Efficient Global Optimization using Multiple Infill ...

AIAA SciTech Forum 8–12 January 2018, Kissimmee, Florida 2018 AIAA Aerospace Sciences Meeting

10.2514/6.2018-0555

Efficient Global Optimization using Multiple Infill Sampling Criteria and Surrogate Models

Downloaded by NORTHWESTERN POLYTECHICAL UNIV. on January 19, 2018 | http://arc.aiaa.org | DOI: 10.2514/6.2018-0555

Yuan Wang1, Zhong-Hua Han2*, Yu Zhang3 and Wen-Ping Song4 National Key Laboratory of Science and Technology on Aerodynamic Design and Research, School of Aeronautics, Northwestern Polytechnical University, Xi’an, 710072, P. R. China

Efficient global optimization (EGO) can be used to dramatically improve the efficiency of a design optimization based on high-fidelity and expensive numerical simulations such as computational fluid dynamics (CFD). In order to further improve the efficiency and take the advantage of parallel computing, this paper propose a parallel infill sampling criterion for EGO. Instead of adding a single sample point in each updating cycle like the original EGO does, the EGO with the proposed method can obtain arbitrary number of new sample points per cycle, which are to be evaluated in parallel. And different from most existing parallel infill criteria, such as kriging believer method, the proposed method employs different infill criteria simultaneously per cycle. Due to combination of various infill criteria, each criterion’s inherent drawbacks may be complemented. The proposed method is verified by a numerical example (two-dimensional Rastrigin function without constraints) and demonstrated with a constrained drag minimization of RAE2822 airfoil parameterized with 18 design variables. The results show that the optimization efficiency is significantly improved compared to the serial EGO and the effectiveness is promoted as well in contrast to the existing parallel infill criteria.

Nomenclature CL or cl CD or cd CM or cm F Ma m n r R R Re S Vkrig w x yS Y Z(·) α β0 σ2

= = = = = = = = = = = = = = = = = = = = =

lift coefficient drag coefficient pitching moment coefficient regression matrix for kriging predictor Mach number number of dimensions number of sample points correlation vector of kriging model correlation matrix of kriging model spatial correlation functions Reynolds number sampling sites for functions kriging predictor vector kriging weights independent variables response values random functions Gaussian random processes angle of attack coefficients of trend model for kriging predictor process variance

1

M.S. candidate, Department of Fluid Mechanics, 127 West Youyi Road, P.O. Box754, Student Member AIAA. Professor, Department of Fluid Mechanics, 127 West Youyi Road, P.O. Box754, Member AIAA. 3 Ph.D. candidate, Department of Fluid Mechanics, 127 West Youyi Road, P.O. Box754, Student Member AIAA. 4 Professor, Department of Fluid Mechanics, 127 West Youyi Road, P.O. Box754, Member AIAA. 1 American Institute of Aeronautics and Astronautics 2

Copyright © 2018 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

I. Introduction


O

ver the past two decades, the efficient global optimization (EGO)1,2,3,4 algorithm based on kriging model5,6 and expected improvement (EI) infill-sampling criterion7,8, has received much attention and gained a great success in aerospace engineering, such as aerodynamic shape optimization9,10,11, structural optimization, and multidisciplinary design optimization of aircraft or spacecraft 4, where high-fidelity and expensive numerical simulations, such as computational fluid dynamics (CFD) or computational solid dynamics (CSD), are often employed. EGO algorithm starts by constucting a rough kriging model 5,6 based on a relatively small set of initial samples. Then, the expected improvement (EI) infill-sampling criterion7,8 is employed to select new sample point(s), which is(are) to be evaluated to refine the model adaptively. This process of adaptive sampling will be repeated until the resulting sample-point sequence converges to the optimal solution. The EGO algorithm soon got popularity in engineering design after its birth, for it enables the searching of global optimum with much less number of expensive function evaluations than the methods such as genetic algorithm (GA), particle swarm optimization algorithm (PSO), etc. Despite the impressive advance of EGO algorithm in recent years, the main challenge is still associated with the prohibitive computational cost when dealing with high-dimensional design optimizations12, such as aerodynamic design of complex aircraft configurations parameterized with many variables. To tackle this problem, many researchers have devoted themselves to improve EGO algorithm to make it more efficient for high-dimensional optimization problems. Among many strategies to improve the optimization efficiency, one of the most promising strategy is to take the advantage of parallel computing13, since the parallel computing techniques has gained significantly development in recent years. However, original EGO algorithm is inherently serial and is unable to utilize parallel computing. Therefore, it is of great urgency to extend EGO to a parallel optimization algorithm to adapt the increasingly engineering needs and there are many methods that have already been developed, such as qEI criterion14, kriging believer (KB)15 method and so on16,17,18,19,20. This paper is motivated by the aspiration of developing a parallel infill criterion for EGO algorithm, to fit the environment of large-scale parallel computation and improve the optimization efficiency dramatically. The core idea of the proposed method is inspired by the work of Liu21,22, which is to employ several infill criteria simultaneously to select multiple new samples. And in order to provide arbitrary number of new samples per cycle, some parallel infill criteria that developed based on a single infill criterion, such as kriging believer method, have to be employed and the obtained new sample points will be added to sample set at the same time. Due to the combination of various infill criteria, each criterion’s inherent drawbacks may be complemented. The proposed method is described in detail and is verified by a numerical example and then applied to an airfoil design problem in transonic viscous flow which are parameterized with 18 design variables on TianHe-1(A), National Supercomputer Center in Tianjin, to exhibit the superiority of the proposed method for complex engineering design and large-scale parallel computing. The remainder of this paper is organized as follows. Section II presents the EGO algorithm assisted by proposed method in general, including DoE23 method, kriging model5,6, core idea of the proposed method of using multiple infill criteria and the termination conditions4 of the algorithm. Section III contains some examples to verify and demonstrate, including a numerical example and an aerodynamic shape optimization cases to minimize the drag coefficient of RAE2822 airfoil. Section IV is for the conclusion.

II. Methodology The aim of this paper is to solve the following constrained optimization problem based on expensive numerical simulations

min. y  x  w.r.t. xl  x  xu s.t.

gi  x   0,

(1)

i  1,

, NC

where y (x) and gi (x) denote the objective and constraint functions, respectively, which are supposed to be evaluated by expensive numerical simulations; NC is the number of constraint functions; xu and xl are the upper and lower bounds of the design variables x , respectively. Note that here we are mainly concerned with the singleobjective optimization, and the extension to multi-objective optimization is beyond the scope of this paper. This section describes the basic steps of the EGO algorithm assisted by the proposed method of using multiple infill criteria, including the design of experiments (DoE), kriging model5,6, infill sampling criteria7,8, and the termination conditions4. 2 American Institute of Aeronautics and Astronautics

A. Design of experiment (DoE) Before constructing the kriging model, design of experiments (DoE)23,24 method is used to generate sample points in the design space. To obtain a kriging model as accurately as possible with a limited number of pre-sampled points, the space-filling design is usually favored. In this paper, Latin hypercube sampling (LHS)23 is employed. B. Kriging model For an m–dimensional problem, suppose we are concerned with the prediction of an expensive-to-evaluate (and unknown) high-fidelity function y : m  . Assume that the high-fidelity function y is sampled at sites nm

S  [x(1) ,..., x( n) ]T 

, x  ( x1 ,..., xm ) 

n

(2)


with the corresponding responses

ys  [ y(1) ,..., y(n) ]T  [ y(x(1) ),..., y(x( n) )]T 

n

(3)

where n is the number of sampling sites. Kriging model treats response as a statistical process as

Y (x)  0  Z (x)

(4)

where  0 is an unknown constant and Z () represents the deviation, which is a stochastic variable with a covariance given by Cov  Z (x), Z (x)   2 R(x, x). (5) Here, R(x, x) is the spatial correlation function, which only depends on the Euclidean distance between x and x . Assuming that the response can be approximated by a linear combination of the observed data y s , the predictor of

y (x) at an untried x can be formally defined as

yˆ(x)  wT ys

(6)

where w  [w ,..., w ] is a vector of weight coefficients associated with sampled data. Then, ys  [ y ,..., y (1)

(1)

( n) T

(n) T

]

is replaced by its corresponding random quantities Ys  [Y (1) ,..., Y ( n) ]T . Besides, we treat yˆ(x) as random and try to minimize its MSE subject to the unbiasedness constraint by Lagrange multipliers method, the predictor for any untried x is given by:

yˆ(x)  0  r T (x) R 1 (yS  0 F)

(7)

:Vkrig

where

F  1, ,1  T

 r :  R(x

n

R : R(x(i ) , x( j ) ) (i )



,



, x) 

i, j



nn

,

(8)

n

i

0  (FT R1F)1 FT R 1yS is a scaling factor. Note that the vector Vkrig only depends on the observed data, and it can be calculated at the model-fitting stage. In addition to the predictor of the unknown y at any untried x , the MSE of the predictor yˆ(x) is proved to be:

MSE  yˆ (x)   2{1.0  rT R1r  (rT R 1F  1)2 FT R 1F}

(9)

which can be used to guide adding new sample points in the optimization process. C. Proposed parallel infill sampling criterion using multiple infill criteria Most existing parallel infill criteria (such as q-EI criterion14 or multi-point PI criterion20) are developed based on a single serial infill criterion. However, it is noted that even the widely-used EI criterion still has its essential 3 American Institute of Aeronautics and Astronautics


drawbacks. For instance, the EI function is highly multi-modal and it is hard to find its global maximum value to guide adding new sample, which may lead to a poor rate of convergence in later stage of EGO. It is obvious that this drawback will be remained in parallel infill criteria based on EI criterion. The basic idea of proposed method in this paper is to employ several infill criteria simultaneously to obtain multiple new sample points to refine the surrogate, which is inspired by Liu21. Since different infill criteria usually generate different new samples, the combination of multiple infill criteria can be a promising approach to get a set of new samples. Furthermore, different infill criteria also have different features when tackling optimization problems, especially the capability to balance local exploitation and global exploration, which is of great significance to obtain a global optimum. Therefore, each criterion’s inherent drawback will be complemented and the results of optimization are likely to be improved. As investigated by Liu22, the MSE criterion3,4,8 performs much worse than the others, therefore the rest four infill criteria employed by Liu ,which are EI3,4,7,8, MSP3,4,8, PI3,4,8, LCB3,4,8, are also used in this paper. In consideration of the finite types of infill criteria, it is obvious that combining a set of q infill criteria together will provide up to q candidate samples, so it may get into trouble when employing Liu’s method in large-scale parallel optimization. In order to overcome this limitation and as an extension, Liu’s method will be coupled with some parallel infill strategies that developed based on a single infill criterion, such as kriging believer (KB)15 strategy, which will be used in this paper. In this way, every infill criterion is able to obtain arbitrary number of new samples and there is no doubt that the total number of new samples selected per cycle is arbitrary as a consequence. D. Termination conditions Several termination conditions can be used in the EGO algorithm, such as the criteria defined based on the distance between samples and EI (or PI, LCB, MSP etc.) difference of their objective responses, the allowable minimum value of EImax, the thresholds about accuracy of surrogate models, and the affordable maximum number of expensive function evaluations. E. Framework of EGO algorithm assisted by the proposed method The EGO algorithm assisted by the proposed method can be summaried as follows: 1) Firstly, choose initial sample points from design space by using a DoE23 method, and evaluate the samples in parallel. Then, the initial kriging models for objective and constraint functions can be constructed based on the initial sample data. 2) Secondly, employ EI, PI, LCB and MSP criteria coupled with KB method (since KB method is invalid for MSP criterion, the number of new sample that generated by MSP criterion is fixed at 1 in this paper) simultaneously and provide multiple candidate sample points to be evaluated in parallel and used to refine the kriging model for objective and constraint functions. (Note that the number of candidate points can be determined arbitrarily according to the computing resources.) 3) Step 2) will be repeated until the termination condition is satisfied.

III. Examples The proposed method has been integrated into an in-house code called “SurroOpt”4. SurroOpt, a surrogate-based optimization code, was developed mainly for academic research and engineering designs driven by high-fidelity, expensive numerical simulations. SurroOpt can be used to solve arbitrary single and multi-objective, unconstrained and constrained optimization problems with continuous and smooth design space. It has built-in modern DoE methods suited for deterministic computer experiments, such as LHS 23, UD24, and Monte Carlo design23. A variety of surrogate models, such as PRSM, kriging and its variants (GEK 25,26, cokriging27, HK28), RBFs, ANN, SVR, etc., were implemented. A couple of infill-sampling criteria and dedicated constraint handling methods were implemented, such as minimizing surrogate predictor (MSP), expected improvement (EI), probability of improvement (PI), mean-squared error (MSE), lower-confidence bounding (LCB) and target searching (TS). Some well-accepted and highly matured optimization algorithms, such as Hooke&Jeeves pattern search, Simplex, BFGS quasi-Newton’s method, sequential quadratic programming (SQP), and single/multi objective genetic algorithms (GAs), are employed to solve the sub-optimization(s), in which the cost function(s) and constraints function(s) are evaluated by the cheap surrogate models. SurroOpt has been paralleled by the message passing interface, which allows the user to run the code with multiple cores to speed up the optimization process. In order to demonstrate the proposed method, firstly, a numerical example is chosen to verify the effectiveness of EGO with the proposed method, and then it is applied to a drag minimization case of RAE2822 airfoil to test its capability on solving aerodynamic shape optimization problem. 4 American Institute of Aeronautics and Astronautics

A. Numerical verification Firstly, the proposed method is tested for a two-dimensional unconstrained global optimization problem with an objective function of 2d-Rastrigin function, which is a highly multi-modal function (see Figure 1 for the landscape of 2d-Rastrigin function). The mathematical formulation is given by

min.

nv   f  x   10  nv    xi 2  10  cos  2 xi      i 1   xi   5.12,5.12 , i  1, , nv , nv  2

(10)


The theoretical optimal solution is at x*  (0,0) and with f (x* )  0 .

Figure 1. Landscape of 2d-Rastrigin function To start the optimization, LHS method is used to selected 4 initial samples and based on which, the initial kriging model is constructed. After that, the sub-optimizations defined by EI criterion, KB method based on EI and the proposed method are solved by a combination of Hooke and Jeeves pattern search and BFGS optimization. It is noted that the termination condition is chosen to be a limitation of number of total samples, which is 1004 here. The optimization results are listed and compared in Table 1, from where we can see that not only the optimization efficiency is improved significantly, but also the optimization quality gains a remarkable improvement when using the proposed method. The convergent histories are drawn in Figure 2, which further confirm the statement we made above. It is noted that EGO with the proposed method is able to obtain a better result than EGO with EI criterion or with KB, especially when the number of candidate samples added per cycle is up to 40, the proposed method brings a dramatically improvement on optimization results within a given number of samples. Table 1. Comparison of the results with different infill methods for 2d-Rastrigin function optimization Number of total Infill-sampling criterion Optimum iterations True optimum EI KB - 4 KB - 10 KB - 25 KB - 40 PM- 4 (EI + LCB + PI + MSP) PM -10 (3-EI + 3-LCB + 3-PI + MSP) PM -25 (8-EI + 8-LCB + 8-PI + MSP) PM -40 (13-EI + 13-LCB + 13-PI + MSP)

0 4.287×10-10 6.514×10-10 4.199×10-10 7.934×10-6 0.053 5.438×10-12 1.460×10-11 3.750×10-11 6.068×10-11

(* “PM” represent “proposed method”)

5 American Institute of Aeronautics and Astronautics

/ 1000 250 100 40 25 250 100 40 25

2

2 EI KB-4 PM-4


0

-2

-2

-4

-4

lg(obj)

lg(obj)

0

-6

-6

-8

-8

-10

-10

-12

0

50

100

150

200

-12

250

EI KB-40 PM-40

0

10

20

30

Iteration

Iteration

Figure 2. Comparison of convergent histories of EGO with EI criterion, KB method and the proposed method for 2d-Rastrigin function optimization (“PM” represent “proposed method”) B. Drag minimization of RAE2822 airfoil in transonic viscous flow The RAE2822 airfoil has been widely used by researchers as a baseline for aerodynamic shape optimization and in order to allow researchers to compare their results, the AIAA aerodynamic design optimization discussion group (ADODG) has defined a series of benchmark problems to test aerodynamic optimization methods, including a drag minimization problem of RAE2822 airfoil in transonic viscous flow (ADODG case 2). Following the instruction of ADODG, the objective is set to minimize the drag coefficient at free-stream condition of Ma  0.734 , Re  6.5 106 , Cl  0.824 subject to a pitching moment and an area constraints, which can be expressed as

min. Cd s.t.

Cl  0.824 Cm  0.092

(11)

Area  Areainitial The airfoil is parameterized with 18 design variables by 8th-order CST method. A special technique called conformal transformation is employed to generate the C-type grid of airfoils to guarantee good uniformity and orthogonality, and the structural grid used for optimization has 512 points in the stream-wise direction and 256 points in the direction normal to the airfoil surface, as Figure 3 sketches. The CFD solver used here is an in-house code called “PMNS2D” which solves the RANS equations with a Spalart–Allmaras turbulence model. To start optimization process, LHS method is used to generate 50 samples to construct initial kriging model, then EI criterion, KB method based on EI and the proposed method are employed to refine the model, respectively. The optimization are terminated when the total number of CFD evaluations reaches 550. The work is carried out at National Supercomputer Center in Tianjin, and the calculations are performed on TianHe-1(A). In order to eliminate the effect of randomness, each optimization case is repeated 10 times with different initial samples. The convergent histories of EGO with EI, KB method based on EI and the proposed method are sketched in Figure 4 and the optimization results are listed in Table 2 in detail. It can be observed that the proposed method and KB method both can improve the optimization efficiency greatly, but as the number of new samples added per cycle increasing, the quality of KB method declines seriously, while the result of the proposed method is much better than KB method when the number of new samples added per cycle is same, and the superiority is increasing along with the number of new samples added per cycle.


0.025

0.025

EI KB-4 PM-4

EI KB-25 PM-25

0.02

Cd

0.02

Cd


Figure 3. Sketch of C-grid for airfoil optimization

0.015

0.015

0.01

0.01 0

50

100

150

0

5

10

15

20

25

30

Iteration

Iteration

Figure 4. Comparison of convergent histories of EGO with EI criterion, KB method and the proposed method for aerodynamic shape optimization of RAE2822 airfoil (“PM” represent “proposed method”) Table 2. Comparison of the results of EGO with EI criterion, KB method and the proposed method for aerodynamic shape optimization of RAE2822 airfoil Best design Worst Average Standard Number of Infill-sampling criterion (cts) design (cts) level (cts) deviation (cts) cycles Baseline airfoil EI KB - 4 KB - 10 KB - 25 PM–4(EI + LCB + PI + MSP) PM–10(3-EI + 3-LCB + 3-PI + MSP) PM–25(8-EI + 8-LCB + 8-PI + MSP)

/ 106.63 106.06 107.35 113.02 106.91 107.25 106.94

/ 108.97 113.10 117.65 175.68 112.31 111.32 117.63

197.29 107.71 108.44 112.46 132.72 109.14 108.62 111.36

/ 0.77 1.89 3.38 22.14 1.76 1.35 2.70

/ 500 125 50 20 125 50 20

(* “PM” represent “proposed method”)

The comparison of the baseline and optimized airfoils’ aerodynamic force coefficients and geometry parameters are shown in Table 3. It can be observed that the drag is reduced by 46.2% and both the aerodynamic and geometry constraints are strictly satisfied. 7 American Institute of Aeronautics and Astronautics

baseline optimum

0.824 0.824

0.0197 0.0106

0.092 0.083

0.07787 0.07788

(increment)

(+0%)

（-46.2%）

(-9.8%)

(+0.01%)

The approximation accuracy of the kriging models at the obtained optimal is shown in Table 4. It can be observed that the relative error for drag coefficient is about 1.254%, which means that the accuracy of the kriging models can be greatly improved in the vicinity of the optimum as optimization proceeding. The pressure contours and surface pressure distributions of the baseline and optimized airfoils are presented in Figure 5 and Figure 6 respectively. It can be observed that for the optimized airfoil, the shock wave on upper suface is weaker, which is the inherent reason of drag reduction. Table 4 Comparison of the predicted and the validated aerodynamic force coefficients at the obtained optimal point for airfoil optimization Kriging prediction CFD validation Relative error Cd Cl

0.010473 0.824514

0.010606 0.825399

1.254% 0.107%

-1.5

0.1

-1 0.05

-0.5

baseline optimized

0

Cp

y/c


Table 3. Comparison of the aerodynamic force coefficients and geometry parameters for baseline and optimized airfoils Cl Cd |Cm| Area

0

0.5

-0.05

baseline optimized

1 -0.1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

x/c

x/c

0.6

0.8

1

Figure 5. Comparison of the shapes (left) and pressure distributions (right) for baseline and optimized airfoils

Figure 6. Comparison of the pressure contours for baseline (left) and the optimized (right) airfoils


IV. Conclusion


In this paper, a parallel infill criterion that uses multiple infill criteria is proposed for EGO algorithm. The core idea of the proposed method is to employ different infill criteria simultaneously and to obtain multiple candidate samples per cycle. In order to get arbitrary number of new samples per cycle, kriging believer (KB) method is coupled into the proposed method. This method is verified by a numerical example and applied to an aerodynamic shape optimization of minimizing the drag of RAE2822 airfoil in transonic viscous flow. The results indicate that the optimization efficiency is dramatically improved compared to original serial EGO, and the effectiveness is promoted as well in contrast to some existing parallel infill criteria, especially for the large-scale parallel environment, which confirms that the proposed method has a great potential for applications on complex engineering design problem and large-scale parallel computing.

Acknowledgments This research was sponsored by the National Science Foundation of China (NSFC) under grant No.11772261 and the Aeronautical Science Foundation of China under grant No.2016ZA53011. The authors would like to thank the National Supercomputer Center in Tianjin for the use of computation resource of TianHe-1(A). The authors also would like to thank Dr. Jun Liu specially for his contribution to the optimization code during his Ph.D.’s study.

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