[American Institute of Aeronautics and Astronautics 12th AIAA/ISSMO Multidisciplinary Analysis and...

Improvement of Airfoil Performances Under Multi

Uncertainties modeled by Tensorial-Expanded Chaos

Collocation

L. Parussini ∗, V. Pediroda † and C. Poloni ‡

Department of Mechanical Engineering, University of Trieste, Via Valerio 10, 34127 - Trieste, Italy

and S. Parashar §

Esteco North America, Livonia, MI, 48152 USA

In this work a multi objective Robust Design of a transonic airfoil has been performed.

For uncertainty quantification of the stochastic process different non-intrusive methodolo-

gies have been compared in order to find the more suitable for our purposes. The main

advantage of non-intrusive formulation is that existing deterministic solvers can be used.

Nomenclature

α Angle of attackΨ set of orthogonal polynomials with respect to the probability density function of the input parametersξ n-dimensional random variableδij Kronecker deltax d-dimensional space variableµ Meanφ stochastic processφi deterministic part of the Generalized Polynomial Chaos series dependent on (x, t)Ψi stochastic part of the Generalized Polynomial Chaos series dependent on (ξ(θ))σ Standard Deviationθ random eventCd Drag coefficientCl Lift coefficientCm Momentum coefficientd space dimensionalityE Expected valuef source termL differential operatorLp Legendre polynomial of order pM Mach numberN number of total terms of the Generalized Polynomial Chaos seriesn uncertainties dimensionalityp order of the expansiont time variableV ar Variance

∗Ph. D. short term researcher, Email: [email protected].†Ph. D. researcher, Email: [email protected].‡Associate Professor, Email: [email protected].§Support Engineer, Esteco North America, Livonia, MI, 48152 USA.

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American Institute of Aeronautics and Astronautics

12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference 10 - 12 September 2008, Victoria, British Columbia Canada

AIAA 2008-5867

Copyright © 2008 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

I. Introduction

The development of a design method, able to find solutions that are as less sensitive as possible to thevariations of those parameters which are affected by uncertainty, is becoming a requirement for industry.

This approach is known as Robust Design and its employment is interesting in many design situations: someparameters are unknown in preliminary design phase, presence of small manufacturing errors or variationsof the operative conditions (fluctuations of the design point).

In order to get a stable design it is necessary to know how the process under study is influenced by therandom parameters. In recent years there has been a great effort to develop uncertainty analysis methodolo-gies applied to computational physics, as uncertainty quantification is necessary in order to obtain reliableresults.

In literature there are several examples of numerical methods to study stochastic phenomena, which canbe divided into two categories: non-intrusive (as Monte Carlo, Stochastic Collocation, Chaos Collocation,Response Surface) and intrusive (as Chaos Polynomials). Given the random design parameters, they allowto compute the stochastic process in terms of expected value and variance. The expected value quantifythe performance and the variance the stability of the process under study, so that in order to find solutionsthat are both stable and performing, a multi objective optimization has to be performed. In this work wecompare different methodologies for uncertainty quantification: Monte Carlo, Adaptive DACE (Design andAnalysis of Computer Experiments) and Tensorial-expanded Chaos Collocation, and next we illustrate adesign method to find solutions that are insensitive to random fluctuations of the design parameters for anaeronautical application. After finding the best strategies, we performed a Multi Objective Optimization bymean Multi Objective Genetic Algorithm to reach the best final design for performances and stability.

II. Multi objective Robust Design

A standard optimization, without considering the uncertainties which affect the phenomenon under study,would find out the absolute maxima as the best solution, but in general in this point the objective functionhas poor stability. To get solutions which are both optimum and stable the concept of Robust Design hasto be introduced.

The main idea is to use a multi objective approach to reach the best possible compromise betweenperformance and stability of design. In the case of Robust Design optimization two different objectives haveto be considered: mean performances and stability of solutions, according to the ideas presented in Wu Liand S. Padula1 . It is interesting to observe that when Robust Design optimization is performed, it is possiblethat the more stable region doesn’t correspond to the more performing one2 . So to perform an optimizationunder fluctuations the best way is to define two different objectives for every function to optimize: its meanvalue and its variance. In mathematical terms it is:

Given φ : ℜd ×ℜn ⇒ ℜm

max E(φi) = µφi(x) =

∫

ξ

φi(x, ξ)p(ξ)dξ min V ar(φi) = σ2φi

(x) =

∫

ξ

[φi(x, ξ) − E(φi)]2p(ξ)dξ

(1)

where φ is the multi objective (in more general terms) function to be maximized, x are the deterministicindependent variables and ξ are the uncertain parameters, modeled by the probability density function p(ξ).

In this way the problem of an optimization under uncertainties becomes a Multi Objective Optimizationproblem where the objectives are the stability and the performance; to solve this problem we need to adoptthe Game Theory3 , which is the best methodology to solve a real multi objective problem without using aweighted function as:

max fw =

m∑

i=1

Wi

(

w(1)i µφi

+ w(2)i σφi

)

(2)

In fact it is tricky to assign a value to Wi, weights of objective functions, w(1)i and w

(2)i , weights of mean

and variance of each objective functions, and this is the reason because it is better to refer to Game Theoryapproach. It is interesting to notice that after the optimization phase, using a Pareto Frontier approach, thedesigner does not get only one solution but a set of solutions (Pareto Frontier) which represents the bestpossible compromise between the objectives.

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After finding the Pareto Frontier, the next phase it is the choice of the best design; the choice s easywhen there are not so many designs in the Pareto Frontier, in this case the designer can easily compare thedifferent solutions and chose the best for his purpose. But when the Pareto Frontier is complex a MultiCriteria Decision Making methodology4 can be adopted to help the designer. In this case, applying thealgorithm, it will possible to realize an automatic ranking of the solutions, simplifying the choice of thedesigner.

It is important to underline that it is possible to face a wide range of problems with Robust Designapproach (small manufacturing process errors5,6 , fluctuations in the operative conditions, unknown inputparameters, etc.). The method is also extendible to more than one function to optimize, for example it ispossible to improve the lift and drag of an airfoil with fluctuations in the flight speed, without the need ofa weighted function to tie the two different performances.

III. Uncertainty quantification methods

Numerical methods to study stochastic phenomena can be divided into two categories: non-intrusiveand intrusive. The main difference is that intrusive methods consist in resolution of a coupled system ofdeterministic equations, non-intrusive methods consist in resolution of a decoupled system of deterministicequations. It is evident the difficulty to design an efficient intrusive solver, both because of computationalcost and because of the obvious handicap to imply an internal modification of the deterministic solver.7

The non-intrusive methodology has a simpler computational management. A remarkable advantage of thisapproach is the deterministic solver represents a black-box and there is no need to modify it. This meansthe non-intrusive method is more versatile than intrusive method. On the base of these considerations, themethods we consider in this work, Tensorial-expanded Chaos Collocation6 and Adaptive DACE8 , are bothnon-intrusive. Next we illustrate the theory of these uncertainty quantification methods.

III.A. Polynomial Chaos methods

Let us consider the following stochastic differential equation:

L(x, t, θ;φ) = f(x, t, θ) (3)

where L is a differential operator which contains space and time differentiation and can be on linear anddepended on random parameters θ; φ(x, t, θ) is the solution and function of the space x ∈ ℜd , time t andrandom parameters θ; f(x, t, θ) is a space, time and random parameters dependent source term.

Under specific conditions9 , a stochastic process can be expressed as a spectral expansion based onsuitable orthogonal polynomials, with weights associated with a particular probability density function. Thefirst study in this field is the Wiener process10,11 , which can be written as a spectral expansion in terms ofHermite polynomials with normal distributed input parameters.

The basic idea is to project the variables of the problem onto a stochastic space spanned by a set ofcomplete orthogonal polynomials Ψ that are functions of random variables ξ(θ), where θ is a random event.For example, the variable φ has the following spectral finite dimensional representation:

φ(x, t, θ) =∞∑

i=0

φi(x, t)Ψi (ξ(θ)) (4)

In practical terms the Eq. (4) divides the random variable φ(x, t, θ) into a deterministic part, the coeffi-cient φi(x, t) and a stochastic part, the polynomial chaos Ψi (ξ(θ)). The basis {Ψi} is a set of orthogonalpolynomials with respect to the probability density function of the input parameters. Following the Askeyscheme12 , it is possible to introduce the Generalized Polynomial Chaos13 . Thanks to this theory, knownalso as Askey-chaos, for certain input parameter distribution there exists the best representation in terms ofconvergence rate. For example, for Gaussian random input we have the Hermite Polynomial Chaos represen-tation, for Gamma distribution the Laguerre representation, for Beta distribution the Jacoby representation,for Uniform distribution the Legendre representation, etc.

In this paper we focus mainly on random inputs with Uniform distribution, so we represent the variableφ(x, t, θ) in terms of Legendre spectral representation, following the Askey scheme. For practical cases, the

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series in Eq. (4) has to be truncated to a finite numbers of terms, here denoted with N . So Eq. (4) becomes:

φ(x, t, θ) =

N∑

i=0

φi(x, t)Ψi (ξ) =

P1∑

p1=0

P2∑

p2=0

. . .

Pn∑

pn=0

φp1p2...pn(x, t)Lp1

(ξ1) Lp2(ξ2) . . . Lpn

(ξn) (5)

where Lpk(ξk) is the Legendre polynomial (see Abramowitz14 for more details) of order pk in terms of the

k-th random variable with Uniform distribution in [−1,+1]. The number of total terms of the series isdetermined by:

N + 1 =

n∏

k=1

(pk + 1) (6)

where n is the uncertainties dimensionality and pk is the order of the expansion respect to the k-th randomvariable. In Eq. (5) a tensorial-expanded representation has been adopted. Other choices are possible6 .

The polynomial basis {Ψj} of Legendre-Chaos forms a complete orthogonal basis, i.e.

〈Ψi,Ψj〉 =⟨

Ψ2i

⟩

δij (7)

where δij is the Kronecker delta and 〈·, ·〉 denotes the ensemble average. This is the inner product in theHilbert space determined by the support of the Uniform variables

〈f(ξ)g(ξ)〉 =

∫

f(ξ)g(ξ)w(ξ)dξ (8)

with weighting function

w(ξ) =1

2n. (9)

What distinguishes the Legendre-Chaos expansion from other possible expansions is that the basis polynomi-als are Legendre polynomials in terms of Uniform variables and are orthogonal with respect to the weightingfunction w(ξ) which has the form of n-dimensional Uniform probability density function of independentvariables.

As an example, for a second order Legendre polynomial expansion with n = 2 we get the following form:

φ(x, t, θ) = φ00(x, t)L0 (ξ1(θ)) L0 (ξ2(θ)) +

φ01(x, t)L0 (ξ1(θ))L1 (ξ2(θ)) + φ02(x, t)L0 (ξ1(θ))L2 (ξ2(θ)) +



φ21(x, t)L2 (ξ1(θ)) L1 (ξ2(θ)) + φ22(x, t)L2 (ξ1(θ))L2 (ξ2(θ)) (10)

where ξ1(θ) and ξ2(θ) are the two random independent variables with Uniform distribution.Substituting the Polynomial Chaos series, given in Eq. (5) for Uniform random inputs, into the stochastic

differential Eq. (3) we obtain:

L

(

x, t, θ;

N∑

i=0

φi(x, t)Ψi (ξ(θ))

)

∼= f(x, t, θ). (11)

The method of Weighted Residuals is adopted to solve this equation. The coefficients φi(x, t) are obtainedimposing the inner product of the residual with respect to a weight function is equal to zero.

If the weight functions are chosen to be the same as the expansion functions Ψi we produce Galerkinmethod. Performing the Galerkin projection on both sides of the equation, the form becomes:

⟨

L

(

x, t, θ;

N∑

i=0

φi(x, t)Ψi

)

,Ψj

⟩

= 〈f(x, t, θ),Ψj〉 j = 0, . . . , N. (12)

If the operator L is non linear, the deterministic set of N + 1 equation is coupled and this form is calledChaos Polynomial7 .

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If we employ Dirac delta function as weight function we produce Collocation method. Using a collocationprojection on both sides of Eq. (12), we obtain:

L (x, t, θj ;φj) = f(x, t, θj) j = 0, . . . , N. (13)

This formulation is a linear system equivalent to solving a deterministic problem at each collocation point;this form is called Chaos Collocation.15 If in Eq. (12) the spectral representation employed is based on thetensorial product of one-dimensional orthogonal polynomials, as that defined in Eq. (5) for Uniform randominput variables, the Chaos Collocation approach will be referred as the Tensorial-expanded Chaos Collocation

method.The expected value and the variance of the stochastic solution φ(x, t, θ)will be:

EPC(φ) = µφ(x) = φ0(x, t, θ) (14)

V arPC(φ) = σ2φ(x) =

N∑

i=1

[

φ2i (x, t, θ)

⟨

Ψ2i

⟩]

. (15)

Here for Uniform random input variables:

〈Ψi,Ψj〉 =⟨

Ψ2i

⟩

δij =1

(2p1 + 1)

1

(2p2 + 1). . .

1

(2pn + 1)δij (16)

where the indices i and j correspond respectively to p1p2 . . . pn and q1q2 . . . qn and δij is the Kroneckeroperator. In fact the inner product for Legendre polynomials of the k-th random variable with uniformdistribution:

∫ +1

−1

1

2Lpk

(ξk) Lqk(ξk)dξk =

1

2pk + 1δpkqk

(17)

The two approaches, Chaos Polynomial and Chaos Collocation, are based on the same theory, but givedifferent numerical representations. In practice the intrusive method consists in resolution of a coupledsystem of deterministic equations, the non-intrusive method consists in resolution of a decoupled system ofdeterministic equations. It is evident the difficulty to design an efficient intrusive solver, both because ofcomputational cost and because of the obvious handicap to imply an internal modification of the deterministicsolver. The non-intrusive methodology has a simpler computational management. A remarkable advantageof this approach is the deterministic solver represents a black-box and there is no need to modify it. Thismeans the non-intrusive method is more versatile than intrusive method.

A still open problem of Chaos Collocation approach is the difficulty to select collocation points: with multidimensional uncertainties the choice is not unique16 . But if we use a tensorial-expanded representation, asthat one defined in Eq. (5), this problem does not exist. The collocation points are unambiguously definedand they are the Gauss quadrature points of the polynomial with order Pk + 1 in each dimension. This isthe reason why we use the Tensorial-expanded Chaos Collocation method: it allows avoiding an arbitrarychoice of collocation points.

In this paper we will examine problems with multi uncertain parameters with Uniform distributionapplying the Tensorial-expanded Chaos Collocation theory. We will exploit a Fictitious Domain solver forits capability to get an accurate solution of those problems with uncertain inputs of geometric kind.

III.B. Response surface: Adaptive DACE

Monte Carlo methods are often used when simulating physical and mathematical systems. Monte Carlomethods are a class of computational algorithms that rely on repeated random sampling to compute theirresults. There is no single Monte Carlo method; instead, the term describes a large and widely-used class ofapproaches. However, these approaches tend to follow a particular pattern: generate inputs randomly fromthe domain, and perform a deterministic computation on them, then aggregate the results of the individualcomputations into the final result. We can compute it by the formulation¿

EMC(φ) = µφ (x) =

∑Ni=1 φ

(

x, ξ(i))

N(18)

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V arMC(φ) = σ2φ (x) =

∑Ni=1

[

φ(

x, ξ(i))

− µφ

]2

N − 1(19)

If the number N of evaluations of function φ is high, the values we get will be accurate. It is evident thatif φ is an analytical function the computation of mean and variance is banal, but if we need of numericsimulations to get the values of function φ, the problem will be computationally expensive.

When a simulation requires too much computational time, the number of needed simulation can bereduced exploiting a response surfaces, in particular DACE model17,18 .

To exploit the DACE formulation, let us consider a function y(x) : ℜk ⇒ ℜ and suppose to know thefunction at n points. y(i) = y(x(i)) is the function value at i-th point. DACE method is based on thefollowing stochastic concept:

d(x(i),x(j)) =

k∑

h=1

Θh

∣

∣

∣x(i) − x(j)

∣

∣

∣

ph

Θ ≥ 0, p ∈ [1, 2] (20)

Corr[

ǫ(x(i)), ǫ(x(j))]

= exp[

−d(x(i),x(j))]

(21)

y(x(i)) = µ̂ + ǫ(x(i)) i = 1, . . . , n (22)

In Eq.(20) it is defined the weighted distance between two known points x(i), x(j) and in Eq.(21) it isdefined the correlation function between extrapolation errors at x(i), x(j) points. In Eq.(22) the stochasticmodel is mathematically described: µ is the mean value of the stochastic process and ǫ(x(i)) is the associatederror which is assumed to have Normal(0, σ2) Distribution; the last term ǫ(x(i)) define the local difference(according to Gaussian Theory) compared to the global mean model.8 Parameters Θh in Eq.(20) can beinterpreted as a measure of the importance of variables x(h); exponents ph are related to the regularity ofthe function in direction h, which increase with the value of the parameter.

DACE method presented by Schonlau18 uses the Best Linear Unbiased Predictor (BLUP) to extrapolatethe value of the function y at the point x∗. Defining the n-dimensional vector of correlations between errorsat point x∗ with r and in all n points of data-base ri(x

∗) = Corr[ǫ(x∗), ǫ(x(i))] then, saving Eq.s(20),(21)the predictor Eq.(22) becomes:

y(x∗) = µ̂ + rT R−1 (y − 1µ̂) (23)

where y = (y(1), ...y(n))T is function database, R is correlation matrix (Rij = Corr[ǫ(x(i)), ǫ(x(j))]) and 1

is a n-dimensional vector of values equal to 1. The value of µ̂ is computed using the Mean Squared Errormethod:

µ̂ =(

1T R−11) (

1T R−1y)

(24)

To estimate the values of Θ̂h and p̂h the following function is maximized:

−n ln

(

σ̂2)

+ ln |R|

2(25)

where

σ̂2 =(y − 1µ̂)R−1 (y − 1µ̂)

n(26)

The Mean Squared Error (MSE) of (x) is computed as:

s2(x∗) = σ̂2[

1 − rT R−1r]

+

(

1 − 1T R−1r)

1T R−11(27)

To maximize Eq.(25) it has been used a genetic algorithm, because the function is characterized by high nonlinearity and so it is important to use a robust optimization algorithm. From preliminary studies it can beasserted that the parameters ph have a low importance to create DACE model, so it is possible to set themequal to ph = 2.

To initialize an extrapolation, we require a systematic way of selecting the set of inputs (called DesignOf Experiments, or DOE) at which to perform a computational analysis. One common choice for generating

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experimental design for computational experiments is the Latin Hypercube19 . Instead of using this techniquewe propose an adaptive arrangement of the initial set of samples (data base) exploiting the value of the MSEgiven in Eq.(27). The value of MSE depends on the correlation of the landscape as well as on the local densityof points. In particular, we consider the behavior of RMSE (Root Mean Squared Error): the RMSE indicatethe accuracy of the prediction and it assumes low values corresponding to the neighborhood of the samplespoints. It is possible to understand that the extrapolation is more precise in regions with high point density.We define the function IER (Index of Error) as follows:

IER =RMSEy(x)

RMSEmax+

y(x) − Ymin

Ymax − Ymin(28)

This coefficient gives a numeric quantification based on two fundamental aspects needed to create the startingdatabase, that is the research of the points in which the function has low information (first term of Eq.(28),and where the function has a higher value (second term of Eq.(28)). Eq.(28) represents the index we use toset the adaptive arrangement of the samples. In fact, we try to exploit the value of RMSE to understandwhere the extrapolation is not accurate, taking care at the same time of the extrapolated value associated(the y(x) function is to be maximized, thus a high value is interesting). For example, a high value of IERindicates that the extrapolation is not accurate or that the function gets a high value; since these pointsare the most interesting, the database will be updated by the evaluation of the function in those points.The Ymax and Ymin values are respectively the highest and lowest values of the extrapolated function, whileRMSEmax and RMSEmin have the same meaning regarding RMSE.

Figure 1. Example of adaptive DACE model: thereal function y(x) (black solid line), the extrapo-lated function (black dotted line), the error indexIER (red solid line), the initial data base (blackdots), the maximum error index (white dot) andthe new point to be added in the database (reddot).

If y(x) is to be minimized, we can substitute Eq.(28) withthe following one:

IER =RMSEy(x)

RMSEmax+

y(x) − Ymax

Ymax − Ymin(29)

In any case, we apply these functions Eq.s(28),(29) in orderto add new input points in the database in following itera-tions, choosing the points of the range where the values ofIER are higher.

In Figure 1 we show, as example, a function y(x) (blacksolid line), the function y extrapolated by DACE (black dot-ted line) and by the initial data base (4 black dots), the er-ror index IER (red solid line), and then the new point to beadded in the database (red dot), for which the error indexis maximum (white dot).

IV. Test

To understand the capability of the Chaos Collocationmethod in comparison with the Monte Carlo techniques and the adaptive DACE methodology we propose asimple two dimensional airfoil example, with interesting connections with the Robust Design idea.

In the common practice, in a single point airfoil optimization, the project point is fixed (e.g. angle ofincidence α = 2, Mach number M = 0.73). Due to not deterministic events (like gusts of wind, atmosphericturbulence, instable conditions of flight, manoeuvre inaccuracy,...), the project point can be consideredslightly fluctuating, consequently it should be rational to consider a range of operating conditions instead ofa single design point (e.g., α = 2 ± 0.5, M = 0.73 ± 0.05).

The relationship between wave drag and free flow velocity is quite non linear for high subsonic designMach numbers, and thus the position of the possible shock waves can change quickly as soon as the operatingconditions slightly changes (α and M): for this reason, by means of the single design point approach it ispossible to find some airfoil shapes which are advantageous corresponding to the design point (low dragresistance) but that are characterized by poor performances in the neighborhood of it20 .

To appraise the capability of the methodologies we calculate by a Navier-Stokes code (Muflo 21 ) the liftand drag coefficient for an airfoil (RAE2822) in N points uniformly distributed in the range of number ofMach and angle of attack under study (α = 2 ± 0.5, M = 0.73 ± 0.05); this model simulates uncertainties,which normally happen on a cruise (see Figure 2).

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0.015

0.02

0.025

0.03

Cd

0.68

0.7

0.72

0.74

0.76

0.78

Mach

1.6

1.8

2

2.2

2.4

Alpha

Cd0.0310.030.0290.0280.0270.0260.0250.0240.0230.0220.0210.020.0190.0180.0170.0160.0150.0140.013

(a) Drag coefficient Cd

0.55

0.6

0.65

0.7

Cl

0.6750.7

0.7250.75

0.775

Mach

1.6

1.8

2

2.2

2.4

Alpha

Cl0.720.710.70.690.680.670.660.650.640.630.620.610.60.590.580.570.560.550.540.530.52

(b) Lift Coefficient Cl

Figure 2. Performances of the RAE2822 calculated by the high fidelity analysis code.

We can compute the mean and variance of the drag and lift coefficients by their definitions:

E(Cd) = µCd=

∫ αmax

αmin

∫ Mmax

Mmin

Cd(α,M)p(α)p(M)dαdM (30)

V ar(Cd) = σ2Cd

=

∫ αmax

αmin

∫ Mmax

Mmin

[Cd(α,M) − E(Cd)]2p(α)p(M)dαdM (31)

E(Cl) = µCl=

∫ αmax

αmin

∫ Mmax

Mmin

Cl(α,M)p(α)p(M)dαdM (32)

V ar(Cl) = σ2Cl

=

∫ αmax

αmin

∫ Mmax

Mmin

[Cl(α,M) − E(Cl)]2p(α)p(M)dαdM (33)

where p(α) = 1/ (αmax − αmin) and p(M) = 1/ (Mmax − Mmin). To solve the integrals of Eqs. (30)-(33) aGauss-Legendre quadrature method has been used with 11-th order of integration. We will consider thecomputed results as referee values.

We can compare the results obtained by the methods illustrated in the previous sections to the refereevalues. In particular we consider the Monte Carlo method on the real model, the Monte Carlo method onthe adaptive DACE model and the Tensorial-expanded Chaos Collocation method. Figures 3 and 4 show theconvergence to the referee values of the results obtained by these methodologies respect to the computationaleffort, which depends on the number of needed simulations. It is evident that Monte Carlo method on thereal surface is computationally expensive with a really slow convergence.

The best accuracy can be obtained by Tensorial-expanded Chaos Collocation method.But we have to remark that by DACE approach we can get good results with a lower number of simu-

lations. Using this approach, the number of needed simulation is just related to the DACE model. Usingthe DACE method, an iterative adaptive Design of Experiments is build, in order to minimize the statisticalerror between the real function and the extrapolated one. It is useful to remember that the advantages of theadaptive methodology increases with the number of free parameters, which in the Robust Design correspondsto the number of uncertain parameters. The values of mean and variance will be computed on the responsesurface by the Monte Carlo Method.

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Design ID

Mea

nC

d

101 10210-18

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

Chaos CollocationDACEMonte Carlo

(a) mean value of drag coefficient Cd

Design IDS

tdC

d101 10210-18

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2


(b) standard deviation of drag coefficient Cd

Figure 3. Convergence of mean and variance of drag coefficient respect to the number of needed simulations.

Design ID

Mea

nC

L

101 10210-18

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2


(a) mean value of lift coefficient Cl

Design ID

Std

Cl

101 10210-18

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2


(b) standard deviation of lift coefficient Cl

Figure 4. Convergence of mean and variance of lift coefficient respect to the number of needed simulations.

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V. Application

V.A. Robust Design of a transonic airfoil

Using the Multi Objective Robust Design theory and Tensorial-expanded Chaos Collocation for uncertaintyquantification, we perform a more realistic optimization case consisting in the design of an airfoil, using asflow solver the Navier-Stokes version of MUFLO and AIRFOIL codes21 , which uses as turbulence model theJohnson-Coakley equations. The upper and lower side of profile are defined by two 10-degree Bezier curves,and the co-ordinates of their control points are the variables of optimization. In total we have 18 designvariables, which represent the position of the control points.

0 0.5 1x

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

y

(a) Mesh

0 0.25 0.5 0.75 1x/c

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

y/c

RAE 2822

Configuration x

Control polygon

BWP

(b) Parameterization

Figure 5. Airfoil mesh with MUFLO (a) and airfoil parameterization using Bezier curves (b).

The uncertainties concern Mach number (M = 0.73 ± 0.05) and the angle of attack (α = 2 ± 0.5).The optimization goal is to find out an airfoil geometry which yields better results respect to the perfor-

mances and the stability, taking in account of the two uncertain parameters (angle of attack and free Machnumber). From a mathematical point of view the optimization problem becomes:

min∆M,∆α

(µCd, σCd

, σCl) max

∆M,∆α(µCl

) (34)

with

µCl≥ µRAE2822

ClµCd

≤ µRAE2822Cd

∣

∣µCm≤ µRAE2822

Cm

∣

∣

σCl≤ σRAE2822

ClσCd

≤ σRAE2822Cd

σCm≤ σRAE2822

Cm(35)

We set six constraints to optimization problem: the new configuration should present values better thanor equal to the original RAE2822 airfoil corresponding to the mean and variance of drag, lift and pitchingmomentum coefficients. The mean and standard deviation of performances of each new airfoil geometry arecomputed by mean of Tensorial-expanded Collocation according to theory explained in Section III.A with3rd order of polynomial expansion, which corresponds to 16 evaluations for each geometry varying the angleof attack and the Mach number.

To perform the multi objective optimization the genetic algorithm (MOGA) of the optimizer modeFRON-TIER22 is employed. The result will be a set of designs belonging to Pareto front.

V.B. Multi Criteria Decision Making

As in many other real-world problems, characterized by multiple objectives, attributes and different types ofmeasures, which have to be satisfied simultaneously, the decision maker needs to articulate his preferencesin terms of tradeoffs among objectives. Whether a multiple criteria decision problem appears naturally in

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life, in engineering design it has to be necessarily23 : i.e. only one design among all the alternatives couldbe putted on production.

Design selection problems are concerned with the evaluation or ranking of a set of available candidatedesigns in terms of multiple attributes and they form one important class of engineering decision problems.Due to the complexity and often huge amount of data, the analysis of an engineering decision problem usuallyrequires the support of a computerized system and this usually requires the mathematical modeling of thedecision problem24 .

In this work, in order to choose the more appropriate airfoil of Pareto frontier, a Multi Attribute DecisionMaking (MADM) method has been used: CODASID24 . This algorithm is based on an extended concordanceanalysis and a modified discordance analysis using raw data represented by a decision matrix and relativeweights (with this method the decision maker may also assign veto threshold values to each attribute: thiskind of information has not been used in this work). The new concordance and discordance analysis are usedto generate three new indices, namely a preference concordance index, an evaluation concordance index anda discordance index. These three indices provide independent measures for evaluation of each alternativedesign and span a new space for ultimate ranking of alternative designs. A distance measure is defined inthe new space to capture the similarities between a feasible design and given reference designs, which may,for example, be the best/least preferred (or ideal/nadir) designs. The basic idea of defining such a distancemeasure originates from the TOPSIS method25 . The new distance measure, however, is more general andable to take into account a limited compensation.

In order to elicit and capture the decision maker preferences and to calculate a priority vector, theAnalytic Hierarchy Process (AHP) coupled with the eigenvector method have been employed26,27 on thepairwise comparison matrix. AHP is a widely used multi-criteria decision analysis method; unlike theconventional methods AHP uses pairwise comparisons, which allows verbal judgments and should enhancethe precision of the result. Skipping the whole theory of AHP, it is worth to note that the comparison matrix,which is used to quantify how much more important a criterion is compared to another one by using a linearscale 1/9, 1/7, . . ., 1, 2, . . ., 9, should be consistent; A = {aij} matrix will have complete consistency, if thefollowing conditions are satisfied:

aij =1

aji

aijajk = aik ∀ i, j, k = 1, . . . , n; i 6= j (36)

Let us remark, the application of the different choices in the preference tables gives after the MCDMapplication different designs.

V.C. Results

In Figure 6 the Pareto Front we have obtained is shown, which represents all the feasible choices. In generala geometry which is good for one objective is no good respect to other objectives. Four geometries belongingto the Pareto Front have been highlighted. ID 447 is the geometry with the maximum µCl

, ID 221 has theminimum µCd

, ID 209 has the minimum σCland ID 228 has the minimum σCd

.To find solutions which are a good compromise of the performances we use the MCDM method4 explained

in the previous section. In Figure 6 the solution selected by MCDM is shown. Giving the same importanceto all four objectives we get the solution ID 204.

The different airfoils have different shapes, especially regarding the suction side of the profile, where theinteractions between geometry and shock waves are bigger. This is an interesting consideration, when weremember that the performances stability of the airfoil is directly correlated whit the shock waves behaviorin transonic case.

In Figure 7(a) the probability density functions of drag coefficient for airfoils ID 204, ID 221 (minimumµCd

) and ID 228 (minimum σCd) are shown, in Figure 7(b) the probability density functions of lift coefficient

for airfoils ID 204, ID 209 (minimum σCl) and ID 447 (maximum µCl

).ID 228 is the geometry with the best value of σCd

but poor performances of µCd. Similar considerations

can be done for the other designs ID 209, ID 221 and ID 447. ID 204 is a geometry with a good value forall the objectives.

In Figure 8 the mean value and the error bars of Pressure Coefficient Cp are shown for airfoil ID 204.Figure 9 shows the drag and the lift coefficient of airfoil ID 204 varying the Mach number and the angle

of attack.

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Mean Lift

Mea

nD

rag

4.5E-01 5.0E-01 5.5E-01 6.0E-01

1.5E-02

1.5E-02

1.6E-02

1.6E-02

1.7E-02

209

477

221

228

204

(a) µCdversus µCl

Std Lift

Std

Dra

g

3.6E-02 3.8E-02 4.0E-02 4.2E-02 4.4E-02 4.6E-02 4.8E-02 5.0E-02

2.0E-03

2.5E-03

3.0E-03

3.5E-03

4.0E-03

4.5E-03

5.0E-03

209

477

221

228

204

(b) σCdversus σCl

Figure 6. Pareto Frontier representation: airfoil ID 447 has the maximum µCl, airfoil ID 221 has the minimum µCd

,airfoil ID 209 has the minimum σCl

and airfoil ID 228 has the minimum σCd. Airfoil ID 204 is selected by MCDM.

Cd

p(C

d)

0.010 0.012 0.014 0.016 0.018 0.020 0.0220.0

50.0

100.0

150.0

200.0

250.0

300.0

350.0

ID 204ID 221ID 228

(a) p(Cd)

Cl

p(C

l)

0.40 0.45 0.50 0.55 0.60 0.65 0.700.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

ID 204ID 209ID 447

(b) p(Cl)

Figure 7. Probability density function of drag and lift coefficients: (a) airfoil ID 204 (MCDM), airfoil ID 221 (minimumµCd

) and airfoil ID 228 (minimum σCd) and ID 302, (b) airfoil ID 204 (MCDM), airfoil ID 209 (minimum σCl

) and airfoil

ID 447 (maximum µCl).

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X/C

Cp

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

1.5

Figure 8. Pressure Coefficient for airfoil ID 204: the mean value and the error bars obtained by means of Tensorial-expanded Chaos Collocation with polynomial expansion order p = 3 are shown.

In Figure 10 the mean and the variance of pressure field around the airfoil ID 204 is shown. In Figure 10(a)the shock wave typical of transonic airfoils is evident. Figure 10(b) shows where the standard deviation ofpressure is higher. This happens in correspondence of the shock wave, highlighting as the uncertain operativeconditions influence the position of the shock wave.

VI. Conclusions

In this paper a comparison among different methodologies for uncertainty quantification has been pre-sented, in order to find the best method for facing the robust design of a transonic airfoil.

Monte Carlo method, DACE model and Tensorial-expanded Chaos Collocation have been considered. Theextreme accuracy of Tensorial-expanded Chaos Collocation has been demonstrated computing the mean andstandard deviation of lift and drag coefficient for an airfoil (RAE2822) with given uniform distribution ofMach number and angle of attack. This model simulates uncertainties, which normally happen on a cruise.

Next the Tensorial-expanded Chaos Collocation method has been employed to compute the objectivefunctions given the airfoil geometry in the shape optimization we have implemented under uncertain operativeconditions.

Acknowledgments

The authors are grateful to ESTECO s.r.l. for the use of modeFRONTIER 4.0.This work was partially supported by the project NODESIM-CFD Non - Deterministic Simulation for

CFD-based Design Methodologies funded by the European Community represented by the CEC, ResearchDirectorate-General, in the 6th Framework Programme, under Contract No. AST5-CT-2006-030959.

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