[American Institute of Aeronautics and Astronautics 6th Symposium on Multidisciplinary Analysis and...

Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.

AIAA Meeting Papers on Disc, 1996, pp. 692-706A9638768, AIAA Paper 96-4055

Multidisciplinary Design Optimization of a supersonic transport aircraftusing a hybrid genetic/gradient-based algorithm

Dolf Bos

Delft Univ. of Technology, Netherlands

AIAA, NASA, and ISSMO, Symposium on Multidisciplinary Analysis and Optimization,

6th, Bellevue, WA, Sept. 4-6, 1996, Technical Papers. Pt. 1 (A96-38701 10-31), Reston,

VA, American Institute of Aeronautics and Astronautics, 1996, p. 692-706

A Multidisciplinary Design Optimization (MDO) procedure allows modifications to subsystems without thiscausing other subsystems to violate their constraints, while simultaneously yielding an optimal design; this is inaccordance with concurrent engineering, which implies incorporating as many relevant disciplines as practicable inan early stage of the design process in order to reduce costly feedbacks. The major problem that can be identifiedwith respect to multidisciplinary optimization is how to obtain the necessary information to guide the optimizationprocedure toward the optimum in a reasonable time, given the fact that a multivariate design evaluation withsufficient accuracy and predictive fidelity is extremely time consuming. The main improvement is to be obtainedby a reduction of the computational time. In the present work this is achieved by replacing the original analysis byan approximate analysis based on regression surfaces. A hybrid optimization method is employed consisting of aglobal genetic search, followed by a local gradient-guided optimization. (Author)

Page 1

96-4055

A96-38768

AIAA-96-4055-CP

Multidisciplinary Design Optimizationof a Supersonic Transport Aircraft

using a Hybrid Genetic/Gradient-Based Algorithm

by Dolf Bos'

Delft University of Technology, Delft, The Netherlands

Abstract

Modern aircraft design is a multidisciplinary processin which a large number of design variables isinvolved. Due to the ever increasing complexityof this process, which causes the number of couplingsbetween the different disciplines to increase as well,it is often hard to predict the impact of changesto a certain subsystem on the system as a whole.This calls for the development of a MultidisciplinaryDesign Optimization (MDO) procedure, which enablesmodifications to subsystems without this causingother subsystems to violate their constraints, whilesimultaneously yielding an optimal design. Thisis in accordance with a modern trend in industryknown as concurrent engineering, which impliesincorporating as many relevant disciplines aspracticable in an early stage of the design processin order to reduce costly feedbacks.

Especially in case of projects where the requirementsimposed on the design are particularly conflicting,like the design of a second generation supersonictransport aircraft, MDO techniques are a necessityto produce a feasible design.

The major problem that can be identified with respectto multidisciplinary optimization is how to obtainthe necessary information to guide the optimizationprocedure toward the optimum in a reasonabletime, given the fact that a multivariate designevaluation with sufficient accuracy and predictivefidelity is extremely time consuming. Since thecalculation effort required for a multidisciplinaryoptimization problem is excessive, regardless ofthe method of optimization employed, the main

improvement is to be obtained by a reduction ofthe computational time. In the present work thisis achieved by replacing the original analysis byan approximate analysis based on regression surfaces.

A hybrid optimization method is employed, consistingof a global genetic search, followed by a local gradient-guided optimization. In this way, the goodcharacteristics of the genetic algorithm (robust)and the gradient-guided algorithm (efficient, exact)are combined.

Notation

areaaspect ratio

a coefficientb approximate coefficientD dragF(v!,v2) Snedecor-distribution with v1 degrees of

freedom of the numerator and v2 degreesof freedom of the denominator

H string schemaISP specific impulse (inverse of specific fuel

consumption)i indexj indexKp lift gradient in potential flowk number of samplesL liftM Mach numbern number of variablesP pressurep number of polynomial coefficientsR range

* Copyright © 1996 by A.H.W. Bos. Published by the AmericanInstitute of Aeronautics and Astronautics, Inc. with permission.

692American Institute of Aeronautics and Astronautics

r residualS area

distances2 mean square about regression (mean square

of the residuals)V velocityW weightX matrixx variabley function response valuey function approximate valuey mean value of y

8 string defining-lengthe (total) pressure ratioX bypass ratioo2 variance

subscripts

0appccrdesegsprsubtowf

at intakeapproachcompressorcruisedesign pointexitgrosssupersonicsubsonictake-offwing-fuselage

EINOX NOX Emissions IndexMCC2 Multiple Correlation Coefficient Squared

Introduction

The major problem concerned with multivariateoptimization is how to obtain the necessaryinformation to guide the optimization proceduretoward the optimum in a reasonable time, giventhe fact that a multivariate design evaluation withsufficient accuracy and predictive fidelity is extremelytime-consuming. The presently known optimizationmethods get this information either by analyzingthe gradients of the objective function and constraintswith respect to the independent design variablesor by performing a very large number of designevaluations.

Because of the large number of design variablesand constraints, the number of required gradientscan become quite large whereas these gradientsmust be recalculated for each improved design untilthe optimum is attained.

Since the calculations for an aircraft design analysisare rather complex and often include iterative solutionprocedures, no closed-form analytic relations areavailable for the gradients needed. Instead, theymust be obtained by finite differencing which requiresmany design evaluations. It was established thatthe accuracy of gradients thus obtained is not veryhigh.

Some methods to decompose either the processof calculating the gradients or the optimizationproblem itself were presented in the 1980's andtested in the present work. Instead of calculatingglobal derivatives by perturbing and recalculatingthe complete design, partial derivatives are generatedby finite differencing of subsystems. These methodsof decomposition are especially well suited for parallelcomputation and it is here that their main advantagelies. The use of methods of decomposition incombination with second-order optimization algorithmswas not completely successful, however. Also, theproblem of inaccurate gradients remains.

The use of non-gradient guided algorithms, notablythe genetic algorithms fiat enjoy a fast rise in popularitytoday, was investigated in the present work as well.The large number of design evaluations requiredand the problem of premature convergence arethe main disadvantages identified for this type ofoptimization. Furthermore, as opposed to gradientmethods, there is no guarantee that the optimumis attained or even will be found at all. On the otherhand, the method is likely to locate the global optimuminstead of converging on a local one. Therefore,the robustness of the method, unless prematureconvergence occurs, is higher than that for gradientguided methods.

Since the calculation effort required for amultivariate/multidisciplinary optimization problemis excessive regardless of the method of optimizationemployed, the main improvement is to be obtainedby a reduction of the computational time, eitherby using expensive high-speed computers or byreducing the time needed for a single design evaluation.


The latter can be achieved either by arranging theanalysis in a smart way or by using a fast and simpleapproximate analysis instead of the slow detailedone. The first option includes minimizing feed-backsthat require iterative solutions, using smart estimatesfor values needed at the beginning of the analysisthat are only known exactly at the end, withoutre-iteration, in short, trading accuracy for speed,within reasonable limits of course. The second optionwas used extensively in the present work.

Two kinds of approximate analysis exist. The firstkind is a local approximation, at the current designpoint, which has to be updated by the exact analysisafter each optimization step. The most simple exampleof this kind of approximate analysis is to use thefirst order derivatives of the current design withrespect to its independent variables, as does everygradient guided optimization algorithm.

The second kind is global approximation by regressionsurfaces, in the present work represented by Second-order polynomials. This concept has many advantages.Not only does it shorten the required computationaltime, also the approximation can be generated inadvance and does not need to be updated afterevery optimization step. The approximate analysismay be used to include results from disciplinesthat are analyzed by separate programs that cannotbe implemented within an optimization algorithmor that simply are too time-consuming to considerusing them in an optimization cycle. Global approxi-mation is also a tool to include analyses that areperformed at a geographically remote location (adifferent department or company). Last but notleast, the use of second-order polynomials yieldsthe relation between the objective function, the con-straints and the independent design variables assimple, closed-form functions. Therefore, gradientinformation is calculated fast and easy.

In the present work a hybrid optimization methodwas employed, consisting of a global search usinga genetic algorithm to decrease the design space,followed by a gradient guided search to locate theexact optimum. The global search uses the exactanalysis in which all redundant iterative loops areomitted, using approximate analysis to include somedisciplines. The gradient guided search uses onlyapproximate analyses generated for the reduceddesign space containing the optimum as obtained

from the global search. In this way the goodcharacteristics of gradient methods (efficient, exact)and global search methods (robust) are combined.

The optimization was performed on a second generationsupersonic transport plane (SST). The main featuresof the analysis method are:

a design sensitive wing weight predictionmethodthe use of a panel method (NLRAERO)to obtain subsonic and supersonic aerodynamiccharacteristics of high-speed configurationsapplication of the Polhamus leading-edgesuction analogy and Carlson attainable thrustconcept to account for partially separatedleading-edge flow and the resulting leading-edge vortices, important for highly sweptwings at angle of attacka procedure to predict the off-designperformance of bypass engines designedfor supersonic flight, including a detailedintake routine to provide optimal intakeefficiency at the design point and to includespillage air and powerplant drag,an estimation of the environmental impactof a second generation SST

Decomposition Methods

Sobieszczanski-Sobieski ([ref.!5]-[ref.24]) was thefirst to address the apparent paradox inmultidisciplinary design between the increasingnumber of design variables and the decreasing freedomto manipulate them because the consequences onother variables could not be overseen by the humandesigner. He introduced the concept of systemdecomposition as a means of optimizing a designwith many variables subject to many constraints.The advantage of dividing the design process intovarious subsystems, while preserving the couplingsbetween them, is that the calculation of the derivativesneeded for a gradient-based optimization (and possiblythe analysis and optimization of the subsystemsthemselves) can be performed concurrently andindependently while taking into account the effectsthat changing a variable in one subsystem has onparameters of other subsystems.

Sobieszczanski-Sobieski introduced two main classes


of decomposition schemes, being the straightforwardhierarchical breakdown and (he network breakdown.It was established that the hierarchical decompositionscheme, although very well suitable for structuraldesign problems, was not applicable to conceptualaircraft design studies. Systems that cannot bedecomposed into a hierarchical tree structure arereferred to as networks and a different approachis needed for such systems. The step from hierarchicto non-hierarchic systems was introduced bySobieszczanski-Sobieski in 1988 ([ref.22], see also[ref.l], [ref.21], and [ref.23]).

If the total analysis is decomposed in subsystemsthat can be treated as blackboxes converting inputto output, the local sensitivity derivatives (beingthe partial derivatives of the output parametersof the blackbox under consideration with respectto the input parameters) can be computed. Thelocal sensitivity derivatives pertaining to eachsubsystem can be analyzed independently andconcurrently. By using the chain rule of differentiation,the local sensitivity derivatives can be transformedinto global sensitivity derivatives that measure theinfluence of the independent design variables onthe parameters of interest (for optimization purposes,only the gradients of the objective function andthe constraints with respect to the independentdesign variables are needed).

Sobieszczanski-Sobieski introduced two differentoptimization procedures based on this method ofSystem Sensitivity Analysis. The most obvious methodis to use the obtained global sensitivity derivativesdirectly in a gradient-guided optimization algorithm.A somewhat more complicated procedure alsodecomposes the optimization problem itself intodifferent suboptimizations, that can be solvedindependently and concurrently too, whilesimultaneously preserving the couplings betweenthe different subsystems. In this way the multivariateoptimization problem is decomposed into severalsmaller problems, that optimize the same objectivefunction while sharing the responsibility to minimizeconstraint violations.

The advantage of this method of ConcurrentSubsystem Optimization in comparison with thesimpler alternative, according to Sobieszczanski-Sobieski, is that it relies on approximate analysisonly in a very limited way, since only information

that represents the couplings between two differentsubsystems is transmitted by means of the sensitivityderivatives, whereas the simpler alternative completelyrelies on first order approximations. However, itwas established that for a truly multidisciplinaryoptimization of a conceptual aircraft design theinterdisciplinary couplings are so many, that mostof the relations are approximated. Furthermore,the method of Concurrent Subsystem Optimizationrequires that a unique subset of the vector ofindependent design variables is assigned to eachsubsystem, which is very difficult for a conceptualaircraft design. This implies that, although the methodof Concurrent Subsystem Optimization in principlecould employ any kind of optimization algorithmdesired, in the practice of multidisciplinary optimizationof a conceptual aircraft design it will degenerateto a first order optimization method, just like thesimpler alternative.

The method of System Sensitivity Analysis wasapplied to the optimization of a medium-rangesubsonic airplane design, analyzed by means ofthe procedures of [ref.25]. The method was linkedto a Sequential Linear Programming routine as wellas to a Sequential Quadratic Programming routine.Combination with the linear programming routineshowed good results, although progress was ratherslow because of the necessity to impose move-limitson the design variables. Combination with the quasi-Newton method proved not so successful, however.

The main advantage of the method of System SensitivityAnalysis is the possibility to carry out the calculationsconcurrently, by parallel computation. If this isnot practicable, the method is far less efficient thanthe brute force method that calculates derivativesby perturbing and recalculating the complete analysis,especially since the method of System SensitivityAnalysis calculates a lot of redundant derivatives,that do provide useful sensitivity information butare not required for the optimization. Furthermore,since the relations in a conceptual airplane designanalysis are usually not available as simple, closed-formrelations, derivatives have to be calculated by finitedifferencing. Using the method of System SensitivityAnalysis involves the risk of cumulatively increasingthe inaccuracies of the derivatives, because everypartial derivative has an error proportional to themagnitude of the perturbation.

695

American Institute of Aeronautics and Astronautics

Using quasi-Newton optimization strategies, impliesthat second-order information is needed for thequadratic direction-finding problem. This informationis obtained by means of a Hessian recursive updatingformula that requires gradient information frompreceding steps. Furthermore, after the search directionhas been calculated, a line search has to be performedto determine the steplength to be taken in the searchdirection. This implies that a large amount of functionevaluations has to be performed by the optimizationmethod. These problems require the analysis tobe coupled directly to the optimization algorithm.If no parallel computation of gradients can be carriedout, the decomposition scheme -for the reasonsmentioned above- should be dropped. In [ref.4]the analysis is coupled directly to the optimizerbecause of the simplicity of the analysis. For a trulymultidisciplinary design problem this is not possible,because the calculation effort would become excessive,which is why special MDO techniques are beingdeveloped in the first place.

Thus -unless Sobieszczanski-Sobieski's method ofConcurrent Subsystem Optimization is used- it seemsthat using second-order methods of optimizationwill force the designer to drop the decomposition.

the accuracy of the approximation, which makesprogress very slow.

An alternative way of approximate analysis is torepresent the relation between a certain parameterand its variables by a single function. In this worksecond-order polynomials are used. To obtain suchpolynomials, it is necessary to perform a numberof design evaluations and create a hypersurfacefit through the obtained points, using the methodof least squares. Such a hypersurface fit is knownas a regression surface.

In the present study, the various relations to beapproximated will be represented by second-orderpolynomials in n variables, therefore:

(1)1=1 j-i

If the number of samples taken is denoted by k,the residual can be expressed as follows:

(2)

Approximate Analysis

A possible solution to the problem described aboveis the use of so-called approximate analysis. Inliterature, methods using linear Taylor expansions,like all linear programming methods do, are referredto as approximate methods too. In the context ofthis work, the term approximate analysis will implyhigher order approximations of accurate but complexmethods of analysis. This enables the complete designanalysis to be incorporated within the optimizationcycle, thus enabling the algorithm to update theHessian and to carry out function evaluations duringthe line search.

Earlier results with approximate analysis werereported in [ref.3], [ref.6], [ref.ll] and [ref.14].However, these methods consisted of either localapproximations or a combination of a local anda global approximation. The disadvantage of suchmethods is that the exact analysis still has to becarried out after every iteration step. Furthermore,move-limits have to be imposed in order to preserve

Written in a slightly different shape:

\0 0 3kl 1 "' akn n akni-l n*l '"

3 i b 1 — v = rk*n(n+3) 'n(n*3) "k k

with:

a a _Y(k) v(k)Ct, - i» d, — A, • •• Akl kn 1 n

(3)

,.. „.

Thus, the desired relation can be written as:


with:

(I)2

<2> Y<2>

((«2

,«2

etc.

The least-squares solution with respect to the residualscan be obtained by solving the following equation:

[x]T[x][b] = [Xf[y] (5)

Therefore, a test is performed to establish whetherthe statistic significance of the regression is acceptable.Departing from the assumption that the residualsof the regression are Normally distributed, r. - N(0,o2),it can be shown ([ref.5]) that a pivoting functionfor the hypothesis that all coefficients except theconstant should in fact be zero is:

-F(p-i,k-p) (8)

[b]=([X]T[X])'1[x]T[y](6)

(this is a very elementary relation, the proof of whichcan be found in many basic textbooks). The matrix[X] is a [k x p] matrix, the vector [y] has dimensionk, whereas the vector [b] has dimension p, wherep is the total number of coefficients (including b0)in the polynomial:

p=(n+l)(n+2)/2.

This is the minimal amount of samples that hasto be generated in order to enable the constructionof a least-squares fit. The best amount of sampleson which to base the regression must be determinedby satisfying a couple of opposing demands onthe goodness-of-fit of the regression. First of all,it is desired that a sufficient percentage of the varianceof the sample data is "explained" by the regression.This percentage is measured by calculating the so-called Multiple Correlation Coefficient Squared,MCC2:

(7)

E(y,-y)2

The problem with this coefficient is, that it can alwaysbe made equal to one, by taking the number ofsamples the same as the number of coefficientsin the regression. The statistical significance of suchan exact fit will be remote, however.

in which the mean square of the residuals, s2, isthe sum of the squares of the residuals dividedby (k-p). If the calculated value for the obtainedregression line exceeds the table value of an F-distribution for, say 5%, then the hypothesis thatthe terms other than the constant should in factbe zero cannot be accepted on statistical grounds,with a 5% chance of being wrong (that is, the first-ordererror is 5%). On the other hand, if the calculatedvalue does not exceed the table value, the zero-hypothesis is accepted and a better regression linemust be found. Nonetheless, there is a chance thatthis is incorrect. This so-called second-order errordepends on the actual value of the F-test. However,the definition of the zero-hypothesis is such thatthe unjust rejection of it is considered the mostserious, and the chance of this happening is lessthan 5%. It is recalled that the statistical analysisof the regression was based on the assumption thatthe residuals are Normally distributed. The assumption

r.~N(0,o2) implies that _l-N(0,l). Since s2 is an1 o

unbiased estimator of o2, the value _i can be examineda

for each sample point to check if the assumptionof the residuals being Normally distributed is arealistic one. As 95% of an N(0,l) distribution liesin the interval (-1.96,1.96), the assumption wouldbe supported (at least could not statistically be rejected)if roughly 95% of the observed values of r. / s wouldbe less than 2.

This second requirement forces the designer to basethe regression on as many samples as possible,

697


whereas the first requirement is best satisfied bybasing the regression on the minimal required numberof samples. Furthermore, even if a regression surfaceis obtained with satisfying values for the MultipleCorrelation Coefficient Squared as well as the F-testof statistical significance, this is by no means sufficientguarantee that the equation is applicable forapproximating purposes. It has, so far, only beenestablished that at least one of the non-constantterms is significant in explaining the observed variancein the true responses and that a sufficient percentageof this variance is explained by the equation. However,the errors in the prediction (the residuals) can stillbe so large that the equation is of no use (theregression surface is in this case merely fitted tothe errors).

Therefore, it is necessary to check as well the standarderror of the regression, s. This value is obtainedby taking the square root of the mean square aboutthe regression (mean square of the residuals, s2).Usually this value is expressed as a percentageof the mean response. It was found that for largeamounts of samples, this value was almostindependent of the number of samples used.

The approximations can be generated independentlyand concurrently prior to the optimization itself.Furthermore, they can be provided by the departmentor group specializing in the discipline underconsideration, thus benefiting from experience. Thisalso implies that analyses may be included thatare carried out by separate programs or at remotelocations or that are too time-consuming to considerusing them in an optimization cycle. In this sensea kind of decomposition is inherent in this methodof global approximation. Since the approximationsare global, they can be used throughout theoptimization process, without the need to be updatedafter every iteration step. But most importantly,gradients can be calculated analytically. This means,that at each iteration step, the optimization routinecan perform function evaluations and gradientcalculations in a very fast and simple way.

In Figure 1 a wing area - wing aspect ratio hyperplaneis shown for the subsonic aircraft testcase mentionedbefore, according to the approximate analysis (above)and to the exact analysis (below).

Although the approximation seems to be quite

satisfying, it is clear that some further fine-tuningis necessary, since obviously slight inaccuraciesin the prediction of critical constraints can causesignificant shifts in the optimal values of the designvariables.

0 , . Feasible regionS Cm2] engine failure * rconstraint cruls. thrust \ constraint

\ constraint

10 12 \ M IE IB 20 22 24 2E 28 30

0 m. climbconstraint

Figure 1 Comparison between approximateand exact analysis

On the other hand, this problem is universal inconstrained multivariate optimization of complexsystems, as any method of analysis will have a certainprediction error. Since the optimum of a constrainedproblem will always lie at the vertex of severalconstraints, slight inaccuracies in the predictionof these constraints might result in a design beinginfeasible and therefore useless upon closer examination

This implies that a design analysis should be asmultivariate and multidisciplinary but simultaneouslyas accurate as possible, without this rendering asingle design evaluation too tirreconsuming. Otherwise,the primary target of MDO and concurrent engineering -the reduction of feedbacks in the design process-cannot be achieved.

Since the calculation effort is the biggest problemin MDO, whereas the accuracy is most important


during the final fine-tuning of the optimum, it isfelt that an acceptable compromise is obtained bysaving computational time at the cost of accuracyduring the global search for the optimum, whilegradually increasing the accuracy of the approximateanalysis during the local search.

Applied to the optimization of an SST, approximationby regression surfaces proved an excellent meansto incorporate the panel method results, not onlybecause the panel program could not be incorporatedin the optimization algorithm because it is verytime-consuming, but also because the results provedto be extremely sensitive to noise, a deadly obstaclefor gradient-guided algorithms. This is illustratedin Figure 2, in which the panel method results ofthe wing-fuselage lift gradient for the final supersonicconfiguration are plotted for a number of differentsweep angles, together with regression equationsbased on different amounts of samples.

Kpu

3.0

CT) regression on 22 aomplesi error = 8X(2) regression on 21 samples* error = 35i

•40° Leading-edge sweep angle 60°

Figure 2 Noisy aerodynamic data andregression

Since noise is a general problem in case of amultidisciplinary aircraft design analysis, becauseoften iterative solution procedures are required,and especially in case of a supersonic aircraft designbecause of numerical discontinuities, approximationby regression surfaces is an excellent means to "iron"an irregular, noisy surface that could not otherwisebe optimized by means of a gradient-guided algorithm.

Direct Search Methods

The main problem concerned with multidisciplinarydesign optimization is, that calculating derivativesof the objective function and the constraints withrespect to the design variables, is a very tirreconsumingtask, while the accuracy of the obtained values inmany cases is not very high. Therefore, it wouldbe logical to consider optimization methods thatdo not need such derivatives. Such methods areknown as direct search methods. The problem withthese methods is, that the information about thecharacter or shape of the design space which isnormally provided by the derivatives, is now notavailable. This information, which is needed to guidethe design towards the optimum, must thereforebe obtained in a different way. Unfortunately, thisoften implies that either an extremely large amountof designs is generated, or that not all the necessaryinformation is available, leading to an improved,but not optimal design.

In the present application, a genetic optimizationmethod was employed ([ref.7], [ref.8], [ref.9], [ref.10],[ref.12] and [ref.13]).

Genetic search algorithms are based on Darwin'stheory of survival of the fittest. According to thistheory, an individual (design) with favorable geneticcharacteristics (values of design variables) will bemost likely to survive and produce offspring (matingof the fittest). In a genetic algorithm, evolution issimulated by generating a population of designs,either at random (primeval soup) or by judgementor intuition, from which a subset is selected to produceoffspring. It is expected that only the best designswill survive and mate again (produce new designs)thus increasing the fitness of the population andhopefully the highest fitness of the population aswell.

Offspring is produced by means of three differentprocesses:

1) reproduction

2) crossover

3) mutation

(selecting candidates forthe crossover process)(exchanging genetic materialbetween mates)(changing genetic materialwithin an individual)


To facilitate this, the values of the design variablesfor each individual (i.e. design) are representedas binary numbers, arranged in long chains whichact as the equivalents of DN A-strings. Reproductiontakes place by picking members from the populationat random, but with a bias towards those memberswhose fitness (measured by some augmented objectivefunction) is relatively high in comparison with theaverage fitness of the population.

Next, the individuals selected for crossover aregrouped into pairs which will act as parents. Twosites on their pseudo-DNA strings are selected atrandom. Subsequently, the sections between thesites are swapped between the two mates (two-pointcrossover). Thus, cross-over is a binary operator;two individuals create two new individuals.

After cross-over has taken place, it is establishedper child, again by quasi-random number, if mutationshould take place. The probability of mutation isusually very small and the process should be regardedas a guard against premature loss of beneficialcharacteristics. If mutation takes place, a site onthe DNA string of the child is selected and the bitunder consideration is inverted. Mutation, therefore,is a unary operator; one individual creating onenew individual.

After all three stages of regeneration are completed,the new population is reduced to its original sizeby letting the least fit individuals die (survival ofthe fittest). In some genetic algorithms the parentsalways die. A special measure, called the elitistoption, is included in this case to preserve the bestindividual in order to avoid deterioration of thehighest fitness in the next generation.

The powerful mechanism that guides the searchtowards the optimum is the combination of highlyfit, short defining-length schemata, the so-calledbuilding blocks ([ref.9]). A schema is a representationof a certain section of a string, e.g. the string 0111000can be represented by different schemata such as:

HI:H2:

n****0

in which the asterix (*) can be either a 0 or a 1.Thus, a schema represents a family of pseudo-DNAstrings and can therefore be interpreted as a

hyperplane in the design space. The defining-lengthof a schema, 8(H), is defined as the difference betweenthe first and the last fixed position and it can beeasily verified lhat S(H1>=5 and 8(H2)=1. The advantageof short defining-length schemata is obvious; theyhave the smallest chance of getting disrupted bycrossover (schemata are relatively unlikely to bedisrupted by mutation) and the best chance of beingpicked for reproduction.

The genetic algorithm of [ref.10] was applied tothe previously mentioned subsonic testcase. Lookingat the bit-representation of the pseudo-DNA strings,the evolution according to the building-block theoremis clearly visible. In Figure 3 the population at thesecond generation (after 100 function evaluations)is shown next to tfie population at the 208"1 generation(after 6000 function evaluations). For clarity, theO's in the strings have been replaced by blanks.The change from a population with a purely chaoticdistribution of genetic material into a populationin which a very distinct pattern among the stringscan be discerned is very striking.

j j ' j isjj;i: :..:!','

Figure 3 Comparison between first and lastgeneration

The advantage of a genetic algorithm over a gradient-guided algorithm is, that a genetic algorithm workswith a population of designs instead of a singledesign, which largely reduces the possibility ofconvergence towards a local optimum (and thereforeincreases the robustness of the method, i.e. theprobability of finding the global optimum). Therefore,the method can be applied to non-convex or noisyoptimization problems. Furthermore, since the methoddoes not require gradient information, it can handlediscrete variables.

Disadvantages of genetic algorithms are the inefficiency(very large number of function evaluations), theinaccuracy (it is very unlikely that the algorithm


will locate the exact optimum) and prematureconvergence (a few designs with superior but farfrom optimal characteristics which quickly willdominate the population).

Especially the large amount of function evaluationsis a major problem, since apparently, the use ofdirect search methods -although no gradients haveto be calculated- does not reduce the calculationeffort. This again suggests the use of approximateanalysis.

The main reason to employ a genetic algorithmin the present work, is the fact that the relationsbetween the objective function, the constraints andthe design variables are very noisy for a supersonictransport aircraft. Furthermore, a genetic optimizationcan be very helpful to provide initial designs thatcomply with the performance constraints. Sincethe feasible design space for the optimization problemspecified in this work (a second generation supersonictransport aircraft) is extremely small (even if fewconstraints are enforced) and the number ofinterdisciplinary couplings is rather large, creatinga feasible design by means of classical designprocedures is a very hard if not impossible task.By defining a penalty function that sums the squaresof violated constraints the genetic algorithm maybe used to find one or more designs that complywith the performance constraints. These designsmay then be used as baseline designs for the gradientguided optimization. Since the gradient-guidedoptimization will exclusively use approximatedinformation, a genetic search prior to the gradient-guided search may enable the designer to decreasethe design space, leading to more accurate regressionsurfaces.

Thus, in the present work, a genetic search isperformed on an analysis from which all redundantiteration loops are removed and in which somecalculations -notably the results from the panelmethod- are approximated by means of regressionsurfaces. In a reduced design space around theoptimum found by the global search, regressionsurfaces are computed for the constraints of interestand a gradient-guided search will be performedto take care of the final fine-tuning of the optimum.

In this way the good characteristics of the globaland the local optimization are combined, whereas

the approximate analysis ensures that the calculationeffort -which is enormous no matter what optimizationprocedure is employed- can at least be carried outwithin reasonable time, be it at the cost of accuracy.

Results pertaining tothe Optimization Method

Global Optimization

The method outlined in this paper, was appliedto the multidisciplinary design optimization of asecond generation supersonic transport aircraft.This design was described by 28 independent designvariables, defining the wing, fuselage and tailplanegeometry, the engine design and the cruise condition.The design was subjected to 43 geometrical constraintsto prevent the occurrence of geometrical impossibleor improbable designs as well as 14 performanceconstraints to specify the aircraft's take-off, climband cruise performance as well as some environmentalaspects like take-off noise, subsonic range, NOXemissions and sonic-boom overpressure (Table 1).

Initial experience with the genetic algorithm showed-as expected- that slow convergence and prematureconvergence are major obstacles. Partly, this is aninherent problem of the use of genetic algorithms,partly it is the result of the very small feasible designspace available for a supersonic transport design.In most cases progress is achieved by minor adjustmentsto the best design encountered, which has beenproduced in an early generation and kept in thepopulation during a large number of generationssince. In these cases, the population consists of alarge number of very bad designs and a singleacceptable one, which is occasionally improvedafter several thousand function evaluations.

Another disadvantage of genetic algorithms is thefact that the choice of the optimization parameters(like the crossover probability and the mutationprobability), as well as the selection procedureemployed, is purely intuitive. No rules are availablethat can guide the user in these choices. Thereforethe process of a successful genetic search involvesa great deal of trial-and-error. The only exceptionis, that in case after a lot of function evaluationsand restarts with new initial populations, the searchhas converged and no further improvement canbe obtained, it sometimes helps to put the mutation


constraint

take-off field length

approach speed

supersonic range

subsonic range

cruise thrust

sea-level rate of climb

ceiling after enginefailure

subsonic longitudinalstability

supersonic longitudinalstability

supersonic directionalstability

thrust at engine failureafter take-off

maximum shock waveoverpressure

NOX emissions

take-off jet velocity

value

< 3500 m.

< 83 m/s

> 9000 km.(5000 nm.)

> 5000 km.

sufficient

25 m/s

4600 m.

sufficient

sufficient

sufficient

sufficient

< 72 Pa.

< 20 g/kgfuel

< 500 m/s

Table 1 Performance requirements

probability at a larger value. If this value is takentoo high, however, the search will degenerate intoa basic random search which will not result in betterdesigns in case of a strongly constrained problem.

It is a particular feature of the optimization of asupersonic transport aircraft that a large amountof incomputable designs is generated, especiallyin case direct search methods are used. For instanceif the aircraft, during the analysis of take-offperformance, runs out of thrust, the take-off fieldlength cannot be computed. Initially, the analysiswas broken off immediately in such cases, and alarge penalty was added to the objective function,

in order to save precious computational time. However,this led quickly to premature convergence becausean occasional design with superior but far fromoptimal characteristics would start to take overthe entire population. This problem was remediedby imposing a proportional instead of a fixed penaltyon incomputable or geometrically improbable designs.A representative value for the parameters that cannotbe evaluated is assumed and the analysis is continued.This method of proportional penalty enables thegenetic algorithm to "learn" and take correctiveaction, whereas information about any favorablecharacteristics of the design under considerationis retained. It was established that this proceduredramatically improved progress of the optimization.

Since the largest gain in objective function reductionis obtained in the first generations, whereas progressusually comes close to a halt as the optimizationproceeds, there is no point in carrying out an excessiveamount of function evaluations. A very effectiveprocedure to further speed up progress provedto be using the best designs, obtained at the pointwhere progress stagnated, as the initial populationfor a new optimization run. The rest of the populationwas generated at random, thus providing for "newblood" in the population. This method proved moresuccessful than increasing the mutation probability.

Local Optimization

Around the (semi-)optimal designs generated bythe genetic algorithm a reduced design space wascreated based on a ±10% perturbance of the (semi-Optimal design variables. Within this reduced designspace, a large amount of samples was generatedfor the purpose of creating regression surfaces.Generally, about 50% of the randomly generateddesigns turned out to be incomputable, and weresubsequently removed from the sample set.

Of the semi-optimal design obtained from the globalsearch, seven variables were fixed, being the fuselagediameter (since the optimizer permanently keptthis value at its lower limit to minimize fuselagewave drag) and the variables that define the enginedesign. This implies that the engine performancedepends only on the off-design Mach number andaltitude and can therefore be analyzed by interpolationfrom a prepared engine data file. This speeds up


the analysis considerably and enables a detailedpoint-to-point mission analysis to be carried outin reasonable time.

Since seven variables are fixed, only 21 remain,implying that the regression surfaces are definedby 253 coefficients and at least that many designshave to be generated (in practice, the regressionwould initially be based on about 400 sample designs).

Although the statistical characteristics of the regressionsurfaces thus obtained were excellent, initially largedifferences showed once the obtained optima wereevaluated with the exact analysis. The cause of thisproblem was identified as follows. Apparently,as indicated by the large amount of incomputabledesigns, there are a lot of "holes" in the design space.Since the incomputable designs were removed fromthe sample set, they are not reflected in the regression.However, the regression surfaces do cover the holesand consequently large errors are possible.

The following remedy was successfully applied.Since most of the problems occurred during thedetailed analysis of the mission, a simplified relationwas used instead, that made it possible to carrythrough the analysis of the design and obtain valuesfor the constraints to be approximated that -althoughnot very accurate- are at least representative forthe design under consideration. In that way anoptimization routine can handle such constraintsand take corrective action. By adding the re-evaluatedoptimum to the sample set, new regression surfaceswith increased accuracy could be generated andthe optimization restarted. By repeating this processseveral times, the errors in the prediction weregradually reduced. Since the optimal designs thusobtained are all situated close to each other, theregression became biased, but also very accuratein this region. In this way designs were obtainedthat showed considerable improvement over thesuboptimal designs generated by the global search.The final design is presented in Figure 4 and Table 2.

Results pertaining to the Designof a Second Generation SST

The most conspicuous feature of the final designis the fact that the wing has a supersonic leading-edge.Although this might seem somewhat peculiar, it

should be remembered that with the present designmethodology the wing is not designed for the cruisecondition, but to yield -in synergy with the otherdesign variables- a design that complies best withall the constraints imposed. Since the design spaceof a supersonic transport aircraft is extremely small,this might be the only way to realize an acceptabledesign (within the limitations of the method ofanalysis).

Figure 4 Final design: main dimensions

performance

wto

%Mcr

L/Dcr

sp,cr

IV

sto

Vapp

430992 kg

784m2

2.03

7.9

3290s

8663km

2910m

74m/s

environment

RSub

AP

Ve

EINOX

7755 km

120 Pa

780 m/s

19 g/kg

powerplant

A0

ec,des

^

3.6m2

16

0.5

Table 2 Final design

It was established, that it is extremely difficult tocomply with all constraints imposed on the design.Especially the limit on the sonic boom overpressureand the take-off jet velocity (implicit take-off noiseconstraint) could not be satisfied in combinationwith the other constraints.

703


The amount of installed thrust necessary to complywith the cruise thrust and take-off requirementturned out to be sufficient to enable excellent climbperformance. Also, the NOX emissions constraintwhich limits the amount of emitted nitric oxidesto a level less than or equal to that of Concordeproved not to be of concern. Furthermore, it wasfound that a subsonic range parameter M.Isp.L/Dcan be obtained that is all but equal to the supersonicvalue, implying that overland stretches flown atsubsonic speed will hardly deteriorate rangeperformance.

The constraints of main concern -after removal ofthe take-off noise and sonic-boom constraint- arethe supersonic range and the subsonic longitudinalstability. This problem is quite well known insupersonic aircraft design and basically comes downto compromising between sufficient subsonic stabilitymargin and minimal supersonic trim drag.

Although the problem of conflicting demands onthe design causing a drastic reduction of the availabledesign space is typical for a supersonic transportaircraft, the problem is aggravated in this workbecause of a number of limitations in the methodof analysis. The most important of these limitationsare found in the mass prediction and the aerodynamicanalysis. The mass prediction -for lack of accuratedata- is based on statistical information obtainedfor subsonic aircraft of the 1970's. Since the useof modern materials is not modeled -mainly becausethe amount of these materials that will be usedin the construction is not known- important weightreductions that might be achievable for a secondgeneration SST are not accounted for. Therefore,die aircraft weight will most likely be overestimatedin the present model. Also, neither the optimizationof the wing camber and twist, nor that of the fuselagearea distribution was accounted for and thereforethe supersonic drag prediction will be conservativetoo. Finally, the conflict between the range constraintand the subsonic longitudinal stability constraintin this work was reduced by alleviating the stabilityconstraint to a minimum dictated by the possibilitiesof artificial stability augmentation and by takingadvantage of the wing sweep to provide for a centerof gravity shift during supersonic cruise. However,a more effective procedure would be the use ofa special trim tank to and from which fuel can betransferred.

Enhancing the method of analysis will thereforecreate additional design space and might thereforeresult in compliance with the take-off noise constraint(or at least provide better values). Since the sonicboom constraint is mainly determined by the weight,cruise Mach number and cruise altitude of the airplane,this criterion will probably never be met.

Conclusions

If an optimization method produces a design thatcomplies with the constraints and provides a bettervalue for the objective function than an earlier obtaineddesign, the method can be deemed successfuL Especiallywhen after many more function evaluations or gradientsearches with different starting points, no furtherimprovement can be realized, the final design -atleast from an engineer's point of view- can be acceptedas the optimum (although a mathematician willwant some proof of this). In this sense, it was shownthat the presently adopted method works and canbe used to generate (semi-)optimal conceptual (aircraft)designs from scratch, without the need to providea baseline design first. The optimization has thereforebecome a practical design tool instead of just a methodto adapt a baseline, designed according to traditionalmethods. This is illustrated in Figure 5 and Table 3,in which the classical design procedure and thepresent method are depicted schematically. Theresults and experiences with the present methodhave further clearly indicated that the most severeconstraints imposed on a design are the limitations(mostly resulting from simplifications) of the analysisprocedure itself.

References

1. Bloebaum, C.L.; Formal and Heuristic System-Decomposition Methods in Multidisciplinary Synthesis;NASA CR-4413, December 19912. Bos, A.H.W.; Multidisdplinanf Design Optimizationof a Second Generation Supersonic Transport Aircraftusing a Hybrid Genetic/ Gradient-guided Algorithm;Ph.D. dissertation, ISBN 90-5623-041-7, Delft Universityof Technology, 19963. Chang, K.J.; Haftka, R.T.; Giles, J.L. and Kao, P.J.;Sensitivity-Based Scaling for Approximating StructuralResponse; published as AIAA 91-0925.


clQBBlc daalqn procedure

Figure 5 Comparison between classic andpresent design procedure

classic designprocedure

sequential

baseline required

human designer inthe loop (slow)

design decisionstaken by humandesigner

few designvariables

present method

concurrentminimization ofconstraint violations

no baseline required

human designer outof the loop (fast)

design decisionstaken by optimizer

many designvariables

Table 3 Comparison between classic andpresent design procedure

4. Consoli, R.D. and Sobieszczanski-SobieskiJ.;Application of Advanced Multidisciplinary Analysisand Optimization Methods to Vehicle Design Synthesis;ICAS 90-2.3.4.5. Draper, N.R. and Smith, H.; Ayylied RegressionAnalysis; John Wiley & Sons, New York, 19666. Fadel, G.M.; Riley, M.F. and Barthelemy, J.M.;Two Point Exponential Approximation Method for

Structural Optimization; Structural Optimization 2,117-124, Springer Verlag 19907. Gage, P. and Kroo, I.; A Role for Genetic Algorithmsin a Preliminary Design Environment; AIAA 93-3933,August 19938. Gage, P. and Kroo, I.; Representation Issues forDesign Topological Optimization by Genetic Algorithms;Department of Aeronautics and Astronautics, StanfordUniversity, Stanford CA 94305, 19959. Goldberg, D.E.; Genetic Algorithms in Search,Optimization and Machine Learning; Addison-WesleyPublishing Company Inc., 198910. Grefenstette, J.J.; A User's Guide to GENESIS;Version 5.0, October 199011. Haftka, R.T.; Combining Global and LocalApproximations; AIAA journal, Vol.29, No.9, September199112. Hajela, P.; Genetic Search-An Approach to the Non-Convex Optimization Problem; AIAA journal Vol.28,No.7, July 199013. Hajela, P.; Genetic Search Strategies in MulticriterionOptimal Design; AIAA 91-1040-CP14. Hutchison, M.; Unger, E.; Mason, W.; Grossman, B.and Haftka, R.; Variable-Complexity AerodynamicOptimization of an HSCT Wing Using Structural Wing-Weight Equations; AIAA 92-021215. Sobieszczanski-Sobieski, ].; A Linear DecompositionMethod for Large Optimization Problems - Blueprintfor Development; NASA TM 83248, 198216. Sobieszczanski-Sobieski, J.; Barthelemy, J.-F.M.and Riley, K.M.; Sensitivity of Optimum Solutionsto Problem Parameters; AIAA journal Vol.20, No.9,September 198217. Sobieszczanski-Sobieski, J.; James, B. and Dovi, A.;Structural Optimization by Multilevel Decomposition;AIAA paper 83-083218. Sobieszczanski-Sobieski, J.; Barthelemy, J.-F.M.and Giles, G.L.; Aerospace Engineering Design bySystematic Decomposition and Multilevel Optimization;ICAS 84-4.7.3.19. Sobieszczanski-Sobieski, J. and Barthelemy, J.M.;Improving Engineering System Design by FormalDecomposition, Sensitivity Analysis and Optimization;International Conference on Engineering DesignICED 85, Hamburg, 26-28 August 198520. Sobieszczanski-Sobieski, J.; James, B.B. and Riley,M.F.; Structural Sizing by Generalized, MultilevelOptimization; AIAA journal Vol25, No.l, January 1987


21. Sobieszczanski-Sobieski, J.; Sensitivity Analysisand Multidisciplinary Optimization for Aircraft Design:Recent Advances and Results; 16th Congress of theInternational Council of the Aeronautical Sciences(ICAS), Jerusalem, Israel, August 28-September 2,1988, Journal of Aircraft Vol.27, No.12, December 199022. Sobieszczanski-Sobieski, J.; Optimization byDecomposition: a Step from Hierarchic to Non-hierarchicSystems; NASA CP-3031 part 1, September 198823. Sobieszczanski-Sobieski, J.; Sensitivity of Complex,Internally Coupled Systems; AIAA journal Vol.28,No.l, January 199024. Sobieszczanski-Sobieski, J.; Bloebaum, C.L. andHajela, P.; Sensitivity of Control-Augmented StructureObtained by a System Decomposition Method; AIAAjournal Vol.29, No.2, February 199125. Torenbeek, R; Synthesis of Subsonk Airplane Design;Delft University Press, Delft, 1982


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