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International Journal of Computational Fluid Dynamics

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Surrogate modelling for characterising theperformance of a dielectric barrier dischargeplasma actuator

Young-Chang Cho , Balaji Jayaraman , Felipe A.C. Viana , Raphael T. Haftka &Wei Shyy

To cite this article: Young-Chang Cho , Balaji Jayaraman , Felipe A.C. Viana , Raphael T. Haftka& Wei Shyy (2010) Surrogate modelling for characterising the performance of a dielectric barrierdischarge plasma actuator, International Journal of Computational Fluid Dynamics, 24:7, 281-301,DOI: 10.1080/10618562.2010.521129

To link to this article: https://doi.org/10.1080/10618562.2010.521129

Published online: 24 Nov 2010.

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Surrogate modelling for characterising the performance of a dielectric barrier discharge plasma

actuator

Young-Chang Choa*, Balaji Jayaramanb, Felipe A.C. Vianac, Raphael T. Haftkac and Wei Shyya

aDepartment of Aerospace Engineering, University of Michigan, Ann Arbor, USA; bT-3 Fluid Dynamics Group, Los AlamosNational Laboratory, Los Alamos, USA; cDepartment of Mechanical and Aerospace Engineering, University of Florida,

Gainesville, USA

(Received 8 November 2008; final version received 22 August 2010)

The dielectric barrier discharge (DBD) plasma actuator offers promising opportunities for flow control because of itsfast response and non-moving parts. In this work, surrogate modelling is adopted to better understand the impact ofthe materials and operational parameters on the actuator performance, and to provide an efficient approach forperformance estimation. The DBD model based on 2-species helium chemistry engages three design variables:operating frequency, polarity (positive/negative) time ratio of the applied voltage and the dielectric constant of theinsulator. Two objectives are identified: the net force generated and the power requirement. Multiple surrogatemodels are used, which identify two branches of the Pareto front with opposite net force direction and substantiallydifferent parametric sensitivities. Global sensitivity analysis indicates that the voltage frequency and polarity ratioare important in different regions of the design space, while the dielectric constant is always important.

Keywords: dielectric barrier discharge plasma actuator; dielectric constant; polarity time ratio; frequency of theapplied voltage; surrogate modelling; Pareto front; global sensitivity analysis

Nomenclature

D Diffusivity of charged particlesde Gap distance between electrodesE Electric field vectorFx Instantaneous x-directional forceFx,S Domain-integrated x-directional

forceFx,ST Domain-integrated and time-

averaged x-directional forcehd Dielectric material thicknesslel Length of lower electrodeleu Length of upper electrodePT Power input per one periodq Charge of one species particlerf Positive to negative time ratio of the

applied waveformr Recombination rate of couples of

particlesS Area of the computational domainSie Creation rate of couples of particlesT Period of the applied voltageu ¼ (ux,uy) Cell-averaged velocity of charged

particlesn Particle number densityVapp Applied voltagee0 Permittivity of vacuumegas Relative permittivity of gas

ed Relative permittivity of the dielectricmaterial

m Mobility of charged particles

Subscripts

e Electroni Ionp Particle species

Acronyms

DBD Dielectric barrier dischargeDOE Design of experimentsFCCD Face-centred composite designGSA Global sensitivity analysisKRG Kriging modelLHS Latin-hypercube samplingPRESS Square root of predicted residual

sum of squaresPRS Polynomial response surface modelRBNN Radial basis neural network modelWAS Parameter-based average surrogate

1. Introduction

The dielectric-barrier discharge (DBD) plasma actua-tor is a flow control device that is comprised of two

*Corresponding author. Email: yccho@umich.edu

International Journal of Computational Fluid Dynamics

Vol. 24, No. 7, August 2010, 281–301

ISSN 1061-8562 print/ISSN 1029-0257 online

� 2010 Taylor & Francis

DOI: 10.1080/10618562.2010.521129

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asymmetrically placed electrodes separated by a di-electric barrier (insulator) and driven by the kilohertzradio frequency AC or pulses with kilo-volt amplitudeas shown in Figure 1. It has been known that thedischarge generates a weakly ionised gas, and chargedparticles influenced by the electric field can delivermomentum to the neutral particles (McDaniel 1964,Mitchner and Kruger 1973, Boeuf 1987). Although theelectric field reverses polarity between the two halfcycles, the resultant neutral flow is an unidirectionalwall-jet-type flow due to the asymmetric geometry andcharged particle dynamics (Roth and Sherman 1998,Shyy et al. 2002, Jayaraman et al. 2008). On the otherhand, the disparity between the positive-going uniformdischarge and the negative-going non-uniform one isregarded as a key factor affecting the efficiency ofmomentum coupling by some researchers (Enloe et al.2004). This is also explained by the stronger filamen-tary discharge during the positive half cycle of theapplied sine wave compared to the negative (Pons et al.2005).

While the DBD-based actuator exhibits potentialas a control device, the performance characteristicssuch as generated force and power consumptiondepend on the type of discharge, applied voltage andmaterial of the insulator. There are several operatingmodes such as streamer or filamentary, glow andcoupled (Gherardi and Massines 2001) in the dielectricbarrier discharge, and a homogeneous low-tempera-ture discharge at atmospheric pressure is of greatinterest among researchers because of its energyefficient non-thermal ionisation (Roth 2001) andversatility of application (Yuan et al. 2006).

In spite of the inherent advantages (no movingparts and vast control potential) of the DBD actuator,little insight is available regarding efficient operatingconditions to accommodate various performanceneeds. With relatively high magnitudes of appliedvoltage (15 kV at 5–10 kHz AC) required for auniform discharge, the force generated by a single

actuator is less than 10 mN/m in air (for example,Porter et al. 2007). The flow velocity induced by asingle actuator is usually less than 5 m/s for pulsedinput voltage (Jukes et al. 2006) as well as sinusoidalvoltage waveform (Forte et al. 2006). In their efforts tounderstand the operating mechanism of the actuator,various researchers have conducted parametric studiesto identify trends and possible optimal conditions. Theeffect of the applied voltage such as waveform (Abeet al. 2007), frequency and amplitude (Forte et al.2006, Roth and Dai 2006, Abe et al. 2007) have beenreported by several groups. The impact of actuatorgeometry and dielectric constant are also assessed byForte et al. (2006) and Roth and Dai (2006),respectively. Gregory et al. (2007) report the inducedvelocity and/or power efficiency with the variation inneutral gas pressure.

Although the key plasma-based phenomena seemto have been captured, the experimental analysis isbounded by the difficulty of measurement, making thenumerical approach crucial to understanding themechanism and physics of the DBD actuator. Re-cently, the numerical approaches have been improvedto simulate the complex actuator discharge mechanismwith qualitative comparison to the experiments. Boeufet al. (2007) use the fluid modelling approach to studythe effect of the negative ions, which is important forair chemistry. Font et al. (2007) compare the particle-in-cell and direct simulation Monte Carlo (PIC-DSMC) and fluid plasma simulations, and analysethe asymmetry in the forward and backward cycles andspatial non-neutrality in oxygen. Likhanskii et al.(2007) demonstrate the use of positive pulses with apositive bias and report an improved performance inforce generation. Roy et al. (2007) use multiple chargedand neutral species for N2/O2 air chemistry simulation,observing the existence of decelerating force down-stream of the powered electrode, which adverselyaffects the actuator performance.

Some features of the relation between the perfor-mance measures and operating variables have beeninferred in previous studies. For example, forcegeneration or induced velocity is directly proportionalto voltage, frequency and rate of voltage change (Forteet al. 2006, Roth and Dai 2006, Baughn et al. 2006,Abe et al. 2007). Materials with larger dielectricconstant (ed ¼ 30 compared to ed ¼ 2) are observedto produce larger forces due to the increase in electricfield strength and plasma volume at the cost ofincreased power (Roth and Dai 2006). In a similarmanner, a thinner dielectric material induces higherflow velocity until the electric field is high enough toproduce filamentary discharge, which causes smallermomentum transfer to the neutral flow (Forte et al.2006). Abdoli et al. (2008) investigate the effect ofFigure 1. Dielectric barrier discharge actuator configuration.

282 Y.-C. Cho et al.

direction and effective region of plasma actuator on theefficiency of airfoil separation control by using con-stant electric field and dimensional analysis. Asreviewed by Moreau (2007), various factors such astype of discharge, number of actuators, geometry andvoltage input have been investigated by many researchgroups with the idea of enhancing the performance ofactuators in flow control applications. However, theperformance of the DBD depends on the interplaybetween material properties such as the dielectricconstant of the insulator, and the operating parameterssuch as frequency and the waveform of the appliedvoltage. Before one can optimise the DBD perfor-mance, it is desirable to gain better insight into theimpact of individual and collective influences of thesematerials and operating variables on the DBDperformance.

Previously, instead of using time-variant highfidelity discharge models, a simplified model represen-tation with linear electric field has been proposed byShyy et al. (2002) to approximate the force fieldgenerated by discharge. The approach of coupling theflow field with reduced order models using body forceshows good agreement with experimental results(Grundmann et al. 2007). The reduced order modelhas been applied to the studies of flow control inairfoils at low Reynolds numbers (Visbal et al. 2006,Jayaraman et al. 2007b), low-pressure turbines (Riz-zetta and Visbal 2007) and bluff-body flows (Rizzettaand Visbal 2008), resulting in separation elimination/delay or significant drag reduction. Corke et al. (2008)propose a lumped-element model which providesboundary conditions for the electric field equationand calculates the time variant force field.

The first-principle-based DBD model is veryexpensive to compute, often requiring an order ofmagnitude longer central processing unit (CPU) timethan that of the associated flow field simulations.However, there is a significant difference in plasma andneutral flow time scales at low Reynolds numbers(Jayaraman and Shyy 2007), which allows us toconduct the DBD simulation without having to linkit to the actual flow simulations. In the present effort,we utilise the hydrodynamic model (Jayaraman andShyy 2007) to facilitate the DBD simulations andgenerate the performance data, and construct surro-gate models (Madsen et al. 2000, Queipo et al. 2005) toestablish reduced order representation of the DBDcharacteristics. The goal is to improve the previouslyproposed linear field model (Shyy et al. 2002, Jayara-man et al. 2007b) for flow control and heat transferapplications.

Surrogate-based techniques (Queipo et al. 2005)have been widely used to support the engineeringdesign problems (Mack et al. 2005, Goel et al. 2006a).

With limited numerical simulations, surrogates enableus to generate reliable approximations to the solutionover a design space and assess the sensitivity andcorrelation among the various parameters. As re-viewed by Shyy et al. (2001), such models offer a globalperspective over the entire design space and canaddress multi-objective issues in a logical manner,accounting for benefit versus penalty of alternativescenarios in quantitative manners. The standardprocedure starts with design of experiments (DOE)which is the identification of design points in the designspace. Once simulation results are obtained for thosepoints, surrogate models can be generated to fit thedata. There are widely used models, namely poly-nomial response surface approximation (Myers andMontgomery 2002, Box et al. 2005), Kriging (Sackset al. 1989, Lophaven et al. 2002), radial basis neuralnetwork (Cheng and Titterington 1994) and weightedaverage of multiple surrogate models (also known asparameter-based surrogate model) (Goel et al. 2006b,Viana et al. 2009). Since the error due to theapproximation is affected by various factors such asthe sampling strategy of design points and the natureof physical system itself, and the cost of fittingsurrogates is usually negligible compared to that ofthe numerical simulations, it is beneficial to usemultiple surrogate models at the same time (Vianaand Haftka 2008). If the accuracy of the constructedsurrogates is not satisfactory and/or there exists aregion of interest, the design space can be refined byrepeating the whole procedure for the refined space. Adesign problem in many cases involves the optimisa-tion of multiple objectives comprised of competingfactors that can be simplified by ignoring insignificantparameters using global sensitivity analysis (Sobol1993, Queipo et al. 2005). With the aid of surrogates, atrade-off curve can be created based on the so-calledPareto front analysis (Miettinen 1999). A solution iscalled a Pareto-optimal solution, if there is no othersolution for which at least one objective has a bettervalue while values of the remaining objectives are thesame or better (Marler and Arora 2004). The optimalsolutions constitute the Pareto front, a (n-1)-dimen-sional hyperplane in n-dimensional design space,providing the information of candidates of the optimalset.

The present study focuses on understanding theeffect of three chosen parameters – waveform andfrequency of the applied voltage and dielectric constantof the insulator – on the DBD actuator performancecharacterised by power input and generated flowdirectional force using surrogate models. The mainobjective is to gain an idea of the impact of thosevariables which have significant interplay. The accu-racy of each surrogate model for this application is

International Journal of Computational Fluid Dynamics 283

also addressed and the surrogate models with reason-able accuracy are shown to contribute to successfullyrefining the design space, resolving the region ofinterest with higher accuracy.

As a first step, we simplify the problem and thecomputational cost is lowered, so that the focus is onapplying the surrogate modelling techniques to under-stand the interplay between various parameters in theDBD actuator. Specifically, the helium gas chemistry isfocused on here. Furthermore, we only investigate theeffect of the above selected parameters while fixingother variables, such as the actuator geometry, that isknown to have significant impact. Error and perfor-mance of the various surrogate models for thisapplication are also discussed in detail. Once asatisfactory modelling approach is established, onecan extend the scope to cover air chemistry as well asother geometric and operating parameters.

2. Modelling of the dielectric barrier discharge

actuator

The set of governing equations used in modelling theDBD actuator are the continuity and momentumequations derived from the Boltzmann equation andthe electric field equation from the Maxwell’s equa-tions. Since the atmospheric pressure is high enough toassume the local thermodynamic equilibrium, the fluidmodel is reasonably accurate and the local electric fielddensity (E/N) can be used to approximate thephenomena related to the collision processes – ionisa-tion/recombination, diffusion and drift – instead ofsolving the energy equation (Jayaraman et al. 2007a).Governing equations are given by Equations (1)–(3)for only two species – Heþ (subscript p ¼ i) andelectron (subscript p ¼ e) – for simplicity in this paper.Sie and r are ionisation and recombination ratecoefficients, and m and D are mobility and diffusivityof charged particles, respectively. q is electric charge ofone species particle, and e0 is permittivity of vacuum.

@np@tþr � ðnpupÞ ¼ neSie � rnine ð1Þ

npmpE�rðnpDpÞ ¼ npup ð2Þ

r � ðedEÞ ¼qini � qene

e0: ð3Þ

Equation (2) is the well-known drift-diffusion equa-tion, which is valid also for ions in high pressure(atmospheric regime) discharge. To solve this set ofequations, the source terms are handled with fourth-order backward differentiation formula and thePoisson equation with the algebraic multigrid method,and the second-order central difference and upwindmethods are employed for the diffusion and convectionterms respectively (Shyy and Sun 1993, Shyy 1994,Jayaraman and Shyy 2007). The charged particledensities and electric field are coupled by solving thePoisson equation between the predictor and correctorsteps where the first-order source splitting is used asnoted in Jayaraman et al. (2007a). The coefficients ofgaseous properties of helium regarding particle colli-sions and ionisation/production are obtained fromMcDaniel (1964), Mitchner and Kruger (1973) andWard (1962).

The computational domain with the actuatorgeometry is presented in Figure 2. The thickness ofthe insulator hd is 0.5 cm, the lengths of upper and lowerelectrodes, leu and lel respectively, are 0.2 cm, and thegap distance de is 0.2 cm. The applied voltage to theupper electrode has a sinusoidal shape with 1 kVamplitude, but the positive-to-negative half–cycle ratiorf can vary. Boundary conditions for the charge speciesat the dielectric surface are set to satisfy the currentcontinuity that allows the accumulation of particles, andonly electrons are allowed to be absorbed in the upperelectrode without the secondary emission. Gas pressureof helium is set as 300mmHg, and the ion temperature is300 K. The electron temperature is calculated as a

Figure 2. Computational domain and applied waveform.

284 Y.-C. Cho et al.

function of the local electric field strength using a localfield approximation approach, which is discussed indetail in Jayaraman and Shyy (2007).

3. Surrogate modelling for design exploration

In order to access their impact on the actuatorefficiency, three design variables – the dielectricconstant of the insulator material ed, frequency of theapplied voltage fv and positive-to-negative half cycleratio rf – are chosen among many possible parametersbecause they are among the key parameters, whichhave non-trivial effects on the resultant force byaffecting both the positive and the negative peakvalues of force evolution and asymmetry in its wave-form (Jayaraman et al. 2008). However, there are otherparameters, which are known to affect the generatedforce and power as well such as the peak value ofsinusoidal voltage and the size of electrode, and theywill be investigated in the future. The bounds for eachdesign variable are based on the choice of materialsand the range of working conditions widely adopted.The objective functions are chosen so as to representthe actuator performance, namely the average x-directional force Fx,ST in the domain and averagepower input to the electrodes PT. The former is thetime-averaged, domain-integrated Lorentzian forceacting on the charged particles, which is assumed tobe equivalent to the body force acting on the neutralgas, especially at atmospheric pressure conditions. Thelatter can be calculated by integrating the charge anddisplacement currents in the volume of the medium(Morrow and Sato 1999). In this study, however, onlythe charge current through the upper electrode isconsidered for simplicity. The definitions and para-meter ranges are presented in Table 1. The current

investigation focuses on identifying the combination ofthe design variables to maximise Fx,ST and minimisePT. However, as will be discussed later, since there arecases which result in Fx,ST less than zero, its absolutevalue is used. For the convenience of minimising bothobjective functions, the objective for the average forceis chosen as minimising 7jFx,STj.

Since the number of sampling points isrestricted by the computational cost, the DOEs needto be chosen carefully. In this study, a two-stagesurrogate modelling process will be presented. In thefirst stage, in order to assess the entire design space, theface-centred composite design (FCCD) combined withLatin-hypercube sampling (LHS) (Mckay et al. 1979)is used in the DOE. Based on the outcome of the globalinvestigation, we develop refined surrogate modelsfocusing on the region of favourable DBD perfor-mance. In this stage, since the main interests are in theinterior portion of the sub-domains, we use thedistance-based design combined with LHS. Thejustification for such approaches rests on the observa-tion that by spreading the design points towardsbounds, FCCD is efficient for second-order design in acuboid design space, and LHS provides space-fillingdata set with an even chance for each design variable.The distance-based design tends to generate evenlydistributed design points in space, thus can covercomplicated design space properly (Myers and Mon-tgomery 2002). In our approach, we first select pointsto characterise the entire design space, followed by arefinement study focusing on the regions of interest.The number of sampling points at each DOE in thisstudy is set to 20 which is double the number ofcoefficients of a second-order polynomial responsesurface representation. The refinement study will bedetailed later.

Table 1. Design variable bounds and objective functions.

Design variables Bounds

ed Dielectric constant of the insulator 2 � ed � 15fv Frequency of the applied voltage (kHz) 5 � fv � 20rf Positive-to-negative half cycle ratio 0.5 � rf � 1.5

Objective functions Definitions

7jFx,STj Time-averaged domain-summedx-directional force (mN/m)

Fx;St ¼ 1T

RT Fx;SðtÞdt

where Fx,S (t) ¼RS Fx(x, y, t) dA

Fx(x, y, t): force density (N/m3) in x-directionPT Power input for one cycle by the

charge current through theupper electrode (W)

PT ¼ 1T

RT IðtÞVappðtÞdt

where, net charge flux,IðtÞ ¼

RleuðqiniðtÞuy;iðtÞ � qeneðtÞuy;eðtÞÞdxS: gas domain area

T: period of the applied voltage

International Journal of Computational Fluid Dynamics 285

Four different surrogate models, namely second-order polynomial response surface (PRS), Kriging(KRG), radial basis neural network (RBNN), and aparameter-based average surrogate (WAS), which isthe weighted combination of the others (Goel et al.2007, 2008a, b), are adopted in this study and relevantfunctions and parameters are presented in Table 2. Theweight of each model in the WAS is set to allow highercontribution to the surrogate with the smaller squareroot of predicted residual sum of squares (PRESS)(Goel et al. 2006b) (Appendix 1).

Main and total effects of variables in the globalsensitivity analysis are computed using the Gaussianquadrature. In order to identify the appropriate designvariables capable of promoting the two objectives(maximising the force and minimising the power), thePareto optimal set is constructed by seeking a set ofpoints that are not dominated by any other points inthe objective function space. Surveying the trade-offpoints on the Pareto front is accompanied by a fewdesign space refinements based on the error statistics ofthe surrogate models in the original design space.

4. Results and discussion

4.1. DBD model assessment

The performance of a DBD actuator is known to beaffected by various factors such as the insulatormaterial, actuator geometry and applied voltage.Furthermore, the fact that most of the experimentsare based on air hinders the validation of the currentnumerical model, which adopts the simplified heliumchemistry. Although the purpose of this study is not aquantitative comparison but a preliminary parametricsurvey, it is necessary to assess the fidelity of the result.In a previous study, the one-dimensional simulation onDBD using the current model is consistent with theother numerical and experimental work that useshelium as the operating gas (Jayaraman and Shyy2007). In addition, some characteristic features ofDBD such as surface charge evolution, dependency offorce on the applied voltage and electric current timeevolution, can be compared for the purpose of aqualitative validation of the current model.

Table 2. Functions and parameters for the surrogatemodels (Appendix 1).

Surrogate model Functions and parameters

Correlation function GaussianKriging Regression model Polynomial

Initial guess of yi 15

Radial basisneural network

Spread, b 0.75Goal (0.025 6 (sample

mean))2

In Table 3, three different models from other studiesare presented for comparison with the current model.Although every model adopts a conventional DBDactuator setup, such as asymmetrically positionedelectrodes and sinusoidal waveform of applied voltage,there exist diversities in geometric dimensions, operatinggas and other parameters. Compared to other models,the current model features a thicker insulator relative tothe electrodes, as well as lower voltage, which results in acomparatively weaker electric field. Also the effect ofnegative ions are absent in the current model due to thechoice of a simplified helium chemistry.

The current model, as presented in Figure 3,predicts the power-law dependency of force generationon the amplitude of the applied voltage. In the figure,the resultant force normal to the surface, Fy of thecurrent model (Jayaraman et al. 2008) is compared tothe experimental work of Van Dyken et al. (2004).Despite the difference in species dynamics betweenhelium and air, as well as the range of applied voltage,the power of 2.7 is comparable with the experimentalresult with the power of 3.1.

Since the electric potential and the actuator dy-namics are characterised by the charge accumulation onthe dielectric surface as well as the applied voltage to theelectrodes, the surface charge evolution represents a keyaspect of the overall mechanism. In Figure 4, the surfacepotential distribution is comparedwith the experimentalresult (Enloe et al. 2008) for four different time instantsthat are evenly spaced over one period of a cycle. Theupper or exposed electrode exists at the left of zero andthe x-coordinate indicates the distance from its edge,which is limited by 6 mm for the current model. Thereare noticeable differences in geometry, frequency,amplitude of applied voltage, and operating gas betweencases summarised in Table 3. Nevertheless, qualitativeobservations between various investigations can bemade. For example, when the applied voltage is nearzero, the accumulated charges, especially electron at t/T ¼ 0 and ion t/T ¼ 0.5, show similar residual potentialdistributions. This demonstrates that the charge transfermechanismpredicted by the currentmodel is reasonable.Furthermore, despite the restricted computationaldomain, the potential evolution especially near theupper electrode edge coincides qualitatively with experi-ment throughout the cycle.

The time history of electric current is compared toUnfer et al. (2008) in Figure 5. The smaller current andabsence of the peaks can be mostly explained by theelectric current calculation which only reflects chargeflux through the upper electrode as defined in Table 1and much lower magnitude of the applied voltage inthe current study. However, there are similaritiesbetween the current time histories, especially theinstants where the maximum current occurs andthe discharge of the positive half cycle ends. Although

286 Y.-C. Cho et al.

Table 3. Parameter comparison for different models.

Comparison Geometry

Operating parameters

Chemistryed fv (Vapp)max

The current model(numerical)

5case for Figure 34 2-specieshelium: Heþ

and e74.5 5 kHz 0.25*2.7 kV

5case for Figure 444.5 20 kHz 1 kV

5case for Figure 5414.7 7.35 kHz 1 kV

Van Dyken et al. (2004),experimental

5case for Figure 34 AirKapton:typically2.9*3.9

5 kHz 1*6 kV

Enloe et al. (2008),experimental

5case for Figure 44 Air6 5 kHz 7 kV

Unfer et al. (2008),numerical

5case for Figure 54 3-species air:positive ion,negative ion,

and e7

4 8 kHz 8 kV

Figure 3. Force dependency on the magnitude of appliedvoltage.

the current model with reduced helium chemistry doesnot capture all the features shown in air applications,the characteristics of the overall discharge process and

evolution of the charged species are in qualitativeagreement with other studies.

It should be noted here, that only a qualitativecomparison is presented and hence no attempt wasmade to match the experimental parameters consis-tently at this time. The current work, however, ismeaningful as a framework to develop an efficient flowcontrol strategy and it can be directly applied to theextended air chemistry and different geometric/operat-ing conditions which are in progress as the next step.Also, the qualitative consistency of the results shedslight on the further design refinement as well ascharacterising parametric correlations in differentchemistries.

4.2. Design exploration

The first level DOE, level 0 using the combination of15 FCCD points and 5 LHS points and the simulationresults of those 20 points are presented in Figure 6.Although the sampled points are well distributed in thedesign space, the response points cluster in some parts

International Journal of Computational Fluid Dynamics 287

of the objective function space as in Figure 6b. As aresult it is hard to get an idea of the objective functiondependency on the design variables with sufficientaccuracy, but some noteworthy features can beobserved from the contours. With higher dielectricconstant of the insulator both power and magnitude offorce, jFx,STj tend to increase. In the context of thecurrent conditions, the lower frequency generally leadsto the lower power. The ratio of the first and secondhalf cycle shows more complex effects and makes ithard to deduce any tendency.

The surrogate models are obtained using thesesampled points and their PRESS errors are presentedin Table 4. Due to the insufficient number of sampledpoints and their complex response, significant PRESSerrors exist at this level especially in the force prediction.For this case the Kriging model shows the best

performance in predicting the force while the para-meter-based average model does the same for power.

In order to explore the objective function distribu-tion corresponding to the design space, a grid with 313

points evenly distributed in the whole design space isemployed and the result using the parameter-basedaverage model is chosen to be presented as Figure 7. Itcan be observed that the Pareto front is notcontinuous, and there are two distinct regions thatare marked with two windows and correspond to thehigher magnitude of force generation. Though theyboth lie in the same side of the force axis due to ouradopting the absolute values of the force, the one withhigher power corresponds to the negative (minusx-direction in Figure 2) force generation, and thelower to the positive. In terms of magnitude, thenegative force generation is larger than the positive.

Figure 4. Electric potential distribution along the dielectric surface. (a) t/T ¼ 0. (b) t/T ¼ 0.25. (c) t/T ¼ 0.5. (d) t/T ¼ 0.75.

288 Y.-C. Cho et al.

By simply changing the direction of the actuator, thiscan be used to maximise the force generation. On theother hand, the region corresponding to the positiveforce generation is accompanied by a better powerefficiency. The power ratio between those two regionsis about 5*6 compared to the force ratio of 2.

The mechanism of the force generation over thetwo half cycles has been a subject of interest withdifferent suggestions such as whether it consists of twoconsecutive positive or positive–negative alternatingpatterns (Shyy et al. 2002). It has been reported bysome researchers that the dominant positive and small

Figure 5. Electric current time evolution. (a) Current model. (b) Reproduced from Unfer et al. (2008).

Figure 6. Design of experiments and their simulated results – level 0. (a) Design of experiments: combination of FCCD (blue)and LHS (red). (b) Design variable contours with simulation result in objective function space: ed, fv and rf.

International Journal of Computational Fluid Dynamics 289

negative force generations for the first and second halfcycles respectively exist for sinusoidal voltage excita-tion (Singh and Roy 2007, Jayaraman et al. 2008) orpulse-mode operation (Porter et al. 2007). Because ofthe different charged species chemistry and otherfactors such as geometry it is difficult to compare thesolution directly. Also in the numerical models thereare other factors such as boundary treatment, domainsize and surface reaction modelling that affect theoverall force generation. In the current study, there arepositive and negative alternating contributions of forcegeneration during the two half cycles, and in a certainpart of design space the negative portion exceeds thepositive, resulting in a negative time-averaged forcegeneration.

In Figure 7, since the distribution of sampledpoints near the Pareto front is too sparse toresolve the regions of interest properly, two windowedregions are used as the constraints for design spacerefinement.

Level 1-1, low power region:70.009 � 7jFx,Stj � 70.005 [mN/m]

0 � PT � 0.02 [W]Level 1-2, high power region:

70.014 � –jFx,Stj � 70.009 [mN/m]0.05 � PT � 0.07 [W]

The design variable constraints corresponding tothese objective function constraints, namely design-space-constraints are generated based on the surro-gate models at level 0. In order to generate theconstraint surfaces, responses of a set of grid pointsuniformly distributed in the design space areobtained by using the surrogate models, and thesurfaces confining the points whose responses satisfythe objective function constraints are specified.Although the parameter-based average surrogatehas a smaller PRESS error in PT as presented inTable 4, the design space confined by its design-space-constraints is included in that of the Krigingmodel, and the refined regions are chosen conserva-tively to cover as much space as possible. Figure 8shows the iso-force and iso-power surfaces andthe design-space-constraint surfaces based on theKriging model (Goel et al. 2006a) (blue is forthe lower bounds and red the upper bounds).

Considering the lower and upper bounds of theobjectives, each level has one refined space alongwith the constraint surfaces. Since these surfaces arecontours of constant force or power, based on theirslopes it can be said that the force generation isrelatively less sensitive to the dielectric constant thanpower.

Since the design space corresponding to level 1–1and 1–2 constraint windows is an irregular shape, it isimpossible to use the DOEs for a rectangularhexahedron or sphere. For the DOEs at the refinedlevel, in order to sufficiently characterise the designspace, the LHS is utilised to generate 5000 points.Then, 20 points are selected by maximising theminimum distance between those points. The designpoints generated by this approach are also shown inFigure 8 along with the constraint surfaces.

The simulated result of design points in each regionis used to generate the surrogate models. With therefinement, model prediction accuracy is improvedboth in relative and absolute measures as shown inTables 5 and 6. It can be seen that the PRESS error forthe polynomial response surface in the refined levels ismuch improved, which means that the physical

Table 4. PRESS errors of the surrogate models – level 0.

Objective functions Kriging Polynomial response Radial basis neural network Parameter-based average surrogate

7jFx,STj 0.0020 (16)a 0.0027 (22) 0.0095 (77) 0.0028 (23)PT 0.0032 (5.0) 0.0033 (5.2) 0.0063 (9.9) 0.0023 (3.6)

a() % ¼ 100 6 PRESS/(Xmax – Xmin), X ¼ 7jFx,STj or PT in level 0.

Figure 7. Design and predicted points and Pareto front bythe parameter-based average surrogate model in level 0.

290 Y.-C. Cho et al.

complexity is adequately captured in these refinedregions.

Using the WAS in level 1–1 and PRS level 1–2which have the best PRESS values, the Pareto front isconstructed again for each data set as shown in Figure9a and b, along with the predicted points by thesurrogate models. It can be verified that the majority ofdesign points 713 in the low power region and 15 inthe high power region – reside in the constraintregions, which suggests that the prediction accuracyof surrogate models based on level 0 is satisfactory,although the projected position of each design pointmay differ by at least as much as the PRESS values.Although there are differences between the branches ofthe Pareto front at level 0 and those at level 1–1 and 1–

2, there is consistency in the shape and orientation ofthe branches.

4.3. Two distinctive regions and Pareto front

To investigate the design variable variations aroundthe Pareto front branches, surrogate models are usedto reconstruct the traces of design variables thatcorrespond to the branches. The traces in the designspace based on the surrogate models in level 0 and level1 are presented in Figure 10a and b, respectively.Compared to the prediction of the level 0 surrogatemodel, the level 1 result shows more coherent tracescorresponding to the Pareto front. In both cases, thehigh and low power branches are separated from each

Table 5. PRESS errors of the surrogate models – level 1–1.

Objective functions Kriging Polynomial responseRadial basis

neural networkParameter-basedaverage surrogate

7jFx,STj 5.261074 (9.5)a 3.761074 (6.7) 2.761074 (4.9) 3.361074 (6.0)PT 1.961074 (1.1) 1.461074 (0.80) 6.961074 (3.9) 1.261074 (0.69)

a() % ¼ 1006PRESS/(Xmax7Xmin), X ¼ 7jFx,STj or PT in level 1–1.

Table 6. PRESS errors of the surrogate models – level 1–2.

Objective functions Kriging Polynomial responseRadial basis

neural networkParameter-basedaverage surrogate

7jFx,STj 1.2 6 1074 (3.8)a 1.1 6 1074 (3.4) 3.4 6 1074 (10.6) 1.1 6 1074 (3.4)PT 2.0 6 1074 (1.1) 0.55 6 1074 (0.31) 9.4 6 1074 (5.3) 0.90 6 1074 (0.51)

a() % ¼ 100 6 PRESS/(Xmax7Xmin), X ¼ 7jFx,STj or PT in level 1–2.

Figure 8. Constraints, contours and design points for refinement in design space. (a) Level 1–1: low power region. (b) Level 1–2:high power region.

International Journal of Computational Fluid Dynamics 291

other by the whole design space in terms of the appliedvoltage frequency.

For each region the most efficient point in forcegeneration, i.e. minimum in 7jFx,STj is separated fromeach other by the whole design space although theyoccur at the same frequency of applied voltage fv. Itcan be observed that when following the Pareto frontline of the high power region, the dielectric constant edis the most influential variable, while for the one of thelow power region, both ed and rf are important.Though it may be inconvenient to vary the materialconstant such as ed to accommodate a desired objectivefunction state, if proper design variables are chosen,this type of information can be used to establish a basis

for the performance of these actuators as effective flowcontrol devices.

One can identify multiple optimal points on thedesign variable bounds – one for fv and the other for fvand rf in Figure 10b. To identify desirable performanceof the actuator, design space can be expanded.However, care needs to be taken to consider thedischarge operating mode that is known to signifi-cantly change the charge densities as well as the powerconsumption. To investigate the phenomena in theseregions in depth and compare them, two pointscorresponding to the minimum 7jFx,STj conditionare selected and the time history of the solution iscompared in Figures 11 and 12. In Figure 11a and b it

Figure 9. Design and predicted points and Pareto front in each refinement level. (a) Level 1–1: parameter-based averagesurrogate model. (b) Level 1–2: polynomial response surface.

Figure 10. Reproduced points corresponding to the Pareto front and two design points with minimum7jFx,STj in low and highpower regions. (a) Level 0. (b) Level 1–1 and 1–2.

292 Y.-C. Cho et al.

can be observed that for the case with lower frequencythat belongs to the low power region, domain-averaged ion number density is higher. This can beexplained by considering the fact that lower frequencyallows more time to generate the particles, which isconsistent with Jayaraman et al. (2008). The electronsaturation instances in these cases – about t/T ¼ 0.8 inthe low power and 0.9 in the high power – coincidewith the start of plateau or second dip in Figure 12,which is also mentioned as one of the key factorsaffecting the solution with frequency.

From the force history result, one can deduce theeffect of the ratio of first and second half cycles rf. Itis considered as an effective parameter for changingthe force history profile and thus adopted as one ofthe design variables in this study. Although positiveforce belongs to the first half cycle and negative to

the second, elongating the period of each part in theapplied voltage source does not necessarily induceincreased force either in positive or negative. Whiledecreasing rf, i.e. increasing the second half cyclecorresponds to the decreased 7jFx,STj point in thehigh power region, increasing rf does not meanincreasing the duration of positive force cycle. Thevalue of rf corresponding to the maximum forcegeneration in the positive x-direction is about 0.8 inthe low power region according to the multiplesurrogate model. The reason is that generating thepositive force is mainly related to the plateau region ofthe second half cycle in Fx,S time history as in Figure12.

This phenomenon is mainly caused by the differ-ence in the amount and evolution of the electron andthe ion clouds that reside on the dielectric surface. In

Figure 11. Domain-averaged particle number density histories for two design points near the Pareto front. (a) Low powerregion: ed ¼ 8.5, fv ¼ 5.0 and rf ¼ 1.0. (b) High power region: ed ¼ 15, fv ¼ 20 and rf ¼ 0.5.

Figure 12. x-directional force and power histories for two design points near the Pareto front. (a) Low power region: ed ¼ 8.5,fv ¼ 5.0 and rf ¼ 1.0. (b) High power region: ed ¼ 15, fv ¼ 20 and rf ¼ 0.5.

International Journal of Computational Fluid Dynamics 293

Figure 13, instantaneous contours of main physicalquantities especially around the upper electrode arepresented at the moment of t/T ¼ 0.95 when the phasechange in applied voltage is about to occur. Jayaramanet al. (2008) note that force generation mainly occursnear wall and electrode region at this instance, which iscaused by the electrons accumulated on the dielectricsurface and strong electric field near the edge ofelectrode. Ion clouds having been repelled from thesurface through the second half cycle compensatethe applied electric field as in Figure 13d, resulting inthe small magnitude of negative force generation in thelater part of the second half cycle, i.e. the plateau

region. On the other hand, in the case of higherfrequency this ion cloud is much weaker and there isstrong electric field near the upper electrode which,with the higher electron density near the wall, results inthe second negative peak at the end of the second halfcycle. In the same study it is pointed out that thedielectric constant affects the asymmetry between thefirst and the second half cycles as well as the amplitudeof generated force. These two competing effectscontribute to the average force of opposite tendencies.For example, by increasing the dielectric constant onecan not only increase the amplitude of force historyand consequently increase the time-averaged force

Figure 13. Solution contour plots at t/T ¼ 0.95 (upper: low power region, lower: high power region). (a) Fx (N/m3). (b) Ex

(V/m). (c) ne (61015 m73). (d) ni (61015 m73).

294 Y.-C. Cho et al.

generation but can also decrease the asymmetrybetween the two half cycles, thus decreasing the time-averaged force generation. As a result, for the lowpower region the efficient force generation in Figure 10occurs at ed ¼ 10.2 which is not on the edge of thedesign variable range, 2.0 � ed � 15.0. However, forthe high power region it occurs at the maximum value,15.0.

According to the present result, the average forcegeneration has more monotonic dependency on thefrequency of applied voltage – at least around thePareto front; for the high power region the higherfrequency induces the larger force, and for the lowerpower region the opposite occurs. But this is becausewe are dealing with the magnitude of the average forcegeneration while ignoring its orientation. As men-tioned earlier the low power region corresponds to

positive force generation, and from the previous study(Jayaraman et al. 2008) it has been also found that thehigher force can be obtained with a lower frequencyfor a parameter set belonging to the low power region.

4.4. Global sensitivity analysis and dependency onparameters

Figure 14 shows the result of the variance-based, non-parametric global sensitivity analysis (Appendix 2) foreach refined level. Compared to the level 1–2, level 1–1shows a stronger parameter correlation, which can beidentified from the difference between total and mainsensitivities. Also, the frequency of applied voltage fvin level 1–1 has a significant effect on both theaverage force and power. On the other hand, inthe high power region (level 1–2) the effect of the

Figure 14. Global sensitivity analysis result. (a) Fx,ST, level 1–1. (b) PT, level 1–1. (c) Fx,ST, level 1–2. (d) PT, level 1–2.

International Journal of Computational Fluid Dynamics 295

positive-to-negative time ratio rf is prominent com-pared to that of fv while the insulator dielectricconstant is always important.

To see the local dependency of the objectivefunctions on each parameter while keeping othersconstant, variations are compared for one parameterwhile fixing boundary values of other parameterswhich are based on the surrogate with best PRESSerror in Tables 4–6 and its predicted parameterconstraints. While rf is kept as 1, the sensitivity onthe applied frequency and dielectric constant is shownin Figure 15. For the refined level 1–1 in Figure 15a,the magnitude of force increases with the dielectricconstant for fv ¼ 5 kHz but decreases for fv ¼ 10kHz. However this trend reverses in level 1–2 as inFigure 15b, i.e. the magnitude of force increases fasterfor larger fv. A similar trend is observed in the effectof fv with constant ed. The average force generationdecreases with larger fv in level 1–1, but although thefrequency variation range is small, the magnitude offorce increases with increasing frequency in level 1–2as shown in Figure 15b. This effect is different from

the other studies discussing the frequency effect onforce generation. For example, it has been reportedthat for the order of several kilohertz frequencyrange, body force linearly increases with frequency(Baughn et al. 2006, Porter et al. 2007) or saturates ataround 2 kHz which is a relatively low frequency(Forte et al. 2006). Along with previously mentionednegative average force generation, this aspect needsfurther examination while the current model isimproved.

The power increases monotonically with the di-electric constant and applied frequency, which isconsistent with the experimental results (Roth andDai 2006) although the average power used in thisstudy does not accurately reflect the overall powerinput to the actuator.

5. Summary and conclusion

Analysis of the parametric effects of the DBD actuatorin helium is attempted by simulating a 2-species fluidmodel with the help of four surrogate models. For

Figure 15. Local dependency of performance on parameters. (a) rf ¼ 1.0, level 1–1. (b) rf ¼ 1.0, level 1–2.

296 Y.-C. Cho et al.

different levels of refinement and different regions indesign space, different surrogates offer the mostaccurate approximation, which justifies the applicationof multiple surrogate models. It has been found thereare multiple branches of the Pareto front whereparametric impacts and performance variables differsignificantly. The degree of correlation and globalsensitivity of design variables are very different in thelow and high power regions. Average force and powershow distinct variation in magnitude and/or direction.Surrogate models combined with the detailed solutionanalysis help understand various parametric depen-dencies as summarised below.

5.1. Dielectric constant

It affects the amount of charged particle clouds abovethe insulator wall during the second half cycle. With asmaller constant, particle clouds thicken, increasing theasymmetry between the two half cycles. As a result thex-directional average force increases. On the otherhand, it also affects the density of the charged particlelayer on the insulator surface. With a larger constant, ahigher electric field is produced, resulting in increase ofthe force magnitude, but along the negative direction.The amount of overall charged particle generation isalso affected by the dielectric constant. With a largerdielectric constant, higher electric field and powerconsumption occur.

5.2. Frequency of applied voltage

It affects the amount of overall charged particlegeneration. With a higher frequency, the dischargeduration decreases and the asymmetry between the twohalf cycles decrease, resulting in the larger x-direc-tional average force along the negative direction.Higher frequency accompanied by high dielectricconstant induces larger power usage.

5.3. Positive-to-negative half cycle time ratio

This also contributes to the overall charged particlegeneration. With a larger ratio, the first half cycledischarge becomes more prominent. As a result, the x-directional average force increases. On the other hand,the amount of charged particle clouds above theinsulator wall during the second half cycle is alsoinfluenced. The larger ratio induces a smaller level ofcharged particle generation, which means insufficientelectric field for the plateau region in the second halfcycle. As a result, the average force decreases.

Surrogate models at each level differ in fitting theoriginal data and the parameter-based average surro-gate provides good accuracy for this application.

Although the surrogate models at the initial level arenot accurate enough, they consistently supply bene-ficial information for the further refinement of thedesign space. The non-polyhedral design-space-con-straints generated by the initial level surrogate modelsprovide the efficient DOEs for the refined regions,resulting in higher accuracy in the regions of interest.

It is interesting that two branches of the Paretofront with different orientations of the average forceexist. Each branch is found to be in one of the tworegions far apart from the other in the original designspace. This information can be used to enhance theperformance of the actuator by devising effectivecontrol variables and understanding their influenceon performance. However, the force generation andpower consumption are also strong functions of otherparameters such as electrode size, insulator thickness,and amplitude of voltage which are invariant in thisstudy, and need to be explored.

Acknowledgement

The present work is supported by the AFRL, under acollaborative centre agreement.

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Appendix 1. Surrogate modelling (Goel et al. 2006a)

Polynomial response surface approximation

The polynomial response surface approximation of y(x) is

yðxÞ ¼Pi

bifiðxÞ

where bi is the estimated value of the coefficient associatedwith the ith basis function fi(x). The coefficient vector b isobtained by minimising the error in the approximation (givenby e(x) ¼ y(x)7y(x)) at k sampled design points in a leastsquare sense as

b ¼ ðXTXÞ�1XTy

where X is the matrix of basis functions and y is the vector ofresponses of the design points.

Kriging

Kriging estimates the value of an objective function y(x) atdesign point x as the sum of a linear polynomial trend modelPi

bifiðxÞ and a systematic departure Z(x) representing low(large scale) and high frequency (small scale) variationsaround the trend model.

yðxÞ � yðxÞ ¼Xi

bifiðxÞ þ ZðxÞ:

The systematic departure components are assumed to becorrelated as a function of distance between the locationsunder consideration. Gaussian function is the mostcommonly used correlation function.

CðZðxÞ;ZðsÞ; hÞ ¼Yi

exp ð�yiðxi � siÞ2Þ:

The parameters bj and yj are obtained using maximumlikelihood estimates. The Kriging software developed byLophaven et al. (2002) is used to construct a Kriging model.

Radial basis neural network

For radial basis neural networks, the objective function isapproximated as a weighted combination of responses fromradial basis functions (also known as neurons).

yðxÞ ¼XNN

i¼1wiaiðxÞ

where NN is the number of neurons, ai(x) is the response ofthe ith radial basis function at design point x and wi is theweight associated with ai(x). Usually, a Gaussian function isused for radial basis function a(x) as

a ¼ e�ðjjs�xjjbÞ2

where b is inversely related to a user defined parameter‘spread constant’ that controls the response of the radialbasis function. The spread constant defines the range of the

International Journal of Computational Fluid Dynamics 299

design space over which a neuron has a response of 0.5 ormore. Small values of the constant can result in the poorresponse in the regions away from the neurons. On the otherhand, large values can cause the low sensitivity of theneurons such that the network may not be able to accountfor high-gradient regions or a larger number of neurons maybe required to adequately characterise the design space.Typically, the spread constant is selected between zero andone. The number of radial basis functions (neurons) andassociated weights are determined by satisfying the userdefined error ‘goal’ on the mean square error inapproximation. The native neural networks Matlab toolbox(Mathworks contributors 2004) is used.

Parameter-based average surrogate

We develop a weighted average surrogate model as

ywasðxÞ ¼XNSM

i

wiðxÞyiðxÞ

where ywas(x) is the predicted response of the weightedaverage of the surrogate models, yi(x) is the predictedresponse by the ith surrogate model, and wi(x) is the weightassociated with the ith surrogate model at design point x.Furthermore, the sum of the weights must be one�PNSM

i¼1 wi ¼ 1�so that if all the surrogates agree, ywas(x)

will also be the same. Weights are determined as follows.

wi ¼w�iPi w�i

;w�i ¼ ðEi þ aEavgÞb

Eavg ¼1

NSM

XNSM

i¼1Ei; b < 0; a < 1

where Ei is the global data-based error measure for ithsurrogate model. In this study, the generalised mean squarecross-validation error (GMSE) (leave-one-out crossvalidation or PRESS in polynomial response surfaceapproximation terminology) is used as global data-basederror measure by replacing Ei with

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiGMSEi

p. By using

subsets with the sample size, k this error measure can becalculated as follows.

GMSE ¼ 1

Ns

XNs

i¼1

�yi � y

ð�iÞi

�2

where yð�iÞi represents the prediction at x(i) using the surrogate

constructed using all sample points except (x(i), yi).In this study a ¼ 0.05 and b ¼ 71 are used. The above

mentioned formulation of weighting schemes is used withpolynomial response surface (PRS) approximations , Kriging(KRG) and radial basis neural networks (RBNN) such that,

ywas ¼ wprsyprs þ wkrgykrg þ wrbmyrbnm:

The SURROGATES toolbox (Viana and Goel 2007) is usedfor the WAS algorithm and the manipulation of all differentcodes.

Appendix 2. Global sensitivity analysis (Sobol 1993,

Goel et al. 2006a)

A surrogate model f(x) of a square integrable objective as afunction of a vector of independent input variables, x indomain [0, 1] is assumedandmodelled as uniformlydistributedrandom variables. The surrogate model can be decomposed asthe sum of functions of increasing dimensionality as

fðxÞ ¼ fo þXi

fiðxiÞ þXi<j

fijðxi; xjÞ þ � � �

þ f12;...;Nðx1; x2; . . . ; xNÞ

where f0 ¼R 10 f dx. If the following condition

Z 1

0

fi1...is dxk ¼ 0

is imposed for k ¼ i1, . . . , is, then the previous decompositionis unique. In context of global sensitivity analysis, the totalvariance denoted as V(f) can be shown to be equal to

VðfÞ ¼XNv

i¼1Vi þ

Xi�i;j�Nv

Vij þ . . .þ V1...Nv

where V(f ) ¼ E((f7f0)2) and each of the terms in represents

the partial contribution or partial variance of theindependent variables (Vi) or set of variables to the totalvariance and provides an indication of their relativeimportance. The partial variances can be calculated usingthe following expressions:

Vi ¼ VðE½ f jxi�ÞVij ¼ VðE½ f jxi; xj�Þ � Vi � Vj

Vijk ¼ VðE½ f jxi; xj; xk�Þ � Vij � Vik � Vjk � Vi � Vj � Vk

and so on, where V and E denote variance and expected valuerespectively. Note that E½ f jxi� ¼

R 10 fidxi and VðE½ f jxi�Þ ¼R 1

0 f 2i dxi. Now, the sensitivity indices can be computedcorresponding to the independent variables and set ofvariables. For example, the first and second ordersensitivity indices can be computed as

Si ¼Vi

VðfÞ ; Sij ¼Vij

VðfÞ

Under the independent model inputs assumption, the sum ofall the sensitivity indices is equal to one. The first-ordersensitivity index for a given variable represents the main effectof the variable, but it does not take into account the effect of theinteractionof the variables.The total contributionof a variableto the total variance is given as the sum of all the interactionsand themain effect of the variable. The total sensitivity indexofa variable is then defined as

Stotali ¼

Vi þP

j;j 6¼i Vij þP

j;j 6¼iP

k;k 6¼i Vijk þ � � �VðfÞ :

Note that the above referenced expressions can be easilyevaluated using surrogate models of the objective functions.

300 Y.-C. Cho et al.

To calculate the total sensitivity of any design variable xi, thedesign variable set is divided into two complementary subsetsof xi and Z (Z ¼ xj, 8j ¼ 1, Nv; j 6¼ i). The purpose of usingthese subsets is to isolate the influence of xi from the influenceof the remaining design variables included in Z. The totalsensitivity index for xi is then defined as

S totali ¼ V total

i�VðfÞ

where

V totali ¼ Vi þ Vi;z

where Vi is the partial variance of the objective with respectto xi and Vi,z is the measure of the objective variance that isdependent on interactions between xi and Z. Similarly, thepartial variance for Z can be defined as Vz. Therefore, thetotal objective variability can be written as

V ¼ Vi þ Vz þ Vi;z

The above expressions can be easily computed usingGauss-Quadrature points for numerical integration ofdifferent partial variance terms. The SURROGATEStoolbox (Viana and Goel 2007) is used for theimplementation.

International Journal of Computational Fluid Dynamics 301