response. Therefore, response surfacemodels for CDO

and K are created using only four of thefive design variables. The quadratic RS models thereforehave 15 terms. A 3m~l full factorial experimental design40

is used for the initial screening of the design space, giving atotal of 81 design points. None of these points are geomet-rically infeasible, so the D-optimality criterion41 is used toselect the 2n = 30 design points to provide the computa-tional experiments.

11


Table 5: Optimal Design Variables (5 VariableProblem).

Figure 10: Optimal Designs from Reduced TermIncremental and Full Term Euler RS Models.

Results from stepwise regression analysis for the fivevariable design are shown in Fig. 9. The RMS error in thelinear theory RS models represents the differences betweenthe linear theory RS model prediction of the cruise dragand the linear theory analysis value. The RMS error in theincremental RS models represents the differences betweenthe incremental RS model prediction of the cruise dragand the actual Euler value. Both errors are computed at30 points randomly selected from the set of feasible designpoints not used in the RS model creation. The regressionanalysis plots show that the RMS error for the incrementalRS models does not have the abrupt increase below seventerms that is present in the linear theory RS models.

The stepwise regression plot demonstrates that the step-wise regression technique can eliminate unimportant termsfrom the response surface models with little or no effect on

C-root

Cttp

ALB,t/cwfuelWwin

WTOGWRange:

(Euler)&Wfuei

WC-TOGW

5 TermIncremental RS

178.0 ft7 A ft71.1°

1.81%309, 800 Ib103, 900 Ib622, 800 Ib

5503 n.mi.-320 Ib

622, 500 Ib

15 TermEuler RS

174.2 ft7.8ft70.6°

1.81%313, 200 Ib103, 900 Ib626, 300 Ib

5544 n.mi.-4010 to

62 1,800 to

the error. Furthermore, it shows that the errors in thereduced term incremental RS models, created using termsfound from stepwise regression analysis on the linear the-ory RS models, do not change significantly as the numberof terms is reduced. This indicates that linear theory anal-ysis does reveal the terms that are important to the Euleranalyses. The five term incremental RS model, which re-quires only 10 design points to be evaluated, is chosen toperform the five variable optimization studies.

The five term incremental RS model requires only 5.3hours to create, as opposed to 16 hours required for thefull term model. The optimal design obtained from theincremental RS models compares well (Fig. 10) to the fullterm Euler RS model. Even more importantly, Euler anal-ysis on the optimal design reveals that the RS model cruisedrag prediction is less than 0.1 count higher than the Eu-ler value. The design variable values for the incrementalRS model optimum fall within their prescribed bounds,demonstrating the success of selecting these bounds basedon the linear theory optimal designs. Table 5 shows thatthe corrected TOGW is only 700 Ib higher than that forthe full term Euler optimum. The corrected TOGW isthe result of adding or subtracting fuel weight to the op-timal designs in order to correct for the range discrepancybetween the RS model and Euler predictions.

6.2. Ten Variable HSCT Design

Because of the lack of dependence on the fuel weight,the RS models for CDO and K are created using only nine ofthe ten design variables. The quadratic RS models there-fore have 55 terms. A face centered central-composite ex-

12



e •3 'Oo

o&60•a T"

——B—— Incremental RS Models---&--- L.T. RS Models

1 'uSi"as

°0 5 10 15 20 25 30 35 40 45 50 55Terms in Polynomial (From Stepwise Regression on L.T. Aero)


Croat

Ctip

6/2

ALB/ALSO(X/CLaX-t

RLEt/cynacelle

Wfuel

wwingWTOGWRange:

(Euler)

AW/ueJ

WC-TOGW


170.4/i9.0ft

72.1ft70.0°18.7°

50.2%2.1

1.91%30.0 f t

306, 000 Ib96, 100 Ib

610,400/6

5542 n.mi.-3715 Ib

606, 300 Ib

55 TermEuler RS

174.3/t7.5ft

72.3ft70.1°26.5°

50.1%2.1

1.82%30.2 f t

301,000/699, 200 Ib

608, 900 Ib

5485 n.mi.1300 Ib

610,300/6

Figure 12: Optimal Designs from Reduced TermIncremental and Full Term Euler RS Models.

perimental design40 is used for the initial screening of thedesign space, giving a total of 531 design points. Noneof these designs are geometrically infeasible, so the D-optimality criterion is used to select 2.5n = 138 designpoints to provide the computational experiments.

Stepwise regression analysis for the ten variable prob-lem (Fig. 11) exhibits similar behavior of the incremen-tal RS models as the number of terms is reduced. TheRMS errors are computed at a 138 randomly selected setof the initial screening experimental design points that arenot selected by the D-optimality criterion to create the RSmodels. As in the five variable problem, linear theory re-sults capture the terms that are important to the Euleranalyses. The 20 term incremental RS model, which re-quires data from 50 design points to create, is chosen forthe ten variable optimization studies. The optimal config-uration is compared to the full term Euler RS models inFig. 12, and the design variables are presented in Table 6.The RS model prediction of the cruise drag is 0.6 countshigher than the Euler value, leading to the 5542 n.mi. ac-tual range. Removing the extra fuel results in a designwith no constraint violations that is 4000 Ib lighter thanthe full term Euler optimum.

13

A sm

all-composite experim

ental design40 is used for

the initial screening of the design space, giving a total of2

14 + 2x19 + 1 = 16, 423 configurations, including 688 ge-om

etrically infeasible configurations.. For some of these

6.4. T

wenty V

ariable HSC

T D

esign

sign is presented in Fig. 13.

The RM

S errors are com-

puted at a 360 randomly selected set of the initial screen-

ing experimental design points that are not selected by the

D-optim

ality criterion to create the RS m

odels. Optim

iza-tion is perform

ed with the 48 term

incremental R

S models.

These models are created using data from

144 experimen-

tal design points. The optimal designs from

the full termEuler and reduced term

incremental R

S models (F

ig. 14)are barely distinguishable from

each other. Table 7 showthe very close agreem

ent in the design variables. In thisregion of the design space, both the Euler and increm

entalR

S models are extrem

ely accurate (within 0.1 drag count).

This is evident in that there is only a five mile discrepancy

between the range estim

ates from the R

S models and those

from Euler evaluations. W

hen the fuel weight is altered to

counter the range variations, the TOG

W for the Euler and

incremental RS m

odels are only 200 Ib different.

COc+CD

1Cfl0)

1o'B

Ico,en*c?t-tC+

cTB53?1<

1CD"o.<?

tational experiments.

c+

CDt-j

3g>_ET

Oqt— iOl

0h-»COa8o5'B

•OO

aCO<r+O

^

§

%c-t-

g"83"B

D-optim

ality criterion is used to select 3. On = 360 out of

c+-

1c-+-

CD"

5'(W

3orl-

O-O

&

gcrt-

sCD

e+

CD

1

HcT

o- p> o.CD fe CDco co cot-» CD i-"

SI!_^ co "O£ ^ 0m >§ SO

f1^'CD i-( 2.

l * l f i* S e>5 " ?&. B S-P= *- £,g 0 0ô. 5' ^si cTP1-S . f c

(W cr COrt- S" CL- s- »R - S . f i -B <TQ BLS g >0co • o

^ O1 &CD !=! CO

S ^ H1 a sB^ CD CD

•oS

1B(Th

£.D.io5o

CO

51t-irt-CD

2-rV

£COIBB'

OTO|-c+

CDFor the fifteenvariable design, a central-com

posite ex-

6.3. Fifteen V

ariable HSC

T D

esign

Figure

M £."w =.. *g

ov* ^<

1 18S. »co — .CD 9

ll»n ooffi -s23. ^§ s -^-x 73

K— * CDOi era< sP 2. „t-l O o5' 3 °s: §CD rh3 H

P , '>i uj. . : ¥ i/ /: % ii

>• itii' i. iiii

t> d

rj

3 > 5 —————————cr1 o

i

ii ]

r =rH G* £en fi2 â;2. c" '5

&i

0I<

0oy>

c7aCrt

fT o3

n'CD CTJ

§ ^B fr

I!issi3 gM tfB or B

iS, O

g.8.£. Hc« CDi-j

3

Optimum (120-Term Euler RS Models)

Optimum (48-Term Incremental RS Models)



Croot

Ctip

6/2

SLE,ALB,ALSO

(z/c/max-tRLEt/crfuSl

rfus2

rfuss

rfus4

I/nacelle

wfuel** wing

WTQGWRange:

(Euler)&Wfuei

WC-TOGW


166. 2 ft7.7ft68.1°

120.4/i69.4°24.2°

49.8%2.1

1.99%5.2ft5.6ft5.6ft5.2ft

28.1ft299,614/6

87, 800 Ib590, 700 Ib

5495 n.mi.450 Ib

591,200/6

120 TermEuler RS

166.2/t7.7ft68.2°

120.3/i69.4°23.8°

49.7%2.1

1.99%5.1ft5.6ft5.6ft5.2ft

28.0ft300, 100 Ib87, 900 Ib

591,500/6

5505 n.mi.-440 Ib

591,000/6 Figure 16: Optimal Design from Reduced TermIncremental RS Models.

•2 .5

gUQ

1.5

'Incremental RS Models

- - L.T. RS Models

V0 25 50 75 100 125 150 175 200Terms in Polynomial (From Stepwise Regression on L.T. Aero)


Since the full term Euler RS models are not present, thisprovides a model with which to compare the errors in the73 term RS models. It is seen that there is no major differ-ence in the cruise drag errors of the 100 term and 73 termincremental RS models.

The optimal design from the incremental RS models isshown in Figure 16, and the optimal design variable valuesare presented in Table 8. The cruise drag prediction forthe optimal design from the incremental RS models is 0.8counts lower than the actual Euler value, which is muchlower than the RMS error of the RS models. Compensat-ing for the range deficiency gives a corrected TOGW of588,000 Ib. Although the error in the RS models is larger,this Euler optimum is 3000 Ib lighter than the 15 variableoptimum.

15



C-root

Ctip

b/2SLE,ALB,ALBO

STB,ATS,(x/c)max.tRLE(t/c)root

(t/C)break

(t/c\ip

' fus*\

/uso

rfus3

r/us4

ynacelle

Ay-nacelle

wfuelwwingWTOGWRange:

(Euler)AWfUel

WC-TOGW


169.5/i7.8ft

67 A ft124.8ft

70.5°30.4°

27.4/i-29.0°51.1%

2.11.99%1.91%1.94%5.2ft5.6ft5.6ft5.2ft

27.7ft6.0ft

293,000/686, 900 Ib

583, 200 Ib

5449 n.mi.4430 Ib

588, 000 Ib

Using the reduced term incremental RS models has pro-vided a means to create RS models for high dimensionalproblems where computing the coefficients of the full qua-dratic RS models is not viable. Figure 17 shows how thepresent method extrapolates to 25 and 30 design variables.The trends indicate that a reduced term response surfacemodel in 25 design variables can be created using the samenumber of terms as a full quadratic model in 15 designvariables.

500

450

400

350

300250

200

150

100

50

a - - Reduced Term RS Models

0 10 20 30Number of Design Variables

Figure 17: Number of terms in RS Models.

6.5. RS Model Accuracy

The RMS errors in the incremental RS models are con-sistently higher than the errors in the optimal designs. In-vestigation led to the discovery that the incremental RSmodel fit through the interior of the design bounding boxis better than that at the vertices, where the points usedto evaluate the RMS errors are located. When evaluatingthe RMS error in the fifteen variable design using a new setof 162 design points scattered through the interior of thedesign box, the RMS error dropped from 1.5 counts to 0.9count. This indicates that using points selected only fromthe vertices of the design box may not provide an accuraterepresentation of the RS model fit.

The RMS errors in the linear theory and incremen-tal RS models steadily increase from satisfactory levelsin the 5 variable to unacceptable levels in the 20 vari-able problem. Preliminary examinations of three meth-ods to improve the accuracy of the RS models have beenmade. The first method involves using a more accuratephysical model to provide the functional form for the dragpolar. With an additional CFD evaluation per design, amore accurate cambered form of the drag polar, CD(X) =CDm(x)+K(x.) [CL-CLm(x)]2, can be evaluated. Whilenot addressing the nonquadratic behavior of the response,this approach was found to reduce the RMS error in theRS models by approximately 0.3 drag count. The non-quadratic nature of the response can be addressed in twoways: reducing the size of the design bounding box andincluding cubic terms in the RS model. Reducing the sizeof the design box has a significant effect on the error; how-ever, care must be taken to ensure that the optimal design

16


lies within the reduced box. Including cubic terms allowsone to maintain the size of the original box, but the com-putational expense of evaluating all the cubic terms makesit unattractive for high dimensional problems. Using themore accurate drag polar along with a "zooming" tech-nique to reduce the size of the design box appears to be apromising approach.

7. ConclusionsA method for efficiently implementing supersonic Eu-

ler analyses in a combined aerodynamic-structural opti-mization of a HSCT configuration has been developed andtested on problems of five, ten, fifteen, and twenty de-sign variables. This method takes advantage of informa-tion obtained from inexpensive lower fidelity aerodynamicanalyses to more effectively create RS models for the Eu-ler solutions. Accuracy of the RS model predictions isenhanced by selecting the functional form of the drag po-lar, CD = CDO (x) +K(x) Ci2, based on conceptual levelaerodynamic models and creating RS models involving theintervening functions CDO (x) and K(x). Creating cor-rection RS models representing the difference between lin-ear theory and Euler values of the intervening functions,ACc0 (x) and AK(x), proves advantageous. IncrementalRS models for the Euler predictions are then created byadding the correction RS models to the full quadratic lin-ear theory RS models. Optimization results from the lineartheory RS models are used to select the design boundingbox within which the optimum from Euler analysis shouldlie. This improves the accuracy of the RS models by al-lowing smaller ranges on the design variables compared tothose required if no information was available on the gen-eral location of the optimal design. Errors in the RS modelcruise drag predictions, based on actual Euler calculations,for the optimal designs range from 0.1 counts to 0.8 counts.

Computational expense is reduced using stepwise re-gression analysis results gained from linear theory analysis.Stepwise regression analysis removes terms from the qua-dratic polynomial RS models that are not important in theevaluation of the response. Since the Mach 2.4 cruise flightregime is predominantly linear, terms that are unimpor-tant to the linear theory RS models are also unimportantto Euler RS models. By removing unnecessary terms, thenumber of CFD analyses required to evaluate the coeffi-cients in the resulting reduced term RS models, and there-fore the computational effort, is reduced. Compared to thecost of creating full term RS models, creating the reducedterm RS models results in savings of 11 out of 16 hours, 47out of 74 hours, 115 out of 192 hours, and 255 out of 392hours of CPU time on a single 75 MHz IP21 processor of aSGI Power Challenge for the five, ten, fifteen, and twenty

variable design problems, respectively. For consistency alltimes are given in terms of single processor performance,however coarse grained parallel computation on the 119node, distributed memory Intel Paragon XP/S at VirginiaTech was used to reduce the computational times by fac-tors of over 45 with 53 processors.

The bulk of the computational effort involved in usingthe RS approach to optimization lies in the initial creationof the RS models (which is trivially parallelized). Afterthey are created, the RS models can be evaluated and usedrepeatedly with minimal computational effort. This is ex-ploited by using more accurate central difference gradientinformation in the optimization, which would be expen-sive in high dimensional problems without the use of RSmodels. In addition, valuable information concerning thecomplex design spaces encountered and design trade-offsand sensitivities can be obtained through the evaluationof simple polynomials.

While the cruise drag errors in the optimal designs arewithin 1.0 count, the RMS errors in the RS models steadilyincrease from the five through the twenty variable prob-lems. The inability to perform calculations at all verticesof the design bounding box and the nonquadratic behaviorof the response are contributing factors. These errors canbe reduced by using the more accurate functional form ofthe drag polar, CD(x) = CDm(x) + (x) [CL - CLm(x.)}2.However, other methods are necessary to bring the RSmodel errors in the twenty variable case to acceptable lev-els. Using the more accurate drag polar along with a"zooming" technique to reduce the size of the design boxappears to be a promising approach. Research is currentlyunder way to investigate the use of these procedures in themultidisciplinary HSCT design problem.

AcknowledgmentsSupport for this research effort was provided through

the NASA Langley Research Center grants NAGl-1160with Mr. Peter Coen as contract monitor and NAGl-1562with Dr. Perry Newman as contract monitor.

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17


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for a High-Speed Civil Transport," Proceedings ofthe Sixth AIAA/NASA/ISSMO Symposium on Multi-disciplinary Analysis and Optimization, AIAA PaperNo. 96-4046, pp. 599-612, Bellevue, WA, September1996.

[11] V. 0. Balabanov. Development of Approximationsfor HSCT Wing Bending Material Weight Using Re-sponse Surface Methodology, Ph.D. thesis, Depart-ment of Aerospace and Ocean Engineering, VirginiaPolytechnic Institute and State University, 1997.

[12] R. S. Sellar, M. A. Stelmack, S. M. Batill,and J. E. Renaud. "Response Surface Ap-proximations for Discipline Coordination in Mul-tidisciplinary Design Optimization," Thirty-seventhAIAA/ASME/ASCE/AHS/ASC Sructures, Struc-tural Dynamics, and Materials Conference, AIAA Pa-per No. 96-1383, Salt Lake City, UT, April 1996.

[13] W. Chen, J. K. Alien, D. P. Schrage, and F. Mistree."Statistical Experimentation Methods for AchievingAffordable Concurrent Systems Design," AIAA Jour-nal, 35, No. 5, May 1997.

[14] R. H. Myers and D. C. Montgomery. Response SurfaceMethodology: Process and Product Optimization Us-ing Designed Experiments, John Wiley & Sons, Inc.,New York, NY, pp. 1-67, 134-174, 297-357, 640-655,1995.

[15] S. A. Powers. "Drag Minimization Using Exact Meth-ods," AIAA Journal, 2: 941-943, 1964, May 1964.

[16] M. Kaufman, V. Balabanov, S. L. Burgee, A. A.Giunta, B. Grossman, W. H. Mason, L. T. Watson,and R. T. Haftka. "Variable-Complexity ResponseSurface Approximations for Wing Structural Weightin HSCT Design," AIAA Paper No. 96-0089, January1996.

[17] D. A. DeLaurentis, C. E. Cesnik, J.-M. Lee,D. N. Mavris, and D. P. Schrage. "A New Ap-proach to Integrated Wing Design in ConceptualSynthesis and Optimization," Proceedings of theSixth AIAA/NASA/ISSMO Symposium on Multidis-ciplinary Analysis and Optimization, AIAA PaperNo. 96-4000, pp. 1835-1843, Bellevue, WA, Septem-ber 1996.

[18] P. J. Rohl, D. N. Mavris, and D. P. Schrage."Combined Aerodynamic and Structural Optimiza-tion of a High-Speed Civil Transport," Thirty-sixthAIAA Structures, Dynamics, and Materials Confer-ence, AIAA Paper No. 95-1222, New Orleans, LA,April 1995.

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[19] R. S. Seller, S. M. Batill, and J. E. Renaud. "ResponseSurface Based, Concurrent Subspace Optimizationfor Multidisciplinary System Design," Thirty-fourthAerospace Sciences Meeting and Exhibit, AIAA Pa-per No. 96-0714, 1996.

[20] I. Kroo, S. Altus, R. Braun, P. Cage, andI. Sobieski. "Multidisciplinary Methods for Air-craft Preliminary Design," Proceedings of the FifthAIAA/USAF/NASA/ISSMO Symposium on Multi-disciplinary Analysis and Optimization, AIAA PaperNo. 94-4325, pp. 697-707, Panama City, FL, Septem-ber 1994.

[21] S. Jayaram, A. Myklebust, and P. Gelhausen."ACSYNT—A Standards-Based System for Paramet-ric Computer Aided Conceptual Design of Aircraft,"AIAA Paper No. 92-1268, 1992.

[22] L. A. McCullers. "Aircraft Configuration Optimiza-tion Including Optimized Flight Profiles," Proceedingsof Symposium on Recent Experiences in Multidisci-plineary Analysis and Optimization, J. Sobieski, com-piler, NASA CP-2327, pp. 396-412, April 1984.

[23] D. L. Knill, V. Balabanov, 0. Golovidov, B. Gross-man, W. H. Mason, R. T. Haftka, and L. T. Wat-son. "Accuracy of Aerodynamic Predictions and ItsEffects on Supersonic Transport Design," MAD Cen-ter Report 96-12-01, Multidisciplinary Analysis andDesign Center for Advanced Vehicles, Virginia Tech,Blacksburg, VA, December 1996.

[24] E. Eminton. "On the Minimization and NumericalEvaluation of Wave Drag," Royal Aircraft Establish-ment Report AERO.2564, November 1955.

[25] R. V. Harris, Jr. "An Analysis and Correlation ofAircraft Wave Drag," NASA TM X-947, 1964.

[26] H. W. Carlson and D. S. Miller. "Numerical Methodsfor the Design and Analysis of Wings at SupersonicSpeeds," NASA TN D-7713, 1974.

[27] H. W. Carlson and R. J. Mack. "Estimation ofLeading-Edge Thrust for Supersonic Wings of Arbi-trary Planforms," NASA TP-1270, 1978.

[28] E. J. Hopkins. "Charts for Predicting Turbulent SkinFriction from the van Driest Method (II)," NASA TND-6945, October 1972.

[29] H. W. Carlson and K. B. Walkley. "Numerical Meth-ods and a Computer Program for Subsonic and Su-personic Aerodynamic Design and Analysis of Wings

with Attainable Thrust Corrections," NASA CR-3808, 1984.

[30] W. D. McGrory, D. C. Slack, M. P. Applebaum, andR. W. Walters. GASP Version 2.2 Users Manual,Aerosoft, Inc., Blacksburg, VA, 1993.

[31] R. L. Barger, M. S. Adams, and R. R. Krish-nan. "Automatic Computation of Euler-Marching andSubsonic Grids for Wing-Fuselage Configurations,"NASA TM 4573, July 1994.

[32] D. L. Knill. Implementing Aerodynamic Predictionsfrom Computational Fluid Dynamics in Multidisci-plinary Design Optimization of a High-Speed CivilTransport, Ph.D. thesis, Department of Aerospaceand Ocean Engineering, Virginia Polytechnic Insti-tute and State University, also found in MAD CenterReport 97-12-01, 1997.

[33] C. B. Craidon. "Description of a Digital ComputerProgram for Airplane Configuration Plots," NASATM X-2074, 1970.

[34] R. L. Barger and M. S. Adams. "Automatic Com-putation of Wing-Fuselage Intersection Lines and Fil-let Inserts with Fixed-Area Constraint," NASA TM4406, March 1993.

[35] Vanderplaats Research & Development, Inc. DOTUsers Manual, Version 4-20, Colorado Springs, CO,1995.

[36] R. T. Haftka and Z. Giirdal. Elements of StructuralOptimization, Kluwer, 3rd edn., pp. 211-219, 1992.

[37] 0. Golovidov. "Variable-Complexity Response Sur-face Approximations for Aerodynamic Parameters inHSCT Optimization," MAD Center Report 97-06-01,Multidisciplinary Analysis and Design Center for Ad-vanced Vehicles, Virginia Tech, Blacksburg, VA, June1997.

[38] A. A. Giunta, J. M. Dudley, R. Narducci, B. Gross-man, R. T. Haftka, W. H. Mason, and L. T. Watson."Noisy Aerodynamic Response and Smooth Approx-imations in HSCT Design," Proceedings of the FifthAIAA/USAF/NASA/ISSMO Symposium on Multi-disciplinary Analysis and Optimization, AIAA Pa-per No. 94-4376, pp. 1117-1128, Panama City, FL,September 1994.

[39] SAS Institute, Inc. JMP Users Guide, Version 3.1,Gary, NC, 1995.

[40] R. Mead. The Design of Experiments, Cambridge Uni-versity Press, New York, NY, pp. 542-548, 1988.

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[41] M. J. Box and N. R. Draper. "Factorial Designs, theXTX | Criterion, and Some Related Matters," Tech-

nometrics, 13, No. 4, pp. 731-742, 1971.

[42] T. J. Mitchell. "An Algorithm for the Constructionof D-Optimal Experimental Designs," Technometrics,16, No. 2, pp. 203-210, 1974.

Appendix A. Design of ExperimentsDesign of experiments (DOE) theory provides a sys-

tematic means of selecting the set of points (called an ex-perimental design) within the m dimensional design spaceat which to perform computational analyses. The 2m ver-tices formed by the upper and lower bounds on the de-sign variables define the design bounding box or hypercubewithin which the experimental design is created. The rangeof each design variable is scaled to span [—1,1] for bothnumerical stability and ease of notation.14 To create theexperimental design, the ranges of the design variables arediscretized at evenly spaced intervals. For example, a 2m

full factorial design is created by specifying each designvariable at two levels: the lower bound (-1) and the up-per bound (1). Therefore, this experimental design con-sists of every vertex in the design bounding box. The typeof experimental design created is denned by the numberof intervals and the distribution of the points on thoseintervals. The choice of experimental design depends onthe dimension of the problem, the computational resourcesavailable, and the type of function to which one wishes tofit the data. Four types of experimental designs are usedin this research: 3m full factorial designs,40 face centeredcentral-composite designs,14'110 small-composite designs,14

and D-optimal experimental designs.41

A 3m full factorial design40 is created by specifying thedesign variables at three levels (—1,0,1) corresponding tothe lower bound, midpoint, and upper bound of the designvariables. A 33 experimental design is presented in Fig. 18.This experimental design provides sufficient information toconstruct quadratic polynomial RS models. However, asthe number of design variables increases, the number ofcomputational experiments required becomes prohibitivelylarge. For example, a 3m full factorial design in twentydimensional space requires 320 ss 3.5xl09 computationalexperiments.

A face centered central-composite design14'40 (CCD)enables resolution of quadratic terms in the RS modelswith fewer computational experiments. It is created bytaking a 2m full factorial design and adding 2m "star"points on the faces of the hypercube and another pointin the middle of the design hypercube. The "star" points

Figure 18: 33 Full Factorial Experimental Design.

correspond to a set of design variables in which all variablesare held at their midpoint value except for a single variablewhich is specified at either its upper or lower bound. As thenumber of design variables is increased, these experimentaldesigns also become prohibitively large. Creating a facecentered CCD in a twenty dimensional space requires 220 +2 • 20 + 1 « l.OxlO6 computational experiments.

The small-composite experimental design14 allows evenfewer computational experiments with which to evaluatequadratic RS models. This experimental design is con-structed in a manner similar to that for the central-compositedesign except that a fractional factorial14 experimental de-sign is used in place of the 2m full factorial design. Thefractional factorial design includes only 2m~13 vertices ofthe m dimensional bounding box (/3 is an integer numbersmaller than m). There is some freedom in the value of/?, however it can not be too large or there will be insuf-ficient data to properly resolve all terms in the quadraticpolynomial. Certain vertices of the design space will nothave any associated data when using this experimental de-sign. While this is not an ideal situation, it is inevitablesince the number of vertices grows exponentially with thenumber of design variables.

The final experimental design used in this study is theD-optimal experimental design.41 Using D-optimal designsallows a flexibility in the number of computational exper-iments that is not allowed in the classical experimentaldesigns. D-optimal designs are also well suited for irregu-larly shaped design spaces, while the classical designs arebest suited to rectangular design spaces. D-optimal de-signs minimize the uncertainty in the polynomial coeffi-cient estimates and in the predicted value of the response.To create a D-optimal experimental design, one selects pdesign points out of q candidate points. In this research,the candidate points are derived from one of the three clas-

20


sical experimental designs described above. An iterativeoptimization method is then used to find the p D-optimalpoints.

For the five and ten variable cases, the JMP39 statis-tical software package is used to provide the D-optimaldesigns. The approach implemented in JMP uses sets ofrandomly selected seed candidate points from the classicaldesign, and the best points in terms of the prediction vari-ance are kept throughout the iterative procedure. Becausethe design points are selected in a quasi-random manner, itis unlikely that the experimental design chosen from JMPis truly a D-optimal set. Limitations in JMP prevent itsuse for the larger design problems. A routine employingMitchell's "k-exchange" method,42 developed by Dr. DanHaim, is used for the fifteen and twenty variable cases.

Appendix B. Parallel ComputingOver one thousand CFD drag solutions are required to

create the RS models and evaluate the errors for the 15variable and 20 variable designs. Performing these calcu-lations on a single processor of the SGI Power ChallengeR8000 machine takes nearly two weeks of wall clock time.This time can be reduced significantly by taking advantageof parallel computing.

A coarse-grained parallelization of the CFD analyseshas been implemented on the 119 node, distributed mem-ory Intel Paragon XP/S at Virginia Tech. While fine-grained parallelization offers potentially better performance,especially for large numbers of nodes, coarse-grained par-allelization is easier to implement and does not requirein-depth knowledge of or modifications to the complexcodes used. The parallel computations are organized in a"master-slave" paradigm, where one processor creates thedirectory structure and input files, distributes the jobs,and checks for their completion. Each individual CFD cal-culation is performed entirely by a single "slave" node.

Two measures of parallel performance are presented:the parallel speedup and efficiency. Speedup representsthe ratio of the serial calculation time to the parallel com-putation time on rip nodes. The parallel efficiency is thespeedup divided by the number of nodes. Ideally, thespeedup equals the number of nodes, bringing the effi-ciency to 1.0; however, this ideal behavior is not realized.File I/O, which is inherently serial, increases with Up andprevents the user from approaching ideal speedup and ef-ficiency for large numbers of nodes. Reading and writingof input files, CFD grid files, and CFD solution files areexamples of the file I/O present in the procedure.

In spite of these detractors, good performance is achievedwhen implementing the CFD calculations in parallel. Fig-

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ure 19 shows the parallel speedup and efficiency obtainedfrom performing 1080 CFD calculations for 360 HSCT con-figurations used in the 15 variable design. When using27 nodes, a speedup of 24.3 (0.90 efficiency) is realized.When using 53 nodes, a speedup of 45.4 (0.86 efficiency)is achieved. Even though a single processor of the IntelParagon is about ten times slower than a single proces-sor of the SGI Power Challenge, significant improvementsin the turn-around time are achieved when using a largenumber of nodes. On 53 nodes, the 1080 Euler calculationsrequire only 2.8 days to complete.

Appendix C. Design Trade-offsParametric studies are invaluable to a designer. These

studies provide information on trade-offs between variousdisciplines and influences, sensitivities to variations in de-sign variables, and effects of perturbations to a chosen de-sign. To perform a single parametric study, 25-50 analysesmay be required. To completely examine a design, manyof these parametric studies would be desired. The com-putational burden involved in performing these studies isgreatly reduced by using results from RS models insteadof data from a large number of CFD calculations since the

21


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expense lies only in the evaluation of the simple quadraticpolynomials. In this section, we examine variations in se-lected design variables to gain insight into the behaviorand relative importance of these parameters.

The optimal designs predicted from linear theory andEuler RS models for the five, ten, and fifteen variabledesign problems show the effects of the higher drag pre-dictions typical of the supersonic Euler solutions. Thethickness-to-chord ratio, t/c, is a direct trade-off (Fig. 20)between aerodynamics and structures. Aerodynamics dic-tates that the wing should be as thin as possible to reducethe drag and therefore the fuel weight. Structural opti-mization, on the other hand, would attempt to increasethe thickness of wing to reduce the wing weight. A com-

promise is met between the fuel weight and wing weight toobtain the minimum TOGW. The effects of replacing thelinear theory aerodynamics with Euler solutions is appar-ent in the fuel weight. The higher drag from Euler analysistranslates to higher fuel weights and a design in which theaerodynamic aspects are more dominant. The optimal de-sign is obtained at a lower t/c value to counter the effectsof the higher drag at the expense of the wing weight.

The linear theory curve for the fuel weight variationwith t/c has a nearly zero slope at t/c = 1.5%. This oc-curs because the fuel weight required to meet the rangeconstraint is not influenced only by the aerodynamics. Asthe wing gets thinner and the wing weight increases, thereis a point where the fuel weight penalty associated withthe increasing weight of the aircraft becomes as impor-tant as the fuel weight benefit from the reduced the drag.This point occurs near t/c = 1.5% when using linear the-ory analysis. When using Euler analysis, the fuel weight isstill dominated by the aerodynamic benefits over the rangeof t/c investigated.

The decreased inboard leading-edge sweep in the opti-mal design from Euler aerodynamics is an interesting oc-currence. This is not a result of aerodynamic-structuraltrade-offs, but rather it is due mainly to a compromisebetween aerodynamic influences. With the semispan, 6/2,and inboard LE length, SLE, > fixed, there is an increasein the size of the outboard section implicit with any in-crease in ALB,. The aerodynamic trade-off (Fig. 21) isbetween the high inboard sweep desired for improved su-personic performance and the size of the outboard sec-tion, which has poor supersonic performance. The nonlin-ear aerodynamic predictions have a relatively larger fuelweight penalty associated with the outboard section thando the linear theory results. This naturally shifts the op-timal ALEI to a lower value.

The wing weight plot in Fig. 21 shows a ALB, compro-mise between structural effects as well. At the lowest wingsweep, the planform takes on a structurally sound shape.However, the large planform area results in extra weight.At the other extreme, the planform area is reduced, butthe design is not as sound structurally. Extra weight isrequired to strengthen the structure. The best design is atrade-off between these two influences.

22


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23

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