AIAA 2001–0922Using Gradients to ConstructResponse Surface Models forHigh-Dimensional DesignOptimization ProblemsHyoung-Seog Chung and Juan J. AlonsoStanford University, Stanford, CA 94305

39th AIAA Aerospace Sciences Meeting and ExhibitJanuary 8–11, 2001/Reno, NV

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics1801 Alexander Bell Drive, Suite 500, Reston, VA 20191–4344

AIAA 2001–0922

Using Gradients to Construct Response SurfaceModels for High-Dimensional Design

Optimization Problems

Hyoung-Seog Chung∗ and Juan J. Alonso†

Stanford University, Stanford, CA 94305

The response surface method (RSM) is one of the most common techniques for build-ing approximation models using values of sampled data alone. In this work, a methodfor using both function values and their gradients with respect to the design variablesin the problem and construct response surface (RS) models has been developed. Thisapproach improves on the efficiency of using the RSM for high-dimensional design opti-mization problems. Provided that gradient information is available through inexpensivealgorithms such as the adjoint method, this approach significantly reduces the large com-putational cost needed for full RS model construction. The minimum number of functionevaluations (CFD analyses, in our case) required for a full quadratic RS model increaseswith the square of the number of design variables, N , effectively preventing their use inhigh-dimensional design optimization. In contrast, the proposed modification to the RSmodel, which incorporates gradient information, can approximate the original functionwith order N function evaluations, without significantly sacrificing the accuracy of theapproximation. After validating the feasibility of the proposed method using simple one-and two-dimensional analytic functions, the approach is applied to the aerodynamic de-sign of a supersonic business jet. The results of this 7-variable design problem indicatesthat the method can be effectively used to construct response surfaces with greatly re-duced computational cost. The advantages of the method become much more significantas the number of design variables increases.

Nomenclatureα weighting factor controlling the contribution of

gradient information for RS construction

� vector of the unknown coefficients in a responsesurface model

b vector of the least squares estimator of β

CD drag coefficient

ε random error

L sum of the squares of the errors

N number of design variables

ns number of sample points (also called sites)

nt number of test sample points to evaluate mod-eling error

RMSub unbiased root mean square error

RS response surface

x scalar component of x

x vector denoting locations (sites) in design space

X matrix of sample sites for RS model

y(.) unknown function

y(.) estimated model function for y(.)

∗Doctoral Candidate, AIAA Member†Assistant Professor, Department of Aeronautics and Astro-

nautics , AIAA MemberCopyright c© 2001 by the authors. Published by the American

Institute of Aeronautics and Astronautics, Inc. with permission.

1. Introduction

COMPUTATIONAL Fluid Dynamics (CFD)methods are recently gaining ground as a design

tool for aerospace systems. The optimization ofaerospace systems is an iterative process that requirescomputational models embodied in complex andexpensive analysis software. This paradigm is wellexemplified by the field of Multidisciplinary DesignOptimization (MDO) which attempts to exploit thesynergism of mutually interacting disciplines in orderto improve the performance of a given design, whileincreasing the level of confidence that the designerplaces on the outcome of the design itself. However,MDO methods greatly increase the computationalburden and complexity of the design process.1,2 Forthis reason, high-fidelity analysis software typicallyused in single discipline designs may not be suitablefor direct use in MDO.3 Faced with these problems,the alternative of using approximation models ofthe actual analysis software has received increasedattention in recent years. A second advantage of usingapproximation models during optimization process isthat they can be used with optimization algorithmswhich do not rely on the computation of sensitivityderivatives.

One of the most common methods for building anapproximate model is the response surface method

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(RSM) in which a polynomial function of varying or-der (usually a quadratic function) is fitted to a numberof sample data points using least squares regression.This method has achieved popularity since it providesan explicit functional representation of the sampleddata, and is both computationally cheap to run andeasy to use. However, response surface models haveseveral key limitations: their accuracy is only guar-anteed within a small trust region, and, by design,they are unable to predict multiple extrema. In ad-dition, the use of these methods is limited to designoptimization problems having only a small number ofdesign variables, since the number of required functionevaluations increases quadratically with the number ofdesign variables. A modified RSM that requires fewerfunctional evaluations must be developed if the ap-proximation method is to be used in a realistic MDOenvironment.

In this research, a modified RSM which uses gra-dient information is developed to reduce the compu-tational cost of constructing traditional RS models.The finite difference method has often been used tocalculate gradient information in a computational en-vironment. This method results in a very high com-putational burden in high-dimensional problems sinceadditional function evaluations are needed for eachnew variable. Therefore, the use of gradient informa-tion based on finite differencing to replace functionevaluations does not result in improvements in com-putational performance. However, recently developedadjoint-based methods7 can be used to compute gradi-ent information of systems governed by PDEs with lowcomputational cost. For a 7-dimensional problem, thecomplete gradient information at a given design pointcan be obtained with the equivalent of two functionevaluations (one for the flow solution and the other forthe adjoint calculation), whereas the finite differenceapproach requires 8 or N+1 (N being the number of de-sign variables) function evaluations. At a given designpoint, a flow solver together with an adjoint methodprovides one function value and seven gradients in thedirection of each design variable with the cost of onlytwo function evaluations. Six flow calculations can besaved with the use of these seven gradients when con-structing a RS model. The main idea of the modifiedRSM is to alleviate the high computational burden ofconstructing RS models for high-dimensional problemsby using the gradient information calculated through arelatively inexpensive method instead of using functionvalues. For the same example problem, the traditionalRS method requires at least 36 flow solutions to con-struct the model. In contrast, the modified RS methodwith gradient information only requires as few as 5function evaluations and 5 adjoint calculations. Oneadjoint calculation takes about same amount of com-putational effort as one function evaluation; therefore,the minimum possible computational cost to construct

a RS model can be reduced to the equivalent of 10 flowsolutions. This means that the required number of flowcalculations for the proposed method is only linearlyproportional to the number of design variables ratherthan quadratically proportional as needed for the orig-inal RSM. The benefit of the modified method willbecome more significant for problems involving largenumbers of design variables.

In our work, the proposed concept was first testedusing simple 1-D and 2-D analytic functions for val-idation and visualization purposes. The tests showexcellent agreement between the original and modi-fied RS models. The investigation of the applicabilityof this method to realistic problems was pursued bylooking at the aerodynamic tailoring of the supersonicwing-body business jet mentioned above. Seven designvariables were used to describe the aerodynamic shapein this test case.

2. Overview2.1 Original RS Method

The response surface method (RSM) develops poly-nomial approximation models by fitting the sampledata using a least squares regression technique. Thetrue response can be written in the following form:

y(x) = f (x) + ε, (1)

where f(x) is an unknown response function and εis the random error. The response surface model ofequation (1) can be written in terms of a series of ob-servations

yi = β0 +k∑

j=1

βjxij + εi, i = 1, 2...ns, (2)

where xij denotes the ith observation of variable xj

and ns is the number of samples.Equation (2) may be written in matrix form as

y = Xβ + ε, (3)

where

y ={

y1 y2 . . . yn

}T, and

X =

1 x11 x12 . . . x1k

1 x21 x22 . . . x2k

. . . .

. . . .

. . . .1 xn1 xn2 . . . xnk

.

β is a (k × 1) vector of the regression coefficients,and ε is an (n × 1) vector of random errors.

The vector of least squares estimators, b, is deter-mined in a way that it minimizes

L =n∑

i=1

ε2i = (y − Xβ)T (y − Xβ). (4)

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This condition simplifies to

XTXb = XTy. (5)

Thus, the least squares estimator of β is

b = (XTX)−1XTy. (6)

The reader if referred to Ref.5 for more details on thedevelopment of the RSM technique.

2.2 Modified RS Method

The modified RS method uses a reduced number ofobservations together with gradient information whenconstructing the vector, y, and the matrix, X, insteadof using the full number of observations required forthe original RS model. Equation (3) can be replacedby

ym = Xmβ + ε, (7)

and b, the vector of least squares estimators of β canbe obtained as

b = (XmTWXm)−1Xm

TWym, (8)

where, for a 2-D problem, ym, Xm and the weightingmatrix, W , can be defined as

ym ={

y1 y2 g11 g21 g12 g22

}T,

Xm =

1 x11 x12 x211 x2

12 x11x12

1 x21 x22 x221 x2

22 x21x22

0 1 0 2x11 0 x12

0 1 0 2x21 0 x22

0 0 1 0 2x12 x11

0 0 1 0 2x22 x21

,

W =

α 0 0 0 0 00 α 0 0 0 00 0 (1− α) 0 0 00 0 0 (1− α) 0 00 0 0 0 (1− α) 00 0 0 0 0 (1− α)

,

where gij denotes the gradient of yi in the directionof variable xj ( ∂yi

∂xj), and α is a weighting factor con-

trolling the contribution of gradient information to theleast squares regression process.

The minimum number of observations needed toconstruct the original RS model is given by (N +1)(N + 2)/2, where N is the number of variables.This number increases quadratically with N , whichprevents the RS method from being used for high-dimensional design applications. However, the pro-posed RS method needs only small fraction of thetotal observations needed for the original RS methodsince it also uses gradient information at the locations

of the observations. This advantage of the modifiedRS method allows it to be effectively incorporatedinto high-dimensional design problems. For 2 variablecases, the original RS model needs at least 6 obser-vations at different design points since there are sixunknown coefficients in the quadratic function that isbeing fitted to the data. When utilizing gradient infor-mation, we can construct the ym vector and Xm ma-trix with only two observations and four gradients, twowith respect to each design variable. We can eliminatethe required CFD analyses to compute the coefficientsof cross product terms in the model equation by usinggradient information obtained from an adjoint-basedmethod. Assuming the gradient information can beobtained cheaply using an adjoint-based method, thisnewly proposed procedure greatly reduces the compu-tational cost of generating accurate RS models.

2.3 Modeling Error Estimation

Each approximation model was constructed basedon the CFD results obtained at ns sample points. Toevaluate the modeling accuracy, another set of CFDcalculations were performed at nt randomly selectedvalidation points and the computed results of CD val-ues were compared with the predictions from the ap-proximation models at the same test locations. Theaccuracy of RS models were compared using the un-biased root mean squared (RMS) error, RMSub, andthe average % error.

The modeling error at each test site is defined asthe difference between the actual result from the CFDanalysis (yi) and the predicted value from the RSmodel (yi).

δi = |yi − yi|, i = 1, ..., nt. (9)

The average % error is defined as

average % error =1nt

nt∑

i=1

(δi × 100.0

yi), (10)

and RMS error is:

RMSub =

√∑nt

i=1 δ2i

nt. (11)

3. Test Problem :Supersonic Business Jet(SBJ) DesignThe design problem in question involves the drag

minimization of a supersonic business jet wing-bodyconfiguration at a specified lift coefficient. The aircraftgeometry and flow conditions were parameterized witha total of 22 potential design variables.

For the initial test case, total of 7 geometric designvariables were used. The chosen design variables rep-resent the radii at three different stations along theaxisymmetric fuselage, (x1 ∼ x3), and the thicknessto chord ratios at four span-wise stations (x4 ∼ x7).

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The airfoil shape for all wing stations was selected asa simple biconvex airfoil of varying thickness. Oncethe design variables were selected, an automatic meshgeneration procedure that was able to handle the ge-ometry variations imposed by the changes in the designvariables was utilized to automatically generate differ-ent sets of meshes needed for the CFD calculations.

The three-dimensional Euler solver, FLO87, devel-oped by Jameson 8,9 was used to calculate aerody-namic coefficients at sample design points chosen byincrementing each variable from the baseline designvalue using a design of experiments approach. Thefree-stream flow conditions were fixed at M∞ = 1.5and the coefficient of lift, based on the wing planformarea was fixed at CL = 0.1. Response surface mod-els were built using drag output data from the Eulersolver, and were incorporated into the nonlinear opti-mization code SNOPT which has been developed byGill, et al.10 to perform realistic design optimizationcalculations.

4. Design Tools4.1 Grid Generator

A grid generator called CH-GRID developed byReuther et al. was used for mesh generation for su-personic business jet wing-body configurations. CH-GRID is a stand-alone form of the C-H type grid gen-erator for the adjoint-based, single-block wing-bodydesign code, SYN87-SB. Figure 2 shows a typical wing-body mesh. A geometry generation engine was con-structed for this work so that the input geometry forCH-GRID can be easily created using an input file thatcontains the values of the design variables describedabove.

4.2 Flow Solver and Gradient Module

The CFD flow solver must meet fundamental re-quirements of accuracy, efficiency, robustness, and fastconvergence to be used in an high-dimensional de-sign optimization problem. The accuracy is importantbecause the approximation model accuracy and theimprovement predicted by the optimization processusing these models can only be as good as the ac-curacy of the flow analysis itself. Efficiency is alsorequired when the number of design variables increasesand the required number of the sample evaluations forconstructing the approximation models increases ac-cordingly. The robustness of the solver, i.e., its abilityto obtain a flow solution for a variety of configura-tion shapes and flow conditions, is particularly criticalfor the construction of sample databases for the ap-proximation models, in which large variations of thedesign variables is allowed. In addition, the benefit ofaerodynamic optimization lies in obtaining the last fewpercentage points in aerodynamic efficiency. In suchcases, the solution must be highly converged such thatthe noise in the figure of merit is well below the level

of realizable improvement. 11

Jameson’s FLO87 code used in this study easily metall of the previously mentioned criteria. FLO87 solvesthe steady three-dimensional Euler equations using amodified explicit multistage Runge-Kutta time step-ping scheme. FLO87 achieves fast convergence withthe aid of multigridding and implicit residual smooth-ing. Also, a driver program called RS87 was developedto utilize multiprocessor computers for analyzing anumber of different configurations simultaneously.

The necessary gradient information for the construc-tion of the response surface models was obtained in themanner described earlier with the use of an adjoint-based method. Details of the adjoint methodologyhave been previously published in the literature andare mathematically involved. These details are omit-ted in this paper, but the reader is referred to11–13 formore details. The particular implementation used inthis work is that embodied in the SYN87-SB solverdeveloped by Reuther and Jameson. The flow solverin this design code is effectively the FLO87 solver de-scribed above.

4.3 Optimization Procedure

Drag minimization of wing-body configurations wasperformed using the SNOPT program. SNOPT usesa sequential quadratic programming (SQP) algorithmthat obtains search directions from a sequence ofquadratic programming subproblems.10 Nonlinearconstraints were imposed on the minimization processby setting bounds for the values of the design variablesand constraining the admissible ranges of wing (wv)and fuselage volume (fv). The optimization problemcan be written as

minx∈Rm

CD(x), (12)

subject to0.8×wvi ≤ wv ≤ 1.2×wvi,0.8×fvi ≤ fv ≤ 1.2×fvi,

xmin ≤ x ≤ xmax,where wvi and fvi are the initial wing and fuselagevolumes.

5. Results and Discussion5.1 Graphical Validation of Modified RS Methodusing Simple Analytic Test Functions

The modified RS technique was applied to approx-imate simple one- and two-dimensional analytic func-tions. The results were compared with those of theoriginal RS method for validation purposes only. Fig-ure 3 shows the results of the one-dimensional valida-tion case. The original RS model was fitted to threesample data points while the modified RS model wasconstructed with only two sample sites and their cor-responding gradients. Although the modified RS is faroff from the actual function when the sample sites areseparated by large distances, ∆x, the two RS models

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become nearly identical as ∆x decreases. With properselection of the optimization algorithm, the modifiedRS can effectively predict local extrema.

For two-dimensional graphical validation of origi-nal and modified RS models, an analytic test functioncalled the peaks function was chosen from MATLABUser’s Guide14 as shown in Figure 4 (a). Results fromthis test case can also be seen in Figure 4. The orig-inal RS model was fitted to 6 sample points aroundthe global minimum of the test function, while themodified RS model used 3 sample sites and their gra-dients in each direction, which make up a total of 9pieces of information (3 function values and 6 gradi-ents) available for surface fitting. Both approximationmodels accurately predicted the global minimum andthe general shape of the test function. These graph-ical examples clearly demonstrate the applicability ofthe modified RS model for simple 1- and 2-dimensionalfunction approximation. Although in these simple testcases, the computational savings are minimal, whenthe number of design variables grows, the use of themodified RS technique becomes compelling.

5.2 Graphical Validation of RS Models for SBJTest Problem

To further investigate their ability to model theresults of the original CFD code, 7-dimensional RSmodels were created for CD of the supersonic busi-ness jet test problem, using sample data obtained fromCFD analysis. For the original RS model a total of 36CFD analyses are necessary to obtain the coefficientsof the functional fit. For the modified RS model, 15sample points together with gradient information atthese points were used. One-dimensional slices of the7-dimensional RS models generated through both theoriginal and modified methods are plotted in Figure5 for the first and fifth design variables correspondingto the radius of a fuselage station ahead of the lead-ing edge of the wing, and to the t/c ratio of the wingstation located right at the side-of-body. Additionalruns of the flow solver are included to get a feeling forthe accuracy of the presented techniques. The originaland modified RS models appear to resolve the functionof interest (coefficient of drag) quite well over a widerange of values of the design variables.

Figure 6 shows two-dimensional slices of originaland modified RS models for the same test case. Asin the one-dimensional surface cut case, the abilityto locate a local minimum and the general shapes oftwo models are quite similar. Through these graphicalcomparisons, the authors have gained confidence onthe fact that the modified RS model can approximatethe response function as accurately as the original RSmodels could.

5.3 Design of Experiments and ModelingAccuracy Comparison

In general, the second-order RS model has the form

y = β0 +N∑

j=1

βjxj +∑

i≤j

βijxixj , (13)

in which there are (N + 1)(N + 2)/2 coefficients to beestimated, where N is the number of variables. Whenconstructing a quadratic model, the design variablesneed to be evaluated at least at 3 locations to esti-mate the coefficients in the model. This leads to a3N factorial design of experiments that requires 3N

data samples. However, the Central Composite De-sign (CCD) introduced by Box, et al.15 has become apopular alternative for second-order RS models. CCDsare first-order (2N ) designs augmented by additionalpoints to allow for the estimation of the second-ordercoefficients.16 Unal, et al.3 found out that CCDenabled the efficient construction of second-order RSmodels with significantly less effort than for a full fac-torial design. Moreover, the fitted model could besuccessfully used within an MDO process with rea-sonable accuracy for cases with four to six designvariables. However, for design problems dealing witha large number of design variables, even CCD becomesunrealistic in the optimization process.

In this work, two design of experiments (DOE)methods for the original RS model were investigatedand the results of their modeling accuracy were com-pared with those of the modified RS model constructedbased on a star design in which only the central pointand the axial points along the each variable directionwere sampled. The first DOE tested for the originalRS was CCD and the second one was the minimumpoint design (MPD). The number of functional evalu-ations needed for MPD is exactly equal to the numberof coefficients in the RS model. Thus, the minimumnumber of CFD calculations needed to construct aRS model for the 7-dimensional design problem was36 and that for CCD was 143 with only one centerpoint. In contrast, only 8 or 15 sample data pointswere needed for the two variations of the star de-sign with the modified RS model. The modified RSmodel was constructed with the CFD results from 15design points selected from the original 36, and theirgradients in each variable direction. The total num-ber of pieces of the information used for the modifiedRS model is 120 (15 function values and 15*7 gradi-ents) which are typically available inexpensively froman adjoint-based sensitivity calculation method suchas the one described above. Figure 7 illustrates eachDOE method mentioned above for a two-dimensionalcase.

The results of a comparison study of these differ-ent design of experiments methods are summarized inTable 1. RS models were constructed based on the

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sample data collected from CCD, MPD and a star de-sign and their modelling accuracy was checked withCFD computations over a different set of 228 testpoints. The results show that the RS model usingCCD generally has the lowest error when comparedwith models using MPD or the star design approaches.However, the difference between them is very small asseen by the results in the Table. Thus, we can concludethat the modified RS model based on the star designusing only axial sample points can approximate theresponse function with similar accuracy to the origi-nal RS models using MPD or CCD, proving that theCFD calculations needed to fit cross product terms inthe response surface approximation function can bereduced and/or eliminated and replaced by gradientinformation at a series of sites, where only one designvariable is changed at a time.

Note that the accuracy of the modified RS modelis even slightly better than that of the original RSmodel using MPD. The result is mainly due to the factthat the total number of data used in the modified RSmethod is much larger since the gradients along allthe design variable directions at each sample point areused in addition to the function values at the samesample points. The result might also be attributed tothe existence of the weighting factor, α in the modifiedRS method. α was chosen in such a way to minimizethe average % error over the set of 228 test points.Therefore, the modified RS model has one more de-gree of freedom to adjust itself to the test data, which,in turn, reduces the error. However, the existence ofthe weighting brings about the problem of selecting thebest value for α. This is not a trivial problem, espe-cially when information about the design space is notavailable. However, in typical design problems, similardesigns are repeated several times. The results fromearlier designs can serve to adjust the value of the αparameter.

For demonstration purposes, one optimization cyclefor each case mentioned above was carried out and theresults are also compared in Table 1. As can be seen,all three models have almost same trends in predictedoptimum value (CD) and the design variables. This re-sult also contributes to the validation of the modifiedRS model for use in realistic design problems. Oneinteresting point is that the results from the 1st op-timization cycle using the CCD based RS model arereally close to those from the 3rd optimization cycleusing the minimum point design as shown in Table2. Therefore, the conclusion is reached that the CCDbased RS model generally has better chances of locat-ing the minimum point in fewer optimization cycles;however, the sequential optimization process using theRS model with MPD has almost the same ability withless computational cost since the number of requiredCFD evaluations for three optimization cycles withMPD is 108 whereas CCD requires 143 computations

for a single design cycle.

5.4 Optimization Results Comparison

The base design point for the SBJ design problemwas chosen to be given by the following values of thedesign variables in the problem: x1=0.45, x2=0.45,x3=0.3, x4∼x7=2.0(%). The design variables aredefined in section 3. Design optimization was per-formed using three different RS models with thefollowing limits on design variables and constraints.The results from three successive optimization cyclesare presented in Table 2.

0.8×wvi ≤ wv ≤ 1.2×wvi,0.8×fvi ≤ fv ≤ 1.2×fvi,

0.35 ≤ x1 ≤ 0.55,0.35 ≤ x2 ≤ 0.55,0.20 ≤ x3 ≤ 0.40,1.60 ≤ x4 ≤ 2.40,1.60 ≤ x5 ≤ 2.40,1.60 ≤ x6 ≤ 2.40,1.60 ≤ x7 ≤ 2.40,

where wvi and fvi are the initial wing and fuse-lage volumes.

Three test cases are chosen: the first consists of op-timization based on the original RS model, the secondis based on the modified RS model with 15 sample de-sign points and their gradients, and the last is basedon the modified RS model with 8 sample design pointsand their gradients. Three optimization cycles wereperformed for each test case.

As shown in Table 2, the predicted optimum designpoint and optimum value of CD for these cases becomenearly identical after the 3rd cycle. The predicted CD

for the original RS model case is 0.0072704 whereasthat for the modified RS model with 15 sample pointsis 0.0072723 with almost same optimum design points.Note that the fuselage radii defined at just forward ofwing (x1) and aft of wing (x3) tends to increase. Thistrend follows the well known supersonic area rule forthe configuration in question. Also, the thickness-to-chord ratios except the one defined at inside of fuselage(x4) all tend to decrease to the variable limit of 1.6%as expected. The reduced cross sectional volume willresult in a reduction in the volume dependent super-sonic wave drag. The total of 5.5 % CD reduction wasachieved for the first case while the reduction for thesecond case was slightly lower at 5.4 %.

5.5 Further Error Estimates of Modified RSModel and Projected Gains for High-DimensionalProblems

The investigation of the effect of the number of sam-ple sites on the accuracy of the modified RS methodis presented in Figure 8. The average percentage er-ror and RMS error over 108 randomly selected samplepoints are plotted against the the number of sample

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points used in the construction of the approximations.The number of sample points used in the construc-tion of the approximation ranges from the minimumamount of 36 to only 5 points. The error is nearly con-stant up to the 15 point case and it increases sharplyfor fewer data points. The case with 15 sample datapoints requires taking the baseline design point ± a∆x deviation from the base along all coordinate direc-tions and the gradients at all of these locations. Thisfigure clearly shows that the additional CFD calcula-tions needed to account for the cross product termsin the RS model can be eliminated with the use ofgradient information. The result illustrates the factthat the required number of sample data or CFD cal-culations for fitting a RS model increases only linearlywith the number of design variables, as compared tothe quadratic increase in the original RS method.

Projected computational savings for the modifiedRS method are plotted in Figure 9. For the more real-istic case of 40 design variables, the required number ofsampling points for the original RS model with MPDis 861, whereas for the modified RS model using gra-dient information obtained from an adjoint method itis only 162. Therefore a computational cost equiva-lent to about 700 CFD analyses can be realized. Thegraph shows the apparent advantage of the method inhigh-dimensional cases.

ConclusionsIn this study, a way to improve the efficiency of

traditional RS method by using gradient informationat the sampling sites was proposed and its applica-bility in a realistic design problem was evaluated for7-dimensional Supersonic Business Jet Design prob-lem. It was demonstrated that the modified RSmethod could generate approximation models as ac-curately and effectively as the original RS methodwith the greatly improved efficiency of the methodin terms of reducing the computational cost for ob-taining sample data. Confidence on the fact that theproposed method can be used efficiently in realistichigh-dimensional design problems such as in complexmultidisciplinary system design was gained. In partic-ular, we are interested in using this type of procedureto extend the use of adjoint methods to more dramaticgeometry changes such as those to be found in plan-form and geometry optimization. For design problemswhere larger areas of the design space are investigated,a procedure such as the one described in this workis likely to be more effective than traditional adjointapproaches where small steps for changes in the con-figuration are required.

Future WorkThe presented study identified the areas of further

investigation. They include (1) the research on a ro-bust method to determine the weighting factor, α, (2)

the testing of the modified RS method in more realis-tic design applications, (3) the investigation on varioustechniques of design of experiments (DOE) most suit-able for the proposed method, and (4) the applicabilitytest of using gradients in other approximation modeltechniques such as Kriging.

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13A. Jameson, N. Pierce and L. Martinelli ”Optimum Aero-dynamic Design Using the Navier-Stokes Equations,” AIAAPaper 97-0101, Jan. 1997.

14MATLAB User’s Guide for Unix Workstations, The Math-Works, Inc. Mass., August 1992, January-February, 1999

15G. E. Box, and K. B. Wilson, “On the experimental Attain-ment of Optimum Conditions,” Journal of the Royal StatisticalSociety Series B, 13, 1-45.

16G. E. Box, and N. R. Draper, Empirical Model Buildingand Response Surfaces, John Wiley, New York, NY, 1987.

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Analytic Test FunctionOriginal RS Model(ORS)Modified RS Model(MRS)Sample Sites for ORS Sample Sites for MRS

(a) Sample Sites x=[0.06, 0.15, 0.24]0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

(b) Sample Sites x=[0.08, 0.15, 0.22]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

(c) Sample Sites x=[0.10, 0.15, 0.20]0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

(d) Sample Sites x=[0.13, 0.15, 0.17]

Fig. 3 Modified RS Model Validation on One-Dimensional Analytic Test Function

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American Institute of Aeronautics and Astronautics Paper 2001–0922

−3−2

−10

12

3

−3

−2

−1

0

1

2

3−6

−4

−2

0

2

4

6

(a) 2-D Analytic Test Function

−2.5−2

−1.5−1

−0.50

−1.5

−1

−0.5

0

0.5

1

1.5−8

−6

−4

−2

0

2

X1

X2

Analytic Test FunctionOriginal RS model

(b) Visualization of Original RS Model on Test Function

−2.5−2

−1.5−1

−0.50

−1.5

−1

−0.5

0

0.5

1

1.5−8

−6

−4

−2

0

2

X1

X2

Analytic Test FunctionModified RS model

(c) Visualization of Modified RS Model on Test Function

−2.5−2

−1.5−1

−0.50

−1.5

−1

−0.5

0

0.5

1

1.5−8

−6

−4

−2

0

2

X1

X2

Original RS modelModified RS model

(d) Comparison of Original and Modified RS Models

Fig. 4 Modified RS Model Validation on Two-Dimensional Analytic Test Function

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American Institute of Aeronautics and Astronautics Paper 2001–0922

0.2 0.3 0.4 0.5 0.6 0.7 0.87

7.5

8

8.5

9

9.5

10

10.5

11x 10

−3

X1 (Fuselage Radius)

CD

Original RS Model CFD Test Results Initial Design Point

(a) Original RS Model 1-D Slice for X1 (all other designvariables fixed)

0.2 0.3 0.4 0.5 0.6 0.7 0.87

7.5

8

8.5

9

9.5

10

10.5

11x 10

−3

X1 (Fuselage Radius)

CD

Modified RS Model CFD Test Results Initial Design Point

(b) Modified RS Model 1-D Slice for X1 (all other designvariables fixed)

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 37.5

7.55

7.6

7.65

7.7

7.75

7.8

7.85

7.9

7.95

8x 10

−3

X5 (t/c of wing at a spanwise station)

CD

Original RS Model CFD Test Results Initial Design Point

(c) Original RS Model 1-D Slice for X5 (all other designvariables fixed)

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 37.5

7.55

7.6

7.65

7.7

7.75

7.8

7.85

7.9

7.95

8x 10

−3

X5 (t/c of wing at a spanwise station)

CD

Modified RS Model CFD Test Results Initial Design Point

(d) Modified RS Model 1-D Slice for X5 (all other designvariables fixed)

Fig. 5 RS Model 1-D Slice Validation for SBJ Design Test Problem

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American Institute of Aeronautics and Astronautics Paper 2001–0922

0.35

0.4

0.45

0.5

0.55

0.35

0.4

0.45

0.5

0.557.5

8

8.5

9

9.5

x 10−3

X1

X2

CD

(a) X1-X2 Surface Slice of Original RS Model (all otherdesign variables fixed)

0.35

0.4

0.45

0.5

0.55

0.35

0.4

0.45

0.5

0.557.5

8

8.5

9

9.5

x 10−3

X1

X2

CD

(b) X1-X2 Surface Slice of Modified RS Model (all otherdesign variables fixed)

0.2

0.25

0.3

0.35

0.4 1.6

1.8

2

2.2

2.47.6

7.65

7.7

7.75

7.8

7.85

7.9

7.95

8

8.05

8.1

x 10−3

X5

X3

CD

(c) X3-X5 Surface Slice of Original RS Model (all otherdesign variables fixed)

0.2

0.25

0.3

0.35

0.4 1.6

1.8

2

2.2

2.47.6

7.65

7.7

7.75

7.8

7.85

7.9

7.95

8

8.05

8.1

x 10−3

X5

X3

CD

(d) X3-X5 Surface Slice of Modified RS Model (all otherdesign variables fixed)

Fig. 6 RS Model 2-D Slice Validation for SBJ Design Test Problem

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American Institute of Aeronautics and Astronautics Paper 2001–0922

5 10 15 20 25 30 35 400

2

4

6

8

10

12

Number of Sampling Data Points for Modified RS Model

Ave

rage

% E

rror

ove

r 10

8 T

est S

ites

(a) Ave. % Error over 108 Random Test Sites vs. Number of Samples

5 10 15 20 25 30 35 400

20

40

60

80

100

120

140

160

180

Number os Sampling Data Points for Modified RS Model

RM

S E

rror

ove

r 10

8 T

est S

ites

(1.0

e−5)

(b) RMS Error over 108 Random Test Sites vs. Number of Samples

Fig. 8 Modeling Accuracy of Modified RS Model vs. Number of Samples

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American Institute of Aeronautics and Astronautics Paper 2001–0922

0 5 10 15 20 25 30 35 400

100

200

300

400

500

600

700

800

900

# of Design Variables

# of

Req

uire

d F

unct

iona

l Eva

luat

ions

for

RS

Required Functional Evaluations for RS

Original RS Reduced RS with Gradient (50% Axial Sites Sampling) Reduced RS with Gradient(100% Axial Sites Sampling) Redyced RS with Gradient, if adjoint is used

0 5 10 15 20 25 30 35 40−100

0

100

200

300

400

500

600

700

800

900

# of Design Variables

# of

Exp

ecte

d S

avin

gs in

Fun

ctio

nal E

valu

atio

n

Expected Savings in Functional Evaluations for RS

Reduced RS with Gradient (50% Axial Sites Sampling) Reduced RS with Gradient(100% Axial Sites Sampling) Reduced RS with Gradient, if adjoint is used

Fig. 9 Projected Gain for Modified RS Method vs. Number of Design Variable

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American Institute of Aeronautics and Astronautics Paper 2001–0922