recent applications of design optimization to rotorcraft|a survey

7/29/2019 RECENT APPLICATIONS OF DESIGN OPTIMIZATION TO ROTORCRAFT|A SURVEY

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RECENT APPLICATIONS OF DESIGN OPTIMIZATION TO ROTORCRAFTA SURVEY

Roberto Celi1

Department of Aerospace EngineeringGlenn L. Martin Institute of Technology

University of Maryland, College Park, Maryland 20742

Abstract

This paper presents a survey of applications of designoptimization to helicopter problems carried out in thelast decade. Helicopter optimization has not yet reachedthe same maturity as structural optimization, and somepotential reasons are discussed first. Next, publishedoptimization studies are reviewed, divided into sensi-tivity analysis studies, applications to rotor dynamicsand aeroelasticity, applications to other helicopter prob-lems, emerging technologies such as genetic algorithmsand simulated annealing, and studies with experimen-tal verification. The usefulness of design optimizationfor helicopter applications will increase with the avail-ability of improved analyses, more efficient algorithmsespecially for sensitivity calculations, and further exper-imental verifications. The formulation of representativeoptimization test problem, including experimental veri-fication, is also recommended.

Introduction

The successful design of a helicopter relies, perhapsto a greater extent than for any other aerospace vehicle,on the tight integration of a variety of aeronautical engi-neering disciplines. An example is the design of the mainrotor system. Rotor blades are slender, flexible beams.Even in normal operating conditions they may undergo

elastic deformations in bending and torsion that can bebeyond the limits of linear beam theories, and thereforemoderately large deformations need to be taken into ac-count. Blade flexibility and related dynamic effects in-fluence not only blade and hub stresses and fatigue life,but also interact with the aerodynamic loading, becausethe deformations of the blades modify substantially theeffective angle of attack of each cross section. In turn,rotor blade aerodynamics is coupled with structural dy-namics because much of the damping in flap and lagbending and in torsion is of aerodynamic origin. Withthe partial exception of hovering flight, the structuraland aerodynamic problems of a helicopter rotor blade

are intrinsically unsteady. Even in steady cruise con-ditions, the blade can encounter airflow velocities thatrange from transonic or slightly supersonic to low speed,

1Associate Professor, Alfred Gessow Rotorcraft Cen-terPresented at the American Helicopter Society 55th An-nual Forum, Montreal, Canada, May 25-27, 1999. Copy-right c1999 by the American Helicopter Society. Inc.All rights reserved.

including stall, and reversed flow. The variations occur

with a period of the order of 200 milliseconds. There-fore, the calculation of aerodynamic forces and structuralloads on a rotor blade is really an integrated aeroelasticproblem.

An even broader integration may be required whenhandling qualities considerations are brought into thedesign. Modern flight control system can modify thenatural characteristics of the response of the helicopterto pilot inputs, in ways that make the helicopter easierto fly and reduce the pilot workload. For example, aflight control system can make the helicopter react to astep input of lateral control with a step variation of rollangle, instead of the more natural step variation of rollrate. Experience has shown that the ability to achievethis kind of tailored handling qualities characteristicscan be limited by adverse dynamic interactions with themain rotor system. This way, the disciplines of flightdynamics and control system design become closely cou-pled with structural dynamics and aerodynamics.

These are but two of the many examples of helicopterengineering problems that require a multidisciplinary ap-proach. This requirement has motivated the develop-ment of comprehensive analysis codes during the lasttwo decades. Such codes attempt to integrate in a singlesoftware program the mathematical models required by

each helicopter engineering discipline [1]-[6]. It is beyondthe scope of the present paper to describe these codes inany detail. They are mentioned here as evidence that thedesign of a helicopter is intrinsically multidisciplinary,and perhaps more so than any other aerospace vehicle.

Design optimization appears to be an ideal method-ology to help simplify helicopter design because it canprovide a systematic way of integrating the various dis-ciplines involved. However, despite considerable activityin this field, helicopter optimization has not yet reachedthe same maturity and acceptance as structural opti-mization. Therefore, the main objectives of the presentpaper are not only to review the current state of the art

for rotorcraft applications of optimization, but also toidentify issues that have impeded its progress.

An extensive survey of multidisciplinary design opti-mization (MDO) applications to aerospace design hasbeen recently presented by Sobieszczanski-Sobieski andHaftka [7]. The present paper is meant to supplementit with a narrower focus on rotorcraft. Research in heli-copter applications of optimization started in earnest inthe early 1980s with pioneering works by Bennett [8],


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Friedmann and Shanthakumaran [9, 10], Davis [11], andPeters et al.[12]. Extensive research in this area was car-ried out throughout the 1980s. This research is reviewedin two papers by Friedmann [13] and Adelman and Man-tay [14]; the first focuses on the use of optimization forvibration reduction, while the second addresses MDO

applications. The present paper will focus primarily ondevelopments in subsequent years.The words Helicopter and Rotorcraft will be used

interchangeably. In fact, with the notable exception oftilt-rotor and tilt-wing aircraft, the rotorcraft of pri-mary interest for optimization applications is the heli-copter. Many practical helicopter optimization problemsare not especially different from those of other aerospaceor ground vehicles. An example could be the minimumweight, structural optimization of a fuselage subject topreassigned harmonic loads, and with stress constraints.Therefore, the present paper will only address those de-sign problems that are unique to the helicopter; in most

cases this implies that the presence of the main rotorsystem is a dominant ingredient in the optimization.

In general, no attempt will be made to extract specificdesign guidelines from the literature reviewed such as, forexample, ply orientations, or airfoil shapes, or planformgeometries of rotor blades. As pointed out later in thepaper, many practical helicopter problems still cannotbe modeled with the desired accuracy, and this affectsnegatively the reliability of the results obtained from op-timization.

The majority of the helicopter optimization studieshave addressed some form of the rotor dynamic behavior,such as aeroelastic stability or vibratory loads. Studies

in other areas will also be reviewed in the present pa-per; however, unless specified otherwise, comments andconclusions will usually refer to dynamics applications.In many helicopter applications the boundary betweenuni- and multidisciplinary optimization is not well de-fined, and there is a lack of consensus within the he-licopter community on this issue. What is a unidisci-plinary aeroelastic optimization to some researchers maybe a multidisciplinary structural/dynamic/aerodynamicoptimization to others. This paper takes the point ofview that such distinction is mostly a matter of seman-tics, and is not too critical in practice.

Challenges for the Applications of Design

Optimization to Rotorcraft Problems

The application of optimization techniques to struc-tural design has been quite successful. A good argumentcan be made that structural optimization is now a rea-sonably mature technology, to the point that optimiza-tion has been added to most major commercial finite

element computer programs [15]. This success can beattributed to three key elements: (i) the availability ofaccurate analyses, (ii) the availability of efficient sensi-tivity analysis techniques, and (iii) the ability to gen-erate accurate, mathematically simple approximationsto many objective functions and behavior constraints of

practical interests; including the availability of suitableintermediate design variables that extend considerablythe range of validity of those approximations. Thus,many structural optimization problems of practical in-terest can be solved with no more than ten completeanalyses of the structure being designed. The successof structural optimization created the expectation thatthe same methodology could be applied directly to he-licopter problems, and achieve the same level of successwith similar computational efficiency. A decade of expe-rience with this class of problems has revealed that suchexpectations were overly optimistic, and that consider-able progress still remains to be made. The principal

obstacle to success is the lack in the three key elementsmentioned above. Each of the three elements is discussedbelow.

Availability of Accurate Analyses

In a 1985 survey of applications of optimization methodsto helicopter design Miura stated that One commonlyexpressed concern about the use of design-optimizationmethods in helicopter design applications is the availabil-ity of adequate analytical techniques [16]. More thana decade later, and despite substantial progress in manyfields, this concern persists. For example, the predictive

capabilities of several flight dynamic simulation modelshave been evaluated by the European research workinggroup Garteur AG06 [17]. The results show that severalareas of weakness remain, such as the underpredictionof torque and power at high speed and the poor predic-tion of cross-coupling response, e.g., the roll response tolongitudinal (pitch) pilot inputs. Another area of greatimportance in helicopter design is that of vibratory ro-tor loads. Ref. [18] reports the results of a workshop onthe correlation of vibratory load predictions from severaladvanced aeroelastic codes, compared with actual flightmeasurements. One of the conclusions of the study isthat On average, codes are not able to predict vibra-tory loads to an accuracy any greater than 50% of the[helicopter] measured loads, and also that In general,the wider incorporation of more advanced [aerodynamicmodels] produces a minor enhancement to the overallgroup correlation standard [18]. Figure 1 shows an ex-ample of results from Ref. [18]. The figure comparesflight test results and 7 different predictions of the 4/revvibratory loads in the cockpit at two different speeds.The predictions with simple momentum inflow models


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appear to be quite poor. Noticeable improvements areobtained by including more sophisticated vortex wakemodels. Indeed, the complexity of the aerodynamic flowfield is an important reason of the difficulty in predict-ing vibratory loads. Ref. [19] presents a review of theproblems associated with the understanding of the vor-

tex dynamics of helicopter rotor wakes. Vortex dynam-ics is responsible for rotor noise and vibratory loads forboth rotor blades and fuselage. The authors state thatWhile these vortex related phenomena have been thesubject of much research, they are still poorly under-stood because they are both difficult to measure as wellas to predict [19].

If the analysis models are not sufficiently accurate, itis inevitable that the results of an optimization based onthem be looked upon critically if not with skepticism.This can dampen the enthusiasm for optimization. Theassessments of the improvements brought about by theuse of optimization could also be called into question.

Comparing the initial and the optimum designs to de-fine percentage improvements must be done very care-fully in any optimization problem, because if the initialdesign is not very good the benefits of optimization willbe overestimated. For a helicopter problem, the uncer-tainty will be further increased if the underlying analysisis not sufficiently accurate.

One of the ways in which Ref. [16] addressed the prob-lem of inadequate predictive capabilities was by suggest-ing that the analysis programs be kept as modular aspossible, so that they could be quickly updated as betteranalyses became available. This way the optimization

program would work with the best technology availableat the time. This suggestion remains valid today.

Availability of Efficient Sensitivity Analysis Tech-

niques

Most practical optimization algorithms require the cal-culation of the gradients of objective function and behav-ior constraints; some also require the matrix of secondderivatives, or Hessian. Gradients and Hessians can becalculated numerically, using finite difference approxima-tions, but this can be computationally expensive becauseat least N + 1 function evaluations are required for thegradient of a function ofN variables. These sensitivitiescan also be obtained analytically, or semi-analytically,that is by first using analytical chain-rule differentiationand then using finite differences for some of the resultingterms; these methods sometimes provide the sensitivitiesfor a fraction of the cost of one function evaluation (be-sides the baseline). Expressions for the sensitivities ofthe solutions of various types of mathematical problemsare available [20]. These include systems of linear andnonlinear algebraic equations, eigenvalue problems, and

systems of ordinary differential equations.

Difficulties in semi-analytical sensitivitiesWhile many methods for semi-analytical sensitivity

calculations can be directly applied to many rotorcraftproblems, some special issues may arise. For example,

consider the following system of linear, time-invariant,homogeneous, ordinary differential equations (ODE):

Mu + Cu + Ku = 0 (1)

The aeroelastic stability of a helicopter rotor in hovercan be obtained from the complex eigenvalue problemassociated with a system like Eq. (1). Then M, C, and Kare the rotor mass, damping, and stiffness matrices. Ifis a complex eigenvalue, u the corresponding eigenvector,and x a design variable, the first-order sensitivity of with respect to x is [20]:

ddx

= 2

uTdM

dx u + uTdC

dx u + uTdK

dx u

2uTMu + uTCu(2)

where , u, M, C and K are already available from thebaseline analysis. The advantage of using Eq. (2), in-stead of finite differences, to calculate the eigenvaluesensitivity depends on the cost of calculating the sensi-tivities dM/dx, dC/dx, and dK/dx of the system matri-ces. In structural optimization problems based on finiteelements these sensitivities can be inexpensive if the de-sign variable x is a cross-sectional dimension or a sectionproperty. In fact, they can often be obtained throughsimple differentiations of the element matrices. Also,the analysis code will likely have a library of finite el-ements, for which such element-level sensitivities couldbe computed a priori once and reused as needed. In ahelicopter problem the calculation of these sensitivitiesmay not be as simple. For example, the damping ma-trix C and the stiffness matrix K will usually containaerodynamic contributions that may require involved it-erative calculations; therefore, it may not be possibleto obtain their sensitivities through simple analytic dif-ferentiations, and finite difference approximations wouldhave to be used. Even more importantly, to set up thecalculations required for dM/dx,dC/dx, and dK/dx it

will be necessary to modify the analysis program, whichcould be a major practical drawback. Currently thereare a few design-oriented helicopter analysis codes thatprovide both the behavior quantities of interest and theirsensitivities with respect to design parameters. All thesecodes have been developed at universities (for example,[21], [22], [46], [24]). No industrial codes of this typehave been reported in the literature.

Difficulties due to problem nonlinearity


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Sensitivity analyses may also be complicated by thenonlinearity of many helicopter problems. Considerfor example the sensitivity of the aeroelastic stabilityeigenvalues in forward flight. These eigenvalues are ob-tained by applying Floquet theory to a system of non-linear ODE with periodic coefficients, linearized about a

steady-state periodic equilibrium position, and are calledcharacteristic exponents [25]. The real part of thecharacteristic exponents indicates the stability of the sys-tem. The sensitivity of the real part i of the i-th char-acteristic exponent with respect to the design parameterx is given by [26]:

didx

=1

T

Ri

2Ri + 2

Ii

dRidx

+Ii

2Ri + 2

Ii

dIidx

(3)

where Ri and Ii are the real and imaginary partsof the characteristic multipliers, which are the complexeigenvalues of a real nonsymmetric matrix [(2)]

called the Floquet transition matrix (FTM) [25]. TheFTM is obtained by integrating the equations of motionover one complete period of variation of the coefficients,and using a special set of initial conditions. The sensi-tivity of of the i-th characteristic multiplier , is givenby [26]:

didx

=dRi

dx+ i

dIidx

=vTi

d [(2)]dx wi

vTi wi(4)

where vi and wi are respectively the left and right eigen-vectors of the FTM [(2)] corresponding to the eigen-value i. In general both vi and wi are complex-valued

vectors. The FTM reflects a linearization of the rotoror coupled rotor-fuselage equations of motion about atrimmed flight condition. In general, a change in the de-sign parameter x will also affect the FTM indirectly, thatis through a change in the equilibrium position aboutwhich the equations are linearized. Therefore, if q isa vector of generalized coordinates defining the equilib-rium position, the sensitivity of the FTM with respectto x will given by [26]:

d [(2)]

dx=

[(2)]

x+

nTk=1

[(2)]

qk

qkx

(5)

The first term in Eq. (5) is the sensitivity of the FTM.The second represents the changes of the FTM due tothe changes in the equilibrium position associated withchanges in x. This term is a direct effect of the nonlinear-ity of the aeroelastic problem, and is expensive to obtainbecause it requires the sensitivity of the FTM to each ofthe trim variables qK. This term may be small enough tobe neglected [26], but if it is not, then the semi-analyticsensitivity of Eq. (5) can be more expensive than simple

finite difference approximations. Setting up the calcula-tions of all the sensitivities in Eq. (5) also requires thatthe internal coding of the aeroelastic analysis be modi-fied rather extensively, and is not a trivial task.

Problems in calculation of sensitivities of inertia and

aerodynamic loadsA practical obstacle to efficient sensitivity calculationsis the need to modify the analysis computer programs.The underlying equations include portions modeling thestructural, inertia, and aerodynamic loads acting on thesystem. The treatment of the structural portion is notvery different from that needed in traditional structuraloptimization. In the most sophisticated models typicallyused the rotor blades are modeled as isotropic or com-posite beams undergoing moderately large elastic deflec-tions, and the fuselages are modeled using traditional fi-nite element techniques. The treatment of blade rotationis similar to that of column buckling, except obviously

for the direction of the loads. Customary structural op-timization techniques are usually appropriate.

Problems begin to arise with the treatment of the in-ertia portion of the equations of motion, especially as theblade modeling becomes more sophisticated, and flightcondition and aircraft configuration become more com-plex. The inertia loads depend on the accelerations ofeach mass point of the system: the mathematical expres-sions for these accelerations can be so lengthy that it maynot be convenient to include all the terms in traditionalmass and damping matrices, but it is necessary to leavemany of them in a generic form as an external nonlinearforcing function [1]. This makes very difficult to obtain

efficient analytical or semi-analytical sensitivities for theinertia loads.

Even more complex can be the treatment of the aero-dynamic terms. Many aerodynamic prediction issues arenot completely understood, and a consensus on the bestmethodology has not yet been reached. The most so-phisticated aerodynamic models include complex vortexwake models, computational fluid dynamics type modelsfor the blade tips, and recursive or state-space formu-lations for the prediction of the unsteady aerodynamiccoefficients of the blade airfoils. Intermediate sensi-tivity analyses for each of these portions will have tobe obtained and coupled together before a global aero-dynamic sensitivity analysis can be set up. The topicappears formidably complex, and very limited work ap-pears to have been done in this direction in the rotorcraftcommunity.

The treatment of the rotor wake can pose special prob-lems. The most accurate rotor wake prediction methodsare those based on the free wake approach, which allowsthe geometry of the vortex wake to evolve under thevelocity field induced by the wake vortices themselves.


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Mathematically, a free wake analysis usually takes theform of a double nested loop [27]. The inner loop ad-

justs the wake geometry for a given distribution of bladecirculation, and produces a converged geometry and in-flow distribution. The outer loop adjusts the blade cir-culation until it becomes compatible with the geometry

produced by the inner loop. Convergence of the doubleloop yields the aerodynamic loads for given blade pitchsettings and blade motions; this is a key ingredient forthe calculation of, for example, hub vibratory loads andblade stresses. If hub loads and blade stresses are re-quired for a steady flight condition, then the double wakeloop needs to be placed inside a third loop that adjustspitch settings and blade motion until a trim condition isreached.

Therefore, the analysis consists of a triple nested loop,and this has important implications if finite difference-based sensitivities are desired. In fact, each of the nestedloops has to converge, and has to do so with an accu-

racy that is consistent with the step size used in thefinite difference approximation. In other words, if thestep size is such that the perturbation of the behaviorquantity of interest is of 0.1%, each of the inner loopsmust converge to within, say, 0.001%, with the conver-gence requirement becoming tighter and tighter goingfrom the outer to the inner loops. These may not betrivial conditions to meet, as some widely used free wakemethods are known not to converge in some flight condi-tions (see discussion in Ref. [27]) and are stopped aftera predefined number of iterations. If the free wake doesnot converge, the sensitivities of the inflow distributioncalculated using finite difference approximations can bemeaningless. This example, and others not reported herefor reasons of space, indicates that the step size for finitedifference-based sensitivities can be an important prac-tical problem. Unfortunately little or no information iscurrently available on the extent of this problem becausemost optimization studies avoid free wake calculations.Relatively large perturbation sizes may have to be usedto absorb convergence problems when the analyses havea multiple nested loop structure.

Availability of Suitable Approximate Analyses

Another key factor of the success of structural optimiza-tion has been the use of a series of techniques that arecollectively known as approximation concepts. Thesetechniques were pioneered by Schmit and his coworkersand include: (i) the construction of high quality explicitapproximations to objective function and behavior con-straints, (ii) design variable linking, and (iii) temporaryconstraint deletion [28].

Approximation of objective function and constraints

The first technique replaces the original problem witha sequence of approximate, faster to solve, optimizationproblems; the sequence of solutions of the approximateproblems converges to the solution of the original prob-lem. The approximate problems can be obtained, forexample, by expanding objective function and behavior

constraints into first or second order Taylor series aboutthe current design. This results in polynomial approxi-mations to objective and constraints, and in an approx-imate optimization problem that can be solved quickly.The solution is a new design, about which objective andconstraints are again expanded in Taylor series, giving anupdated approximate optimization problem. This tech-nique has been extensively used in helicopter problems.It is sometimes possible to increase the range of validityof the local approximations by performing the expan-sions in terms of intermediate, or intervening designvariables. For example, a linear Taylor series expansionof a displacement constraint for a statically determinate

beam is not very accurate if carried out in terms of crosssectional dimensions, but is exact if reciprocal sectionproperties are used instead [20]. Even if exact lineariza-tion cannot be achieved, a judicious use of interveningvariables can reduce the linearity of the problem.

No systematic attempt has been made to identify suit-able intermediate variables in typical helicopter prob-lems. For example, it is not known whether suitablecombinations of blade cross sectional dimensions or sec-tion properties could increase the linearity of constraintson aeroelastic stability, rotating and nonrotating hubloads, or blade stresses. Most optimization studies withthese constraints, and which use approximations such as

Taylor series expansions, require move limits on the de-sign variables. For handling qualities optimization it hasbeen shown [29] that a first-order Taylor series expan-sion in terms of selected stability derivatives can essen-tially linearize a constraint based on the moderate am-plitude criteria of the ADS-33 Handling Qualities Spec-ifications [30].

Design variable linking

Design variable linking of some form is often appliedin many helicopter optimization studies. Linking re-duces the number of design variables, which is espe-cially important since most optimization studies includefinite difference calculations of the sensitivities of someor all the constraints. Helicopter optimization problemshave rarely exceeded the number of 50 design variables,with most studies in the literature in the 10-30 variablesrange.

Temporary constraint deletion

The usefulness of constraint deletion varies greatlywith the type of problem. For most optimization prob-


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lems involving hub loads, blade stresses, and aeroelasticstability the benefits are modest. It is true that in manyproblems only some harmonics and components of rotorloads and stresses appear in objective function or behav-ior constraints, and that only a few modes are potentiallycritical for aeroelastic stability. On the other hand, most

of the computational effort often goes into the calcula-tion of aerodynamic, structural, and inertia loads, whichusually must be carried out even if only a few constraintsare retained. For example, deleting aeroelastic stabilityconstraints on the flap and torsion modes, which tend tobe very stable, and retaining only those in lag, does notreduce the computational effort because the stability ofall modes is obtained as a single procedure. Constraintdeletion may be effective if the constraints are defined,for example, for different flight conditions. Then onlythe most critical flight conditions could be retained atany given step.

Applications of Design Optimization to

Rotorcraft Problems

This section reviews some recent applications of de-sign optimization to rotorcraft problems. The referenceswill be divided into five groups: those with a primaryorientation toward rotor dynamics, divided into (i) sen-sitivity analysis studies and (ii) complete optimizations,(iii) those with a different focus, such as handling quali-ties or preliminary design, or otherwise multidisciplinary,(iv) those that, regardless of the disciplines covered, usesome newer optimization technologies such as genetic

algorithms or simulated annealing algorithms, and (v)those that report experimental evaluations of designs ob-tained from numerical optimization. This distinction isto some extent artificial, and therefore several referencescould have been legitimately included in more than onegroup.

Sensitivity Analyses

Semi-analytical expressions for hub loads sensitivityhave been presented by Lim and Chopra [31]. In Ref. [31]the aeroelastic analysis is based on a finite elementmethod in time, and therefore the hub loads arise fromthe solution of a system of nonlinear algebraic equations,both in hover and in forward flight. The sensitivitiesare obtained from this system using chain-rule differen-tiation. Another methodology for hub loads sensitiv-ity has been presented by Spence [32]. In this studythe aeroelastic analysis is based on a quasilinearizationmethod. The various algebraic expressions that make upthe mathematical model are not expanded symbolically,but are assembled numerically as part of the solutionprocess. This numerical formulation of the equations

of motion is ideally suited for the derivation of semi-analytical sensitivities using chain-rule differentiation.

The calculation of the sensitivities of the Floquet sta-bility eigenvalues has been addressed by several stud-ies in the period covered by the present article. Limand Chopra have provided the first such method [33],

based on a semi-analytical approach. Expressions for theeigenvalue sensitivities are derived by chain-rule differ-entiation of the basic equations describing the Floquetstability analysis. These expressions contain the sen-sitivities of the element matrices, which are calculatedanalytically. The same method was later derived with adifferent approach by Shih et al. [26], who also includednonlinear terms due to the change in equilibrium po-sition about which the aeroelastic system is linearized(see Eq. (4)). It was shown that these terms, which areexpensive to compute, have only a small effect on thesensitivity. Ref. [26] also showed that a finite differencecalculation of the eigenvalue sensitivities may yield in-

correct results if some eigenvalues occur in closely spacedclusters.

Lu and Murthy [34] perform a modal coordinate trans-formation, based on the complex eigenvectors of the ho-mogeneous system. Thanks to this transformation, thesensitivities with respect to each design parameters arecalculated independently, instead of in a coupled way asin Refs. [33] and [26]. Another semi-analytical methodwas presented by Venkatesan et al. [35]. The methodis formulated for the case of hover, which is governedby stability equations with constant coefficients. Theeigenvalue sensitivity equations contain the sensitivities

of the element matrices, which are calculated using fi-nite difference approximations. For all the stability sen-sitivity methods outlined above, the aerodynamic modelis limited to quasi-steady aerodynamics and uniform orlinearly varying inflow. Except for Ref. [26] the iner-tia model is limited to a hub that is fixed or undergoessmall amplitude motions. Ref. [36] takes the same anal-ysis methodology of Ref. [35] as a starting point. Despiteits title, however, it does not offer any real details on thestability sensitivity methodology.

A completely different approach has been proposed byWalsh and Young [37], based on the use of the automaticdifferentiation code ADIFOR [38]. This code transformsany computer program into a new program that alsocomputes sensitivities of given dependent variables withrespect to given independent variables. The method hassome similarities with the differentiation generated bysymbolic manipulation software, but it operates directlyon Fortran code. In Ref. [37] ADIFOR is used with theCAMRAD/JA [39] code. The code size increased fromabout 83,500 to about 198,000 lines. The sensitivitiescalculated by the resulting code agreed very well with


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those calculated using finite differences and, being es-sentially analytical derivatives, did not require the selec-tion of a finite difference step size. The CPU time in-creased compared with finite difference sensitivities; thiswas attributed to the treatment by ADIFOR of the freewake portions of the code. More recently, ADIFOR has

been applied to a version of the TECH-01 comprehen-sive analysis code [3]. An unpublished report on thisstudy [40] confirms excellent agreement of the sensitivi-ties with those calculated using finite difference approx-imations, and shows results of a successful optimizationcarried out with the resulting code. The CPU time wasabout 6 times that required for finite difference calcu-lations; this was tentatively attributed to the large sizeof the resulting code (the original analysis program con-sisted of over 140,000 lines).

When the solution process is iterative, such as for thesolution of a system of nonlinear algebraic equations,ADIFOR generates code that merges the iteration for

the baseline solution with an iteration for the sensitivi-ties. It has been shown that, for a large class of prob-lems, if the baseline solution converges the derivativesare also likely to converge [41]. However, the convergenceof the derivatives may be slower than that of the base-line solution; for a group of test problems the ADIFOR-differentiated algorithm required 2 to 5 times more CPUtime than with finite differences [41]. This may helpexplain the increase in CPU times observed in the ro-torcraft studies mentioned before. Automatic differen-tiation can become a very useful tool in helicopter op-timization. The differentiation of entire comprehensiveanalyses is probably too ambitious at this time. On theother hand, ADIFOR-type codes could efficiently gen-erate the sensitivities of portions of the mathematicalmodel, and complement manual derivations and imple-mentations of design-oriented analyses.

Aeroelastic and Dynamic Optimizations

Applications to helicopters

Chopra and his coworkers have carried out a seriesof aeroelastic optimization studies ([22], [42], [43]). InRefs. [22] and [42] the purpose of the optimization was toreduce vibratory loads and dynamic stresses by tailoringthe structural couplings induced by the composite rotorblade spar. Three different combinations of hub loadswere used as objective functions. The design variableswere the ply angles for several different one-cell [42] andtwo-cell ([22],[43]) box-beam blade spars. Constraintswere placed on the aeroelastic stability and on the bladenatural frequencies. The gradients of objective functionand constraints were computed using a semi-analyticalmethod, more efficient than finite difference approxima-

tions. The optimizer was the feasible direction basedcode CONMIN [44]. The results indicate that signifi-cant reductions in vibratory loads and blades stressescan be obtained, through the intermodal couplings thatare generated by the laminate ply angles that are cho-sen by the optimizer. The optimization was carried out

at one value of the advance ratio , i.e., = 0.3, butload and stress reductions were observed for the opti-mized design also at all the other advance ratios consid-ered ( = 0.15 0.4). A major contribution of thesestudies was that the optimization was repeated with arefined aerodynamic model including a free wake model.In this case the aerodynamic portion of the sensitivitieswas computed using finite differences. The wake geome-try was kept constant during the sensitivity calculations,which probably removed the numerical problems men-tioned in a previous section of the present paper, andassociated with the double nested circulation/geometryloop and with the problem of difficult convergence of the

wake model used in the study. No convergence problemsor multiple solutions were reported in these studies.

Yuan and Friedmann ([46], [23], [45]) have carried outa structural optimization study, for vibration reduction,of a composite rotor blade with a swept tip. The ob-

jective function was a weighted average of the N/rev os-cillatory hub load components. The ply orientations inthe horizontal and vertical walls of the composite bladecross section, tip sweep and anhedral were the designvariables. Frequency placement and hover aeroelasticstability constraints were placed on the design. Objec-tive function and behavior constraints were expanded in

linear Taylor series expansion, with the latter using amixed, conservative approximation [20]; semi-analyticalsensitivities were used [35]. The optimizer was the feasi-ble direction based code DOT [47]. The aeroelastic anal-ysis was based on a moderate deflection finite elementmodel, with special provisions for the modeling of theswept tip and of arbitrary, anisotropic cross sections; uni-form inflow and quasi-steady airfoil aerodynamics wereused. Optimization proved very effective at reducing vi-bratory loads. A parametric study in Ref.[45] showedthat the hub shears and moments, plotted as a functionof the ply orientation of the horizontal wall, had localminima and extensive flat areas. Hover aeroelastic

stability also showed local minima. No such complica-tions appeared for the loads as a function of tip sweep.The initial designs were typically selected based on theresults of the this parametric study; this is probably thereason why no convergence problem or multiple minimawere reported.

Peters and coworkers have also addressed rotor dy-namics optimization. In Ref. [48] the objective wasweight minimization subject to frequency placement con-


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straints (up to 30), and to a lower bound on autorota-tion inertia. The design variables were cross-sectionaldimensions, some blade geometry parameters, and 2fiber orientation variables. The composite blade hadan anisotropic cross-section. One interesting result ofthe study is the presence of local minima, which were

strongly dependent on the details of the frequency place-ment constraint set. This confirms the results of theparametric study of Ref. [45].

A state-space representation of unsteady wake dynam-ics is the finite-state wake model, developed by He andPeters. They have also developed analytical sensitivi-ties for this model [24], coupled it with sensitivity equa-tions for the remainder of the aeroelastic model, andused it in a study for the minimization of a weightedaverage hub vibratory loads and required rotor power.This is an extension of a previous study [49], based ona less sophisticated aerodynamic model. A total of 63design variables were used, describing spanwise distri-

butions of blade chord and twist, flange and web thick-nesses of the box beam-shaped structural cross section,internal lumped masses and leading edge weights. Be-havior constraints were placed on rotor autorotationalinertia, solidity, chordwise position of the center of mass,and placement of natural frequencies. CONMIN [44] wasused to perform the optimization.

Rotor dynamics optimization studies have been per-formed by Chattopadhyay et al. ([50]-[54]). The objec-tive was a weighted average of blade weight and N/revvertical shear [50] or of selected vibratory load compo-nents ([51], [54]). Design variables were, depending on

the study: blade stiffnesses or cross-sectional dimensionsof an internal box beam spar, taper ratio, root chord,root mass radius of gyration, twist, and nonstructuralweights. In Ref. [50] constraints were placed on autorota-tional inertia, blade frequency placement, and centrifu-gal stresses; in [51] constraints on blade weight, rotorthrust, and components of vibratory loads not includedin the objective function were added; in Ref. [54] an ad-ditional constraint was placed on the chordwise positionof the section center of mass, to guard against aeroe-lastic instabilities. The optimizer was CONMIN [44],the analysis typically CAMRAD [55]; objective functionand constraints were expanded in first order Taylor series

with finite difference gradients.A multilevel decomposition approach was followed in

Ref. [56]. A three-level procedure was used. Level 1was an aerodynamic performance optimization, with thepower coefficient as the objective function, and span-wise chord and twist distributions as design variables.In level 2 some components of the vibratory hub loadswere minimized; the constraints included autorotationalinertia and aeroelastic stability; the spanwise flap and

lag stiffness distributions and the values of nonstructuralweights were the design variables. Level 3 was a weightminimization using the cross sectional dimensions of abox beam spar as design variables. Through the use ofconstraints, level 3 implemented a design recovery phasethat attempted to match the stiffnesses obtained from

level 2.The research conducted at NASA Langley Research

Center on multidisciplinary helicopter optimization hasbeen described in several papers including, for the pe-riod covered by the present survey, Refs. [14], and [57]-[59]. An integrated aerodynamic/dynamic optimizationprocedure for rotor blade optimization is presented byWalsh et al. [60]. This study extended a previous one onperformance optimization alone [61]. The design vari-ables defined chord and twist distribution of the blade,its bending and torsion stiffness distribution and size andspanwise location of three tuning masses. The designwas for a 1/6th-scale rotor model. Three flight condi-

tions were considered, namely: hover, forward flight, anda maneuver simulated by increasing the rotor thrust.Behavior constraints were placed on: (i) performance,enforcing upper bounds on required power and airfoildrag coefficients around the azimuth, requiring that therotor be trimmable, and placing a lower bound on theblade chord; (ii) dynamics, with upper bounds on bladeweight and lower bounds on autorotational inertia, andconstraints on the placement of natural frequencies. Theobjective function was a weighted average of the powersrequired in the three flight conditions and of the N/revvertical hub shear. Aeroelastic stability was not con-sidered in the optimization. The optimization was car-ried out as a sequence of linear approximate problemswith move limits on the design variables. The aeroelas-tic analysis for forward flight and maneuvers was CAM-RAD/JA [39], used with a uniform inflow model. Theoptimizer was CONMIN [44].

Two design procedures were considered. In the first aperformance optimization was performed first, with onlypowers in the objective functions, blade geometry as de-sign variables, and performance constraints; a dynamicsoptimization was performed next, based on the remain-der of objective, design variable and constraints. Thesecond design procedure was an integrated optimization

that solves the performance and dynamics portions si-multaneously; in this case there were 19 design variablesand 80 behavior constraints. The optimized designs werefeasible; the one obtained from the integrated optimiza-tion appeared slightly superior. It is interesting that theoptimum designs were not too different from a referenceblade optimized without the use of formal numerical op-timization techniques.

In Ref. [62] the analysis was replaced by a neural


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network. In Refs. [63] and [64] structural optimizationwas added to the formulation of Ref. [60], and the in-tegrated problem was solved using a two-level decom-position methodology conceptually based on those ofRefs. [65] and [66]. In Ref. [64] the upper level opti-mization adjusted blade planform, stiffnesses, and tun-

ing masses to minimize a weighted average of the powersrequired in three flight conditions and of the N/rev ver-tical hub shears in forward flight, subject to the sameconstraints as in [60] plus upper bounds on the averageaxial strains. The lower level adjusted the dimensionsof the cross section (wall thicknesses and lumped areasrepresenting longitudinal stringers) to minimize the dif-ference between the stiffnesses corresponding to thesedimensions and those required by the upper level opti-mization, that is [64]

F =

EIzz (EIzz)

(EIzz)

2+

EIxx (EIxx)

(EIxx)

2

+

GJ (GJ)

(GJ)

2

(6)

The coordination between upper and lower level was im-plemented by adding a special constraint to the upperlevel problem, namely

g = FU (1 + )FL0 0 (7)

where FL0 is the optimum of F, Eq. (6), at every cy-cle, FU is an estimate of the change in FL0 that wouldbe caused by a given change in the upper level designvariables, and is a specified tolerance. This constraint

penalizes the upper level if it generates stiffnesses thatthe lower level cannot match, and which therefore couldnot be obtained in practice. CONMIN [44] was usedto solve both upper and lower level. The two-level op-timization proved efficient and more robust than thesingle-level one, because it managed to generate feasi-ble designs from an infeasible starting point even if thesingle-level version did not. The lower level optimizationproblem was reformulated in Ref. [67] using a responsesurface methodology; in this case, the objective functionof the lower level problem was replaced by a quadraticTaylor series in terms of the upper level design variables.

Refs. [68] and [69] describe an optimization for vibra-tion reduction conducted by Callahan and Straub. Thedesign variables defined the spanwise mass and stiffnessdistribution, structural twist and tip sweep, for a to-tal of up to 30 design variables. Several combinationsof vibratory loads at the hub and the pilot seat wereused as objective functions. The analysis was CAM-RAD/JA [39], used with uniform inflow for improvedefficiency; however, a validation with a free wake modelwas also carried out. The optimizer was CONMIN [44].

Vibration minimization at the hub did not necessarilyresult in vibration minimization at the pilot seat. Usinguniform inflow produced vibration reductions that oftenwere optimistic, compared with those based on the freewake analyses. It was also noted that the optimiza-tion program is only as good as its core analysis model.

Any deficiencies therein will certainly be reflected in theoptimization process [69].

Applications to tilt-rotor aircraft

Research on the multidisciplinary optimization of ro-tor and wing of a tilt-rotor aircraft has been describedby Chattopadhyay and her coworkers in a series of pa-pers. Typical design variables defined the chord, twist,and sweep distribution of the rotor blades, the thicknessdistribution of the cross section of the wing box beam,and wing root chord and taper ratio; side constraintswere placed on some design variables. One ([70],[71]),

two [75], or three ([72], [73]) flight conditions were con-sidered. The behavior constraints fixed the hover andcruise thrusts and power during the optimization, placedlower bound on the lowest frequency rotor mode and onthe autorotational inertia of the rotor, upper bounds onthe blade weight, lower bound on high speed aeroelas-tic stability, and upper bounds on the wing root stresses(the specific constraints vary slightly in the different ref-erences). In Ref. [70] the objective function was thepropulsive efficiency in axial flight; in [71] it was theweight of the box beam placed inside the rotor blade;in [74] the objective functions attempted to maximize thepropulsion efficiency and minimize total weight; in [75]

it was the weight of engine, transmission, and fuel, ob-tained from preliminary design-type sizing equations.In [72], [73], and [74], several constraints were absorbedinto the objective functions, which in turn were collapsedinto a single objective function using the Kreisselmeier-Steinhauser (KS) function [20], so that the optimizationbecame unconstrained. The gradients were calculatedusing finite differences. The optimization was carriedout as a sequence of approximate problems obtained us-ing linear Taylor series expansions [70], or a two-pointexponential approximation [76] with move limits on thedesign variables ([71]-[75]). The optimizer was CON-MIN [44]. The analysis was CAMRAD [55], used withuniform inflow and quasi-steady aerodynamics ([71], [73],[74]), or CAMRAD/JA ([70], [75]). In Ref. [72] a sim-pler analysis limited to aerodynamic performance wasused, intended as the top level of a possible multilevelapproach to the optimization. Subsequent references de-scribe the multilevel formulation of the problem; becausethe lower level subproblems are solved using simulatedannealing, these references are reviewed in a later sectionon Emerging Optimization Technologies.


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Other Helicopter Applications of Optimization

Flight dynamics

An interactive optimization software facility has beendeveloped by Tischler et al. [77]. The code addresses thedesign of flight control systems, formulated as a multiob-

jective optimization problem. The design variables arethe gains and other parameters of the control system.The constraints originate from handling qualities speci-fications. For helicopter applications the ADS-33 crite-ria [30] are used. A sequential quadratic programmingalgorithm is used. The objective function is formulatedin minimax form:

F(X) = min (max ifi(X)) 1 i m, i 0 (8)

where X is the vector of design variables, and fi(X) isthe i-th objective function; the i are weights that canbe changed interactively to determine a trade-off among

the various objectives. Each of the functions fi(X) canbe a performance criterion to minimize or a constraintto satisfy. With this form of objective function the opti-mizer concentrates on one objective function at a time,that is the least satisfactory performance criterion or themost violated constraint, with the choice also influencedby the weights j . A sophisticated graphical user inter-face supports the decisions of the analyst. The aircraft ismodeled as a linearized system of differential equations,which is derived outside the code itself.

Unpublished reports indicate that local minima havebeen encountered. The constraints can be implementedas hard or soft, depending on whether they must

definitely be satisfied or are desirable values that theoptimizer should try to achieve. They can be adjustedinteractively and, together with the weights j in theobjective function, Eq. (8), give the designer flexibilityin carrying out the optimization. The optimization pro-gram acts as a designers assistant; the designer inter-actively adjusts the problem and the path taken to reachthe optimum. The concept of having the human designershare the optimization loop with traditional mathemat-ical programming methods is an important contributionof this research. It can be argued that this approachdenies an important benefit claimed for numerical opti-mization, namely that the optimization algorithm doesnot carry the same biases of the designer, and thereforeit is free to find solutions that the designer might nothave conceived [78]. On the other hand, many practi-cal design problems do have some uncertainty in theirformulationfor example, a constraint might be due toa requirement that can be waived or relaxed by thecustomerand therefore a flexible formulation such asthat of Ref. [77] can represent more accurately a practi-cal design situation.

A more traditional approach has been followed inRefs. [29], [79] and [80], that describe the multidisci-plinary optimization of rotor and flight control systemwith aeroelastic stability and handling qualities con-straints. The handling qualities constraints enforced asubset of the specifications of ADS-33 [30], included con-

straints on pole placement, on the shape of selected fre-quency responses to pilot inputs, and on the shape of thetime histories of the response to selected pilot inputs.The design variables were the flap and lag stiffnesses ofthe rotor and the flap-lag elastic coupling factor, pluscontrol system gains and other parameters of the feed-forward loop. The objective function was a weighted av-erage of the swashplate control displacements and ratesfor two preassigned maneuvers. The optimization wascarried out at one velocity, or advance ratio, and at twovelocities simultaneously. In the latter case, the gainsof the control system at each speed were separate de-sign variables, and all the constraints were separately

imposed at each speed. The maximum number of designvariables and constraints was 15 and 48 respectively. Asequential quadratic programming algorithm was usedto solve the problem. The gradients of objective func-tion and behavior constraints were calculated using finitedifference approximations. The analysis was based on acoupled rotor-fuselage model with rigid blades with off-set hinges and root springs.

The study concluded that designing rotor and flightcontrol system simultaneously, rather than the rotor firstand the control system next, may lead to designs that re-quire lower control effort. Two problems were identified

in Ref. [79], namely the large computer time required,of the order of 10-20 hours on typical workstations, andthe lack of robustness of the design, which would be-come infeasible at speeds other than those used in theoptimization. The first problem is mostly due to theconstraints on the time histories of the response to pilotinputs, and was addressed in Ref. [29]. The gradients oftwo such constraints were calculated from the time his-tories of approximate, low order linearized models thatmatched the frequency response of the full size linearizedmodel over a preassigned frequency range. The tech-nique reduced the additional cost of calculating thesegradients by two orders of magnitude. The resultingoptimization achieved good reductions of the objectivefunction, although not as large as with the original fullnonlinear gradients. The second problem identified inRef. [79] was addressed in Ref. [80], namely the infea-sibility in off-design conditions, by adding a robustnessconstraint. A robust performance constraint proved toostringent, and a relaxed stability constraint had to beimplemented instead. The robustness of the design wasgreatly enhanced, and all the off-design constraint viola-


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tions were greatly reduced or eliminated.

Aerodynamics

Ref. [81] outlines the use of optimization at Aerospa-tiale (now Eurocopter France) and ONERA around1990. The optimization was primarily aerodynamic:

the design variables defined the airfoil shape, the ob-jective function was the cx coefficient at a Mach numberM=0.77 (advancing blade), the constraints included up-per bounds on the absolute value of the pitching momentcoefficient at M=0.77 and lower bounds on the thickness-to-chord ratio. The optimizer used was CONMIN [44].A multidisciplinary flavor was obtained by moving fromairfoil to blade design, and by adding performance andloads considerations. The design variables described theblade geometry and the spanwise distribution of the air-foils. A weighted average of the power required at mod-erate and high speed was minimized. Reductions of 3-5% in required power were reported [81]. Additional

details of the airfoil design optimization are provided inRefs. [82, 83]. This work evolved in the optimization pro-gram ORPHEE, an application of which is described inRef. [84]; although the underlying aeroelastic code wascapable of more sophisticated modeling, rigid blades anduniform inflow were used in the study, which focused pri-marily on aerodynamics and airfoil design.

Another aerodynamic/performance optimization ca-pability is described by Moffitt et al. in Refs. [85]and [86]; the study addresses the aerodynamic optimiza-tion of a variable diameter tilt rotor in hover and cruiseflight. The analysis consisted of a hover free wake modelbased on an iterative relaxation scheme. The sensitivi-

ties were obtained using finite difference approximations,starting from the baseline solution, perturbing each de-sign variable in turn, and letting the relaxation continueuntil convergence was achieved again. Re-convergencewas achieved very rapidly. The optimization was carriedout using a sequential linear programming technique. Atotal of 48 design variables were used, namely chord,twist, and sweep at 16 radial stations. Improvementsin hover and cruise efficiency could be obtained withoutcomplex variations of planform geometry.

Preliminary and conceptual design

In Ref. [87] Hajek outlines the use of optimization inthe preliminary design phase at MBB Helicopter Divi-sion (now Eurocopter Deutschland) around 1990. Thedisciplines involved were: aerodynamics, mass estima-tion, engine modeling, and cost modeling. Vibrations,acoustics, loads, and handling qualities were not con-sidered. In general terms, the design variables were:mission and empty weight, rotor size, cabin size, en-gine power and transmission rating, and several mea-sures of performance. The analysis was based on semi-

empirical or empirical relationships, or very simple the-oretical analyses. Objective functions included powerrequired, fuel consumption, and a productivity parame-ter related to payload, empty weight, and cruise speed.No precise information on the solution algorithm wasprovided; the technique appears to be sequential linear

programming. No illustrative results are presented.Finally, optimization has been applied to the con-

ceptual design of an advanced civil tiltrotor by Schle-icher [88]. The study was conducted using a helicoptersizing code, coupled with a Fletcher-Reeves type opti-mizer. This suggests that the optimization was uncon-strained; however, no further details on the formulationof the optimization problem are provided.

Emerging Optimization Technologies

Genetic algorithms

Recently there has been a growing interest in geneticalgorithms (GA) for helicopter multidisciplinary opti-mization. In a series of papers [89]-[92] Hajela and Leehave explored their use for rotor blade design. In thesealgorithms the design variables are expressed as stringsof 0s and 1s; these bits can be the exponents of 2in a binary representation, or denote with a particularcombination one of a number of possible discrete valuesfor the variable, much as a pointer to a look-up tablewould. Obviously, for continuous variables, the longerthe binary string the more precise the value of the vari-able. The strings corresponding to each variable are then

joined together to form a longer binary string that de-

fines the entire design [20]. For a 42 design variableexample presented in Ref. [92], and a relatively coarseprecision, this longer design string was composed of 179zeros and ones. The GA starts with a certain numberof initial design strings with random combinations of 0sand 1s. The algorithm then proceeds by generating newdesigns with bit operations devised to mimic mathemat-ically some biological evolutionary phenomena, such asmating, crossover, and mutation. Through these opera-tions the design evolves toward the optimum.

Genetic algorithms can easily deal with integer anddiscrete variables. Additionally, they have a betterlikelihood to identify global minima than conventionalgradient-based algorithms. In fact, if the initial designsspan a sufficiently large portion of the design space, sub-sequent designs may cluster in groups around differentlocal minima, one of which might be the actual globalminimum. A third advantage is that they can be easilyimplemented on parallel computers, since the evolution-ary steps may be applied to independent pairs of designs.The main drawbacks of GA are that the size of the prob-lem increases rapidly if many design variables need to be


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determined precisely (and therefore must be representedwith long binary strings), that convergence can be slowsince the properties of the feasible region are not ex-ploited directly, and that intermediate designs may beinfeasible.

A basic GA algorithm is proposed in Ref. [89]; multi-

stage searches are proposed to increase efficiency. Thismeans that the optimization is repeated several times,starting from coarser representations (which requireshorter design strings) and progressively increasing theprecision of the formulation. In subsequent papers [90]-[92] the optimization is decomposed into subproblems,with a structure in which the subproblems optimizationrepresents an inner loop, and the system-level optimiza-tion an outer loop. The subproblems are identified bya neural network, which determines the relationshipsbetween inputs (design variables) and outputs (objec-tive and constraint values), and therefore which groupsof design variables and objective/constraints are mostclosely coupled. In Ref. [92] the objective function was aweighted average of vibratory hub loads. The constraintscovered the power required in hover and forward flight,the figure of merit of the main rotor, an autorotationindex related to blade mass, weight, and failure stresses.The design variables included blade geometric quantitiesand the cross-sectional dimensions of the blade spar, as-sumed to vary in discrete steps to simulate the numberof layer of a composite laminate. The 42 design variableswere represented with a total of 179 bits, for a precisionof about two significant digits. A multi-body dynamicmodel was used for the rotor blade, together with an un-

steady aerodynamic model. The training of the neuralnetwork used to decompose the problem required from1450 to 12,800 function evaluations, depending on theneural network algorithm and the desired accuracy. Af-ter 70 system level optimizations a reasonable conver-gence was obtained.

Additional optimization studies using genetic algo-rithms have been carried out by Crossley, Wells, et al.In Ref. [93] number of blades, airfoil sections, solidity,twist, tip speed, disk loading and taper were the designvariables for an acoustic optimization study. The designstring length was of 27 bits. A more recent study [94]focuses on the acoustic design of tilt-rotors using geneticalgorithms. In Ref. [95] the objective was to design arotor airfoil. The blade was assumed to be frozen atthe azimuth angles of 90, 180, and 270 degrees, and wasanalyzed with a fixed-wing type, steady aerodynamicmodel. Structural constraints were simulated by plac-ing lower bounds on the airfoil thickness. The focus wason conceptual design in Refs.[95]-[98], in which the ob-

jective was power and weight minimization subject toa variety of performance and mission constraints. Two

integer, two discrete, and five continuous design vari-ables resulted in a 28-bit design string. A typical in-dustry conceptual design code was used for the analy-sis. From about 1500 to about 4500 function evaluationswere needed for convergence. The design variables in-cluded truly discrete variables such as the number of ro-

tors and of rotor blades, the presence or absence of a wingand of a propeller, and the number of engines. Therefore,this is clearly an optimization problem which would havebeen impossible to solve with gradient-based optimiza-tion methods. Rotor design for minimum weight andpower required were studied in Refs. [99] and [100] usingthe same design variables of Ref. [93]. Upper boundson local aerodynamic angles of attack and Mach num-ber at 90 and 270 degrees, and a lower bound on thethrust were appended to the objective functions using aquadratic exterior penalty function. In [101] the sameconstraints and objective functions are collapsed into asingle objective function using the KS function.

The problem of Ref. [92] could be solved with conven-tional gradient-based methods. It would be interestingto compare conventional solutions with those obtainedwith genetic algorithms, to assess advantages and disad-vantages of this technique. Genetic algorithms requirea large amount of function evaluations, and can pro-vide only low or moderate precision unless lengthy designstrings are used, which would further increase the size ofthe optimization problem. Therefore, they seem suitableonly for small problems and for problems in which lim-ited accuracy is required, and in both cases conventionalgradient-based techniques are likely to be much more effi-cient. On the other hand, if the feasible region for a given

problem is really heavily nonconvex or even disjoint, andif objective function and constraint gradients have tobe calculated using finite differences, then conventionaltechniques may indeed turn out to be more expensive.In fact, in this case it would be prudent to repeat the op-timization starting from different designs, to increase thelikelihood of locating a global minimum. The best ap-proach might be to perform an initial search with geneticalgorithms, or some other probabilistic search method,select some promising configurations, and then optimizethese further using conventional gradient-based meth-ods.

Simulated annealing

Another emerging optimization technique is the simu-lated annealing algorithm, which mimics mathematicallysome physical processes that occur in the solidificationof metals and the formation of crystals [20]. Like geneticalgorithms, simulated annealing is a probabilistic designmethod, does not require derivatives, can deal with in-teger or discrete variables and with nonconvex designspaces, and for problems with local minima can increase


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the likelihood of finding a global minimum. Also likegenetic algorithms, it can be much more expensive thanconventional gradient-based methods when the proper-ties just mentioned are not required in the optimization.Simulated annealing has been applied by Chattopadhyayet al. to high-speed prop-rotor design [102]. The algo-

rithm is composed of the following steps [102]:1. The current design is X, the corresponding value of

the objective function is F(X).

2. Perturb the current design: the perturbed design isXnew.

3. IfF(Xnew) F(X) then Xnew becomes the currentdesign because it is as good as or better than X.

4. If F(Xnew) > F(X) then Xnew is a worse design,but accept it anyway if a random number P, with0 P 1, drawn for this design is greater thanan acceptance probability Pacc defined as:

Pacc = exp(1/T) (9)

5. Repeat the previous steps until convergence; reduceT as the algorithm proceeds.

Accepting a design even if it is not an improvement al-lows the algorithm to jump out of a local minimum,which a gradient-based algorithm usually would not beable to do. As T in Eq. (9) decreases, so does the prob-ability Pacc of accepting a worse design, and the methodsettles in the search of the minimum. No informationon the shape of the feasible region is used, which causes

the method to be less efficient than gradient-based al-gorithms. Because the design can be perturbed witharbitrary methods, the design variables need not be con-tinuous, and discrete or integer variables can be includedin the optimization.

Simulated annealing was also used by Chattopadhyayand her coworkers in the lower level subproblem of a mul-tilevel optimization approach to prop-rotor design ([103]-[105]). In Ref. [103] the upper level was an aerodynamicperformance optimization problem, in which the designvariables govern the blade geometry. The lower level wasa structural optimization problem, in which the designvariables were the ply angles of the box beam spar of

the blade. It was assumed that the ply angles could onlytake discrete values, which justified the use of simulatedannealing. Refs. [104] and [105] extended the work ofRef. [103] to include takeoff performance in the formula-tion of the upper level optimization problem.

Experimental verifications

Refs. [106] and [107] present the results of the opti-mization of a four-bladed articulated rotor for vibration

minimization. The design variables are the spanwisemass and stiffness distribution. The objective function isa weighted average of the hub loads in the fixed system.Some constraints on the static and dynamic behavior ofthe blade are appended to the objective function. InRef. [106] only N/rev harmonics were considered in the

optimization, in Ref. [107] both N/rev and 2N/rev wereconsidered. Other constraints ensure that the blades canactually be manufactured. The airloads were held con-stant during the optimization. Predicted and actual vi-bratory loads of the optimized rotor are not comparedin the paper, and therefore it is impossible to determinethe role played by possible inaccuracies of the analy-sis. An experimental comparison between the loads ofthe baseline and of the optimized, Mach-scaled modelrotor shows a reduction of 4/rev hub loads from 45%to 77%, depending on the specific load component andflight speed [106]; further reductions could be obtainedby including the 2N/rev components [107].

Another optimization study with experimental valida-tion has been presented by Leconte and Geoffroy [108].The objective was to minimize the moment due to 3/revinplane shears for a 4-bladed articulated rotor, using thespanwise mass and stiffness distribution and weight andposition of two nonstructural masses as design variables.A finite element model for the anisotropic compositecross section was used, with three-dimensional aerody-namics and quasi-steady airfoil characteristics. The op-timizer was CONMIN [44], and the finite difference gra-dients were used. Behavior constraints were placed onsome 4/rev hub load components and frequency place-

ment. The solution was sought at a specified value ofthrust coefficient and advance ratio, but it was verifiedover a broader range of thrust and speed values. Theresults show a significant decrease of the 3/rev inplanemoment; a vector plot of the contributions from all themodes shows that this was achieved by reducing mostmagnitudes and altering all the phases for each mode.Both the baseline and the optimized blades were builtand tested in a wind tunnel. Figure 2 shows a compar-ison between the in-plane moments for the baseline andthe optimized rotor; both the calculated and the experi-mental results are shown. The figure clearly indicates areduction of the objective function in the optimized con-

figuration at all advance ratios. The agreement betweentheoretical and experimental values appears to be rea-sonably good. Numerical robustness tests indicate thatthe solution is basically insensitive to variations of10%of the design variables at the optimum.

Tuning masses were used by Pritchard et al. to reducevibratory hub loads in Ref. [109]. Objective functionand some of the behavior constraints were formulated totry to minimize the loads without incurring an excessive


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weight penalty. Magnitude and spanwise locations of thetuning masses were the design variables. Additional be-havior constraints placed upper and lower bounds on theblade natural frequencies. The code CAMRAD/JA [39]was used; objective function and behavior constraintswere expanded in linear Taylor series, and the resulting

sequential linear programming problem was solved usingCONMIN [44]. The theoretical results were comparedwith tests of a Mach-scaled, 1/6-th scale four-bladed UH-60 Blackhawk rotor, in which the vibratory loads weremeasured for varying spanwise positions of one bladetuning mass. It is interesting to note that the vibratoryloads for the baseline and the optimized rotor were notpredicted very accurately at any of the advance ratios:the underprediction ranged from a factor of 2 to about5. Nevertheless, predicted and actual spanwise positionsof the tuning mass for minimum 4/rev hub shear werein reasonably good agreement. Furthermore, the ratiosbetween the baseline and the optimum shears were pre-

dicted with less than 3% error.

Weller and Davis carried out an extensive experimen-tal program to validate the predictions of a blade designoptimization methodology [110, 111]. This methodologyis based on the minimization of vibration indices that in-clude blade natural frequencies, generalized forcing func-tions, and generalized inertias of given blade modes, andwhich require very simple analyses for their calculation(i.e., not sophisticated aeroelastic or comprehensiveanalyses). These metrics are discussed in greater detailin Ref. [112]. In Refs. [110] and [111] the optimizationwas performed with the ADS-1 code [113]. The tests

were carried out on a family of 1/5-th scale, Froude-scaled rotors. The simple, modal-based metrics provedcapable of reducing blade vibratory loads (frequencyplacement criteria alone, on the other hand, could not re-duce loads consistently). Figure 3, taken from Ref. [112],shows magnitude and phase of the various modal contri-bution to the total 4/rev vertical shear and hub mo-ments. The figure indicates that the reduction in vibra-tory loads comes from the reduction in magnitude of allthe components, rather than from phase shifts that re-duce the vector sum of the components without reducingtheir magnitude.

The same designs have been recently studied with acomprehensive analysis code coupled to an optimizer byGanguli et al. [114]. Quasi-steady aerodynamics and lin-ear inflow modeling underpredicted the vibratory loadssignificantly; free wake modeling was judged critical forhub loads predictions. Figure 4 compares analytic pre-dictions and experimental results for the baseline andthe optimized rotor. At medium and high advance ra-tios the error of the analytical predictions is as large asthe improvements brought about by the optimization.

Nevertheless, the optimization does generate a designthat has consistently lower hub shears than the baseline.Another interesting conclusions is that the modal-basedapproach can generate designs that are almost as good asthose obtained witha much more sophisticated analysis,and with a much lower computation al effort. However,

Ref. [114] warns that this may be due to the lack ofstrong aeroelastic interactions in a Froude-scaled modelrotor: in Mach-scaled or full scale rotors the modal-basedprocedure may not be as powerful.

It is encouraging to note that the use of formal opti-mization techniques leads to improved real designs evenif the predictions may not have the desired accuracy.This also indicates that the connection between accuracyof the predictions and accuracy of the optimum may bemore subtle than it would appear at first glance. Con-sider Fig. 5 as a hypothetical, qualitative description ofthe effects of the accuracy of the predictions on the qual-ity of the final optimum. The figure shows an objective

function as a function of a design variable (or of the movealong a given search direction for problems with more de-sign variables). The solid line and the dashed lines rep-resent respectively the exact solution and two theoreticalpredictions. The prediction marked P1 would gener-ally be considered much worse than that marked P2.However, using the prediction P1 one would obtain theprecise optimum (even if the corresponding value of theobjective function would be quita inaccurate), whereasthe better prediction P2 would yield a clearly worseoptimum. Whether Fig. 5 is at all representative of ac-tual helicopter optimization problems is unknown at thistime, because this fundamental aspect of the optimiza-

tion is not easy to study, and has never been up to now.

Concluding Remarks

As this review demonstrates, there is considerable ac-tivity in applications of design optimization to helicopterproblems. The accuracy of the underlying analyses re-mains an important problem, because optimization re-sults obtained using analyses of insufficient accuracyhave only very limited value. However, as the reliabilityof predictive analytical tools continues to increase, therole of optimization in actual helicopter design will likelybe expanded. In the meantime, one should inject somerealism in evaluating the actual results of an optimiza-tion. Perhaps each design optimization study should in-clude an evaluation of the robustness of the optimum,possibly based on optimum design sensitivity analysismethodologies [20], to help determine how sensitive theoptimum is to modeling inaccuracies.

No algorithmic silver bullet has emerged to make thesolution of helicopter optimization problems as computa-


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tionally efficient and robust as for traditional structuraloptimization problems. It is currently unrealistic to ex-pect to find an optimum for the cost of 10-15 analyses.Efficiency can increase if the gradient information is cal-culated analytically or semi-analytically rather than byfinite differences, but this requires modification of the

analysis computer programs, which may pose a road-block in industrial applications. Efficiency could also in-crease with judicious use of approximation concepts, butadditional research is needed to further specialize thistechnique to helicopter problems. Future advances incomputer hardware could make these issues partially ir-relevant, especially if large scale parallelism can be prac-tically implemented. On the other hand, future researchmay show that accurate predictions require substantiallymore complex and time consuming analyses than currentones. Therefore, there is still much value in continuingresearch in more efficient sensitivity calculations and ap-proximation concepts.

The traditional gradient-based optimization methodswill probably remain the algorithmic mainstay of he-licopter design optimization for the foreseeable future.Emerging technologies such as simulated annealing andgenetic algorithms currently seem too inefficient forproblems of realistic complexity. They may play a role inproblems with truly discrete design variables, and couldprove useful for a preliminary search of the design space,to be completed with traditional algorithms. Noncon-vexity and local minima are considered traditional ene-mies by many practitioners of design optimization, andmuch research is directed toward algorithms with global

convergence properties. It should be kept in mind, how-ever, that local minima can represent alternate design so-lutions. In practice these designs may prove better thanthe theoretical global optima, because of criteria not con-sidered in the formulation of the original optimizationproblem. Future research should help determine whenthe nonconvexity of the feasible region is truly an intrin-sic negative feature of the optimization problem to betackled with globally convergent algorithms, when it iscaused by a poorly chosen formulation of the problem,or when it simply reflects the presence of more than oneacceptable design solution.

It is not easy to verify experimentally that a designobtained using optimization techniques is truly the op-timum design. In fact, one would have to build andtest not only the design corresponding to the theoreticaloptimum, but also enough designs in its neighborhoodto verify that theoretical and actual optima are suffi-ciently close. It is easier to simply verify that the use offormal optimization techniques leads to a better design,because one only needs to build and test the baselineand the optimum design, and compare the two. This

has been done in a few, very interesting studies in thearea of rotor vibration reduction. Often there were somesignificant discrepancies between the predicted and themeasured loads on the optimum designs. This confirmsthat several basic aspects of the problem are not yet fullyunderstood. On the other hand, in all such studies the

designs did somehow improve through the use of formaloptimization techniques, and this is very encouraging.Researchers in helicopter applications of optimization

face a complex multidisciplinary problem, with severalpossible choices of design variables, objective functions,behavior and side constraints, analysis models, sensitiv-ity formulations, approximation concepts, optimizationalgorithms, not to mention the many types of resultsthat can be generated and presented. Therefore, the he-licopter community should agree on a small number ofrepresentative test cases, augmented by judicious exper-imental verification, which researchers in the field shouldadopt. Studies such as those of Refs. [17] and [18] have

been exceptionally useful because they have provided arealistic assessment of the state of the art, and clearlyindicated where additional work was needed. Helicopterdesign optimization would greatly benefit from similarwork that judiciously bridges the gap between theoryand practice.

References

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[2] Johnson, W., Technology Drivers in the Develop-ment of CAMRAD II, Proceedings of the AHSAeromechanics Specialist Conference, San Fran-cisco, CA, Jan 1994, pp. PS.3-1PS.3-14.

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[18] Hansford, R. E., and Vorwald, J., Dynam-ics Workshop On Rotor Vibratory Loads Predic-tion, Journal of the American Helicopter Society,Vol. 43, No. 1, Jan 1998, pp. 76-87.

[19] Leishman, J. G., and Bagai, A., Challenges inUnderstanding the Vortex Dynamics of HelicopterRotor Wakes, Paper AIAA 96-1957, Proceedings

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[20] Haftka, R. T., and Gurdal, Z., Elements of Struc-tural Optimization, 3rd Ed., Kluwer AcademicPublishers, 1991.

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[24] He, C.-J., and Peters, D. A., Optimum Ro-tor Interdisciplinary Design With a Finite StateAeroelastic Modeling, Mathematical and Com-

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recent applications of design optimization to rotorcraft|a survey

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