67.preference-basedmultiobjectiveparticleswarm preference-b

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Preference-B 1311 Part F | 67.1 67. Preference-Based Multiobjective Particle Swarm Optimization for Airfoil Design Robert Carrese, Xiaodong Li A significant challenge to the application of evo- lutionary multiobjective optimization (EMO) for transonic airfoil design is the oſten excessive number of computational fluid dynamic (CFD) simulations required to ensure convergence. In this study, a multiobjective particle swarm op- timization (MOPSO) framework is introduced, which incorporates designer preferences to pro- vide further guidance in the search. A reference point is projected onto the Pareto landscape by the designer to guide the swarm towards so- lutions of interest. The framework is applied to a typical transonic airfoil design scenario for robust aerodynamic performance. Time- adaptive Kriging models are constructed based on a high-fidelity Reynolds-averaged Navier– Stokes (RANS) solver to assess the performance of the solutions. The successful integration of these design tools is facilitated through the ref- erence point, which ensures that the swarm does not deviate from the preferred search trajectory. A comprehensive discussion on the proposed optimization framework is provided, highlighting its viability for the intended design application. 67.1 Airfoil Design....................................... 1311 67.1.1 Airfoil Design Architecture .......... 1311 67.1.2 Intelligent Optimization: PSO ...... 1312 67.1.3 Multiobjective Optimization ....... 1313 67.1.4 Surrogate Modeling ................... 1316 67.2 Shape Parameterization and Flow Solver ................................... 1317 67.2.1 The PARSEC Parameterization Method .................................... 1317 67.2.2 Transonic Flow Solver ................. 1318 67.3 Optimization Algorithm........................ 1319 67.3.1 The Reference Point Method ....... 1319 67.3.2 User-Preference Multiobjective PSO: UPMOPSO ........................... 1320 67.3.3 Kriging Modeling ....................... 1322 67.3.4 Reference Point Screening Criterion ................................... 1323 67.4 Case Study: Airfoil Shape Optimization .. 1323 67.4.1 Pre-Optimization and Variable Screening .............. 1324 67.4.2 Optimization Results .................. 1325 67.4.3 Post-Optimization and Trade-Off Visualization ........ 1326 67.4.4 Final Designs ............................ 1327 67.5 Conclusion........................................... 1329 References ................................................... 1329 67.1 Airfoil Design Airfoil design originates from an understanding of the fundamental physics of flight, where the aim is to identify or conform to the best possible shape for the given operating requirements. It has evolved from the use of wind tunnel catalogs and traditional cut-and- try methods to automated computational frameworks. While automated frameworks effectively simplify the design process, success is still largely dependent on the fidelity of the computational methods, as well as the experience of the designer in formulating the prob- lem [67.1]. This section is devoted to a discussion of airfoil design optimization architecture. The con- cepts that are especially applicable to this study are introduced, laying the foundations for the proposed methodology. 67.1.1 Airfoil Design Architecture The direct method of airfoil design, pioneered by the work of Hicks and Henne [67.2], refers to the philos- ophy of using mathematical optimization methods to identify the optimal shape that achieves the prescribed

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Page 1: 67.Preference-BasedMultiobjectiveParticleSwarm Preference-B

Preference-B1311

PartF|67.1

67. Preference-Based Multiobjective Particle SwarmOptimization for Airfoil Design

Robert Carrese, Xiaodong Li

A significant challenge to the application of evo-lutionary multiobjective optimization (EMO) fortransonic airfoil design is the often excessivenumber of computational fluid dynamic (CFD)simulations required to ensure convergence. Inthis study, a multiobjective particle swarm op-timization (MOPSO) framework is introduced,which incorporates designer preferences to pro-vide further guidance in the search. A referencepoint is projected onto the Pareto landscape bythe designer to guide the swarm towards so-lutions of interest. The framework is appliedto a typical transonic airfoil design scenariofor robust aerodynamic performance. Time-adaptive Kriging models are constructed basedon a high-fidelity Reynolds-averaged Navier–Stokes (RANS) solver to assess the performanceof the solutions. The successful integration ofthese design tools is facilitated through the ref-erence point, which ensures that the swarmdoes not deviate from the preferred searchtrajectory. A comprehensive discussion on theproposed optimization framework is provided,highlighting its viability for the intended designapplication.

67.1 Airfoil Design....................................... 131167.1.1 Airfoil Design Architecture .......... 131167.1.2 Intelligent Optimization: PSO ...... 131267.1.3 Multiobjective Optimization ....... 131367.1.4 Surrogate Modeling ................... 1316

67.2 Shape Parameterizationand Flow Solver ................................... 131767.2.1 The PARSEC Parameterization

Method .................................... 131767.2.2 Transonic Flow Solver................. 1318

67.3 Optimization Algorithm........................ 131967.3.1 The Reference Point Method ....... 131967.3.2 User-Preference Multiobjective

PSO: UPMOPSO ........................... 132067.3.3 Kriging Modeling ....................... 132267.3.4 Reference Point Screening

Criterion ................................... 132367.4 Case Study: Airfoil Shape Optimization . . 1323

67.4.1 Pre-Optimizationand Variable Screening .............. 1324

67.4.2 Optimization Results.................. 132567.4.3 Post-Optimization

and Trade-Off Visualization ........ 132667.4.4 Final Designs ............................ 1327

67.5 Conclusion........................................... 1329References ................................................... 1329

67.1 Airfoil Design

Airfoil design originates from an understanding of thefundamental physics of flight, where the aim is toidentify or conform to the best possible shape for thegiven operating requirements. It has evolved from theuse of wind tunnel catalogs and traditional cut-and-try methods to automated computational frameworks.While automated frameworks effectively simplify thedesign process, success is still largely dependent onthe fidelity of the computational methods, as well asthe experience of the designer in formulating the prob-lem [67.1]. This section is devoted to a discussion

of airfoil design optimization architecture. The con-cepts that are especially applicable to this study areintroduced, laying the foundations for the proposedmethodology.

67.1.1 Airfoil Design Architecture

The direct method of airfoil design, pioneered by thework of Hicks and Henne [67.2], refers to the philos-ophy of using mathematical optimization methods toidentify the optimal shape that achieves the prescribed

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1312 Part F Swarm Intelligence

Define objectivefunction(s)

Defineconstraints

Initializecandidate shape(s)

Geometrical shapeparameterization

Computationalflow solver

Convergenceobtained?

Optimal shape

Perturbcandidate shape(s)

No

Yes

Optimizationalgorithm

Fig. 67.1 Generalized processflowchart for direct airfoil shape op-timization

design criteria. The generalized framework for an aero-dynamic shape optimization process is demonstrated inFig. 67.1. The success of the direct approach is essen-tially dependent on three main components within thedesign loop:

� Shape parameterization.All design strategies sharethe common requirement that the geometry is rep-resented by a finite number of design variables.A method to mathematically parameterize shapesis, therefore, required so that modifications can bemade via direct manipulation of the design vari-ables. The number of design variables is directlyproportional to the geometrical degrees of freedomand, therefore, governs the dimensionality of theproblem.� Computational flow solver. The objective functionis obtained from the flow solver and it is, therefore,up to the discretion of the designer to appropriatelyformulate the objective and constraint functions,such that they reflect the design and operating re-quirements. The choice of the flow solver ultimatelygoverns the overall fidelity and efficiency of the op-timization process, since repeated evaluations of theobjective function are required for each candidateshape.� Optimization algorithm. The responsibility of theoptimizer is to iteratively determine the shape modi-fications required to satisfy the objective, whilst ad-hering to any shape or performance constraints. Theoptimizer should be robust and applicable to a wideoperational spectrum, yet efficient to guarantee con-vergence with the least computational expense.

The integration of high-fidelity flow solvers andflexible parameterization methods for numerical op-

timization is still a computationally challenging andintensive undertaking. The extension to multiple objec-tives leads to a more generalized problem formulation,yet significantly increases the computational cost ofconvergence. While all elements of the design loop in-fluence the efficiency of the process, arguably the mostimportant element is the optimizer itself. The followingsection introduces the optimization paradigm adoptedin this study, derived from the field of computationalswarm intelligence.

67.1.2 Intelligent Optimization: PSO

The formation of hierarchies within groups of animalsis a naturally occurring phenomenon and is simple tocomprehend. Even humans have the intuitive tendencyto appoint leaders (e.g., political leaders, military gen-erals, etc.). Another interesting phenomenon, which ismore difficult to perceive, is the self-organized behav-ior of groups where a leader cannot be identified. Thisis known as swarming and is evident from the flock-ing behavior of birds or fish moving in unison. Theincreasingly cited field of computational swarm intel-ligence focuses on the artificial simulation of swarmingbehavior to model a wide range of applications, includ-ing optimization [67.3].

Particle swarm optimization (PSO) is the stochas-tic population-based technique described by Kennedyand Eberhart [67.4] in accordance with the principlesof swarm intelligence. The PSO architecture was de-rived from a synthesis of the fields of social psychologyand engineering optimization. As was eloquently statedby the authors in their original paper [67.4]:

Why is social behavior so ubiquitous in the animalkingdom? Because it optimizes. What is a good way

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Preference-Based Multiobjective Particle Swarm Optimization for Airfoil Design 67.1 Airfoil Design 1313Part

F|67.1

to solve engineering optimization problems? Mod-eling social behavior.

The dynamics of the swarm are modeled on thesocial-psychological tendency of individuals to learnfrom previous experience and emulate the success ofothers. Similar to most evolutionary techniques, theswarm is initialized with a population of random indi-viduals (particles) sampled over the design space. Theparticles navigate the multi-dimensional design spaceover a number of iterations or time steps. Each particlemaintains knowledge of its current position in the de-sign space. This is analogous to the fitness concept ofconventional evolutionary algorithms (EAs). Each par-ticle also records its personal best position, which iswhere the particle has experienced the greatest success.Aside from recording personal information, each par-ticle also tracks the position of other members in theswarm. This level of social interaction between parti-cles is coined the swarm topology. Particles may eitherbe confined to share information only with their imme-diate neighbors, or they may be encouraged to sharetheir experiences with the entire swarm. Utilizing thisinformation, each particle adjusts its position in the de-sign space by accelerating towards the successful areasof the design space. The absence of selection is com-pensated by this use of leaders to guide the swarm toconverge to the most successful position. In this way,a solution which initially performs poorly may possiblybe on the future road to success.

PSO has steadily gained popularity as a global op-timization technique [67.3]. Its increasing use in theliterature is due to its simple and straightforward imple-mentation (despite its intricate origins) and its efficientand accurate convergence rates [67.5].

67.1.3 Multiobjective Optimization

Airfoil design problems are often characterized by sev-eral interacting or conflicting requirements, which mustbe satisfied simultaneously. In the case of an airfoiloperating within the transonic regime, airfoil shape op-timization is performed to limit shock and viscous drag(Cd) losses, and reduce shock-induced boundary layerinstability at the design Mach number (M) and liftcoefficient (Cl). This often occurs at the expense of ex-cessive pitching moments (Cm) due to aft loading andperformance degradation under off-design conditions.To facilitate adequate performance over a wide opera-tional spectrum requires a search algorithm capable ofhandling multiple conflicting objectives.

Let S 2Rn denote the design space and let xDfx1; x2; : : : ; xng 2 S denote the decision vector withlower and upper bounds xmin and xmax, respectively. Thegeneric unconstrainedmultiobjective problem (MOP) isthus expressed as,

min f .x/D ff1.x/; : : : ; fm.x/g ; (67.1)

where fi.x/ W Rn ! R is the i-th component of the ob-jective vector and m is the number of objectives. Thedefinition of the optimum must be redefined since inthe presence of conflicting objectives, improvement inone objective may cause a deterioration in another. It isoften necessary to identify a set of trade-off solutions,which can all be considered equally optimal. A solutionis termed non-dominated or Pareto optimal (after thenineteenth century Italian economist Vilfredo Pareto) ifthe value of any objective cannot be improved withoutdeteriorating at least one other objective. The candidatesolutions are defined as a and b 2 S. The candidate adominates the candidate b (denoted by a� b) if,

8jD 1; : : : ;m fj.a/� fj.b/^9j W fj.a/ < fj.b/ :

(67.2)

The concept of dominance is illustrated in Fig. 67.2.The shaded area denotes the area of objective vectorsdominated by a. A decision vector a� is, therefore, non-dominated or Pareto optimal if there is no other feasibledecision vector a¤ a� 2 S such that f .a/� f .a�/. ThePareto front is the set of objective vectors which cor-respond to all non-dominated solutions. Multiobjectivealgorithms aim to identify the closest approximation tothe true Pareto front, while ensuring a diverse Paretooptimal set.

f1(a) f1(b)

Dominance regionof solution a

f1

f2

f2(b)

f2(a)

Fig. 67.2 Illustration of dominance on a two-objectivelandscape

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1314 Part F Swarm Intelligence

Techniques for Solving MOPFrom a design perspective, the primary aim of mul-tiobjective optimization is to obtain Pareto optimalsolutions which are in the preferred interests of the de-signer, or best suit the intended application. Methodsfor solving MOPs are, therefore, characterized by howthe designer preferences are articulated. As suggestedby Fonseca and Fleming [67.6], there are three genericclasses of methods for solving multiobjective problems:

� A priori methods. The preferences of the designerare expressed by aggregating the objective functionsinto a single scalar through weights or bias, ulti-mately making the problem single objective.� A posteriori methods. The algorithm first identifiesa set of non-dominated solutions, subsequently pro-viding the designer greater flexibility in selectingthe most appropriate solution.� Interactive methods. The decision making and op-timization processes occur at interleaved steps, andthe preferences of the designer are interactively re-fined.

The a priori strategy requires the designer to indi-cate the relative importance of each objective beforeperforming the optimization. A notable method thatfalls into this category is the weighted aggregationmethod, which is a fairly popular choice for airfoil de-sign applications due to its simplicity and capability ofhandling many flight conditions [67.7–9]. Despite itspopularity, there are recognized deficiencies with thisstrategy [67.10]. The prior selection of weights doesnot necessarily guarantee that the final solution willfaithfully reflect the preferred interests of the designer,and varying the weights continuously will not neces-sarily result in an even distribution of Pareto optimalsolutions, nor a complete representation of the Paretofront [67.11].

Alternatively the a posteriori methods provide max-imum flexibility to the designer to identify the most pre-ferred solution, at the expense of greater computationaleffort. Generally, these methods involve explicitly solv-ing each objective to obtain a set of non-dominatedsolutions, a concept which is ideal for population-basedevolutionary algorithms [67.12–14]. While these meth-ods are computationally more complex, researchers inaerodynamic design are realizing the benefits of evolu-tionary multiobjective optimization (EMO), especiallyif there is a certain ambiguity in selecting the final de-sign [67.15–17]. However, it poses the challenge ofidentifying and exploiting the entire Pareto front, which

may be impractical for design applications due to theexcessive number of function evaluations.

While conventional EMO techniques may be com-putationally demanding, Fonseca and Fleming [67.12]argue that their most attractive aspect is the intermedi-ate information generated, which can be exploited bythe designer to refine preferences and improve conver-gence. These interactive methods involve the progres-sive articulation of preferences, which originates fromthe multicriteria decision making literature [67.18]. Theoptimization and decision making processes are in-terleaved, exploiting the intermediate information pro-vided by the optimizer to refine preferences [67.6].

Handling Multiple Objectives with PSOPSO has been demonstrated to be an effective toolfor single-objective optimization problems due to itsfast convergence [67.5]. It has also gained rapid pop-ularity in the field of multi-objective optimization(MOO) [67.19]. Since PSO is a population-based tech-nique, it could ideally be tailored to identify a numberof trade-off solutions to a MOP in one single run, sim-ilar to EMO techniques. Comprehensive surveys onextending PSO to handle multiple objectives have beenprovided by Engelbrecht [67.20], and more recently bySierra andCoello Coello [67.19]. It was established thatthe primary ambiguity in specifically tailoring PSO tohandle multiple objectives was the selection of guidesfor each particle to avoid convergence to a single so-lution. The selection process for particle leaders must,therefore, be restructured, to encourage search diversityand to ensure that non-dominated solutions found dur-ing the search are maintained.

Initial attempts to design a multiobjective particleswarm optimization (MOPSO) algorithm were moti-vated by the archive strategy by [67.21]. Coello Coelloand Lechuga [67.22] incorporated the concept of Paretodominance in PSO by maintaining two independentpopulations: the particle swarm and the elitist archive.Non-dominated solutions are stored in the archive andsubsequently used as neighborhood leaders. The objec-tive space is separated into hypercubes, which serveas a particle anti-clustering mechanism. Solutions insparsely populated hypercubes have a higher selectionpressure to be leaders, and solutions in densely pop-ulated hypercubes are removed if the archive limit isexceeded. This initial approach was later extended byMostaghim and Teich [67.23], who studied the conceptof -dominance and compared it to existing clusteringtechniques for fixing the archive size, with favorable re-sults.

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Fieldsend and Singh [67.24] addressed the compu-tational complexity of maintaining a restricted archive,by incorporating the dominated tree method. This datastructure allows for an unrestricted archive size, whichinteracts with the population to define global leaders.A turbulence operator (similar to the concept of mu-tation in EA) was also implemented, where swarmmembers were randomly displaced on the design spaceto reduce the probability of premature stagnation. Inthe non-dominated sorting particle swarm optimization(NSPSO) algorithm of Li [67.25], the non-dominatedsorting mechanisms of non-dominated sorting geneticalgorithm (NSGA-II) are incorporated. The popula-tion and the personal best position of each particle areconsolidated to form one single population, and thenon-dominated sorting scheme is utilized to rank eachsolution. Global guides are selected based on particleclustering, where a niching or crowding distance met-ric is used to further classify non-dominated solutions.Li later proposed the maximinPSO algorithm [67.26],which does not use any niching method to maintaindiversity.

Sierra and Coello Coello [67.27] proposed an eli-tist archive incorporating the -dominance strategy tomaintain global leaders for the swarm. A crowding dis-tance operator is employed to classify non-dominatedsolutions and maintain uniformity. The crowding dis-tance operator is also used to limit the number ofcandidate leaders after each population update, simpli-fying the mechanism to control the set of candidateleaders. A turbulence operator is implemented to en-courage diversity, whereby particles are randomly mu-tated. A similar approach by [67.28] was developed inparallel (although this method does not implement -dominance), where the crowding distance was used toboth define the global guides and truncate the size of thearchive. The proposed algorithm is primarily influencedby the two latter studies.

Preference-Based OptimizationThe concept of interactive optimization has led toan increasing interest in coupling classical interactivemethods to EMO as an intuitive way of reflecting thedesigner preferences and identifying solutions of inter-est to the designer. This has led to the developmentof the preference-based optimization philosophy, whichprovides the motivation for the current study. Compre-hensive surveys on preference-based optimization areprovided by Coello Coello [67.29] and Rachmawatiand Srinivasan [67.30]. The first recorded attempt atincorporating preferences within an evolutionary mul-

tiobjective optimization framework was made by Fon-seca and Fleming [67.31] using the goal programmingapproach. Goal programming [67.11] is an ideal ap-proach to indicate desired levels of performance foreach objective, since they closely relate to the finalsolution of the problem. Goals may either representtarget or ideal values. Fonseca and Fleming later ex-tended the approach where an online decision makingstrategy was proposed based on goal and priority in-formation [67.6]. A goal programming mechanism foridentifying preferred solutions for MOP was also pro-posed by [67.32]. While the reported frameworks drawon the preferred interests of the designer to aid theoptimization process, the goal programming approachis computationally complex, and there is no means ofspecifying any relation or trade-off between the objec-tives [67.30].

Thiele et al. [67.33] proposed another variant ofinteractive evolutionary multiobjective optimization.A coarse representation of the Pareto front is initiallypresented to the designer. The most interesting regionsare subsequently isolated, on which the algorithm con-tinues to focus on exclusively. This proposal effectivelyremoves the necessity to predefine target values for eachobjective and provides the designer with a means ofisolating the preferred trade-offs. However, it is a two-stage approach requiring an initial approximation to thePareto front, which may be unnecessarily expensive.The integration of other classical preference articula-tion methods has also been proposed in the literature.A reference point-based evolutionary multiobjectiveoptimization framework was proposed by [67.34]. Thecrowding distance operator of the NSGA-II algorithmwas modified to include the reference point informationand the extent of the preferred region was controlled by -dominance. Deb and Kumar also experimented withthe use of other classical preference methods, such asthe reference direction method [67.35] and the lightbeam search method [67.36].

Recently, the use of interactive methods has alsobeen integrated within PSO frameworks. Wickramas-inghe and Li [67.37] integrated the reference pointmethod to both the NSPSO [67.25] and maxim-inPSO [67.26] algorithms. Significant improvement inconvergence efficiency was highlighted, and it wasdemonstrated that final solutions are of higher rele-vance to the designer. Wickramasinghe and Li [67.38]later extended their approach to handle MOP, by re-placing the dominance criteria entirely with the simplerdistance metric. It was conclusively demonstrated thatwithout the use of the reference point, obtaining a final

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1316 Part F Swarm Intelligence

set of preferred solutions solely through conventionaldominance-based techniques is improbable.

67.1.4 Surrogate Modeling

The most prohibiting factor of design optimizationis the cost of evaluating the objective and constraintfunctions. For high-fidelity airfoil design, function eval-uations may very well be measured in hours. Thiscomputational burden ultimately questions the practi-cality of performing an optimization study, and is oftenalleviated by simply reducing the level of sophistica-tion of the solver. This consequently reduces the fidelityof the final design, which is undesirable. Another miti-gating strategy which has steadily gained popularity indesign is the use of inexpensive surrogates or metamod-els [67.39]. These models emulate the response of theexpensive function at an unobserved location, based onobservations at other locations. Surrogate models arenot specifically optimization methods, but rather theymay ideally be used in lieu of the expensive functionto extract information from the design space during theoptimization process.

The insightful texts by Keane and Nair [67.39] andForrester et al. [67.40] provide a detailed account of theuse of surrogates in design. The most common use is toconstruct a curve fit of an expensive function landscape,which can be used to predict results without recourse tothe original function. This is supported by the assump-tion that the inexpensive surrogate will still be usefullyaccurate when predicting sufficiently far from observeddata points [67.40]. Figure 67.3 illustrates the use of

Original functionObserved pointsSurrogate

0 0.2 0.4 0.6 0.8 1

f (x)

x

20

15

10

5

0

–5

–10

Fig. 67.3 Constructing an interpolation-based surrogate tofit a one-dimensional function

a surrogate to fit the one-dimensional multi-modal func-tion, based on four sample observations. It is importantto note, however, that the original function landscapecould potentially represent any deterministic quantity ofthe design space. Rather than exactly emulating the re-sponse of a high-fidelity flow solver, the surrogate may,in fact, be used to bridge the gap between flow solversof varying fidelity [67.40]. Alternatively, a surrogatemay be used to interpret or filter noisy landscapes, soas to eliminate the adverse effects of flow solver con-vergence or grid discretization. Surrogates may also beused for data mining and design space visualization.Such methodologies are applied to extract useful infor-mation about the relationship between the design spaceand the objective space, allowing informed decisions tobe made, which could simplify a seemingly complexproblem.

For the aforementioned uses of surrogate model-ing, the common requirement is to replicate the func-tion relationship between the variable inputs and theoutput quantity of interest. This is typically achievedby sampling the design space using the exact func-tion to sufficiently model the underlying relationshipwithin the allowable computational budget. Whetherthe aim is to locally model the design space surround-ing an existing design or tune a surrogate to repli-cate the global design space is entirely dependent onthe formation of the sampling plan [67.39]. The con-struction of a surrogate model in either case shouldideally make use of a parallel computing structure.A suitable surrogate model Of of the precise objec-tive function f should then be constructed to fit thedataset.

There are a multitude of popular techniques forconstructing surrogates in the literature. For a com-prehensive review of different methods, the reader isreferred to (among others) [67.39–42]. The selectionof the surrogate model is dependent on the informationthat the designer is attempting to extract from the designspace. Polynomial response surfaces and radial basisfunctions are fairly popular techniques for constructinglocal surrogates, especially if some level of regressionis desirable. Techniques such as Kriging or supportvector machines are more ideally suited to global op-timization studies, since they offer greater flexibility intuning model parameters and provide a confidence in-terval of the predicted output. Neural networks requireextensive training and validation, yet have also beena popular technique for design applications, notably inaerodynamic modeling [67.43] and visualization tech-niques [67.44, 45].

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Preference-Based Multiobjective Particle Swarm Optimization for Airfoil Design 67.2 Shape Parameterization and Flow Solver 1317Part

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67.2 Shape Parameterization and Flow Solver

It was established in Sect. 67.1.1 that the shape param-eterization method essentially governs the dimension-ality of the problem and the attainable shapes, whereasthe objective flow solver dictates the overall fidelity ofthe optimum design. In this section, we present a dis-cussion on these elements of the design loop to be usedin conjunctionwith the developed optimizer for the sub-sequent design process.

67.2.1 The PARSEC Parameterization Method

Geometry manipulation is of particular importance inaerodynamic design. The selection of the shape param-eterization method is an important contributing factor,since it will effectively define the objective landscapeand the topology of the design space [67.46]. If theaim of the optimization process is to improve on anestablished design, then perhaps local parameterizationmethods, which offer a greater number of geometri-cal degrees of freedom, are desirable. However, thelarge number of variables may cause the convergencerate for global design applications to deteriorate. Thedevelopment of efficient parameterization models has,therefore, been given significant attention, to increasethe flexibility of geometrical control with a minimumnumber of design variables.

For certain applications, it is possible to make use offundamental aerodynamic theory to refine the param-eterization method, such that the design variables re-late to important aerodynamic or geometric quantities.A common method for airfoil shape parameterizationis the PARSEC method [67.47]. It has the advantageof strict control over important aerodynamic features,and it allows independent control over the airfoil geom-etry for imposing shape constraints. The methodology

zxxLO

xxxUP

zLO

xLO

xUPzTE

αTE

�TEΔzTErLE

zUP

Fig. 67.4 Airfoil representation via the PARSEC method

is characterized by 11 design variables (Fig. 67.4), in-cluding leading edge radius (rLE), upper and lowerthickness locations .xUP; zUP; xLO; zLO/ and curvatures.zxxUP ; zxxLO/, trailing edge direction .˛TE/ and wedgeangle .ˇTE/, and trailing edge coordinate .zTE/ andthickness .zTE/. The shape function is modeled viaa sixth-order polynomial function

zk D6X

nD1

an;k xn�

12

k ; (67.3)

where .x; z/ are the shape coordinates and k denoteseither the upper (suction) or lower (pressure) airfoilsurface. The coefficients an are determined from the ge-ometric parameters. A modification by Jahangirian andShahrokhi [67.48] was introduced to provide additionalcontrol over the trailing edge curvature. For supercrit-ical transonic airfoils, this is beneficial to reduce theprobability of downstream boundary layer separation,which results in increased drag values. A new vari-able ˛TE was introduced, which directly influencesthe additional curvature of the trailing edge. The modi-fication decouples the trailing edge parameterization byfirst defining a smoother upper surface contour and thenconstraining the lower surface to intersect the trailingedge coordinate. Figure 67.5 illustrates the modifica-tion to the trailing edge curvature. The modification isapplied to the upper and lower surfaces as follows

ız D L tan˛TE

2��

h1C � x� � �1� x�

� i; (67.4)

where the constants �, �, and � are set to 0:8, 2, and 6,respectively. The modification is applied over the entire

LUP

LLO

ΔαTE

ΔZUP

ΔZLO

Fig. 67.5 Additional trailing edge curvature via the modi-fied PARSEC method

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Table 67.1 PARSEC parameter ranges for transonic optimization

Description Variable Lower bound Upper boundLeading edge radius rLE 0.0063 0.0151Trailing edge direction ˛TE 0.2405.�/ 0.0026.�/

Trailing edge wedge angle ˇTE 0.0655 0.2618Upper-crest abscissa xUP 0.3170 0.5250Upper-crest ordinate zUP 0.0497 0.0683Upper-crest curvature zxxUP 0.5135.�/ 0.2393.�/

Lower-crest abscissa xLO 0.2835 0.3418Lower-crest ordinate zLO 0.0603.�/ 0.0478.�/

Lower-crest curvature zxxLO 0.2535 0.8405Trailing edge curvature ı˛TE 0.0080.�/ 0.3696

surface, such that LUP D LLO D c, where c is the airfoilchord length.

Table 67.1 presents the upper and lower bound-aries for the subsequent optimization case study. Theseboundaries have been selected based on a thoroughscreening study involving a statistical sample of a num-ber of benchmark airfoils.

67.2.2 Transonic Flow Solver

The optimization process is ultimately dependent onthe selection of the flow solver, since it is the mostcomputationally expensive component, and repeatedevaluations of the objective and constraint functionsare required for each candidate shape. However if theflow solver is not sufficiently accurate, the optimizationprocess will converge to shapes that exploit the numer-ical errors or limitations, rather than the fundamentalphysics of the problem. For this reason, it is desirableto maintain the correct balance between solution accu-racy and computational expense, which is dictated bythe flow regime. For certain problems where the aero-dynamic flow field is well behaved, it may be sufficientto consider more robust linear solvers. However forhigh-fidelity design it is prudent to consider non-linearand more computationally demanding solvers, to ensurethat optimized shapes provide the anticipated perfor-mance requirements in flight.

The general purpose finite volume code ANSYSFluent is adopted in this study. A pressure-based nu-merical procedure is adopted with third-order spatialdiscretization to capture the occurring flow phenomena.The momentum equations and pressure-based continu-ity equation are solved concurrently, with the Courant–Friedrichs–Lewy number set at 200. The one-equation

–0.2 0 0.2 0.4 0.6 0.8 1 1.2x–coordinate

0.6

0.4

0.2

0

–0.2

–0.4

–0.6

Fig. 67.6 C-type grid for transonic simulation

Spalart–Allmaras turbulence model [67.49] is selected,and turbulent flow is modeled over the entire airfoilsurface. The C-type grid (as represented in Fig. 67.6)stretches 25 chord lengths aft and normal of the airfoilsection. Resolution of the C-grid is 460� 65, providingan affordable mesh size of approximately 30 000 ele-ments. The first grid point is located 2:5� 10�4 unitsnormal to the airfoil surface, resulting in an average y-plus value of 120. In the interest of robust and efficientconvergence rates, a full multi-grid (FMG) initializa-tion scheme is employed, with coarsening of the gridto 30 cells. In the FMG initialization process, the Eulerequations are solved using a first-order discretization toobtain a flow field approximation before submission tothe full iterative calculation.

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67.3 Optimization Algorithm

The proposed algorithm was primarily motivated by thestudies of Wickramasinghe and Li [67.38]. The prin-cipal argument is that for most design applications,to explore the entire Pareto front is often unneces-sary, and the computational burden can be alleviated byconsidering the immediate interests of the designer. InSect. 67.1.3, a discussion on the benefits of preference-based optimization was provided. Drawing on theseconcepts, a preference-based algorithm is proposed,where a designer-driven distancemetric is used to scalarquantify the success of a solution. The multiobjectivesearch effort is coordinated via a MOPSO algorithm.The swarm is guided by a reference point, which is anintuitive means of articulating the preferences of thedesigner and can ideally be based on an existing ortarget design. This section provides a comprehensivediscussion on the proposed algorithm, highlighting itsviability for the intended domain of application.

67.3.1 The Reference Point Method

In this research, the swarm is guided by a referencepoint to confine its search focus exclusively on thepreferred region of the Pareto front as dictated by thepreferences of the designer. Introducing the preferredregion provides the designer flexibility to explore otherinteresting alternatives. This hybrid methodology is ad-vantageous for navigating high-dimensional and multi-modal landscapes, which are typical of aerodynamicdesign problems. Furthermore, inherently consideringthe preferences of the designer provides a feasiblemeans of quantifying the practicality of a design.

The Reference Point Distance MetricThe reference point method has been integrated intoMOO algorithms, notably by Deb and Sundar [67.34]and Wickramasinghe and Li [67.37, 38]. These stud-ies highlight the benefits of incorporating preferenceinformation via the reference point in terms of con-vergence. Guided by the information provided by thereference point, the swarm can simultaneously identifymultiple solutions in the preferred region. This providesthe designer flexibility to explore several preferred de-signs, while alleviating the computational burden ofidentifying the entire Pareto front. A reference pointdistance metric following the work of Wickramasingheand Li [67.37] is proposed. This metric provides an in-tuitive criterion to select global leaders and assists theswarm to identify only solutions of interest to the de-

signer. The distance of a particle x to the reference pointNz is defined as

dz.x/D maxiD1Wm

f.fi .x/� Nzi/g : (67.5)

A solution a is, therefore, preferred to solution b ifdz.a/ < dz.b/. This condition is an extension of the con-dition f .a/� f .b/, therefore, the distance metric may, infact, substitute the dominance criteria entirely [67.38].Using this distance metric, the swarm is guided to pre-ferred regions of the Pareto optimal front. Figure 67.7illustrates the search directions of the algorithm whenguided by a reference point, and the correspondingpreferred design as a direct result of minimizing the dis-tance metric dz.

The distinguishing feature of the reference pointdistance metric over the mathematical Euclidean dis-tance is that solutions do not converge to the referencepoint, but on the preferred region of the Pareto frontas dictated by the search direction. This is illustratedin Fig. 67.8. All solutions are non-dominated and lieon the circular arc surrounding the reference point Nz,and thus the Euclidean distance to the reference pointis equal. However, since solution i has the smallestmaximum translational distance to the reference pointcompared to any other solution, it is considered pre-ferred. The definition of the reference point distancealso suggests negative values. If the distance of thepreferred solution dz.z0/ < 0, then it can simply be con-sidered that the reference point is dominated or z0 �Nz. Since the designer generally has no prior knowl-edge of the topology and location of the Pareto front,

z' 1

z' 2

z–1

z– 2

f1

f2

Fig. 67.7 Illustration of the search direction governed bythe reference point

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1320 Part F Swarm Intelligence

a reference point may be ideally placed in any fea-sible or infeasible region, as is shown in Fig. 67.7.It is, therefore, the consensus that the reference pointdraws on the experience of the designer to expressthe preferred compromise, rather than specific targetvalues or goals. Similarly, the reference point dis-tance metric ranks or assesses the success of a particleas one single scalar, instead of an array of objectivevalues.

Defining the Preferred RegionAs is demonstrated in Fig. 67.7, if there is no con-trol over the solution spread the swarm will explorethe preferred search direction and converge to the sin-gle solution z0 as dictated by the reference point Nz.The advantage of maintaining a population of parti-cles provides the designer the possibility to explorea range of interesting alternatives within a preferredregion of the Pareto front. The aim is, therefore, to iden-tify a set of solutions surrounding the intersection pointz0. A threshold parameter ı > 0 is defined, such thata solution x is within the preferred region if the follow-ing conditional statement is true

dz.x/ � dz.z0/C ı : (67.6)

Figure 67.9 illustrates the preferred region for a bi-objective problem. The extent of the solution spread isproportional to ı and evidently as ı ! 0, the designeris interested in determining only the most preferredsolution z0. Conversely, as ı !1, the designer is in-terested in determining all solutions along the Paretofront, and thus the influence of the reference point loca-tion diminishes.

dz(i + 1)

d z(i

– 1

)

d z(i

)

i

i + 1

i – 1

z–

f1

f2

Fig. 67.8 Illustration of the reference point distance forsolutions with equal Euclidean distance

67.3.2 User-Preference Multiobjective PSO:UPMOPSO

The proposed algorithm combines the searching pro-ficiency of PSO and the guidance of the referencepoint method. The swarm is guided by the user-definedreference point to confine its search to focus exclu-sively on the identified preferred region of the Paretofront. While the concept of the reference point is fairlyintuitive, ensuring that the swarm is guided by thisinformation to identify preferred solutions is more am-biguous. The algorithm function is consolidated inAlgorithm 67.1 and further described in the subsequentsteps.

Algorithm 67.1 The UPMOPSO algorithm1: OBTAIN user-defined preferences2: INITIALIZE swarm3: EVALUATE fitness and distance metric4: ASSIGN personal best

z'

∀dz ≤ dz(z' ) + δ1

a)

z–

f1

f2

z'

∀dz ≤ dz(z' ) + δ2

b)

z–

f1

f2

Fig. 67.9a,b Definition of the preferred region via the pa-rameter ı. (a) ı1 D 0:01, (b) ı2 D 0:001

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5: CONSTRUCT archive6: tD 17: repeat8: SELECT global leaders9: UPDATE particle velocity10: UPDATE particle position11: ADJUST boundary violation12: EVALUATE fitness and distance metric13: UPDATE personal best14: UPDATE archive15: tD tC 116: until tD tmax OR fmax

OBTAIN User-Defined PreferencesThe designer stipulates the reference point Nz and thecorresponding solution spread ı to define the loca-tion and extent of the preferred region. For airfoildesign applications, designers can exploit the exist-ing domain knowledge to determine the most feasi-ble performance compromise for the desired operatingconditions.

INITIALIZE the ParticlesA swarm of N particles is required to navigate the de-sign space S bounded by xmin and xmax. To safeguardagainst magnitude and scaling issues, all variables arenormalized into the unit cube, such that SD Œ0; 1�n. Thei-th particle in the swarm is characterized by the n-dimensional vectors xi and vi, which are the particleposition and velocity, respectively. These vectors arerandomly initialized within the unit cube at time tD0. The particle personal best position is recorded asthe particle position, such that pi D xi. The particlesare then evaluated with the objective functions and fit-ness is assigned. The reference point distance metric iscomputed for each particle to measure the individualpreference value.

UPDATE Archive and SELECT Global LeadersA secondary population of non-dominated solutionsin the form of an elitist archive is maintained attime, t. The non-dominated solutions identified bythe particles are appended to the archive. A non-dominated sorting procedure is applied, where allmembers pertaining to local inferior fronts are omit-ted. The archive serves as a mutually accessiblememory bank for the particles of the swarm. Eachmember is a potential candidate for global leader-ship of the particles during the subsequent velocityupdate.

Defining the global leaders ultimately governs thedirection of the search. The swarm should efficientlynavigate the design space such that the search effort islocally focused within the preferred region and providesa uniform spread of solutions. Since all members of thearchive are mutually non-dominated, a ranking proce-dure is necessary to distinguish the most appropriatecandidates for leadership from the remaining members.At each time step t, the most preferred solution z0.t/is recorded. The subset of members Xg.t/ selected forglobal leadership satisfy the condition of (67.6), suchthat

Xg.t/ 2 dz.Q.t//� dz.z0.t//C ı : (67.7)

Since not every member will initially satisfy this condi-tion, the number of candidate leaders may fluctuate overtime. This condition provides the necessary selectionpressure for particles to locally focus the search effortwithin the preferred region, avoiding the unnecessarycomputational effort of exploring undesired regions ofthe design space. Each swarm particle is randomly as-signed a leader to promote diversity in the search. In thecase where all non-dominated solutions satisfy the con-dition of (67.7), additional guidance through a crowdingdistance metric (as described in [67.27]) is provided topromote a uniform spread.

As the particles are guided to converge to the pre-ferred region, the number of identified non-dominatedsolutions will steadily increase. To avoid this numberescalating unnecessarily and to maintain high com-petitiveness within the archive, there is a restriction(denoted by Kmax) on the number of solutions permit-ted for entry. If the number of members K > Kmax,the newest solution is permitted entry, and the exist-ing least preferred member is removed. If all archivemembers exist within the preferred region, the mostcrowded solutions are removed. This ensures that solu-tions in densely populated regions are removed in favorof solutions which exploit sparsely populated regions,to further promote a uniform spread.

UPDATE Particle PositionThe update equations of PSO adjust the position ofthe i-th particle from time t to tC 1. In this algo-rithm, the constriction type 1 framework of Clerc andKennedy [67.50] is adopted. In their studies, the authorsstudied particle behavior from an eigenvalue analysis ofswarm dynamics. The velocity update of the i-th par-ticle is a function of acceleration components to boththe personal best position, pi, and the global best po-

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1322 Part F Swarm Intelligence

sition, pg. The updated velocity vector is given by theexpression,

vi.tC 1/D �fvi.t/CR1Œ0; '1�˝ .pi.t/� xi.t//

CR2Œ0; '2�˝ .pg.t/� xi.t//g : (67.8)

The velocity update of (67.8) is quite complex and iscomposed of many quantities that affect certain searchcharacteristics. The previous velocity vi.t/ serves asa memory of the previous flight direction and preventsthe particle from drastically changing direction and isreferred to as the inertia component. The cognitive com-ponent of the update equation .pi.t/� xi.t// quantifiesthe performance of the i-th particle relative to past per-formances. The effect of this term is that particles aredrawn back to their own best positions, which resem-bles the tendency of individuals to return to situationswhere they experienced most success. The social com-ponent .pg.t/� xi.t// quantifies the performance of thei-th particle relative to the global (or neighborhood) bestposition. This resembles the tendency of individuals toemulate the success of others.

The two functions R1Œ0; '1� and R2Œ0; '2� returna vector of uniform random numbers in the range Œ0; '1�and Œ0; '2�, respectively. The constants '1 and '2 areequal to '=2 where ' D 4:1. This randomly affects themagnitude of both the social and cognitive components.The constriction factor � applies a dampening effect asto how far the particle explores within the search spaceand is given by

�D 2=j2� ' �p'2 � 4'j : (67.9)

Once the particle velocity is calculated, the particle isdisplaced by adding the velocity vector (over the unittime step) to the current position,

xi.tC 1/D xi.t/C vi.tC 1/ : (67.10)

Particle flight should ideally be confined to the feasibledesign space. However, it may occur during flight thata particle involuntarily violates the boundaries of thedesign space. While it is suggested that particles whichleave the confines of the design space should simplybe ignored [67.51], the violated dimension is restrictedsuch that the particle remains within the feasible designspace without affecting the flight trajectory.

UPDATE Personal BestThe ambiguity in updating the personal best using thedominance criteria lies in the treatment of the case

when the personal best solution pi.t/ is mutually non-dominated with the solution xi.tC 1/. The introductionof the reference point distance metric elegantly dealswith this ambiguity. If the particle position xi.tC 1/ ispreferred to the existing personal best pi.t/, then thepersonal best is replaced. Otherwise the personal bestis remained unchanged.

67.3.3 Kriging Modeling

Airfoil design optimization problems benefit from theconstruction of inexpensive surrogate models that em-ulate the response of exact functions. This sectionpresents a novel development in the field of preference-based optimization. Adaptive Kriging models are in-corporated within the swarm framework to efficientlynavigate design spaces restricted by a computationalbudget. The successful integration of these design toolsis facilitated through the reference point distance met-ric, which provides an intuitive criterion to update theKriging models during the search.

In most engineering problems, to construct a glob-ally accurate surrogate of the original objective land-scape is improbable due to the weakly correlated designspace. It is more common to locally update the predic-tion accuracy of the surrogate as the search progressestowards promising areas of the design space [67.40].For this purpose, the Kriging method has receivedmuch interest, because it inherently considers confi-dence intervals of the predicted outputs. For a completederivation of the Kriging method, readers are encour-aged to follow the work of Jones [67.41] and Forresteret al. [67.40]. We provide a very brief introduction tothe ordinary Kriging method, which expresses the un-known function y.x/ as,

y.x/D ˇC z.x/ ; (67.11)

where xD Œx1; : : : ; xn� is the data location, ˇ is a con-stant global mean value, and z.x/ represents a localdeviation at the data location x based on a stochasticprocess with zero-mean and variance �2 following theGaussian distribution. The approximation Oy.x/ is ob-tained from

Oy.x/D O C rTR�1.Y� 1 O/ ; (67.12)

where O is the approximation of ˇ, R is the correla-tion matrix, r is the correlation vector, Y is the trainingdataset of N observed samples at location X, and 1 isa column vector of N elements of 1. The correlation

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matrix is a modification of the Gaussian basis function,

R.xi; xj/D exp

nXkD1

�kjxik � xjkj2!

; (67.13)

where �k > 0 is the k-th element of the correlation pa-rameter � . Following the work of Jones [67.41], thecorrelation parameter � (and hence the approximationsO and O�2) are estimated by maximizing the concen-trated ln-likelihood of the dataset Y, which is an n-variable single-objective optimization problem, solvedusing a pattern search method. The accuracy of the pre-diction Oy at the unobserved location x depends on thecorrelation distance with sample points X. The closerthe location of x to the sample points, the more confi-dence in the prediction Oy.x/. The measure of uncertaintyin the prediction is estimated as

Os2.x/D O�2

1� rTR�1rC .1� 1TR�1r/2

1TR�11

�(67.14)

if x� X, it is observed from (67.14) that Os.x/ reducesto zero.

67.3.4 Reference Point Screening Criterion

Training a Kriging model from a training dataset is timeconsuming and is of the order O.N3/. Stratified sam-pling using a maximin Latin hypercube (LHS [67.52]is used to construct a global Kriging approximationŒX; Y�. The non-dominated subset of Y is then storedwithin the elitist archive. This ensures that candidatesfor global leadership have been precisely evaluated (orwith negligible prediction error) and, therefore, offer nofalse guidance to other particles. Adopting the conceptof individual-based control [67.42], Kriging predictionsare then used to pre-screen each candidate particle after

the population update (or after mutation) and subse-quently flag them for precise evaluation or rejection.The Kriging model estimates a lower-confidence boundfor the objective array as

fOf1.x/; : : : ; Ofm.x/glb D ŒfOy1.x/�!Os1.x/g ; : : : ; fOym.x/�!Osm.x/g� ; (67.15)

where ! D 2 provides a 97% probability for Oflb.x/ tobe the lower bound value of Of .x/. An approximation tothe reference point distance, Odz.x/, can thus be obtainedusing (67.5). This value, whilst providing a means ofranking each solution as a single scalar, also gives anestimate to the improvement that is expected from thesolution. At time t, the archive member with the highestranking according to (67.5) is recorded as dmin. The can-didate x may then be accepted for precise evaluation,and subsequent admission into the archive if Odz.x/ <dmin. Particles will thus be attracted towards the areas ofthe design space that provide the greatest resemblanceto Nz, and the direction of the search will remain consis-tent.

As the search begins in the explorative phase and theprediction accuracy of the surrogatemodel(s) is low, de-pending on the deceptivity of the objective landscape(s)there will initially be a large percentage of the swarmthat is flagged for precise evaluation. Subsequently,as the particles begin to identify the preferred regionand the prediction accuracy of the surrogate model(s)gradually increases, the screening criterion becomesincreasingly difficult to satisfy, thereby reducing thenumber of flagged particles at each time step. To re-strict saturation of the dataset used to train the Krigingmodels, a limit is imposed of N D 200 sample pointswhere lowest ranked solutions according to (67.5) areremoved.

67.4 Case Study: Airfoil Shape Optimization

The parameterization method and transonic flow solverdescribed in the preceding section are now integratedwithin the Kriging-assisted UPMOPSO algorithm foran efficient airfoil design framework. The framework isapplied to the re-design of the NASA-SC(2)0410 airfoilfor robust aerodynamic performance. A three-objectiveconstrained optimization problem is formulated, withf1 D Cd and f2 D�Cm for M D 0:79, Cl D 0:4, andf3 D @Cd=@M for the design range M D Œ0:79; 0:82�,Cl D 0:4. The lift constraint is satisfied internally within

the solver, by allowing Fluent to determine the an-gle of incidence required. A constraint is imposed onthe allowable thickness, which is defined through theparameter ranges (see Table 67.2) as approximately9:75% chord. The reference point is logically selectedas the NASA-SC(2)0410, in an attempt to improve onthe performance characteristics of the airfoil, whilst stillmaintaining a similar level of compromise between thedesign objectives. The solution variance is controlledby ı D 5� 10�3.

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1324 Part F Swarm Intelligence

The design application is segregated into threephases: pre-optimization and variable screening; opti-mization and; post-optimization and trade-off screen-ing.

67.4.1 Pre-Optimizationand Variable Screening

Global Kriging models are constructed for the aero-dynamic coefficients from a stratified sample of N D100 design points based on a Latin hypercube de-sign. This sampling plan size is considered suffi-cient in order to obtain sufficient confidence in theresults of the subsequent design variable screeninganalysis. Whilst a larger sampling plan is essentialto obtain fairly accurate correlation, the interest hereis to quantify the elementary effect of each vari-able to the objective landscapes. The global Krigingmodels are initially trained via cross-validation. Thecross-validation curves for the Kriging models areillustrated in Fig. 67.10. The subscripts to the aero-dynamic coefficients refer to the respective angle ofincidence.

It is observed in Fig. 67.10 that the Kriging mod-els constructed for the aerodynamic coefficients are ableto reproduce the training samples with sufficient confi-dence, recording error margin values between 2 to 4%.It is hence concluded that the Kriging method is veryadept at modeling complex landscapes represented bya limited number of precise observations.

To investigate the elementary effect of each de-sign variable on the metamodeled objective landscapes,we present a quantitative design space visualizationtechnique. A popular method for designing prelimi-nary experiments for design space visualization is thescreening method developed byMorris [67.53]. This al-gorithm calculates the elementary effect of a variable xiand establishes its correlation with the objective space fas:

a) Negligibleb) Linear and additivec) Nonlineard) Nonlinear and/or involved in interactions with xj.

Table 67.2 NASA-SC(2)0410 airfoil results for the formu-lated objectives

Airfoil Machnumber,M

f1 f2 f3

NASA-SC(2)0410 0.79 0.008708 0.1024 0.189625

0.005 0.015 0.025 0.035 0.045

a) Drag coefficient Cd79

Drag coefficient Cd79

0.045

0.035

0.025

0.015

0.005

0 0.05 0.1 0.15 0.2 0.25

b) Moment coefficient Cm79

Moment coefficient Cm79

0.25

0.2

0.15

0.1

0.05

0

0 0.02 0.04 0.06 0.08 0.1

c) Drag coefficient Cd82

Drag coefficient Cd82

0.1

0.08

0.06

0.04

0.02

0

Fig. 67.10a–c Cross-validation curves for the constructedKriging models. (a) Training sample for Cd at M D 0:79.(b) Training sample for Cm at M D 0:79. (c) Training sam-ple for Cd at M D 0:82

In plain terminology, the Morris algorithm mea-sures the sensitivity of the i-th variable to the objectivelandscape f . For a detailed discussion on the Morris al-

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a) b)

c)

rLE

αTE

�TE

xUP

zUP

zxxUP

xLO

zLO

zxxLO

ΔαTE

d)

Fig. 67.11a–d Variable influenceon aerodynamic coefficients (sub-scripts refer to the operating Machnumber). (a) Drag Cd79 . (b)MomentCm79 . (c) Drag Cd82 . (d) dz

gorithm the reader is referred to Forrester et al. [67.40]and Campolongo et al. [67.54]. Presented here are theresults of the variable screening analysis using the Mor-ris algorithm for the proposed design application.

Figure 67.11 graphically shows the results obtainedfrom the design variable screening study. It is immedi-ately observed that the upper thickness coordinates havea relatively large influence on the drag coefficient forboth design conditions. At higher Mach numbers theeffect of the lower surface curvature zxxLO is also sig-nificant. It is demonstrated, however, that the variableszxxLO and ˛TE have the largest effect on the momentcoefficient – variables which directly influence the aftcamber (and hence the aft camber) on the airfoil. Thesevariables will no doubt shift the loading on the airfoilforward and aft, resulting in highly fluctuating momentvalues.

Similar deductions can be made by examining thevariable influence on dz shown in Fig. 67.11d. Thevariable influence on dz is case specific and entirely de-pendent on the reference point chosen for the proposedoptimization study. Since the value of dz is a meansof ranking the success of a multiobjective solution asone single scalar, variables may be ranked by influ-ence, which is otherwise not possible when consideringa multiobjective array. Preliminary conclusions to thepriority weighting of the objectives to the referencepoint compromise can also be made. It is observed thatthe variable influence on dz most closely resembles theplots of the drag coefficients Cd79 and Cd82 , suggesting

that the moment coefficient is of least priority for thepreferred compromise. It is interesting to see that thetrailing edge modification variable˛TE is of particularimportance for all design coefficients, which validatesits inclusion in the subsequent optimization study.

67.4.2 Optimization Results

A swarm population of Ns D 100 particles is flown tosolve the optimization problem. The objective space isnormalized for the computation of the reference pointdistance by fmax�fmin. Instead of specifying a maximumnumber of time steps, a computational budget of 250evaluations is imposed. A stratified sample of N D 100design points using an LHS methodology was used toconstruct the initial global Kriging approximations foreach objective. A further 150 precise updates were per-formed over t� 100 time steps until the computationalbudget was breached. As is shown in Fig. 67.12a, thelargest number of update points was recorded duringthe initial explorative phase. As the preferred regionbecomes populated and Os! 0, the algorithm triggersexploitation, and the number of update points steadilyreduces.

The UPMOPSO algorithm has proven to be verycapable for this specific problem. Figure 67.12b fea-tures the progress of the highest ranked solution (i. e.,dmin) as the number of precise evaluations increase. Thereference point criterion is shown to be proficient in fil-tering out poorer solutions during exploration, since it

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1326 Part F Swarm Intelligence

0 10080604020

a)

Time steps (t)

14

12

10

8

6

4

2

0

100 250200150

b)

No. of precise evaluations

0.4

0.3

0.2

0.1

0

–0.1

Fig. 67.12a,b UPMOPSO performance for transonic air-foil shape optimization. (a) History of precise updates.(b) Progress of most preferred solution

is only required to reach 50 update evaluations within15% of dmin, and to reach a further 50 evaluations within3%. Furthermore, no needless evaluations as a result ofthe lower-bound prediction are performed during theexploitation phase. This conclusion is further comple-mented by Fig. 67.13a, as a distinct attraction to thepreferred region is clearly visible. A total of 30 non-dominated solutions were identified in the preferredregion, which are shown in Fig. 67.13b.

67.4.3 Post-Optimizationand Trade-Off Visualization

The reference point distance also provides a feasi-ble means of selecting the most appropriate solutions.For example, solutions may be ranked according tohow well they represent the reference point compro-mise. To illustrate this concept, self-organizing maps

0.2

0.1

0

f3

f1

f2

a)

1

0.5

0

0.03

0.02

0.01

0.1

0.12

0.08

0.06

f3

f1 × 10–3

f2

b)

3

0.2

0.1

0

–0.112

10

8

6

Fig. 67.13a,b Precise evaluations performed and the re-sulting non-dominated solutions. (a) Scatter plot of allprecise evaluations. (b) Preferred region 250 evaluations

(SOMs) [67.44] are utilized to visualize the interac-tion of the objectives with the reference point com-promise. Clustering SOM techniques are based ona technique of unsupervised artificial neural networksthat can classify, organize, and visualize large sets ofdata from a high to low-dimensional space [67.45].A neuron used in this SOM analysis is associatedwith the weighted vector of m inputs. Each neuron isconnected to its adjacent neurons by a neighborhoodrelation and forms a two-dimensional hexagonal topol-ogy [67.45]. The SOM learning algorithm will attemptto increase the correlation between neighboring neuronsto provide a global representation of all solutions andtheir corresponding resemblance to the reference pointcompromise.

Each input objective acts as a neuron to the SOM.The corresponding output measures the reference pointdistance (i. e., the resemblance to the reference pointcompromise). A two-dimensional representation of thedata is presented in Fig. 67.14, organized by six SOM-ward clusters. Solutions that yield negative dz values

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Table 67.3 Preferred airfoil objective values with measureof improvement

Airfoil f1 f2 f3NASA-SC(2)0410 0.008708 0.1024 0.189625Preferred design 0.008106 0.0933 0.168809% Improvement 6.9 8.8 10.9

indicate success in the improvement over each aspi-ration value. Solutions with positive dz values do notsurpass each aspiration value but provide significantimprovement in at least one other objective. Each ofthe node values represent one possible Pareto-optimalsolution that the designer may select. The SOM chartcolored by dz is a measure of how far a solution deviatesfrom the preferred compromise. However, the conceptof the preferred region ensures that only solutions thatslightly deviate from the compromise dictated by Nz areidentified. Following the SOM charts, it is possible to

NASA-SC(2)0410Preferred airfoil

NASA-SC(2)0410Preferred airfoil

0 0.2 0.4 0.6 0.8 1

a) z-coordinate

x-coordinate

0.08

0.06

0.04

0.02

0

–0.02

–0.04

–0.06

0 0.2 0.4 0.6 0.8 1

b) Pressure coefficient, Cp

x-coordinate

–1

–0.5

0

0.5

1

1.5

0.008 0.009 0.01 0.011 0.012 0.013

a)

0.004 0.006 0.008 0.009 0.011 0.013

b)

0 0.002 0.004 0.006 0.008 0.01

c)

–0.001 0 0.001 0.002 0.003 0.004

d)

Fig. 67.14 SOM charts to visualize optimal trade-offs between thedesign objectives. (a) f1, (b) f2, (c) f3, (d) dz

visualize the preferred compromise between the designobjectives that is obtained. The chart of dz closely fol-lows the f1 chart, which suggests that this objectivehas the highest priority. If the designer were slightlymore inclined towards another specific design objec-tive, then solutions that perhaps place more emphasison the other objectives should be considered. In thisstudy, the most preferred solution is ideally selected asthe highest ranked solution according to (67.5).

67.4.4 Final Designs

Table 67.3 shows the objective comparisons with theNASA-SC(2)0410. Of interest to note is that the mostactive objective is f1, since the solution which providesthe minimum dz values also provides the minimum f1value. This implies that the reference point was sit-

Fig. 67.15a,b The most preferred solution observed bythe UPMOPSO algorithm. (a) Preferred airfoil geometry.(b) Cp distributions for M D 0:79, Cl D 0:4 J

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1328 Part F Swarm Intelligence

a) b)

Fig. 67.16a,b Pressure contours for design condition ofM D 0:79, Cl D 0:4. (a) NASA-SC(2)0410, (b) Preferred airfoil

NASA-SC(2)0410Preferred airfoilMost robust solution

0.77 0.78 0.79 0.8 0.81 0.82 0.83

Drag coefficient, Cd

Design mach number, M

0.02

0.018

0.016

0.014

0.012

0.01

0.008

Fig. 67.17 Drag rise curves for Cl D 0:4

uated near the f1 Pareto boundary. Of the identifiedset of Pareto-optimal solutions, the largest improve-ments obtained in objectives f2 and f3 are 36:4 and91:6%, respectively, over the reference point. The pre-ferred airfoil geometry is shown in Fig. 67.15a incomparison with the NASA-SC(2)0410. The preferredairfoil has a thickness of 9:76% chord and main-tains a moderate curvature over the upper surface.A relatively small aft curvature is used to gener-

ate the required lift, whilst reducing the pitchingmoment.

Performance comparisons between the NASA-SC(2)0410 and the preferred airfoil at the design condi-tion of M D 0:79 can be made from the static pressurecontour output in Fig. 67.16, and the surface pressuredistribution of Fig. 67.15b. The reduction in Cd is at-tributed to the significantly weaker shock that appearsslightly upstream of the supercritical shock position.The reduction in the pitching moment is clearly visiblefrom the reduced aft loading. Along with the improve-ment at the required design condition, the preferredairfoil exhibits a lower drag rise by comparison, as isshown in Fig. 67.17. There is a notable decline in thedrag rise at the design condition of M D 0:79, and thedrag is recorded as lower than the NASA-SC(2)0410,even beyond the design range. Also visible is the solu-tion that provides the most robust design (i. e., min f3).The most robust design is clearly not obtained at theexpense of poor performance at the design condition,due to the compromising influence of Nz. If the designerwere interested in obtaining further alternative solutionswhich provide greater improvement in either objective,it would be sufficient to re-commence (at the currenttime step) the optimization process with a larger valueof ı, or by relaxing one or more of the aspiration val-ues, Nzi.

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67.5 ConclusionIn this chapter, an optimization framework has beenintroduced and applied to the aerodynamic design oftransonic airfoils. A surrogate-driven multiobjectiveparticle swarm optimization algorithm is applied to nav-igate the design space to identify and exploit preferredregions of the Pareto frontier. The integration of allcomponents of the optimization framework is entirelyachieved through the use of a reference point distancemetric which provides a scalar measure of the preferredinterests of the designer. This effectively allows for thescale of the design space to be reduced, confining it tothe interests reflected by the designer.

The developmental effort that is reported on hereis to reduce the often prohibitive computational costof multiobjective optimization to the level of prac-tical affordability in computational aerodynamic de-sign. A concise parameterization model was consid-ered to perform the necessary shape modifications inconjunction with a Reynolds-averaged Navier–Stokesflow solver. Kriging models were constructed basedon a stratified sample of the design space. A pre-optimization visualization tool was then applied to

screen variable elementary influence and quantify itsrelative influence to the preferred interests of the de-signer. Initial design drivers were easily identified andan insight to the optimization landscape was obtained.Optimization was achieved by driving a surrogate-assisted particle swarm towards a sector of specialinterest on the Pareto front, which is shown to be aneffective and efficient mechanism. It is observed thatthere is a distinct attraction towards the preferred regiondictated by the reference point, which implies that thereference point criterion is adept at filtering out solu-tions that will disrupt or deviate from the optimal searchpath.

Non-dominated solutions that provide significantimprovement over the reference geometry were iden-tified within the computational budget imposed andare clearly reflective of the preferred interest. A post-optimization data mining tool was finally applied tofacilitate a qualitative trade-off visualization study. Thisanalysis provides an insight into the relative priority ofeach objective and their influence on the preferred com-promise.

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