Efficient Global Optimization Applied to Multi-Objective Deign Optimization of Lift Creating Cylinder Uing Plama Actuator Maahiro Kanazaki 1, Takahi Matuno 2, Kengo Maeda 2 and Mituhiro Kawazoe 2 1 Graduate Scool of Sytem Deign, Tokyo Metropolitan Univerity, 6-6 Aahigaoka, Hino, Tokyo 191-0065, Japan 2 Graduate School of Engineering, Tottori Univerity, 4-101 Koyama-Minami, Tottori-hi, Tottori 680-8553, Japan May 15, 2015 Abtract A Kriging baed genetic algorithm (GA) wa employed to optimize the parameter of the operating condition of plama actuator (PA). In thi tudy, the multi-objective problem around a circular cylinder wa conidered. The objective function are the lift maximization and the drag minimization. Two PA were intalled on the upper and the lower ide of the cylinder. Thi problem wa imilar to the airfoil deign, becaue the circular ha potential to work a airfoil due to the control of flow circulation by the PA with four deign parameter. The aerodynamic performance wa aeed by wind tunnel teting to overcome the diadvantage of time-conuming numerical imulation. The developed optimization ytem explore the optimum waveform of parameter for AC voltage by changing the waveform automatically. Baed on thee reult, optimum deign and global deign information were obtained while dratically reducing the number of experiment required compared to a full factorial experiment. An analyi of variance and a catter plot matrix were introduced for deign knowledge dicovery. According to the dicovered deign knowledge, it wa found that kana@tmu.ac.jp 1
duty ratio for two PA are an important parameter to create lift while reducing drag. Keyword: Plama actuator; Genetic algorithm; Efficient global optimization; Experimental evaluation. 1 Introduction Plama actuator (PA, hown in Fig. 1) are flow control device that utilize atmopheric preure dicharge [7][8]; they have gained attention in recent year, becaue their advantage of being fully electronically driven with no moving part and having a imple tructure and a fat repone are potentially ideal for application to ubonic flow control. Such active flow control device have potential to control of the circulation around arbitrary object and produce the lift-creating object even if it i not airfoil geometry. In thi tudy, the deign problem i defined a the optimization of lift creation and drag reduction via flow circulation controlled by the PA. A circular cylinder model i ued a a model and two PA are intalled. Thu, the objective function conidered in thi paper are the maximizing lift and the minimizing drag around the circular cylinder. A multi-objective genetic algorithm (MOGA)-baed efficient deign technique wa employed with wind tunnel teting to efficiently find the optimum deign. Through the deign cae, the applicability of the preent wind tunnel teting to the multiobjective/ multi-parameter deign problem wa alo invetigated. Deign problem are often olved by GA baed on numerical imulation, uch a computational fluid dynamic (CFD) [5]. However, there are everal difficultie with olving the flow field around PA. Firt, the accuracy of exiting imulation method i till inufficient. Second, the computational cot i very high for deign technique uch a GA. Several day are needed to acquire the reult for each cae, wherea the actual flow phyic finihe in a few econd. In MOGA baed efficient deign technique, Kriging urrogate model wa applied to repreent the input/output relationhip in the experimental data to reduce the experimental cot. Thi optimization technique, which i called efficient global optimization (EGO) [1][3][4], enable the optimization of global parameter in a mall number of experiment while imultaneouly obtaining information on the deign pace. The EGO baed on Kriging urrogate model can find efficiently near-global optimum. In thi tudy, Kriging urrogate model baed GA perform optimization during a wind tunnel 2
experiment in real time. The deign ytem i automated developing the interface between the optimization and the wind tunnel teting. Figure 1: Schematic of plama actuator. 2 Overview of Active Flow Control by Mean of Plama Actuator In thi reearch, a PA coniting of an expoed electrode and inulated electrode wa ued. A nonconductor wa placed between the two electrode, and AC voltage wa applied. Fig. 1 how the etup; thi type of PA i called a ingle dielectric barrier dicharge (SDBA) PA. The flow around the PA can be controlled by changing the number and location of PA and the waveform of the AC voltage. Thu, the optimal technique for olving the deign problem ha to handle many parameter to acquire the bet flow control. Generic home-tyle AC voltage ha a waveform with a contant frequency. However, everal tudie have reported that pule width modulation (PWM) i effective for flow control of PA. PWM i a drive ytem that turn the AC voltage on or off, a hown in Fig. 2. The frequency of on/off i defined a the "modulation frequency" (f mod ) and i expreed by following equation: f mod = 1 T 1 [Hz] (1) where T 1 i the time of one cycle and T 2 i the time the AC voltage i on. The ratio of T 2 to T 1 i defined a the duty ratio, which i an important parameter for PWM. The duty ratio (D cycle ) i expreed by the following equation: D cycle = 100 T 2 T 1 [%] (2) 3
Figure 2: Power upply by mean of pule width modulation (PWM). 3 Deign Method: Efficient Global Optimization 3.1 Efficient Global Optimization (EGO) The optimization procedure (Fig. 4) for PA deign conit of the following tep. Firt, N deign ample are elected by Latin hypercube ampling (LHS) [1][3][4][6], which i a pace filling method, and then aeed for the contruction of an initial Kriging urrogate model. Second, an additional deign ample i added, and the deign accuracy i improved by contructing a Kriging model baed on all N +1 ample. Note that the additional ample i elected by uing expected improvement (EI) maximization [1][3][4][6]. GA i applied to olve thi maximization problem. Thi proce i iterated until the improvement of the objective function become negligible. Through the deign procedure propoed in thi paper, all ample are evaluated by the wind tunnel teting. Each technique of the optimization procedure i decribed in detail in the following ection. 3.1.1 Kriging Model The Kriging model expre the value y(x i ) at the unknown deign point x i a y(x i ) = µ + ɛ(x i ) (i = 1, 2,..., m) (3) where m i the number of deign variable, µ i a contant of the global model, and ɛ(x i ) repreent a local deviation from the global model. The correlation between ɛ(x i ) and ɛ(x j ) i trongly related to the ditance between the correponding point, x i and x j. In the Kriging model, the local deviation at an unknown point x i expreed uing tochatic procee. Specifically, a number of deign point are calculated a ample point and 4
then interpolated uing a Gauian random function a the correlation function to etimate the trend of the tochatic proce. 3.1.2 Expected Improvement Once the model are contructed, the optimum point can be explored uing an arbitrary optimizer. However, it i poible to mi the global optimum deign, becaue the approximate model include uncertainty. Therefore, thi tudy introduced EI value a the criterion. Thi tudy olve the lift maximization problem, then EI for maximization problem can be calculated a follow: ( ) ( ) fmax ŷ fmax ŷ E[I(x)] = (f max ŷ) Φ + φ (4) EI for maximization problem can be calculated a follow: (ŷ ) (ŷ ) fmin fmin E[I(x)] = (ŷ f min ) Φ + φ where f max and f min are the maximum and the minimum value among ample point, repectively. i root mean quare error (RMSE) and ŷ i the value predicted by Eq. 3 at an unknown point x. Φ and φ are the tandard ditribution and normal denity, repectively. EI conider the predicted function value and it uncertainty, imultaneouly. Therefore, by electing the point where EI take the maximum value, a the additional ample point, robut exploration of the global optimum and improvement of the model can be achieved imultaneouly a hown in Fig. 4 becaue thi point ha a omewhat large probability to become the global optimum. In thi tudy, the maximization of EI i carried out uing GA expreed a following ection. 3.1.3 Genetic Algorithm GA (Fig. 5(a)) wa firt propoed by Holland in the early 1970 [2] and are baed on the evolution of living organim with regard to adaptation to the environment and the paing on of genetic information to the next generation. GA can find a global optimum becaue they do not ue function gradient, which often lead to an exact local optimum. Thu, GA i a robut and effective method that can handle highly nonlinear optimization problem involving nondifferentiable objective function. Owing to thi advantage, GA were applied to thi experimental ytem. The GA ued in thi tudy [5] (5) 5
utilize a real-coded repreentation, the blended croover (BLX-α), and the uniform mutation. The election probability of individual for the croover and mutation i expreed a follow: prob = c(1 c)rank1 1.0 (6) where rank i the value of fitne ranking among the population. In BLX-α, children are generated in a range defined by the two parent a hown in Fig. 5(b). The range i often extended equally on both ide a determined by the parameter α. 3.2 Knowledge Dicovery Technique 3.2.1 Analyi of Variance (ANOVA) In thi tudy, Kriging model baed ANOVA [4][1][3][6] i employed to invetigate the effect of the deign variable to objective function. Varinace of an urrogate model can be calaulated a, µ i (x i ) ŷ(x 1,, x n )dx 1,, dx i 1, dx i+1,, dx n µ (7) where the total mean µ i calculated a µ ŷ(x 1,, x n )dx 1,, dx n (8) The proportion of the variance attributed to the deign variable x i to the total variance of the model can be expreed a: [µi (x i )] 2 dx p [ŷ(x1 x n ) µ] 2 (9) dx 1 dx n The value obtained by Eq. (9) indicate the enitivity of an objective function to the variance of a deign variable. 3.3 Scatter Plot Matrix (SPM) The olution and the deign pace of the multivariable deign problem obtained by EGO are oberved by the SPM [9] which i one of the data mining, becaue the Kriging model cannot be viualized directly when the deign problem ha over four attribute value. SPM arrange two-dimenional catter plot like a matrix among the objective function and the deign variable and facilitate the invetigation of the deign problem invetigation. Each of 6
the row and column i aigned attribute value uch a deign variable, objective function, and contraint value. The diagonal element how mutual ame plot. Therefore, it can be aid that the SPM how catter plot on the upper triangular part of the matrix and the correlation coefficient on the lower triangular part a additional information. modefrontier TM ver. 4. 4. 2 i employed in thi tudy. Figure 3: Optimization procedure baed on wind tunnel evaluation. Figure 4: Improvement of the global model by expected improvement (EI) maximization. Figure 5: Schematic illutration of genetic algorithm (GA). 7
4 Experimental Setup 4.1 Wind Tunnel and Model The wind tunnel experiment were conducted in the ubonic cloed-return wind tunnel of the Aerodynamic Laboratory at Tottori Univerity. The wind tunnel ha a cloed tet ection with a 0.70 m 1.0 m cro-ection and 2.0 m length (Fig. 6). A two-dimenional circular model (105 mm in diameter) wa ued a hown in Fig. 7. Model wa placed on a flat plate and mounted to a upport connected to a ix-component external balance for meaurement of the aerodynamic force and moment. The output of the balance wa amplified and acquired with a data acquiition board (National Intrument PXI-8106). The output ignal contain noie from the atmopheric dicharge of the plama actuator. To eliminate thi effect, the clean portion of the ignal, during which the dicharge did not appear, wa extracted and ued a a clean portion of the data. 4.2 PA and It Power Supply In thi tudy, two PA were intalled on the urface of the model. PA#1 and PA#2 were intalled with mount angle of θ1 = 85.0 and θ2 = 85.0, repectively, a hown in Fig. 7. The reference waveform of a high-voltage AC input wa amplified by a olid-tate high-power amplifier; the input power wa increaed up to 400.0 W with amplitude of 70.0 Vpp. A highvoltage tranformer wa ued to achieve an AC input with amplitude of up to 30 kv at a frequency of 5.0-15.0 khz. The voltage and current of the AC input were monitored by an ocillocope along with the reference waveform. 4.3 Integration of Experiment Sytem Figure 8 how the chematic illutration of the developed ytem. EGO i executed in the worktation and receive the experimental data via LabVIEW R from the balance in the wind tunnel. The condition of the AC voltage can be automatically et during the optimization proce baed on balance meaurement. 5 Formulation In thi tudy, multi-objective/ multi-parameter deign problem which ha four deign variable wa conidered and the lift creation and drag reduction effect due to circulation control by PA wa invetigated. The objective 8
Figure 6: Tet ection of the wind tunnel. Figure 7: Circular cylinder model and the location of plama actuator. Figure 8: Schematic diagram of the integrated optimization ytem. 9
function wa maximization of the lift coefficient (C l ) and the minimization of the drag coefficient (C d )around the circular cylinder model. Thi deign problem can be expreed a follow: { Maximize Cl Minimize C d (10) The flow velocity wa et to 10.0 m/. Eq. 10 can be written for the preent deign problem a follow: ( ) ( ) Maximize EI Cl = (Cl max ŷ) Φ Clmax ŷ + φ Clmax ŷ Minimize EI Cd = (ŷ Cd min ) Φ ( ŷ Cdmin ) + φ ( ŷ Cdmin ) (11) where Cl max and Cd min are repectively the maximum C l and the minimum C d among ample point, repectively. The deign problem expreed in Eq. (10) wa olved by changing four parameter (f mod, D cycle1, D cycle2, φ) related to the AC voltage waveform. In thi cae, two PA are applied different D cycle ; D cycle1 and D cycle2, for each deign and the difference between D cycle1 and D cycle2 i decided by a phae difference φ. The deign pace i defined a follow: 30.0 f mod 200.0 [Hz] 0.0 D cycle1 50.0 [%] 0.0 D cycle2 50.0 [%] 90.0 φ 90.0 [deg.] (12) φ i the phae difference between PA#1 and PA#2. Conequently, the time lag can be expreed a φ/f mod. 6 Reult 6.1 Deign Exploration Reult In thi ection, the deign problem expreed by Eq. (10) i dicued. To contruct the initial Kriging model, 15 ample were obtained by LHS. To acquire additional ample, the iland GA wa executed with the following pecification: BLX-0.5 (α = 0.5), four ubpopulation, 16 individual for each ubpopulation(64 individual generated in total) and 64 generation. The EGO proce will be topped after ten or more additional ample how better function value than that of initial ample [6]. 10
After the objective function wa converted, even additional ample were obtained, for a total of 22 ample deign. Figure 9 how the hitory of C l value for the ampling proce. According to the hitory, the objective function converged well with a mall number of ample. Without EGO, a full factorial deign of over 1000 ample would be needed to find the global optimum. The propoed ytem reduce the cot of the wind tunnel teting by over 99 6.2 Deign Knowledge by Analyi of Variance Figure 10(a) how the main effect and the two-way interaction of the deign variable for objective function for C l. According to Fig. 10(a), f mod and D cycle2, which define the driving condition of PA on the lower ide of the cylinder, ha a predominant influence on C l. In addition, two-way interaction of f mod D cycle2 i alo effective to C l. Thee reult ugget that the circulation which create aerodynamic lift around the model i decided by duty ratio PA on the lower ide. Figure 10(b) the main effect and the two-way interaction of the deign variable for objective function for C d. According to Fig. 10(a), f mod which define the driving condition of eaxh PA on the cylinder, ha a predominant influence on C d. It i reaonable reult becaue higher f mod create higher volume force which can reduce the flow eparation. A thi reult, the drag i affected by f mod. 6.3 Viualization of Deign Problem by SPM Figure 11 how the viualization reult obtained by SPM, which how the catter plot for all parameter combination. Plot colored by red repreent deigne which achive higher aerodynamic performance. According to Fig. 11, higher f mod and D cycle2 are alway required for higher C l and lower C d. In addition, C d and f mod how the high correlation (-0.882.) Thi reult ugget that the lower C d can be carried out with the higher f mod. 11
(a) (b) (c) Figure 9: Solution of the degin problem (a)deign reult, (b)progreion of objective function with ample number for the C l and (c)progreion of objective function with ample number for the C d. 12
(a) (b) Figure 10: ANOVA reult.(a)effect of deign variable of the deign variable for C l and (b)effect of deign variable of the deign variable for C d. Figure 11: Viualization of the deign problem uing SPM. 13
7 Concluion Aerodynamic control performance of plama actuator wa optimized uing wind tunnel tet-baed EGO. In thi tudy, the lift-creating cylinder uing plama actuator wa conidered. Thi problem wa that the circulation around a circular cylinder model wa controlled to maximize the lift around the model. In addition, thi tudy alo conidered the drag minimization around the cylinder, thu the deign problem wa formulated a the multiobjective problem. The optimization technique i firtly integrated in the operating ytem of the wind tunnel experiment to enable automation of the data-acquiition/optimization proce. Uing the developed ytem, multiobjective deign problem (lift maximization/ drag minimization) wa olved. After everal additional ample are obtained, the analyi of variance and the catterplot matrix i employed for the knowledge dicovery. Uing thee technique, it i found that duty ratio and modulation frequency for the plama actuator intalled on the lower urface have the dominant effect for thi problem. It i alo found that the higher modulation frequency i required for the plama actuator to minimize the drag. Reference [1] R. J. Donald, S. Matthia, and J. W.William. Efficient global optimization of expenive black-box function. Jounal of Global Optimization, 13:455 492, 1998. [2] J. H. Holland. Adaptation in natural and artificial ytem. Univerity of Michigan Pre Ann Arbor, 1975. [3] S. Jeong, M. Murayama, and K. Yamamoto. Efficient optimization deign method uing kriging model. Jounal of Aircraft, 42(2):413 420, 2005. [4] M. Kanazaki and S. Jeong. High-lift airfoil deign uing kriging baed moga and data mining. Korea Society for Aeronautical and Space Science International Jounal, 8(2):28 36, 2007. [5] M. Kanazaki, S. Obayahi, and K. Nakahahi. Exhaut manifold deign with tapered pipe uing divided range MOGA. Engineering Optimization, Taylor & Franci, 36(2):149 164, 2004. [6] M. Kanazaki, Y. Yokokawa, M. Murayama, T. Ito, S. Jeong, and K. Yamamoto. Nacelle chine intallation baed on wind tunnel tet uing effi- 14
cient deign exploration. Tranaction of Japan Society and Space Science, 51(173):146 150, Nov. 2008. [7] T. Matuno, H. Kawazoe, and T. C. Corke. Forebody vortex control on high performance aircraft uing pwm-controlled plama actuator. 2008. [8] T. Matuno, H. Kawazoe, and R. C. Nelon. Aerodynamic control of high performance aircraft uing puled plama actuator. 2009. [9] Akira Oyama. Deign innovation with multiobjective deign exploration. http://flab.eng.ia.jaxa.jp/monozukuri/mode/englih/index.html, 2011. 15