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A surface parametric control and global

optimization method for axial ﬂow compressor

blades

Jinxin CHENG, Jiang CHEN

*

, Hang XIANG

School of Energy and Power Engineering, Beihang University, Beijing 100083, China

Received 29 October 2018; revised 28 November 2018; accepted 6 March 2019

Available online 11 June 2019

KEYWORDS

Aerodynamic optimization;

Bezier surface;

Compressor;

Global optimization;

Surface parametric control

Abstract An aerodynamic optimization method for axial ﬂow compressor blades available for

engineering is developed in this paper. Bezier surface is adopted as parameterization method to con-

trol the suction surface of the blades, which brings the following advantages: (A) signiﬁcantly reduc-

ing design variables; (B) easy to ensure the mechanical strength of rotating blades; (C) better

physical understanding; (D) easy to achieve smooth surface. The Improved Artiﬁcial Bee Colony

(IABC) algorithm, which signiﬁcantly increases the convergence speed and global optimization abil-

ity, is adopted to ﬁnd the optimal result. A new engineering optimization tool is constructed by

combining the surface parametric control method, the IABC algorithm, with a veriﬁed Computa-

tional Fluid Dynamics (CFD) simulation method, and it has been successfully applied in the aero-

dynamic optimization for a single-row transonic rotor (Rotor 37) and a single-stage transonic axial

ﬂow compressor (Stage 35). With the constraint that the relative change in the ﬂow rate is less than

0.5% and the total pressure ratio does not decrease, within the acceptable time in engineering, the

adiabatic efﬁciency of Rotor 37 at design point increases by 1.02%, while its surge margin 0.84%,

and the adiabatic efﬁciency of Stage 35 0.54%, while its surge margin 1.11% after optimization, to

verify the effectiveness and potential in engineering of this new tool for optimization of axial com-

pressor blade.

Ó2019 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is

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

1. Introduction

The aerodynamic design of axial ﬂow compressors is the ﬁrst

key step in the design of aeroengines. The goal of aerodynamic

design is to achieve ‘‘three high performance”of ‘‘high efﬁ-

ciency, a high pressure ratio, and a high surge margin”, but

the ‘‘three high performance”is often interrelated and contra-

dictory. The traditional aerodynamic design method of axial

ﬂow compressors is to combine one-dimensional design and

*Corresponding author.

E-mail address: chenjiang27@buaa.edu.cn (J. CHEN).

Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

Chinese Journal of Aeronautics, (2019), 32(7): 1618–1634

Chinese Society of Aeronautics and Astronautics

& Beihang University

Chinese Journal of Aeronautics

cja@buaa.edu.cn

www.sciencedirect.com

https://doi.org/10.1016/j.cja.2019.05.002

1000-9361 Ó2019 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/).

analysis, two-dimensional design and analysis, quasi three-

dimensional design, and three-dimensional analysis, and its

disadvantage is that it highly relies on expert experience and

can only be compared and improved in a limited number of

scenarios. In order to improve compressor design performance

with as little manual intervention as possible, optimization

methods have been introduced into aerodynamic design of

compressors since the 1980s.

The key to the aerodynamic optimization of axial ﬂow com-

pressor is twofold: one is geometrical parameterization; the

other is the optimization algorithm. Geometrical parameteri-

zation is a method for controlling the geometrical deformation

of a blade or a ﬂow channel according to several parameters in

an optimization process. The traditional method of compres-

sor blade parameterization is based on changes to radial sec-

tions. One change involves the geometry of each section

being held constant while the relative position changes; that

is, the axial and circumferential offsets of the section constitute

the bow and sweep of the blade.

1–3

The other approach is to

change the shape of central arced curve and thickness distribu-

tion in each section,

4–7

or to change the suction or pressure

side proﬁle of each section by means of free curves,

8–10

so as

to change the blade geometry. The disadvantages of the

method mentioned above include a large number of optimiza-

tion parameters and non-smooth surface. In 2003, Burguburu

and le Pape

11

used a Bezier surface to parameterize a single-

row transonic axial ﬂow rotor. This surface parametric method

has the advantages including fewer design parameters, better

physical understanding, smooth surface and easy to guarantee

the mechanical strength, and it is an important development

direction in blade parameterization currently. In the aerody-

namic design of axial ﬂow compressors, aerodynamic opti-

mization algorithms have developed from local optimization

to global one. In 1983, Sanger

12

combined optimization tech-

nology with compressor aerodynamic designs for the ﬁrst time,

using the general purpose control algorithm (a local optimiza-

tion algorithm), which began the era of aerodynamic compres-

sor optimization design. In 1999, Koller and Monog

13

used an

optimization algorithm, which combined local optimization

with global one, to optimize the subsonic proﬁle, increasing

the incidence range of the compressor blades. After 2000, Ashi-

hara

14

and Oyama

15

et al. adopted genetic algorithms to opti-

mize three-dimensional (3D) compressor blades, improving the

compressor’s efﬁciency and surge margin at its design point.

During the last two decades, artiﬁcial intelligence algorithms,

as a global optimization algorithm, have been widely used in

aerospace and other engineering ﬁelds. New artiﬁcial intelli-

gence algorithms, including the artiﬁcial ant colony algo-

rithm

16

(proposed in 1992), particle swarm algorithm

17

(proposed in 1995) and Artiﬁcial Bee Colony (ABC) algo-

rithm

18

(proposed in 2005), have developed rapidly. The

ABC algorithm has a unique advantage in terms of searching

ability among the intelligent algorithm.

19

It performs both glo-

bal and local searches in each iteration, combining the advan-

tages of global optimization (accuracy) and rapid convergence

(speed), making it suitable for constrained, multi-parameter,

and multi-objective global optimization of complex systems,

such as the aerodynamic design of multi-stage axial ﬂow com-

pressors. However, the ABC algorithm does not make full use

of the overall food source information in its search process,

and it can be further improved in terms of search capability

and convergence speed.

3D aerodynamic optimization of axial ﬂow compressor is a

typically High-dimensional, time-consuming-Expensive and

Black-box (HEB) problem. The core of the engineering opti-

mization for HEB problems is to adopt kinds of strategies to

achieve the relatively better optimal result,

20

at expense of

the precision of optimal solution, within limited time by using

the existing computing resources. Currently used optimization

strategies include decomposition method,

21–23

surrogate model

method,

24–26

non-gradient optimization method

27–28

and

direct optimization method.

29–31

Ref.

23

points out that no opti-

mization strategy is completely superior, and different combi-

nation optimization strategies should be adopted for different

situations.

This study develops a new tool for engineering optimization

of axial compressor blades. The tool adopts the efﬁcient

Improved ABC (IABC) algorithm to operate ﬁnite number

of iterations of bee colony (non-gradient optimization

method), and utilize the Bezier surface parametric control

method to modify the suction surface of blades (decomposition

method), which signiﬁcantly reduced design variables, and

employs the strategy of using coarse mesh in the optimization

process whilst ﬁne mesh to calculate the optimal solution

(direct optimization strategy), combining with the veriﬁed

CFD simulation method, to form a new engineering global

optimization method. This method is applied to optimizing

the aerodynamic performance of a single-row transonic rotor

(Rotor 37) and a single-stage transonic axial ﬂow compressor

(Stage 35) in order to verify the effectiveness, exploring the

acceptable engineering optimization approach for multi-stage

axial ﬂow compressor with constraint and multi-parameter.

2. Surface parameterization control method

The essence of the surface parametric control method is to add

a Bezier surface on the original one, using the control points of

the Bezier surface as the optimization variables to control the

blade geometry in the optimization cycle, as shown in Fig. 1.

In Fig. 1, LE and TE refer to the leading edge and trailing

edge. During the adding process, the four vertices of the Bezier

surface correspond to the four vertices of the original one. The

high-order continuity of each point on the Bezier surface

32

ensures the smoothness of the optimized surface, which can

reduce the ﬂow loss caused by the partial roughness of the

blade.

Fig. 1 Surface parameterization diagram.

A surface parametric control and global optimization method 1619

As shown in Fig. 2, the Bezier surface parameterization

process can be divided into the following three steps:

Step 1. To parameterize arc length for each section of the

original surface. Since the Bezier surface is a unit surface in

the computational domain, the original blade proﬁle that

needs to be optimized is ﬁrst transformed into the unit plane

in the calculation domain through arc length parameterization,

so that points in the physical and computational domains can

be mapped one by one. Arc length parameterization is

achieved by

ni;j¼Pi

m¼1lm

Lj

ð1Þ

gi;j¼Pj

n¼1ln

Li

ð2Þ

where i21;np

;j21;ns

ðÞ

;npis the number of points of the

section; nsis the number of sections; Ljis the total arc length

of the jth section line in the radial direction; lmrepresents the

chord length of the mth segment of the jth radial section line;

lnis the chord length of the nth segment of the ith radial section

line; Liis the total arc length of the ith radial section line; ni;j

and gi;jare the horizontal and vertical coordinate after param-

eterization of arc length on the unit plane, respectively.

Step 2. The Bezier surface function is used to calculate the

variation of each point in the computation domain of the orig-

inal surface. The Bezier surface function is deﬁned as

R¼X

n

k¼0X

m

l¼0

Pk;lBm

lvðÞ

()

Bn

kðuÞð3Þ

Bn

kuðÞ¼Cn

kukð1uÞnkð4Þ

Cn

k¼n!

nk

ðÞ

!k!0knð5Þ

In Eq. (3),Rrepresents the distance of movement of each

point in the computational plane that is perpendicular to the

direction of the computational plane; P

k,l

refers to the control

point of the Bezier surface, and the number of control points is

mþ1ðÞnþ1ðÞ;Bm

lvðÞ;Bn

kuðÞare Bernstein function deter-

mined by Eq. (4), where vand uare the two coordinate axes

of the Bezier surface in computational domain, respectively,

and their range of variation is [0,1]; Cn

kis the number of com-

binations determined by Eq. (5).

Step 3. The Rvalue is combined with the direction of move-

ment of each point of the original blade proﬁle to obtain a new

one.

3. IABC algorithm

3.1. Basic principles of ABC algorithms

The ABC algorithm is a global optimization algorithm, which

is inspired by the process of bees looking for food source, as

shown in Fig. 3.

The bee colony involved in the optimization process con-

sists of three kinds: employed bees, onlooker bees and detector

bees. In the initial phase, half of the bee population are

employed bees and the other half are onlooker bees. The loca-

tions of food sources collected by employed bees correspond to

a set of feasible solutions, and the food source concentration

represents the ﬁtness of a feasible solution. The process of

the ABC algorithm is as follows:

Step 1. Initialize the food source. First, a feasible solution

of ND-dimensional vectors X

i

(i= 1,2,,N) is randomly gen-

erated, where Dis the number of optimization variables. The

formula for randomly generating the initial food source is as

follows:

Xj

i¼Xj

min þRand 0;1ðÞXj

max Xj

min

ð6Þ

where j21;2;;D

fg

stands for a component of the D-

dimensional solution vector; Xj

irepresents the jth component

of the ith individual; Rand(0,1) is a random number between

(0,1); Xj

min and Xj

max denote the smallest and largest feasible

solutions of the jth component of the Nfeasible solutions,

respectively.

Step 2. Calculate the initial food source concentration; that

is, the ﬁtness of the feasible solution of the group, record the

best value, and sort according to the ﬁtness value. The bees

whose ﬁtness value of the food sources rank the ﬁrst half serve

as employed bees, while the remaining bees serve as onlookers.

Step 3. Each bee adopts Eq. (7) to explore other food

sources near the original food source, and update the ﬁtness

value. If the new food source has higher ﬁtness value, it will

replace the original one. If it is lower, the bee will explore

new food source next time.

Vj

i¼Xj

iþuj

iXj

iXj

k

ð7Þ

where Vj

istands for the jth component explored by the ith

employed bee; j21;2;;D

fg

;k21;2;;N

2

;i21;2;

f

;N

2g, and k–i;uj

istands for a random number between

[1,1].

Step 4. In accordance with the Russian Roulette Law, the

onlooker bees select a food source with probability propor-

tional to the ﬁtness of food source explored by employed bees,

then the onlooker bees explore new food sources in the vicinity

of the existing sources. If the new food source has a higher ﬁt-

ness value, onlooker bees become employed bees and replace

the original food source with the new one. If the opposite is

true, onlooker bees do the next exploration. The onlooker bees

select the food source with the probability calculated by

Fig. 2 Bezier surface parameterization process.

1620 J. CHENG et al.

P¼fXi

ðÞ

PNe

m¼1fXm

ðÞ ð8Þ

where Nestands for the number of employed bees, that is, N

2;P

is the probability of choosing a food source; f(X

i

) and f(X

m

)

stand for the ﬁtness of the ith and mth individuals,

respectively.

Step 5. If the number of explorations by employed bees or

onlooker bees exceeds a certain limit and no food source with

higher ﬁtness is found, the employed bee or the onlooker bee

will be transferred to a detector bee, and the detector bee will

regenerate a new food source according to Eq. (6). After the

new food source is generated, the detector bees are converted

to employed bees.

Step 6. Record the best food source so far, and skip to Step

2 until the out-of-loop condition is met, and output the best

food source location.

3.2. IABC algorithm principles

In the ABC algorithm, both onlooker and employed bees use

Eq. (7) to search for new food sources. However, this formula

only uses local random information of the total bee colony,

which leads to the lack of capabilities in global exploration

and exploitation. Eq. (7) is modiﬁed in the IABC algorithm,

employed bees adopting Eq. (9) which contains the informa-

tion of global optimum point to search for new food sources

whilst onlooker bees adopting Eq. (10) which contains the

information of local optimum point to search for new food

source. Better utilizing the comprehensive information of the

last generation by Eqs. (9) and (10) is the core cause for

improving performance of the IABC algorithm.

Vj

i¼Xj

iþlj

iXj

iXj

k

þ1:5kj

iXj

best Xj

i

ð9Þ

Vj

Neib

best ¼Xj

Neib

best þnj

iXj

Neib

best Xj

k

ð10Þ

where Vj

iin Eq. (9) stands for the new location of a food source

found by employed bees; lj

iand kj

iare both random numbers

between [1,1]; Xj

best stands for the location of the best food

source ﬁtness among the total food source information;

ðXj

NeibÞbest means the location of the food source with the best

ﬁtness in the neighborhood; ðVj

NeibÞbest represents the new food

source location explored by onlooker bees. The distance used

in the exploration of local best points in Eq. (10) is the Cheby-

shev distance, as per Eq. (11):

di;tðÞ¼lim

q!þ1 ðX

D

j¼1

Xj

iXj

t

qÞ

1=q

ð11Þ

where j21;2;;Dfg;and di;t

ðÞ

represents the Chebyshev

distances between Xiand Xt.

The neighborhood of X

i

is determined by Eq. (12), and the

formula shows that when the Chebyshev distance between X

i

Fig. 3 Flowchart of a standard ABC algorithm.

A surface parametric control and global optimization method 1621

and X

t

is less than the product of the neighborhood radius r

and the average Chebyshev distance, X

t

is in the X

i

neighbor-

hood, otherwise it is not.

di;tðÞrmdit2S

di;tðÞ>rmditRS

ð12Þ

where mdiis the average Chebyshev distance between X

i

and

the total onlooker bee colony, Sis the area adjacent to X

i

,r

is the radius of the adjacent area, and experience shows that

when r= 1, the algorithm has the best convergence.

XNeib

ðÞ

best is calculated by

fit XNeib

ðÞ

best

¼max fit XNeib

ðÞ

1

;fit XNeib

ðÞ

2

;;fit½XNeib

ðÞ

S

ð13Þ

where ﬁt() is the individual ﬁtness of each bee.

3.3. IABC algorithm veriﬁcation

To verify the superiority of the IABC algorithm compared to

general Genetic Algorithm (GA) and ABC algorithm, two

benchmark functions (sphere and Griewank functions) are uti-

lized. Both benchmark functions have a variable dimension of

30, a function minimum of 0, a sphere function range of

[5.12, 5.12], and a Griewank function variable of [600.0,

600.0]. We set the number of bee colonies for the two functions

to 200, the number of iterations to 200, and the maximum

number of exploration to 100. Table 1 and Fig. 4 show the

result that both the convergence speed and global optimization

ability of IABC algorithm have a signiﬁcant increase compared

to GA and ABC algorithm.

4. Engineering global optimization for Rotor 37 and Stage 35

based on an IABC algorithm and surface parametric control

method

In order to verify effectiveness and superiority of the surface

parametric control method and the IABC optimization algo-

rithm in the aerodynamic optimization of an axial ﬂow com-

pressor, a global optimization platform combining surface

parametric control method, the IABC optimization algorithm

with 3D CFD numerical simulation was constructed, ﬁrstly

applying it on Rotor 37, a single-row transonic rotor, and sec-

ondly Stage 35, a single-stage transonic compressor as the opti-

mization object for verifying.

Rotor 37, with design speed of 17188.7 r/min and 36 blades,

is one of four high-pressure compressor inlet stages designed

by NASA’s Glenn Research Center in the 1970s, and it has

geometric data and detailed experimental data that can be

found in Ref.

33

Stage 35, with design speed of 17188.7 r/min, 36 rotor

blades and 46 stator blades, is a low-aspect-ratio single-stage

transonic axial ﬂow compressor developed by NASA’s Glenn

Research Center in 1978, and the compressor has detailed geo-

metric and experimental data available in Ref.

33

4.1. Optimization method

4.1.1. Blade parameterization

Optimum blade obtained in traditional way controlling the

stacking line has characteristic of bow and sweep. A large tor-

que will be produced for rotors when rotating at high speed,

Fig. 4 Comparison of performance of ABC and IABC algorithms.

Table 1 Comparison of test results among three algorithms.

Function Minimum value Number of steps to converge

GA ABC IABC GA ABC IABC

Sphere 3.82 10

3

3.77 10

4

1.36 10

11

182 170 60

Griewank 5.3 10

1

2.35 10

1

9.173 10

9

194 122 83

1622 J. CHENG et al.

difﬁcult to guarantee the mechanical strength of the blade.

34

Bezier surface parametric control method hardly change the

characteristic of bow and sweep for blades, leading to over-

coming this shortcoming.

The key problem of aerodynamic optimization of compres-

sors is to solve the contradiction between design space and

time-consumption. The design space increases exponentially

with the increase of the design variables, even suffering from

‘‘dimension curse”,

35

leading to the failure of optimization.

So the requirement in the practical engineering is to minimize

the number of design variables under the condition of guaran-

teeing the sufﬁcient design space.

Under normal circumstances, compared to the pressure sur-

face of the blade, the loss caused by airﬂow on the suction sur-

face is the main source of efﬁciency reduction. Therefore, only

the suction surface of the blade is selected as the optimized sur-

face in the optimization cycle to achieve the purpose of dimen-

sionality reduction. The blade suction surface is parameterized

using a 6 3 order Bezier surface. Considering that the geom-

etry near leading and trailing edge has a non-negligible inﬂu-

ence on ﬂow ﬁeld, as shown in Fig. 5, seven control points

are set in the ndirection, with positions at the leading and trail-

ing edges (0% and 100%) and at the 10%, 30%, 50%, 70%

and 90% positions whist four points are set in the gdirection,

with position at the 0%, 20%, 50% and 100%. Due to the

guarantee that the ﬁrst derivative of the connection point,

which is between the suction side and pressure side at the lead-

ing and trailing edge, is continuous, the ﬁrst two points (n1;n2)

and the last two points (n6;n7) of each radial height are set as

the ﬁxed points. Taking into account the mechanical strength

of the rotor components, the points at the 0% radial height

is set as ﬁxed points. The black and red points indicate the

ﬁxed point and the active point in the optimization process,

respectively, so only nine points per blade are selected for opti-

mization, reducing the computing costs signiﬁcantly.

4.1.2. Optimization algorithm

To reduce computing costs of 3D aerodynamic optimization for

compressors in terms of algorithm, the traditional method is to

adopt adjoint algorithm and surrogate model. Adjoint algorithm

was introduced to aerospace ﬁeld by Jemeson in 1988

36,37

and

then this method was also applied in the optimization of turbo-

machinery, obtaining lots of scientiﬁc research result.

38–40

The

advantage of adjoint method is that the time spent on optimiza-

tion is independent of the number of design variables, but the

disadvantage is only local optimization. The essence of surrogate

model is to obtain approximate function of the time-consuming

ﬂow ﬁeld simulation by sample training, reducing the computing

cost signiﬁcantly. Two disadvantages of the surrogate model are

as follows: one is that when the design variables gradually

increase, the number of samples trained to obtain approximate

functions increases rapidly, which increases the calculation cost

and even exceeds the scope allowed by the engineering; the other

one is that in the reﬁned optimization of the original blade with

small performance improvement space, the error result is likely

to occur due to the limited accuracy of the approximation

function.

The new optimization method proposed in this paper

adopts IABC algorithm for optimization. The number of bees

in the algorithm can increase correspondingly with the increase

of optimization variables. The number of iteration steps can be

set artiﬁcially and depends on the actual needs. In this way, the

optimized solution (not the optimal solution) can be found

globally within the engineering allowed time cost, which over-

comes the disadvantages of adjoint algorithm and surrogate

model algorithm. Considering the time cost, based on the exist-

ing experience, this optimization sets the colony to iterate for 3

times to obtain the relative optimized solution.

4.1.3. Optimization objective function and constraints

In order to save computing cost, according to experience, we

set the point which has relative high back pressure as the opti-

mization condition to achieve the goals that both the adiabatic

efﬁciency and the surge margin could improve. The objective

function in the practical optimization process was set as

follows:

max f¼eff

The constraint is as follows:

f¼eff if TPRTPRori

TPRori

<0:5%and mass massori 0

f¼minus Otherwise

(

xL

i6xi6xU

i

where eff means the adiabatic efﬁciency; TPR and TPRori are

the total pressure ratio and the original total pressure ratio

in the optimization process, respectively; mass and massori

are the ﬂow rate and the original ﬂow rate in the optimization

process, respectively; minus is a very small value which is set

artiﬁcially; xiis the optimization variables; xL

iand xU

iare the

upper and lower limit of optimization variables, respectively.

We set the adiabatic efﬁciency under the optimization con-

dition as the optimization goal, and set a relative pressure ratio

change of not more than 0.5% and the ﬂow rate not decrease

as the strong constraint condition. In the optimization process,

we set the objective function to the minus value when the solu-

tion does not match the constraints, thus eliminating this posi-

tion of food source. For the purpose of increasing the number

of feasible solutions, the exploring times of bees can be

increased properly when the strong constraint occurs.

4.1.4. Optimization process

The optimization process is shown in Fig. 6. Firstly, the pro-

gram reads the geometric optimization variables of the blade,

Fig. 5 Distribution of control points on Bezier surface.

A surface parametric control and global optimization method 1623

then initializes the food source, obtains the initial solution, and

then calculates the ﬁtness of each initial solution. The calcula-

tion of ﬁtness consists of blade geometry generation, grid gen-

eration and ﬂow ﬁeld calculation. At this point, if the ﬁtness

reaches the condition of an exit loop, the optimization is ﬁn-

ished and the optimized blade geometry is output. Otherwise,

the IABC algorithm is used for optimization exploration, so

as to give a new feasible solution and complete an iteration.

The loop continues until the exit condition is met (i.e., it con-

verges or reaches the maximum number of iterations), and

thus the ﬁnal optimal blade is obtained.

4.2. Numerical simulation

4.2.1. Numerical methods

Flow ﬁeld calculation is the premise of performance evaluation

in the optimization process. The numerical calculation tool is

the Fine Turbo module in NUMECA_V9.1, and Spalart-

Allmaras (S-A) one-equation model is selected as the turbulent

model. The fourth-order explicit Runge-Kutta method is used

to solve the equation for time advancement, and the central

difference scheme and artiﬁcial viscosity are used to control

the false oscillations near the shockwaves and eliminate other

minor oscillations. The convergence is accelerated by a local

time-step multi-grid technique and implicit residual error

method.

The inlet boundary conditions are: total pressure

101325 Pa, total temperature 293.15 K, and incoming ﬂow

axial direction. The solid wall is a no-slip boundary condition.

The outlet boundary conditions are the given back-pressure,

and the back-pressure at the outlet is gradually adjusted from

the jam point to the near stall point during the calculation pro-

cess. The near stall point is the highest back-pressure point

before the divergence, and the blockage point is the minimum

back-pressure point before the divergence is calculated.

4.2.2. Numerical method check

Stage 35 with detailed aerodynamic experimental data is

adopted at design speed to verify numerical method. As shown

in Fig. 7(a), the experimental highest pressure ratio is 3.7% lar-

ger than the simulated one, whilst the blockage mass ﬂow is

1.4% smaller. Fig. 7(b) shows that the experimental highest

efﬁciency is 1.1% greater than the simulated one, whilst the

surge margin is 8.44% higher. According to the analysis,

although there is a relative error between the CFD calculation

and the experiment, the curves obtained by these two

approaches have the same trend, and relatively good calcula-

tion accuracy is obtained, which ensures the reliability of the

ﬂow ﬁeld calculation in the aerodynamic optimization cycle.

4.2.3. Grid generation and grid independence veriﬁcation

Stage 35 is also used for grid independence veriﬁcation. The

grids are generated automatically by the Autogrid5 module

in NUMECA_V9.1. We set the ﬁrst layer near-wall grid spac-

ing as 0.001 mm, to guarantee the near-wall grid y

+

5. The

tip clearance of the rotor blade was set as 0.4 mm, the hub

clearance of the stator blade as 0.4 mm, and the grid topology

as 4HO.

Four sets of grids are adopted for the Stage 35 reference

blade, and all grid qualities meet the calculation requirements.

A comparison of the ﬂow ﬁeld calculation results of the four

sets of grids is shown in Fig. 8. The grid numbers of the rotor

blade with the four sets of grid templates (Mesh 1, Mesh 2,

Mesh 3 and Mesh 4) are 320000, 680000, 1020000 and

1840000, and the stator blade 340000, 750000, 1100000 and

1880000. The difference between the ﬂow ﬁeld calculation

results of the third and fourth sets of grids is very small, so

the third set of grids can meet the requirement of grid

independence.

For the three-dimensional optimization of the compressor,

in order to ensure a certain calculation accuracy in the opti-

mization process, and at the same time save the computational

cost, this paper uses the ‘‘rough grid optimization, ﬁne grid

veriﬁcation”method used in Ref.

41

: ﬁrst using the second set

of grids to optimize, and then using the third set of grids for

ﬂow ﬁeld analysis. This method saves about 1/3 of the time.

4.3. Rotor 37 optimization veriﬁcation

Rotor 37 is optimized by the new optimization method, and

nine design variables are set. According to experience, the

design space of each variable is speciﬁed to be [0.05, 0.1],

where the negative values represent the direction of expansion

at the suction surface of the blade, while the positive values are

the direction of adduction at the suction surface of the blade.

The Rotor 37 blade tip clearance is set to 0.36 mm and the grid

number to 680000. The ﬂow ﬁeld will be calculated after

obtaining the optimized blade using 1.02 million grids. We

Fig. 6 Flowchart of optimization process.

1624 J. CHENG et al.

set the scale of the bee colony in IABC algorithm to 50, the

maximum number of iteration to 3, and the maximum number

of exploitation to 3.

The computer used to run the optimization process has an

Intel Core i7 3.07 GHz processor and 2 GB RAM. In the case

of parallel computation using 5 CPUs, each generation of the

bee colony optimization takes 8 h and a total of three opti-

mization iterations are performed. Hence, the total runtime

is 24 h, within the acceptable range of engineering. The history

of optimization cycle is shown in Fig. 9, and f

opt

f

ori

means

the difference between the optimal efﬁciency and the original

efﬁciency after each iteration.

4.3.1. Comparison and analysis of optimization results

As shown in Table 2, at the design point and design speed,

after optimization, the ﬂow is increased by 1.82%, the adia-

batic efﬁciency by 1.02%, and the surge margin by 0.84%

whilst keeping the total pressure ratio almost constant.

Fig. 10 shows a comparison of the calculated aerodynamic

performance of Rotor 37 before and after optimization at the

design speed. From Fig. 10(a), it can be observed that under

the same pressure ratio, the ﬂow of the optimized blade is

larger than that of the original blade in the full ﬂow range,

with the maximum difference not exceeding 2.0%, within the

acceptable range of engineering. In Fig. 10(b), the adiabatic

efﬁciency of the optimized blade is higher than that of the

Fig. 7 Comparison of performance of Stage 35 between experiment and simulation.

Fig. 8 Grid independence veriﬁcation.

Fig. 9 History of optimization cycle of Rotor 37.

A surface parametric control and global optimization method 1625

original blade in the full ﬂow range, and the maximum differ-

ence is 1.2%.

As shown in Fig. 11, compared with the original blade, the

suction surface of the optimized blade has an outward expan-

sion at the trailing edge, and with the increase of the radial

height, the extent of the outward expansion area in the chord

direction increases, and the thickness of the blade in outer

expanding area increases, while in other areas, the suction sur-

face of optimized blade is adducted and the blade thickness is

reduced.

Fig. 12 shows the variation of each point of the optimized

blade relative to the original one on the computational plane,

where the positive value represents the adduction direction and

the negative value represents the outward expansion direction.

The intersection area of the range above 50% radial height and

20%–50% range of chord length direction has the largest

adduction amplitude of the blade. Since the geometric varia-

tion of the blade’s suction surface is very small, the amount

of change before and after optimization is multiplied by 10,

as shown in Fig. 13, so that the difference of geometry of the

section at different heights (h/H) can be seen more clearly.

The optimized airfoil is adducted at the hub section and is

‘‘S-shaped”at the middle and tip sections, whose inﬂection

point at the middle and tip section is located at about 70%

of the chord length from the leading edge, and the other area

is adducted except for the expansion near the trailing edge.

4.3.2. Comparison and analysis of ﬂow ﬁeld before and after

optimization

The ﬂow ﬁeld before and after optimization is analyzed in

combination with Figs. 14–16. As the positive incidence angles

of the incoming airﬂow at each section in the radial direction

of the optimized blade are all smaller than the original one

Fig. 10 Comparison of aerodynamic performance before and after optimization of Rotor 37.

Table 2 Comparison of Rotor 37 performance at design point and design speed before and after optimization.

Rotor 37 Mass ﬂow (kg/s) Total pressure ratio Adiabatic eﬃciency Surge margin (%)

Original 20.99 2.010 0.8656 18.06

Optimal 21.38 2.012 0.8758 18.90

Relative change (%) +1.82 +0.10 +1.02 +0.84

Fig. 11 Comparison of geometry of suction surface before and

after optimization of Rotor 37. Fig. 12 Distribution of changes on suction surface of Rotor 37.

1626 J. CHENG et al.

(Fig. 14), the accelerated distance of the airﬂow at the leading

edge of the optimized blade shortens, and the intensity of obli-

que shockwave near the leading edge decreases, and thus the

loss of shockwave and that caused by interaction between

the shockwave and the boundary layer decrease. According

to the analysis in Figs. 13,15(a), (b) and 16(a), the suction sur-

face of the optimized blade at the hub section is slightly

adducted, which slows down the airﬂow acceleration at the

suction side, slightly decreasing the intensity of the channel

shockwave, thus decreasing the loss of shockwave slightly.

And the deceleration of airﬂow acceleration also pushes back

the channel shockwave position slightly, reducing the range

of airﬂow separation caused by the boundary layer after the

shockwave, and thus the separation loss near the trailing edge

of the hub area of the optimized blade is slightly reduced. As

shown in Figs. 13,15(c)–(f) and 16(b)–(c), compared to the

hub section, the middle and tip one have more obvious

adducted proﬁles at the suction side, which locate ahead of

70% chord length from leading edge, and thus the reduction

of the loss, including the shockwave loss caused by the decel-

eration of airﬂow acceleration, the loss caused by the interac-

tion between the shockwave and boundary layer and the

separation loss, are more obvious.

Due to the reduction of separation areas of the boundary

layer and the degree of airﬂow separation, which are at the

hub, middle, and tip sections of the optimized blade, the air-

ﬂow mixing downstream of the blade’s trailing edge weakens,

which is more observable at the middle and tip sections, and

hence the entropy increase, which is in the middle and tip

region of the S3 cross-section downstream of the optimized

blade, signiﬁcantly decreases, as shown in Fig. 17.

The limiting streamline on the suction surface before and

after optimization are compared in Fig. 18. As seen in

Figs. 16(b), (c) and 18, the shockwave position is pushed back

at the middle and tip sections of the optimized blade, which

reduces the separation area of airﬂow after the shockwave,

and thus the separation line of the optimized blade is pushed

back signiﬁcantly. And, with increasing radial height, this phe-

nomenon becomes increasingly observable.

4.4. Stage 35 optimization veriﬁcation

The new optimization method is veriﬁed by a single-stage tran-

sonic axial compressor Stage 35. The Bezier surface parametric

control method is applied to the suction surface of the rotor

and stator blade respectively, with a total of 18 design vari-

ables. The design space of the optimization variables of the

suction surface of rotor and stator blade is set to [1.0, 0.5],

and the negative values represent the expansion direction of

the blades’ suction surfaces, while positive values represent

the adduction direction.

The rotor blade tip clearance is set to 0.4 mm, and the sta-

tor blade root clearance is 0.4 mm. The rotor blade grid num-

ber is set to 680000, and the stator blade grid number 750000,

which meets the requirement of grid quality. The ﬂow ﬁeld will

be calculated with 1.02 million grids for rotor blade and 1.1

million grids for stator blade after the optimal blade is

obtained. We set the scale of the bee colony in IABC algorithm

to 80, maximum number of iteration to 3, and maximum num-

ber of exploitation to 3.

With the same computer conﬁguration as Rotor 37 veriﬁca-

tion, in this case of parallel computation using 8 CPUs, each

generation of the bee colony optimization takes 24 h and a

total of three optimization iterations are performed. Hence,

the total runtime is 72 h, within the acceptable range of engi-

neering. The history of optimization cycle is shown in Fig. 19.

Fig. 13 Comparison of geometry of hub, middle and tip sections before and after optimization of Rotor 37 (multiply magnitude of

change by 10).

Fig. 14 Comparison of incidence angle distribution of incoming

airﬂow in radial direction before and after optimization of Rotor

37.

A surface parametric control and global optimization method 1627

4.4.1. Comparison and analysis of optimization results

Table 3 compares the performance of design points at the

design speed before and after optimization. After optimiza-

tion, the ﬂow of the design points increased by 0.45%, the adi-

abatic efﬁciency increased by 0.54%, whilst the surge margin

expanded by 1.11%.

The aerodynamic performance of the Stage 35 compressor

is compared at the design speed before and after optimization,

Fig. 15 Comparison of relative Mach distributions at hub, middle and tip sections before and after optimization of Rotor 37.

Fig. 16 Comparison of static pressure distributions at hub, middle and tip sections before and after optimization of Rotor 37.

1628 J. CHENG et al.

as shown in Fig. 20.InFig. 20(a), the optimized blade ﬂow is,

on average, 0.5% greater than that of the original blade within

the full operating range, with the same total pressure ratio con-

dition. In Fig. 20(b), the optimized blade adiabatic efﬁciency is

0.3% higher than the average of the original one in the full

ﬂow range. Compared with the original blade, the optimized

blade ﬂows more smoothly under high back pressure, which

improves the surge margin of the optimized blade.

From the comparison of the Stage 35 dynamic and static

blade three-dimensional geometry before and after optimiza-

tion shown in Fig. 21, it can be seen that optimized Rotor

35 blade is adducted in the range of 15% or more in the radial

direction. The adduct region gradually increases from a range

of 35% chord length from the leading edge and 15% of the

radial height, to a range of 80% chord length from the leading

edge at the blade tip (Fig. 21(a)). The Stator 35 suction surface

is expanded as a whole (Fig. 21(b)).

Fig. 22 shows the change of the suction surface of the blade

before and after optimization of the dynamic and stationary

blades. Fig. 23 shows the comparison of the changes of the

hub, middle and tip of the Stage 35 before and after optimiza-

tion (magniﬁcation of the geometric change by 10 times). As

can be seen from Figs. 22(a) and 23(c), the maximum extent

of adduction of Rotor 35 blade locates at 35% from the lead-

ing edge at the tip, and the maximum extent of external expan-

sion locates at 30% from the trailing edge at the middle of the

blade. As can be seen from Fig. 22(b), the maximum area of

the expansion is 45% of the chord length from the leading edge

at the tip, which is veriﬁed in Fig. 23(d)–(f).

4.4.2. Comparison and analysis of ﬂow ﬁeld before and after

optimization

As shown from Fig. 24, the positive incidence angle of incom-

ing airﬂow of the optimized rotor and stator blade is smaller

than that of the original one from hub to tip, and thus the

Fig. 17 Comparison of entropy distributions at S3 section before and after optimization of Rotor 37.

Fig. 18 Comparison of limiting streamlines on suction surface

before and after optimization of Rotor 37.

Fig. 19 History of optimization cycle of Stage 35.

Table 3 Comparison of Stage 35 performance at design point and design speed before and after optimization.

Stage 35 Mass ﬂow (kg/s) Total pressure ratio Adiabatic eﬃciency Surge margin (%)

Original 20.19 1.816 0.8342 6.59

Optimal 20.28 1.819 0.8396 7.70

Relative change (%) +0.45 +0.16 +0.54 +1.11

A surface parametric control and global optimization method 1629

accelerated distance, where air ﬂows around the leading edge

of the optimized rotor and stator blade, shortens. Therefore

the intensity of oblique shockwave weakens there, which is ver-

iﬁed in Fig. 25(b) and (c), reducing the loss of shockwave and

that of interaction between the shockwave and the boundary

layer. According to Figs. 23(b), (c) and 26(c)–(f), the proﬁles

of the middle and tip sections of the rotor blade are S-

shaped, and the proﬁle of the ﬁrst half is adducted, which

slows the airﬂow acceleration at the suction side of the middle

and tip sections of the optimized rotor blade, and thus the

intensity of channel shockwave weakens, leading to the reduc-

tion of the loss of shockwave, whilst the shockwave position is

pushed back. Consequently, the air separation area near the

trailing edge due to an adverse pressure gradient decreases,

reducing the loss of air separation, and thus the low-entropy

area downstream of the trailing edge, shown in Fig. 27(a)

and (b), increases, decreasing the loss of mixing airﬂow there.

Fig. 28 shows the comparison of the limiting streamlines on

the suction surface of the rotor and stator blades before and

after optimization of Stage 35. It can be seen from Fig. 28 that

the separation forms of the suction surface are all open sepa-

ration before and after the optimization. Since the suction

sides of the middle and tip sections of the optimized rotor

blade are S-shaped and the proﬁle near the leading edge is

adducted, the shockwave position pushes back, and then the

separation position of the boundary layer caused by the

inverse pressure gradient pushes back, thus pushing back

the separation line of the optimized rotor blade. Therefore

Fig. 20 Comparison of aerodynamic performance before and after optimization of Stage 35.

Fig. 21 Comparison of geometry of suction surface on Rotor 35

and Stator 35 blades before and after optimization of Stage 35.

Fig. 22 Distribution of changes on suction surfaces of Rotor 35 and Stator 35 blades.

1630 J. CHENG et al.

Fig. 23 Comparison of geometry of hub, middle and tip sections of Rotor 35 and Stator 35 blades before and after optimization

(multiply magnitude of change by 10).

Fig. 24 Comparison of radial distribution of incoming airﬂow incidence angles of Rotor 35 and Stator 35 blades before and after

optimization.

Fig. 25 Comparison of static pressure at hub, middle and tip sections of blades before and after optimization of Stage 35.

A surface parametric control and global optimization method 1631

the separation loss is reduced. The separation on the suction

surfaces of stator blade is angular separation both before

and after optimization. Since the suction side of the tip area

of the optimized stator blade has the greatest expansion, a

pressure gradient that points to the middle of the blade is

formed. Thus, the radial migration of airﬂow in the region

from the tip to the middle of the blade enhances, slightly

increasing the open separation area, leading to the slight

increase of the ﬂow loss. However, compared with the loss

reduction of shockwave, and that of the interaction between

the shockwave and boundary layer, and that of the separation

of the boundary layer, the increased loss of airﬂow, caused by

the increased open separation area, is small.

5. Conclusions

Combining Bezier surface parametric control method and

IABC algorithm with veriﬁed CFD numerical simulation, a

new engineering global optimization method is constructed,

and a single-row transonic rotor (Rotor 37) and a single-

stage transonic axial compressor (Stage 35) are used to verify

this method. The conclusions are as follows:

(1) The Bezier surface parametric control method is success-

fully applied to the optimization of Rotor 37 and Stage

35, and only 9 optimization variables are used to control

each blade, signiﬁcantly reducing the number of opti-

mization variables, which saves a lot of calculation costs.

Moreover, the ﬁne adjustment of suction surface of

blades is well realized, with integrity and smoothness.

(2) In the IABC algorithm, the employed bees and the

onlooker bees utilize the global and local optimal infor-

mation when exploring the food source, respectively,

replacing the original way of randomly acquiring the

information, and the overall exploration information

of the food source can be better utilized. Compared with

the general GA and ABC algorithms, the capability of

global optimization and convergence speed of the IABC

algorithm are signiﬁcantly improved.

(3) The surface parametric control and global optimization

methods are successfully applied to the aerodynamic opti-

mization of the transonic axial ﬂow compressor Rotor 37

and Stage 35, and the optimized results are obtained in

shorter time (24 h and 72 h, respectively), which is easy

to accept in engineering. After optimization, under the

Fig. 26 Comparison of relative Mach number at hub, middle and tip sections of blades before and after optimization of Stage 35.

1632 J. CHENG et al.

constraint that the ﬂow rate is not reduced and the pres-

sure ratio is basically unchanged, the adiabatic efﬁciency

of Rotor 37 at design point increases by 1.02%, whilst the

surge margin increases by 0.84%; the adiabatic efﬁciency

of Stage 35 at design point increases by 0.54%, whilst the

surge margin increases by 1.11%.

(4) The new optimization tool composed of the Bezier sur-

face parametric control and global optimization meth-

ods presented in this paper has the advantages of

saving a lot of calculation cost and fast global optimiza-

tion for aerodynamic optimization of 3D blade of axial

compressor, which leads to broad application prospects

for optimization of compressor blades.

Acknowledgements

This study was supported by the National Natural Science

Foundation of China (No. 51576007) and Civil Aircraft Spe-

cial Research of China (No. MJZ-016-D-30).

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