Pressure distribution guided supercritical wing optimization

Abstract

Pressure distribution is important information for engineers during an aerodynamic design process. Pressure Distribution Oriented (PDO) optimization design has been proposed to introduce pressure distribution manipulation into traditional performance dominated optimization.In previous PDO approaches, constraints or manual manipulation have been used to obtain a desirable pressure distribution. In the present paper, a new Pressure Distribution Guided (PDG) method is developed to enable better pressure distribution manipulation while maintaining optimization efficiency. Based on the RBF-Assisted Differential Evolution (RADE) algorithm, a surrogate model is built for target pressure distribution features. By introducing individuals suggested by sub-optimization on the surrogate model into the population, the direction of optimal searching can be guided. Pressure distribution expectation and aerodynamic performance improvement can be achieved at the same time. The improvements of the PDG method are illustrated by comparing its design results and efficiency on airfoil optimization test cases with those obtained using other methods. Then the PDG method is applied on a dual-aisle airplane’s inner-board wing design. A total drag reduction of 8 drag counts is achieved.

Full text

Pressure distribution guided supercritical wing
optimization
Runze LI, Kaiwen DENG, Yufei ZHANG, Haixin CHEN
*
School of Aerospace Engineering, Tsinghua University, Beijing 100084, China
Received 4 April 2018; revised 13 April 2018; accepted 15 May 2018
KEYWORDS
Aerodynamic optimization;
Man-in-loop;
Pressure distribution;
RBF-assisted;
Supercritical wing
Abstract Pressure distribution is important information for engineers during an aerodynamic
design process. Pressure Distribution Oriented (PDO) optimization design has been proposed to
introduce pressure distribution manipulation into traditional performance dominated optimization.
In previous PDO approaches, constraints or manual manipulation have been used to obtain a desir-
able pressure distribution. In the present paper, a new Pressure Distribution Guided (PDG) method
is developed to enable better pressure distribution manipulation while maintaining optimization
efficiency. Based on the RBF-Assisted Differential Evolution (RADE) algorithm, a surrogate model
is built for target pressure distribution features. By introducing individuals suggested by sub-
optimization on the surrogate model into the population, the direction of optimal searching can
be guided. Pressure distribution expectation and aerodynamic performance improvement can be
achieved at the same time. The improvements of the PDG method are illustrated by comparing
its design results and efficiency on airfoil optimization test cases with those obtained using other
methods. Then the PDG method is applied on a dual-aisle airplane’s inner-board wing design. A
total drag reduction of 8 drag counts is achieved.
Ó2018 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/).
1. Introduction
Optimization design is more and more widely used to gain
practical industrial designs in recent years. Multiple types of
optimization algorithms have been well developed for more
efficient and capable industrial optimal searching. However,
although there have been many efforts using optimization
algorithms to improve aerodynamic performances, practical
design via optimization is still limited, and ‘‘cut-&-try”is still
heavily relied on in the aircraft industry.
1
On one hand, even
though multi-objective and multi-constraint optimization has
become popular,
2,3
the complexity of industrial considerations
makes it difficult to define all engineering-needed objectives
and constraints for optimization algorithms.
4,5
On the other
hand, engineers find their experiences, considerations, and
judgments difficult to be introduced into an automatic opti-
mization design process.
1,4,6–8
Therefore, in order to gain an
engineering-acceptable design, optimization design not only
needs a robust and flexible algorithm to endure a large amount
of objectives and constraints, but also needs ways to transfer
*Corresponding author.
E-mail address: chenhaixin@tsinghua.edu.cn (H. CHEN).
Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
Chinese Journal of Aeronautics, (2018), xxx(xx): xxx–xxx
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.2018.06.021
1000-9361 Ó2018 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/).
Please cite this article in press as: LI R et al. Pressure distribution guided supercritical wing optimization, Chin J Aeronaut (2018), https://doi.org/10.1016/j.
cja.2018.06.021

those in-brain requirements into what the optimization algo-
rithm can accept or process.
Experienced supercritical wing designers usually do not
seek a wing with the highest lift/drag ratio, but a design that
best compromises the performances of different disciplines
and different flight conditions. What’s more, they emphasize
much on design robustness, such as the drag divergence Mach
number, the buffet onset lift coefficient, etc. Since both the
performance and robustness are essentially the outcomes of
flow structures as well as their evolutions, the flow physics is
relatively clear. Engineers tend to judge a design through flow
patterns and details. Currently, pressure distribution is the
most cared flow structure in supercritical wing design. Many
rules or criteria on pressure distribution have been proposed.
For instance, the shock should be properly located to get a
good robustness, and the aft loading of an airfoil should not
be too large or else the nose-down pitching moment could be
unacceptable.
9–11
By realizing these explicit rules of hint experiences on pres-
sure distribution, a designer could improve a wing’s design
point performances while achieving preferable off-design and
multi-disciplinary properties. This is basically what they do
in ‘‘cut-&-try”. From the 1980s, with the development of
inverse design methods
12–14
researchers tried to gain a design
by realizing a desired pressure distribution. However, due to
the difficulties of ‘‘designing”a physically-existing pressure
distribution, even for a simple geometry at a specific flow con-
dition,
11,15
the application of these methods in the industry is
still limited.
Zhang et al.
16
studied the drag, moment, and especially
robustness of three categories of typical pressure distributions
for supercritical airfoils, i.e., shock-free, double shock, and
weak shock. The weak shock pressure distribution was evalu-
ated as the best. Furthermore, other suggestions on pressure
distribution have been proposed, such as that the shock loca-
tion should range from 45% to 55% chord length for a
single-aisle civil aircraft, the aft-loading should compromise
the lift generation and a mild pitching moment, etc. To realize
these suggestions, methods were developed to induce the
pressure distribution to approach a desired pattern during
optimization design. Some of them have been proven effective
in industrial design. Optimal searching is no longer driven only
by performances, but also by pressure distributions. Such
methods are generalized in the present paper as Pressure
Distribution Oriented (PDO) optimizations.
The PDO method has also been applied to a dual-aisle air-
plane wing design.
17
The cruise performance and robustness
were improved while the proposed weak shock pressure distri-
bution was also achieved. The shock location was pushed
slightly downstream that of a single-aisle civil aircraft to fit a
higher Reynolds number and different cruise lift coefficients.
Both the single- and dual-aisle design studies have shown that
the location of shock wave is a critical factor for a supercritical
wing’s balance of performance and robustness.
Since the PDO method is characterized by the ability of
manipulating pressure distributions, it can be consequently
used to study the performance of a specified type of pressure
distribution. By using the PDO method, Zhang et al.
18
achieved supercritical natural laminar airfoils with different
pressure distributions characterized by assigned favorable
pressure gradients and shock locations. Their gains on laminar
friction reduction and penalties on the wave drag and robust-
ness were then systematically compared.
There are two types of PDO optimization developed in
previous studies according to their methods of manipulating
pressure distributions. The first one can be called Pressure
Distribution Constrained (PDC) method.
16–18
Constraints
are set to rule out or punish designs with an unsatisfying pres-
sure distributions shape. It essentially posts restrictions on the
optimal search direction. The optimization efficiency and
global optimal searching capability are inevitably deteriorated.
The other method is a manual or ‘‘man-in-loop”
1,16
one.
Engineers can guide an optimization’s pressure distribution
trend by manipulating the population. They need to introduce
external individuals, which have the expected pressure distri-
bution characteristics, into the population, or they need to
eliminate unsatisfactory individuals from the population. This
method demands a large amount of human labors and
experiences.
In this paper, a new approach of using the pressure distri-
bution expectation to guide optimization is developed. Instead
of manual population control, the so-called Pressure Distribu-
tion Guided (PDG) method uses a surrogate model to search
potential individuals which best satisfy the expectation on
pressure distribution and also have excellent performances.
These individuals are introduced into the population of an
evolutionary optimization algorithm. In this way, the opti-
mization process is automatically guided to approach the
expected pressure distribution feature, while the diversity of
the population during the optimization is well preserved. To
more rationally define the ‘‘expectation”on the pressure distri-
bution, several physical or empirical relations are also studied.
Unlike the PDC method, the pressure distribution is pur-
sued by ‘‘guidance from good individuals”instead of ‘‘punish-
ing bad individuals”.
As a new branch of PDO optimization, in the present
paper, the idea of the PDG method is firstly introduced. It is
then tested and compared with previous methods by airfoil
cases. Proven by results that it has better optimal searching
efficiency and pressure distribution manipulation capabilities,
the PDG method is applied to the supercritical wing design
of a dual-aisle civil airplane.
2. Optimization and modeling methods
2.1. RBF assisted differential evolution algorithm
An evolutionary optimization algorithm can use the assistance
of surrogate models to improve efficiency.
19
In the present
study, an RBF (radial basis function) Assisted Differential
Evolution (RADE) algorithm is used as the primary optimiza-
tion algorithm.
20
The basic flow chart of the RADE algorithm
is outlined in Fig. 1(a), in which kis the index of the current
generation and Pis short for population. The dash box con-
tains the main optimization process of Differential Evolution
(DE) optimization, and the upper part outside the box shows
the RBF surrogate model’s behavior. By utilizing computed
individuals’ information, the RBF surrogate model can obtain
approximation of CFD results. Then an optimal search is
conducted on the RBF response surface to find individuals
that are most likely to produce excellent objectives. Those
2 R. LI et al.
Please cite this article in press as: LI R et al. Pressure distribution guided supercritical wing optimization, Chin J Aeronaut (2018), https://doi.org/10.1016/j.
cja.2018.06.021

individuals are added into the current candidate population to
participate in the main DE optimization process. Since the
search direction of an evolutionary algorithm is mainly based
on the candidate population and the sorting strategy, this will
form guidance to the search direction. In RADE, the surrogate
model is not a substitution to an accurate CFD analysis. For
individuals from both the main DE optimization process and
the RBF surrogate model, a CFD objective evaluation is fully
conducted.
In an original RADE algorithm, the optimal search on the
RBF surrogate model shares the same objectives and con-
straints as those of the main DE process. The purpose of the
surrogate model is only to accelerate the main optimization
process. The whole method can be used as a black-box
multi-objective multi-constraints optimizer. The PDC and
man-in-loop pressure distribution manipulation can both be
directly realized on the RADE. The optimization can be
directed to achieve the expected pressure distribution through
manually adding/eliminating individuals by people, or through
automatically eliminating individuals by constraints on the
pressure distribution shape.
There are drawbacks in these methods. For manual
population manipulation, the basic idea is to manually judge
all calculated individuals’ pressure distributions and add or
delete individuals to reconstruct the current population for
evolutionary operators. On the contrary, the constraints of
PDC optimization limit the optimization direction by eliminat-
ing non-feasible individuals, and then the population will meet
the constraints eventually. Both methods alter the candidate
population according to an engineer’s judgment, for example,
their consideration on pressure distribution. However,
eliminated individuals might have good objective function
values, while artificially-introduced individuals usually do not
have good enough performances to survive in the sorting pro-
cedure. Therefore, the optimization efficiency often suffers
badly. If the introduced individuals could have competitive
performances, the efficiency should be able to be maintained.
Furthermore, proposed pressure distribution features may
compromise the performance and robustness, and hence the
method should only provide attempts on these features instead
of forcing the entire population achieving them.
The present PDG method is based on a modification to the
RADE algorithm. The surrogate model contains the mapping
between pressure distribution characteristic parameters and
the design variable space. Objectives and constraints can be
set different from those of the main DE process. A multi-
objective optimal search on the RBF surrogate model is
constructed to find potential individuals that have both good
performances and expected pressure distribution features.
These potential individuals are introduced to the DE’s
Fig. 1 Flow charts of the RADE algorithm and PDC method and the PDG method.
Pressure distribution guided supercritical wing optimization 3
Please cite this article in press as: LI R et al. Pressure distribution guided supercritical wing optimization, Chin J Aeronaut (2018), https://doi.org/10.1016/j.
cja.2018.06.021

candidate population to guide the optimization process,
instead of forcing the entire population like the PDC method
does. The constraints on the pressure distribution in the DE
can therefore be relaxed or eliminated in the PDG process.
The realization of the PDG method is shown in Fig. 1(b).
2.2. Modeling and numerical method
The Class Shape Transformation (CST) method is widely used
in wing design due to its capability of constructing a smooth
airfoil with a few design variables.
18
In the present paper, a
6th-order Bernstein polynomial is used as the shape function
for both the upper and lower surfaces of an airfoil. The num-
ber of design variables for an airfoil is 14. The ranges of the
variables are the same as those in Ref.
18
. The wing surface is
constructed by interpolation with several span-wise distributed
airfoils.
The aerodynamic performances of airfoils and wings are
evaluated by an in-house solver,
21,22
which is a cell-centered
finite volume compressible Reynolds Average Navier-Stokes
(RANS) solver with multiple types of numerical schemes and
turbulence models. The solver has been validated by various
test cases and successfully applied in many industrial design
activities.
16–18,21–23
In this paper, it employs the MUSCL
scheme for the reconstruction, Roe’s scheme for the spatial dis-
cretization, the lower-upper symmetric Gauss-Seidel method
for the time advancing, and the Shear Stress Transport
(SST) model for the turbulence modeling. The accuracy of
the solver has been validated by our previous study.
23
3. Method validation
3.1. Pressure distribution consideration of a supercritical wing
For a supercritical wing, the pressure distribution contains var-
ious information of aerodynamic characteristics. Since the off-
design performances are essentially the outcome of the flow
structure evolution among different flight conditions, it is
not necessary to evaluate the drag divergence Mach number
or buffet onset point through CFD calculations of a series of
flight conditions. Pressure distribution manipulation and per-
formance optimization on several typical conditions can
suffice.
1,17
On the other hand, many important characteristics are
directly associated with certain flow patterns, such as the wave
drag and the shockwave strength. Some experience of pressure
distribution has been gathered to gain a fine compromise
among lift, drag, moment, shockwave stability, trailing edge
separation characteristic, and geometry considerations.
16
However, features are different for different flight conditions
and different aircraft planforms. A dual-aisle aircraft usually
has a higher flight Mach number and a higher Reynolds num-
ber than those of a single-aisle aircraft, and the shockwave is
downstream with a higher strength. Therefore, the wave drag
is a major party in drag reduction, and the off-design charac-
teristic is more non-linear. In this paper, several pressure
distribution considerations
16
of a wing of a dual-aisle aircraft
at a flight Mach number of 0.85 are taken in the following
optimizations. The pressure distribution considerations of an
airfoil are transformed according to the cosine rule of a swept
wing.
23
These considerations can be expressed as follows:
(1) The pressure coefficient of the suction peak of a wing
cannot be lower than 1.0; otherwise, the flow acceler-
ates too much around the leading edge, which will make
the aerodynamic performance like that of ‘‘peaky”
airfoils.
24
(2) The slope of the pressure plateau on the upper surface
(from the suction peak to the shockwave front) shall
be between 0.2 and 0.5 to ensure an appropriate length
of the suction platform for providing enough lift.
(3) The pressure plateau shall be as smooth as possible to
avoid undesirable robustness issues.
(4) The slope of the pressure recovery zone near the trailing
edge of a wing is kept lower than 3.0 to prevent flow
separation.
(5) The aft loading shall not be too large, or else the nose-
down pitching moment will be too large.
3.2. Relation between shockwave strength and drag coefficient
The wave drag is directly related to the shockwave strength.
Although the shockwave strength may have different
definitions in previous studies, it can be roughly described by
the pressure rise between the two sides of the shockwave
DCp, as shown in Fig. 2. According to the Oswatitsch
theorem,
25
the wave drag of an airfoil can be described as in
Eq. (1) under the assumption of isentropic flow before the
shockwave, where Ma1is the Mach number at the wave front.
CD;wave /m4þe
e¼0:3to0:4
m¼Ma2
11¼0
8
>
<
>
:
ð1Þ
The pressure coefficient can be defined as
Cp¼2=cMa2
1

p=p11ðÞ, where cis the specific heat ratio,
Ma1is the freestream Mach number, and p=p1is the ratio of
static pressure normalized by the freestream value. Then DCp
can be described as in Eq. (2) using the isentropic relation
and the Rankine-Hugoniot equation.
DCp42þc1
ðÞ
Ma2
1

cþ1ðÞ
2c1
c1Ma2
1
mþc
cþ1m2
 ð2Þ
Fig. 2 Definitions of Err and DCp.
4 R. LI et al.
Please cite this article in press as: LI R et al. Pressure distribution guided supercritical wing optimization, Chin J Aeronaut (2018), https://doi.org/10.1016/j.
cja.2018.06.021

Then the wave drag can be expressed as in Eq. (3), where S
is a coefficient associated with the airfoil shape and the flight
condition.
CD;wave SDC4þe
pð3Þ
A large number of supercritical airfoils through optimiza-
tion are gathered to verify the relation. The airfoils are evalu-
ated by RANS. The free stream Mach number is 0.742, and the
Reynolds number is 10 million. The lift coefficients of the
airfoils are kept the same (CL¼0:787). The airfoils all have
similar flow patterns, which are a single shockwave and a
smooth suction plateau, and all airfoils have normalized chord
length, i.e. X20;1½. The smoothness function Err is defined
as Cpfluctuations on the suction plateau, which is the area
of the blue shadow region in Fig. 2. The pressure distribution
in Fig. 2 has a smoothness function of Err ¼0:012. The drag
coefficients of the sampled airfoils which have small fluctua-
tions (Err <0:012) are shown in Fig. 3. A curve fitting relation
between the drag coefficient CDand the shockwave strength
DCpis shown as the red curve in Fig. 3, where the coefficient
Sequals to 0.05, and CD0is the smallest drag coefficient for
non-shockwave airfoils.
The sampled airfoils fit the curve with a less than 0.1 deriva-
tion of DCp. Therefore, the shockwave strength is correlated
with the total drag and can be used as an effective auxiliary
parameter for drag reduction. It is also a parameter to distin-
guish between different pressure distribution types. As shown
in Fig. 3, when DCpis less than 0.3, the drag coefficient penalty
caused by the shock wave is smaller than 0.0005, and the drag
reduction caused by shockwave weakening is negligible when
DCpis less than 0.2. Moreover, it demonstrates that airfoils
with the same total drag coefficient can have different shock-
wave strengths, which reveals that while keeping the total drag
unchanged, the shock strength can be adjusted to get a better
robustness. Therefore, if the shockwave strength is used as an
objective or sub-objective in a drag reduction optimization, it
could have the potential to increase the overall optimization
efficiency or robustness.
3.3. Single-objective PDG optimization
Pressure distribution features can be used as objectives in the
main optimization process, or in the sub-optimization process.
In this section, single-objective optimizations of a supercritical
airfoil with different optimization settings are compared to
demonstrate the acceleration effect of sub-optimization. The
free stream Mach number is 0.742, the Reynolds number is
10 million, and the lift coefficient is kept 0.787 during the opti-
mization. The design variables are 14 CST parameters.
Five settings are studied in this case, as shown in Table 1,
where RLE is the airfoil leading edge radius. Opt1 and Opt2
are baseline PDC optimizations using DE and RADE algo-
rithms, respectively. Opt3 and Opt4 directly use the shockwave
strength as an overall objective of RADE. Opt1 has no sub-
optimization process. Opt2, Opt3, and Opt4 have the same
objectives and constraints in both the main optimization pro-
cess and the sub-optimization; therefore, Opt3 and Opt4 are
still called PDC optimizations. Opt5 uses the modified RADE
with different objectives and constraints in the main optimiza-
tion process and the sub-optimization, which is a demonstra-
tion of the present PDG method. According to the relation
between the shockwave strength and the drag coefficient in
Section 3.2,DCpis used as a sub-objective to guide the opti-
mization generating airfoil designs for an inboard wing, of
which DCpis expected to be smaller than 0.2. Pressure distribu-
tion considerations described in Section 3.1 are also applied as
constraints in the main optimization.
The population sizes of all the cases are 32, and 32 genera-
tions are carried out to compare the efficiency and effective-
ness. All the cases have the same initial population, and the
averaged drag coefficients of the final generation are used for
comparison, as shown in Table 2. The convergence histories
are shown in Fig. 4.Table 2 shows that Opt2 has a slightly bet-
ter final result than that of Opt1, and a result that has a similar
performance to the final result of Opt1 can be achieved by
Opt2 with only around 600 individuals, as shown in Fig. 4,
Fig. 3 Relation between total drag and shockwave strength.
Table 1 Optimization settings of 5 cases.
Item Opt1 Opt2 Opt3 Opt4 Opt5
Method PDC PDC PDC+obj PDC+obj+const PDG
Algorithm DE RADE RADE RADE RADE with sub-optimization
Objectives Main optimization CDCDCD,DCpCD,DCpCD
Sub-optimization CDCD,DCpCD,DCpCD,DCp
Constraints RLE >0.0105 >0.0105 >0.0105 >0.0105 >0.0105
CD<0.0130
Pressure distribution guided supercritical wing optimization 5
Please cite this article in press as: LI R et al. Pressure distribution guided supercritical wing optimization, Chin J Aeronaut (2018), https://doi.org/10.1016/j.
cja.2018.06.021

where ID is the index of each individual. Therefore, the RBF-
assisted method could improve the efficiency of the
optimization.
Fig. 5 shows two typical pressure distributions of the results
which have an unsatisfactory performance. The strong shock-
wave airfoil (Fig. 5(a)) whose CDequals to 0.01071 is the initial
design. Although reducing the shockwave strength can reduce
the wave drag, shock-free and weak-shock airfoils are not the
only possibilities for reducing the shockwave. Due to a lack of
proper constraints, Opt3 wastes part of its ability to search in a
wrong direction, i.e., airfoils with massive upper surface sepa-
ration or double-shockwave airfoils (Fig. 5(b)). Moreover,
excessive constraints can have negative effects on the main
process, as shown in Opt4, and the overall efficiency is low.
Opt5 uses the shockwave strength as a sub-objective to provide
an alternative searching direction for the main process.
Because the computation cost for searching on the RBF is neg-
ligible, the efficiency of the main optimization process is not
depressed. Table 2 shows that Opt5 has the best final result
in all of the five optimizations. Fig. 4 shows that a result sim-
ilar to the final result of Opt2 can be achieved by Opt5 with
around 500 individuals. Therefore, the optimization is acceler-
ated by the sub-optimization, and the desired pressure distri-
bution feature is achieved.
Fig. 6 shows the final result of Opt5, of which CDequals to
0.00958. The pressure distribution type is shock-free. The pres-
sure distribution satisfies all of the constraints in the optimiza-
tion, but the robustness is not good according to Ref.
16
. This
case is only a demonstration of the PDG optimization.
Multi-point optimization is carried out in the next section to
improve the robustness.
In summary, the PDG optimization is achieved by a combi-
nation of the main optimization process and sub-optimization.
As shown by the results of Opt3 and Opt4, the pressure distri-
bution features used as main process objectives might lead to
undesired results, while adding more constraints in the main
optimization process leads to a low efficiency. The PDG opti-
mization, i.e., Opt5, uses sub-objectives and sub-constraints to
utilize the pressure distribution expectation, and the efficiency
is not compromised. Moreover, the sub-optimization only
serves as a source of potential individuals to the main process.
Therefore, the sub-optimization can handle flexible expecta-
tions of engineers without interfering the entire optimization
efficiency.
3.4. Multi-point pressure distribution guided optimization
Different methods of pressure distribution oriented optimiza-
tion along with a baseline optimization are compared in this
section. Cases are used to demonstrate the PDG method’s
effectiveness of pressure distribution manipulation and
optimization. A multi-point airfoil optimization is applied
based on the RADE algorithm. The airfoil is expected to have
a shockwave location at 50% chord length at the cruise condi-
tion. This case is to demonstrate the pressure distribution con-
trol ability of the present method. In order to make the
comparison simple and clear, only the cruise drag (CD0) is used
as the objective of the main optimization process. Drag
divergence and buffet onset control are maintained through
Table 2 Final results of 5 cases.
Case Opt1 Opt2 Opt3 Opt4 Opt5
Average CD0.00978 0.00973 0.00989 0.00988 0.00962
Fig. 4 Convergence histories of the five cases.
Fig. 5 Typical pressure distributions with an unsatisfied
performance.
6 R. LI et al.
Please cite this article in press as: LI R et al. Pressure distribution guided supercritical wing optimization, Chin J Aeronaut (2018), https://doi.org/10.1016/j.
cja.2018.06.021

constraints. Flight conditions are shown in Table 3. The objec-
tives and constraints of 4 different settings can be seen in
Table 4.
Opt1 is the baseline optimization. Opt2 is a demonstration
of manual population manipulation. Since the final effect can
be highly influenced by different engineers’ involvement, a
fixed strategy is programed into the optimization sorting strat-
egy to mimic the manual manipulation. The basic idea of man-
ual manipulation is to select individuals for the DE operators
based on personal judgement of performance and pressure dis-
tribution considerations. In this case, individuals with a low
drag and a single shockwave in all 3 flight conditions as well
as a small value of XSW0 0:5
jj
are more likely to be selected.
The strategy for Opt2 is shown in Fig. 7. Opt3 applies con-
straints of the shockwave position XSW0 0:5
jj
<0:05. Opt1,
Opt2, and Opt3 share the same objectives and constraints in
the main optimization process and the sub-optimization.
Opt4 uses the PDG method, which means more objectives
and constraints are applied in the sub-optimization to guide
the search direction of the optimization. According to the rela-
tion between the shockwave strength and the drag coefficient
in Section 3.2, the shockwave strength at the cruise condition
is constrained by DCp0<0:3 in the sub-optimization.
Fig. 8 shows the convergence histories of the optimizations.
Fig. 8(a) is the objective of the cruise condition. Fig. 8(b) is the
pressure distribution consideration, i.e., the distance between
the shockwave and the half chord length. Fig. 8(c) and (d)
show the drag increments for the drag divergence condition
and the buffet onset point. The dash lines in Fig. 8(b)–(d) show
the critical constraint values. Table 5 shows some statistical
results of the methods. Fig. 9(a) shows the individuals within
the requested shock wave region, i.e., 45–55%, and only the
ones meeting all the constraints are included. A typical
optimized result is shown in Fig. 9(b). It is a weak-
shockwave airfoil with a cruise drag of 0.00988.
As shown by the results in Fig. 8(b), compared to Opt1, all
the pressure distribution oriented methods are able to get
shockwaves in the requested region. The results of Opt1 are
almost equally distributed in different shock locations. By con-
trast, the results of Opt2 to Opt4 are getting near 50% chord
length when the optimization proceeds. Table 5 shows that
Opt2 and Opt3 have fewer results satisfying Opt1 constraints
than those of Opt1, whereas Opt2 and Opt3 have more results
satisfying Opt3 constraints. Therefore, although manual popu-
lation manipulation and constraints in Opt3 and Opt4 have a
better efficiency generating results with shockwaves in the
requested region, the ability of generating results satisfying
basic constraints, i.e., Opt1 constraints, is compromised. By
contrast, Opt4 has more results satisfying Opt1 and Opt3 con-
straints, and these results have shockwaves near the required
50% chord length location.
According to the results of Opt2 and Opt3, we can see that
the constraint in the main optimization process is more
objective than manual manipulation, but also more inflexible.
When the critical value for XSW0 0:5
jj
constraint is small, the
constraint may eliminate most individuals in the early stage of
the optimization, and a convergence is hard to achieve. How-
ever, if a loose constraint is applied, the final results cannot
satisfy the expectation of the shockwave location.
Opt4 applies more objectives and constraints in the
sub-optimization to guide the searching direction. The
Fig. 6 Pressure distribution of Opt5’s final result.
Table 3 Flight conditions of three design points.
Parameter Cruise
(Condition 0)
Drag divergence
control
(Condition 1)
Buffet onset
control
(Condition 2)
Ma 0.742 0.752 0.742
C
L
0.787 0.787 0.850
Re 10
7
10
7
10
7
Re 10
7
10
7
10
7
Table 4 Four different optimization settings.
Item Opt1 Opt2 Opt3 Opt4
Method RADE Man-in-loop PDC PDG
Algorithm RADE RADE RADE RADE with sub-optimization
Objectives Main optimization CD0CD0CD0CD0
Sub-optimization CD0,XSW0 0:5jj,DCp0
Constraints RLE >0.0105 >0.0105 >0.0105 >0.0105
CD1CD0<0.0007 <0.0007 <0.0007 <0.0007
CD2CD0<0.0012 <0.0012 <0.0012 <0.0012
Single shock Active Active
XSW0 0:5
jj <0.05 <0.05
Pressure distribution guided supercritical wing optimization 7
Please cite this article in press as: LI R et al. Pressure distribution guided supercritical wing optimization, Chin J Aeronaut (2018), https://doi.org/10.1016/j.
cja.2018.06.021

sub-optimization provides promising individuals for the main
process. Since the sub-constraints are stronger and the shock-
wave strength along with XSW0 0:5jjare also used as sub-
objectives, the individuals provided by the sub-optimization
have a higher possibility to reduce the drag and meet the con-
straints synchronously. Moreover, because the drag divergence
and buffet characteristics are associated with shockwave-
induced boundary layer separation, the sub-constraints about
the shockwave strength in these flight conditions can help
improving the robustness, and then accelerating the main opti-
mization process. Therefore, the PDG method could increase
the optimization efficiency and generate more desirable results.
Fig. 7 Flow chart with an additional sorting strategy for Opt2.
Fig. 8 Convergence histories of the 4 optimization settings.
8 R. LI et al.
Please cite this article in press as: LI R et al. Pressure distribution guided supercritical wing optimization, Chin J Aeronaut (2018), https://doi.org/10.1016/j.
cja.2018.06.021

4. Method application
Previous studies
16,17
have indicated that a weak-shockwave
pressure distribution has a better balance between the cruise
performance and the robustness than those of double-
shockwave and shock-free pressure distributions on a single-
aisle airplane. However, since a dual-aisle airplane has a differ-
ent planform and a different flight condition, the details of
preferable features are unclear, especially for the inboard wing.
The inboard wing has a much longer chord length than that of
the outboard wing, and hence a small modification of the
inboard wing can have a significant impact on the span-wise
load distribution and a strong influence on the wing perfor-
mance. Meanwhile, the inboard wing’s pressure distribution
can directly influence the outboard wing due to the cross flow,
and the interference of the wing, fuselage, and nacelle/pylon
makes the non-linear phenomenon even more severe than on
a single-aisle airplane. As from the discussion of the
CDDCprelation in Section 3.2, once DCpfor an airfoil is
lower than 0.3, the drag increment caused by a shockwave is
insignificant. Therefore, a weak shockwave of DCp¼0:3 can
be a good balance between cruise drag and robustness.
For a typical supercritical airfoil, once the shockwave
strength DCpis fixed, the section lift coefficient is mainly
dependent on the suction plateau and the shockwave location.
A higher suction peak causes a strong pressure recovery on the
suction plateau, and may induce unsatisfying robustness. A
further downstream shockwave location tends to have a
slightly stronger pressure recovery behind the shockwave,
which may increase the risk of shockwave-induced boundary
layer separation, but it also provides more lift. It is especially
important for the inboard wing, because an increment of the
lift coefficient of inboard wing sections can lead to a large
reduction of the outboard wing load, which is good for
improving the buffet characteristics. Therefore, when the
shockwave strength DCpis kept lower than 0.3, a further down-
stream shockwave location and a lower suction peak on the
inboard wing may result in a more excellent lift/drag ratio
and robustness.
A dual-aisle airplane is used as the test case for the inboard
wing optimization based on the PDG method. The cruise
Mach number is 0.85, the Reynolds number is 40 million,
and the lift coefficient is 0.48. The baseline wing was designed
in our previous study,
23
which mainly focused on the outboard
wing optimization. The baseline wing has an excellent robust-
ness and a satisfying weak-shockwave pressure distribution on
the outboard part. However, there is a strong shock on the
inboard wing. In this paper, it is used to study different
inboard wing pressure distributions and the influence on the
overall performances. The inboard wing geometry is interpo-
lated by 3 airfoils generated by the CST method. The locations
are shown as Sections 1–3 in Fig. 10. The design variables are
the 42 CST variables of the 3 airfoils, and the computational
mesh is the same as that in our previous study.
23
The black
solid and dash lines in Fig. 10 show the shockwave location
of the baseline wing and the expected inboard wing shockwave
location, respectively.
Table 5 Statistics of the 4 optimization settings.
Configuration Individuals satisfying
Opt1 constraints
Individuals satisfying Opt3 constraint
Individual amount
(average XSW0 0:5
jj
)
Individual amount
(XSW0 0:5
jj
<0:02)
MinimumCD0
Opt1 132 29 (0.034) 8 (28% of 29) 0.00987
Opt2 87 53 (0.030) 12 (23% of 53) 0.00989
Opt3 89 39 (0.028) 13 (33% of 39) 0.00987
Opt4 290 190 (0.023) 92 (48% of 190) 0.00988
Fig. 9 Final results of the 4 optimization settings.
Pressure distribution guided supercritical wing optimization 9
Please cite this article in press as: LI R et al. Pressure distribution guided supercritical wing optimization, Chin J Aeronaut (2018), https://doi.org/10.1016/j.
cja.2018.06.021

The PDG method is used to manipulate the pressure distri-
bution while reducing the cruise drag. Optimization settings
are listed in Table 6. The cruise drag is set as the main objective
and sub-objective. The distance between the actual shockwave
location and the proposed position as well as the shockwave
strength are also adopted as sub-objectives. The pressure dis-
tribution considerations mentioned in Section 3.1 are used as
constraints, as well as the constraints about the geometry
and the shockwave strength. The optimization population in
each generation has a size of 40 individuals, and total 25 gen-
erations are carried out to gain final results. The convergence
history is shown in Fig. 11(a). Results with and without pro-
posed features are used to compare the performances. Four
typical results in Fig. 11(b) are selected to compare the perfor-
mances and pressure distributions. The pressure distributions
are shown in Fig. 12. The characteristics of the pressure distri-
butions are shown in Table 7. The average values in the table
are based on the values of Sections 1–4 in Fig. 10.
Design4 in Table 7 is the final result of the optimization. It
has the minimum drag coefficient of all feasible results and a
short distance to the proposed shock location. Although it
has a weak double shock on the 17.5% half-span section, it
does not violate the constraints of the main optimization pro-
Fig. 10 Wing section locations and shockwave locations.
Table 6 Settings of dual-aisle airplane inboard wing optimization.
Item Main optimization Sub-optimization
Objective
Cruise drag CDCruise drag CD
Summation of shockwave strengths PDCp;i
Summation of shockwave location requirements
PXSW XSW;proposed


i
Summation of absolute pressure coefficients of suction peak
PCp;suc


i
Constraint
Amount of double shock
sections
<2 <1
Pitching moment coefficient <0.065 <0.065
Leading edge radius >critical values for 3 sections,
respectively
>critical values for 3 sections, respectively
Maximum thickness >critical values for 3 sections,
respectively
>critical values for 3 sections, respectively
Note: Constraints about considerations in Section 3.1 are also applied.
Fig. 11 Optimization history and results.
10 R. LI et al.
Please cite this article in press as: LI R et al. Pressure distribution guided supercritical wing optimization, Chin J Aeronaut (2018), https://doi.org/10.1016/j.
cja.2018.06.021

cess. Design3 is also a typical result for drag reduction. How-
ever, it cannot satisfy the constraint of the shockwave location,
and its suction peak is high. Design2 shows that when the dou-
ble shock in Design4 is eliminated, the drag reduction quantity
is limited. If a lower suction peak is applied, i.e., in Design1,
the cruise drag is compromised.
Fig. 13 shows the shockwave locations and span-wise loads
of the wings. Basically, a further downstream shockwave has a
larger inboard wing load (Design4); consequently, the maxi-
mum lift coefficient on the outboard wing is reduced, which
usually means a weak shockwave strength (see Fig. 12,
17.5%, 32.7%, and 45.0% sections) and a potentially better
buffet characteristic.
Fig. 14(a) shows the Cmvs CLcurves of the 4 results.
Fig. 14(b) shows the CDvs Ma curve. Inflection in a
Cmvs CLcurve can be used to define the buffet onset
26
.
The CDvs Ma curve is used to determine the drag divergence
Mach number. It can be seen that designs with a larger inboard
wing load have a larger buffet onset lift coefficient. Although
Design4 has a similar cruise drag and a similar drag divergence
performance to those of Design3, it has a better buffet charac-
teristic. Through the present optimization with the PDG
method, the proposed shockwave location and the pressure
distribution expectations of a dual-aisle airplane are achieved.
The result shows that the optimized design has a low drag and
good buffet and drag divergence performances.
Fig. 12 Wing section pressure distributions of typical results.
Table 7 Cruise performances of the baseline and typical results.
Parameter Baseline Design1 Design2 Design3 Design4
CD0.02468 0.02427 0.02416 0.02389 0.02388
Average distance to proposed shock location 0.028 0.035 0.023 0.063 0.022
Average Cpof suction peak 0.83 0.80 0.83 0.86 0.83
Have double shock sections No No No No Yes
Average DCp0.38 0.32 0.30 0.31 0.27
Pressure distribution guided supercritical wing optimization 11
Please cite this article in press as: LI R et al. Pressure distribution guided supercritical wing optimization, Chin J Aeronaut (2018), https://doi.org/10.1016/j.
cja.2018.06.021

5. Conclusions
The pressure distribution of a supercritical wing contains com-
prehensive information about its performance and robustness.
Pressure distribution oriented optimization can help achieve
more engineering-applicable designs. This paper has developed
a pressure distribution guided method of PDO optimization to
introduce the considerations of pressure distribution into the
optimization process. The work of this paper can be summa-
rized as follows:
(1) The PDG method is developed based on an RBF-
assisted differential evolution algorithm. The sub-
optimization on the RBF response surface can have
objectives and constraints different from those of the
main optimization process. By generating candidate
individuals more likely to have both satisfactory pres-
sure distribution features and performances, the sub-
optimization enables the PDG method to better manip-
ulate the pressure distribution while maintaining the
aerodynamic optimization efficiency.
(2) The relation between the shockwave strength and the
drag coefficient is theoretically and statistically discussed
in this paper. The result shows that the shockwave influ-
ence on the drag coefficient is insignificant when the
shockwave strength is lower than 0.3.
(3) Single- and multi-point optimizations of supercritical
airfoils are carried out to study the efficiency of the
PDG method. The results show that setting the shock-
wave strength as the objective for the sub-optimization
can accelerate the drag optimization process, and the
PDG method is able to more effectively achieve the pres-
sure distribution expectation.
(4) The PDG method is applied on the inboard wing design
of a dual-aisle airplane. A proposed shockwave location
of the inboard wing is achieved through different pres-
sure distribution manipulation methods. The perfor-
mance and robustness of the final result and some
other typical results are compared. The results show that
the PDG method can gain designs with the proposed
pressure distribution while maintaining the aerodynamic
optimization efficiency. The results also show that a
weak-shockwave pressure distribution is a satisfactory
balance between cruise drag and robustness.
Acknowledgements
This work was co-supported by the National Key Basic
Research Program of China (No. 2014CB744806) and
Tsinghua University Initiative Scientific Research Program
(No. 2015Z22003).
References
1. Li RZ, Zhang YF, Chen HX. Evolution and development of
‘‘man-in-loop”in aerodynamic optimization design. Acta Aerody-
namica Sinica 2017;35(4):529–43 [Chinese].
2. Fonseca CM, Fleming PJ. An overview of evolutionary algorithms
in multi-objective optimization. Evol Comput 1995;3(1):1–16.
3. Cliff SE, Reuther JJ, Saunders DA, Hicks RM. Single-point and
multipoint aerodynamic shape optimization of high-speed civil
transport. J Aircraft 2001;38(6):997–1005.
4. Yao W, Chen X, Luo W, Tooren MV, Guo J. Review of
uncertainty-based multidisciplinary design optimization methods
for aerospace vehicles. Prog Aerosp Sci 2011;47(6):450–79.
5. Obayashi S, Takanashi S. Genetic optimization of target pressure
distributions for inverse design methods. AIAA J 1996;34(5):
881–6.
Fig. 13 Shockwave locations and span-wise lift distributions.
Fig. 14 Buffet and drag divergence characteristics.
12 R. LI et al.
Please cite this article in press as: LI R et al. Pressure distribution guided supercritical wing optimization, Chin J Aeronaut (2018), https://doi.org/10.1016/j.
cja.2018.06.021

6. Berci M, Toropov VV, Hewson RW, Gaskell PH. Multidisci-
plinary multifidelity optimisation of a flexible wing aerofoil with
reference to a small UAV. Struct Multidiscip Optim 2014;50
(4):683–99.
7. Buckley HP, Zhou BY, Zingg DW. Airfoil optimization using
practical aerodynamic design requirements. J Aircraft 2010;47
(5):1707–19.
8. Kumar A, Keane AJ, Nair PB, Shahpar S. Robust design of
compressor fan blades against erosion. J Mech Des 2006;128
(4):864–73.
9. Dang T, Isgro V. Euler-based inverse method for turbomachine
blades Part 1: two-dimensional cascades. AIAA J 1995;33
(12):65–75.
10. Harris CD. NASA supercritical airfoils: a matrix of family-related
airfoils. Washington, D.C.: NASA; 1990. Report No.: NASA
Report 90N16710.
11. Kim HJ, Rho OH. Aerodynamic design of transonic wings using
the target pressure optimization approach. J Aircraft 1998;35
(5):671–7.
12. Garabedian P, McFadden G. Design of supercritical swept wings.
AIAA J 1982;20(3):289–91.
13. Bell RA, Cedar RD. An inverse method for the aerodynamic
design of three-dimensional aircraft engine nacelles. Washington,
D.C.: NASA; 1991. Report No.: NASA Report 92N13958.
14. Malone JB, Vadyak J, Sankar LN. Inverse aerodynamic design
method for aircraft components. J Aircraft 1987;24(1):8–9.
15. Takahashi S, Obayashi S, Nakahashi K. Inverse design optimiza-
tion of transonic wings based on multi-objective genetic algo-
rithms. AIAA J 1999;37(12):1656–62.
16. Zhang Y, Chen H, Fu S, Zhang M, Zhang MH, Liu TJ. A
practical optimization design method for transport aircraft
wing/nacelle integration. Acta Aeronautica et Astronautica Sinica
2012;33(11):1993–2001.
17. Zhao T, Zhang YF, Chen HX. Multi-objective aerodynamic
optimization of supercritical wing with substantial pressure
constraints. 53rd AIAA aerospace sciences meeting. Reston: AIAA;
2015.
18. Zhang Y, Fang X, Chen H, Fu S. Supercritical natural laminar
flow airfoil optimization for regional aircraft wing design. Aerosp
Sci Technol 2015;43:152–64.
19. Jin Y. Surrogate-assisted evolutionary computation: Recent
advances and future challenges. Swarm Evol Comput 2011;1
(2):61–70.
20. Deng K, Chen H. A hybrid aerodynamic optimization algorithm
based on differential evolution and RBF response surface. 17th
AIAA/ISSMO multidisciplinary analysis and optimization confer-
ence. Reston: AIAA; 2016.
21. Zhang YF, Chen HX, Fu S. Improvement to patched grid
technique with high-order conservative remapping method. J
Aircraft 2011;48(3):884–93.
22. Li Z, Zhang YF, Chen HX. A low dissipation numerical scheme
for implicit large eddy simulation. Comput Fluids 2015;117:233–46.
23. Zhao T, Zhang YF, Chen HX, Chen Y, Zhang M. Supercritical
wing design based on airfoil optimization and 2.75D transforma-
tion. Aerosp Sci Technol 2016;56:168–82.
24. Pearcey HH. The aerodynamic design of section shapes for swept
wings. Adv Aeronaut Sci 1962;3:277–322.
25. Paparone L, Tognaccini R. Computational fluid dynamics-based
drag prediction and decomposition. AIAA J 2003;41(9):1647–57.
26. Liebeck RH. Design of the blended wing body subsonic transport.
J Aircraft 2004;41(1):10–25.
Pressure distribution guided supercritical wing optimization 13
Please cite this article in press as: LI R et al. Pressure distribution guided supercritical wing optimization, Chin J Aeronaut (2018), https://doi.org/10.1016/j.
cja.2018.06.021