multidisciplinary aerospace design …mln/ltrs-pdfs/nasa-aiaa-96-0711.pdf1 american institute of...

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American Institute of Aeronautics and Astronautics

MULTIDISCIPLINARY AEROSPACE DESIGN OPTIMIZATION:SURVEY OF RECENT DEVELOPMENTS

Jaroslaw Sobieszczanski-Sobieski*

NASA Langley Research CenterHampton, Virginia 23681([email protected])

Raphael T. Haftka†

University of FloridaGainesville, Florida 32611

([email protected])

Abstract

The increasing complexity of engineering systemshas sparked increasing interest in multdisciplinaryoptimization (MDO). This paper presents a survey ofrecent publications in the field of aerospace whereinterest in MDO has been particularly intense. The twomain challenges of MDO are computational expenseand organizational complexity. Accordingly the surveyis focused on various ways different researchers use todeal with these challenges. The survey is organized bya breakdown of MDO into its conceptual components.Accordingly, the survey includes sections onMathematical Modeling, Design-oriented Analysis,Approximation Concepts, Optimization Procedures,System Sensitivity, and Human Interface. With theauthors’ main expertise being in the structures area, thebulk of the references focus on the interaction of thestructures discipline with other disciplines. Inparticular, two sections at the end focus on two suchinteractions that have recently been pursued with aparticular vigor: Simultaneous Optimization ofStructures and Aerodynamics, and SimultaneousOptimization of Structures Combined With ActiveControl.

1. Introduction

The term “methodology” is defined by Webster'sdictionary as "a body of methods, procedures, workingconcepts, and postulates, etc." Consistent with this

_____________________________*Multidisciplinary Research Coordinator, Fellow, AIAA†Professor, Associate Fellow, AIAA

Copyright © by the American Institute of Aeronautics and Astronautics, Inc.No copyright is asserted in the United States under Title 17, U. S. Code. The U.S. Government has a royalty-free license to exercise all rights under thecopyright claimed herein for Government Purposes. All other rights arereserved by the copyright owner.

definition, multidisciplinary optimization (MDO) canbe described as a methodology for the design ofsystems where the interaction between severaldisciplines must be considered, and where the designeris free to significantly affect the system performance inmore than one discipline. Using this definition,structural optimization of an aircraft wing to preventflutter will not be considered multidisciplinaryoptimization. For this case, the interaction ofaerodynamics and structures is present only at theanalysis level, and the designer does not attempt tochange the aerodynamic shape of the wing.

The interdisciplinary coupling inherent in MDOtends to present additional challenges beyond thoseencountered in a single-discipline optimization. Itincreases computational burden, and it also increasescomplexity and creates organizational challenges forimplementing the necessary coupling in softwaresystems.

The increased computational burden may simplyreflect the increased size of the MDO problem, with thenumber of analysis variables and of design variablesadding up with each additional discipline. A case oftens of thousands of analysis variables and severalthousands of design variables, reported in Berkes (90)for just the structural part of an airframe design,illustrates the dimensionality of the MDO task one hasto prepare for. Since solution times for most analysisand optimization algorithms increase at a superlinearrate, the computational cost of MDO is usualIy muchhigher than the sum of the costs of the single-disciplineoptimizations for the disciplines represented in theMDO. Additionally, even if each discipline employslinear analysis methods, the combined system mayrequire costly nonlinear analysis. For example, linearaerodynamics may be used to predict pressure

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distributions on a wing, and linear structural analysismay be then used to predict displacements. However,the dependence of the pressures on the displacementsmay not be linear. Finally, for each disciplinaryoptimization we may be able to use a single-objectivefunction, but for the MDO problem we may need tohave multiple objectives with an attendant increase inoptimization cost.

In MDO of complex systems we also faceformidable organizational challenges. The analysiscodes for each discipline have to be made to interactwith one another for the purpose of system analysis andsystem optimization. The kind and breadth ofinteraction is affected by the MDO formulation.Decisions on the choice of design variables and onwhether to use single-level optimization or multileveloptimization have profound effects on the coordinationand data transfer between analysis codes and theoptimization code and on the degree of humaninteractions required. The interaction between themodules in the software system on one side and themultitude of users organized in disciplinary groups onthe other side may be complicated by departmentaldivisions in the organization that performs the MDO.

One may discern three categories of approaches toMDO problems, depending on the way theorganizational challenge has been addressed. Two ofthese categories represent approaches that concentratedon problem formulations that circumvent theorganizational challenge, while the third deals withattempts to address this challenge directly.

1. The first category includes problems with twoor three interacting disciplines where a single analystcan acquire all the required expertise. At the analysislevel, this may lead to the creation of a new disciplinethat focuses on the interaction of the involveddisciplines, such as aeroelasticity or thermoelasticity.This may lead to MDO where design variables inseveral disciplines have to be obtained simultaneouslyto ensure efficient design. The past two decades havecreated the discipline of structural control, with analystswho are well versed in both structures and controlsystem analysis and design. There has also been muchwork on simultaneous optimization of structures andcontrol systems (e.g., Haftka, 90). Most of the papersin this category represent a single group of researchersor practitioners working with a single computerprogram, so that organizational challenges wereminimized. Because of this, it is easier for researchersworking on problems in this category to deal with someof the issues of complexity of MDO problems, such as

the need for multiobjective optimization (e.g., Guptaand Joshi, 90, Rao and Venkayya, 92, Grandhi et al.,92, and Dovi and Wrenn, 90).

2. The second category includes works where theMDO of an entire system is carried out at theconceptual level by employing simple analysis tools.For aircraft design, the ACSYNT (Vanderplaats, 76,Jayaram et al., 92) and FLOPS (McCullers, 84)programs represent this level of MDO application.Because of the simplicity of the analysis tools, it ispossible to integrate the various disciplinary analyses ina single, usually modular, computer program and avoidlarge computational burdens. Gallman et al. (94), Gatesand Lewis (92), Lavelle and Plencner (92), Morris andKroo (90), Dodd et al. (90), Harry (92), Reddy et al.(92), and Bartholomew and Wellen (90) provideinstances of this approach. As the design processmoves on, the level of analysis complexity employed atthe conceptual design level increases uniformlythroughout or selectively (Adelman et al., 92, presentsan example of the latter). Therefore, some of thesecodes are beginning to face some of the organizationalchallenges encountered when MDO is practiced at amore advanced stage of design process.

3. The third category of MDO research includesworks that focus on the organizational andcomputational challenges and develop techniques thathelp address these challenges. These includedecomposition methods and global sensitivitytechniques that permit overall system optimization toproceed with minimum changes to disciplinary codes.These also include the development of tools thatfacilitate efficient organization of modules or that helpwith organization of data transfer. Finally,approximation techniques are extensively used toaddress the computational burden challenge, but theyoften also help with the organizational challenge.

The present review emphasizes papers that belongto the third category. The survey is organized by theMDO breakdown into its conceptual componentssuggested in Sobieszczanski-Sobieski (95).Accordingly, the survey includes sections onMathematical Modeling and Design-oriented Analysis,Approximation Concepts, Optimization Procedures,System Sensitivity, Decompositio, and HumanInterface (and an Appendix on the Design Space Searchalgorithms). With the authors main expertise being inthe structures area, the bulk of the references focus onthe interaction of the structures discipline with anotherdiscipline such as structures and electromagneticperformance (Padula et al., 89). In particular, two

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sections at the end focus on two such interactions thathave recently been pursued with a particular vigor:Simultaneous Optimization of Structures andAerodynamics and Structures Combined With ActiveControl. This emphasis on structures reflects also theroots of aerospace MDO in structural optimization andthe central role of structures technology in design ofaerospace vehicles.

2. MDO Components

This section comprises references grouped by theMDO conceptual components defined as proposed inSobieszczanski-Sobieski (95).

2.1 Mathematical Modeling of a System

For obvious pragmatic reasons, softwareimplementation of mathematical models of engineeringsystems usually takes the form of assemblages of codes(modules), each module representing a physicalphenomenon, a physical part, or some other aspect ofthe system. Data transfers among the modulescorrespond to the internal couplings of the system.These data transfers may require data processing thatmay become a costly overhead. For example, if thesystem is a flexible wing, the aerodynamic pressurereduced to concentrated forces at the aerodynamicmodel grid points on the wing surface has to beconverted to the corresponding concentrated loadsacting on the structure finite-element model nodalpoints. Conversely, the finite-element nodal structuraldisplacements have to be entered into aerodynamicmodel grid as shape corrections.

The volume of data transferred in such couplingsaffects efficiency directly in terms of I/O cost.Additionally, many solution procedures (e.g., GlobalSensitivity Equation, Sobieszczanski-Sobieski, 90)require the derivatives of this data with respect todesign variables, so that a large volume of data alsoincreases computational cost. To decrease these costs,the volume of data may be reduced by variouscondensation (reduced basis) techniques. For instance,in the above wing example one may represent thepressure distribution and the displacement fields by asmall number of base functions defined over the wingplanform and transfer only the coefficients of thesefunctions instead of the large volumes of the discreteload and displacement data. An example of suchcondensation for supersonic transport design wasreported in Barthelemy et al. (92) and Unger et al.(92).

In some applications, one may identify a cluster ofmodules in a system model that exchange very largevolumes of data that are not amenable to condensation.In such cases, the computational cost may besubstantially reduced by unifying the two modules, e.g.,August et al. (92) or merging them at the equationlevel. A heat-ransfer-structural-nalysis code is anexample of such merger as described in Thornton (92).In this code, the analyses of the temperature fieldthroughout a structure and of the associated stress-strainfield share a common finite-element model. This lineof development was extended to include fluidmechanics in Sutjahjo and Chamis (94).

Because of the increased importance ofcomputational cost, MDO emphasizes the tradeoff ofaccuracy and cost associated with alternative modelswith different levels of complexity for the samephenomena. In single-discipline optimization it iscommon to have an “analysis model” which is moreaccurate and more costly than an “optimization model”.In MDO, this tradeoff between accuracy and cost isexercised in various ways. First, optimization modelscan use the same theory, but with a lower level ofdetail. For example, the finite-ment models used forcombined aeroelastic analysis of the high-speed civiltransport (e.g., Scotti, 95) are much more detailed thanthe models typically used for combined aerodynamic-structural optimization (e.g., Dudley et al., 94) .

Second, models used for MDO are often lesscomplex and less accurate than models used for a singledisciplinary optimization. For example, structuralmodels used for airframe optimization of the HSCT(e.g., Scotti, 95) are substantially more refined thanthose used for MDO. Aircraft MDO programs, such asFLOPS (McCullers, 84) and ACSYNT (Vanderplaats,76), Jayaram et al., 92) use simple aerodynamicanalysis models and weight equations to estimatestructural weight. Similarly Livne et al. (92) use anequivalent plate model instead of a finite-elementmodels for structures-control optimization of flexiblewings.

Third, occasionally, models of differentcomplexity are used simultaneously in the samediscipline. One of them may be a complex model forcalculating the discipline response, and a simpler modelfor characterizing interaction with other disciplines.For example, in many aircraft companies, the structuralloads are calculated by a simpler aerodynamic modelthan the one used for calculating aerodynamic drag

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(e.g., Baker and Giesing, 95). Finally, models ofvarious levels of complexity may be used for the sameresponse calculation in an approximation procedure orfast reanalysis described in the next two sections.

Recent aerospace industry emphasis on economicswill, undoubtedly, spawn generation of a new categoryof mathematical models to simulate man-madephenomena of manufacturing and aerospace vehicleoperation with requisite support and maintenance.These models will share at least some of their inputvariables with those used in the product design toaccount for the vehicle physics. This will enable one tobuild a system mathematical model encompassing allthe principal phases of the product life cycle:formulation of desirements, product design,manufacturing, and operation. Based on such anextended model of a system, it will be possible tooptimize the entire life cycle for a variety of economicobjectives, e.g., minimum cost or a maximum return oninvestment, as forecasted in Tulinius (92) . There areseveral references that bring the life cycle issues intothe MDO domain; examples are Korngold and Gabriele(94), Fenyes (92), Bearden et al. (94), Brockman et al.(92), Current et al. (90), Shupe and Srinivasan (92),Briggs (92), Claus (92), Godse et al. (92), Dolvin (92),Eppinger et al. (94), Lokanathan et al. (95), Marx et al.(94), Niu and Brockman (95), Kirk (92), and Yeh andFulton (92). Schrage (93) discussed the role of MDO inthe Integrated Product and Process Development(IPPD), also known as Concurrent Engineering (CE),and surveyed references on the subject.

Mathematical modeling of an aerospace vehiclecritically depends on an efficient and flexibledescription of geometry. This subject is addressed inSmith and Kerr (92).

2.2 Design-Oriented Analysis

The engineering design process moves forward byasking and answering "what if" questions. To getanswers to these questions expeditiously, designersneed analysis tools that have a number of specialattributes. These attributes are: selection of the variouslevels of analysis ranging from inexpensive andapproximate to accurate and more costly, "smart"reanalysis which repeats only parts of the originalanalysis affected by the design changes, computation ofsensitivity derivatives of output with respect to input,and a data management and visualization infrastructurenecessary to handle large volumes of data typicallygenerated in a design process. The term "Design-oriented Analysis" introduced in Storaasli and

Sobieszczanski (73) refers to analysis procedurespossessing the above attributes.

The data management and visualizationinfrastructure, e.g., (Herendeen et al., 92) is a vast fieldbeyond the scope of this survey. Sensitivity analysis isdiscussed in section 2.4, and the issue of the selectionof analysis level was discussed in the previous section,and will be returned to in the next section onapproximations.

An example of a design-oriented analysis code isthe program LS-CLASS developed by Livne andSchmit (90), Livne et al. (92, 93) for the structures-control-aerodynamic optimization of flexible wingswith active controls. The program permits thecalculation of aeroservoelastic response at differentlevels of accuracy ranging from a full model to areduced one based on vibration modes. Additionally,various approximations are available depending on theresponse quantity to be calculated.

A typical implementation of the idea of smartreanalysis has been reported in Kroo and Takai (88a, b)and Gage and Kroo (92). The code (called PASS) is acollection of modules coupled by the output-to-inputdependencies. These dependencies are determined andstored on a data base together with the archivalinput/output data from recent executions of the code.When a user changes an input variable and asks for newvalues of the output variables, the code logic uses thedata dependency information to determine whichmodules and archival data are affected by the changeand executes only the modules that are affected, usingthe archival data as much as possible. One may addthat such smart reanalysis is now an industry standardin the spreadsheets whose use is popular on personalcomputers. It contributes materially to the fastresponse of these spreadsheets.

2.3 Approximation Concepts

Direct coupling of a the design space search code(DSS) to a multidisciplinary analysis may beimpractical for several reasons. First, for any moderateto large number of design variables, the number ofevaluations of objective function and constraintsrequired by DSS is high. Often we cannot afford toexecute such a large number of exact MDO analyses inorder to provide the evaluation of the objective functionand constraints. Second, often the different disciplinaryanalyses are executed on different machines, possibly atdifferent sites, and communication with a central DSSprogram may become unwieldy. Third, some

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disciplines may produce noisy or jagged response as afunction of the design variables (e.g., Giunta et al., 94).If we do not use a smooth approximation to theresponse in this discipline we will have to degrade theDSS to less efficient nongradient methods.

For all of the above reasons, most optimizations ofcomplex engineering systems couple a DSS toeasy-to-calculate approximations of the objectivefunction and/or constraints. The optimum of theapproximate problem is found and then theapproximation is updated by the full analysis executedat that optimum and the process repeated. This processof sequential approximate optimization is popular alsoin single-discipline optimization, but its use is morecritical in MDO as the principal cost control measure

Most often the approximations used in engineeringsystem optimization are local approximations based onthe derivatives. Linear and quadratic approximationsare frequently used, and occasionally intermediatevariables or intermediate response quantities (e.g.,Kodiyalam and Vanderplaats, 89) are used to improvethe accuracy of the approximation. For example,instead of a Taylor series in the variables, structuralresponse has been approximated by a Taylor series inthe reciprocals of all the variables or some of them(Starnes and Haftka, 79). Similarly, instead ofapproximating eigenvalues directly, we canapproximate the numerator and denominator of theRayleigh quotient that defines them (e.g., Murthy andHaftka, 88, Canfield, 90, Livne and coworkers, e.g.,Livne et al., 93, and 95). Li and Livne (95) haveexplored extensively various approximations forstructural, control and aerodynamic response quantities.A procedure for updating the sensitivity derivatives in asequence of approximations using the past data wasformulated for a general case in Scotti (93).

Global approximations have also been extensivelyused in MDO. Simpler analysis procedures can beviewed as global approximations when they are usedtemporarily during the optimization process, with moreaccurate procedures employed periodically during theprocess. For example, Unger et al. ( 92) developed aprocedure where both the simpler and moresophisticated models are used simultaneously duringthe optimization procedure. The sophisticated modelprovides a scale factor for correcting the simpler model.The scale factor is updated periodically during thedesign process. Because this approach employs modelsof variable complexity it was dubbed `variablecomplexity modeling (VCM). For example, Unger etal. ( 92) applied the procedure to aerodynamic drag

calculation for a subsonic transport, while Hutchison etal. ( 94) applied the procedure to predict the drag of ahigh-speed civil transport (HSCT) during theoptimization process. Similarly, Huang et al. ( 94)employed structural optimization together with a simpleweight equation for predicting wing structural weight incombined aerodynamic and structural optimization ofthe HSCT. Traditional derivative-based approximationcan be combined with such global VCMapproximations by using a derivative-based linearapproximation for the scale factor (Chang et al., 93) .

Another global approximation approach that isparticularly suitable for MDO is the response-surfacetechnique. This technique replaces the objective and/orconstraints functions with simple functions, oftenpolynomials, which are fitted to data at a set ofcarefully selected design points. Neural networks aresometimes used to function in the same role. Thevalues of the objective function and constraints at theselected set of points are used to “train” the network.Like the polynomial fit, the neural network provides anestimate of objective function and constraints for theoptimizer that is very inexpensive after the initialinvestment in the net training has been made.

Response surface techniques are not commonlyused in single-discipline optimization because they donot scale well to large number of variables. For MDO,response surface techniques also provide a convenientrepresentation of data from one discipline to otherdisciplines and to the system. Since design points arepreselected rather than chosen by an optimizationalgorithm, it may be possible to plan and coordinate thesolution process by different modules with less tightintegration than required with derivative-basedmethods. In fact, this feature has motivated the use ofresponse surfaces even for single-disciplineoptimization when the analysis program is not easy toconnect to an optimizer (e.g., Mason at al., 94).

Indeed, response surface techniques have recentlygained popularity as a simple way to connect codesfrom various disciplines, or more generally, to facilitatecommunication between specialists on the design team.In this sense, these techniques are becoming one of themeans to meet the organizational challenge of MDO.For example, Tai et al. (95) have used response surfacetechnique to couple a large number of disciplinaryanalysis programs for the design of a convertiblerotor/wing concept. Giunta et al. (95) have usedresponse surfaces to combine aerodynamic andstructural optimization. They have also takenadvantage of the inherent parallelism of response

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surface generation to employ extensively parallelcomputation.

Additionally, the discrete models employed invarious disciplines can occasionally generatediscontinuities in response due to the effect of shapechanges on grids (e.g., Giunta et al., 94). Multileveldesign schemes can produce similar phenomena due tochanges in sets of critical constraints at lower-leveloptimizations. MDO procedures, which use manymodules and often resort to multilevel techniques areparticularly vulnerable to the occurrence of suchdiscontinuities. Traditional derivative-basedapproximation techniques can become useless in suchcircumstances, while response surface techniquessmooth the design space and proceed without difficulty.For smooth response quantities where derivatives canbe calculated cheaply, we can employ response surfacesbased on these derivatives.

Finally, the global nature of response surfaceapproximation means that they can be repeatedly usedfor design studies with multiple objective functions anddifferent optimization parameters for gradual buildingof the problem database (e.g., Wujek et al., 95).Furthermore, they permit visualizations of the entiredesign space. These features have been usedextensively by Mistree and his coworkers (e.g., Mistreeet al., 94).

2.4 System Sensitivity Analysis

In principle, sensitivity analysis of a system mightbe conducted using the same techniques that becamewell-established in the disciplinary sensitivity analyses(see surveys, Haftka and Adelman, 89, Adelman andHaftka, 93, Barthelemy et al., 95, for automaticdifferentiation, and Bischof and Knauff, 94, and Altuset al., 96 for application examples). However, in mostpractical cases the sheer dimensionality of the systemanalysis makes a simple extension of the disciplinarysensitivity analysis techniques impractical inapplications to sensitivity analysis of systems.

Also, the utility of the system sensitivity data isbroader than that in a single analysis. In design of asystem that typically engages a team of disciplinaryspecialists these data have a potential of constituting acommon vocabulary to overcome interdisciplinarycommunication barriers in conveying information aboutthe influence of the disciplines on one another and onthe system. Utility of the sensitivity data for tracinginterdisciplinary influences was illustrated by an

application to an aircraft performance analysis inSobieszczanski-Sobieski (86).

An algorithm that capitalizes on disciplinarysensitivity analysis techniques to organize the solutionof the system sensitivity problem and its extension tohigher order derivatives was introduced inSobieszczanski-Sobieski (90a) and (90b). There aretwo variants of the algorithm: one is based on thederivatives of the residuals of the governing equationsin each discipline represented by a module in a systemmathematical model, the other uses derivatives ofoutput with respect to input from each module.

So far operational experience has accumulated onlyfor the second variant. That variant begins withcomputations of the derivatives of output with respectto input for each module in the system mathematicalmodel, using any sensitivity analysis techniqueappropriate to the module (discipline). The module-level sensitivity analyses are independent of each other,hence, they may be executed concurrently so that thesystem sensitivity task gets decomposed into smallertasks. The resulting derivatives are entered ascoefficients into a set of simultaneous, linear, algebraicequation, called the Global Sensitivity Equations(GSE), whose solution vector comprises the systemtotal derivatives of behavior with respect to a designvariable. Solvability of GSE and singularity conditionshave been examined in Sobieszczanski-Sobieski (90a).It was reported in Olds (94) that, in some applications,errors of the system derivatives from the GSE solutionmay exceed significantly the errors in the derivatives ofoutput with respect to input computed for the modules.

The system sensitivity derivatives, also referred toas design derivatives, are useful to guide judgmentaldesign decisions, e.g., Olds (94), or they may be inputinto an optimizer (e.g., Padula et al. (91). Applicationof these derivatives extended to the second order in anapplication to an aerodynamic-control integratedoptimization was reported in Ide et al. (88).

A completely different approach to sensitivityanalysis has been introduced in Szewczyk and Hajela(94) and Lee and Hajela (95) It is based on a neural nettrained to simulate a particular analysis (the analysismay be disciplinary or of a multidisciplinary system).Neural net training, in general, requires adjustments ofthe weighting coefficients in the net internal algorithmuntil a correlation of output to input is obtained that is asatisfactory approximation of the output to inputdependency in the simulated analysis over a range of

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interest. The above references show that the weightingcoefficients may be interpreted as a measure of thesensitivity of the output with respect to input. In otherwords, they may be regarded as the derivatives ofoutput with respect to input averaged over the range ofinterest

In Sobieszczanski-Sobieski et al. (82) andBarthelemy and Sobieszczanski-Sobieski (83) theconcept of the sensitivity analysis was extended to theanalysis of an optimum, which comprises theconstrained minimum of the objective function and theoptimal values of the design variables, for sensitivity tothe optimization constant parameters. The derivativesresulting from such analysis are useful in variousdecomposition schemes (next section), and inassessment of the optimization results as shown inBraun et al. (93).

2.5 Optimization Procedures with Approximations and Decompositions

Optimization procedures assemble the numericaloperations corresponding to the MDO elements(Sobieszczanski-Sobieski, 95) into executablesequences. Typically, they include analyses, sensitivityanalyses, approximations, design space searchalgorithms, decompositions, etc. Among theseelements the approximations (Section 2.3) anddecompositions most often determine the procedureorganization, therefore, this section focuses on thesetwo elements as distinguishing features of theoptimization procedures.

The implementation of MDO procedures is oftenlimited by computational cost and by the difficulty tointegrate software packages coming from differentorganizations. The computational burden challenge istypically addressed by employing approximationswhereby the optimizer is applied to a sequence ofapproximate problems.

The use of approximations often allows us to dealbetter with organizational boundaries. Theapproximation used for each discipline can begenerated by specialists in this discipline, who cantailor the approximation to special features of thatdiscipline and to the particulars of the application.When response surface techniques are used, thecreation of the various disciplinary approximations canbe performed ahead of time, minimizing the interactionof the optimization procedure with the variousdisciplinary software.

In addition to approximations, it is desirable tohave flexibility in selection of different searchtechniques for different disciplines and different phasesof optimization. By the same token, one should be ableto choose among various types of sensitivity analysisbecause in some disciplines derivatives are readilyavailable, while in others they may not be available ormay not even exist. Examples of references thatillustrate evolution of the use of approximations inoptimization procedures are Schmit and Farshi (74),Fleury and Schmit (80), Vanderplaats (85),Sobieszczanski-Sobieski (82), and Stanley et al. (92),whose focus was on response surface approximationbased on the Taguchi arrays. A procedure toaccommodate a variable complexity modeling inapplication to a transport aircraft wing was described inUnger et al. (92). An application of approximations inrotorcraft optimization was reported in Adelman et al.(91).

Decomposition schemes and the associatedoptimization procedures have evolved into a keyelement of MDO (Gage, 95, Logan, 90). One importantmotivation for development of optimization procedureswith decomposition is the obvious need to partition thelarge task of the engineering system synthesis intosmaller tasks. The aggregate of the computationaleffort of these smaller tasks is not necessarily smallerthan that of the original undivided task. However, thedecomposition advantages are in these smaller taskstending to be aligned with existing engineeringspecialties, in their forming a broad workfront in whichopportunities for concurrent operations (calendar timecompression) are intrinsic, and in making MDO verycompatible with the trend of computer technologytoward multiprocessing hardware and software.

Much of the theory for decomposition hasoriginated in the field of the Operations Research, e.g,Lasdon (70) and more recently Cramer and Dennis(94). In parallel, several approaches have emergedfrom applied research and engineering practice ofoptimization applied to large problems both withindisciplines and in system optimization. This surveyfocuses primarily on the latter.

Many decomposition schemes (Bloebaum et al.,93) are possible, but they all have in common thefollowing major operations that together constitute asystem synthesis: system analysis including disciplinaryanalyses, disciplinary and system sensitivity analyses,optimizations at the disciplinary level, and optimizationat the system level (the coordination problem). Even

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though the coordination problem is now regarded as thekey element in decomposition, it was absent in some ofthe early optimization procedures, e.g., Sobieszczanskiand Loendorf (72), and Giles and McCullers (75).However, true MDO presupposes taking advantage ofinterdisciplinary interactions, hence a need forcoordination in optimization procedures that implementdecomposition.

Three basic optimization procedures havecrystallized for applications in nonhierarchic aerospacesystems. The simplest procedure is piece-wiseapproximate with the GSE used to obtain thederivatives needed to construct the system behaviorapproximations in the neighborhood of the design point.In this procedure only the sensitivity analysis part of theentire optimization task is subject to decomposition andthe optimization is a single-level one encompassing allthe design variables and constraints of the entiresystem. Hence, there is no need for a coordinationproblem to be solved. This GSE-based procedure hasbeen used in a number of applications, e.g.,Sobieszczanski-Sobieski et al. (88), Barthelemy et al.(92) Coen et a l . (92) , Consoli andSobieszczanski-Sobieski (92), Hajela et al. (90), Abdi etal. (88), Dovi et al. (92), Padula et al. (91), andSchneider et al. (92). The cost of the procedurecritically depends on the number of the couplingvariables for which the partial derivatives arecomputed.

Disciplinary specialists involved in a designprocess generally prefer to control optimization in theirdomains of expertise as opposed to acting only asanalysts. This preference has motivated developmentof procedures that extend the task partitioning tooptimization itself and enable one to organize thenumerical process to mirror the existing humanorganization. A procedure called the ConcurrentSubspace Optimization (CSSO) introduced inSobieszczanski-Sobieski (89) allocates the designvariables to subspaces corresponding to engineeringdisciplines or subsystems. Each subspace performs aseparate optimization, operating on its own uniquesubset of design variables. In this optimization, theobjective function is the subspace contribution to thesystem objective, subject to the local subspaceconstraints and to constraints from all other subspaces.The local constraints are evaluated by a locallyavailable analysis, the other constraints areapproximated using the total derivatives from GSE.Responsibility for satisfying any particular constraint isdistributed over the subspaces using "responsibility"coefficients which are constant parameters in each

subspace optimization. Postoptimal sensitivity analysisgenerates derivatives of each subspace optimum to thesubspace optimization parameters. Following a roundof subspace optimizations, these derivatives guide asystem-level optimization problem in adjusting the"responsibility" coefficients This preserves thecouplings between the subspaces. The system analysisand the system– and subsystem level optimizationsalternate until convergence.

In Bloebaum (91) and (92), the above system-leveloptimization was reformulated by using an expertsystem comprising heuristic rules to adjust theresponsibility coefficients, to allocate the designvariables, and to adjust the move limits. In Korngold etal. (92), the algorithm was extended to problems withdiscrete variables. A variant of the algorithmintroduced in Wujek et al. (95) allows the variablesharing between the subspaces, and it bases the system-level optimization on a response surface functions forthe objective and each of the constraints. Thesefunctions are fitted to all the design points that havebeen generated in all the previous subspaceoptimizations (including the intermediate ones). The"responsibility" coefficients are not used, and thesystem-level optimization is being solved in the spaceof all the design variables. A trained Neural Netalgorithm was used instead of the response surfacefitting in the system-level optimization in Sellar et al.(96). A number of application examples for the CSSOvariants have been reported in Wujek et al. (96) andLokanathan et al. (96). The procedure has also beenimplemented in a commercial software described inEason et al. (94 a,b) and Nystrom et al. (94).

The degree to which CSSO is expected to reducethe system-optimization cost strongly depends onproblem sparsity. To see that, consider that in theextreme case, when every design variable affects everyconstraint directly, each subspace optimization problemmay grow to include all the system constraints in theSobieszczanski-Sobieski (88) version and all the systemvariables and all the system constraints in the Wujek etal. (95) version.

Another procedure proposed in Kroo et al. (94) andKroo (95) is known as the Collaborative Optimization(CO). Its application examples for space vehicles are inBraun and Kroo (95) and Braun et al. (95 a,b) and foraircraft configuration, in Sobieski and Kroo (96). Thisprocedure decomposes the problem even further byeliminating the need for a separate system and systemsensitivity analyses. It achieves this by blending thedesign variables and those state variables that couple

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the subspaces (subsystems or disciplines) in one vectorof the system-level design variables. These variablesare set by the system level optimization and posed tothe subspace optimizations as targets to be matched.Each subspace optimization operates on its own designvariables, some of which correspond to the targetstreated as the subspace optimization parameters, anduses a specialized analysis to satisfy its own constraints.The objective function to be minimized is a cumulativemeasure of the discrepancies between the designvariables and their targets. Optimization in subspacesmay proceed concurrently; each of them is followed bya postoptimum sensitivity analysis to compute theoptimum sensitivity derivatives (Sobieszczanski-Sobieski et al., 82 and Barthelemy and Sobieszczanski-Sobieski, (83) with respect to the parameters (targets).The ensuing system-level optimization satisfies all theconstraints and adjusts the targets so as to minimize thesystem objective and to enforce the matching. Thisoptimization is guided by the above optimumsensitivity derivatives.

As in the other procedures, the subspaceoptimizations, their postoptimum sensitivity analyses,and the system-level optimization alternate untilconvergence. Similar to CSSO, the systemoptimization sparsity is critical for the CO to be able toreduce the system optimization cost because, reasoningby the extremes again, in case of everything influencingeverything else directly, each subspace problem wouldhave to include all the design variables (but still onlythe local constraints). The CO procedure is in thecategory of simultaneous analysis and design (SAND)because the system analysis solution and the systemdesign optimum are arrived at simultaneously at the endof an iterative process. A different implementation ofthe SAND concept is described in Hutchison et al. (94).

Each of the above procedures applies also tohierarchic systems. A hierarchic system is defined asone in which a subsystem exchanges data directly withthe system only but not with any other subsystem.Such data exchange occurs in analysis of structures bysubstructuring. A concept to exploit this in structuraloptimization was formulated in Schmit andRamanathan (78) and generalized in Sobieszczanski-Sobieski (82) and (93). (It was shown in the latter howthe hierarchic decomposition derives from theBellman’s optimality criterion of the DynamicProgramming.) The concept was also contributed to byKirsch, (e.g., Kirsch, 81) It was demonstrated inseveral applications, including multidisciplinary ones,e.g., Wrenn and Dovi (88) and Beltracchi (91). Oneiteration of the procedure comprises the system analysis

from the assembled system level down to the individualsystem components level and optimization thatproceeds in the opposite direction. The analysis datapassed from above become constant parameters in thelower level optimization. The optimization results thatare being passed from the bottom up include sensitivityof the optimum to these parameters. The coordinationproblem solution depends on these sensitivity data. Asshown in Thareja and Haftka (90), one may encounternumerical difficulties in that solution whendiscontinuities occur in the optimum sensitivityderivatives. Recently, a variant of this procedure wasdeveloped (Balling and Sobieszczanski-Sobieski, 95)which differs in the way the local and systemconstraints are treated.

It was shown in Balling and Sobieszczanski-Sobieski (94) that the above procedures may beidentified as variants of the six fundamental approachesto the problem of a system optimization. This referenceoffers also a compact notation for describing a complexprocedure without using a flowchart, and it assesses thecomputational cost of the fundamental approaches as afunction of the problem dimensionality.

The current practice relies on the engineer's insightto recognize whether the system is hierarchic, non-hierarchic, or hybrid and to choose an appropriatedecomposition scheme, Logan (90). For a largeunprecedented system this decision may be difficult.Motivated by this, a formal description of the inter-module data flow in the system model has beendeveloped. This led to techniques, e.g., Rogers (89)and Steward (91), for visualization of that flow in theso-called n-square matrix format and for identificationof the arrangements of the modules in computationalsequences that maximize user-defined measures ofefficiency. Examples of such maximization usingheuristics and/or formal means, such as geneticalgorithms, were reported in Rogers et al. (96), Altus etal. (95), Johnson and Brockman (95), Jones (92),Bloebaum (96), and McCulley and Bloebaum (94).Identification of such sequences results also in a cleardetermination of the system as hierarchic, non-hierarchic, or a hybrid of the two. A code described inRogers (89) is a tool useful in the above; Grose (94)and Brewer et al. (94) are application examples. Analternative to the above approach to decomposition ismentioned in the Appendix on Design Space Search. Itis a transformation of the design space coordinatesystem that identifies orthogonal subspaces described inRowan (90).

Independently of its use for optimization,decomposition has also been used as a means, based

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predominantly on the directed graph approach, todevelop a broad workfront in a human organization orin a production plant and in design to allocate functionsamong components in complex systems.Representative to this category of decomposition areEppinger et al. (94), Pimmler and Eppinger (94), andKusiak et al. (94) An extension of this approach to theuse of hypergraphs and a model-based partitioning wasintroduced in Papalambros and Michelena (95) andMichelena and Papalambros (95). It was shown inCramer and Huffmall (93) that computational burden inMDO applications may be alleviated withoutdecomposition by a judicious exploitation of sparsity ofthe matrices involved.

2.6 Human Interface

MDO is, emphatically, not a push-button design.Hence, the human interface is crucially important toenable engineers to control the design process and toinject their judgment and creativity into it. Therefore,various levels of that interface capability is prominentin the software systems that incorporate MDOtechnology and are operated by industrial companies.Because these software systems are nearly exclusivelyproprietary no published information is available forreference and to discern whether there are any unifyingprinciples to the interface technology as currentlyimplemented.

However, from personal knowledge of some ofthese systems we may point to features common tomany of them. These are flexibility in selectingdependent and independent variables in generation ofgraphic displays, use of color, contour and surfaceplotting, and orthographic projections to capture largevolumes of information at a glance, and the animation.The latter is used not only to show dynamic behaviorslike vibration but also to illustrate the changes in designintroduced by optimization process over a sequence ofiterations. Development has already started in the nextlevel of display technology based on the virtual realityconcepts. In addition to the engineering data display,there are displays that show the data flow through theproject tasks, the project status vs. plans, etc. Onecommon denominator is the desire to support theengineer's train of thought continuity because it is wellknown that such continuity fosters insight thatstimulates creativity. The other common denominatoris the support the systems give to the communicationamong the members of the design team.

One optimization code, usable for MDO purposesand available to general public, is described in

Parkinson et al. (92). This code informs the user on theoptimization progress by displaying the values ofdesign variables, constraints, and objective functioncontinually from iteration to iteration.

The above features support the computer-to-usercommunication. In the opposite direction, users controlthe process by a menu of choices and, at a higher level,by meta-programming in languages that manipulatemodules and their execution on concurrently operatingcomputers connected in a network, e.g., code FIDO inWeston et al. (94). One should mention at this point,again, the nonprocedural programming introduced inKroo and Takai (88). This type of programming maybe regarded as a fundamental concept on which to basedevelopment of the means for human control ofsoftware systems that support design. This is sobecause it liberates the user from the constraints of aprepared menu of preconceived choices, and itefficiently sets the computational sequence needed togenerate data asked for by the user with a minimum ofcomputational effort.

A code representative of the state of the art wasdeveloped by General Electric, Engineous, (94), andLee et al. (93), for support of design of aircraft jetengines. The distinguishing feature of the code calledEngineous is interlacing of the numerical and AItechniques combined with an intrinsically interactiveoperation that actively engages the user in the process.Similar emphasis on the user interaction is found inBohnhoff et al. (92). Examples of other codes thatprovide MDO features to support design process are inKisielewicz (89), Volk (92), Woyak, Malone, et al.(95), and Brama and Rosengren (90).

3. Simultaneous Aerodynamic and Structural Optimization

One of the most common applications ofmultidisciplinary optimization techniques is in the fieldof simultaneous aerodynamic and structuraloptimization, in particular for the design of aircraftwings or complete aircraft configurations. The reasonfor this prominence is that the tradeoff betweenaerodynamic and structural efficiency has always beenthe major consideration in aircraft design: slendershapes have lower drag but are heavier than the stubby,more draggy shapes. The bi-plane wings thatdominated early aircraft configurations were theconcession of the aerodynamicists to the need forstructural rigidity. Only after advances in structuraldesign and structural materials permitted buildingmonoplanes with enough wing rigidity, were we able to

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take advantage of the superior aerodynamic efficiencyof monoplanes.

The interaction between aerodynamics andstructures in the form of aeroelastic effects, such as loadredistribution, divergence, flutter, and loss of controlsurface effectiveness, has given rise to the discipline ofaeroelasticity. Structural optimization was oftencoupled with aeroelastic constraints. Occasionally,structural optimization was even used to improveaerodynamic efficiency (e.g., Haftka, 77) andFriedmann, 92). However, here we focus our attentiononly on studies where both the aerodynamic andstructural design were optimized simultaneously. Thereader interested in work on aeroelastic optimization isreferred to Shirk et al. (86), Haftka (86), and Friedmann(91).

The trade-off between low drag and low structuralweight for aircraft wings is affected by two mechanismsof interaction between aerodynamic and structuralresponse. First, structural weight affects the requiredlift and hence the drag. Second, structural deformationschange the aerodynamic shape. The second effect iscompensated for by building the structure, so that thestructural deformation will bring it to the desiredshape. This so called jig-shape approach nullifies mostof the second interaction when the deformations of theaircraft structure are approximately constant throughmost of the flight time. This is the case for manytransports. For fighter aircraft, structural deformationsduring various maneuvers can adversely affectaerodynamic performance, and the jig shape can correctonly for the most critical maneuver or cruise conditions.Similarly, for very long range or high-speed transports,where the weight and cruise conditions can vary a lot,the jig-shape correction will only partially compensatefor the adverse effects of structural deformations.

If the effects of structural deformation onaerodynamic performance are assumed to be correctedby the jig shape, the interaction between theaerodynamicist and structural designer becomesone-sided. The aerodynamic design affects all aspectsof the structural design, while the structural designaffects the aerodynamic design primarily through asingle number—the structural weight. This asymmetryin the mutual influence of aerodynamic and structuraldesigns means that the problem can be treated as a two-level optimization problem, with the aerodynamicdesign at the upper level and the structural design at thelower level. However, this means that for eachaerodynamic analysis one has to do a structuraloptimization, which makes sense because the cost of

structural analysis is usually much lower than the costof the aerodynamic analysis. For example,Chattopadhyay and Pagaldipti (95) employ such anapproach for a high-speed aircraft with a Navier Stokesmodel for the aerodynamic and a box beam model ofthe structure. Similarly, Baker and Giesing (95)demonstrated this approach with an Euler aerodynamicsolver and a large finite-element model optimized bythe ADOP program.

Taking advantage of the asymmetric interactionbetween structures and aerodynamics presents anenormous saving in computational resources becausewe do not need to calculate the large number ofderivatives of aerodynamic flow with respect tostructural design variables. Further savings are realizedby taking advantage of this asymmetry to generatestructural optimization results for a large number ofaerodynamic configurations and fit them with ananalytical surface usually called the “weight equation”(see Torenbeek, 92, for references on the subject). Thestructural weight of existing aircraft can also be usedfor the same purpose. McCullers developed a transportweight equation based on both historical data andstructural optimization for the FLOPS program(McCullers, 84) . This equation was used forincluding structural weight considerations inaerodynamic optimization of a high-speed civiltransport configurations by Hutchison et al. ( 94).

At the conceptual design level, structural weighthas traditionally been estimated by algebraic weightequations and similar algebraic expressions foraerodynamic performance measures such as drag. Thesimplicity of these expressions allowed designers toexamine many configurations with a minimum ofcomputational effort. More recently, such tools havebeen combined with modern optimization tools. Forexample, Malone and Mason ( 91) optimized transportwings in terms of global design variables, such as wingarea, aspect ratio, cruise Mach number and cruisealtitude, using simple algebraic equations for structuraland other weights and aerodynamic performance. Thesame authors (92) then used such models also toexamine the effect of the choice of objective function(maximize range, minimize fuel weight, etc.)

At the preliminary design level, numerical modelsof both structures and aerodynamics are employed.Early studies of combined aerodynamic and structuraloptimization relied on simple, usually one-dimensionalaerodynamic and structural models and a small numberof design variables (e.g., McGeer, 84, and referencestherein), so that computational cost was not an issue.

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Rather, the goal was to demonstrate the advantages ofthe optimized design. For example, Grossman et al.(88) used lifting line aerodynamics and beam structuralmodels to demonstrate that simultaneous aerodynamicand structural optimization can produce superiordesigns to a sequential approach. Similar models wereused by Wakayama and Kroo (90) and (94) whoshowed that optimal designs are strongly affected bycompressibility drag, aeroelasticity, and multiplestructural design conditions. Gallman et al. ( 93) usedbeam structural models together with vortex latticeaerodynamic to explore the advantages of joined-wingsaircraft.

Modern single-disciplinary designs in bothaerodynamics and structures go beyond such simplemodels. Aerodynamic optimization for transports isoften performed with three-dimensional nonlinearmodels (e.g., Euler equations, Korivi et al., 94) Whilestructural optimization is performed with large finite-element models. For example, Tzong et al. (94)performed a structural optimization with staticaeroelastic effects of a high-speed civil transport usinga finite-element model with 13,700 degrees of freedomand 122 design variables. The validity of resultsobtained with simple models is increasingly questioned,and there is pressure to perform multidisciplinaryoptimization with more complex models. However,because of the asymmetric interaction betweenaerodynamics and structures, discussed above, theemphasis in multidisciplinary optimization is onimproved aerodynamic models (e.g., Giesing et al., 95).Thus, modern conceptual design tools such as FLOPS(McCullers, 84) or ACSYNT (Vanderplaats, 76), andMason and Arledge, 93) incorporate aerodynamic panelmethods at the same time that they use algebraic weightequations to represent structural influences on thedesign.

Computational efficiency becomes an issue whenthe complexity of the aerodynamic and structuralmodels and the number of design variables increase.Borland et al. (94) performed a combinedaerodynamic-structural optimization of a high-speedcivil transport using a large finite-element model andthin Navier Stokes aerodynamics. However, they wereable to afford only 3 aerodynamic variables along with20 structural design variables. Chattopadhyay andPagaldipti (95) used parabolized Navier Stokesaerodynamic model and beam structural model, butwith only four aerodynamic variables. Similarly, Bakerand Giesing (95) used Euler code for aerodynamics anda detailed finite-element analysis, but with only two

aerodynamic design variables representing theaerodynamic twist distribution.

One of the major components of the computationalcost is the calculation of cross-sensitivity derivativessuch as the derivatives of aerodynamic performancewith respect to structural sizes and derivatives ofstructural response with respect to changes inaerodynamic shape. Grossman et al. (90) reduced theinteraction front between the aerodynamic andstructural analysis in the Global Sensitivity Equationapproach (Sobieszczanski-Sobieski 90) to substantiallylower the computational cost of calculating suchderivatives. Automated derivative calculations,obtained by differentiating the computer code used forthe analysis, may also help, as demonstrated by Ungerand Hall ( 94).

Additional savings in computational resourceswere achieved by the use of variable complexitymodeling techniques (see Section 2.3). For example,Dudley et al. (94) used structural optimization toperiodically correct the prediction of algebraic weightequations. Using this approach they have optimized ahigh-speed civil transport using 26 configuration designvariables and 40 structural design variables. However,because they did not calculate and use derivatives of theweight obtained by structural optimization with respectto configuration design variables, the performance ofthe procedure was not entirely satisfactory. It ispossible that there is no need to couple structuraloptimization tightly with aerodynamic optimization.Instead, as done by McCullers (84) in FLOPS,structural optimizations may be performed ahead oftime to obtain improved weight equations for the classof vehicles under consideration (see also Haftka et al.95).

Of course, limiting structural influences to weightequations may not always work, in particular, whenaerodynamic performance is important for multipledesign conditions whose structural deformations arevery different. Then a completely integrated structuraland aerodynamic optimization may be necessary forobtaining high-performance designs. In such cases wemay want to tailor the structure so that structuraldeformation will help aerodynamic performance underthe multiple flight conditions. However, an alternateapproach is to use control surfaces to compensate forstructural deformations for multiple flight conditions.In that case, the aerodynamicist can still assume thataerodynamic performance will not be compromised bystructural deformations. The structural designer will

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have to design the jig shape, the structure, and thecontrol surface deflections simultaneously to minimizestructural weight while safeguarding aerodynamicperformance. Miller (94) employed this type of anapproach in a study of an active flexible wing. Anotherexample where such approach may be necessary is asupersonic transport design for efficient flight in boththe supersonic and subsonic speed regimes (AIAA 91),the latter required by sonic boom restrictions.

In the past few years there has been also a lot ofprogress for combined aerodynamic and structuraloptimization of rotor blades, and in the requisiteanalytical advances, the latter illustrated by He andPeters (92), Lim and Chopra (91), and Kolonay et al.(94). Work in this direction started with structuraloptimization subject to aerodynamic constraints (e.g.,Yuan and Friedmann, 95). However, there is also muchwork which includes both aerodynamic and structuraldesign variables. Callahan and Straub (91) used a codecalled CAMRAD/JA to design rotor blades forimproved aerodynamic performance and reducedstructural vibration with up to 17 design variables.Walsh et al. (92) has integrated the aerodynamic anddynamic design of rotor blades, and Walsh et al. (95)have added structural optimization to the formercapability, using a multilevel approach. As in the caseof fixed wing optimization, the multilevel approach wasaided by the relative simplicity of the structural model.However, unlike fixed wings, rotor blades naturallylend themselves to inexpensive, beam structuralmodels. Chattopadhyay and McCarthy, (93a, b) haveexplored the use of multiobjective optimization forsimilar integration of aerodynamics, dynamics andstructures for the design of rotor blades. Otherexamples of applications in rotor blade design weregiven in Celi (91) and Chattopadhyay et al. (91).

Additional examples of optimization that accountsfor interaction of aerodynamics and structures inflexible wing design may be found in Rais-Rohani et al.(92), Yurkovich (95), Scotti (95), and Rohl et al. (95).Interaction of aerodynamics and structures occurs alsoin the emission, transmission, and absorption of noisegenerated by propulsion and by the airframe movingthrough the air. This interaction has spawned thediscipline of structural acoustics, e.g., Lamancusa (93)and Pates (95), whose approach is based on theboundary finite elements.

4. Simultaneous Structures and Control Optimization

Another common application of multidisciplinaryoptimization is in simultaneous design of a structure

and a control system. A typical aeronauticalapplication is active flutter suppression, and typicalspace structure application is the suppression oftransient vibration triggered by transition from Earthshadow to sunlight.

Past practice has been sequential so that thestructural layout and cross-sectional dimensions weredecided first, and a control system was addedsubsequently to eliminate or alleviate any undesirablebehavior still remaining. Occasionally, when it wasknown in advance how effective the control systemwould be in reducing a particular behavior constraint,violation the structural design was carried out first tosatisfy the above constraint partially, and the design ofthe control system followed to achieve the fullsatisfaction of the constraint. Iterations ensued if thecontrol system design was unable to satisfy its share ofthe constraint. An example of this approach wasreported in Sobieszczanski-Sobieski et al. (79) in whichstructural sizing was used to provide flutter-freeairframe of a supersonic transport up to the divingvelocity (VD), and an active flutter suppressionprovided the required 20 % velocity margin beyondVD.

The sequential practice is deficient because it doesnot accommodate general objective function orfunctions, nor does it enable one to explicitly tradestructural stiffness, inertia, and weight for the activecontrol system effort and weight. These deficienciesare remedied by simultaneous optimization of thestructure and the system for control of its behavior.Haftka (90) surveys various simultaneous formulationsranging from ad hoc ones to those in the multiobjective(pareto-optimal) category. In general, one expects thesimultaneous approach to generate designs whosestructural weight and control effort are less than thoseachievable under the sequential approach. Even thoughthere is no doubt as to the superiority of the integratedapproach, still the integrated structures-controloptimizations on record typically use a compositeobjective function that is a weighted sum of thestructural mass and the control effort, with theweighting factors set by subjective judgment. This is sodespite availability of tools that are ready for a lesssubjective approach under which the airframe masscould be traded off for the mass of the control system,the latter including the mass of the requisite powergeneration equipment.

When all the objectives cannot be converted to asingle one, such as mass, a pareto optimization is calledfor. A full pareto-optimal optimization would seek to

14


generate a locus of all the pareto-optimal solutions.Unfortunately, this locus can have discontinuities andbranches, as demonstrated by Rakowska et al. (91) and(93). Tracing the entire locus may be very expensive,even though homotopy techniques, Rakowska et al.(93), may alleviate this burden. However, even lessambitious simultaneous optimization approaches arecomputationally expensive. That has motivated anumber of papers that addressed that problem.Reducing the mathematical model dimensionality wasproposed in Karpel (92) as a means to savecomputational cost in synthesis of aeroservoelasticsystems. To alleviate cost of the particularly expensiveunsteady aerodynamics, Hoadley and Karpel (91) usedapproximate surrogate for a full unsteady aerodynamicanalysis in the optimization loop combined withinfrequent repetitions of the full analysis. In the samevein, Livne (90) reported on the accuracy ofreduced-order mathematical models for calculation ofeigenvalue sensitivities in control-augmentedstructures. Chattopadhyay and Seeley ( 95) havedeveloped an efficient simulated annealing approach forthe solution of the multiobjective optimization problemfor rotorcraft applications.

The dimensionality of the design space of anactively controlled structure dramatically increases forcomposite structures when fiber orientations areincluded among the design variables, and even more sowhen overall shape variables, e.g., the wing sweepangle, are added. Livne (89) gave a comprehensiveintroduction to optimization of that category applied toaircraft wings and continued it in Livne (92). Thecomputational cost issue was addressed in context ofthe above applications in Livne and Friedmann (92) andin Livne (93). Livne and Wei-Lin (95) assert that thesensitivity analysis and approximation concepts inaerodynamics and airframe structures, the lattermodeled by equivalent stiffness plate, have progressedto a point where a realistic wing/control shapeoptimization with active controls and aeroservoelasticconstraints begins to appear to be within reach.

In space applications, recent thrusts have entailedsynthesis of structure-control systems designed tomaintain pointing accuracy, shape control of reflectivedishes (both transmitting and receiving), andelimination of vibrations that might be induced byexternal influences, e.g., a thermal excitation by sun, orthe spacecraft maneuvers. Many of these applicationsinvolve locations of sensors and actuators that aredetermined by discrete variables, therefore, searchtechniques capable of handling discrete variables havebeen added to the toolbox. An example is the use of

modified simulated annealing for a combinatorialoptimization melded with a continuous variableoptimization of honeycomb infrastructures forspaceborne instrumentation in Kuo and Bruno (91). Amultiobjective optimization was used in Milman andSalama (91) to generate a family of designs optimizedfor competing objectives of disturbance attenuation andminimum weight in spaceborne interferometers.Design for the global optimum has been achieved bythe use of a homotopy approach by Salama et al. (91) indevelopment of families of the actively controlled spaceplatform designs for conceptual trade studies. Briggs(92) reported on adding optics to control-structuresoptimization, and in Flowers et al. (92) that type ofoptimization was extended to a multibody system.Further application examples may be found in Harn etal. (93), Hirsch et al. (92), Padula et al. (93), Park andAsada (92), Sepulveda and Lin (92), Suzuki andMatsuda (91), Suzuki (93), Weisshaar et al. (86), andYamakawa (92).

A system sensitivity analysis based on the GlobalSensitivity Equations (GSE) (see section 2.5 on SystemSensitivity Analysis) was introduced in the control-structure optimization by Sobieszczanski-Sobieski et al.(88). Padula et al. (91) reported on an optimization ofa lattice structure representative of a generic large spacestructure in which they used a GSE-based systemsensitivity analysis. They integrated a finite-elementmodel of the structure, multivariable control, andnonlinear programming to minimize the total weight ofthe structure and the control system under constraints ofthe transient vibration decay rate. Fifteen designvariables governed the structural cross-sections and thecontrol system gains. James (93) extended optimizationdimensionality in the above application to 150 designvariables, 12 for the control system gains and theremainder for the structural member cross sections.

Actuator placement on a large space structure is atypical example of an MDO discrete problem. Padulaand Sandridge (92) presented a solution using aninteger programming code and a finite-elementstructural model. Alternatives to the above approachwere given by Kincaid and Barger (93) who used a tabusearch and by Furuya and Haftka (93) who applied agenetic algorithm. An evaluative discussion andcomparison of several methods applicable tooptimization of structures with controls was given inPadula and Kincaid (95). The comparison included thesimulated annealing, tabu search, integer programming,and branch and bound algorithms in the context ofapplications ranging into 1500+ design variables in

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space structures, as well as in airframes. An acousticstructural control was the case in the latter.

An unusual case of optimization of a large spacestructure to make its dynamic behavior deliberatelydifficult to control was reported in Walsh (87) and (88).The object was a long lattice beam mounted to protrudeout of the Space Shuttle orbiter and having its naturalvibration frequencies spaced very closely in order tochallenge the system identification algorithm and thecontrol law synthesis in the design of the controlsystem. Potential of a relatively recent technology ofneural nets to control structural dynamics of elasticwings was demonstrated in Ku and Hajela (95).

5. Concluding Remarks

The survey revealed that in aerospace, MDOmethodology has transcended its structural optimizationroots and is growing in scope and depth towardencompassing complete sets of disciplines required byapplications at hand. It has broadened its utility beyondbeing an analysis and optimization engine to includefunctions of interdisciplinary communication. It hasalso formed a symbiosis with the heterogeneouscomputing environments for concurrent processingprovided by advanced computer technology.

The two major obstacles to realizing the fullpotential of MDO technology appear to be the twinchallenges posed by very high computational demandsand complexities arising from organization of the MDOtask. To deal with these twin challenges the majoremphasis in MDO research has been on approximationand decomposition strategies. Both hierarchical andnon-hierarchical decomposition techniques have beenproposed to deal with the organizational challenge.Response surface approximations are emerging as auseful tool for addressing both the computational andorganizational challenges.

In general, there are still very few instances inwhich the aerospace vehicle systems are optimized fortheir total performance, including cost as one of theimportant metrics of such performance. However, avigorous beginning in that direction has been evident inthe number of references devoted to mathematicalmodeling of manufacturing and operations, and to theuse of these models in optimization. In addition,despite the very well-known fact that engineeringdesign is intrinsically multiobjective, there is a dearthof papers addressing the very formulation of thatmultiobjective problem, the structure-controloptimization being a case in point.

For the human interface, the MDO developers andusers seem to have arrived at a consensus that thecomputer-based MDO methodology is an increasinglyuseful aid to the creative power of human mind whichis the primary driving force in design. The once-popular notions of automated design have been notablyabsent in the surveyed literature, nor were there anyexpectations expressed that AI techniques will changethat in the foreseeable future.

The survey leaves no doubt that the MDO theory,tools, and practices originate in the communities ofmathematicians, software developers, and designerswhom these products ultimately serve. Therefore, its isremarkable that there is little evidence of closecollaboration among these three groups that have beento a large extent working apart missing on valuablecross-fertilization of ideas and understanding of needsand opportunities. Only recently there wereindications of increasing interest in the threecommunities in working more closely together (e.g.,AIAA 91). In a similar vein, there has been almost nointeraction of the aerospace multidisciplinaryoptimization research with other engineering researchcommunities; it would be beneficial to increase theawareness of similar research in fields such as chemicalengineering and electrical engineering.

If one were to end on a speculative note, it is likelythat future similar surveys will find a number of papersdevoted to virtual design and manufacturing built on thefoundation laid out by the works included herein.

Acknowledgment

The work of the second author was partiallysupported by NASA grant NAG1-1669.

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Appendix: Design Space Search

Theoretical and applied mathematics of the designspace search is a very large area of endeavor beyond thescope of this survey. (A recent review is in Frank et al.92.) Nevertheless, a few remarks pertaining directly tothe use of search algorithms in optimization of largesystems are included below.

In large-scale optimization it is now customary toconnect search algorithm to an approximate analysis(See section on Optimization Procedures.) rather than tothe full-fledged analysis of the problem. This gave riseto a notion that the efficiency of that algorithm does nothave much impact on the total computational cost ofoptimization as long as the algorithm solves theapproximate problem because the bulk of thecomputational effort and cost is in the full analysis.However, this notion is not entirely true for at least tworeasons. When the number of design variables goes upinto the range of thousands, the computational cost ofsearch itself goes up superlinearly and begins to mattereven relatively to the full analysis cost. Even moreimportantly, the algorithm memory requirements drivenprimarily by the size of the Jacobian of the constraintsgo up as a product of the number of constraints and thenumber of design variables, and may quickly exceedthe fast memory capacity forcing the operation into atime-consuming mass storage communications.

If there are many more constraints than designvariables and the constrained minimum is defined byonly a few active constraints, then a search techniquethat clings to the constraint boundary is efficient. Theusable-feasible directions algorithm is an example.However, if the constrained minimum lies at a full, ornearly-full, vertex of the feasible space, then such asearch technique may be forced into small steps movingfrom one constraint boundary intersection to the next.In that situation an interior point methods may beexpected to move through the design space over alonger distance. This advantage of the interior pointmethods has been recognized in linear programing, e.g.,Polyak (92). A broader discussion of the interior pointmethods is given in Nesterov (94) and Nash (94).

Even though the gradient-guided search is moreefficient that the one based on the zero-th orderinformation only; still the computational cost ofgradients is of concern. Hence, a continued researchhas been pursued into the zero-th order methods. It hasresulted in improvements in the pattern searchalgorithms, such as those reported in Torczon (93) and

Dennis (91). A related development was described inRowan (90) in which the search is assisted bytransformation of the design space coordinates. Thattransformation results in a set of subspaces that aremutually orthogonal, hence each may be searchedindependently. This results in a process amenable toconcurrent processing. It may also be regarded as aform of decomposition based entirely on mathematicalproperties of the design space, in contrast todecomposition based on physical insight used inpreviously described optimization by decomposition.Search efficiency may also be improved by tailoring itsmathematics to the special features of the problem asshown in Arora (92).

Genetic Algorithms (GA) offer another alternativeto gradient-guided search, e.g., Hajela et al. (92). AnGA algorithm treats a set of design points in the designspace as a population of individuals that producesanother set of points as a generation of parents producesthe next generation of children. A GA algorithmcomprises a mechanism for pairing up the design pointsinto the pairs of parents for transfer of the parentcharacteristics to children and for mutations thatoccasionally endow children with features absent ineither parent. The mechanism favors probabilisticalcreation of children that are better than parents in termsof the objective function and satisfaction of constraints.The mutation mechanism in GA is particularlyimportant to prevent the process from ending up in alocal minimum. It was also demonstrated in Gage etal. (95), on an example of an aircraft wing design thatthis mechanism may be used to create new designs withfeatures that were absent not only in the pair parents butanywhere in the entire parent generation. This amountsto extending design space by adding new variables andis entirely beyond the capability of gradient-directedsearch. References Gage and Kroo (92 and 95), Gageet al. (95), Gage (95), and King et al. (91) provide otherexamples of applications and discussion of issues thatarise in the use of GA.

The capability of escape from local minima is onecharacteristic that Simulated Annealing (SA) class ofsearch algorithms, e.g., Khalak et al. (94) and Kuo et al.(91) has in common with GA. In SA the search israndom, and acceptance of a new design worse than theprevious one is occasionally and probabilisticallyallowed to provide for such escape.

Both the GA and SA techniques generate a largenumber of calls to analysis, hence, their usability islimited by the cost of analysis. In this regard they areinferior to methods that generate and exploit search

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directions in the design space, for instance Snyman etal. (87).Despite that, judging by the number of thereported applications, they are gaining popularity inMDO applications probably because they are verysimple to couple with the analysis modules, and they donot incur the cost of computing the derivatives.However, the extent to which these algorithms can becombined with approximations as the gradient-guidedones can, is a question open to further investigation.

An alternative to numerical search of design spacefor constrained minima is an optimality criteriaapproach. It comprises two elements: the optimalitycriteria appropriate to the case at hand and an algorithmfor transformation of the design variables to achievesatisfaction of the criteria. This approach wassuccessful mostly in structural engineering where thecriteria of fully stressed design and uniform strainenergy density indeed ensure that design is at, or near, aconstrained optimum. In multidisciplinary systemswhose various parts and aspects are governed bydifferent physics, it is difficult to identify a common,physics-based optimality criterion. However, if thesystem is dominated by structures, e.g., optimization ofairframes with aerodynamic loads, the optimalitycriteria method may be successfully extended fromstructures to encompass the entire system, as shown in areview provided in Venkayya (89).

multidisciplinary aerospace design …mln/ltrs-pdfs/nasa-aiaa-96-0711.pdf1 american institute of...

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