JOURNAL OF AIRCRAFTVol. 39, No. 3, MayJune 2002

Lessons Learned During Solutions of MultidisciplinaryDesign Optimization Problems

Surya N. PatnaikOhio Aerospace Institute, Brookpark, Ohio 44142

andRula M. Coroneos,Dale A. Hopkins, and Thomas M. Lavelle

NASA John H. Glenn Research Center at Lewis Field, Cleveland,Ohio 44135

Optimization research atNASAGlennResearch Center has addressed design of structures, aircraft, andaircraftengines. Four issues were encountered during the solution of the multidisciplinaryproblems. Strategy adapted fortheir resolution is discussed. Design optimization produced an inef cient design for an engine component. Thede ciency was overcome through animation andweight was reduced by 20%. Infeasibility encountered in aircraftand engine designproblemswas alleviated by cascadingmultiple algorithms.The cascade strategy reduced aircraftweight and produced a feasible design. Pro le optimization converged to an irregular shape for a beam. A regularshape was restored through engineering intuition, but the optimum weight was not changed. The subproblemoptimization for a cylindrical shell converged to a design that might be dif cult to manufacture. This issue remainsa challenge. The issues are illustrated through design of an engine component, synthesis of a subsonic aircraft,operation optimization of a supersonic engine, design of a wave-rotor-topping device, pro le optimization of abeam, and design of a cylindrical shell. A multidisciplinary problem might not yield an optimum solution inthe rst attempt. However, the combined effort of designers and researchers can bring design optimization fromacademia to industry.

I. Introduction

O PTIMIZATION researchat NASAGlenn ResearchCenter hasaddressed the structural design of airbreathingpropulsionen-gines and space station components, aircraft synthesis, as well asperformance improvement of the engines. The accumulated mul-tidisciplinary design activity is collected under a test bed entitledCOMETBOARDS. 1 It was observed that such problems might notbe amenable to a solutionin the rst attempt.Solution,however, canbe obtained through an augmentation of available tools and strate-gies. While generating solutions to the multidisciplinaryproblemsthrough the test bed, several issues were encountered. This paperlists four issues and presents the strategies that were employed fortheir resolutions:1) The optimization process produced a local inef cient design

for a reardivergent ap of a downstreammixingnozzle.An augmen-tation of animation improved the design. The weight was reducedby 20%.The optimumstructureexhibiteda desirablebreathingtypeof mode shape.2) Solutions obtained for aircraft and engine problems, using

a single optimization algorithm, encountered infeasibility eventhough the values of the merit functions were in the vicinity ofthe correct solutions.The infeasibilitywas alleviated through a cas-cadingofmultiple algorithms.The cascadestrategy reducedaircraftweight by about 1%.3) The shape optimizationof a beamproducedan irregularshape.

The regular shape could be restored through engineering intuition.However the optimumweight remained the same for both solutions.

Received 27 January 2001; revision received 11 October 2001; acceptedfor publication 12 December 2001. This material is declared a work of theU.S. Government and is not subject to copyright protection in the UnitedStates. Copies of this paper may be made for personal or internal use,on condition that the copier pay the $10.00 per-copy fee to the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; includethe code 0021-8669/02 $10.00 in correspondence with the CCC.Senior Engineer, Structures and Acoustics Branch, MS 49-8. Associate

Fellow AIAA.Mathematician, Computational Sciences Branch.Structures and Acoustics Division. Senior Member AIAA.Aerospace Engineer, Aeropropulsion Analysis Of ce.

4) For a cylindrical shell the subproblem solution strategy con-verged to a local design that could be dif cult to manufacture.Res-olution of this issue is still a challenge. This paper expounds uponthe lessons learned in solvingmultidisciplinaryproblemswith littleemphasis on the algorithmor analysismethod. The paper is dividedinto ve subsequent sections. An outline to the testbed COMET-BOARDS is given in Sec. II. A descriptionof the illustrativeexam-ples is given in Sec. III. The next section describes the four issues:the local solutionand animation, infeasibilityand the cascade strat-egy, irregular design and intuition, and substructure solution andmanufacturability.Discussionsand conclusionsare given in Secs. Vand VI.

II. COMETBOARDS TestbedThe design optimizationtest bed COMETBOARDS can evaluate

the performance of different optimization algorithms and analy-sis methods while solving a problem. It is a research test bed butnot a commercial code. The acronymCOMETBOARDS stands forComparative Evaluation TestBed of Optimization and AnalysisRoutines for the Design of Structures. The scope of the test bedhas been expanded to include the design of structures, the synthesisof aircraft, and the operation optimization of airbreathing propul-sion engines. COMETBOARDS has three different analysis meth-ods and 12 optimization algorithms. It has a modular organizationwith a soft coupling feature, which allows quick integrationof newor user-suppliedanalyzersand optimizerswithout any change to thesource code. The COMETBOARDS code reads information fromdata les, formulates design as a sequence of subproblems, andgenerates the optimum solution. COMETBOARDS can be used tosolve a large problem, de nable through multiple disciplines, eachof which can be further broken down into subproblems. Alterna-tively, it can improve an existing system by optimizing a small por-tion of a large problem.Other unique featuresof COMETBOARDSinclude design variable formulation, constraint formulation, sub-problem strategy, global scaling technique, analysis approximationthrough neural network and linear regression method, use of se-quential and parallel computational platforms, and so forth. Thespecial features and unique strengths of COMETBOARDS assistconvergence and reduce the amount of CPU time required to solve

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PATNAIK ET AL. 387

Fig. 1 Organization of the multidisciplinary design optimization testbed COMETBOARDS.

dif cult optimizationproblemsof the aerospace industry.COMET-BOARDS has been successfully used to solve the structural designof the InternationalSpaceStation components,the designof thenoz-zle components of an airbreathing engine, and airframe and enginesynthesis for subsonic and supersonic aircraft,mixed- ow turbofanengine,wave-rotor-toppedengine,andso forth.Themodularorgani-zation of COMETBOARDS is depicted in Fig. 1. Brief descriptionsof some of its modules follow.For scaling and constraint formulation a multidisciplinarydesign

problemcan have a distorted design space because its variables andconstraintscan varyoverawide range.For example,an enginethrustdesign variable,which is measured in kilopounds,is immensely dif-ferent from its bypass ratio, which is a small number. Likewise, thelandingvelocityof an aircraftmeasuredin knotsand landingor take-off eld lengthsmeasured in units of thousandsof feet differ both inmagnitude and in units of measure. This module provides a schemeto reduce the distortionby scaling the design variables, the objectivefunction,and the constraintssuch that their relativemagnitudesdur-ing optimization calculations are around unity. The constraints arereformulated to alleviate redundancy without affecting the prob-lem de nition. Constraint formulation alleviates redundancy andreduces their number.2 The cascade algorithm employs more thanone optimizer to solve a complex design problem when individualmathematical programmingmethods experience dif culty.3

The module Analyzers-Structure,Aircraft, Engine-Neural net-works and Regression approximationshouses three different typesof analysis methods. For structural analysis the methods avail-able are COSMIC NASTRAN,4 MSC/NASTRAN,5 MHOST,6

Analyze/Danalyze codes,7 and IFM/Analyzers.8 Aircraft analysiscan use the FLOPS code.9 The NEPP code10 is employed for air-breathing propulsion engine cycle analysis. Neural network11 andregression techniques12 can be employed to approximate analysis.A problemcan utilize any one of three analyzers:1) an originalana-lyzer, for example FLOPS code, or one of the two derivedanalyzersbasedon 2) a neuralnetworkor 3) a linearregressionapproximation.The module Engine operations in Fig. 1 refers to the perfor-

mance optimizationof airbreathingpropulsionengines for multipleoperation points.13 Aircraft synthesis refers to the airframe andengine integration for subsonic and supersonic aircraft.3

The module Structural designSubproblem strategy and Par-allel computational environment refers to design of structuresthroughregularoptimizationor a subproblemstrategy.This strategyis availablein sequentialandparallelcomputationalenvironments.14

Multiple disciplines refers to the solution of a problem, whichis de ned through different disciplines.COMETBOARDS can ac-commodateseveraldisciplines,eachofwhich can be furtherdividedinto subproblems. Subproblem strategy is an attempt to alleviateconvergencedif culties that can be encounteredduring the solutionof a large optimization problem. In this strategy the large problemis replaced by a sequence of overlappingmodest subproblems.Thesolution to the large problem is obtained by repeating the solutionto the set of subproblems until convergence is achieved.

Fig. 2 Cargo-bay support system of the international space station(dimensions are in inches.)

Substructure strategy, a key component of COMETBOARDS,is further illustrated through the design of a cargo-bay support ofthe International Space Station. The support structure is fabricatedout of four plates, a cluster of plates referred to as a box, and vebeams, as shown in Fig. 2. For the purpose of design, the supportsystem was divided into four active and one passive substructures.The rst substructure was the closed box (FGHKIJ) consisting of ve plates. Its nite element model was obtained by discretizing itinto 72 (QUAD4 ) shell elements. The second substructure was atrapezoidalplate (FHEC), and its nite element model had 37 shellelements. The third and fourth substructureswere triangular plates(GHE and GHD) with 12 nite elements each. The fth substruc-ture containedthe ve connectingbeamsBD, BG, AD, AG, andAF.The beam designs were not changed. These passive variables didnot participate in the design calculations but were retained duringreanalysis. Two independent nite analysis codes LE HOST andMSC/NASTRAN were used to verify the nite element model ofthe support structure. The nite element model with a total of 133QUAD4 shell and 20 beam elements was considered adequate fordesign optimizationbecauseboth analyzersproducedan acceptablelevel of accuracy for stress, displacement, and frequency. The sub-structureswere clusterednext to obtain a set of subproblems.A sub-problemcontainsa substructurealong with some or all of its neigh-boringsubstructures.The foursubstructureswere clusteredto obtainthe following four subproblems: subproblem 1substructures 3and 4; subproblem 3substructures 1 and 2; subproblem 2substructures1 and 4; and subproblem4substructures2 and 3.Notice the couplingbetweensubproblems:substructure1 is com-

mon to subproblems 2 and 3. Likewise, substructure 2 is commonto subproblems 3 and 4, and so forth. Adequate coupling betweensubstructures is required for convergence. Inappropriate couplingcan increase the amount of computation and/or encounter conver-gence dif culties. At this time substructure coupling is decided in-tuitively. However, it might be possible to automate substructurecoupling through the gradient scheme developed in Ref. 15. TheCOMETBOARDS testbedcan optimizea systemthat canbede nedin terms of 100 optimization subproblems.14

In the module Problem formulationand solution information isread from data les, the design is cast as a sequence of optimizationsubproblems, and the solution is obtained.The COMETBOARDS testbed is written in the Fortran 77 lan-

guage except for the neural network algorithm, which is written in

388 PATNAIK ET AL.

the CCC language. The test bed is available in the Unix operatingsystem in workstations and Cray computers.One dozen mathematical programming algorithms are available

in COMETBOARDS. A list of the algorithms in no particular or-der follows: 1) method of feasible direction,16 2) modi ed feasibledirection method,17 3) reduced gradient method,18 4) generalizedreducedgradientmethod,19 5) sequenceof unconstrainedminimiza-tion technique,20 6) sequential linear programming,16 7) sequentialquadratic programmingmethod,21 8) IMSL optimization routine,22

9) NPSOL package of NAG library,23 10) 16 different versions ofoptimality criteria methods,24 11) a genetic algorithm,25 and 12)fully utilized design algorithm.15

Different algorithms employ different strategies to calculate thesearch direction and the step length. The comparative performanceof the algorithms in solving a set of 45 problems is reported inRef. 26. The performance of six algorithms in solving a set ofmedium and large structural problems is depicted in a bar chartin Figs. 3a and 3b, respectively.Examples with design variables inthe rangeof 20 to 39with about200 behaviorconstraintsare referredto as medium problems (see Fig. 3a). Examples with more than 40independent design variables and several hundred constraints arereferred to as large problems (see Fig. 3b). The success rate of dif-ferent optimization algorithms for 10 large structural problems isalso depicted in a Venn diagram in Fig. 3c. Cascade solutions forthe problem are given in a table (see insert Fig. 3e). The successof an algorithm is represented by unity, which is the normalizedvalue of the merit function (see Figs. 3a and 3b). A normalizedvalue of less than unity (an infeasiblesolution) or greater than unity(an over-design condition) represents underperformance.Most op-timizers available in the test bed solved at least one-third of theexamples, but none of the optimizers could successfullysolve all ofthe problems. Every structural problem could be solved by at leastone of the six different optimization algorithms.However, even the

Fig. 3 Comparative evaluation of optimization algorithms.

most robustoptimizerencountereddif culty in generatingoptimumsolutions for aircraft and engine problems.

III. Illustrative ExamplesFor the purpose of illustration, we have selected six problems:

problem1Structuraldesignof an enginecomponent,problem2Synthesisof a subsonicaircraft,problem3Operationoptimizationof a supersonicengine,problem4Design of a wave-rotor-toppingdevice, problem 5Pro le optimization of a cantilever beam, andproblem 6Design of a cylindrical shell.Augmentation of animation into optimization is illustrated

throughproblem1. Problems 2, 3, and 4 illustrate the cascade strat-egy. Problem 5 addresses shape optimization. The subproblem so-lution strategy is illustrated through problem 6. Brief de nitions ofthe problems follow.

A. Problem 1: Structural Design of an Engine Component

A mixed- ow turbofan engine exhaust nozzle referred to as adownstreammixing nozzle for a High Speed Civil Transport air-craft to operate at a cruise speed of Mach 2.4 and in a range of5000 nautical miles is shown in Fig. 4a. It is fabricated out of rearand forward divergent aps, rear and forward sidewalls, bulkheads,duct extensions, six disk supports, and other components. The de-sign complexityof the nozzle is increasedwith ightMach number,pressure ratio, temperature gradient, dynamic response, and degra-dation of material properties at elevated temperature. The ap ismade of Rene 125 material with a Youngs modulus of 30.4 mil-lion psi, a Poisson ratio of 0.3, a density of 0.308 lb/in.,3 and anallowablestrengthof 117 ksi. The ap shown in Fig. 4b has a lengthof 96 in. and a width of 72 in. It is supported by two variable-depthedge beams with a maximum depth of 14 in. A grid with a spac-ing of 12 12 in. and a depth of 2 in. stiffens the ap. MHOSTand MSC/NASTRAN analyzers were used for the static as well as

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Fig. 4 Design of a rear divergent ap of a downstream mixing nozzle.

the dynamic analyses of the ap. A nite element model with 5593degrees of freedom and 946 QUAD4 elements was used for anal-ysis because both methods produced acceptable values for stress,deformation, and frequency. For this problem the thickness of theedge beams, stiffeners, and skin panel was consideredas the designvariables.The ap was designedfor minimumweight for vonMisesstress, local buckling,displacement,and frequencyconstraints.Theproblem had 946 stress and 946 instability constraints, as well asfour displacements and one-frequencyconstraints.

B. Problem 2: Synthesis of a Subsonic Aircraft

The system synthesis capability of COMETBOARDS is il-lustrated through a subsonic aircraft to operate at Mach 0.85cruise speed. Solution of this problem required a soft couplingof COMETBOARDS to NASA Langleys aircraft analysis codeFLOPS. The FLOPS analyzer encompasses several disciplines:weight estimation, aerodynamic analysis, engine cycle analysis,propulsion data interpolation,mission performance, air eld lengthrequirement, noise footprint calculation, and cost estimation.2729The objective of the synthesis problem was to determine an opti-mum airframe-enginedesign combination for a set of active designvariables under speci ed engine and aircraft behavior parametersto minimize a composite merit function that could be generated asa linear combination of weight, mission fuel, lift-to-drag ratio, andNOx emission. The design variables consideredwere engine thrust,wing area, engine design point turbine entry temperature, overallpressure ratio, bypass ratio, and fan pressure ratio. Constraintswereimposed on mixed approach climb thrust, second segment climbthrust, landing approach velocity, takeoff eld length, jet velocity,and compressor discharge temperature. For the subsonic aircraftmodel the gross takeoffweightwas consideredas themerit function.

C. Problem 3: Operation Optimization of a Supersonic Engine

The operationoptimizationof airbreathingpropulsionsupersonicengine required a soft coupling of the NASA Engine PerformanceProgram (NEPP) analyzer to COMETBOARDS. The engine-cycleNEPP code can con gure and analyze almost any type of gas tur-bine engine that can be generated through the interconnection ofa set of standard physical components such as propellers, inlets,combustors, compressors, turbines, heat exchangers, ow splitters,nozzles,and others.An enginecan be designedfor different typesofhydrocarbonjet fuels, cryogenicfuel, and slurries.For their thermo-dynamic analysisbuilt-in curve ts generatedfrom empirical resultshave been incorporated into the NEPP code. A description of the

NEPP codewith typical input les for a set of engine con gurationscan be found in Refs. 30 and 31.The engine operation optimization problem, with its associated

design variables (such as the engine shaft speed, the wave-rotorspeed, the ow area, the geometrical parameters of the ducts, etc.)and constraints (imposed on pressure ratios, surge margins, tem-perature limits, and entrance Mach number, etc.), was cast as asequence of nonlinear optimization subproblems with thrust as themerit function. Engine operation design became a sequence of in-terdependent problems, or one optimization subproblem for eachoperating point. The optimization process typically adjusted a fewengine parameters. The dif culty in the engine problem did not liewith thenumberof activedesignvariables,but itwas associatedwithitsmultipleoperating-pointcharacter,constraintvalidityranges,andthe iterative nature of engine cycle analysis. The most reliable indi-vidual optimizationalgorithmavailablein COMETBOARDS couldnot producea satisfactoryfeasibleoptimum solution for this engineproblem because of the large number of operation points in the ight envelope, the diversity of the constraint types, and the overalldistortionof the design space.However,COMETBOARDS uniquefeatureswhich includeda cascadestrategy,variableand constraintformulations, and scaling devised especially for dif cult multidis-ciplinary applicationssuccessfully optimized the performance ofsubsonic and supersonic engine concepts. Even when started fromdifferentdesign points, the combinedCOMETBOARDS andNEPPcodes converged to about the same global optimum solution. Thisreliableand robustdesign tool eliminatedmanual interventionin thedesign of the airbreathing propulsion engines and eased the cycleanalysis procedures.Engine design is illustrated through a supersonicmixed- ow tur-

bofan engine problem. It was con gured with 15 components andwas designed for a ight envelope with 122 operating points. Thedesign required the solution of a sequence of 122 optimization sub-problems.The objectivewas to maximize the net thrust of the super-sonic engine at each operating point, accounting for an installationdrag. Each subproblemhad three independent design variables and22 behavior constraints.

D. Problem 4: Design of a Wave-Rotor-ToppingDevice

A high-bypass-ratiosubsonicwave-rotor-toppedturbofan engineis made of 16 components that are mounted on two shafts with 21 ow stations. It was modeled with standard components that in-clude an inlet and a splitter, then branching off to a compressor, aduct, and a nozzle. The main ow proceeded through a fan, a duct,

390 PATNAIK ET AL.

a high-pressurecompressor, a duct, a high-pressure turbine, a low-pressure turbine, a duct, and a nozzle.The componentsmounted onthe rst shaft included the fan, the compressor along the secondary ow branch, the low-pressure turbine, and a load. The second shaftcarried the high-pressurecompressor along the main ow, the high-pressure turbine, and a load. The four-portwave-rotor (with burnerinlet and exhaust,compressorinlet,and turbineexhaust)was locatedbetween the high-pressure compressor and the high-pressure tur-bine. The engine operating conditionwas speci ed by a 47-mission ight envelope,with altitude in the range (sea level to 40,000 ft) andspeed between 0.00.85 Mach. To examine the bene ts that accruefrom the wave-rotor device, the engine was optimized consideringseveral of its baseline variables and constraints,not explicitly asso-ciated with the wave rotor, as passive. The design objective was tomaximize the net engine thrust at each of the 47 operatingpoints. Ithad two designvariables:heat added to thewave rotor and thewave-rotor speed. Its 16 behaviorconstraints included the correctedspeedratio for the compressor and the fan, the unmixed wave-rotor tem-perature, the surge margin on the compressor, and the fan pressureratio for the turbine.

E. Problem 5: Cantilever Beam

Calculation of an optimum pro le or depth along the lengthis illustrated considering the cantilever beam shown in Fig. 5.The beam is made of aluminum with a Youngs modulus E D10 106 psi, a Poissons ratio D 0:3, and a weight density D0:1 lb/in.3 It is 240 in. long and 6 in. deep. The beam weight wasthe merit function, and stress and displacement were its behaviorconstraints.For the purposeof analysis, the beamwasmodeledwithsix 20-node-hexahedralelements of the IFM/Analyzers code.32 The nite elementmodelwith 160 displacementdegreesof freedomwasused for analysis because this model adequatelypredicted the stressand displacement responses of the beam. The pro le optimizationproblem had nine depth design variables, and its nine constraintsincluded eight stress and one displacement limitations.

F. Problem 6: Cylindrical Shell

A cylindrical shell with rigid diagrams under two line loadsis shown in Fig. 6. It is made of steel with a Youngs modulusE D 30 106 psi, a Poissons ratio D 0:3, and a weight density D 0:289 lb/in3. It has a radius of 100 in. and length of 200 in. Be-cause of symmetry, only one-eighthof the structurewas consideredfor design optimization. A nite element model with 100 QUAD4elements of the MHOST analyzer was considered adequate to pre-dict its response. Its design was cast as an optimization problemwith weight as the objective function, and constraintswere imposedon stress and displacement. The thickness of the cylinder alongits length and circumference were considered as design variablesthrough a pro led depth link factor to provide a heavier designunder the load. The one-eighth shell model was divided into foursubstructures along its length. The substructureswere clustered toobtain three and four subproblems for sequential and parallel com-putations, respectively. To avoid convergence dif culty in parallelcalculations,an additional (fourth) subproblemwas required.

IV. Issues in Multidisciplinary Design OptimizationThe four issues1) local solution and animation, 2) infeasi-

bility and cascade strategy, 3) irregular design and intuition, and

Fig. 5 Pro le optimizationofa cantileverbeam for stress anddisplace-ment constraints.

Cylindrical shell

Optimum pro le along cylinder length

Fig. 6 Substructure optimization of a cylindrical shell.

4) substructure solution and manufacturabilityare addressed inthis section.

A. Local Solution and Animation

The design of the ap or problem1, which is depicted in Fig. 4b,was obtainedusingthreeoptimizationalgorithms.All threemethodsproduced the same optimum weight of 1448.2 lb. At the optimumthe frequency at 40 Hz and displacement at 1 in. were the activeconstraints.Stress and local instabilitywere passive constraints. Itsanimation at the 40-Hz frequency was examined, and one frame isdepicted in Fig. 4c. The animation exhibited a local frequencycon-dition. The edge beams vibrated with signi cant amplitude, whilethe response of the bulk of the structure was insigni cant. In otherwords, only a small part of the structure carried a major portion ofthe load. In this design the edge beams became more failure pronethan other parts of the structure.The con guration of the ap was modi ed through an exami-

nation of the animation of a series of designs. The nal modi edcon gurationwas obtained by increasing the depth of a single edgestiffener to 6 in. from its original 3-in. depth. This con gurationwas optimized. The increase in the material for this stiffener wascompensated for by a reduction in the thickness of the edge beamby 58% between the two con gurations.The dynamic animation ofthe ap and the von Mises stress distribution are shown in Figs. 4dand 4e, respectively. The structure vibrated in a breathing type ofmode, or the entire ap responded as a single unit. The optimumdesign also displayed a full stress condition for a major portion ofthe ap as shown in Fig. 4e. The optimumweight at 1204.7 lb was20% lighter than the original con guration.Animation-assistedop-timization reduced the weight and improved the overall design ofthe ap.

B. Infeasibility and Cascade Strategy

Individualoptimizationalgorithmsencountereddif culty in gen-erating solutions to aircraft and engine problems.A useful strategythat combined the strength of more than one optimizer was con-ceived. This cascade strategy, using a number of optimizationalgo-rithmsone followedby anotherin a speci ed sequence,was foundtobe superior to the component optimizers (see Fig. 3d). The cascadealgorithm was employed to solve the 10 large structural problemsreferenced in Sec. II. The success rate of the cascade strategy andthe individual algorithms is shown in Figs. 3c and 3e. The cascadestrategy solved all 10 problems.Cascade solutions to the subsonic aircraft, the supersonicmixed-

ow turbofanengine,and the subsonicwave-rotor-toppedengineareshown in Fig. 7. The subsonic aircraft system design problem wassolved using a three-optimizer cascade algorithm: optimizer 1, fol-lowed by optimizer 2, and optimizer 1 again. The cascade solution,

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a) Optimizer 1 solution

b) Optimizer 2 solution

c) Subsonic aircraft

d) Mixed- ow turbofan engine

e) Wave-root-topped subsonic engine

Fig. 7 Cascade solutions for aircraft and engine problem.

along with solutions obtained from individual algorithms, is de-picted in Fig. 7 and Table 1.Optimizer 1, when used alone, converged to a heavy infeasible

solution at 202,005 lb for the aircraft weight (see Fig. 7a). Like-wise, optimizer 2 alone also produced a heavy design at 202,854 lb,(see Fig. 7b). Optimizer 1 required a takeoff eld length of 6282 ftagainst an available 6000-ft runway. Optimizer 2 overestimated theexcess fuel requirementat 8150 gallons against a desired amount of5000 gallons. The convergence rate of the two algorithms differedproducing solutions that were 1 and 2% overweight, respectively,for the two algorithms (see Figs. 7a and 7b). The two solutions,although close to the optimum, were not attractive to industry be-cause they violated the safetymargins.A cascade algorithmcreatedfrom the same two optimizers (optimizer 1-optimizer 2-optimizer1) successfullysolved the problemwith a feasibleoptimumsolutionat 199,818 lb for the aircraft weight (see Fig. 7c).Solution of the mixed- ow supersonic and wave-rotor-topped

subsonicenginesalso requiredthecascadestrategy.A two-optimizercascade (optimizer 3 followed by optimizer 2) successfully solvedthe supersonic engine problem (see Fig. 7d). The solution tothe wave-rotor-enhanced subsonic engine (see Fig. 7e) required athree-optimizer cascade algorithm: optimizer 1 followed by opti-mizer 2 and optimizer 1 again.Aircraft (FLOPS) and engine (NEPP) analyzers are nonlinear

codes that incorporate multiple disciplines. The analysis assump-tions, dependent on altitude, Mach number, and engine power set-ting, can be challenged. The NewtonRaphson iterations duringengine-balancingmay not converge, leading to an engine stallcondition with zero thrust, or the NEPP code producing a NaN(not a number) for some constraints. A single optimization algo-rithm can terminate abruptly, or hit a constraint boundary withoutany improvement to the merit function.To overcome the dif cultiesin aircraft and engine analysis codes, we employed two compet-ing approximators: linear regression and neural network methods.Even then, the cascade algorithm was required because individual

algorithms encountered dif culty.13 The nature of engine and air-craft problems required the strength of multiple algorithms or thecascade strategyevenwith the analysis approximators,which, how-ever, was not expected.

C. Irregular Design and Intuition

The pro le optimization of the cantilever beam, or problem 5,was attempted with uniform upper and lower bounds for the designvariablesat 0.5 and 15 in., respectively (see Table 2). The optimiza-tion problemwas solved using optimizer4. The solutionobtained isshown in Fig. 5 andTable 2.The optimumweightwas 1697.5 lb, andthe root stresswas the only active constraint.An odd-shapedpro leshown in Fig. 5 was obtained. The pro le was peculiar because thefree end, corresponding to a zero stress condition, had a depth of5.14 in. insteadof an anticipatedlowerbounddepthof 0.5 in.The op-timum solutionmost likely challenged linear structural analysis as-sumptions, and the optimization iterations encountereddif culties.The situation did not improve when a different optimization algo-rithm or a cascade strategywas employed.The problemwas solvedsuccessfullywith manual intervention. Instead of uniform upper orlower bounds and a single optimization algorithm, the design wascast as a sequence of subproblems. Upper and lower bounds andinitial designthrough engineering intuitionwere progressivelychanged for the subproblems.This procedureproduceda convergedsolution that is shown in Fig. 5. It had the same minimum weightof 1697.5 lb, which was identical to the odd-shaped design. Itsroot stress and tip displacementwere the active constraints.For thisproblemoptimizationalgorithmsconvergedto two distinct local so-lutionswith equalminimumweight. Industrywill be more inclinedto accept the monotonically pro led beam over the odd-shapeddesign.The ap design and beam pro le calculations represent typi-

cal problems of the aerospace industry. Neither problem could besolved satisfactorily without manual intervention. The dif culties

392 PATNAIK ET AL.

Table 1 Solution to the subsonic aircraft design problem using a cascade strategy

Optimizer 1 Optimizer 2solution solution

Cascade solution (as percent of (as percent of(optimizer 123) cascade solution) cascade solution)

optimum Infeasible and feasible butParameters designa heavy heavy

Weight, lb 199,818.00 1.01 1.02Design variablesWing aspect ratio (from 5 to 12) 8.33 0.94 1.19Engine thrust (from 20,000 to 70,000 lb) 31,228.21 0.94 0.99Wing area (from 1000 to 3000 ft2/ 1,936.05 1.04 1.00Quarter-chord sweep angle of the wing 11.63 0.93 2.32(from 0 to 45 deg)

Wing thickness to chord ratio (from 0.05 to 0.15) 0.09 0.89 1.22Engine design point turbine entry temperature, R 3,014.29 0.93 0.94Overall pressure ratio (from 10 to 40) 39.12 0.83 0.86Bypass ratio (from 0.1 to 15.0) 5.65 0.61 0.68Fan pressure ratio (from 1.1 to 3.5) 1.83 1.16 1.14

ConstraintsApproach velocity (not to exceed 125 kn) 118.75 0.98 1.0Takeoff eld length (not to exceed 6000 ft) 6,000.00 6,282.00a 6,000.00a

Landing eld length (not to exceed 6000 ft) 5,460.00 0.98 1.0Mixed approach thrust (normalized 97,000.00 1.02 1.0with respect to 100,000 lb)

Second segment climb thrust (normalized 93,000.00 1.03 0.98with respect to 100,000 lb)

Compressor discharge temperature, R 1,416.20 0.93 0.95Excess fuel (normalized with respect to 5000 gal) 5,000.00 2,150.00a 8,150.00a

aActual values.

Table 2 Intermediate and nal solutions for the beam pro le

Intermediate solutiona Final solutionb

Design Lower Initial Upper Optimum Design Lower Initial Upper Optimumvariable bound design bound design variable bound design bound design

1 0.5 10.0 15.0 14.68 1 10.0 12.0 13.0 11.722 0.5 10.0 15.0 2.98 2 9.0 8.0 12.0 9.003 0.5 10.0 15.0 5.04 3 6.0 7.0 10.0 6.224 0.5 10.0 15.0 6.27 4 5.0 6.0 9.0 6.175 0.5 10.0 15.0 5.41 5 4.0 5.0 8.0 5.026 0.5 10.0 15.0 0.50 6 3.0 4.0 7.0 3.677 0.5 10.0 15.0 2.49 7 2.0 3.0 6.0 2.008 0.5 10.0 15.0 2.67 8 1.0 2.0 5.0 1.329 0.5 10.0 15.0 5.14 9 0.5 1.0 4.0 0.50

aOnly the stress constraint at the root was active. bTip displacement and root stress were the active constraints.

encountered to some extent justify the reluctance of the aerospaceindustryto acceptadvancedoptimizationmethods,abandoningtheirtime-tested design rules. Neither mathematical programming algo-rithms nor the traditionaldesign rules could produceoptimumhard-ware con gurationsthat couldbemanufactured.Their combination,however, was a winner.

D. Subproblem Solution andManufacturability

A nonmanufacturablesolution obtained in the subproblem strat-egy is illustrated considering the design of a cylindrical shell prob-lem, or problem 6. Solution to the problem was obtained using:1) regular optimization, where the entire structure was consideredas a single problem, and 2) subproblemsolution,wherein four over-lappingsubstructureswere used.The optimumpro les for the cylin-drical shell obtainedusing regularand subproblemstrategiesare de-picted in Fig. 6. The two optimum weights obtainedwere 1161.95and 1154.1 lb for regular and subproblem strategies, respectively.The differenceof 0.676% in the solutions could be considered neg-ligible especially for the problem complexity. The depth differedsubstantiallybetween the two solutions.At the crown the optimumdepths of 1.322 and 2.471 in. obtainedby the twomethods variedby53%. At the optimum the regular optimization and the subproblemstrategy produceda differentnumber of active constraints.The sub-problem strategy produced only active stress constraints, whereas

both stress and displacement constraintswere active for the regularoptimization. The pro le obtained using regular optimization wasmore uniform than that generated through subproblem optimiza-tion. To verify the existance of two different designs (one obtainedfrom the subproblem strategy and the other from the regular op-timization), we resolved the regular optimization case by settingthe initial design equal to the optimum solution that was generatedfrom the subproblem strategy. The solution converged to the initialdesign, con rming the existence of the two local solutions withabout the same minimum weight. For this problem the attractive-ness of subproblemstrategyhas been challengedbecausethe designthus obtained is more dif cult to manufacture compared to the reg-ular solution.The authors are not aware of a scheme to alleviate thislimitation of the subproblemsolution strategy.

V. DiscussionDiscussion is given under multiple design solutions and the con-

vergence of algorithms.

A. Multiple Design Solutions

More than one optimum solutioncan be obtained for engineeringdesign problems. For example, different solutions were obtainedfor the nozzle ap, beam pro le, cylindricalshell, subsonicaircraft,and engine problems. The values of merit function changed little

PATNAIK ET AL. 393

between the different solutions.The intermediatedesigns, however,could became nonmanufacturable or their safety could be ques-tioned. These limitations were alleviated through animation andengineering intuition. The utilization of animation improved thedesign of the nozzle ap, and manipulation of the design boundspromoted the manufacturabilityof the beam.Philosophically,we can arrange designmethods into a spectrum.

Experience occupies the bottom strata. Optimization methods be-long to the upper spectrum. Popular design rules occupy the cen-tral spectrum. Design improved as we moved from the lower toupper spectrummethods but complexity increased. The design op-timization method is in existence for about half a century, yet itsfull potential has yet to be exploited by industry. The situation canbe improved by merging design optimization with the engineeringknowledgecontained in the broad spectrummethods. This goal canbe achieved when proponents of the optimization method and in-dustrial designers work in tandem.

B. Convergence of Optimization Algorithms

Consider the convergenceof the subsonic aircraft weight by twodifferent algorithms, shown in Figs. 7a and 7b. Convergence wasmonotonic by both methods even though the rate differed, as ex-pected. A similar trend was observed for the engine problems.Convergence at times oscillated about a mean solution until themaximum iteration limit was reached. Redundancy of the activeconstraints, their continuity, and convexity may have created thissituation. Engineering design problems might not satisfy some ofthe basic requirements that form the foundation of mathematicalprogramming methods. For example, the stress and displacementconstraints of structural problems are by nature redundant.2 Thespeci ed range of the constraints of engine and aircraft problemsare susceptible to violation during optimization calculations. Op-timum solution to multidisciplinary problems of the industry haveto be obtained utilizing available analysis and optimization capa-bilities. The cascade strategy was found successful in generatingreliable solutions for structures, aircraft, and engine problems. Thecascade strategy was required even when neural network and re-gression approximators were used to approximate constraints andmerit functions of the engineering design problemsthis howeverwas not expected.

VI. ConclusionsAn animation-assistedoptimization successfully solved the ap

design problem. The nal design was lighter. The entire structureresponded as a single unit in the fundamental mode. Cascading ofmultiple algorithms solved aircraft and engine problems. The de-sign infeasibilitywas alleviated.The optimum weight was reducedthrough a redistribution of the parameters of the design. The beampro le problemwas solved throughengineeringintuition.A regularshapewas obtained,but therewas no change in the optimumweight.Generatinga regularmanufacturabledesignusing a subproblemop-timization scheme still remains a challenge. The process producesdifferent designs with about the same optimum weight. Bringingoptimization methods to their rightful industrial environment fromthe academic plane would require the combined effort of designersand optimization researchers. Industry can bene t from the designoptimization methods.

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