GLOBAL OPTIMIZATION ISSUES FOR TRANSONIC AIRFOIL DESIGN

The objective of this research is to explore issues for global optimization of airfoil shapes in the transonic flow region by comparing a Genetic Algorithm GA with a Gradient-based optim

GLOBAL OPTIMIZATION ISSUES FOR TRANSONIC AIRFOIL DESIGN

Howoong Namgoong1, William A Crossley2, and Anastasios S Lyrintzis3

School of Aeronautics and Astronautics

Purdue University West Lafayette, Indiana 47907-1282

ABSTRACT

Airfoil design is very important for aircraft, because the airfoil can determine, to a large extent, the aircraft’s performance With the development of CFD techniques, many optimization algorithms have been applied to airfoil shape design, but the use of local optimization tools (gradient-based methods) may risk missing the best designs The objective of this research is to explore issues for global optimization of airfoil shapes in the transonic flow region by comparing a Genetic Algorithm (GA) with a Gradient-based optimization Method (GM) To determine the local optimum characteristics of the airfoil shape design space, different airfoil shapes are used as base or starting airfoils The results showed that the GA generated nearly the same shape regardless of the different initial base airfoils, which suggests that these shapes are approaching the global optimum However, the GM produces a different solution for each of the base airfoil shapes, suggesting that these results are local optima Comparisons of the generated airfoil shapes, the airfoil shapes’ performance, and the computational effort expended as a result of the GA and

GM methods are also presented

NOMENCLATURE

d

C Coefficient of drag

1

d

C Coefficient of drag at design Mach number1

2

d

C Coefficient of drag at design Mach number2

l

C Coefficient of lift

1

l

C Design lift coefficient

i

f Shape functions

M Free stream Mach number

tE Time for one evaluation of object function

x Airfoil coordinate

k

x Control point for shape functions

ω Weighting factor for multipoint design

i

ξ Design variables

1 INTRODUCTION

In the early stages of the design process, the search for optimal airfoil shapes encompasses a broad range of possibilities Many different optimization strategies and techniques have been applied for aerodynamic design problems If the aerodynamic design space is smooth enough (e.g continuous first derivatives), Gradient-based Methods (GM) usually have performance advantages over their global optimization counterparts As an example, in the adjoint1 approach the gradient information at a single design point can be obtained with the equivalent of two flow calculations2 However, the aerodynamic performance of an airfoil is very sensitive to the surface geometry, and it is difficult to guarantee the convexness of the objective functions used in airfoil optimization Moreover, in transonic airfoil design, the objective function itself may be discontinuous due to shock waves3 One of the well-known concerns of using gradient-based

1 Graduate Research Assistant, School of Aeronautics and Astronautics, Student Member AIAA

2 Associate Professor, School of Aeronautics and Astronautics, Senior Member AIAA

3 Professor, School of Aeronautics and Astronautics, Associate Fellow AIAA

optimization techniques is that they conclude their

search at a point where some form of the Kuhn-Tucker

conditions are satisfied; however, these conditions only

describe a local optimum point For airfoil design, this

means that an airfoil shape found by a gradient-based

optimizer is likely the locally optimal solution nearest

to the initial airfoil shape Few papers have addressed

this problem of encountering locally optimum shapes in

the airfoil shape design space

Recently, the Genetic Algorithm (GA) has emerged

as viable (although more costly) alternative for airfoil

optimization (Refs 4, 5, 6), because of its global nature

However, direct comparisons between the two

approaches (a GA and a GM) have been sparse Thus,

the main focus of this paper is to investigate whether or

not the design space has numerous local minima

through the comparison of drag improvement of GA

and GM method If numerous local optimum points

occur, global optimization techniques will be an

appropriate way to design airfoils despite the penalty of

computational time increases Further, parallel and / or

distributed computing can mitigate some of the

computational expense associated with the GA

2 DESIGN VARIABLES

To pose the airfoil shape optimization problem, the

design variables that control the geometrical shape of

airfoils are needed In the approach used for this paper,

shape functions7 are added to a baseline airfoil shape

The design variables are multipliers that determine the

magnitude of the shape function as it is added to the

baseline shape Figure 1 depicts the shape functions

and their individual effects on a baseline NACA 0012

airfoil

The y-coordinate position of the upper and lower

surface of the airfoil are then described using the

following equation:

∑ ξ +

)

( x y x BaseAirfoil f x

(ξi : Design Variables, f i : Shape Functions, i =1, 16)

Eight shape functions are applied to the upper surface,

and these same eight functions are applied to the lower

surface, for a total of 16 shape functions

2 , 1 ], ) 1 ( sin[

)

( x = π − x ( ) k =

5 , 4 , 3 ], [

sin )

( x = 3 π x ( ) k =

8 , 7 , 6 ], sin[

)

( x = π x ( ) k =

2 , 1 , ) 1 ln(

) 5 0 ln(

)

8 , 7 , 6 , 5 , 4 , 3 , ) ln(

) 5 0 ln(

)

The control points for all shape functions used here are

as follows:

94 0 , 87 0 , 8 0 , 6 0 , 4 0 , 2 0 , 13 0 , 06 0

=

k

x

3 OBJECTIVE FUNCTION FORMULATION

For a single point optimization, the objective function is chosen as the drag coefficient at specified Mach number that has required value for lift coefficient In the approach to be used in this paper, the lift curve slope of an airfoil shape is calculated using two flow solutions, and then the angle of attack corresponding to the lift is predicted This angle of attack is used as an input for the flow solver

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

x

-0.075 -0.05 -0.025 0 0.025 0.05 0.075 0.1

Figure 1 Shape functions (top), NACA 0012 with each

shape function applied (bottom)

Single Point Objective Function

Maximize:

1

* 100 )

( xi Cd

Subject to:

1

l

Airfoils designed for a single flow condition

generally have poor performance in flow at other than

the design condition Some of the off-design

performance problems of single point optimization are

presented in Refs.7 and 8 It is recognized that the

single-point problem may not be highly applicable for

general airfoil shape design, but this simple formulation

allows investigation of the design space

Multi Point Objective Function

In the case of a two-point shape optimization, the

problem formulation is shown below utilizing a

weighting factor (ω) in the objective function

Maximize:

]

* ) 1 (

* [

* 100 )

(

2

d

x

Subject to:

1

l

In this formulation, the intent is to minimize the

drag coefficient at two different Mach numbers The

choice of the weighting factors affects the contribution

of each term in the objective function The weighting

factors of used by Drela8 in his discussion of airfoil

optimization appear to provide a rational objective

function A constraint is imposed to ensure the desired

lift coefficient is obtained in both Mach number

conditions

4 EVALUATION OF OBJECTIVE FUNCTION

Because the transonic regime is of great interest for

airfoils and airfoil shape design, an Euler-code is used

as the objective function evaluator to account for the

effect of shocks in the transonic region The scheme

that is used in this research is the Implicit Upwind

Finite Volume Scheme suggested by Roe9 Because this

is an inviscid Euler code, the drag predicted for the

airfoil is essentially the wave drag

Figure 2 shows the C-type grid system used for

solution of the Euler equations The grid size is 181×30

110 grid points are located on the surface of airfoil A

comparison with other grids showed that this grid

arrangement and resolution is a compromise between accuracy and efficiency

Y -1.5

-1 -0.5 0 0.5 1 1.5

Figure 2 C-grid scheme using 180 × 30 points Figure 3 shows the drag decrease with the increase

in number of surface grid points for the NACA0012

airfoil at a Mach number in the subsonic region (M =

0.4) Figure 4 compares the Euler solution at the design point with the AGARD10 experimental data to show the accuracy of solver A stronger shock closer to the airfoil trailing edge is predicted, as expected from an Euler solver

Grid points on surface

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

C d

Figure 3 “Residual” subcritical Euler drag coefficient as

a function of grid points on airfoil surface (NACA 0012

airfoil, M=0.4, α=5.00)

0 0.25 0.5 0.75 1

X

-1.5

-1

-0.5

0

0.5

1

1.5

Cp

Experiment Euler Analysis RAE2822 [M=0.74, α =3.19]

Figure 4 Pressure coefficient distribution of Euler

prediction vs published experiment (RAE 2822 airfoil,

M=0.74, α=3.19°)

During their execution, the present optimization

routines (GA or GM) using the shape function

multiplier variables can generate airfoil shapes that

violate geometric constraints, such as crossing upper

and lower surfaces In that case a large penalty is added

to the objective function without evaluation of the

airfoil drag

The convergence tolerance of the Euler-code is not

varied during the iteration, and the maximum residual is

reduced about four orders of magnitude An upper limit

of 2000 flow solver iterations is set for each airfoil

evaluation Each function evaluation required a wall

clock time of approximately 4 minutes (= tE) on a Linux

cluster machine that has AMD Athalon 1.2GHz CPU

5 PARALLEL GENETIC ALGORITHM

Since its first descriptions, the GA has been applied

to many engineering optimization problems.11,12 Based

on Darwin’s “survival of the fittest” concept, the GA

performs optimization tasks by “evolving” a population

of highly fit designs over many generations A GA has

the ability to search highly multimodal, discontinuous

design spaces The GA also locates designs at, or near,

the global optimum without requiring a good initial

design point

The GA represents design variables as strings of

binary numbers, which serve as chromosomes

Initially, the GA randomly generates a population of

individuals After decoding the chromosome of each

individual into the corresponding design variables, each

design is analyzed to determine a fitness value

Individuals with better fitness values are considered

“more optimal”, so this fitness value must reflect both the objective of the design problem and any constraints imposed upon the design

The GA employs selection, crossover and mutation operators to perform its search The selection routine performs the survival of the fittest function that allows better individuals to survive and serve as parents for the next generation of designs Crossover combines portions of chromosomes from the surviving parent designs to form the next generation of designs; combining features of good designs on average, but not always, results in better designs This gives the GA its optimization-like capability The mutation operator is used quite infrequently, as in nature, but this operator can mutate a binary bit in a chromosome to its opposite value (e.g “0” to “1”), which may introduce beneficial design traits that did not exist in the current population

If the mutated trait is poor, the design with this mutation will be unlikely to survive This process transforms an initial population of randomly selected designs into a population of individuals that have

“adapted” to their environment by becoming “more optimal” Additional details of the genetic algorithm can be found in several texts, like Ref 8

However, using a GA for design optimization is computationally expensive To overcome the computational time problem, the GA is adapted to a coarse-grain parallel implementation In this research,

a Master-Slave type parallelization is applied to convert

a serial GA into a parallel program A MIMD-type IBM SP2 and a Linux-Cluster machine were used for calculation following the basic approach of Ref 13 The GA algorithm is inherently parallelizable, because for each airfoil out of total airfoil population (which is usually several hundreds) the objective function evaluation (i.e the Euler solver) can be done in parallel independently of other airfoils To illustrate this, Figure

5 shows the total wall-time for the parallel GA with the different number of processors The communication time is about 38.8% of wall-time when using 30 CPUs The total computational time for the test function evaluations in the Linux cluster machine (with nodes connected via a 100base-T Ethernet) decreases nearly exponentially as the processor number increases

10 20 30 40

Number of CPUs

500

1000

1500

2000

2500

3500

4500

Ideal Speed-Up Parallel GA

Figure 5 Speed-up of parallel GA, computational wall

time vs number of CPUs

6 GRADIENT-BASED OPTIMIZATION

METHOD

If the objective function and constraints provide a

unimodal convex domain and are also differentiable, a

gradient-based optimization method can find the global

optimum solution However, it is very hard to prove

convexness and differentiability for general engineering

design problems14 The transonic airfoil design problem

is also difficult to characterize as convex and unimodal

For this effort, the method of feasible directions, as

provided by the CONMIN subroutines,15 is used as the

gradient-based optimizer In this research, different

initial airfoil shape designs are used to check the

consistency of optimum points found by the

gradient-based method This can give some indication of

multiple local minima appearing in the design space

7 RESULTS AND DISCUSSION

7-1 Single-Point Optimization Results

For the single-point optimization problem, the

objective is to minimize the drag when the free stream

velocity is M=0.74 while producing a lift coefficient

C l=0.733 Three base airfoils are chosen from the

database of Ref 16; these airfoils are the NACA 0012,

the RAE 2822, and the Whitcomb supercritical airfoil

The NACA 0012 is a subsonic, symmetric airfoil, while

the RAE 2822 and Whitcomb are cambered airfoils

originally designed to reduce wave drag in transonic

flight conditions

A summary of this single-point optimization

problem is presented as follows:

1 =

= l

l C

C

 Base airfoils a) NACA0012 b) RAE2822

c) Whitcomb Super Critical Airfoil

7-1-1 Genetic Algorithm Results

The input values for GA are described below The population size and mutation rate were selected using empirically derived guidelines for GAs using tournament selection and uniform crossover Seven bits represent each of the 16 shape function multipliers, for

a total chromosome length of 112 bits

 Population size: 448

 Resolution: 7 bit

 Design Variables: 16

 Variables limits: −0.01≤x i ≤0.01

 Mutation probability: 0.0022

The algorithm is based on an elitist reproduction strategy, where members of the population that are evaluated most fit are selected for reproduction Using the shape function approach to represent changes in the airfoil shape requires a base airfoil, so all

of the GA runs are associated with one of the base airfoils However, the initial generation of the GA is generated randomly, so the GA’s search does not begin with the base airfoil The best airfoil shape encountered

in selected generations during a parallel GA run is shown in Figures 6-8 along with the base airfoil sections In each case, the GA was allowed to run for

90 generations with no other stopping criteria

The pressure distribution plots in Figures 6-8 show that the newly designed airfoils do not have strong shock waves and maintain the specified design lift coefficient

Figure 9 presents the convergence history of the

GA In Figure 9, although the starting points are different, the fitness values are converging to about the same value as the generation increases This suggests

X

-0.05

0

0.05

0.1

0.15

0.2

0.25

NACA0012 Generation 1 Generation 10

Generation 30

X

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

Cp

Generation 1 Generation 10

Generation 30

Base Airfoil [NACA0012]

Figure 6 Best airfoil shapes (left) and pressure coefficient distributions (right) in selected generations of the GA using the

NACA 0012 base airfoil

X

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

RAE2822 Generation 1 Generation 10

Generation 30

X

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

Cp

Generation 1 Generation 10

Generation 30

Base Airfoil [RAE2822]

Figure 7 Best airfoil shapes (left) and pressure coefficient distributions (right) in selected generations of the GA using the

RAE 2822 base airfoil

X

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Whitcomb Generation 1 Generation 10

Generation 30

X

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

Cp

Generation 1 Generation 10

Generation 30

Base Airfoil [Whitcomb SuperCritical]

Figure 8 Best airfoil shapes (left) and pressure coefficient distributions (right) in selected generations of the GA using the

Whitcomb supercritical base airfoil

that each run is approaching the same drag performance

for the optimal airfoil shape

Generation

-1.2

-1.1

-1

-0.9

-0.8

-0.7

-0.6

-0.5

Base Airfoil (RAE2822) Base Airfoil (NACA0012) Base Airfoil (Whitcomb)

Figure 9 Best fitness value convergence history for all

three GA runs

The final best airfoil shapes are compared in Figure

10 The upper surfaces of the airfoils are very close to

each other, whereas there are some differences in the

lower surfaces Because the objective is to minimize

the wave drag, the upper surface is more important than

lower surface for a lifting airfoil If we add the pitching

moment as a constraint then the lower surface would be

also important The similarity of the upper surface

shapes suggests that the GA is indeed approaching the

same “optimal” airfoil shape, and this final shape seems

to be independent of the base airfoil

X

-0.1

-0.05

0

0.05

0.1

0.15

Base Airfoil (NACA0012), Generation 90

Base Airfoil (Whitcomb), Generation 90

Figure 10 Best airfoil shapes from generation 90 of all

three GA runs

In Figure 11 the pressure coefficients of the best airfoils after 90 generations are compared The upper surface pressure contours exhibit a very similar shape, whereas the lower surface has some variety, but still keeps the same design lift coefficient None of these pressure distributions indicate a strong shock; hence, the low predicted values of Euler drag

X

-1.5 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1

Cp

Base Airfoil NACA0012 (90 Generation)

Base Airfoil Whitcomb (90 Generation)

Figure 11 Pressure coefficient distributions for best airfoils from 90th generations of GA runs

7-1-2 Gradient-Based Optimization Results

CONMIN uses the method of feasible directions to perform its search through the design space To provide

an initial design for the search, setting all shape function multipliers to zero gives the base airfoil Figure 12 shows the convergence history of the CONMIN program using the NACA0012 base airfoil as the starting point The number of iterations for CONMIN to meet its convergence criterion is eight

1 2 3 4 5 6 7 8

Iteration Number -1.85

-1.8

-1.75

-1.7

-1.65

Base Airfoil [NACA0012]

Figure 12 Convergence history for CONMIN with

NACA 0012 as the starting shape

Figure 13 shows the change of airfoil shape and the

change of pressure coefficient during the iterations

from the base NACA 0012 airfoil shape Only small

changes to the shape are made during the search The

airfoil is modified to reduce the shock of airfoils, but a

substantial shock still remains upon convergence

Comparing with the GA solution (Figure 10) the

CONMIN results show some reduction of the shock

strength on the upper surface, but the GA results show a

much higher reduction of the shock strength

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

X -0.05

0 0.05 0.1 0.15 0.2 0.25 0.3

NACA0012 Iteration 1

Final Iteration

X

-1.5 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1

Cp

NACA0012 Iteration 1 Iteration 6 Final Iteration

Figure 13 Airfoil shape designs (top) and pressure coefficient distributions (bottom) generated during CONMIN search using NACA 0012 base airfoil

In the case of RAE2822 and Whitcomb Super Critical Airfoils, CONMIN converged to almost the same airfoils as the base airfoils (Figures 14-15) These results were expected because both the RAE2822 and the Whitcomb airfoils were designed for transonic flow

0 0.25 0.5 0.75 1

X

-0.05

0

0.05

0.1

0.15

0.2

0.25

RAE2822 Final Iteration

Figure 14 Initial RAE 2822 airfoil shape and final

shape generated by CONMIN

X

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Figure 15 Initial Whitcomb airfoil shape and final shape

generated by CONMIN

Using the gradient-based search method, a different

converged solution results from each different initial

airfoil This supports the notion that the transonic

airfoil design space has many local optimum design

points, and the GM simply finds the local optimum

nearest to the initial airfoil shape

7-2 Multi Point Optimization Results

A two-point design case was tried to investigate the

effects of the objective function selection The

weighting factor in equation (9) is ω = 1 3, and the

design Mach numbers are M=0.68 and M=0.74 while

keeping the design lift coefficient equal to 0.733 for

both Mach numbers

Figure 16 shows the result of two-point optimization using the GA and CONMIN The pressure coefficient distributions for these two airfoils are also plotted in Figure 17 to investigate the effect of two-point design The GA results also showed less shock wave in both design Mach number

X

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

GA Multipoint (M=0.68)

GM Multipoint (M=0.68)

X

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

GA Multipoint (M=0.74)

GM Multipoint (M=0.74)

Figure 16 Pressure coefficient distributions for

two-point objective function results at M=0.68 (top) and

M=0.74 (bottom)

0 0.25 0.5 0.75 1

X -0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Base airfoil (NACA0012)

GM multipoint (CONMIN)

GA multipoint (90 Generation)

Figure 17 GA and CONMIN results for two-point

objective function formulation

7-3 Computational Efficiency Comparison

Table 1 compares the drag results and

computational time from each method All times are

multiplied by tE (= time needed for one function

evaluation using the Euler method) In the case of the

NACA0012 base airfoil, the GA result has much

smaller drag coefficient than GM result However, the

GA needs about 250 times more computational time

than the GM if a serial code is employed When a

parallel GA with 45CPUs is used the difference reduces

to only about 5 times more than the GM Even though

the GM used in this research is not the best method to

solve this airfoil optimization problem, it is reasonable

to say that the GA’s penalty for higher computational

cost can be alleviated using parallelization

8 CONCLUSION

We have developed a GA based airfoil optimization

strategy based on shape functions We have

investigated transonic airfoil design using the GA and

the GM, and employing an Euler solver for the shock

wave drag prediction In case of the GM, we arrived at

different converged solutions with the change of initial base airfoils This result shows that the design space of transonic airfoil has numerous local optimum points However, when we used GA as an optimization tool we were able to obtain a quite similar solution (at least in the upper surface, which is the important one in this case) even though we started from totally different airfoils Therefore, this study verified that the GA can

be used as a robust global optimization technique, even though the transonic airfoil design space has several local optima In addition, our results showed that the increased CPU time needed for the GA can be addressed with a parallelization strategy, which made it possible to use an Euler solver for fitness evaluations with fast turnaround times

ACKNOWLEDGMENT

The first author was support by a Purdue Research Foundation (PRF) grant The calculations were performed on a 104-node cluster acquired by a Defense University Research Instrumentation Program (DURIP) grant

REFERENCES

1 Reuther J J et al., ”Constrained Multipoint Aerodynamic Shape Optimization Using an Adjoint Formulation and

Parallel Computers,” AIAA Journal of Aircraft, Vol.36, No1,

1999, pp.51-86

2 Jameson A., ”Re-engineering the Design Process Through

Computation,” AIAA Journal of Aircraft, Vol.36, No1, 1999,

pp.36-50

3 Obayashi, S., and Tsukahara, T., “Comparison of Optimization Algorithms for Aerodynamic Shape Design,”

AIAA Journal, Vol.35, No.8., Aug 1997, pp 1413-1415.

4 Holst L.T., and Pulliam H T., “Aerodynamic Shape Optimization Using Real-Number-Encoded Genetic Algorithm,” AIAA Paper 2001-2473, 2001

Table 1 Comparisons of drag values and computational costs for GA and GM runs

Base airfoil Method Drag (Cd) Number of function

evaluation

Parallel Computational time with

45 processors NACA 0012

RAE2822

Whitcomb