GLOBAL OPTIMIZATION ISSUES FOR TRANSONIC AIRFOIL DESIGN
Howoong Namgoong*, William A. Crossley†, and Anastasios S. Lyrintzis‡
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
Cd
Coefficient of drag
C d1
Coefficient of drag at design Mach number1
C d2
Coefficient of drag at design Mach number2
Cl
Coefficient of lift
C l1
Design lift coefficient
fi
Shape functions
M
tE
x
xk
Free stream Mach number
Time for one evaluation of object function
Airfoil coordinate
Control point for shape functions

Weighting factor for multipoint design
i
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
*
Graduate Research Assistant, School of Aeronautics and Astronautics, Student Member AIAA
†
Associate Professor, School of Aeronautics and Astronautics, Senior Member AIAA
‡
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.
1
f2
f1
f3
f5
f4
f6
f7
f8
0.9
0.8
0.7
0.6
0.5
0.4
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.
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1
0.075
0.05
y
0.025
0
-0.025
2. DESIGN VARIABLES
-0.05
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.
-0.075
0
0.5
0.75
1
x
Figure 1 Shape functions (top), NACA 0012 with each
shape function applied (bottom).
e(k )  ln( 0.5) ln( 1  xk ) ,
e(k )  ln( 0.5) ln( xk ) ,
The y-coordinate position of the upper and lower
surface of the airfoil are then described using the
following equation:
y( x)  y( x) Base Airfoil   i f i ( x)
0.25
k  1,2
(5)
k  3,4,5,6,7,8 (6)
The control points for all shape functions used here are
as follows:
x k  0.06, 0.13, 0.2, 0.4, 0.6, 0.8, 0.87, 0.94
(1)
(i: Design Variables, fi: Shape Functions, i =1, 16)
3. OBJECTIVE FUNCTION FORMULATION
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.
f k ( x)  sin[ (1  x)e ( k ) ], k  1,2
(2)
f k ( x)  sin 3 [x e ( k ) ], k  3,4,5
(3)
f k ( x)  sin[ x e ( k ) ], k  6,7,8
(4)
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.
2
Single Point Objective Function
arrangement and resolution is a compromise between
accuracy and efficiency.
Maximize:
F ( xi )  100 * Cd1
1.5
(7)
1
Subject to:
C l  C l1 , x imin  x i  x imax
(8)
0.5
0
X
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.
-0.5
-1
-1.5
-1
-0.5
0
0.5
1
1.5
2
Y
Multi Point Objective Function
Figure 2 C-grid scheme using 180 × 30 points.
In the case of a two-point shape optimization, the
problem formulation is shown below utilizing a
weighting factor (  ) in the objective function.
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.
Maximize:
F ( xi )  100 * [ * C d1  (1  ) * C d 2 ]
(9)
Subject to:
C l  C l1 , C l  C l2 , x imin  x i  x imax
(10)
0.04
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.
0.035
0.03
Cd
0.025
0.02
0.015
0.01
4. EVALUATION OF OBJECTIVE FUNCTION
0.005
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.
0
0
20
40
60
80
100
120
Grid points on surface
Figure 3 “Residual” subcritical Euler drag coefficient as
a function of grid points on airfoil surface (NACA 0012
airfoil, M=0.4, =5.00)
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
3
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.
-1.5
-1
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.
Cp
-0.5
0
0.5
1
1.5
Experiment
Euler Analysis
RAE2822 [M=0.74,=3.19]
0
0.25
0.5
0.75
1
X
Figure 4 Pressure coefficient distribution of Euler
prediction vs. published experiment (RAE 2822 airfoil,
M=0.74, =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.
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.
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
4
originally designed to reduce wave drag in transonic
flight conditions.
Computational wall-time (Sec.)
4500
4000
3500
3000
2500
Ideal Speed-Up
Parallel GA
A summary of this single-point optimization
problem is presented as follows:
2000
1500

C l  C l1  0.733


M = 0.74
Base airfoils
a) NACA0012
b) RAE2822
c) Whitcomb Super Critical Airfoil
1000
500
10
Number of CPUs
20
7-1-1. Genetic Algorithm Results
30 40
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.
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 gradientbased method. This can give some indication of
multiple local minima appearing in the design space.





Population size: 448
Resolution: 7 bit
Design Variables: 16
Total Chromosome: 112
Variables limits:  0.01  xi  0.01




Mutation probability: 0.0022
Elitism
Tournament selection
Uniform crossover
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.
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
Cl=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
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.
5
0.3
-1.5
-1.25
0.25
-1
0.2
-0.75
NACA0012
Generation 1
Generation 10
Generation 20
Generation 30
Generation 90
Y
0.1
-0.5
Cp
0.15
-0.25
0
0.05
0.25
0
Generation 1
Generation 10
Generation 20
Generation 30
Generation 90
0.5
-0.05
0.75
0
0.25
0.5
0.75
1
1
X
0
0.25
0.5
X
Base Airfoil [NACA0012]
0.75
1
Figure 6 Best airfoil shapes (left) and pressure coefficient distributions (right) in selected generations of the GA using
the NACA 0012 base airfoil.
0.3
-1.5
-1.25
0.25
-1
0.2
-0.75
Y
0.1
-0.5
Cp
RAE2822
Generation 1
Generation 10
Generation 20
Generation 30
Generation 90
0.15
-0.25
0
0.05
0.25
0
Generation 1
Generation 10
Generation 20
Generation 30
Generation 90
0.5
-0.05
0.75
1
0
0.25
0.5
0.75
1
0
0.25
X
0.5
X
Base Airfoil [RAE2822]
0.75
1
Figure 7 Best airfoil shapes (left) and pressure coefficient distributions (right) in selected generations of the GA using
the RAE 2822 base airfoil.
-1.5
0.3
-1.25
0.25
-1
0.2
-0.75
Y
0.1
-0.5
Cp
Whitcomb
Generation 1
Generation 10
Generation 20
Generation 30
Generation 90
0.15
-0.25
0
0.05
0.25
0
Generation 1
Generation 10
Generation 20
Generation 30
Generation 90 Base Airfoil [Whitcomb SuperCritical]
0.5
-0.05
0.75
1
0
0.25
0.5
0.75
1
X
0
0.25
0.5
X
0.75
1
Figure 8 Best airfoil shapes (left) and pressure coefficient distributions (right) in selected generations of the GA using
the Whitcomb supercritical base airfoil.
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
that each run is approaching the same drag performance
for the optimal airfoil shape.
6
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.
-0.5
-0.6
-1.5
-0.8
-1.25
-0.9
-1
-1
-0.75
-1.1
-0.5
Cp
Fitness (-Cd*100)
-0.7
-1.2
Base Airfoil (RAE2822)
Base Airfoil (NACA0012)
Base Airfoil (Whitcomb)
20
40
Generation
60
0
0.25
80
0.5
Figure 9 Best fitness value convergence history for all
three GA runs.
Base Airfoil NACA0012 (90 Generation)
Base Airfoil RAE2822 (90 Generation)
Base Airfoil Whitcomb (90 Generation)
0.75
1
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.
0
0.25
0.5
X
0.75
1
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.
0.15
0.1
-1.6
0.05
-1.65
0
-1.7
-Cd*100
Y
-0.25
-0.05
-1.75
-1.8
-0.1
Base Airfoil (NACA0012), Generation 90
Base Airfoil (RAE2822), Generation 90
Base Airfoil (Whitcomb), Generation 90
0
0.25
0.5
0.75
-1.85
1
Base Airfoil [NACA0012]
X
Figure 10 Best airfoil shapes from generation 90 of all
three GA runs.
1
2
3
4
5
Iteration Number
6
7
8
Figure 12 Convergence history for CONMIN with
NACA 0012 as the starting shape.
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,
7
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.
0.3
0.25
0.2
Y
0.15
RAE2822
Final Iteration
0.1
0.05
0.3
0
0.25
-0.05
0.2
0
0.25
0.5
0.75
1
X
Y
0.15
NACA0012
Iteration 1
Iteration 6
Final Iteration
0.1
Figure 14 Initial RAE 2822 airfoil shape and final
shape generated by CONMIN
0.05
0.3
0
0.25
-0.05
0.2
0
0.1
0.2
0.3
0.4
0.5
X
0.6
0.7
0.8
0.9
1
0.15
Y
-1.5
Whitcomb
Final Iteration
0.1
-1.25
0.05
-1
0
-0.75
Cp
-0.5
-0.05
-0.25
0
0
0.5
0.75
1
X
0.25
0.5
Figure 15 Initial Whitcomb airfoil shape and final shape
generated by CONMIN
NACA0012
Iteration 1
Iteration 6
Final Iteration
0.75
1
0.25
0
0.25
0.5
0.75
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.
1
X
Figure 13 Airfoil shape designs (top) and pressure
coefficient distributions (bottom) generated during
CONMIN search using NACA 0012 base airfoil.
7-2. Multi Point Optimization Results
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.
A two-point design case was tried to investigate the
effects of the objective function selection. The
weighting factor in equation (9) is   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.
8
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.
-1.4
GA Multipoint (M=0.68)
GM Multipoint (M=0.68)
-1.2
-1
-0.8
-0.6
Y
-0.4
0.3
-0.2
0
0.25
0.2
0.2
0.4
0.6
Y
0.15
0.8
Base airfoil (NACA0012)
GM multipoint (CONMIN)
GA multipoint (90 Generation)
0.1
1
0
0.25
0.5
0.75
1
X
0.05
-1.4
GA Multipoint (M=0.74)
GM Multipoint (M=0.74)
0
-1.2
-0.05
-1
-0.8
0
0.25
0.5
0.75
1
-0.6
X
-0.4
Y
Figure 17 GA and CONMIN results for two-point
objective function formulation.
-0.2
0
7-3. Computational Efficiency Comparison
0.2
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.
0.4
0.6
0.8
1
0
0.25
0.5
Method
NACA 0012
GM
RAE2822
GM
Whitcomb
GA
GM
GA
GA
1
Figure 16 Pressure coefficient distributions for twopoint objective function results at M=0.68 (top) and
M=0.74 (bottom)
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
Table 1 Comparisons of drag values and computational costs for GA and GM runs
Base airfoil
0.75
X
Drag (Cd)
0.00586
0.01642
0.00576
0.01602
0.00614
0.01085
9
Number of function
evaluation
40320
160
40320
157
40320
79
Parallel
Computational time with
45 processors
~ 896 * tE
160 * tE
~ 896 * tE
157 * tE
~ 896 * tE
79 * tE
Quagliarella, D. and Della Cioppa, A., ”Genetic Algorithms
Applied to the Aerodynamic Design of Transonic Airfoils,”
AIAA Paper 94-1896-CP, 1994
5
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.
Oyama, A. “Multidisciplinary Optimization of Transonic
Wing Design Based on Evolutionary Algorithms Coupled
with CFD Solver,” ECCOMAS, 2000
6
Hager, J. O., Eyi, S., Lee, K. D., “Two-point transonic
airfoil design using optimization for improved off-design
performace,” AIAA Journal of Aircraft, Vol.31, No5, 1994,
pp. 1143-1147.
7
Drela, M., “Pros & Cons of airfoil optimization”, Frontiers
of Computational Fluid Dynamics-1998,(D. A. Caughey and
M. M. Hafez, eds.), World Scientific Publishers, 1998,
pp.363-381.
8
Roe, P. L., “Approximate Riemann solvers, parameter
vectors and difference schemes,” Journal of Computational
Physics, 43,1981, pp. 357-72.
9
ACKNOWLEDGMENT
“Experimental Data base for Computer Program
Assessment”, AGARD advisory report, No.138, 1979
10
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.
11
Goldberg, D. E., Genetic Algorithms in search optimization
and machine learning, Addison-Wesley, MA, 1989
Hajela, P., “Genetic Search - An Approach to the
Nonconvex Optimization Problem,” AIAA Journal, Vol. 28,
No.7, Jul. 1990, pp. 1205-1210.
12
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