parameter classification using adjoint derived ...with many design variables for drag reduction of a...
TRANSCRIPT
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: [email protected]
* 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
[1] A. Jameson. Aerodynamic design via control theory. Journal of Scientific Computing, 3:233–260, September 1989.
[2] M.B. Giles and N.A. Pierce. `An introduction to the adjoint approach to design'. Flow, Turbulence and Combustion, 65(3-4):393-415, 2000.
[3] A. Le Moigne and N. Qin. Variable-fidelity aerodynamic optimization for turbulent flows using a discrete adjoint formulation. AIAA Journal,42(7):1281–1292, 2004.
[4] W. S. Wong, A. Le Moigne and N. Qin Parallel Adjoint-based Optimization of a Blended Wing Body Aircraft with Shock Control Bumps, The Aeronautical Journal, Vol.111 No.1117, 2007, pp165-174.
[5] A Le Moigne, N Qin, Airfoil profile and sweep optimization for a blended wing-body aircraft using a discrete adjoint method, The Aeronautical J, Vol.110, No. 1111, 2006,pp589-604.
[6] A. Jameson. Optimum aerodynamic design using CFD and control theory. AIAA paper 95-1729, AIAA 12th Computational Fluid Dynamics Conference, San Diego, CA, and June 1995.
[7] Salim Koc, Hyoung-Jin Kim, Kazuhiro Nakahashi. Aerodynamic Design Optimization of Wing-Body Configurations, 43rd AIAA Aerospace Sciences Meeting and Exhibit, AIAA 2005-331, 2005.
[8] A. Jameson. Automatic design of transonic airfoils to reduce the shock induced pressure drag. In Proceedings of the 31st Israel Annual Conference on Aviation and Aeronautics, Tel Aviv, pages 5–17, February 1990.
[9] Yoshikazu Makino,Toshiyuki Iwamiya. Fuselage Shape Optimization Of A Wing-Body Configuration With Nacelles. AIAA 2001-2447, 19th Applied Aerodynamics Conference, Anaheim, CA, 11-14 June 2001.
[10] G. Kuruvila, Shlomo Ta’asan,M.D. Salas. Airfoil Design and Optimization by the One-Shot Method. 33rd Aerospace Sciences Meeting and Exhibit, AIAA 95–0478, 1995
[11] Antony Jameson, An Investigation of the Attainable Efficiency of Flight at Mach One or Just Beyond, 45th AIAA Aerospace Sciences Meeting and Exhibit, AIAA 2007-37, 2007
[12] J. Reuther, J.J. Alonso, M.J. Rimlinger, and A. Jameson. Aerodynamic shape optimization of supersonic aircraft configurations via an Adjoint formulation on parallel computers. AIAA paper 96-4045, 6th AIAA/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Bellevue, WA, September 1996.
[13] Sangho Kim, Kaveh Hosseini, Kasidit Leoviriyakit, Antony Jameson.Enhancement of Adjoint Design Methods via Optimization of Adjoint Parameters, AIAA 2005-448, 43rd AIAA Aerospace Sciences Meeting and Exhibit, Nevada ,2005.
[14] Giles, M. B. and Pierce, N. A., “An Introduction to the Adjoint Approach to Design,” Flow, Turbulence and Combustion, Vol. 65, 2000, pp. 393–415.
.