A Competitive Game Approach for Multi Objective Robust Design Optimization

A. Clarich§ , C. Poloni*, V. Pediroda±,

Department of Energetica-University of Trieste, Trieste, Italy, 34127

Abstract

This paper describes an application of Robust Design methodology in the transonic airfoil design. It has been observed that, minimizing the drag at a single design point (Mach number and angle of attack fixed), it is possible to find solutions characterized by poor off-design performances (over-optimizing problem). For this reasons, the stability of the performances inside the range of operative conditions is an important objective in the design. Once the operative conditions are defined (range of Mach number and angle of attack), a Multi Objective approach is needed; in particular, two are the objectives to be optimized: the mean performances inside the range of operative conditions (optimise mean value of the aerodynamic coefficients) and the stability of the solution (minimize variance of the coefficients).

In this Multi Objective optimization problem, we have applied a competitive Game Strategy, based on Nash equilibrium, combined with a particular mono-objective algorithm, the Simplex. The players are in charge of different objectives, corresponding to the two objectives, that have to be optimized by the Simplex algorithm. Since the variables space is split between the two players, each player influences the choices of the other one in the course of the optimisation, until an equilibrium point, corresponding to the best compromise between the objectives, is found.

About the optimization test case, the range of operative conditions is Mach=0.73±0.05 and angle of attack 2°±0.5, and the original RAE2822 airfoil is parameterized. To reduce the high number of CFD analysis based on Navier-Stokes equations, a statistic extrapolation method, based on an adaptation of DACE, is used to define the required response surfaces.

According to our results, the methodology seems to be a promising approach which offers a new possibility to the designer, in particular when a good compromise of performance and stability is required, with cheap computational resources.

Nomenclature

E( fi( x , u )) = mean value of function fi of the variables x in the range of fluctuating control parameters u

? ?( fi( x , u )) = standard deviation of function fi in the range of fluctuating control parameters u ???????????????????????? mean of the stochastic process in DACE extrapolation ?(xi) = extrapolation error in the point xi defined by a Gaussian distribution of Normal type (0,? ?) RMSE = Root Mean Squared Error of the extrapolation errors

I. Introduction

n most industrial applications, some design operative parameters are not precisely known or it is impossible to fix a constant value. For example, some uncertainties could characterize some geometric entities (lengths, relative positions, angles,

etc.) that are related to the case studied. Many times, the opera tive conditions are not fixed, but there is the presence of fluctuations: in turbo-machinery, it is the case of the mass flow rate and the inlet pressure, whereas, in aeronautics, it is the case of the flight speed, the angle of attack, the air temperature, etc.

For these reasons, in all these cases the design parameters can be specified by the mean value and its variance, following the classic Gaussian theory9.

I

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In the case of fluctuations of the operative conditions, it is important to achieve the stabili ty of the solution, because a traditional optimization approach could tend to a problem of "over-optimization" (fig.1, left), giving high performances in correspondence of the design point, but with poor off-design characteristics.

Many numerical methods have been developed to optimize the design under uncertainty of the input

parameters2,3 , and commonly, the performance and the stability are joined in an unique objective function to be maximised.

On the contrary, our approach4 consists in the use of a mu lti objective algorithm to reach the best possible compromise between performance and stability of the design. With reference to figure 1 (right), it is possible to note that the function presents an absolute extreme and a relative extreme respectively corresponding to the coordinates x1 and x2; in this case, the uncertainties could be represented by the tolerance ? of the input parameter x. Obviously, a standard optimization, that does not consider the fluctuations, would find out the point x1. On the contrary, a robust design optimization would find both the point x1, that corresponds to the highest mean value of f(x), and the point x2, that corresponds to the highest value of stability of the function inside the tolerance range ? of the input parameter x.

We can express the two objective functions in mathematical terms as follows:

? ? Variance)()(),()(

Mean value)(),()(

,...,1)(min

,...,1)(max

22

2

udupfEuxff

udupuxffE

nif

nifE

iii

ii

i

i

??

??

?

?

?

?

?

(1)

where fi( x , u ) is the ith function of the variable x to be maximized and u is the vector of the design control parameters that are subjected to the fluctuations and uncertainties; p(u ) is the probability distribution of the

uncertain parameter u . To apply Robust Design in the optimization of a transonic airfoil, the use of a Navier-Stokes solver, even in a

two-dimensional approach, could be very expensive, because of the large number of computations required, due to the definitions of average values and variance (Eq. 1). For this reason, a mathematical methodology, that allows to extrapolate accurately the function values updating a database of CFD calculations, has been proposed; this methodology is called adaptive DACE9.

In previous works4, the algorithm used to solve the mu lti-objective optimisation problem was MOGA (Multi-Objective Genetic Algorithm), that has shown its good efficiency, but that has also revealed some limits due to the large number of computation required to obtain a good Pareto front, i.e. the set of not-dominated solutions that represent the best compromise for the two objectives (performance and stability).

For this reason, in this paper we describe a different optimisation approach, based on Game Theory5,6. The variables and the objectives are divided between two players, and the result is an equilibrium point (Nash

Single point

Robust design

x

F(x)

? ?

X1 X2

Figure 1. Comparison of robust design optimisation with single point optimization

equilibrium) that represents the best compromise of the two (contrasting) objectives. Even though the solution may be only a point of the Pareto front obtained by MOGA, that may offer more than one solution to the designer’s choice, the Nash approach has the great advantage, particularly important in a Robust Design problem where the computational cost is heavy, of a higher convergence speed7,8 .

We first describe briefly the DACE theory for the extrapolation of the design functions, then we introduce the Game Theory for the multi-objective optimisation, and finally we show the results obtained applying this approach in the Robust Design optimisation of a transonic airfoil.

II. Adaptive D.A.C.E. Response Surface Methodology

Originally developed and used in mining engineering and geostatistics data, the Kriging method is an approach for curve fitting and response surface approximation. In the 1980s, some statisticians have developed Design and Analysis of Computer Experiments (D.A.C.E.) for deterministic computer-generated data based on the Kriging method9. The Kriging method used in this study is based on the D.A.C.E. approach.

Suppose we have evaluated a deterministic function of k variables at n points. We denote the i-sampled point by xi=(x1

i,….,xki) and the associated function value by yi=y(xi), for i=1,..,n. The Kriging (D.A.C.E.) technique is based

on the following stochastic process model:

]2,1[,0,),(1

???? ??

k

hh

Phjk

ikh

ji Phxxxxd ?? (2)

)],(exp[)](),([ jiji xxdxxCorr ???? (3)

),...1(,)()( nixxy ii ??? ?? (4) The Equation (2) is the weighted distance formula between the sample points xi and xj, and Eq. (3) is the

correlation between the errors corresponding to the points xi and xj. Equation (4) is the model we use in the stochastic process approach: ? is the mean of the stochastic process,

?(xi) is defined by a Gaussian distribution of Normal type (0,??); the latter term is the result of a stationary Gaussian random function that creates a localized deviation from the global model10. The parameter ?h in the distance formula Eq. (2) can be interpreted as a measure of the importance or “activity” of the variable xh. The exponent ph is related to the smoothness of the function in coordinate direction h, ph=2 corresponding to most smooth functions. The stochastic process model in Eqs. (2-4) is essentially a generalized least squares (GLS) model11 , with a simple set of regressors (just a constant term) and a special correlation matrix, that has unknown parameters and depends on the distances between the sampled point.

The Kriging approximation presented by Schonlau11 uses the best linear unbiased predictor (BLUP) of y at the point at which we are predicting, x*. Let r denote the n-vector of correlations between the error term at x* and the error at the previously sampled points. That is, element i of r is ri(x*)=Corr[?(xi),??(xi)] , computed using the formula for the correlation function in Eqs. (2) and (3). The estimated model of Eq. (4) can be expressed by the BLUP of y(x*):

)ˆ(ˆ)( 1* ?? IyRrxy T ??? ? (5) where y=(y1,…yn)T denote the n-vector of observed function values, R denotes the n?n matrix whose (i,j) entry

is Corr[?(xi),??(xj)], and I denotes an n-vector of ones. The value for ? is estimated using the generalized Least Squares method as follows:

yRIRI TY 111 )1( ????? (6) The estimation of ? h and ph and hence an estimation of the correlation matrix are obtained by the maximization

of a Likelihood Function12. The mean squared error (MSE) of y(x) can be derived as:

IRIrRII

rRrIs T

TT

1

21122 )(

][ ?

?? ?

??? ? (7)

Equation (7) provides an estimation of the Variance of the stochastic process component of the Kriging approximation.

Earlier studies imply that, including the parameter ph as a part of maximum likelihood estimation, doesn’t help to improve very much the Kriging approximation; thus, in the current study, ph=2 is used for all the design variables.

A. Adaptive DACE

Before to build an extrapolation, we require a systematic means of selecting the set of inputs (called Design Of Experiments, or DOE) at which to perform a computational analysis. One common choice for generating experimental design for computational experiments is the Latin Hypercube13. Instead of using this technique, we propose an adaptive arrangement of the initial set of samples (data base), based on the value of the MSE (Eq. 7). The value of MSE depends on the correlation of the landscape as well as on the local density of points.

More precisely, we consider the behaviour of RMSE (Root Mean Squared Error): the RMSE indicates the accuracy of the prediction and it assumes low values corresponding to the neighbourhood of the samples. It is clear to understand that the extrapolation becomes more precise in regions with higher point density. We define the functions IEA (Index of Absolute Error) and IEAN (Index of Absolute Error Normalized) as follows:

RMSE y(x) IEA ? (8)

)-RMSE/(RMSERMSE |)Y-)/(Y-Y(y(x)|IEAN minmaxy(x)minmaxmin ?? (9) Eqs. (8) and (9) represent the index we use to set the adaptive arrangement of the samples. In fact, we use the

RMSE to understand where the extrapolation is not accurate, taking care at the same time of the extrapolated value associated (the y(x) function is to be maximised, thus a high value is more interesting). For example, a high value of IEA or IEAN indicates that the extrapolation is not accurate or that the function is higher; these points are the most interesting, and thus the database will be updated by the evaluation of the function in those points. Equation (9) has the same meaning of Eq. (8) but is normalized.

The Ymax and Ymin values are respectively the highest and lowest values of the extrapolated function, whereas RMSEmax and RMSEmin have the same meaning regarding RMSE. If y(x) is to be minimised, we can substitute eqs. (8) and (9) with the following ones:

/RMSEy(x) IEA ? (10)

)-RMSE/(RMSERMSE |)Y-)/(Y-Y(y(x)|IEAN minmaxy(x)minmaxmax ?? (11)

In any case, we apply these functions (Eqs. 8-11) in order to add new input points in the database in following iterations, choosing the points of the range where the values of IEA or IEAN are higher.

In fig.2 below we show, as example, a function F(x) , the function F extrapolated by DACE with the initial data base (4 circle points), the error index (that may be a IEA or IEAN function), and then the new point, for which the error index is maximum, to be added in the database.

F real

F extrapolated

Error index

Data base

New point in data base

Figure 2. Definition of the new points in the database.

III. Game Theory Approach for Multi-Objective Design Optimisatio n

In a competitive game, independently from the field of application, there are two or more contrasting objectives to be reached, whereas the space of the variables represent the different strategies that can be played.

We identify two players that divide the objectives and the search space (fig.3). The player 1, red, have to minimise the function f1 considering as variables only the x vector, whereas the y variable is fixed. At the same time, the player 2, green, have to maximise the function f2 considering as variables the vector y and fixing the x vector; both the two players apply a Simplex algorithm.

After a certain number of Simplex iterations, each player finds the best configuration (and set of variables) for its objective, and then the search continues by a new step, in which the variables that are fixed for each player are updated to the values found by the other player in the previous step (for instance, the value x0 for player 2 is replaced by x1 that is the best individual found by player 1 in the first step).

It means that each player have to optimize his variables following its objective, but the variables that are optimised by the other player influence his search. The solution is an equilibrium point, that occurs when the choices of the two players do not change in following steps. This choice represent the best compromise for the two objectives: it is a unique solution, but this solution depends on the way the variables space has been split between the two players. For this reason, it is important to know, or at least to guess, which variables are most significant for each objective.

Previous works in the aerodynamic field 8 has proved the efficiency of this methodology, especially in terms of low number of configurations required; in addition, an adaptive methodology that allows to change the variables from one player to another in the course of the optimisation, accordingly to statistical influence of each variable, is still been developing15.

Best Y=Y1 is found

SIMPLEX1 is run (Player 1) Obj.: min. f1 variables X, Y0 fixed

SIMPLEX2 is run (Player 2) Obj.: min f2. Variables Y, X0 fixed

Evaluation of f1 Evaluation of f2

n1 steps n2 steps

Best X= X1 is found

SIMPLEX1 is run (Player 1) again Optimise X with Y fixed to Y1

SIMPLEX2 is run (Player 2) again Optimise Y with X fixed to X1

.

.

.

.

.

. A converged optimized solution (XN , YN) is found Figure 3. Description of Nash/Simplex algorithm.

IV. Application of Game Theory-Robust Design in 2D Transonic Airfoil Design A. Definition of test case, parameterisation and optimisation strategy

In the common practice, for a single point airfoil design optimization, the design point is fixed (e.g. angle of incidence ??????Mach number M?????). Due to not deterministic events (like gusts of wind, atmospheric turbulence, instable conditions of flight, manoeuvre inaccuracy, ...), the design point can be considered slightly fluctuating, consequently it should be rational to consider a range of operating conditions instead of a single project point (e.g., ??? ???????? ?????????????

The relationship between wave drag and free flow velocity is quite non linear for high subsonic design Mach numbers, and thus the position of the possible shock waves can change quickly as soon as the operating conditions slightly changes (? ?and ? ): for this reason, by the single design point approach, it is possible to find some airfoil shapes which are advantageous corresponding to the project point (low drag resistance) but that are characterized by poor performances in the neighbourhood of it1.

A way to obtain solutions that maintain good performances at nearby off-design point is to consider the stability of the solution as objective function in addition to the mean value of the performance.

The objective functions described in Eq. (1) can be applied in this case, discretized and approximated as follows:

)1(

))()(()(min

)()(min

1

1

?

??

?

?

?

?

?

n

xCxCx

n

xCxC

n

iDDi

CD

n

iDi

D

?

(12)

The )(xCDi values are the drag coefficients calculated in function of the x variables (geometric parameterisation) and for n differe nt points, selected by uniform distribution in the range of M and ? ??? ? ? ???????? ? ? ???????????In this way, we calculate, for each configuration defined by the x variables, the mean value of the drag and its standard deviation, that is an index of stability. Both the two functions are to be minimised, using the Nash/Simplex algorithm described in section 3.

About the parameterisation, we modify an initial baseline configuration that corresponds to the supercritical airfoil RAE2822 designed by the Royal Aircraft Establishment. The upper and lower side of the profile are defined by two 7-degree Bèzier curves, and the co-ordinates of their control points become the variables of the optimisation (fig.4).

The split of the variables between the two players of Nash has been defined accordingly to an hypothesis. We suggest that, since the position of shock wave in the upper side of the profile is particularly influenced by the small variations of Mach and alpha, the player in charge of the drag stability objective will work on the variables relative

Player 1 (drag stability)

Player 2 (drag mean)

Figure 4. Detail of the mesh around the airfoil profile and assignment of the geometric variables to the Players.

to the upper side of the profile, whereas the player in charge of the mean drag objective will work on the lower side variables. In other words, we think that the upper side of the profile influences more directly the stability of the performances (that is, we suppose, the stability of the shock wave position).

As we will see in the next section, this choice is very important for the efficiency of the optimisation; of course, different solutions are possible, and maybe the best one will be the application of an adaptive Nash algorithm, that re-distribute the variables between the players in the course of the optimisation15.

About the flow solver and mesh generator (fig.4, left), we have used a Navier-Stokes CFD code based on Johnson-Coakley turbulence model, and in particular the MUFLO and AIRFOIL codes14.

The mesh is a structured C-type of about 30,000 cells, the Reynolds number is set to Re=1.5 106, whereas the Mach number and angle of attack are defined in the ranges previously described.

In this two -objective optimization we set 5 further constraints: the thickness is fixed to be higher than 12% of the chord length, the value of the RAE2822; in addition, the new configuration should present values better than or equal to the original RAE2822, relatively to the mean drag and lift coefficients, and to the standard deviation of the same coefficients.

Finally, we report (fig.5) the drag coefficient chart in function of Mach and alpha relatively to the original RAE2822; on the left there are the real values (obtained by 121 CFD computations), whereas on the right there is the comparison of this surface with the one extrapolated by adaptive DACE, through 9 total training points (see paragraph 2.A).

In table 1 we report the lift and drag (mean and standard deviation) values relative to the real RAE2822 airfoil and to the data extrapolated by adaptive DACE with a different number of training point.

Since the relative errors are less than 1% using 9 training points, we have decided to use this extrapolation method in the optimisation.

Table 1. Comparison of real and extrapolated mean and deviation performances. RAE2822 Cl mean Cd mean Sigma Cl Sigma Cd % err ? cl % err ? cd Training

points Real 0.677 0.173 2.24 2.00 RS 0.671 0.179 1.71 1.80 23.7% 10.0% 5 RS 0.675 0.179 2.28 1.90 1.8% 5.0% 7 RS 0.674 0.176 2.21 2.00 1.3% <1% 9 RS 0.675 0.176 2.26 2.00 <1% <1% 11

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

2.22.4

Alpha

Cd0.0310.030.0290.0280.0270.0260.0250.0240.0230.0220.0210.020.0190.0180.0170.0160.0150.0140.013

0.0 15

0.0 2

0.0 25

0.0 3

Cd

0.6 75

0 .70.7 25

0 .750.7 75

Mach1.5

1 .75

2

2 .2 5

2.5

Alpha

err0.00050.000450.00040.000350.00030.000250.00020.000150.00015E-050

real surface extrapolated surface

Figure 5. Real RAE2822 CD(M,? ) surface and comparison with DACE extrapolated surface.

B. Results of the optimisation

Figure 6 shows the minimization of the two objective functions, the mean value of CD and its standard deviation, during the optimization, that has been performed by the Nash-Simplex algorithm described in section 3.

Each player has required for the convergence 5 steps, and each step is characterised by about 18 Simplex iterations, that is 10 plus the initial n+1 starting set (number of variables n being 7).

The total number of designs is thus about 2*5*18=180, that is very interesting for a two-objectives optimisation of 14 variables and 5 constraints.

Using a double CPU PC machine, since for the computation of one configuration we require about 1 hour of time (DACE needs about 9 points for the calculation of CD response surface by an error less than 1%), the total time required for the two simultaneous players is 90 hours, about 4 days.

The reduction of the two objectives is about 5% for the mean and 13% for the deviation (fig.6); the comparison between the original and optimised geometries (left) and pressure fields (right) is shown in fig.7.

Finally we report in fig.8 the CD surface of the best configuration compared with the original one. On the right there is the chart relative to the percentage difference and, as it is possible to see, in all the range considered of the parameter Mach and alpha, the difference is greater than zero. This means that the new airfoil is characterised by better performances in the whole range considered, with peaks of about 10% of improvement (around M=0.75 and alpha=2.5), and a mean improvement of 5%.

0.0172 0.0060

0.0163

0.0052

Figure 6. Objective functions during the optimisation.

Figure 7. Best geometry and relative pressure field, compared with original RAE2822.

V. Conclusion

This paper shows an example of application of Game Theory in the Multi Objective Robust Design Optimisation of an airfoil.

The aim of the design is to produce an airfoil whose aerodynamic coefficients are optimised not only in the design point, but that conserves good performances for light variations of the operative conditions, such as angle of attack and Mach number, since during the mission there may be several causes of perturbation. The problem thus becomes a multi-objective one, since there is to minimize the mean value of drag in the range described for the operative conditions, and there is also to minimize the standard variation of the coefficient, in order to guarantee the stability of the aerodynamic performances. To calculate the mean value and the standard deviation for the performances of each configuration, it is necessary to calculate the response surface of drag in the range of the operative conditions, and in order to reduce CPU time we have adopted a particular efficient Extrapolation Methodology, the adaptive DACE.

The algorithm used for the optimization is a combination of Game Theory and Simplex, particularly efficient to find, as compromise of contrasting objectives, a configuration (Nash equilibrium) that optimise both the two objectives. The objectives and the variable space are divided between two players, following a preliminar hypothesis.

According to our results, the methodology seems to be a promising approach, which offers a new possibility for the design, finding out solutions that are characterized by more stable performances.

Acknowledgments

A.C. Author in particular wishes to thank Prof. Jacques Periaux for the precious help and support during his Phd stage at Dassault Aviation, and for the important contribution of him in the application of Game Theory in the industrial design.

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Figure 8. Cd response surface (left) and percentage difference between the original and best one (right).

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