A surface parametric control and global
optimization method for axial ﬂow compressor
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
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-
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/).
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
E-mail address: email@example.com (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
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.
The other approach is to
change the shape of central arced curve and thickness distribu-
tion in each section,
or to change the suction or pressure
side proﬁle of each section by means of free curves,
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
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
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
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-
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-
(proposed in 1992), particle swarm algorithm
(proposed in 1995) and Artiﬁcial Bee Colony (ABC) algo-
(proposed in 2005), have developed rapidly. The
ABC algorithm has a unique advantage in terms of searching
ability among the intelligent algorithm.
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,
at expense of
the precision of optimal solution, within limited time by using
the existing computing resources. Currently used optimization
strategies include decomposition method,
non-gradient optimization method
direct optimization method.
points out that no opti-
mization strategy is completely superior, and different combi-
nation optimization strategies should be adopted for different
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
ensures the smoothness of the optimized surface, which can
reduce the ﬂow loss caused by the partial roughness of the
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
;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
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
refers to the control
point of the Bezier surface, and the number of control points is
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
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= 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
min þRand 0;1ðÞXj
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
min and Xj
max denote the smallest and largest feasible
solutions of the jth component of the Nfeasible solutions,
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.
istands for the jth component explored by the ith
employed bee; j21;2;;D
2g, and k–i;uj
istands for a random number between
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.
where Nestands for the number of employed bees, that is, N
is the probability of choosing a food source; f(X
) and f(X
stand for the ﬁtness of the ith and mth individuals,
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.
iin Eq. (9) stands for the new location of a food source
found by employed bees; lj
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;
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):
where j21;2;;Dfg;and di;t
represents the Chebyshev
distances between Xiand Xt.
The neighborhood of X
is determined by Eq. (12), and the
formula shows that when the Chebyshev distance between X
Fig. 3 Flowchart of a standard ABC algorithm.
A surface parametric control and global optimization method 1621
is less than the product of the neighborhood radius r
and the average Chebyshev distance, X
is in the X
hood, otherwise it is not.
where mdiis the average Chebyshev distance between X
the total onlooker bee colony, Sis the area adjacent to X
is the radius of the adjacent area, and experience shows that
when r= 1, the algorithm has the best convergence.
best is calculated by
¼max fit XNeib
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
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.
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.
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
182 170 60
Griewank 5.3 10
194 122 83
1622 J. CHENG et al.
difﬁcult to guarantee the mechanical strength of the blade.
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
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
then this method was also applied in the optimization of turbo-
machinery, obtaining lots of scientiﬁc research result.
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
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
The constraint is as follows:
f¼eff if TPRTPRori
<0:5%and mass massori 0
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
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
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
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
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
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.
: 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
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
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
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
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
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
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.
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.
This study was supported by the National Natural Science
Foundation of China (No. 51576007) and Civil Aircraft Spe-
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