Zho Ke,Go Zhenghong,Hung Jingto,Li Qun

aThe First Aircraft Institute of AVIC,Xi’an 710089,China

bNational Key Laboratory of Aerodynamic Design and Research,Northwestern Polytechnical University,Xi’an 710072,China

cChina Aerodynamics Research and Development Center,Mianyang 621000,China

Aerodynamic optimization of rotor airfoil based on multi-layer hierarchical constraint method

Zhao Kea,*,Gao Zhenghongb,Huang Jiangtaoc,Li Quana

aThe First Aircraft Institute of AVIC,Xi’an 710089,China

bNational Key Laboratory of Aerodynamic Design and Research,Northwestern Polytechnical University,Xi’an 710072,China

cChina Aerodynamics Research and Development Center,Mianyang 621000,China

Rotor airfoil design is investigated in this paper.There are many difficulties for this highdimensional multi-objective problem when traditional multi-objective optimization methods are used.Therefore,a multi-layer hierarchical constraint method is proposed by coupling principal component analysis(PCA)dimensionality reduction and ε-constraint method to translate the original high-dimensional problem into a bi-objective problem.This paper selects the main design objectives by conducting PCA to the preliminary solution of original problem with consideration of the priority of design objectives.According to the ε-constraint method,the design model is established by treating the two top-ranking design goals as objective and others as variable constraints.A series of bi-objective Pareto curves will be obtained by changing the variable constraints,and the favorable solution can be obtained by analyzing Pareto curve spectrum.This method is applied to the rotor airfoil design and makes great improvement in aerodynamic performance.It is shown that the method is convenient and efficient,beyond which,it facilitates decision-making of the highdimensional multi-objective engineering problem.

1.Introduction

It is,as is universally accepted,difficult to conduct optimization design for rotor airfoil to improve the performance of helicopter as it involves many con flictive objectives and constraints.During the whole flight process of helicopter,blades of rotor have to work in an extremely complex aerodynamic condition associated with a large spectrum of flow region.Especially for forward flight,the Mach number around the tip of the advancing blade reaches up to transonic regime,leading up to a shock wave/boundary layer interaction.At the same time,to maintain helicopter roll stability,the local lift of retreating side needs to be high enough to balance the advancing blade in high dynamic pressure,so the retreating blade has to work at low speed and high angle of attack,which may cause local flow separation in the outer region.All the fight conditions request that the airfoils have a high maximum lift coefficient at the low and moderate subsonic Mach state with a small zero-lift drag coefficient and high drag divergence property at transonic state at the same time.Beyond that,a small pitching moment is essential to reduce the torque and control loads.In addition,the rotor airfoil is also required to have a high lift-to-drag ratio in the hover state.So the rotor airfoil design is a multi-objective multi-constraint problem.1,2

Compared to the airfoil offixed-wing aircraft,research of rotor airfoil design evolves slowly,mainly because of the complex design requirements.However,as the main element of the rotor blade,the performance of helicopter is determined by the airfoils to a great extent.Therefore,with the further development of aeronautic technology,efficient design of highperformance rotor airfoil is possible and particularly important to improve the overall performance of the helicopter.3In the early years,symmetrical airfoil was often used as rotor airfoil due to various reasons.Until the 1970s,with the increase of helicopter flight speed,airfoil had become the key obstacle to improving performance of the helicopter.After twenty years’research conducted by NASA,ONEAR,Boeing and other research institutions,a series of advanced airfoils were developed,such as OA series,VR series,and TsAGI series,4–6and the performance of helicopter was substantially improved.Recently,Vu et al.7made use of genetic algorithm and the two-dimensional viscous panel method,XFOIL,to optimize rotor blade airfoils within a single-objective/multipoint formulation,considering forward flight and hover conditions.Massaro and Benini8proposed a multi-objective approach for rotor airfoil optimization under a fixed condition using a framework of integrated GA and gradient-based algorithms.At home,recently Yang et al.9did some work for rotor airfoil with average method.Wang et al.10did some optimization work for helicopter airfoil considering the dynamic stall characteristic.However,the above work considered only part ofrequirements,so only thespecificperformancewas improved.

If all the design requirements are considered,a highdimensional multi-objective problem will be resulted in,which has to be solved by relevant algorithm.In recent years,with the rise and development of intelligent optimization techniques,research on solving multi-objective optimization problem has become a hot spot.Evolutionary algorithms treat the entire solution set as the evolution group,and search Pareto optimal solution set in a parallel manner.It becomes the best way to solve complex engineering problems with multiobjective constraints.Currently,however,the problem with more than four objectives is intractable enough for evolutionary optimization.11–14Thus,two or three optimization objectives are often involved in general engineering optimization.With the increase of the objective number,dimension of Pareto optimal front surface increases and even worse,and the number of Pareto optimal frontier points grow exponentially,which will greatly increase the algorithm’s time and space complexity.At the same time,the number of non-dominated solutions leapsseverely.Fora fixed scale external group,outstanding individuals in the evolutionary process may not be preserved so that the whole search process will slow down.Traditional optimization methods,such as NSGA II,15,16are ill in handling this kind of problems.In addition,with the increase of the objective number,the visualization of optimization results becomes difficult,which hinders the selection of optimal results for further decision-making.Although there are several methods in the auxiliary display area,it is at the expense of large-scale calculations.To solve this problem,extensive research has been carried out,which is mainly divided into two aspects:(A)improving the optimization algorithm to make it more suitable for high-dimensional optimization problem by defining loose Pareto dominant mechanism,increasing selection pressure of individuals,and thus speeding up the convergence of the algorithm.However,it is still a problem whether these improvements are suitable for engineering.Moreover,even if we can get the optimal solution set,the calculation is too expensive and it is difficult to show optimized results for further decision-making.(B)Reducing the highdimensional multi-objective optimization problem to a lowdimensional optimization problem by introducing dimension reduction method in mathematical analysis.But such method is still at the theoretical level and can be rarely used in complex engineering applications.Therefore,it is of great theoretical and practical significance to develop a method for these problems.

Therefore,in this paper,a multi-layer hierarchical constraint(MHC)method is proposed referring to e-constraint method17to translate the complex optimization problem into a bi-objective optimization problem.The paper is organized as follows:Section 2 gives the rotor airfoil design requirements and sets up the many-objective optimization model.The principal component analysis—non-dominated sorting genetic algorithm II(PCA-NSGA II)method and multilayer hierarchical constraint method are given in Section 3.In Section 4,different methods are compared for rotor airfoil design.A discussion of the main findings concludes the paper in Section 5.

2.Design criteria and model for rotor airfoils

The airfoils experience drastically different conditions within a single blade revolution.Especially for forward flight,the rotor airfoil works inside a broad range of Mach numbers and angles of attack,so the advanced rotor airfoil design often meets conflicting design requirements. Specifically, the advanced rotor airfoil design should satisfy the following requirements:

(1)A high maximum lift coefficient CLmaxunder condition of Ma=0.3–0.5 to postpone the separation of retreating blade stall and reduce blade vibration at high-speed.

(2)High drag divergence Mach number(CL=0)and low transonic drag coefficient to reduce noise and the power requirement for forward flight.

(3)High lift-to-drag ratio characteristics(Ma=0.5–0.6,CL=0.6)to ensure the rotor hover efficiency.

(4)Very low zero-lift pitching moment coefficient Cm0to reduce the blade torsion and manipulate load of the control system.

Considering all the above design requirements,the following design objectives and constraints,as shown in Table 1,can be obtained.

Referring to Table 1,the following optimization model can be constructed:

Table 1 Design objectives and conditions for rotor airfoil.

Superscript ‘0” represents the aerodynamic coefficients of initial airfoil.f2represents objectives of maximum drag divergence Mach number and minimum drag coefficients.CD2,CD3and CD4representthedragcoefficient at Ma=0.800,Ma=0.825 and Ma=0.845 respectively.CD1and Cm1represent the drag and moment coefficients at hover state.Cm2represents the moment coefficient at Ma=0.845.t represents thickness of airfoil,x is design variable vector.xland xuare the lower and upper boundary of design variable vector.

3.Multi-objective optimization

Eq.(1)is a typical multi-objective constrained optimization problem.Because the design objective is more than three,it was called many-objective optimization problem mathematically.For such problems,with the increase of objective number, the computational resources increase, and the convergence characteristic of the optimization algorithm gets worse.Meanwhile,the optimization result is a hyper-surface which is a difficulty for further decision-making.

There are two general ideas to handle these problems.First one is to improve the algorithm to enhance the quality of search results and the efficiency;another one is to reduce the dimension of objectives.Dimensionality reduction is more mature and feasible than the first method.Here the dimensionality reduction algorithm based on PCA is introduced to solve the Eq.(1).18

3.1.PCA-NSGA II

For high-dimensional multi-objective optimization problem,the objective dimension reduction is a practical and efficient way.In recent years,there have been severalhighdimensional multi-objective dimensionality reduction methods.Through comparative analysis,dimensionality reduction algorithm based on principal component analysis method proposed by Deb and Saxena18is widely applied.Because it is simple and mature and can obtain excellent results for highdimensional problems,the PCA dimensionality reduction algorithm is studied in this paper.

Multi-objective dimension reduction method based on PCA is divided into the following steps:

Step 1.Initialize iteration counter I=0,the initial objective set M=Ø,threshold cut TC=0.97.

Step 2.For all objectives in set M,random initialize the populations,carry out multi-objective optimization,and get a set of Pareto solutions.

Step 3.Perform PCA analysis to optimization results Q,eliminate redundant objectives using pre-specified threshold TC,and get a new set of objectives:specific implementation strategies are as follows:

(1)Normalize the objective vector,calculate the correlation matrix R(i,j)and its eigenvector,V(i,j),extract the first and second principal components by PCA analysis.

(2)For the first feature vector,select objectives corresponding to the most positive and most negative elements into M.

(3)For the next feature vector,check threshold TC;if the threshold is satisfied,then end,otherwise,check eigenvalues:If the eigenvalue＜0.1,select the objective corresponding to the element with the maximum absolute value|max(V(i,j))|into M;Otherwise,let P=max(V(i,j)),N=|min(V(i,j))|.If all elements of the eigenvector are greater than 0,select the objective corresponding the largest element into M;If all elements of the eigenvector are less than zero,select all the objectives into M;If(P＜N),then execute as the following two situations:If(P≥0.9N),then select the objectives corresponding P and N into M;Else choose the objective corresponding N into M.If(P＞N),then execute as the following two situations:If(N≥0.8P),then select the objectives corresponding P and N into M;Else select the objective corresponding P into M.

(4)Reduce the number of objectives further by using the correlation coefficients of the non-redundant objectives found in item 2 above,investigate if there still exists a set of objectives having identical positive or negative correlation coefficients with other objectives and having a positive correlation among themselves,and retain the one which was chosen the earliest(corresponding to the largest eigenvalue)by the PCA analysis;

Step 4.If M=M(I-1),stop and output optimal solution set,otherwise set I=I+1 and return the second step.

In order to eliminate the influence of bad samples and noise point to the analysis results,the robust PCA19method is used for dimension reduction.

For high-dimensional multi-objective problem,dimensionality reduction through PCA analysis can distinguish redundant objectives and improve the efficiency of optimization process,which lays a foundation for further decisions.However,the design objectives are generally less than 10 for engineering optimal design problem.The engineering design problem containing 4–7 objectives can be called the moderate high-dimensional problem.For these problems,the ratio of redundant objectives is lower,which would limit the effectiveness of PCA-NSGA II algorithm.Secondly,if the redundant objectives are the design goals,which we precisely concern about for a real problem,they will be directly discarded by PCA-NSGA II algorithm.Then this aspect of design products cannot be guaranteed.Again,if design objectives are still more than four after selection of PCA algorithm,it is still difficult for further optimization and analysis.Therefore,it seems urgent and important to develop a practical method,which can be used for engineering moderate high-dimensional multi-objective design.

3.2.Multi-layer hierarchical constraint method

Learning from ε-constraint method and preserving two objectives of the original problem with other objectives treated as variable constraints,the original high-dimensional multiobjectiveoptimization problem isconverted to alowdimensional problem.A set of different Pareto curves can be obtained by changing the constraint value.It is very convenient for the further decision in low-dimensional Pareto curve sets.This method can be classified into the ‘Decision-making-Optimization-Decision-making”mode which is different from‘Optimization-Decision-making” direct optimization method.Preliminary decision was made before the optimization based on previous experience.The optimization model of this method is as follows:

Lemma.The solution of above optimization problem is weak Pareto optimal solution of the original multi-objective problem.

Proof.Let x*∈X be the solution to the above problems,assume that it is not weak Pareto optimal solution of the original problem,and then there exists some other x∈X such that fi（x）＜ fi（x*）for all i=1,2,...,k.

It means that for all j=1,2, ..., k, j≠l, m,fj（x）＜ fj（x*）≤ εjexists,and thus x ∈ X is a feasible solution to the above problems,and also fl（x）≤ fl（x*）,fm（x）≤ fm（x*）,which is contradictory to the assumption that x*∈X is solution of the above problem.So x*∈X must be weak Pareto solution of the original multi-objective problem. □

From the lemma,the solution obtained by the method described above is a weak Pareto solution set of highdimensional problems.It means that stringent solution may also exist under the constraints,so it can only guarantee that the remaining objectives are optimal solution.The optimal solution of constraints may be covered.Therefore,the most critical issue in this method is to select the design objectives and constraints.We propose a multi-layer optimization approach as follows:first analyze the design objectives before the optimization;then classify and grade objectives according to their preference and correlation;finally treat the one with the most negative correlation and preference as objectives;then the strong Pareto optimal solution can be ensured only for primary objectives.The principle of the method is shown below.

Fig.1 shows a schematic of the optimization method described above.εi，εi+1，εi+2are different constraint vectors.It can be seen that a series of Pareto curves is formed on objective plane by adjusting the constraint vector.Decision-making can be fast and intuitive by analyzing the Pareto curve for further analysis.Each curve corresponds to a bi-objective problem which can be solved efficiently by the existing multi-objective algorithm.Thus the whole high-dimensional problem is solved.At the same time,the user can adjust constraint value based on experience or optimization progress status to further improve the efficiency of design.

The method can be employed according to the following steps:

(1)Carry out the PCA analysis on objectives and sort the design objectives by their importance.PCA can extract the major objective and reflect the relationship among samples.The sequence is based on the user’s engineering experience and requirements to the problem and physi-cal and mathematical relationships of the objectives.The purpose of sequence is to add a priori knowledge of the user to the optimization process.

Fig.1 Principle of MHC method.

(2)Treat the strongest contradictory and two top-ranking optimization objectives as selected objectives with other objectives treated as constraints,and then the biobjective optimization problem is established.

(3)Set constraints range based on design experience and initialize optimization model parameters.

(4)Parallelly optimize every sub-optimization for every set of variable constraints.

(5)Assess and analyze the optimization results of each group;if the optimal solution was obtained,stop searching,otherwise adjustconstraintfactor(successive approximation can be carried out by bisection method),return to the optimization problem,go back to step(3),and continue to optimize(repeat the process until the optimal solution was obtained).

(6)The optimal solution set is selected by user from the results.

Fig.2 shows the flowchart of multi-layer hierarchical constrained method.

3.3.Optimization design system

For a specific optimization problem,the parameterization methods,optimization search algorithm,surrogate-model,CFD analysis tools and the appropriate management framework need to be coupled to set up an optimization design system.In this paper,a well-organized optimization design system contained in all the above aspects in house is adopted.

The free form deformation(FFD)20,21parametric method is used for airfoil shape parameterization.In order to finely express the airfoil shape,the 22 design variables were adopted.This parameter method can be used for perturbation and deformation of any shape.Figs.3 and 4 show the control framework and the deformation of airfoil shape.

Fig.2 Flowchart of multi-layer hierarchical constraint method.

Accurate assessment of the airfoil is the basis of shape optimization design.In this paper,a multi-block grid Reynoldsaveraged Navier-Stokes(RANS)equations method is used for airfoil flow simulation and aerodynamic characteristic evaluation.The shear stress transport(SST)turbulence model22and ROE schemes23are adopted.Fig.5 shows the computing grid used in the paper.An O-C mesh topology is used where O grid is within the boundary layer mesh and C grids are for outside zone.Grid number is 70000.In order to validate this method,the flow around SC1095 airfoil was calculated.24Fig.6 shows the comparisons of lift coefficient CLbetween CFD result and experimental data.The comparisons of drag coefficients CDvarying with Mach number were shown in Fig.7.The dash line is the boundary of the ten wind tunnel data,and it can be seen that the CFD result is located in the central region of the experimental data.At the high Mach number,the strong shock wave appears in the flow,so the error increases.From the comparison,it can be seen that the CFD code used in this paper is reliable.

In order to improve the efficiency of optimization,Kriging25,26approximation model,which is often used in aerodynamic design optimization,was chosen to evaluate the airfoil.The model uses the stochastic process for space prediction,which is a robust method and has a good approximation for nonlinear problem.The Latin hypercube sampling27was used to select samples.1000 samples were produced in order to ensure design accuracy in this paper.

In this paper,NSGA II was used for multi-objective optimization.Due to the ingenious mechanism of NSGA II,the algorithm is computationally more efficient than other early algorithms and has broad applications.It is also the benchmark algorithm for the performance comparisons of optimization algorithm.

4.Results and discussion

In order to verify the feasibility of the proposed algorithm,direct multi-objective optimization,PCA-NSGA II methods and MHC method are used for the rotor airfoil design.The OA309 airfoil was used as the initial airfoil for optimization.

4.1.Direct multi-objective optimization

Firstly,the NSGA II optimization algorithm was directly used to solve optimization problem Eq.(1);since there are many design variables and objectives,reference to Deb and Saxena,15the following parameters were settled to ensure convergence and diversity of the solution:population size 1000,generations 1000,crossover probability(Pcross)0.9,distribution parame-ter(for crossover)10,mutation probability(Pmut)0.1,distribution parameter(for mutation)100.

Fig.3 FFD lattice and original airfoil.

Fig.4 Movement of control point and deformation of airfoil.

Fig.5 Computational grid for SC1095 airfoil.

Fig.6 Comparison of lift coefficient between CFD result and experimental data.

It takes six hours to get evolutionary convergence solutions.It is impossible to make a direct and proper selection because the Pareto forefront is a high-dimensional hyper surface.

4.2.PCA-NSGA II methods

Fig.7 Comparison of drag coefficients varying with Mach number between CFD result and experimental data.

The redundant objectives were selected based on the above high-dimensionaloptimization resultsthrough thePCA method described in Section 3.1.A set of PCA analysis results is given in Tables 2–7.According to the PCA-NSGA II algorism,the cumulation of the first three eigenvalues exceeds the threshold TC,so the corresponding eigenvector was selected.For the first eigenvector V1,the two objects f5and f6were selected;for the second eigenvector V2,the object f3was selected;for the third eigenvector V3,the two objects f2and f4were selected.For the selected objectives f2,f3,f4,f5and f6,according to correlation matrix,it is shown that f4and f6have strong positive correlation.Since the eigenvalues off6are larger,the objective f6was retained.After the first round of PCA dimensionality reduction,the four objectives were selected:f2,f3,f5and f6.

For the second round of PCA dimensionality reduction,the cumulation of the first three eigenvalues exceeded the threshold TC,so the corresponding eigenvector was selected.For the first eigenvector V1,the two objects f2and f5were selected;for the second eigenvector V2,the object f6was selected;for the third eigenvector V3,the two objects f2and f3were selected.For the selected objectives f2,f3,f5and f6,according to correlation matrix,it is shown that there is no strong positive correlation objective,and at last the objective cannot be reduced.

A new set of Pareto front was produced by conducting optimization based on these objectives.The PCA result for new objectives was shown in Table 3.It can be seen that no redundant object exists,and the dimension cannot be reduced,so the ultimate non-redundant objectives are f2,f3,f5and f6(underlined elements in the table).PCA analysis is consistent with the physical mechanism,suggesting the objectives for highdimensional PCA analysis are reasonable.

The PCA-NSGA II result and four-objective optimization Pareto results were compared in Fig.8 to show the effect of objective dimensionality reduction to the optimization results.It can be seen that the Pareto front of objective No.4 is more anterior and centralized than No.6,which indicates that the quality of the optimization results has significant improvement after dimensionality reduction.

4.3.Multi-layer hierarchical constraint method

By analyzing the PCA results,it is shown that the objective f2,f3,f5and f6should be retained.However,it is still difficult to assess the optimization algorithm and further analysis for the four-objective optimization.Then the specific priority of objectives must be obtained to further determine the primary design objectives and variable constraint.Table 8 shows the priorities for the objectives herein.The lowest priority number indicates the highest priority objective.

Table 2 Correlation matrix of PCA-NSGA II to six objectives.

Table 3 Eigenvectors of PCA-NSGA II to six objectives.The underline indicates this element meets the algorithm requirements.

Table 4 Eigenvalues of PCA-NSGA II to six objectives.

In this paper,the two objectives f2and f5will be selected as design objectives according to Table 8,and f3and f6are chosen as variable constraints.The MHC design model is established as follows:

An optimization design process shown in Fig.9 is established according to multi-layer hierarchical constraint optimization(MHCO)method described in Section 3.2.It can be seen that a series of two-dimensional Pareto curve spectrum about variable constraints can be obtained by reasonably changing variable constraint value.The current constraint value can be chosen based on the former design constraints and the design result to further improve the design efficiency.The range of variable constraints can be obtained according to the design requirements and the single objective design results.The constraint zone in this paper is ε1(0.9–1.05)and ε2(0.7–1.1).

Fig.10 shows comparison of Pareto front of direct multiobjective optimization,PCA-NSGA II and MHC method.It can be seen that the Pareto solutions of MHC method are more centralized and anterior than the other two methods.Thus the solution of MHC method is better than the other two methods.At the same time,the results are twodimensional curves,and it is easy to further make a decision because other characteristics are fixed by the constraints and only two objectives vary.

Two-dimensional Pareto curve spectrum obtained by MHC method is shown in Fig.11.A set of Pareto curve meeting the design requirements is selected through analysis and decisionmaking,and the optimized airfoil is chosen by evaluating Pareto optimal results(Fig.12),the corresponding constraints are ε1=0.98 and ε2=0.75,after evaluation,the optimization airfoil was selected,which is compared with the optimization results of NSGA II-6 and PCA-NSGA II(Fig.13).

Table 5 Correlation matrix of PCA-NSGA II to four objectives.

Table 6 Eigenvectors of PCA-NSGA II to four objectives.The underline indicates this element meets the algorithm requirements.

Table 7 Eigenvalues of PCA-NSGA II to four objectives.

4.4.Comparison of results

Fig.13 shows the geometry of the airfoil before and after optimization.NSGA II-6 represents the airfoil designed by direct optimization method,PCA-NSGA II is the airfoil designed by PCA dimensional reduction method,and MHC-NSGA II is the airfoil designed by multi-layer hierarchical constraint method.It can be seen that the lower surface of optimized airfoil is flatter than the original airfoil,which ensures high-speed drag divergence characteristics of the airfoil.This is because the angle of attack of airfoil is negative under the conditions of high Mach and zero lift;the lower surface becomes the upper surface.Meanwhile,there is a bump on the rear upper surface of the airfoil to reduce high-speed shock wave,which is similar to shock control bump.

Fig.8 Comparison of Pareto front between six-objective and four-objective optimization.

Table 8 Prioritiesforsix-objective airfoil design problem.

Fig.9 Optimization design flowchart for rotor airfoil.

Fig.10 Comparison of Pareto optimal results among three methods.

Lift coefficient curves of the airfoil at Ma=0.4 is shown in Fig.14,and it can be seen from the figure that the maximum lift coefficient increases after optimization;the maneuverability characteristics is better than the original airfoil.Fig.15 shows the comparison of lift-to-drag ratio L/D between two airfoils at the hover state;the drag characteristics of optimized airfoil is significantly better than the original airfoil,while the maximum lift-to-drag ratio increases and is exactly near the rotor hover condition.The lift-to-drag ratio is better than the original airfoil over the entire range.Fig.16 shows comparison of the pressure coefficient Cpdistribution between two airfoils at hover state;from the figure,it can be seen that the adverse pressure gradient of upper surface becomes weaker to improve the stall characteristics and reduce the drag.

Fig.11 Pareto curves spectrum obtained by MHC method.

Fig.12 Pareto curve selected from curves spectrum for further decision-making.

Fig.13 Comparison of airfoil geometries between baseline and optimized airfoil.

Fig.14 Comparison of lift coefficients between baseline and optimized airfoil.

Fig.15 Comparison of lift-to-drag ratios between baseline and optimized airfoil.

Fig.16 Comparison of pressure coefficient distributions between baseline and optimized airfoil at hover state.

Fig.17 Comparison of drag coefficients varying with Mach number between baseline and optimized airfoil.

Fig.18 Comparison of moment coefficients varying with Mach number between baseline and optimized airfoil.

Fig.17 shows comparison of the drag divergence characteristics of the airfoil.It can be seen that the zero-lift drag coeff icient of airfoil declines visibly,while drag divergence Mach number increases,which will improve the forward speed of the helicopter.

Fig.18 shows the zero-lift moment curves of the airfoils;it can be seen from figure that the moment characteristics are significantly improved after optimization,which provides a favorable space for manipulation and structural design of the helicopter.

Table 9 shows the comparison of aerodynamic characteristics between the airfoils before and after optimization.MaDd0is the drag divergence Mach number when CLis zero,Cm0-mddis zero-lift moment coefficient at the drag divergence Mach number MaDd0,CD0-mddis zero-lift drag coefficient at the drag divergence Mach number MaDd0.It can be seen that the performance of airfoil is improved after the multiobjective optimization.The drag divergence characteristic and lift-to-drag ratio under hover condition are greatly improved,and the moment coefficient is less than that of the baseline airfoil.The maximum lift coefficient is slightlyimproved.The optimization result shows that the best result is obtained by the MHC-NSGA-II method.

Table 9 Comparison of aerodynamic performance between baseline airfoil and optimized airfoil.

5.Conclusion

In this paper,a multi-layer hierarchical constrained method is developed for helicopter airfoil optimization design.The following can beconcluded from thepresentationsand applications:

(1)According to the requirements and characteristics of the rotor airfoil design,a high dimensional multi-objective optimization design model was established.

(2)A multi-layer hierarchical constrained method was proposed,and traditional NSGA II method and PCA based on objective dimension reduction method were also used for comparison.It is shown that the method proposed in this paper is more appropriate and feasible for moderate scale many-objective engineering problems.

(3)A group of outstanding helicopter airfoils was obtained.The drag divergence Mach number,moment coefficient,zero-lift drag coefficient at high Mach number and liftto-drag ratio are all better than the OA309 airfoil.

Acknowledgements

This work was supported by the National Natural Science Foundation of China(No.11402288 and 11372254)and the National Basic Research Program of China (No.2014CB744804).

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Zhao Ke is an engineer in the First Aircraft Institute of AVIC.He received the Ph.D.degree from Northwestern Polytechnical University in 2015.His main research interest lies in flight vehicle design and computational fluid dynamics.

Gao Zhenghong is a professor in Northwestern Polytechnical University.Her main research interest lies in flight vehicle design and flight control.

21 December 2015;revised 12 May 2016;accepted 27 May 2016

Available online 21 October 2016

Multi-layer hierarchical constraint method;

Multi-objective optimization;

NSGA II;

Pareto front;

Principal component analysis;

Rotor airfoil

Ⓒ2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is anopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*Corresponding author.

E-mail addresses:zhaokecfd@163.com(K.Zhao),zgao@nwpu.edu.cn(Z.Gao).

Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.cja.2016.09.005

1000-9361Ⓒ2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).