Multiobjective Conceptual Studies For Aerospace Design IMA

Multiobjective Conceptual Studies

For Aerospace Design

IMA Mathematical Modeling Workshop 2007

Mentor: Natalia AlexandrovNASA Langley Research Center

Qiang Chen Derek DalleUniversity of Delaware University of Minnesota twin cities

Chad Griep Jingwei HuUniversity of Rhode Island University of Wisconsin-Madison

Jahmario Williams Zhenqiu XieMississippi State University Purdue University

August 20, 2007

1 Introduction

Designing affordable, efficient, quiet supersonic passenger aircraft has beenunder investigation for many years. Obstacles to designing such aircraft arealso many, both in fundamental physics and in computational science andengineering. The problem of design is multidisciplinary in nature and thegoals of the constituent disciplines that govern the behavior of an aircraftare often at odds. In particular, aircraft that yields low sonic boom maynot be attractive aerodynamically, while aerodynamically optimized aircraftmay produce unacceptable sonic boom. One of the essential difficulties inusing direct optimization methods to design for low boom and low drag isin modeling the design problem. For instance, it is not clear what objectivefunctions to use. The present study investigates a number of variables andobjectives in subsonic and supersonic conceptual design spaces with theultimate aim of identifying promising design models or optimization problemformulations.

1

2 MOTIVATION

2 Motivation

The general design optimization problem may be stated as

minx{f1(x, u1(x)), ..., fN (x, uN (x))} (1)subject to x ∈ S,

where, fi : Rn → R, i = 1, ...N are the objectives of the design problem,x ∈ S ⊂ Rn is the vector of design variables, and ui are the state variablesor outputs obtained from the disciplinary analyses via some computationalprocedure (e.g., solving the equations governing some aspects of the systembehavior). In this definition, S is the feasible set defined via general linearand nonlinear constraints and bounds on the design variables.

State-of-the-art conceptual design does not view the design problem inthis way. We briefly discuss traditional design methods to motivate potentialusefulness of formal multiobjective nonlinear optimization in aircraft design.

Traditionally, the first step in the design process is the establishmentof requirements. Once a conceptual configuration has been developed thatmeets all requirements, an engineer may try to improve the conceptual de-sign via optimization, subject to satisfying the requirements. The manyobjectives of the problem are usually aggregated into a weighted objective

F (x) =n∑

i=1

wifi(x),

where wi are the weights that signify the designer’s view of a particularobjective’s importance. A few objectives are aggregated at a time and mostof the design variables are fixed, leaving only a few to serve as optimizationvariables. The low dimensionality is maintained to facilitate visualizationand decision making. Once a few variables are studied, they are fixed, andthe next set is investigated.

This step-by-step, “reduced” optimization allows few degrees of freedomto the design and relies heavily on the designer’s intuition and expertise. Bythe end of the conceptual stage, the major features of the design are locked.Subsequent optimization at the preliminary stage allows for relatively minorvariations. As a result, the minimization process may lead to an improve-ment, but the improved design will be based on the original design and maymiss a radically different design with a better performance.

In addition, it is possible that no design meets the requirements. Inthis case, traditional design methods perform parametric studies that sug-gest promising ways to relax the requirements. We would like to investigate

IMA Workshop, August 2007 2

3 APPROACH

whether formal multiobjective optimization with computed trade-offs mayassist designers in identifying designs that are closest to meeting the require-ments in some norm. For example, one could minimize the function

F (x) =n∑

i=1

(fi(x)− yi)2

where the requirements are denoted by yi. Performing this minimizationcould provide useful information about which requirements could be relaxedand by how much.

Another aim of formal optimization is producing radically new aircraftdesigns. The traditional stage of designing an aircraft that meets require-ments relies largely on experience. When attempting to develop a type ofaircraft with no predecessors, the design process becomes more difficult. Weconjecture that using formal optimization methods to develop and analyzea large assortment of aircraft could lead to revolutionary aircraft configura-tions.

A key application of nonlinear optimization to aircraft design under de-velopment now is the quiet supersonic transport. While the potential payoffof civil supersonic transport has been accepted, there are many technicalobstacles. A primary obstacle to practical implementation of a civil super-sonic aircraft is the sonic boom, which is a pressure wave transmitted to theground by any aircraft flying at a velocity greater than the speed of sound.Heard on the ground, sonic boom can be very loud and potentially damagingto structures, property, livestock, and humans. As a result, the FAA doesnot allow commercial supersonic aircraft to fly over land. Because the costof a supersonic transport would be prohibitively expensive to justify onlyover-water flights, many researchers (e.g., [1], [6], and [7]) are attempting todesign a supersonic transport that produces an acceptable sonic boom.

A number of difficulties in modeling the boom and designing for bothlow boom and good aerodynamic performance have prevented practical de-signs of quiet supersonic transport to-date. Numerical optimization is beinginvestigated as a technique to assist designers in building a viable supersonicaircraft.

3 Approach

In this study we investigate some aspects of modeling the design problem.In particular, we study the functional dependence of some of the majordesign objectives on several salient design variables. Because we had no


3 APPROACH 3.1 FLOPS – the Analysis Tool

access to sonic boom signature modeling software, we have applied nonlinearoptimization methods only to the subsonic case, in order to gain experienceduring this workshop. We have also investigated the general relationshipsbetween the same design variables and possible objective functions for thesupersonic case, but without explicitly considering the sonic boom signature.

We use several computational environments: stand-alone FORTRAN 90coupled with FLOPS (see below) and NPSOL; MATLAB with FLOPS andNPSOL; and Mathematica with FLOPS.

3.1 FLOPS – the Analysis Tool

We use the Flight Optimization System (FLOPS) [4] to evaluate the func-tions related to aircraft performance. FLOPS is a multidisciplinary systemof computer programs for conceptual and preliminary design and evaluationof advanced aircraft concepts. It consists of nine primary modules:

1. Weights: Statistical/empirical equations predict the weight of eachitem in a group weight statement.

2. Aerodynamics: A modified version of the empirical drag estimationtechnique provides drag polars for performance calculations.

3. Engine cycle analysis: Internally generated engine deck consisting ofthrust and fuel flow data at a variety of Mach-altitude conditions.

4. Propulsion data scaling and interpolation: Using an engine deck inputor generated by the engine cycle analysis module, provides propulsiondata requested by the mission performance module or the takeoff andlanding module.

5. Mission performance: Using the weights, aerodynamics, and propul-sion system data, calculates performance; for supersonic aircraft, sonicboom overpressures are computed along the aircraft track.

6. Takeoff and landing: Computes the engine takeoff field length, thelanding field length, and associated values.

7. Noise footprint: Using the take off and climbout profile, computesnoise footprint contour data or noise levels.

8. Cost analysis: Computes life-cycle cost for subsonic transport aircraft.


3 APPROACH 3.2 Optimization Tools

9. Program control: FLOPS may be used to analyze point design, para-metrically vary certain design variables, or optimize a configurationwith respect to design variables.

3.2 Optimization Tools

We employ three optimization tools. FLOPS internal optimization code usesthe KS function or Fiacco-McCormick penalty function with DFP or BFGSalgorithm. We use this capability for computing baseline solutions.

FLOPS handles multiobjective formulations via a single formulation –multiobjective function synthesis. To investigate alternative formulations,we use NPSOL [3] (general purpose nonlinear programming software) andMathematica (see section 5.2).

An initial difficulty in using general purpose optimizers for a specializedproblem instead of special purpose optimizers is that much tuning of the codeparameters may required to obtain a satisfactory solution. We encounteredthis difficulty because of the short project duration. For example, in thesubsonic case, FLOPS was able to find better solutions than NPSOL (seetable). It is likely that after some tuning, NPSOL would have identifiedbetter solutions, but we could not investigate this option for lack of time.However, bootstrapping solutions between NPSOL and FLOPS allowed usto find even better solutions than either FLOPS or NPSOL alone.

Method GW

FLOPS 213554NPSOL 221495NPSOL+FLOPSa 211920twice NPSOL+FLOPSb 210046

ameans using FLOPS optimizer to find aninitial value for NPSOL

bmeans to do the step twice

3.3 Parametric Studies

Our main investigation tool was parametric studies of the sensitivity ofseveral objectives in FLOPS to variation in design variables. Table 1 showsthe variables and objective functions used in the study. FLOPS optimizes acomposite objective function

OBJ = OFG*GW + OFF*FUEL + OFM*VCMN*(Lift/Drag) + OFR*Range


3 APPROACH 3.3 Parametric Studies

Table 1: Variables and FunctionsOFG Objective function weighting factor for gross weightOFF Objective function weighting factor for mission fuelGW Ramp wight, lbFUEL Fuel consumedFAROFF Takeoff field lengthAR Wing aspect ratioTHRUST Maximum rated thrust per engine, lbSW Reference wing area, sq ftSWEEP Quarter-chord sweep angle of the wing, degreesTCA Wing thickness-chord ratio

Table 2: Configuration variablesVARIABLE NAME VALUE

WING ASPECT RATIO AR 8.1080THRUST PER ENGINE, LBF THRUST 47500REF WING AREA, SQ FT SW 2272WING 1/4 CHORD SWEEP, DEG SWEEP 31.5WING T/C TCA 0.10950

+ OFC*Cost + OSFC*SFC + OFNOX*NOx+ OFNF*(Flyover Noise) + OFNS*(Sideline Noise)+ OFNFOM*(Noise Figure of Merit)+ OFH*(Hold Time for Segment NHOLD).

For visualization and simplicity, we focus on

OBJ = OFG*GW + OFF*FUEL,

and assume that the other weights are zero.For the purposes of this study, we are interested in the configuration

variables outlined in Table 2. Some of the design variables are shown inFigure 1.

AR is the Wing aspect ratio, which is an airplane’s wing span b dividedby its standard mean chord (c̄). The wingspan of an airplane is the distancefrom the outer wingtip to the inner wingtip. Chord refers to the distance be-tween the leading edge and trailing edge of a wing, measured in the directionof the normal airflow. AR is then calculated as AR = b

c̄ = b2

S . Informally, a


4 SUBSONIC CASE

Figure 1: The pictures for some design variables

high aspect ratio indicates long, narrow wings, whereas a low aspect ratioindicates short, stubby wings. The taper ratio of the wing is TR = tip chord

root chord .

4 Subsonic Case

In the subsonic design problem, we investigate the relative merits of severalmultiobjective problem formulations. The objectives are the gross weight,the fuel required to fly a mission, and the maximum takeoff field length.

4.1 Optimality for Single Objective

To determine the viability of single-objective optimization, we solved theindividual single-objective problems. The shape of the functions is depictedin Figure 2. They are smooth and locally convex.

4.2 Pareto Efficiency

While optimization problems with single objective focus on the design space,problems with multiple objectives focus on the function space. The salientnotion is that of Edgeworth-Pareto optimality (for an overview see [8])

Because some of the multiple objectives conflict (except in trivial cases)and because the objectives may be incommensurable, there is, in general,no single solution that is a minimum for all the objectives. Pareto optimalpoints are designs where all objectives cannot be improved simultaneously,i.e., if any of the objectives are improved, it is at the cost of making at leastone other objective worse.


4 SUBSONIC CASE 4.2 Pareto Efficiency

Figure 2: Subsonic parametric study for fuel, and gross weight

Figure 3: A typical Pareto curve for two convex functions


4 SUBSONIC CASE 4.3 Multiobjective Parametric Study

The Pareto optimal configurations form a curve known as Pareto frontier.A typical Pareto frontier for two convex functions is depicted in Figure 3.In multiobjective design, the decision maker is presented with a part or theentire Pareto frontier and must exercise expert judgment and/or use othercriteria to select a final design.

4.3 Multiobjective Parametric Study

The major objectives considered in aircraft design include gross weight, fuelusage, life-cycle cost, and others. FLOPS optimizes aggregated weightedobjectives to find individual points on the Pareto frontier. However, theformulation of optimizing a convex combination of the objectives can onlyfind solutions on the convex part of the Pareto frontier. And, in general,the convex part of the frontier is not guaranteed to include all solutions.

Figure 4 depicts the Pareto curve generated by plotting individual so-lutions computed in FLOPS and NPSOL, where the weight OFG runs para-metrically from 0 to 1, and OFF = 1 - OFG. The independent design variablesare SW, THRUST, AR, SWEEP and TCA.

Figure 4: Optima found by FLOPS with different weights on GW and FUEL

4.4 Alternative Problem Formulations

In general, once a Pareto frontier is generated, a decision maker has to useadditional criteria to arrive at a final design. We would like to investigatealternative formulations that would assist the designer in this task.

Among many alternative formulations, in this study, we investigate two.


5 SUPERSONIC CASE

Our first choice is a minimizer of the weighted sum

F (x) =N∑

i=1

ωifi(x)− yi

yi, (2)

where∑N

i=1 ωi = 1, and ωi ≥ 0 are the weights and yi 6= 0 are the targetvalues for the objectives. Our other choice is the formulation

F (x) =N∑

i=1

∣∣∣∣fi(x)− f∗i

f∗i

∣∣∣∣p, (3)

where p = 1 or 2, and f∗i are the individual optima of the single-objective

optimization problems (the so called ideal vector). The choice may alsodepend on values of other functions, e.g., supersonic boom in supersonicdesign.

5 Supersonic Case

The term sonic boom refers to the shocks caused by the supersonic flightof aircraft (e.g., Concorde, Mach 2.03) and the Space Shuttle (up to Mach27). Sonic booms generate great amounts of sound energy. Typically theshock front may approach 167 megawatt per square meter, and may exceed200 decibels. Thunder is a type of natural sonic boom, created by the rapidexpansion of heated air [9].

The effect of the sonic boom on an observer depends on the distancebetween the observer and the aircraft producing the sonic boom. The an-noyance factor for the observer tends to depend on the sharpness of theboom (so called rise time) and the amplitude of the sound wave (loudness).

Traditional methods for sonic boom minimization are based on inversedesign (e.g., [6]). Figure 6 (the left-hand side) depicts the process of sonicshocks coalescing and propagating to the ground in a classical “N-wave”.The picture on the right depicts the results of shaping the aircraft to “smear”the shock in time and space and to prevent the coalescence. The resultingshock should be much weakened and acceptable to the observer. To achievethis aim, traditional methods assume an ideal shape of the overpressuresignature and seek aircraft shapes that would satisfy this signature.

This technique, while generally successful, has a number of difficulties.In particular, the inverse design problem may be ill-posed mathematicallyand the target signature is idealized. An alternative approach that would


5 SUPERSONIC CASE 5.1 Multiobjective Parametric Study

Figure 5: Sonic boom (Wikipedia)

attempt to avoid these difficulties, if successful, would replace the inversedesign procedure with direct design optimization. Namely, one would pa-rameterize the potential signature, extract the parameters that could serveas design variables and use them, along with shape design variables, in mul-tiobjective design optimization of a supersonic aircraft.

Ideally, we would have liked to investigate parameterization of the boomsignature and the dependence of the parameters on configuration shape vari-ables. However, since the only measure of the boom in FLOPS is the over-pressure measured at a single point directly under the passing aircraft, wecannot infer the shape of the signature. Therefore, we studied minimizationformulations for the gross weight and takeoff field length as functions of thereference wing area and thrust and merely observed the value of the boomas an additional objective that discriminates between two otherwise equallyacceptable designs.

5.1 Multiobjective Parametric Study

Because the modeling of sonic boom propagation is not feasible during theworkshop for lack of both time and software, we could not apply formal op-timization to boom minimization. Instead, we used optimization to attemptminimizing boom indirectly.

The dependence of gross weight and fuel burn on wing area and thrust



Figure 6: Sonic boom minimization theory

is shown in Figure 7. The figure shows that gross weight correlates withfuel burn. The relationship between the gross weight and maximum takeofflength is somewhat different (see Figure 12).

Here we use NPSOL to generate Pareto curves of objectives GW, FUELand FAROFF, while the maximum boom during the breaking of the soundbarrier is also taken into consideration.

The Pareto curve for the supersonic case (Figure 8) is generated byoptimizing OFG * GW + OFF * FUEL in NPSOL, while OFG varies from 0 to1 and OFF = 1 - OFG. The independent variables are SW and THRUST. For eachPareto optimal point, we use FLOPS to compute the value of boom. Thesmall square on the graph represents the minimum value of the maximumboom.

Another Pareto curve, with different initial values of SW and THRUST anddifferent fixed value for AR, SWEEP and TCA, is given in Figure 9.

Because GW and FUEL are strongly correlated, we turn to studying thefunction— maximum takeoff length, FAROFF—that should be inversely corre-lated with GW. We wish to minimize both GW and FAROFF. Minimizing FAROFF



Figure 7: Supersonic parametric study for fuel and growth weight

Figure 8: Supersonic Pareto curve

Figure 9: Another Pareto curve


5 SUPERSONIC CASE5.2 Weight vs. Takeoff Length for the Supersonic Case

involves increasing SW and THRUST (i.e., bigger wings and bigger engines im-ply shorter takeoff length), which implies a heavier airplane, i.e., larger GW.Hence GW and FAROFF represent typical conflicting objectives. We discussthis in more detail in the next subsection.

5.2 Weight vs. Takeoff Length for the Supersonic Case

We also conduct a computational study within the Mathematica (WolframResearch) framework. We use Mathematica optimization tools that, in turn,call FLOPS to compute function evaluations.

Mathematica has two built-in numerical optimization functions, NMinimizeand FindMinimum. The first function performs constrained optimization us-ing derivative-free algorithms. These techniques, such as differential evolu-tion and random search, are robust but extremely slow. The other optimizer,FindMinimum, uses derivative-based techniques and is fast, but it can onlysolve unconstrained optimization problems. Therefore some additional pro-gramming was required to perform minimizations in a reasonable amount oftime.

To take constraints into account, we implemented a penalty method,using information from infeasible designs to help in the optimization process.Specifically, if the problem statement is to minimize f(x) subject to gk(x) ≤uk, with k = 1, 2, . . . ,m with several constraints, then we minimize

F (x) ={

f(x) gk(x) ≤ uk

Y + Σ(gk(x)− uk)2 gk(x) > uk(4)

where Y is a large number that will discourage the algorithm from settlingon a point that does not meet the constraints. The situation for the single-objective, single-variable case is illustrated in Figure 10. Information fromrejected aircraft designs guides the algorithm to a feasible design. When thealgorithm starts with an infeasible design, it quickly finds feasible ones.

The following table shows how the minimum gross weight changed whendifferent constraints were applied to the runway length. Notice that whenthe runway length was restricted to 9000 ft, the gross weight actually de-creased. One conjecture is that that the Mathematica optimizer ran intoan additional constraint. A more likely explanation is that Mathematicaoptimizer reached the part of the design space for which the FLOPS modelsmay not be valid (or validated). As a rule, as the runway constraint de-creased, the wing area increased, and the other design variables remainedapproximately constant.



Figure 10: Illustration: altering the function so that the constrained mini-mum is actually the global minimum.

Gross weight Takeoff runway length Landing runway length203667 lbs 9181 ft 7800 ft202953 lbs 9000 ft 7755 ft204935 lbs 8000 ft 7146 ft205902 lbs 7000 ft 6547 ft

The list below shows the results from Mathematica using several differentapproaches to multiobjective optimization.



1. Optimization for gross weight alone.Gross weight: 203669 lbs Fuel: 42649 lbs

AR: 8.1089 THRUST: 33828 lbsSW: 1091.6 ft2 SWEEP: 19.92◦

TCA: 0.09242. Optimization for fuel alone.

Gross weight: 2230664 lbs Fuel: 30078 lbsAR: 26.093 THRUST: 24547 lbsSW: 1004 ft2 SWEEP: 36.19◦

TCA: 0.10953. Optimization for the sum of the two weights.


TCA: 0.09054. Optimization using the global criterion method.


TCA: 0.1095

The results indicate that increased use of formal optimization has poten-tial, but further research is required. Some of the issues that need furtherstudy are as follows. The gross weight obtained in the weighted-sum ap-proach actually produced a lower gross weight than the scheme that wasonly trying to minimize the gross weight. The reason for this is that theoptimization scheme depends strongly on the initial value (especially if theinitial point is in the basin of attraction for the solution) and on the boundconstraints. When optimizing the weighted sum, however, the optimizationquickly took a step toward the design far from the initial design. Whenthe design variables from item 3. are taken as an initial point, and thegross weight is optimized again, the result is a gross weight of 193148 lbs.Although the optimization programs written during this workshop are farfrom perfect, they do provide an efficient method to find some good, if notquite optimal, designs. Some results can be seen in the Figures 11 and 12.

For completeness, we also show Pareto information for GW and FAROFFgenerated via NPSOL and MATLAB graphics. Figure 13 shows the Paretoset, with two-variable optimization SW and THRUST.



(a) Thrust=46200 lbs. (b) Wing area=7100 sq ftGross weight vs. wing area Gross weight vs. thrust

(c) Gross weight vs. thrust and wing area)

Figure 11:



(a) Thrust=46200 lbs. (b) Wing area=7100 sq ftTakeoff length vs. wing area Takeoff length vs. thrust

(c) Takeoff length vs. thrust and wing area

Figure 12:

Figure 13: Pareto set of GW against FAROFF


7 ACKNOWLEDGMENTS

6 Concluding Remarks

Traditional design methods face a number of serious difficulties. They arethe relatively low dimensionality of the design space, the need for labori-ous variable-by-variable investigation of design alternatives, the difficultyin making decisions in a highly multiobjective and multidisciplinary envi-ronment, the need to rely almost exclusively on experiential and heuris-tic procedures. Despite these difficulties, traditional methods have beenhighly successful in the design of conventional aerospace vehicles. However,arguably, further progress and changing airspace environment will requirerevolutionary designs for which expert knowledge may not exist and forwhich the designer will have to search much larger design spaces. And thisis where mathematical modeling should come in.

Together with modeling of the physical processes that govern the behav-ior of an aircraft, modeling of the design problem (i.e., the design optimiza-tion problem formulation) plays an important role in enabling model-drivendesign. Unfortunately, little attention is usually paid to appropriate formu-lation of the design problem.

In this project, our aim was to ”get a feel” for the process of problemformulation and the effect of the formulation on the resulting solution. To-ward that end we conducted parametric studies of functional dependencesbetween objectives of interest and the salient design variables in typicalconceptual models for designing a subsonic and a supersonic airplane.

While conducting our computational experiments, we have acquired someunderstanding of the complexity of the process, as well as the various depen-dencies of functions on the variables of interest in aeronautical design. Someof our results were unexpected. For instance, for a different variable inter-val, the parametric study of fuel burn and gross weight produced a differentpicture 14. We cannot explain it with certainty now, but we conjecture thatthis effect has to do with varying validity of the FLOPS models in differentparts of the design space.

In another example (a five-variable Pareto study), the solutions were alsodistributed not as expected, but were clustered around the second point (thethicker of the two points in Figure 15). Further research is indicated.

7 Acknowledgments

Team 1, supersonic and subsonic design would like to thank our mentor,Natalia Alexandrov for her boundless enthusiasm, inspiring us to explore


7 ACKNOWLEDGMENTS

Figure 14: Singularity for supersonic cases

Figure 15: Optimization results by varying 5 design variables


REFERENCES REFERENCES

real problems. We also thank IMA for all the excellent support. We thankeveryone who participated in making this workshop running great. It isdefinitely great and a fun learning experience for all of us.

References

[1] Alonso, J.J., Kroo, I.M., Jameson, A. “Advanced Algorithms for Designand Optimization of Quiet Supersonic Platform”, 40th AIAA AerospaceSciences Meeting and Exhibit, AIAA Paper 2002-0144, Reno, NV, Jan-uary 2002

[2] Carlson, H.W., Maglieri, D.J.; ”Review of Sonic Boom Generation The-ory and Prediction Methods”, J. Acoust. Soc. Amer., 51, pp. 675-685(1972)

[3] P. E. Gill, W. Murray, M. A. Saunders, AND M. H. Wright, User’sguide for ”NPSOL (Version 5.0): A Fortran package for nonlinear pro-gramming”, Stanford University, 1995.

[4] L. A. (Arnie) McCullers, Swales Aerospace, Flight Optimization SystemUser’s Guide, Release 7.20, 23 May 2007

[5] Patnaik, S.,Brook Park, R. Coroneos, J. Guptill, D. Hopkins and W.Haller. “A Subsonic Design Optimization with Neural Network and Re-gression Approximators” AIAA-2004-4606. 10th AIAA/ISSMO Multi-disciplinary Analysis and Optimization Conference, Albany, New York,Aug. 30-1, 2004

[6] Seebass, R., Argrow, B.; ”Sonic Boom Minimization Revisited”, AIAAPaper 98-2956

[7] Shepherd, K.P., Sullivan, B.M.; ”A Loudness Calculation ProcedureApplied to Shaped Sonic Booms”, NASA Technical Paper 3134, 1991

[8] Steuer, R.E., ”Multiple Criteria Optimization: Theory, Computation,and Applications”, John Wiley & Son, Inc. 1986

[9] Wiki Website: http://en.wikipedia.org/wiki/Main Page


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Multiobjective Conceptual Studies For Aerospace Design IMA