Parameter Classification using Adjoint Derived Sensitivities for Aerodynamic Shape Optimization for Transonic Aircraft

Wenhua Wu *,**, Zhaolin Fan**, Dehua Chen **, Ning Qin***, XiaoyongMa*,**, Xinwu Tang*,** Corresponding author: 619677947@qq.com

* State Key Laboratory of Aerodynamics. CARDC, Mianyang Sichuan, China

** High Speed Institute of CARDC, China. *** University of Sheffield, Sheffield S1 3JD, UK

Abstract: In this paper, a new global optimization algorithm based on adjoint sensitivity derivatives and parameter classification is proposed for aerodynamic shape optimization. The classification of the design parameters makes it possible to avoid the optimization trapped at local minima in the design space, achieving better potential for global optimum in the optimization process. The method is demonstrated for the optimization of a transonic transport aircraft, showing significant improvement as compared to the gradient based Sequential Quadratic Programming algorithm. Keywords: Transonic aircraft, Aerodynamic optimization, Adjoint solution, Drag reduction.

1 Introduction

Higher and higher aerodynamic performance is required in modern aircraft design, thus it relies

more and more on automatic design procedures based on Computational Fluid Dynamics(CFD) and

high performance computing platforms, new and efficient design algorithms. Two requirements must

be met to find the best shape from numerous possible shapes that meet the design constraints. First, a

good shape parameterization method with enough parameters to construct a large design space that

include the best shape. Secondly, the searching algorithm must be able to efficiently search the

possible shapes in the large design space to find the best shape.

Usually, two types of optimization algorithms are used in aerodynamic shape optimization:

global optimization algorithms which are independent on sensitivity derivatives, local optimization

algorithms that dependent on sensitivity derivatives.

Global optimization algorithms include response surface algorithms, genetic algorithms, particle

swarm algorithms, etc. The global minimum of a limited design space with several parameters can be

found by these algorithms. It is not suitable for the optimizations of a large design space with very

large number of design parameters because the computational cost is unacceptable when the number

of design parameters is large.

On the other hand, optimization algorithms based on sensitivity derivatives are a more

effective multi-parameter optimization algorithm with higher searching efficiency.

Aerodynamic shape optimization with thousands of design parameters can be carried out with adjoint

solver derived sensitivity derivatives.

The focus of CFD applications has shifted to aerodynamic design since the introduction of the

adjoint method for Aerodynamic Shape Optimization (ASO) by Jameson [1,13] in 1989. The adjoint

method is extremely efficient since the computational expense incurred in the calculation of the

complete gradient is effectively independent of the number of design variables. Thus the ASO with

large number of design parameters can be successfully realized combining the adjoint method with

and gradient-based searching algorithms [1,6,8,11,12,13]. The method becomes a popular choice for design

problems involving fluid flow and has been successfully used for the aerodynamic design of complete

aircraft configurations. Many others research groups have developed adjoint code for aerodynamic

shape optimization, these codes are used in 2D airfoil, 3D wing and finally complete aircraft

configurations optimization [2, 3, 4, 5, 7,9,10] .

Considering the importance of aerodynamic design and optimization, it is surprising that the

development of the adjoint based optimization software and the usage of the technique in engineering

has not been more widely used since Jameson’s introduction of the method 1989.This may be due to

the limitations and shortcomings of the adjoint method [2]. The complexity and the difficult to build

the code was discussed by Giles [2]. In this paper, another and important shortcoming that prevent the

adjoint from more engineering application, i.e. the optimize algorithm used with the adjoint method,

is addressed. .

It is well known that, generally, an aerodynamic optimization problem may have more than one

local optimum. For example, Makino and Iwamiya[9] find two local minima in drag reduction

optimization for the aerodynamic shape of NEXST (National Experimental Supersonic Transport).

Taking a practical approach, Jameson noted that ‘there is a possibility of more than one local

minimum, but in any case this method will lead to an improvement over the original design’.[8]

In this paper, the sensitivity derivatives from the adjoint solution are made use of to address the

problem of trapped local minima. A new optimization algorithm which can achieve potentially global

best shape for multi-parameter shape optimization is proposed based on the characteristics of the

design parameters and sensitivity derivatives. The new method is then used in a shape optimization

with many design variables for drag reduction of a transonic aircraft, showing significant

improvement in the optimized design.

2 The numerical method 2.1 Flow solver

The flow solver is constructed by Osher vector flux splitting scheme based on the Riemann

fundamental approximate solutions and the finite volume method. MUSCL [3,4,5] discrete scheme is

used in space and implicit or explicit difference scheme in time. This solver is high accurate, stable,

so it can suppress the oscillation of solutions to meet the several numerical requirements of the solver

used in optimization.

KW-SST turbulence model is used in the flow computation.

2.2 Sensitive derivatives solve method

Sensitive derivative is the derivative of object function with respect to the design parameters, it

can be used to compute the value of the design parameters for better aerodynamic shapes.

The most popular and simple sensitive derivative compute method is the finite difference

method, but the computational load is large. For example, if the number of design parameters is n, the

n+1 times of numerical calculation is needed for a sided difference scheme, 2n times is needed for

central difference scheme to get all the sensitivity derivatives. The first step of optimization must

complete 1001 times of numerical calculation with an optimization of 1000 design parameters, and

spends too much time. So this method is usually only used when there’s not as many design

parameters.

The sensitive derivative solution method based on Adjoint include following steps: solve the

flow field, solve the Adjoint equation and get the Adjoint operators, calculate the grid derivatives,

calculate the partial derivative of the objective function with respect to the design parameters,

calculate the partial derivative of the objective function with respect to the grid and the partial

derivative of the flow field residues with respect to design parameters and etc, finally, we get sensitive

derivatives through some algebra calculation of these results. By this method, all sensitivity

derivatives can be calculated with only one flow analysis. Adjoint operators are same for a certain

aerodynamic shape, that’s to say, they need to be re-calculated only when the aerodynamic shape

changes. So in each optimization step, it only needs to solve the Adjoint equation and the Navier-

Stokes equations once, which greatly improves the calculation efficiency, particularly effective for

optimization problem with large amount of variables. Take an optimization problem with as many

design parameters as 1000 for example, it only needs to solve the N-S equations once and the Adjoint

equations once, whose complexity and time-consuming is almost similar to the N-S equations, and

then all the sensitivity derivatives is computed by some algebra calculation.

The objective function can be expressed as:

F=F(Q*(β),X*(β), β) (1)

The components of the formula are as allows: Q is the flow field variable, *

indicates the flow field variable is convergent; X is the vector composed of the grid variables; β is the

vector of design variables.

The differential form of equation (1) is as follows:

* *

( ) ( )t t

k k k k

dF F dQ F dX Fd Q d X dβ β β β

∂ ∂ ∂= + +∂ ∂ ∂

(2)

When we solve it with the Adjoint equations, adding the Adjoint vector λ, and equation (2)

becomes:

* *

* *

( ) ( )

[( ) ( ) ]

t t

k k k k

t t t

k k k

dF F dQ F dX Fd Q d X d

R dQ R dX RQ d X d

β β β β

λβ β β

∂ ∂ ∂= + +∂ ∂ ∂

∂ ∂ ∂+ + +

∂ ∂ ∂

(3)

Also written as:

*

*

[( ) ]

[( ) ]

t t

k k

t t t

k k k

dF F R dQd Q Q d

F R dX F RX X d

λβ β

λ λβ β β

∂ ∂= +

∂ ∂

∂ ∂ ∂ ∂+ + + +

∂ ∂ ∂ ∂

(4)

To avoid solving the flow field repeatedly when we solve dQ/dβk , let:

( )t tF RQ Q

λ∂ ∂

= −∂ ∂

After solving the Adjoint vector λ, we can calculate the sensitivity derivatives by the following

equation:

*

[( ) ]t t t

k k k k

dF F R dX F Rd X X d

λ λβ β β β

∂ ∂ ∂ ∂= + + +

∂ ∂ ∂ ∂ (5)

We uses the discrete method in this paper. First we solved the Adjoint equations, got the Adjoint

vector λ, and then calculated the sensitivity derivatives.

3 Shape parameterization and grid update In this study, the Bezier-Bernstein method is used to parameterize the aerodynamic shape. the

advantages of this method is that aerodynamic shape can be accurately described with less parameters, and

adjust precisely and effectively while maintain the smoothness of the surface at the same time , which is of

special importance for the parameterization of aerodynamic shape of airliner.

By this method, the shape is divide into certain number of curves with a number of control points on

each curves, and the locations of these points (the coordinates) is used as the design parameters. The

number of the control points in each curve depends on the complexity of the surface and the optimization

sophistication requirement.

For a two-dimensional curve, it can be expressed as:

2 ,

0( ) ( )

N

K N kk

S u B u P=

=∑ (6)

S2 (u) =x (u)/y (u), PK=Px/Py are the control points of the Bezier-Bernstein curve. And among the

Bernstein polynomials, u represents the reference curve length, N represents the number of the control

points, namely the number of design parameters. Px 、Py represent the vertical and horizontal coordinates

of the control points. Take the wing for example, only Y coordinates need to be changed during the

optimization, so the design variable is the Y coordinate of the control point.

3.1 Method for the mesh deformation

Multi-block structured grids of are used in this research. The grid number is 12 million.

The aerodynamics shape of transonic aircraft studied in this paper is well designed and optimized by

several traditional methods, so its aerodynamic performance is high. Thus the supercritical wing shape of

the aircraft will not be changed hardly during the optimization. The algebraic method is chosen to do the

grid deformation: first, move the grid points of the surface to new locations, and then gradually pass this

change to the out boundary. In the transmission process, adjust the displacement of the point

proportionately according to the point location to ensure that the grid’s outer boundary of the block

remains unchanged. By this method, the topological structure of grid keeps same and the grid keeps

similar, thereby restrain the numerical error caused by the grid changes and improved the accuracy of

optimization results.

4 The new optimization algorithm for the shape optimization of the transonic aircraft 4.1 The aerodynamic characteristics of the original shape

The original shape and the grid distribution of the aircraft is shown in Fig 1.The surface pressure

coefficient of the original shape is shown in Figure 2.

The aerodynamic characteristics and surface pressure coefficient distribution of the initial shape

are shown in Figure 2. The aerodynamic force coefficients of the plane: lift coefficient: CL = 0.5, lift-

drag ratio: K = 17.53, the drag coefficient: CD = 0.02853.

4.2 The shape parameterization and the distribution of design parameters

The wing of the transonic aircraft is parameterized and optimized while the other parts of the

aircraft keep same during optimization, but the objective function, such as drag coefficient is

calculated on the whole plane, so the influences of the nacelle, fuselage, pylons, etc are all took into

account in optimization, this is very important for a detailed optimization of a high performance

aerodynamic shape. The initial shape of the aircraft and the grid is shown in Figure 1. The wing is

divided into 11 sections. The location of the each control surface is shown in Figure 3.

Fig.1 The original shape and surface grid of the shape

Fig.2 The original surface pressure distribution of the shape

Fig.3 The position of control Sections

There are each 5 control sections on the left and right of the nacelle pylons. The control

parameters of each section distribute as: 8 on the upper surface, 8on the lower surface, total 16

parameters distributed uniformly along the flow direction. For the control section which has same

position as pylon, there are only 8 design parameters on the top surface, therefore the total number of

design parameters is 168.

The position of the control sections

These control surfaces connected with each other in a certain way to form a complete wing. In

order to accurately depict the original shape, the superposition method is used in this paper, namely

the relative displacement of the new wing to the prototype is parameterized, and then the

displacement is added to the original wing to form the new shape. By this method, the original wing is

composed when all the design parameters are 0. The changes of the shape caused by the changes of

design parameters are shown in Fig4. it shows the change of the control section when two parameters-

the last parameter in the lower surface of the No.8 control surface (parameter No.128) and the second

last one in the upper surface (parameter No.119)-are set 0.01 or -0.01. It can be seen from the figure,

positive value of the control parameter makes the wing surface near it move above, negative value

control parameter does the opposite. The design parameter can only affect a region near it, the farther

away from the parameter, the smaller impact it can impose on.

Fig.4 Variation of the wing section shape caused by the design parameters

4.3 Characteristics and classification of the design parameters

The initial aim of the study is to achieve further performance improvement for a high

performance aerodynamic shape by detailed shape optimization with hundreds of parameters based on

Adjoint technique. In the early 2010, the shape is optimized with 168 parameters by the ADJOPT

optimization design platform which is built based on the Adjoint operator and SQP algorithm. the

drag was reduced by 0.9 drag units while the lift remaines the same.

It is not easy to reduce the drag of the aerodynamic shape that optimized several times by other

optimization methods even by 0.9 drag counts , this implies that multi-parameter optimization method

can explore the aerodynamic potential of a high-performance aerodynamic shape more. However, if

we can reduce the drag of the aerodynamic shape more by multi-parameter optimization?

X

Y

0 0.2 0.4 0.6 0.8 1

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

para119_-0.01para128_0.01para128_-0.01orig

As is mentioned before, the SQP algorithm is based on the sensitivity derivatives and it can only

find the local minima other than the global minima, so it is not very good for multi-minima

optimization.

Is multi-parameter shape optimization studied in this paper a multi-minima one? Is there any

way to change the algorithm to improve aerodynamic performance more? Usually, we take several

different optimization starting points at random, to see if we can get any optimal solution results

better than the existing ones. By this way, we may get better optimization results some times, but it’s

of no use often. It has been proven by the attempts in this study. Another five different shapes are

optimized by the same method to find better shapes, all six optimized shapes are compared in table 1,

it can be seen from the from that no better shapes are found, the four of the optimized results are

worse than the optimized shape from original shape, one have the same result as the first one.

Shape 1(origin

a shape)

2 3 4 5

cd_pri 0.02853 0.0288 0.02862 0.0292 0.02970

cd_opt 0.02844 0.02865 0.02844 0.0287 0.02869

Table 1 the optimized result from different shape

If some laws exist in the relations between the design parameters and aerodynamic forces, we

can study the relationship among the objective function, the constraint conditions and the design

parameters to find some characteristics of relationships. Then new the optimization algorithm may be

developed based on these features, to improve the effectiveness and efficiency of the multi-parameter

optimization. Kuruvila, Salas classify the design parameter to low frequency and high frequency

shape perturb parameters and optimize the low frequency design variable in coarse grid and high

frequency design variable in fine grid in the airfoil optimization with excellent results[10].

Based on this idea, how these 168 design parameters affect the aerodynamic characteristics of

the whole plane need to be studied at first. Part of the design parameters of some typical control

section is studied carefully to find some laws. The selected design parameters are changed step by

step, then the new shapes with respect to the new value of the parameters are present. The

aerodynamic characteristics of all these shapes are calculated. The relationship between the main

aerodynamic characteristics of the and some typical design parameters are shown in the figures from

Fig5 to Fig9.These design parameters are on the control surfaces of No.7, NO.8, and No.9. The 168

design parameters are classified to four categories after studying and analyzing the curves and the

relevant data.

The first category parameters are mainly located on the rear of the lower surface of the wing. The

absolute value of the sensitive derivatives of the objective function with respect to these parameters is

large. The lift, drag, lift-drag ratio and pitch moment are all very sensitive to these design parameters.

Lift, drag, moment, and lift-drag ratio change almost linearly as the parameters changes from -0.03 to

0.03. it can be seen form Fig 5, No.111, 112,128 design parameters all belong to this category. Design

parameters in this category can be called strong linear hypersensitive single minima parameter

(SLHSE).

The design parameters of the second category are located in the upper surfaces of control

sections. The sensitivity derivatives of these design parameters are still large, but smaller than that of

first category. Meanwhile, the lift and the moment changes linearly as the design parameters increase,

the slope of the curve changes in some place, but the objections still have only one minima. The drag-

parameter curve looks like parabolic curve, with also only one minima in the studied range.

The lift-drag ratio has a similar relationship with the design parameters as drag does. In Fig 5,

design parameters No.97, 98,115,149,150 belong to this category. Design parameters of this category

can be called weak linear single extremum design parameters (WLSEP)

The design parameters of the third category are located at the front part of the down surface of

the wing section. One of the main features of these parameters is the objective function has a

significant multi-minima characteristic within the study range. Another feature is that the sensitive

derivative of the design parameters changes rapidly or even opposites the sign as the design

parameters changes, while the objective function, such as lift, drag and moment changes little. In

Fig5, design parameters No.123, 124 belong to this category. Design parameters of this category can

be called as nonlinear insensitive multiple extremum parameters. (NLIMEP)

The main feature for design parameters of the fourth category is that the absolute value of the

sensitivity derivatives are relatively large, the lift, drag, lift-drag ratio and the moment are all sensitive

to these parameters, meanwhile the sensitivity derivatives are also sensitive to the change of these

design parameters: they change rapidly or even opposites sign, which lead to great changes in the

objectives such as lift, drag and moment. Within the study range of parameters, the objective

equations have a significant multi-extremum characteristic. Design parameters of this category can be

called as strong nonlinear hypersensitive multiple extremum parameter (SNHMEP）.

beta

Cd

-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04

0.0272

0.0274

0.0276

0.0278

0.028

0.0282

0.0284

0.02869798115149150111112123124128

beta_serial number =

Fig.5. Variation of drag coefficient with each design parameters

Fig.6 Variation of lift coefficient with each design parameters

Fig 7. Variation of pitch moment coefficient with design parameters

beta

Cl

-0.02 0 0.02 0.04

0.48

0.49

0.5

0.51

0.52 9798115149150111112123124128

beta_serial number =

beta

Cm

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

-0.36

-0.35

-0.34

-0.33

-0.32

-0.31 9798115149150111112123124128

beta_serial number =

Fig 8 Variation of lift /drag ratio with each design parameters

Fig.9 Variation of lift /drag ratio with parameters on upper surface

The parameter changes shown in Fig 5 are under the situation when all the other design

parameters are put as zero. Although the first and second categories of design parameters are single-

extremum when other design parameters are zero, but what if other design parameters aren’t zero?

Are they still single-extremum? We selected two typical design parameters to study this problem: No.

125 on the lower surface and No.116 on the upper surface.

To be more specific, one is the fifth control parameter of the lower surface of N0.8 control

surface, the other is the fourth control parameter of the upper surface of this control surface. We

represented the No.125 control parameter as beta1, and the No.116 as beta2. These two parameters

mainly influence the central part of the wing shape. Fig10 to Fig14 show how the drag, lift and the

lift-drag ratio change with these two design parameters. It’s obvious in the figure that the lift, drag

and the lift-drag all have a linear relationship with beta1, namely the design parameter No.125. This

discipline almost has nothing to do with beta2: in the set of curves with different beta2, the

relationship between beta1 and aerodynamic characteristics remain unchanged.

beta

k

-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.0417.4

17.5

17.6

17.7

17.8

17.9

18

18.1

18.2 9798115149150111112123124128

beta_serial number =

beta

k

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

17.5

17.6

17.7

17.8

17.9

189798115149150

beta_serial number =

The lift, drag and moment are still single-minima function of beta1, the sensitivity derivatives to

beta1 remained almost same. Figure 15 - Figure 17 show how the No.123 and No.124 design

parameters affect the lift, drag and lift-drag ratio, and the drag shows multi-minima characteristic with

the change of No.123 design parameter, its’ sensitive derivative changes relatively great, even

opposite sign. There’s little change for the drag in the whole process, which is within 2 drag units. So

the design parameter No.123 is a typical design parameter of the third category (WLSEP).

Fig.10 Variation of drag with beta1 （beta1=125，beta2=116）

Fig.11 Variation of lift with beta1 （beta1=125，beta2=116）

beta1

Cd

-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.040.028

0.0282

0.0284

0.0286

0.0288

0.029 -0.03-0.025-0.02-0.015-0.01-0.00500.0050.010.0150.020.0250.03

beta2=

beta1

Cl

-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04

0.492

0.494

0.496

0.498

0.5

0.502

0.504

0.506

0.508-0.03-0.025-0.02-0.015-0.01-0.00500.0050.010.0150.020.0250.03

beta2=

Fig.12 Variation of lift /drag coefficient with beta1（beta1=125，

beta2=116）

Fig.13 Variation of drag coefficient with beta2（beta1=125，beta2=116

）

Fig.14 Variation of lift /drag coefficient with beta2（beta1=125，

beta2=116）

beta1

k

-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.0417.1

17.2

17.3

17.4

17.5

17.6

17.7

17.8

17.9 -0.03-0.025-0.02-0.015-0.01-0.00500.0050.010.0150.020.0250.03

beta2=

beta2

Cd

-0.02 0 0.02 0.04

0.0281

0.0282

0.0283

0.0284

0.0285

0.0286

0.0287

0.0288

0.0289

-0.03-0.025-0.02-0.015-0.01-0.00500.0050.010.0150.020.0250.03

beta1=

beta2

Cl

-0.02 0 0.02 0.04

0.492

0.494

0.496

0.498

0.5

0.502

0.504

0.506

0.508-0.03-0.025-0.02-0.015-0.01-0.00500.0050.010.0150.020.0250.03

beta1=

Fig.15 Variation of drag coefficient with beta(beta1=123，beta2=124）

Fig.16 Variation of Lift coefficient with beta 1（beta1=123，beta2=124）

Fig.17 Variation of lift/drag ratio with beta 123（ beta1=123，

beta1

Cl

-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04

0.497

0.498

0.499

0.5

0.501

0.502

0.503-0.03-0.025-0.02-0.015-0.01-0.00500.0050.010.0150.020.0250.03

beta2=

beta1

k

-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.0417.64

17.66

17.68

17.7

17.72

17.74

17.76

17.78-0.03-0.025-0.02-0.015-0.01-0.00500.0050.010.0150.020.0250.03

beta2=

beta2=124）

Among the total amount of design parameters, the first category takes up about 25%, the second

about 50%, the third about 20%, the fourth though is not included in this study, we think it may not

takes up more than 5%. From the optimization point of view, design parameters of the first and

second category have only one minima within the study range of this paper. Therefore, the

optimization algorithm based on sensitivity derivatives can be applied to these design parameters.

Design parameters of the third and fourth category, the NLIMEP and the SNHMEP, are not fit

for the optimization algorithm based on sensitivity derivatives, because the objective functions such

as the drag, the lift-drag ratio, etc, are multi-minima function of these design parameters, besides,

these parameters will affect on the optimization process greatly and even break it.

To avoid the bad affect of the multi-minima design parameters, we don’t take these parameters

into account during the optimization process based on the sensitivity derivatives. That’s to say, we

only consider the design parameters of the first and the second category when using the SQP method.

As to the nonlinear multi-minima design parameters of the third and fourth category, we use the

global optimization method.

Although the multi-minima parameters take up less than 25% of the total design parameters, but

there are nearly 40 in the example studied in this paper. There will be a huge computational load

using the global optimization method to deal with these design parameters.

In our study, the design parameters of the third and fourth categories only have nonlinear

coupling relationship with the design parameters of its adjacent control surfaces which means the

global optimization of the multi-minima parameters can be optimized separately. So the global

optimization uses only the design parameters on three adjacent control surfaces together, which limits

the number of control parameters to no more than 15 for each global optimization process. It can meet

the design requirements using the particle swarm optimization algorithm.

The new optimization algorithm is designed based on the information from the research above,

classify the parameters according to their features, using different optimization algorithm for design

parameters of different features. Use SQP for single-minima design parameters, and use particle

swarm algorithm for multi-minima ones, or just keep them unchanged.

The classification method is based on the sensitive derivatives. During the ADJOINT based

optimization, the sensitive derivatives are calculated at each step. Then these design parameters are

classified according to the size and sign of the sensitive derivatives. For example, if the derivative of a

certain parameter is relatively small, and changes sign for at least twice, then that parameter must be

multi-minima, which needs to be got rid of from the parameter of SQP optimization process; if the

derivative is big and changes sign more than twice, then it is used in the particle swarm optimization

method. During the optimization process, we adjust the design parameters until the results converge.

5 Application and verification

In order to verify the validity of this optimization method, we compared this optimization

algorithm and the Sequential Quadratic Programming optimization algorithm, the model used is a

shape which is designed and optimized by classical method, and has pretty good aerodynamic

performance.

Three optimization algorithm are compared in this paper: the first method uses the SQP

searching algorithm, the optimization result is not good; the second method, named as MPSQP,

classifying the parameters and do not use multi-minima ones in optimization, increases the effect

dramatically; the third method, named REPSQP, based on the second one, using the particle swarm

optimization method to the multi-minima parameters. That is, after every step of the SQP, use the

particle swarm optimization for multi-minima design parameters, and then use the results for the next

step of Adjoint optimization.

The optimization processes of the three optimization algorithms are shown in Fig 18. It can be

seen that, the drag reduction of simple SQP algorithm is the least, the optimization continues only 9

steps, the drag decreased to 0.02844, with 0.00009 decreased. That’s because the optimization cannot

continue when it comes to a certain local optimum, which the program taken as the optimal solution.

The second algorithm carried out 19 steps, the optimized drag is 0.02813, with 0.0004 decreased. The

third algorithm conducted 41steps; the optimized drag is 0.02785, with 0.00068 decreased.

Fig.18 Optimize history with different optimize algorithm

6 Conclusions

In this paper, we discussed the technological shortcomings for the multi-parameter optimization

based on ADJOINT operators—it is not good for multi-minima optimization. Then we assume that

we may develop a new searching algorithm to get global best shape in ADJOINT based multi-

parameter optimization of a transonic aerodynamic shape. The Bezier-Bernstein method is used to

parameterize the wing of the aerodynamic shape. Then the affect these typical design parameters have

on the aerodynamic characteristics of the whole plane are studied. The design parameters can be

Step

CD

0 10 20 30 40

0.0279

0.028

0.0281

0.0282

0.0283

0.0284

0.0285

sqpmpnsqprepsqp

Optomize history

classified in four categories: SLHSEP, WLSEP, NLIMEP and SNHMEP. Different searching

algorithm to different types of design parameters is used based on their features for better

optimization results. Then the REPSQP method is proposed in the paper, the method uses SQP

algorithm for single minima design variables, and uses particle swarm algorithm for multi-minima

design variables.

The REPSQP method is compared with SQP method in the optimization of the aerodynamic

shape of a airliner, which is already well designed and have high aerodynamic performance. The

results shows that REPSQP method performs much better than SQP method and get much better

aerodynamic shape, partially avoids the shortcomings of the multi-parameter optimization technology

based on Adjoint operators. The drag of the original shape is 0.02853, the SQP method decreases the

drag by 0.0009 and makes it 0.02844; the REPSQP method decreases the drag by 0.00068 and makes

it 0.02785. It is obvious that, in this study, the REPSQP method works better than the ordinary SQP

method.

The multi-parameter optimization method based on parameters classification proposed in this

paper shows a good performance on the multi-parameter optimization for the large transonic aircraft.

It also can possibly enforce the role that the Adjoint operator optimization technology plays in the

complex engineering aerodynamic shape optimization. This method best fit for those problems with a

larger part of single-minima design parameters. More researches need to be done to verify if it is

suitable for any other kinds of aerodynamic shape optimization.

The multi-parameter optimization for aerodynamic shape has great potential in improving the

aerodynamic characteristics. In order to explore these aerodynamic potential, the breakthrough needs

to be made in the optimization algorithm and the computational accuracy, and the relationship

between the design parameters and the optimization object need to be further studied. As far as this

research, it’s very hard to develop an algorithm that can solve all the problems. What we can do is to

develop different optimization algorithm according to different problems.

1

3 Conclusion and Future Work ICCFD7 will be held on the Big Island of Hawai’i at the beautiful Mauna Lani Bay Hotel. Situated amidst mountains and the sea on the sunny Kohala Coast on the tropical island of Hawai’i, the venue offers a gorgeous backdrop. We wholeheartedly invite you to attend and make reservations while reduced conference rates are available. Aloha. References

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