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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 ■i T ■;/ r MUlT I't. J' ': T T V ori'^ncos Corp. J -' -^ n .s--^-,,/ n n ,i fi^. r I n ^ 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 Space Administration 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 payload = 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 12. Sponsoring Agency Name and Addreis 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 20. Security Classlf. (of this page) 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