Variable-fidelity optimization with design space reduction

Chinese Journal of Aeronautics, 2013,26(4): 841–849

Chinese Society of Aeronautics and Astronautics & Beihang University

Chinese Journal of Aeronautics [email protected] www.sciencedirect.com

Variable-fidelity optimization with design space reduction Mohammad Kashif Zahir, Gao Zhenghong

*

School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China Received 10 May 2012; revised 19 June 2012; accepted 19 October 2012 Available online 2 July 2013

KEYWORDS Airfoil optimization; Curse of dimensionality; Design space reduction; Genetic algorithms; Kriging; Surrogate models; Surrogate update strategies; Variable fidelity

Abstract Advanced engineering systems, like aircraft, are defined by tens or even hundreds of design variables. Building an accurate surrogate model for use in such high-dimensional optimization problems is a difficult task owing to the curse of dimensionality. This paper presents a new algorithm to reduce the size of a design space to a smaller region of interest allowing a more accurate surrogate model to be generated. The framework requires a set of models of different physical or numerical fidelities. The low-fidelity (LF) model provides physics-based approximation of the high-fidelity (HF) model at a fraction of the computational cost. It is also instrumental in identifying the small region of interest in the design space that encloses the high-fidelity optimum. A surrogate model is then constructed to match the low-fidelity model to the high-fidelity model in the identified region of interest. The optimization process is managed by an update strategy to prevent convergence to false optima. The algorithm is applied on mathematical problems and a two-dimensional aerodynamic shape optimization problem in a variable-fidelity context. Results obtained are in excellent agreement with high-fidelity results, even with lower-fidelity flow solvers, while showing up to 39% time savings. ª 2013 Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA. Open access under CC BY-NC-ND license.

1. Introduction With advances in computational fluid dynamics (CFD) and computer hardware, CFD has now become an integral part of the aircraft design process. The high-fidelity (HF) aerodynamic data it provides has contributed to cutting aerodynamic design cost and time scales by reducing the number of required * Corresponding author. Tel.: +86 29 88495971. E-mail addresses: [email protected] (M.K. Zahir), zgao@nwpu. edu.cn (Z. Gao). Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

wind tunnel tests.1 However, major benefits can be achieved if CFD is included in the conceptual design phase, where as many as tens of thousands of analyses must be performed, and global optimization can play a key role. Given the high cost of CFD and optimization, a prominent area of research today is to find ways to reduce the computational time while retaining the high fidelity of the analysis. In the area of aerodynamic optimization, the variable-fidelity (VF) (also called multi-fidelity) method has quickly grown in popularity.2–22 Variable-fidelity and other model management methods have been developed to solve optimization problems that involve simulations with high computational expense.9,11 In many engineering design problems, differing levels of fidelity can model the system of interest. Higher-fidelity models typically incorporate more detailed physics and are computationally expensive to evaluate than lower-fidelity (LF) models. Lower-fidelity models are typically much cheaper to evaluate,

1000-9361 ª 2013 Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA. Open access under CC BY-NC-ND license. http://dx.doi.org/10.1016/j.cja.2013.06.002

842 but designs produced by using these models neglect important physical effects included in more expensive higher-fidelity models. In aircraft design, the Navier–Stokes and Euler equations are examples of two computational models with different fidelities, where the latter is obtained by removing the viscosity terms from the Navier–Stokes equations. Variable-fidelity optimization (VFO) has emerged as an attractive method of performing both high-speed and highfidelity optimization and the past three decades have seen rapid increase in its development and usage.2–23 These algorithms attempt to leverage information from computationally inexpensive low-fidelity models to reduce the time required to converge to the optimum of high-fidelity functions. This is usually accomplished by using a low-fidelity solver plus a correction term – the difference between high-fidelity and lowfidelity solvers – modeled by a surrogate model calibrated at selected sample points in the design space.16 A variety of methods have been used for generating these surrogate models including Kriging, 4,7–9,11,14,15,18,24–30 radial basis functions (RBFs),28,31 neural networks,4,9,15,31,32 and support vector regression (SVR).33,34 Insightful reviews of surrogate models and variable-fidelity methods have appeared in the literature.11,19,23,24,35 Advanced engineering systems, like aircraft, are defined by tens or even hundreds of design variables. Building an accurate surrogate model for use in such high-dimensional optimization problems is a difficult task. In essence, a surrogate model is a data-fit and is only accurate in the region where it is adequately trained. Intelligent techniques of generating sampling plans (also called design of experiments or DoE) – a sparse set of points where the surrogate will be trained – exist to achieve uniform coverage of the design space. However, if the problem being dealt with has many dimensions, the number of training points required for reasonable uniform coverage of the design space rises exponentially – the so-called curse of dimensionality.23,36 A surrogate-based optimizer may converge to a local optimum, or worse a false optimum, due to inaccuracies of the surrogate model.25,37 While it is easy to control the range of validity of the surrogate model in gradient-based optimization algorithms by using ad hoc move limits or a trust-region framework, it is not straightforward in global optimization schemes like genetic algorithms (GAs).38 This issue has been addressed by other researchers in the past. Ratle39 uses a heuristic convergence criterion to determine when the approximate model must be updated. The basic idea is that the convergence of the search process should be stable and therefore, the change of the best solution should not be larger than a user-defined value. This, however, relies on the assumption that the first sets of data points are weakly correlated with the global optimum of the original problem, which is not necessarily true for highdimensional systems. Others perform on-line learning of the approximate model based on a prescribed generation delay.2,40 Another concept of evolution control is applied by Jin et al.37 where the surrogate model and the original fitness function are both used in tandem during the evolutionary process based on a fixed32 or adaptive17,37,41 criterion. One way to solve this problem is to limit the range of design variables so that the shape being modeled is sufficiently simple to be approximated from very sparse data.36 This begs the question: what should these limits be? This paper presents a technique of intelligently narrowing down a search space to

M.K. Zahir, Z. Gao a smaller region of interest using low-fidelity methods. The surrogate model developed in this small region is found to be very accurate. It is then combined with several update strategies and used in a variable-fidelity optimization context to predict the global optimum in a larger design space. The method is demonstrated on a two-dimensional (2D) aerodynamic design optimization problem with good accuracy. The remainder of the paper is organized as follows. Section 2 describes the design space reduction (DSR) technique. Update strategies for the surrogate model are discussed in Section 3. Optimization of mathematical functions is performed in Section 4, and a more complex 2D aerodynamic optimization problem is introduced in Section 5 along with the analytical methods and tools used. Airfoil optimization results are presented in Section 6 followed by conclusions in Section 7. 2. Design space reduction For a surrogate model to be useful in an optimization context, it is important that the surrogate model is accurate at the sequence of iterates generated by the search algorithm as it converges towards the true optimum. How the model performs at other points in the parameter space is of no concern in this specific context.38 This observation provides the pretext for development of the design space reduction technique. In previous research, the authors have pointed out the importance of selecting a low-fidelity solver capable of predicting the aerodynamic behavior that is consistent with the high-fidelity solver.2,42 It is reasoned that such a low-fidelity solver will converge towards the region of the high-fidelity optimum – the desired region of interest. This region can be determined by examining the search trajectory of the low-fidelity solver. An accurate surrogate can be created in this small region of interest and thereafter be used for variable-fidelity optimization. The design space reduction algorithm proceeds as follows: (1) the optimization is initially performed using a genetic algorithm (GA) coupled to a low-fidelity solver in a large design space; (2) the search trajectory is analyzed to identify the reduced design space; (3) this is followed by another optimization using the low-fidelity solver and the surrogate model in a variable-fidelity context. The flowchart of the complete variable-fidelity framework with design space reduction (DSR– VFO) is shown in Fig. 1. Fig. 2 shows a sample design space from a low-fidelity optimization run on a 10-variable problem. Since a GA progressively converges towards the optimum, only the population members for the last five generations are analyzed. Three methods are considered for selecting a small region for generating the surrogate model: (1) The extreme minimum and maximum values of each design variable. (2) A normal distribution fit to the design variables with 95% confidence levels as the bounds. (3) A 5% region around the low-fidelity optimum point. The upper and lower bounds are defined by a region that forms a 5% locus around this point. Fig. 2 shows the design space produced by the above methods. The location of the high-fidelity optimum is also shown for reference. All methods produce regions smaller than the

Variable-fidelity optimization with design space reduction

843

Fig. 2 Reduced design spaces obtained from a low-fidelity solver. The location of the high-fidelity optimum is also shown.

in the larger region defined by Method 1. The high-fidelity optimum may still lie outside this region in one–two dimensions as will be seen later. Positive results are obtained even in this scenario. The design space reduction technique uses the narrow region defined by Method 3 for creation of the surrogate model and the larger region defined by Method 1 for performing the optimization. It is reasoned that the very narrow region of the design space chosen to generate the surrogate model is small enough to yield an accurate surrogate model representation of the actual function. However, using a smaller design space for the surrogate model and a larger one for the optimization is non-intuitive and, principally, incorrect in a variable-fidelity context. The surrogate model cannot be expected to provide correct predications outside the training space. Several surrogate update strategies are devised to rectify this problem. These are discussed next. 3. Design space reduction surrogate model update strategies Three update strategies are considered for updating the design space reduction surrogate model during the course of the optimization: (1) update after a fixed generation interval – a legacy method from previous research2,42; (2) training of all points outside the surrogate creation space at every generation; (3) a combination of the above methods. 3.1. Update strategy 1

Fig. 1 Flowchart of the complete variable-fidelity framework with design space reduction (DSR–VFO).

original design space. Methods 2 and 3 produce very similar regions, while Method 1 produces a larger region. It is also clear that the true (high-fidelity) optimum may not completely fall within the regions defined by Methods 2 and 3 but may fall

This is a very simple method. The variable-fidelity and highfidelity predictions of the optimum point are checked after a fixed generation interval. If they are within a user-defined tolerance, the optimization is continued; otherwise, the surrogate model is updated by performing high-fidelity evaluations on the GA population and refitting the surrogate before continuing the optimization. It is possible that points outside the original training space would be trained during the update step. However, if the optimum point lies within the surrogate space and is within the tolerance, any erroneous prediction outside the surrogate creation space would not be updated and the

844

M.K. Zahir, Z. Gao

optimization may easily yield a false optimum. Moreover, this method involves use of the surrogate model outside its creation space (at least until it is updated) and any positive results obtained can only be regarded as fortuitous. It is clear that a more consistent method is required.

Table 1

Parameters used for the artificial error.

Hartman

a

A

x

3 6

[4,4,4] [4,4,4,4,4,4]

0.5 0.5

10 25

3.2. Update strategy 2 In this method, all points outside the surrogate creation space are trained at every generation. However, to keep the number of high-fidelity evaluations to a minimum, the surrogate space is expanded as soon as a particular point is trained, so any point that is outside the original creation space but within the updated space is not trained. This method circumvents the dubious use of the surrogate model outside its training (creation) space. However, updating the surrogate space using the first occurrence of an outside point assumes that the surrogate is capable of making accurate predictions in that ‘‘expanded-outside space’’. This may not always be true. This leads directly to the definition of a third strategy. 3.3. Update strategy 3 Update Method 3 is a combination of Methods 1 and 2. All points outside the surrogate creation space are trained at every generation and the surrogate space is expanded as soon as a particular point is trained. The variable-fidelity and high-fidelity predictions of the optimum point are checked after a fixed generation interval. If they are within a user-defined tolerance, the optimization is continued; otherwise, the surrogate model is updated by performing high-fidelity evaluations on the GA population and refitting the surrogate before continuing the optimization. In case the GA population is trained, it implies that the surrogate needs further training at points where the tolerance limits have been violated. The surrogate space is, therefore, shrunk to a restricted state so that it only encloses the training points (including points trained in this generation). Any new point xmin < x < xmax will be trained at the next generation. Here xmin and xmax are defined by: 

xmin ¼ maxðLB; maxðcurrent populationÞÞ xmax ¼ minðUB; minðcurrent populationÞÞ:

ð1Þ

where LB and UB are the current lower and upper bounds of the surrogate creation space. This method circumvents the dubious use of the surrogate model outside its training (creation) space and also allows subsequent updates. It is the chosen update strategy for the DSR– VFO framework. 4. Mathematical optimization problems The DSR–VFO framework is applied on a few mathematical test problems to test its abilities. The test problems chosen are the three- and six-dimensional Hartman problems described by Dixon and Szego¨.43 These test problems are smooth and computationally cheap to evaluate, and are merely used for testing the new framework.

Low-fidelity models of these functions are artificially constructed by adding a simple analytical error function to them20: ! n X ð2Þ FLF ðxÞ ¼ FðxÞ þ A sin x ðxi  aÞ2 i¼1

This corresponds to an undulating error radially symmetric about the point a, with amplitude A and frequency x. The parameters a, A and x used for both Hartman functions are shown in Table 1. Fig. 3 shows the reduced design spaces for the Hartman problems. The high-fidelity optimum, obtained from running a GA search with a population size of 30 and a maximum of 100 generations, is also shown. The high-fidelity optimum lies completely within the reduced optimization space (Method 1) for the Hartman-6 function, but lies outside the first dimension (i = 1) for the Hartman-3 function. Kriging surrogates36,44 trained in the reduced design space obtained from Method 3 are used in the DSR–VFO framework to yield the optimization results shown in Table 2. The variable-fidelity optimum of the Hartman-3 function is better than that obtained by the high-fidelity optimization directly, and the variable-fidelity optimum of the Hartman-6 function is within 1% of the high-fidelity value. The DSR–VFO technique is applied to a more complex engineering problem in the next section. 5. Aerodynamic optimization problem The DSR–VFO framework is used to find the optimum of a 2D aerodynamic shape optimization problem at a transonic Mach number of 0.729 and a fully turbulent Reynolds number of 6.5 · 106. The objective is to minimize the drag coefficient, CD at a constant lift, CL = 0.6 ± 0.01, subject to thickness and pitching moment constraints. The RAE2822 is used as the initial airfoil to start the optimization. The optimization problem is stated as follows: Minimize: CD ðxÞ subject to : xl 6 x 6 xu 0:59 6 CL 6 0:61 Cm P 0:122 ðt=cÞmax P 12:11%

ð3Þ

The constant lift constraint is maintained by varying the angle of attack, a, of the airfoil as described in a previous work [2]. Other constraints are imposed by adding penalty terms to the fitness function:     fðxÞ ¼ CD jCL ¼const þ max 0; ðt=cÞ  ðt=cÞmax þ max 0; Cmmin  Cm ð4Þ

Variable-fidelity optimization with design space reduction

845 arbitrary three-dimensional geometries – encompassing a very large design space with a relatively few scalar parameters. In fact, it captures the entire design space of smooth airfoils, axisymmetric bodies, and nacelles, and is capable of producing the round nose and sharp trailing edge required of airfoils. The CST representation is thoroughly described in Ref.45 and only the final equations are mentioned here. Defining normalized coordinates w = x/c, f = z/c, and c as the chord length, the upper and lower surfaces of the airfoil, fupper and flower , are defined as: pffiffiffiffi 8 z fupper ðwÞ ¼ wð1  wÞSu ðwÞ þ w ucTE > > > > n > X > > > xui Si ðwÞ > Su ðwÞ ¼ < i¼1 pffiffiffiffi z > flower ðwÞ ¼ wð1  wÞSl ðwÞ þ w lTE > c > > > n > X > > > Sl ðwÞ ¼ xli Si ðwÞ :

ð5Þ

i¼1

where zuTE and zlTE are the upper and lower surface ordinates at the trailing edge (TE) of the airfoil; Si is the term of a Bernstein polynomial of order n, yielding n + 1 terms of the form: 8 ni i > < Si ðwÞ ¼ Ki w ð1  wÞ ; i ¼ 0; 1; . . . ; n   n n! > ¼ : Ki  i!ðn  iÞ! i

ð6Þ

The coefficients xui and xli in Eq. (5) are the 2n + 2 design variables. Ten CST variables (five for each surface of the airfoil) are used in this study. The upper and lower bounds for low-fidelity optimization are: RAE2822  0:1 6 xi 6 RAE2822 þ 0:2 Fig. 3 Reduced design spaces of the Hartman 3 and Hartman 6 functions. HF optima are also shown.

Table 2 Comparison of DSR–VFO and high-fidelity optimization results of the Hartman functions. Hartman

FHF(x)

FVF(x)

3 6

3.8432 3.3202

3.8546 3.2984

where ðt=cÞ is the minimum allowable thickness of 12.11% (the (t/c)max of the initial airfoil) and Cmmin is 0.122, the moment coefficient obtained from high-fidelity optimization in previous studies.2 The class-function/shape-function (CST) scheme42,45 is used for geometric parameterization. This is a recently developed geometric transformation technique capable of representing round-nose/sharp-aft-end airfoil geometries as well as other classes of geometries exactly by analytic well behaved and simple mathematical functions having easily observed physical features.45,46 The method is capable of representing a large class of geometries – including two-dimensional airfoils, axisymmetric bodies or nacelles, as well as rather

ð7Þ

where RAE2822 is the vector of CST parameter values for the initial RAE2822 airfoil and x is the combined vector of design variables [xu, xl]. An indigenously developed CFD code capable of being run in Reynolds averaged Navier–Stokes (RANS) and Euler modes – using the LU-SGS time stepping scheme, the Roe upwind scheme, and multi-grid acceleration – is used to perform all computations. A dense grid (216 · 44) RANS solver is used for high-fidelity evaluations; the grid size is selected after obtaining good agreement in the surface pressure distribution and aerodynamic coefficients at transonic conditions.47 Euler and RANS solvers with several grid sizes are used to produce a suite of LF solvers. All solvers use an automatically generated C-type mesh extending 20 chord lengths downstream from the trailing edge. The first grid line is 2 · 106 units from the airfoil surface for the RANS solvers and 0.01 units for the Euler solvers. The k–x turbulence model is used for turbulence modeling in the RANS solutions. A Kriging surrogate36,44 trained at points obtained from a Sobol sequence48 is used to provide the correction between the high-fidelity and low-fidelity solvers as described in the authors’ previous work.47 Surrogate accuracy is checked by calculating the RMSE (Root Mean Square Error) and R2 for a validation dataset generated using a uniform Latin hypercube sample consisting of ntest = nvar · 10 points.47 The number of sample points used to create the variablefidelity surrogate are such that the R2 > 0.9. For brevity,

846

M.K. Zahir, Z. Gao

Fig. 4 Reduced design spaces obtained from the low-fidelity solvers. The location of the high-fidelity optimum is also shown.

the Kriging equations are not mentioned here. Forrester el al.36 describe more detail about Kriging. The Kriging model in this study uses a constant regression term and a Gaussian correlation model. The p term is 2 for all dimensions and h is optimized in the range 102 6 hi 6 200, i = 1,2, . . ., nvar.

Fig. 5 State of the design space for the DSR–VFO solvers with update Method 3 at the end of the optimization. All trained and untrained points are shown along with the locations of the highfidelity and variable-fidelity optimum points.

Variable-fidelity optimization with design space reduction The surrogate model is created using the surrogate toolbox49 for MATLAB. Optimization is performed using MATLAB’s GA implementation in the global optimization toolbox. A population size of 30 and a uniform crossover probability of 0.8 are used. The maximum number of generations is set to 100. All simulations are performed in parallel on a cluster of 14 nodes. The variable-fidelity and high-fidelity lift coefficient (CL), drag coefficient (CD), and pitching moment coefficient (Cm) are required to agree within 0.01 for CL and Cm and 0.0001 for CD every 10 generations, or else the entire GA population is evaluated with the high-fidelity solver, and the surrogate model refitted, before continuing the optimization. Optimization is terminated once the cumulative change in the fitness function value over 50 generations is less than or equal to 106, or when 100 GA generations are completed (whichever comes sooner). 6. Results and discussion Three low-fidelity solvers are used in this study – Euler 60 · 20, Euler 160 · 24, and RANS 160 · 24 (the numbers indicate the grid size). The reduced design spaces, obtained by analyzing the last five generations of the low-fidelity optimization runs, are shown in Fig. 4. The high-fidelity optimum is also shown. It is observed that the high-fidelity optimum

Table 3

lies completely within the reduced optimization space (Method 1) for the RANS-160 · 24 solver, and is outside the reduced optimization space for the other low-fidelity solvers in one or two dimensions (the low-fidelity optimum is outside in dimensions 5 and 10 in Fig. 4(a); and outside in dimension 5 in Fig. 4(b)). Fig. 5 shows all design points that are or are not trained at the time of convergence of the optimization using update strategy 3. The variable-fidelity and high-fidelity optimum points are also shown for reference. The optimization results are shown in Table 3 and compared to the low-fidelity, highfidelity, and variable-fidelity (without DSR)42 optimization results in Table 4. All variable-fidelity solvers produce optimums close to the high-fidelity optimum with 17%–39% time savings. This is in stark contrast to previous work where the variable-fidelity Euler 60 · 20 and variable-fidelity Euler 160 · 24 solvers failed to even converge within the allotted number of generations.42 This clearly shows that the DSR surrogate method allows use of lower-fidelity low-fidelity solvers to produce accurate results in less time. It may appear to be surprising that the aerodynamic coefficients obtained using the variable-fidelity and high-fidelity solvers (shown in Table 3) are the same. This is a direct result of the update strategy that requires the variable-fidelity and high-fidelity results to agree within a given tolerance (0.01 for CL and Cm and 0.0001 for CD).

The DSR–VFO results using surrogate update strategy 3. The HF optimization results are also shown for reference.

Configuration

VF, Euler 60 · 20, 156 samples VF, Euler 160 · 24, 156 samples VF, RANS 160 · 24, 100 samples HF (direct optimization)

Table 4

847

VF optimum result

ðct Þ

HF evaluation at VF optimum point

(%)

Surrogate creation time (Sims + fitting) (h)

Optimization time (including Surrogate update) (h)

Total time (h)

Hours saved (compared to direct optimization)

max

CL

CD

CL

CD

Cm

k

0.60

0.0059

0.60

0.0059

0.121

100.96

12.12

0.357

0.811

1.454

1.359

0.60

0.0058

0.60

0.0058

0.119

102.87

12.11

0.396

0.811

1.705

1.108

0.60

0.0059

0.60

0.0059

0.121

101.16

12.11

0.305

1.219

2.332

0.481

0.60

0.0058

0.121

103.39

12.11

2.813

Comparison of low-fidelity, high-fidelity and variable-fidelity (without DSR), and DSR–VFO optimization results.

Solver configuration

Lift-to-drag ratio, k, evaluated by the high-fidelity solver at the VF/LF optimum

Optimization time (h)

LF

VF old42

DSR–VFO (Update strategy 3)

LF

VF old42

DSR–VFO (Update strategy 3)

VF-Euler 60 · 20 156 samples

93.764

86.6

100.96

0.286

2.057

1.454

VF-Euler 160 · 24 156 samples

99.234

83.632

102.87

0.497

2.273

1.705

VF-RANS 160 · 24 100 samples

96.05

100.54

101.16

0.808

1.709

2.332

HF, RANS 216 · 44 (direct optimization)

103.39

2.81

848

M.K. Zahir, Z. Gao computation time, and with lower-fidelity low-fidelity solvers which indicates that the method is very practical for engineering applications.

References

Fig. 6 Comparison of optimum geometries obtained with highfidelity and the DSR–VFO solver with update Method 3. The original RAE-2822 airfoil is also shown for reference.

Fig. 7 Comparison of surface pressure coefficient distributions on the optimum airfoils obtained using high-fidelity and the DSR– VFO solvers with update Method 3.

The DSR–VFO optimized geometries are shown in Fig. 6 and the corresponding surface pressure coefficient Cp distributions are shown in Fig. 7. All VF optimized geometries resemble the high-fidelity results especially the VF-NS 160 · 24 solver geometry which, almost, exactly matches the · optimum. The pressure distributions also show a shock-free flow on the upper surface of the airfoils. 7. Conclusions

(1) Well-selected low-fidelity solvers can be used to determine a reduced design space for developing a more accurate surrogate model and performing optimization. (2) The DSR surrogate model developed is accurate and does not lead the optimization towards incorrect and false optimums. (3) Variable-fidelity optimization with a surrogate model created in a reduced design space (DSR–VFO) yields an optimum close to the high-fidelity optimum with less

1. Oyama A, Obayashi S, Nakahashi K, Nakamura T. Aerodynamic optimization of transonic wing design based on evolutionary algorithm. In: Third international conference on nonlinear problems in aviation and aerospace; 2000. 2. Zahir MK, Gao ZH. Variable fidelity surrogate assisted optimization using a suite of low fidelity solvers. Open J Optim 2012;1(1):8–14. 3. Xia L, Gao ZH. Application of variable-fidelity models to aerodynamic optimization. Appl Math Mech 2006;27(8):1089–95. 4. Zhang D, Gao ZH, Huang L, Wang M. Double-stage metamodel and its application in aerodynamic design optimization. Chin J Aeronaut 2011;24(5):568–76. 5. Alexandrov NM, Gumbert CR, Green LL, Newman PA, Lewis RM. Approximation and model management in aerodynamic optimization with variable-fidelity models. J Aircr 2001;38(6):1093–101. 6. Balabanov V, Venter G. Multi-fidelity optimization with highfidelity analysis and low-fidelity gradients. AIAA-2004-4459; 2004. 7. Forrester AIJ, Bressloff NW, Keane AJ. Optimization using surrogate models and partially converged computational fluid dynamics simulations. Proc R Soc A, Math, Phys Eng Sci 2006;462(2071):2177–204. 8. Forrester AIJ, So´bester A, Keane AJ. Multi-fidelity optimization via surrogate modelling. Proc R Soc A, Math, Phys Eng Sci 2007;463(2088):3251–69. 9. Gano S, Renaud J, Sanders B. Hybrid variable fidelity optimization by using a kriging-based scaling function. AIAA J 2005;43(11):2422–33. 10. Haftka RT. Combining global and local approximations. AIAA J 1991;29(9):1523–5. 11. Huang D, Allen T, Notz W, Miller R. Sequential kriging optimization using multiple-fidelity evaluations. Struct Multidiscip Optim 2006;32(5):369–82. 12. Hutchinson MG, Unger ER, Mason WH, Grossman B, Haftka RT. Variable-complexity aerodynamic optimization of a high speed civil transport wing. J Aircr 1994;31(1):110–6. 13. Kampolis IC, Giannakoglou KC. A multilevel approach to singleand multiobjective aerodynamic optimization. Comput Methods Appl Mech Eng 2008;197(33):2963–75. 14. Keane AJ. Wing optimization using design of experiment, response surface, and data fusion methods. J Aircr 2003;40(4):741–50. 15. Leary SJ, Bhaskar A, Keane AJ. A knowledge-based approach to response surface modelling in multifidelity optimization. J Global Optim 2003;26(3):297–319. 16. Lehner SG, Lurati LB, Smith SC, Bower GC, Cramer EJ, Crossley WA, et al., Advanced multidisciplinary optimization techniques for efficient subsonic aircraft design. In: 48th AIAA aerospace sciences meeting including the new horizons forum and aerospace exposition. Berlin: Springer, 2010. 17. Lim D, Ong Y-S, Jin Y, Sendhoff B. Evolutionary optimization with dynamic fidelity computational models. Advanced intelligent computing theories and applications with aspects of artificial intelligence. Berlin: Springer-Verlag; 2008. pp. 235–242. 18. Nelson A, Alonso J, Pulliam T. Multi-fidelity aerodynamic optimization using treed meta-models. In: 25th AIAA applied aerodynamics conference; 2007. 19. Queipo NV, Haftka RT, Shyy W, Goel T, Vaidyanathan R, Tucker PK. Surrogate-based analysis and optimization. Prog Aerosp Sci 2005;41(1):1–28.

Variable-fidelity optimization with design space reduction 20. Rajnarayan D, Haas A, Kroo I. A multifidelity gradient-free optimization method and application to aerodynamic design. In: 12th AIAA/ISSMO multidisciplinary analysis and optimization conference; 2008. 21. Robinson TD, Willcox KE, Eldred MS, Haimes R. Multifidelity optimization for variable-complexity design. In: 11th AIAA/ ISSMO multidisciplinary analysis and optimization conference; 2006. 22. Booker AJ, Dennis JE, Frank PD, Serafini DB, Torczon V, Trosset MW. A rigorous framework for optimization of expensive functions by surrogates. Struct Multidiscip Optim 1999;17(1):1–13. 23. Simpson TW, Toropov V, Balabanov V, Viana FAC. Design and analysis of computer experiments in multidisciplinary design optimization: a review of how far we have come – or not. In: 12th AIAA/ISSMO multidisciplinary analysis and optimization conference; 2008. 24. Chung H-S, Alonso JJ. Comparison of approximation models with merit functions for design optimization. In: AIAA/USAF/ NASA/ISSMO symposium on multidisciplinary analysis and optimization; 2000. 25. Jones DR. A taxonomy of global optimization methods based on response surfaces. J Global Optim 2001;21(4):345–83. 26. Jones DR, Schonlau M, Welch WJ. Efficient global optimization of expensive black-box functions. J Global Optim 1998;13(4):455–92. 27. Obayashi S, Jeong S, Chiba K. Multi-objective design exploration for aerodynamic configurations. In: 35th AIAA fluid dynamics conference and exhibit; 2005. 28. Polynkin A, Toropov V. Multiple mid-range and global metamodel building based on linear regression. In: 50th AIAA/ASME/ ASCE/AHS/ASC structures, structural dynamics, and materials conference 17th AIAA/ASME/AHS adaptive structures conference; 2009. 29. Sacks J, Welch WJ, Mitchell TJ, Wynn HP. Design and analysis of computer experiments. Stat Sci 1989;4(4):409–23. 30. Zhong X, Ding J, Li W, Zhang Y. Robust airfoil optimization with multi-objective estimation of distribution algorithm. Chin J Aeronaut 2008;21(4):289–95. 31. Su W, Gao Z, Zuo Y. Application of RBF neural network ensemble to aerodynamic optimization. In: 46th AIAA aerospace sciences meeting and exhibit; 2008. 32. Jin Y, Olhofer M, Sendhoff B. On evolutionary optimization with approximate fitness functions. In: Whitley D, editor. Proceedings of the genetic and evolutionary computation conference GECCO. Las Vegas: Morgan Kaufmann; 2000. p. 786–93. 33. Clarke S, Griebsch J, Simpson T. Analysis of support vector regression for approximation of complex engineering analyses. J Mech Des 2005;127:1077–87. 34. Gunn SR. Support vector machines for classification and regression. ISIS Technical Report; 1998. 35. Forrester AIJ, Keane A. Recent advances in surrogate-based optimization. Prog Aerosp Sci 2009;45(1):50–79. 36. Forrester AIJ, So´bester A, Keane AJ. Engineering design via surrogate modelling: a practical guide. Wiley; 2008.

849 37. Jin Y, Olhofer M, Sendhoff B. Managing approximate models in evolutionary aerodynamic design optimization. IEEE Congr Evol Comput 2001;1:592–9. 38. Ong YS, Nair PB, Keane AJ. Evolutionary optimization of computationally expensive problems via surrogate modeling. AIAA J 2003;41(4):687–96. 39. Ratle A. Accelerating the convergence of evolutionary algorithms by fitness landscape approximation. In: Eiben A, Ba¨ck T, Schoenauer M, Schwefel H-P, editors. Parallel problem solving from nature. Berlin, Heidelberg: Springer; 1998. p. 87–96. 40. El-Beltagy MA, Nair PB, Keane AJ. Metamodeling techniques for evolutionary optimization of computationally expensive problems: promises and limitations. In: Daida J, Eiben AE, Banzhaf W, editors. Genetic and evolutionary conference. Morgan Kaufmann; 1999. p. 196–203. 41. Jin Y. A comprehensive survey of fitness approximation in evolutionary computation. Soft Comput 2005;9(1):3–12. 42. Zahir MK, Gao ZH. Increased computational power and challenges to variable-fidelity solvers. In: Fourth international conference on computational intelligence and communication networks, CICN2012. Mathura, Uttar Pradesh, India 2012 in press. 43. Dixon LCW, Szego¨ GP. The global optimization problem: an introduction. In: Dixon LCW, Szego¨ GP, editors. North-Holland; 1978. p.1–15. 44. Rasmussen CE, Williams CKI. Gaussian processes for machine learning. The MIT Press; 2006. 45. Kulfan BM. Universal parametric geometry representation method. AIAA J 2008;45(1):142–58. 46. Kulfan BM. A universal parametric geometry representation method – ‘‘CST’’. In: 45th AIAA aerospace sciences meeting and exhibit; 2007. 47. Zahir MK, Gao Z. Variable fidelity surrogate assisted optimization using a suite of low fidelity solvers. In: International conference on applied and engineering mathematics (AEM2011), World congress on engineering and technology (CET); Shanghai, China; 2011. 48. Sobol’ IM. Distribution of points in a cube and approximate evaluation of integrals. Math Phys 1967;7:784–802. 49. Viana FAC. Surrogates toolbox user’s guide. Version 2.0 ed.; 2010. Available from: http://fchegury.googlepages.com,. Mohammad Kashif Zahir is a Ph. D. student in the School of Aeronautics at Northwestern Polytechnical University. He received his B.S. degree from University of Virginia, USA and M.S. degree from National University of Sciences and Technology, Pakistan. His main research interests are aircraft design and surrogate-based optimization. GAO Zhenghong is a professor at Northwestern Polytechnical University. Her main research interests are flight vehicle design and flight control.