Efficient Multi-point Aerodynamic Design Optimization Via Co-Kriging

Transcription

1 Efficient Multi-point Aerodynamic Design Optimization Via Co-Kriging David J. J. Toal and Andy J. Keane 2 University of Southampton, Southampton, SO7 BJ, United Kingdom Multi-point objective functions are often employed within aerodynamic optimizations to prevent a reduction in off-design performance. However, this typically results in the need for a significant number of simulations at a variety of design conditions in order to calculate the objective function. The following paper attempts to address this problem through the application of a multi-level co-kriging model within the optimization process. A large number of single point design simulations are augmented by a smaller number of multi-point simulations. The technique is shown to result in surrogate models as effective as those produced using a traditional multi-point process when optimizing a transonic airfoil, but with a reduction in the total number of simulations. Nomenclature C d = Drag coefficient C wtd d = Weighted drag coefficient C m = Pitching moment coefficient C p = Pressure coefficient C = Co-kriging covariance matrix d = Differences between expensive and cheap data K = Total number of design conditions n = Total number of sample points p = Hyperparameter governing smoothness r = Correlations between known and unknown points R = Correlation matrix w = Design point weighting x = Vector of design variables X = Matrix of design points y = Vector of objective function values Z(x) = Gaussian process θ = Hyperparameter governing correlation σ 2 = Variance µ = Mean ϕ = Concentrated log-likelihood λ = Regression constant ρ = Scaling parameter Subscripts c = Cheap data d = Difference between cheap and expensive data e = Expensive data Research Fellow, School of Engineering Sciences, Member AIAA. 2 Professor of Computational Engineering, School of Engineering Sciences

2 I. Introduction Optimization has become an important element in the design and development of modern civil airliners. It is the continual goal of aircraft designers to improve performance, reducing drag, weight, fuel consumption and cost. Aerodynamic optimization aims to reduce the drag of an aircraft thereby increasing its efficiency and reducing fuel consumption. This can be achieved in a number of ways through, for example, modifications to a wing s planform or airfoil section or through adjustments to nacelle and pylon geometries. However, a design condition must be selected for such optimizations. For a large civil airliner this could be the cruise condition, but optimization of a geometry for a single flight condition risks a degradation in performance at off-design conditions. An alternative approach is to consider a series of important design points and optimize a design for each of these in parallel through the application of an objective function based on a predetermined weighting of drag coefficients, C wtd d = K i= w i C (i) d, () where the weighted drag coefficient, CD wtd, is calculated from K drag coefficients at K design conditions. Such weightings can be predefined by the designer based on experience and the relative importance of each of the considered design conditions. This multi-point design technique is popular throughout the literature where there are numerous examples of its application to the optimization of airfoils, wings and whole aircraft. For example, Painchaud-Ouellet et al.[] optimized a 2-D airfoil for a weighted combination of drag at three different design conditions. Nemec et al.[2] considered the optimization of B-spline based parameterizations of single- and multi-element airfoils employing a multi-point scheme. Szmelter[3] optimized a transonic wing for a combination of minimum drag and deviation from a target pressure at three different lift coefficients. Epstein and Peigin considered the multi-point optimization of an airfoil[4], a business jet wing[5], a civil airliner wing[6] and a blended wing body aircraft[7]. Multipoint optimization has also been employed in high speed designs with Kim et al.[8] considering the multi-point optimization of a supersonic fighter wing and Cliff et al.[9] considering the shape optimization of a high-speed civil transport. While effective at preventing off-design performance degradation, the utilization of a multi-point design strategy has a significant drawback. With each additional design condition considered the total number of simulations required for each geometry increases. This can be further hampered if, as is so often the case, the performance evaluation at each design point requires an iteration to achieve a required lift coefficient. The use of a direct optimization technique such as a genetic algorithm is therefore hampered considerably by the associated simulation cost and typically only local gradient descent[ 3, 9], response surface methods[8, ] and methods employing a reduced order model[5 7] are practical. Although the application of a surrogate model of the objective function can considerably reduce the total number of computational fluid dynamics (CFD) simulations required in a search, a simulation is still required for each design condition. The following paper aims to address this problem through the application of co-kriging to the construction of surrogate models for multi-point design. Co-kriging, a variation of standard kriging, is capable of utilizing multiple levels of information about an objective function. Such data can come from varying simulation fidelities[, 2] or even from wind tunnel experiments[3, 4]. Applying this surrogate modeling technique, the strategy proposed within this paper aims to augment the weighted drag coefficient from a few designs considered at all flight conditions with drag data for a large number of designs at a single flight condition, thereby producing a surrogate model of similar accuracy but with fewer CFD simulations. The following paper commences with a brief discussion of both kriging and co-kriging. The proposed multi-point optimization strategy is then described in detail and the two-dimensional airfoil parameterizations used to demonstrate the process are presented. A two variable subset of one of these parameterizations is then used to demonstrate the process graphically. Finally the performance of the strategy is demonstrated with respect to two optimizations of a two-dimensional airfoil. 2

3 II. Kriging Based Response Surfaces A. Kriging Surrogate models have grown in popularity over recent years and have been applied to a wide variety of optimization problems. Kriging[5, 6] is a particularly popular method of constructing surrogate models as it can effectively represent a wide variety of responses whilst providing a useful error estimate of the predictor. The interested reader can find comprehensive reviews of kriging and other surrogate modeling techniques in Jones[7], Simpson et al.[8], Queipo et al.[9], Wang and Shan[2] and Forrester and Keane[2]. Although the current paper is mainly concerned with the application of co-kriging, it is necessary to first briefly review the standard kriging process upon which it is based; a more detailed description of kriging can be found in Jones[7]. If two points, x i and x j are close together within a design space then Kriging assumes the resulting objective functions, y(x i ) and y(x j ) will be similar. This can be modeled statistically by assuming that the correlation between two sets of random variables, Y (x i ) and Y (x j ) is given by, ( ) d R ij = Corr [Y (x i ), Y (x j )] = exp θ l x il x jl p l, (2) where θ l and p l represent the hyperparameters of the l th variable. The θ hyperparameter determines the rate at which the correlation decreases and p determines the degree of smoothness in each coordinate direction. These hyperparameters, are chosen via a maximization of the concentrated log likelihood function[7], where the optimal variance and mean are, and l= ϕ = n 2 ln(ˆσ2 ) ln( R ), (3) 2 ˆσ 2 = n (y ˆµ)T R (y ˆµ) (4) ˆµ = T R y T R. (5) This concentrated likelihood function is dependent only on the correlation matrix and hence on the hyperparameters which are tuned in an attempt to maximize this function. The maximization of the likelihood function can be an expensive and rather difficult optimization problem requiring a large number of O(n 3 ) factorizations of the correlation matrix. Efforts have recently been made by Toal et al. [22, 23] to employ a hybrid strategy which utilizes an adjoint of the likelihood to accelerate the hyperparameter optimization. Based on the linear algebra results presented by Giles [24], the adjoint of the correlation matrix can be shown to be[23], with the derivatives of the likelihood then, R = 2ˆσ 2 R T (y ˆµ) T (y ˆµ) T R T 2 R T, (6) ϕ θ l = ln ij θ l x il x jl p l R ij Rij, (7) and ϕ p l = ij θ l x il x jl p l ln x il x jl R ij Rij. (8) Given a known set of hyperparameters the kriging predictor is, y(x ) = ˆµ + r T R (y ˆµ), (9) where r denotes a vector of correlations between the unknown point, x, and the previous sample points, x i. 3

4 B. Co-Kriging In numerous optimization problems it is possible to obtain information about an objective function from a variety of different sources. In the case of aerodynamic or structural optimizations information may be obtained from a variety of simulation fidelities from simple empirical equations to complex finite element, or volume, simulations. It makes sense to exploit all of the available information in order to construct a more accurate surrogate model for the purposes of optimization. A multi-level surrogate typically couples a large amount of relatively cheap data to a much smaller amount of expensive data. Coupling data levels in this manner enhances the accuracy of the surrogate model of the expensive data. However, to make use of any available cheap data a correction process must be formulated which models the differences between the cheap and expensive data. Co-kriging is one such method of correlating multiple sets of data and is presented now employing the notation of Forrester et al.[25]. Using the auto-regressive model of Kennedy and O Hagan[26], the expensive code is approximated by multiplying the cheap code by a scaling factor, ρ, plus a Gaussian process representing the difference between the cheap and expensive codes, Z e (x) = ρz c (x) + Z d (x). () If X e and X c represent the known expensive and cheap data points respectively, where it is assumed that a cheap evaluation has been carried out at the location of every expensive evaluation, then the covariance matrix C is, C = ( σ 2 c R c (X c, X c ) ρσc 2 R c (X c, X e ) ρσc 2 R c (X e, X c ) ρ 2 σc 2 R c (X e, X e ) + σd 2R d(x e, X e ) ). () The correlations of Eq. are of the same form as that of standard kriging, Eq. 2, except that there are now two correlations and therefore twice as many hyperparameters to determine. A set of θ and p hyperparameters must be determined for both the Gaussian process of the cheap data and the difference in the cheap and expensive data. The scaling parameter, ρ, must also be determined. As cheap data is considered independent of the expensive data the hyperparameters, θ c and p c, can be determined in an identical manner to standard kriging. To calculate the remaining hyperparameters the difference between the expensive and cheap data at known locations is first calculated, The concentrated log-likelihood for the expensive data is therefore, d = y e ρy c (X e ). (2) ϕ d = n e 2 ln(ˆσ2 d) + 2 ln( R d(x e, X e ) ), (3) with variance and mean given by Eqs. 4 and 5 where R, y and n have been replaced by their equivalent values in the difference model, R d (X e, X e ), d and n e. As with the cheap data, an adjoint of the hyperparameters can be used to accelerate the likelihood maximization. The adjoint of the correlation matrix is calculated as per Eq. 6 with the relevant terms again being replaced by those from the difference model. Given this adjoint the derivatives with respect to θ d and p d can be calculated as per Eqs. 7 and 8. The adjoint of the vector of differences between the cheap and expensive data can be found using another of the results of the second quadratic form[24] which simplifies to, ( ) d = R d (d ˆµ d), (4) with the derivative of the likelihood with respect to ρ therefore, i= ˆσ 2 d ne ϕ ρ = y ci di. (5) 4

5 The co-kriging predictor is derived in a similar manner to that of traditional kriging. The data set is augmented by an unknown point which produces the following augmented covariance matrix, ( ) C c T C = c ρ 2ˆσ c 2 + ˆσ d 2, (6) where c is a column vector of the covariances of the unknown point with the known points. prediction of the expensive function at a new point is, A y e (x ) = ˆµ + c T C (y ˆµ), (7) where ˆµ is, ˆµ = T C Y T C, (8) and Y is a vector containing both the known cheap and expensive data. Co-kriging has been used quite successfully throughout the literature to aid the construction of surrogate models for optimization. Forrester et al.[27] applied co-kriging in conjunction with partially converged CFD simulations, while Laurenceau and Sagaut[28] and Won and Ray[29] employed gradient information within a special implementation of the presented co-kriging formulation. Although co-kriging has been presented here for two levels of data the process can be extended further to multiple levels[26]. Finally it should be noted that each of the kriging and co-kriging models employed within the strategies presented within the current paper have been tuned via a hybridized particle swarm employing the adjoint of the relevant likelihood function[23]. III. Proposed Multi-point Design Strategy As previously discussed, a multi-point design optimization process involves a series of simulations at a variety of design conditions. The multi-point design of a civil airliner could, for example, consider flight conditions during cruise, climb and take-off[7, ], while the multi-point design of a supersonic fighter could consider the maximum flight speed, cruise and high angles of attack[8]. Typically, such optimizations involve the minimization of an objective function defined by the weighted sum of the drag coefficients at each of the considered flight conditions, Eq.. Hence, for every objective function evaluation a series of CFD simulations is required. If a large number of flight conditions are considered this can result in each objective function evaluation becoming extremely expensive. Although in the modern world of high performance computing such evaluations can be carried out in parallel, any reduction in overall computational effort is advantageous, allowing optimizations to be completed faster or indeed permitting more complex optimizations. The proposed optimization strategy aims to significantly reduce the total simulation costs incurred during multi-point optimization through the application of co-kriging. Rather than constructing a single surrogate model of the weighted drag coefficient, a model is first constructed using the drag coefficient from only one of the flight conditions. This forms the cheap data in the previously presented co-kriging formulation. The surrogate constructed using this data is then augmented by a smaller subset of design points which have been simulated at all necessary flight conditions and for which the true weighted drag coefficient is therefore known. This data forms the expensive data in the co-kriging model. The resulting co-kriging model therefore aims to build an accurate surrogate of the weighted drag coefficient with the aid of additional information from a larger number of single flight condition drag coefficients. Such an approach has a number of advantages. First, and most importantly, the total number of CFD simulations is reduced quite significantly as each design point is no longer simulated at every flight condition. An optimization requiring analysis at three flight conditions, where only 2 out of a total of 5 design points are considered at every flight condition saves 4% of the CFD simulations over a traditional strategy. This saving increases further when more flight conditions are considered or if fewer multi-point analyzes are carried out. 5

6 The process can also be applied to any aerodynamic constraints. Pitching moment, at each flight condition for example, is often considered as a constraint in a multi-point airfoil design. A series of co-kriging models could be constructed where pitching moment data from one design point is augmented with a smaller amount of data from each of the other desired flight conditions. By employing such a strategy the reduction in the amount of true data for some flight conditions does not seriously affect the prediction of pitching moment at these conditions hence a constrained optimization can be carried out as normal. As an identical geometry is generated and used at all flight conditions the accuracy of a surrogate model of geometric constraints would remain similar to that of a traditional optimization strategy. The representation of pitching moment constraints via co-kriging may result in a reduction in surrogate modeling costs. While a co-kriging model of the objective function increases hyperparameter tuning costs as two optimizations are required, the inclusion of constraints negates this effect as a prediction of the pitching moment at the cheap flight condition is required during the overall optimization, the construction of its surrogate is a necessity; however, this surrogate also forms the first level of each of the subsequent co-kriging models of the remaining constraints. The hyperparameters of the surrogate representing the differences between the cheap and expensive data remain unknown but as there is significantly less data the cost of the O(n 3 ) factorization of the associated correlation matrices reduces. This therefore reduces the total cost of the hyperparameter optimization when compared to constructing multiple traditional kriging models of the constraints. A constrained multi-point optimization involving a series of flight conditions is therefore not only cheaper in terms of the required number of CFD simulations but also in terms of the surrogate model construction time. The failure of a single simulation at any flight condition during the calculation of the weighted drag coefficient can cause a significant problem. The simplest, and perhaps least helpful, method of dealing with such cases is to neglect that design point from the surrogate model. This of course negates any information successfully gathered at the other flight conditions. If the drag coefficient was successfully calculated for the cheap flight condition this data could be readily used to enhance the accuracy of the first level of the co-kriging model. Alternatively, imputation[3] could be employed and the value of a missing drag coefficient predicted based on a surrogate model of that flight condition. Using traditional multi-point design this would require the construction of a separate surrogate model for each flight condition and would prove to be an expensive process. Constructing a co-kriging model for the purposes of imputation is much cheaper as the first data level has been constructed already and, as with the constraints, only a subset of data points is used in tuning the second data level. Fig. A flowchart of the multi-point co-kriging design optimization process In summary the proposed multi-point optimization strategy, shown graphically in Figure, commences as any surrogate modeling strategy does, with the definition of an appropriate sampling plan. An appropriate space filling subset of this plan is then selected[25] and defines the design points analyzed at all necessary flight conditions with the larger sampling plan analyzed at a single flight condition. The geometries corresponding to each of the design points are then simulated at the desired conditions in parallel, thereby making effective use of any available compute cluster. With the performance data obtained the multi-point design objective function can be calculated and used to construct a co-kriging model. Any other required surrogate models of constraints or 6

7 other objectives are also constructed at this stage. These models are then searched for a series of update points which are then evaluated at single or multiple design conditions. The selection of these update points could be based solely on the model s predictor, the probability of improvement or expected improvement. With the performance at the update points calculated the process then repeats until a desired stopping criterion, such as a fixed budget of evaluations, is reached. IV. Airfoil Design Cases A. NURBS Based Parameterization In order to investigate the performance of the proposed multi-point optimization strategy a series of test design problems is required. Only two-dimensional airfoil optimizations are considered within this paper so as to permit both a larger number of investigations and to allow meaningful averaging of the results. Original RAE 2822 Aerofoil NURBS Parameterisation NURBS Control Polygon Fig. 2 NURBS based parameterization of the RAE-2822 airfoil The first test case is based upon a 3 variable parameterization of the RAE-2822 airfoil using Non-Uniform Rational B-Splines (NURBS). Both the upper and lower surfaces are represented as separate NURBS curves which have been optimized to fit the original airfoil as per the method of Lépine et al.[3]. The NURBS curves, control polygon and original airfoil are presented in Figure 2. The parameterization permits the movement of the control points which represent the polygon of Figure 2. The leading and trailing edge control points remain fixed while the control points governing the leading edge curvature are permitted to move only vertically to maintain curvature continuity at the leading edge. The remaining control points are permitted to move both vertically and horizontally resulting in a parameterization with a total of 3 variables. The above geometry is analysed using the viscous Garabedian and Korn (VGK) solver. VGK comprises the generation of a 6 3 conformal mapping and a finite difference solution of the full potential equations coupled with integral methods for both laminar and turbulent boundary layers[32]. A comparison between a VGK simulation of the RAE-2822 airfoil and experimental data taken from Cook et al.[33] at Mach.725, a Reynolds number of and fixed lift coefficient of.658 is presented in Figure 3. VGK predicts the majority of the upper and lower surface pressure very well with the exception of an under prediction of the shock strength and the exact shock position. The experimental data indicates a drag coefficient of.7 at these flight conditions while VGK predicts a drag coefficient of.3, an error of approximately 4.2%. Pitching moment is also predicted reasonably well, with VGK predicting a coefficient of and the experimental data indicating a pitching moment of -.9, an error of 4.9%. 7

8 .5.75 C p.75 Experimental VGK Fluent x/c Fig. 3 Comparison of a VGK and Fluent simulation of the RAE-2822 airfoil at Mach.725, Reynolds no and fixed lift coefficient of.658. VGK is an attractive solver for the investigation of airfoil design optimization techniques due to its speed. A single airfoil simulation using VGK will take approximately one second on a desktop computer. This allows for a considerable number of aerofoil simulations to be carried out at a wide variety of flight conditions in order to assess the performance of the proposed co-kriging based multi-point design optimization process. B. Free-form Deformation Based Parameterization While VGK is an extremely useful tool it does suffer from instabilities when attempts are made to simulate an airfoil at high transonic conditions. To this end a second design case is considered employing Reynolds-Averaged Navier-Stokes (RANS) simulations. While capable of simulating airfoils over a much wider range of design conditions than VGK, each analysis takes approximately 2 minutes to complete therefore a different parameterization with a smaller number of design variables is considered. The baseline RAE-2822 airfoil was meshed using a 33 6 Cartesian grid and enclosed within a lattice of 35 free-form deformation[34] control points. Deformations applied to this control lattice also therefore deform both the airfoil geometry and the mesh nodes. A similar process has been previously employed by Huyse et al.[35] in the optimization of 2D airfoil sections, Lassila and Gianluigi[36] in the inverse design of the NACA 2 airfoil and by Widhalm et al.[37] in the optimization of a flying wing. The airfoil leading and trailing edges are coincident with the edges of the control lattice, as shown in Figure 4(a). By fixing all of the control points along each edge of the control lattice the airfoil is fixed within the computational domain and the validity of the mesh is ensured by preventing grid intersections. (a) (b) Fig. 4 Plots of a) the original RAE-2822 aifoil with control lattice and b) a deformed airfoil and control lattice The control points on the 2 nd and 4 th lines of the control lattice are permitted to move vertically 8

9 by up to ±% of the chord. This equates to a total of design variables which can control the airfoil thickness and camber distributions. An example deformation of the baseline RAE-2822 airfoil is presented in Figure 4(b). Simulations of the deformed mesh are carried out within the commercial CFD package Fluent with adaptive meshing to aid the capturing of shockwaves. Figure 5(a) shows the Cartesian grid for the baseline RAE-2822 airfoil. This has then been deformed using the lattice control points of Figure 4(b) creating the deformed mesh of Figure 5(b). After completion of the CFD simulation at Mach.85 and an angle of attack of the adapted grid presented in Figure 5(c) is produced which captures the shockwaves on the upper and lower surface better than the baseline mesh. Such adaption is useful as it allows the computational domain to adjust itself independently of both the initial flow conditions and the initial mesh. (a) (b) (c) Fig. 5 Plots of a) the original structured mesh around the RAE-2822 airfoil, b) an example of a deformed geometry and mesh, c) the same mesh after adaption All of the CFD simulations carried out using Fluent employ the Spalart-Allmaras turbulence model with a total of 4,5 iterations and mesh adaption after, 75,,,,25,,5 and 2,5 iterations. A single simulation takes approximately 2 minutes when utilizing two processor cores. Figure 3 also compares the experimental pressure distributions of the RAE-2822[33] airfoil with those obtained from Fluent at the same Mach number and lift coefficient. Fluent predicts a drag coefficient of.2 and a pitching moment coefficient of -.97 which equate to errors of approximately 4.7% and.9% respectively over the experimental results. As one can observe from Figure 3 the Fluent simulation results in a rather similar prediction of the pressure distribution to that produced by VGK. Fluent however, predicts the upper surface shockwave slightly closer to the experimental result. 9

10 V. Two Variable Multi-point Example To demonstrate the basic multi-point co-kriging process, consider a two variable subset of the NURBS parameterization presented in Section IV A. Considering only two variables provides a simple test for the process and allows for a graphical comparison of the resulting surrogate models. Variables and 2 of the 3 variable parameterization are considered as these constitute two of the most active variables in the design problem. This is perhaps unsurprising considering that these variables adjust the y-coordinates of the 5 th and 6 th knots of the control polygon representing the upper airfoil surface, in the approximate region of the shockwave. Table Flight conditions for the multi-point design optimization of an airfoil using VGK. Mach No. C l Weighting (w) C m Lower Bound The weighted drag coefficient in this example is constructed from a combination of four different flight conditions which are presented in Table along with their relative weightings. A traditional kriging model was constructed of the weighted drag coefficient using a DOE of 5 points from a random Latin Hypercube. This required a total of 6 flight condition evaluations, four for each of the 5 different airfoil designs. A co-kriging model was also generated using a DOE of the same 5 points at flight condition one and a four point optimal subset for which the performance at the other three design points was evaluated and hence the true weighted drag coefficient known. The surrogate model of the single design point was therefore augmented by a few points for which the true weighted drag coefficient is known. Constructing such a model requires a total of 27 flight condition evaluations, a saving of 55% over that of the traditional kriging model. A grid of true weighted drag coefficients was also generated to provide a dataset by which the accuracy of the resulting kriging and co-kriging models could be determined. Table 2 Comparison of kriging and co-kriging based surrogate models of the weighted drag and pitching moment responses for a four point design case. Cd wtd C m C m2 C m3 C m4 Strategy r 2 RMSE r 2 RMSE r 2 RMSE r 2 RMSE r 2 RMSE Kriging Co-Kriging This process was repeated a total of 5 times, with a different 5 point DOE generated each time. Both the r 2 correlation and the root mean squared error (RMSE) were calculated for each of the resulting surrogate models and averaged, these results are presented in Table 2. As the results of Table 2 indicate there is a small reduction in the accuracy of the surrogate model when co-kriging is employed with the r 2 correlation reducing from.873 to.846 and the RMSE increasing from to Even with this apparent reduction in global accuracy the model could still be considered more than adequate for the purposes of optimization. Figure 6 presents graphically the true response along with examples of the kriging and co-kriging models. Figure 6(d) graphically demonstrates the similarities of the kriging and co-kriging models of Figures 6(b) and 6(c). As one can observe, the co-kriging model is quite similar to the kriging model over the majority of the design space. The only exception being in the region of x =, x 2 = where there is very little information about the true response due to failures of the corresponding VGK simulations. Prediction of the pitching moment coefficient was also considered although with a slightly modified methodology. Rather than predicting a weighted combination, as per surrogates of the weighted drag coefficient, the co-kriging model attempts to precisely predict the pitching moment at a defined

11 flight condition. The lowest level of the co-kriging model therefore consists of a 5 point DOE of the pitching moment at Mach.7 and C l =.45 with the second level of the model consisting of a subset of four points where the pitching moment is known at the desired flight condition. This therefore results in the generation of one kriging model and three co-kriging models C d wtd. C d wtd..8.5 x 2 x x 2 x.5 (a) (b).4 x 3 C d wtd x 2 x.5.5 x 2 x.5 (c) (d) Fig. 6 Surface plots of a) the true, b) the kriging and c) the co-kriging predictions of an example two variable design space and d) the absolute difference between the kriging and co-kriging models Once again both the mean r 2 correlation and RMSE are presented in Table 2 and demonstrate that there is very little difference between the models of pitching moment generated via kriging and co-kriging. Thus a series of reasonably accurate response surfaces of both the objective function and constraints have been constructed for a considerable reduction in the number of flight condition evaluations. VI. Thirty Variable Airfoil Optimization In order to discern the impact of the presented surrogate modelling strategy on an actual optimization the optimization of the complete 3 variable NURBS parameterization is now considered for a weighted drag coefficient based upon the four design points presented in Table. The traditional kriging based optimization was commenced with an initial 5 point sampling of the weighted drag coefficient response using a random Latin Hypercube. A further 3 objective function evaluations were used as updates to the surrogate model in regions of interest. Updates were applied to the model in optimal regions as indicated by the kriging model s prediction of the objective function. A genetic algorithm was used to search for these update points and a KMEANS clustering analysis was performed to provide update points per cycle. The alternate hyperparameter tuning strategy of Toal et al.[38] was employed where the hyperparameters of the kriging model were retuned after every other batch of updates was evaluated. The best 5 points were used in the tuning of the hyperparameters while the complete database was used in the predictor. This maintained the overall tuning cost to an acceptable level throughout the optimization. A kriging model of the pitching moment was constructed for each of the four flight conditions

12 and these were queried in the constraint evaluations when update points were searched for. A one pass penalty function was used to maintain the pitching moment coefficient constraints shown in Table. The multi-point co-kriging optimization employed a 5 point sampling plan of the Mach.7 and C l =.45 flight condition and an optimal 3 point subset of points for which the true weighted drag coefficient was calculated. A further 3 update points were permitted at the first flight condition with a further 6 evaluations of the true weighted drag coefficient permitted. Once again a genetic algorithm with one pass penalty function was used to search the surrogate model s prediction for update points with points returned after a KMEANS clustering analysis. Of these points, the best two were evaluated at all design conditions while the remaining eight were evaluated only at the first flight condition. This resulted in a total of 72 flight condition evaluations a 6% reduction over the 8 flight condition evaluations used in the traditional kriging strategy. Once again an alternate hyperparameter tuning strategy was employed and a restriction of 5 points for the purposes of tuning maintained. Table 3 provides an overview of the results of these optimizations, and once again these results have been averaged over 5 optimizations, each commencing from a different initial Latin Hypercube. In terms of the mean optimal weighted drag coefficient there is little difference between the optimization methodologies, with the traditional kriging approach achieving a mean value of while co-kriging achieves a slightly reduced mean value of The variance between the final designs is a similar case with the co-kriging approach resulting in a standard deviation of.5 4 and traditional kriging resulting in a standard deviation of Table 3 Summary of optimization results for the 3 variable airfoil optimization. Total no. of Optimum Cd wtd Total tuning % Improvement flight condition evals. Mean Std. Time (hrs) Kriging, % 2.2 Co-Kriging %.4 Extended Co-Kriging % x Kriging Co kriging Extended Co kriging C D wtd Flight Condition Evaluations Fig. 7 Mean optimization history of the traditional kriging methodology and Co-kriging methodology when applied to the 3 variable multi-point airfoil design problem. Both of the presented optimization strategies could therefore be considered to result in a broadly similar improvement over the baseline RAE-2822 airfoil. However, it should be stressed once again that in terms of the CFD analyzes carried out, the co-kriging methodology used only 4% of that 2

13 used by the traditional kriging strategy and only 2% of the true weighted drag calculations. This is demonstrated graphically in Figure 7 where a quite clear improvement in the rate of convergence of the multi-point co-kriging methodology over traditional kriging can be observed. The co-kriging optimization strategy employed above utilized a proportion of the available update budget in the evaluation of additional infill points for the lowest level of the co-kriging model. Given the large number of variables in the problem and no prior knowledge of the accuracy of the initial co-kriging model, this was deemed an appropriate strategy as it allowed the accuracy of the surrogate to be improved in multiple locations as update points were evaluated while reducing the overall number of simulations. Alternatively it could be assumed that the first level of the cokriging model requires no further updates other than those evaluated as part of complete weighted drag coefficient evaluations. Such a strategy could result in approximately four evaluations of the true weighted drag coefficient in each batch of update points whilst maintaining an overall reduction of 6% in the total number of flight condition simulations. An increase in the total number of true weighted drag evaluations might therefore improve the optimal designs found by the multi-point co-kriging strategy beyond that of the traditional kriging approach. Table 3 and Figure 7, for example, presents the performance and optimization history when the co-kriging optimization is extended beyond that used in the original optimization. By continuing the optimization for a further 3 batches of update points, where again only two points are evaluated at each of the four design points, the co-kriging optimizations surpass the traditional kriging optimizations. The extended co-kriging optimizations, for example, achieve an average 3.3% improvement in weighted drag coefficient to traditional kriging s 3.26% improvement. This is all the more impressive when it s observed that this is achieved at a 33.3% reduction in the total number of airfoil analyzes. The variances between the final designs for each of the three optimizations presented in Table 3 are relatively large due to the poor convergence of a number of the optimizations. After a number of iterations these optimizations fail to improve upon the current best design which manifests as the presented variances. The stalling of these optimizations could be due to a number of factors. As update points are selected based on the surrogate models prediction of the weighted drag coefficient the optimization may become trapped in a local minimum. Expected improvement[39], for example, may better balance exploitation and exploration. The optimization may also become stalled by the presence of constraint violations and the distortive effect of noise in the design space. The total time spent tuning the hyperparameters of the co-kriging optimization has been reduced by approximately 6.6% over that of the traditional kriging optimization. As suggested in Section III there is a reduction in the tuning time due to the consideration of the pitching moment constraints as a series of co-kriging models based upon a common kriging model of the response at flight condition one. This results in a series of smaller matrix inversions in the calculation of the concentrated likelihood for the pitching moment models of flight conditions, two, three and four. This improvement however is somewhat countered by the re-tuning of the initial level of the co-kriging model predicting the weighted drag coefficient. Naturally as the extended co-kriging optimization employs twice the number hyperparameter optimizations the total tuning time is approximately twice that of the basic co-kriging approach. The designer must therefore carefully consider whither or not the reduction in cost of the CFD simulations is out weighed by the increase in tuning cost before considering such a strategy. VII. Variable RANS Optimization The 3 variable optimization presented previously considered a series of four design points in a mildly transonic flow regime. As discussed, VGK is limited to a maximum Mach number of approximately.725, so in order to study the multi-point co-kriging process at higher transonic flow conditions a variable RANS optimization employing the FFD based parameterization presented in Section IV B is now considered. Table 4 presents the design points considered for this particular optimization. As the optimization is carried out for a fixed lift coefficient a series of sub-analyzes are required to attain the desired lift. This results in a significant additional overhead with the evaluation of a single design condition taking approximately.7 hours. To reduce the computational burden the multi-point optimization considers only three design points. While at first this appears rather elementary it should be 3

14 observed that compared to the previous examples the design points extend much further into the transonic regime where the effect of shockwaves can have a significant impact on the response of the design space. A design point at a relatively low Mach number is also included to extend the range of design points further and to more closely emulate a typical multi-point design process[4 7, ]. The co-kriging model is therefore attempting to recreate a more complex response and the current example represents a significant test for the design methodology. Table 4 Airfoil design points for the variable RANS optimization. Mach No. C l Weighting (w) C m Lower Bound Once again the co-kriging approach is compared directly to a traditional kriging based strategy. An initial kriging model is constructed from a 5 point Latin hypercube based design of experiments with a further update points permitted. As with the previous test case, the surrogate model is searched using a genetic algorithm and a K-MEANS clustering routine extracts potential update points. The weighted drag coefficients at these points are evaluated in parallel and used to update the model. Pitching moment constraints are once again enforced and the maximum thickness is now constrained to a minimum of 2.% chord, which corresponds to that of the baseline RAE airfoil. The surrogate models constructed of pitching moment and maximum thickness include designs which both meet and violate the constraints and a one pass penalty function ensures that these constraints are met as the genetic algorithm searches for updates. An alternate hyperparameter tuning strategy is again employed. The co-kriging strategy employs a 5 point sample plan of the drag at Mach.75 with an optimal 3 point subset having the true weighted drag coefficient evaluated. As with the traditional kriging approach, the co-kriging model is searched using a genetic algorithm with a one pass penalty function to extract potential update points. Based on the predicted objective function value for these points, the weighted drag coefficient is evaluated for the best three with the remainder evaluated at Mach.75. Both levels of the co-kriging model are therefore updated as the optimization progresses with the strategy using approximately 4% less CFD simulations than traditional kriging. The increase in the number of true weighted drag evaluations compared to the 3 variable optimization of Section VI is to attempt to better deal with the more complex response of the design space which results from considering such a wide variety of design conditions spanning both the transonic and low speed regimes. Table 5 Results for the variable RANS multi-point optimization. Kriging Co-Kriging Extended Co-Kriging Opt. No. Cd wtd % Improvement Cd wtd % Improvement Cd wtd % Improvement A total of 3 optimizations were carried out each commencing from a different initial sampling plan, the results for each of these are presented in Table 5 with the individual optimization histories presented in Figure 8. Based on the results of Table 5 the traditional kriging strategy offers a larger degree of improvement over the baseline airfoil than the co-kriging strategy. The design resulting from the best performing kriging approach, for example, achieves an 8.3% improvement while the best performing co-kriging design achieves only a 5.7% improvement. It should be emphasized however, that the co-kriging approach has employed 4% less CFD evaluations than traditional kriging which equates to only 4% of the complete weighted drag 4

15 calculations. This is illustrated graphically in Figure 8 where, in a manner similar to the 3 variable optimizations of Figure 7, the co-kriging approach demonstrates a faster rate of convergence towards an optimum. After 6 true objective function evaluations all of the co-kriging based optimizations have out performed each of the kriging based optimizations Kriging Co Kriging Extended Co kriging C wtd D Multi point Design Evaluations Fig. 8 Optimization histories for the variable RANS optimizations using kriging and cokriging. By continuing the co-kriging optimizations for a further 3 weighted drag calculations it can be observed that the co-kriging approach now produces better designs than kriging on two out of three of the optimizations. The first and third optimizations now result in a 2.2% and 9.9% improvement respectively. Increasing the number of updates in this manner, however, reduces the efficiency savings over the traditional kriging approach. In this case the extended co-kriging optimization offers only a 4% reduction in the total number of CFD evaluations. However, the optimization convergence histories of Figure 8 show that after only 3 additional true weighted drag evaluations the extended co-kriging optimizations have already out performed the traditional kriging approach. This equates to a 25% saving in the number of CFD evaluations to obtain an equivalent or better design than traditional kriging. As discussed previously in Section VI a more careful consideration of the updating strategy used and the division of resources between the single and multi-point evaluations may further improve the efficiency of the presented co-kriging based strategy. Figure 9 presents the geometry and pressure distributions at design points one and two for the best designs resulting from the kriging, co-kriging and extended co-kriging optimizations. Each of the optimizations has resulted in a similar overall reduction in camber along the length of the airfoil, however, the local camber of the best co-kriging and extended co-kriging designs has not been reduced quite as much close to the leading edge as that of the best kriging design. While similar to the best co-kriging design around the leading edge, the best of the extended co-kriging design results in a slight increase in thickness at 25% and 6% chord. Even though these three designs are broadly similar, the resulting pressure distributions differ quite substantially. This highlights the difficulties of optimizing airfoils at transonic conditions where relatively small geometric changes can result in significant changes in both shockwave position and strength. The best kriging design exhibits a double shock on the upper surface at Mach.75, with a relatively strong shockwave around the 2% chord point and a secondary weaker shock in the same region as the upper surface shock on the original RAE-2822 airfoil. At Mach.8 the same geometry results in both a reduction in the strength of the upper surface shockwave and a 5% movement aft. The lower surface shockwave at Mach.8, however, remains in a similar position to the original airfoil but increases slightly in strength. At Mach.75 the best co-kriging design exhibits a forward movement of the upper surface 5

16 shockwave which remains at a similar strength to the shockwave on the RAE At Mach.8 this design exhibits a reduction in the upper surface shock but, compared to the kriging design, moves only slightly aft. The lower surface shockwave also moves slightly aft but increases in strength y/c C p x/c x/c (a) (b).5 C p.5 RAE 2822 Kriging Design Co Kriging Design Extended Co Kriging x/c (c) Fig. 9 Pressure distributions and a) geometry of the best designs resulting from the kriging, co-kriging and extended co-kriging optimizations at b) design point one and c) design point two The best design resulting from the extended co-kriging optimizations results in the complete removal of the upper surface shockwave at Mach.75. At Mach.8 the upper surface shockwave has moved aft to approximately the same location as is observed with the best kriging design and reduced in strength over the baseline design. Unlike either the best kriging or co-kriging designs, the lower surface shockwave has been moved aft by a more substantial amount, by approximately 5% chord. Each of the presented design optimizations attempts to trade-off a large reduction in drag at Mach.8 with a smaller reduction at Mach.75 whilst simultaneously attempting to limit the impact this has on the drag at Mach.2. Both the best kriging and co-kriging designs offer a substantial reduction in drag over the original airfoil at Mach.8 but at the expense of very little improvement at Mach.75. At Mach.8 the best kriging design reduces the drag coefficient from.33 to.9, while the drag at Mach.75 reduces by a single drag count to.7. Likewise the best co-kriging design results in drag coefficients of. and.27 at Mach.75 and.8 respectively. However, while the best kriging design offers improved drag in the transonic flow regime this is at the expense of an increase in drag at Mach.2. The drag at this flow condition increases from.57 to.9. While the drag coefficient also increases at this flight condition for the best co-kriging design it is rather less severe and increases to.68. The best extended co-kriging design results in a slightly smaller reduction in drag at Mach.8, reducing to.2, but, due to the removal of the shockwave, offers a larger reduction in drag at Mach.75, reducing to.96. The design is also penalized less at Mach.2 with the drag increasing only four counts to.6. Although the final designs presented here are perhaps not as efficient as those found within the literature for the same initial airfoil[4, 4], the parameterization used in this optimization is much more restrictive than those used within the literature. Nadarajah and Jameson[4], for example, 6