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```NASA Contractor Report 4328

Aircraft Design for
Mission Performance Using
Nonlinear Multiobjective
Optimization Methods

Augustine R. Dovi and Gregory A. Wrenn

CONTRACT NASI -19000
OCTOBER 1990

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NASA Contractor Report 4328

Aircraft Design for
Mission Performance Using
Nonlinear Multiobjective
Optimization Methods

Augustine R. Dovi and Gregory A. Wrenn
Lockheed Engineering & Sciences Company
Hampton, Virginia

Prepared for

Langley Research Center

under Contract NASI -19000

NASA

National Aeronautics and

Oflice of Management

Scientific and Technical
Information Division

1990

CONTENTS

IV
V

1

LIST OF TABLES

LIST OF HGURES

SUMMARY

INTRODUCTION 1

GENERAL MULTIOBJECTIVE OPTIMIZATION 2

Formulation of the Mission/Performance Optimization Problem 3

Description of the Analysis System for Mission Performance 5

DESCRIPTION OF OBJECTIVE FUNCTION FORMULATION METHODS 6

Envelope Function Formulation (KSOPT) 6

Global Criterion Formulation 7

Utility Function Formulation Using a Penalty Function Method 7

RESULTS AND DISCUSSION 8

Single Objective Function Optimization 8

Parametric Results of the Design Space 10

Multiobjective Optimization 10

Comparison of Two Objective With Single Objective Designs 10

Comparison of Three Objective With Single Objective Designs 1 1

Comparison with Overall Best Single Objective Designs 1 1

CONCLUSIONS 11

REFERENCES 13

PM;fc //' . _l NUra«WAUl iLANi ^ PRECEDSMG PAGE BLArrX ^'0T FILMED

LIST OF TABLES

Table la: Multiobjective Cases

Table lb: Single Objective Cases

Table 2: Single Objective Etesign Results

Table 3: Two Objective Design Results

Table 4: Three Objective Design Results

Table 5: Best Single Objective Results

16
16
17
18
19
20

IV

LIST OF FIGURES

Figure 1 Objectives, Design Variables and Constraints 21

Figure 2 Mission Profile 21

Figure 3 FLOPS Primary Modules 22

Figure 4 Single Objective Optimization Change From Initial Conditions 22

Figure 5 Ramp Weight as a Function of Aspect Ratio and Thickness Ratio 22

Figure 6 Mission Fuel as a Function of Aspect Ratio and Thickness Ratio 22

Figure 7 Mach (L/D) as a Function of Aspect Ratio and Thickness Ratio 22

Figure 8 Two Objective Optimization Compromise From Single 23
Objective Cases

Figure 9 Three Objective Optimization Compromise From Single 23
Objective Cases

Figure 10 Two Objective Optimization Compromise From Best Single 23
Objective Cases

Figure 1 1 Three Objective Optimization Compromise From Best Single 23
Objective Cases

SUMMARY

A new technique which converts a constrained optimization problem to an
unconstrained one where conflicting figures of merit may be simultaneously considered has
been combined with a complex mission analysis system. The method is compared with
existing single and muWobjective optimization methods. A primary benefit from this new
method for multiobjective optimization is the elimination of separate optimizations for each
objective, which is required by some optimization methods. A typical wide body transport
aircraft is used for the comparative studies.

INTRODUCTION

Aircraft conceptual design is the process of determining an aircraft configuration
which satisfies a set of mission requirements. Engineers within several diverse disciplines
including but not limited to mass properties, aerodynamics, propulsion, structures and
economics perform iterative parametric evaluations until a design is developed.
Convention limits each discipline to a subset of configuration parameters, subject to a
subset of design constraints, and typically, each discipline has a different figure of merit.

Advanced design methods have been built into synthesis systems such that
communication between disciplines is automated to decrease design time^-^. Each
discipline may select its own set of design goals and constraints resulting in a set of
thumbprint and/or carpet plots from which a best design may be selected. In addition, the
conceptual design problem has been demonstrated to be very amenable to tiie use of formal
matiiematical programming methods, and these algorithms have been implemented to
quickly identify feasible designs^-^-^.

The purpose of this report is to investigate the use of multiobjective optimization
methods for conceptual aircraft design where conflicting figures of merit are considered
simultaneously. Three multiobjective methods6.7.8 have been combined with a complex
mission analysis system^. Trade-offs of the methods are compared with single objective
results. In addition parametric results of the design space are presented. The aircraft
chosen for this investigation is a typical wide body transport.

GENERAL MULTIOBJECTIVE OPTIMIZATION

The constrained multiobjective optimization problem stated in conventional
formulation is to

minimize Fk(X), k = 1 to number of objectives (1)

such that,

gj(X) < 0, j = 1 to number of constraints

and

x^i < xi < x"i i = 1 to number of design variables

where,

X = {xi,x2,x3,...Xn}T n = number of design variables

The fundamental problem is to formulate a definition of Fk(X), the objective vector,
when its components have different units of measure thereby reducing the problem to a
single objective. Several techniques have been devised to approach this problem'^. The
methods selected for study in this report transform the vector of objectives into a scalar

function of the design variables. The constrained minimum for this function has the
property that one or more constraints will be active and that any deviation from it will cause
at least one of the components of the objective function vector to depart from its minimum,
the classic Pareto-minimal solution^.lO. One should add that multiobjective optimization
results are expected to vary depending on the method of choice since the conversion
method to a single scalar objective is not unique.

Formulation of the Mission/Performance Optimization Problem

The purpose of the optimization is to rapidly identify a feasible design to perform
specific mission requirements, where several conflicting objectives and constraints are
considered. The aircraft type selected for this study is a typical wide body transport,
figure 1, in the 22680 kg weight class^^ The aircraft has three high-bypass ratio turbofan
engines, with 6915 newtons thrust each. The mission requirements are

design range

=

7413.0 km

cruise Mach number

=

0.83

cruise altitude

=

11.9 km

=

42185.0 kg

number of passengers

and

crew

=

256

The primary and reserve mission profiles are shown in figure 2.

The design variables considered, figure 1, are aspect ratio (AR), area (Sw), quarter
chord sweep (A) and thickness to chord ratio (t/c) of the wing, where the initial values
chosen for all cases are

Xo =

AR
A

11.0
361.0 m2
35.0 deg
0.11

The objectives to be minimized or maximized for this investigation include

Fi (X) = ramp weight (minimize)

F2(X) = mission fuel (minimize)

F3(X) = lift to drag ratio at constant cruise Mach number (maximize)

F4(X) = range with fixed ramp weight (maximize)

The functions to be maximized were formulated as negative values so that they
could be used with a minimization algorithm. These objectives are first optimized for
feasible single objective designs. The objectives are then considered simultaneously for
multiobjective designs. Tables la and lb list fourteen cases, six multiobjective and eight
single objective, along with the unconstrained objective function formulation used for each.

Each of the three formulations use the Davidon-Fletcher-Powell variable metric
optimization method to compute the search direction for finding a local unconstrained
minimum of a function of many variables 12.

The inequality behavioral constraints used in each case are

gl(X) = lower limit on range, (1853.2 km)

g2(X) = upper limit on approach speed, (280.0 km/hr)

g3(X) = upper limit on takeoff field length, (2700.0 m)

g4(X) = upper limit on landing field length, (2700.0 m)

g5(X) = lower limit on missed approach climb gradient thrust, (3458.0 newtons)
g6(X) = lower limit on second segment climb gradient thrust, (3458.0 newtons)
g7(X) = upper limit on mission fuel capacity (fuel capacity of wing plus fuselage)

where the constraint functions gj are written in terms of computable functions stated as
demand(X) and capacity. These functions provide the measure of what a design can
sustain verses what it is asked to carry

gj(X) = demand(X)/capacity - 1 (2)

In addition, side constraints were imposed on wing sweep and wing area in the form of
upper and lower bounds.

Description of the Analysis System for Mission Performance

The Flight Optimization System (FLOPS) is an aircraft configuration optimization
system developed for use in conceptual design of new transport and fighter aircraft and the
assessment of advanced technology^. The system is a computer program consisting of
four primary modules shown in figure 3: weights, aerodynamics, mission performance,
and takeoff and landing. The weights module uses statistical data from existing aircraft
which were curve fit to form empirical wing weight equations using an optimization
program. The transport data base includes aircraft from the small business jet to the jumbo
jet class. Aerodynamic drag polars are generated using the empirical drag estimation
technique ^3 in the aerodynamics module. The mission analysis module uses weight,
aerodynamic data, and an engine deck to calculate performance. Based on energy
considerations, an optimum climb profile is flown to the start of the cruise condition. The

cruise segment may be flown for maximum range with ramp weight requirements specified;
optimum Mach number for maximum endurance; minimum mission fuel requirements; and
minimum ramp weight requirements. Takeoff and landing analyses include ground effects,
while computing takeoff and landing field lengths to meet Federal Air Regulation (FAR)
obstacle clearance requirements.

DESCRIPTION OF OBJECTIVE FUNCTION FORMULATION METHODS

Envelope Function Formulation (KSOPT)

This algorithm is a new technique for converting a constrained optimization
problem to an unconstrained one^ and is easily adaptable for multiobjective optimization ^^
The conversion technique replaces the constraint and objective function boundaries in n-
dimensional space with a single surface. The method is based on a continually
differentiable function ^^^

K
KS(X)=J-loge£ efkW

P k=i (3)

where fk(X) is a set of K objective and constraint functions and p controls the distance of
the KS function surface from the maximum value of this set of functions evaluated at X.
Typical values of p range from 5 to 200. The KS function defines an envelope surface in
n-dimensional space representing the influence of all constraints and objectives of the
mission analysis problem. The initial design may begin from a feasible or infeasible
region.

Global Criterion Formulation

The optimum design is found by minimizing the normalized sum of the squares of
the relative difference of the objective functions. Single objective solutions are first
obtained and are referred to as fixed target objectives. Computed values then attempt to
match the fixed target objectives. Written in the generalized forai

K

F*(X) = X

k=l

fI(X)-Fi,(X)

. fI(X)

(4)

where fJ is the target value of the kth objective and Fk is the computed value. F* is the
Global Criterion performance function"^. The performance function F* was then minimized
using the KSOPT formulation described earlier.

Utility Function Formulation Using a Penalty Function Method

The optimum design is found by minimizing a utility function stated as

K
F*(X) = XwkFk(X)

k=i (5)

where wk is a designers choice weighting factor for the kth objective function, Fk, to be
minimized. This composite objective function is included in a quadratic extended interior
penalty function ^6 jhis function is stated in generalized form as

F(X,rp) = F*(X)-rpXGj(X)

j=i (6)

and

Gj(X) =

1
gj(X) I forgj(X)>e

2£ - gj(X) I for gj(X) < e

a

where the rp ^ Gj (X) term penalizes F ( X,rp), the performance function in proportion

to the amount by which the constraints are violated and e is a designers choice transition
parameter. The value of the penalty multiplier, rp, is initially estimated based on the type of
problem to be solved and is varied during the optimization process. The penalty multiplier,
Tp, is made successively smaller to arrive at a constrained minimum.

RESULTS AND DISCUSSION

Single Objective Function Optimization

Single objective results for two of the methods are presented, the envelope function
KSOPT and the classic penalty function PF methods. Single objective cases were run to
establish a base line for comparison of multiobjective performance. In addition, target
objectives are obtained for the Global Criterion Method. Final optimization values are
presented in table 2 for both methods. Both techniques converged to very similar designs
for all cases listed in table lb. Greatest modifications from the initial design are seen in lift
to drag ratio (L/D), cases 9 and 13 and range, cases 10 and 14.

Lift to drag was modified by increasing the aspect ratio and wing area thus
minimizing the wing loading (W/S). Thrust requirements (TAV) increased due to the larger
ramp weight. In addition, the wing was made thinner and unswept. The KSOPT method

converged to a 23% higher IVD verses the PF method. This is typically due to the way
constraint boundaries are followed.

Range improvements, cases 10 and 14, were accomplished by unsweeping the
wing to the lower limit allowed and wing volume was adjusted to carry the maximum fuel
load with reserves at the penalty of increased ramp weight. In addition, the optimizers
reduced wing thickness, area and aspect ratio from initial values. Wing loading was kept
at a minimum. KSOPT again produced a slightly better design compared with the PF
method.

To minimize mission fuel requirements, cases 8 and 12, the aspect ratio was
increased, and the wing area and was decreased. In addition, the wing was unswept and
made thinner. This design improved aerodynamic performance by over 20% from the
initial value while ramp weight increased slightly. The PF method converged to a slighdy
better design for this case.

Ramp weight, cases 7 and 11, was decreased by unsweeping the wing to the lower
limit of 22.0 degrees. Aspect ratio is essentially unchanged from the initial condition
design point. The wing thickness was decreased, along with a decrease in area.
Aerodynamic performance was not penalized significantly from the initial design value.
KSOPT produced a slightly lower ramp weight.

The chart in figure 4 compares the final design objective's percent change from the
initial design point.

Parametric Results Of The Design Space

Point designs, obtained parametrically, for minimum ramp weight, minimum
mission fuel and maximum Mach (L/D) are shown in figures 5 through 7. Wing aspect
ratio and thickness to chord ratio were varied, while other design variables were set to
optimum values given in table 2, Case 8, Case 7 and Case 9, respectively. The design
space is shown with the most critical constraints or criteria governing the design. To arrive
at the optimum point designs shown by traditional parametric trade studies over 256
evaluations would have been required.

Multiobjective Optimization

Multiobjective optimization considers all conflicting design objectives and
constraints simultaneously to meet mission specifications. Three methods are compared,
the envelope function KSOPT, the Penalty Function (PF) method and Global Criterion
(GC) method. Feasible designs were obtained for two objectives, table 3, and three
objectives, table 4, satisfying all constraints.

Comparison Of Two Objective With Single Objective Design

Figure 8 shows the percent deviation or compromise from each method's single objective
design. KSOPT treated ramp weight and mission fuel equally where the PF and GC
methods favored ramp weight, preferring to pay a larger penalty for mission fuel. This
behavior is expected with the PF and GC methods since the ramp weight is larger in
magnitude giving this objective greater influence. This effect could have been eliminated
by judicious normalization or weighting.

10

Comparison Of Three Objective With Single Objective Design

Figure 9 shows the percent deviation or compromise from each methods single objective
design. KSOPT traded aerodynamic efficiency (L/D) and ramp weight to keep fuel
requirements down. The PF method weighted L/D to a greater extent since the weighting
coefficient wk was 10,000, with small penalties in ramp weight and mission fuel. The GC
penalty behavior is similar to the two objective results in that the ramp weight was weighted
more over mission fuel and aerodynamic efficiency. The overall compromise is lowest for
the PF method.

Comparison With Overall Best Single Objective Designs

The best single objective design results are listed in table 5 along with objectives and
methods. Since L/D was not part of the objective function set, figure 10, two objective
compromised results behaved very similar to figure 8. Three objectives, figure 11, caused
the design space to be more constrained. KSOPT again traded ramp weight and L/D to
keep mission fuel requirements down. The PF method traded in a similar way but
compromised L/D to a greater extent. The GC method gave more priority to ramp weight
because of its magnitude. The overall compromise of KSOPT and PF were about the same
at 26.3 and 24.4 percent respectively and the GC method 40.4 percent.

CONCLUSIONS

A typical wide body subsonic transport aircraft configuration was used to
investigate the use of three multiobjective optimization methods, 1) an envelope of
constraints and objectives, KSOPT, 2) a Penalty Function and 3) the Global Criterion.
The methods were coupled with a complex mission performance analysis system. The

11

optimizer used with all three methods is the Davison-Fletcher-Powell variable metric
method for unconstrained optimization. Multiobjective compromised solutions were
obtained for two and three objective functions. Feasible designs for each objective were
also obtained using single objective optimization as well. The initial value design variable
vector Xo and the constraints gi through g-j were the same for all cases in this comparative
study.

The KSOPT method was able to follow constraint boundaries closely and
considered the influence of all constraints and objectives in a single continuously
differentiable envelope function. KSOPT defines the optimum such that the function
component with the greatest relative slope dominates the solution. The PF method also
produced feasible designs similar to the KSOPT final designs for single objective
optimization. This method, however, weights the individual objective functions in the
multiobjective cases.

The GC method is usually applied to multiobjective problems but may be used in
the single objective problem if a target objective is supplied. This would be equivalent to
imposing an upper or lower bound on the performance function. The GC method has a
disadvantage in resource requirements, requiring separate single objective optimizations to
provide target objectives.

Computational effort has been measured in functional evaluations, shown in the
tables of results. They are defined as the number of calls to the analysis procedures from
the optimization procedures. Function evaluations are very similar for single objective
cases except for mission fuel using KSOPT. This deviation is due to the methods
implementation, convergence criteria and the way constraint boundaries are followed. The
multiobjective table shows the GC method with the least functional evaluations, however

12

single objective function evaluations must be included with these values thereby making it
the most costiy in terms of number of analyses.

All of the methods produced feasible solutions within the design space. Attributes
of the methods, such as ease of use, data requirements and programming should also be
considered when evaluating their performance along with computational efficiency. Many
cases have been compared, too numerous to report herein, where initial design variables
were changed up to 40 percent above and below the initial values given in this report.
KSOPT continued to perform in a robust manner compared to the penalty function method.
Producing similar final designs within 1 percent of the mean. Based on the results of this
study and the above considerations, KSOPT is tiius concluded to be a viable general
metiiod for multiobjective optimization. Finally, one should add that multiobjective
optimization results are expected to vary depending on the method of choice.

REFERENCES

1 . Radovcich, N. A., "Some Experiences in Aircraft Aeroelastic Design Using
Preliminary Aeroelastic Design of Stioictures [PADS]", Part 1, CP-2327, April 1,
1984, pp. 455-503.

2. Ladner, F. K., Roch, A. J., "A Summary of the Design Synthesis Process; SAWE
paper No. 907 presented at the 31st Annual Conference of the Society of
Aeronautical Weight Engineers; Atlanta, Georgia, 22-25 May 1972.

3 . Piggott, B. A. M.; and Taylor, B. E., "Application of Numerical Optimisation
Techniques to the Preliminary Design of a Transport Aircraft", Technical Report -
71074, (British) R. A. E., April 1971.

13

4. SUwa, Steven M. and Arbuckle, P. Douglas, "OPDOT: A Computer Program for
the Optimum Preliminary Design of a Transport Airplane", NASA TM-81857,
1980.

5 . McCuUers, L. A., "FLOPS - Flight Optimization System", Recent Experiences in
Multidisciplinary Analysis and Optimization, Part 1, CP-2327, April 1984, pp.
395-412.

6 . Wrcnn, Gregory, A., "An Indirect Method for Numerical Optimization Using the
Kreisselmeier-Steinhauser Function", NASA CR-4220, March 1989.

7 . Rao, S. S., "Multiobjective Optimization in Structural Design with Uncertain
Parameters and Stochastic Processes". AIAA Journal, Vol. 22, No. 11, November
1984, pp. 1670-1678.

8 . Fox, Richard L., "Optimization Methods for Engineering Design". Addison-
Wesley Publishing Company, Inc., Menlo Part, CA, 1971, pp. 124-149.

9 . Zadeh, L. H., "Optimality and Non-Scalar-valued Performance Criteria", IEEE
Transactions on Automatic Control, Vol. AC-8, No. 1, 1963.

10. Pareto, V., "Cours d'Economie Politiques Rouge, Lausanne, Switzerland, 1896.

1 1 . Loftin, Laurence K., Jr., "Quest for Performance the Evolution of Modem
Aircraft". NASA SP-468, 1985, pp. 437-452.

14

12. Davidon, W. C, "Variable Metric Method for Minimization", Argonne National
Laboratory, ANL-5990 Rev., University of Chicago, 1959.

13. Feagin, R. C. and Morrison, W. D., Jr., "Delta Method An Empirical Drag
Buildup Technique", NASA CR-151971, December 1978.

14. Sobieski-Sobieszczanski Jaroslaw; Dovi, Augustine R., and Wrenn, Gregory, A.,
"A New Algorithm for General Multiobjective Optimization". NASA TM- 100536,
March 1988.

15. Kreisselmeier, G., Steinhauser, G., "Multilevel Approach to Optimum Structural
Design", Journal of the Structural Division, ASCE, ST4, April, 1975, pp. 957-
974.

16. Cassis, Juan H., Schmit, Lucian, A., "On Implementation of the Extended Interior
Penalty Function", International Journal for Numerical Methods in Engineering,
Vol. 10, 3-23, 1976, pp. 3-23.

15

Table la
Multlobjective Cases

Case Number

kSOf'T Multlobjectives

1
2

Fi (X) and F2 (X)
F] (X) and F2 (X) and F3 (X)

Penalty Method

Weighted Composite

Multiobjectives

3
4

Fi(X)+F2(X)
Fi (X) + F2 (X) + 10,000.00 F3 (X)

Global Criterion Method
Target Objectives

5
6

FT (X) = 201629.0 kg and
FT (X) = 60954.0 kg
FT (X) = 201629.0 kg and
FT(X) = 60954.0 kg and
FT (X) = M(28.1)

Table lb
Single Objective Cases

Case Number

KSOPT Single objectives

7

Fi(X)

8

F2(X)

9

F3(X)

10

F4(X)

Penalty Method
Single Objectives

11

Fi(X)

12

F2(X)

13

F3(X)

14

F4(X)

16

Table 2
Single Objective Design Results

Xo

Initial

Conditions

Final Values

Misson Fuel
(minimiu)

Final Values
Ramp Weiehl

(minimiz«)

Final Values
M«sh (UD)
fma^iimizc)

Final Values

Ranee

(maximize)

Case 8
aSQZL

Case 12
£E

Case 7
KSOPT

Case 11
E£

Case 9
KSOFT

Case 13
£E

Case 10
KSOPT

Case 14
EE

Desifn

Vari»bl?s

AR, xj

Sw, X2, m^
Sweep, X3,

11.00
361.0

18.20
304.0

18.94
295.0

11.35
281.1

11.10
281.4

22.13
381.0

22.14
361.0

10.68
331.0

10.39
361.0

35.00

26.16

27.62

22.00

22.22

30.21

36.39

22.00

22.22

deg

t/c, X4

0.11

0.091

0.0913

0.0996

0.0989

0.087

0.107

0.099

0.098

Objective
Functions

"^TiOfSg 207729.0 219248.0 220155.0 201629.0 201763.0 256156.0 239332.0 219248.0 219248.0

*^T2°CX)."kg 67136.0 60954.0 60728.0 66891.0 66981.0 62791.0 62882.0 79492.0 79040.0

*^F^^) .83 (19.34) .83 (24.50) .83 (24.76) .83 (18.92) .83 (18.78) .83 (28.09) .83 (22.86) .83 (19.25) .83 (19.31)

F4(Xrtai 7413.0 7413.0 7413.0 7413.0 7413.0 7413.0 7413.0 8974.0 8922.0

Constraints
Si
82
«3
S4
85
86
87

Other

Quantities

Span, (b), m

UD

W/S

TAV

Function
Evaluations

63.0
19.34
117.80
0.327

l.O

-1.0

-1.0

-1.0

-1.0

-1.0

-.210

-.203

-.0307

-.0140

-.0327

-.0332

-.0636

-.0699

-.0701

-.0910

-.0601

-.0181

-.114

-.129

-.410

-.0781

-.112

-.165

-.0227

-.00166

-.0326

-.444

-.0479

-.0628

-.0626

-.0870

-.568

-.554

-.00227

-.532

-.565

-.601

-.391

-.403

-.459

-.451

-.217

-.211

-.475

-.509

-.110

-.112

-.0249

-.000754

-.0293

-.0266

-.102

-.114

-.126

-.0635

74.3
24.50
147.60
0.318

483

70.5
24.76
152.70
0.309

255

56.5
18.92
147.00
0.337

158

56.5
18.78
146.80
0.337

146

91.8
28.09
137.70
0.280

213

89.36
22.86
135.9
0.284

242

59.40
19.25
136.00
0.310

190

61.2
19.31
129.80
0.310

180

17

Table 3
Two Objective Design Results

Design Variables

AR, xj

14.51

Sw, X2, m2

289.0

Sweep, X3, deg

24.50

t/C, X4

0.0948

Objective Functions

Ramp Weight,

Fi (X), kg

206268.0

Mission Fuel,

F2 (X). kg

62353.0

M(L/D).F3(X)

.83 (21.77)

Range, F4 (X), km

7413.0

Constraints

81

-1.0

82

-.0352

83

-.116

84

-.0334

85

-.537

86

-.385

87

-.0341

Other Ouantities

Span, (b), m

64.7

L/D

21.77

w/s

146.20

T/W

0.329

Ramp Weight and Mission Fuel ('minimize')
Case 1 Case 3 Case 5

KSQEE EE Global Criteria

Function Evaluations

325

10.28
369.0
22.17
0.0946

205499.0

65803.0
.83 (19.84)
7413.0

-1.0
-.148
-.349
-.161
-.529
-.263
-.254

58.9
19.84
114.0
0.331

121

12.31
282.0
22.00
0.0958

202360.0

64647.0
.83 (19.97)
7413.0

-1.0
-.0334
-.120
-.0334
-.481
-.281
-.0243

55.5
19.97
146.80
0.336

98

18

Table 4

Three Objective

Design Results

Ramo Weieht and Mission Fuel (m

inimize^

and M fL/D) fmaximize)

Case 2

Case 4

Case 6

KSOPT

EE

Global Criteria

Pggign VariaMgs

AR, X]

16.87

15.49

11.64

Sw. X2, m2

365.0

291.0

286.0

Sweep, X3, deg

26.39

22.12

24.20

t/C. X4

0.083

0.089

0.099

Obiective Functions

Ramp Weight,

Fi (X). kg

228716.0

210065.0

202162.0

Mission Fuel,

F2 (X), kg

62041.0

61564.0

65980.0

M{L/D).F3(X)

.83 (25.09)

.83 (22.81)

.83 (19.29)

Range, F4 (X), km

7413.0

7413.0

7413.0

Constraints

gi

-1.0

-1.0

-1.0

82

-.0956

-.0305

-.0405

83

-.171

-.0921

-.134

84

-.0961

-.0267

-.0416

85

-.599

-.543

-.466

86

-.465

-.402

-.249

87

-.118

-.0839

-.0485

OHi?r Ouanpti??

Span, (b), m

78.4

63.4

54.4

IVD

25.09

22.08

19.29

W/S

128.50

147.60

144.60

T/W

0.297

0.323

0.336

Function Evaluations

62

174

73

19

Best Sin

igl<

Table 5
; Objective Results

Cas?

Objective

Method

Final Value

12

Fuel

PF

60728.0 kg

7

Weight

KSOFl'

201629.0 kg

9

VD

KSOFl

28.09

20

Sw, /IR,
t/c

OBJECTIVES

Ramp Weight (Minimum)

Mission Fuel (Minimum)

Mach (L/D) (Maximum)

Range (Maximum)

DESIGN VARIABLES
Sw, t/c, AMR

Figure 1. Objectives, Design Variables and Constraints

PRIMARY MISSION PROFILE

Tixi oui.

Tikccff,

Chmb It maximum Ciuise-climb tl desigiuted
nie-of-climb to long- | Mach number

nnge cnjue altiiude

Descend,
Iconsunt

CLidle
thiust

RESERVE PROnLE DOMESTIC OPERATIONS

Figure 2. Mission Profile

21

RAMPWEIGHT AS A FUNCTION OF
ASPECT RATIO AND THICKNESS RATIO

Figure 3, FLOPS Primary Modules

SINGLE OBJECTIVE OPTIMIZATION
CHANGE FROM INITIAL CONDITIONS

? S ?! 2

u u u u

2 2 2 2

u u t> u

Figure 5

MISSION FUEL AS A FUNCTION OF
ASPECT RATIO AND THICKNESS RATIO

Figure 6

MACH (L/D) AS A FUNCTION OF ASPECT
RATIO AND THICKNESS RATIO

Figure 4

22

TWO OBJECTIVE OPTIMIZATION
COMPROMISE FROM SINGLE OBJECTIVE CASES

Figure 8

10-

^ ■ WEIGHT

8-

CO

n FUEL

Ui

X

B 4-
a

a.

2-

"

1

i

O

K50PT PF GLOBAL
CASE 1 CASE Z CASE 5

TWO OBJECTIVE OPTIMIZATION COMPROMISE
FROM BEST SINGLE OBJECTIVE CASES

10

6-

u "l"

■ WEIGHT (CASE 7) ^

D FUEL (CASE 12) "^

KSOPT

-^ "^ (N

PF

Figure 10

GLOBAL

THREE OBJECTIVE OPTIMIZATION
COMPROMISE FROM SINGLE OBJECTIVE CASES

40-

30'

20-

KSOPT
CASE 2

■ WEIGHT
D FUEL
B L/D

PF
CASE 4

Figure 9

GLOBAL
CASE 6

THREE OBJECTIVE OPTIMIZATION COMPROMISE
FROM BEST SINGLE OBJECTIVE CASES

■ WEIGHT (CASE 7)
D FUEL (CASE 12)
H L/D (CASE 9)

KSOPT

GLOBAL

Figure 11

23

(WNSA

Report Documentation Page

1. Report No
NASA CR-4328

2. Government Accession No

4. Title and Subtitle

Aircraft Design for Mission Performance Using
Nonlinear Multiobjective Optimization Methods

3. Recipients Catalog No.

5. Report Date

October 1990

6. Performing Organization Code

7. Author{s)

Augustine R. Dovi and Gregory A. Wrenn

9. Performing Organization Name and A()dress

Lockheed Engineering & Sciences Company
Hampton, VA 23666

NASA Langley Research Center
Hampton, VA 23665

15. Supplementary Notes

8. Performing Organization Report No.

10. Work Unit No.

505-63-01

11. Contract or Grant No.

NASI- 19000

13. Type of Report and Period Covered

Contractor Report

14. Sponsoring Agency Code

Langley Technical Monitor: Jaroslaw Sobieski

16. Abstract

A new technique which converts a constrained optimization problem to an
unconstrained one where conflicting figures of merit may be simultaneously
considered has been combined with a complex mission analysis system. The
method is compared with existing single and multiobjective optimization methods
A primary benefit from this new method for multiobjective optimization is the
elimination of separate optimizations for each objective, which is required by
some optimization methods. A typical wide body transport aircraft is used for
the comparative studies.

17. Key Words (Suggested by Author(sl)

constrained
optimization

multiobjective optimization
conceptual design

19. Security Classif. (of this report)

Unclassified

18. Distribution Statement

Unclassified - Unlimited

Subject Category 05

Unclassified

NASA FORM 1626 OCT 88

21 No. of pages
32

22. Price
AOj

For sale by the National Technical Information Service, Springfield, Virginia 22161-2171

NASA -Langley, 1990

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