TRANSONIC AIRFOIL DESIGN BY THE INVERSE METHOD
USING VARIABLE-FIDELITY MODELLING
Slawomir Koziel, Leifur Leifsson and Stanislav Ogurtsov
Engineering Optimization & Modeling Center, School of Science and Engineering
Reykjavik University, Menntavegur 1, Reykjavik, Iceland
Keywords: Inverse airfoil design, Target pressure distribution, Surrogate models, Variable-resolution modelling,
Response surface modelling.
Abstract: The paper presents an improved optimization algorithm for the inverse design of transonic airfoils. Our
approach replaces the direct optimization of an accurate, but computationally expensive, high-fidelity airfoil
model by an iterative re-optimization of two different surrogate models. Initially, for a few design iterations,
a corrected physics-based low-fidelity model is employed, which is subsequently replaced by a response
surface approximation model. The low-fidelity model is based on the same governing fluid flow equations
as the high-fidelity one, but uses coarser discretization and relaxed convergence criteria. A shape-preserving
response prediction technique is utilized to align the pressure distribution of the low-fidelity model with that
of the high-fidelity one. This alignment process is particularly suitable since the inverse design aims at
matching a given target pressure distribution. Our algorithm is applied to constrained inverse airfoil design
in inviscid transonic flow. A comparison with the basic version of the optimization algorithm, exploiting
only a physics-based low-fidelity model, is also carried out. While the performance of both versions is
similar with respect to their ability to match the target pressure distribution, the improved algorithm offers
substantial design cost savings, from 25 to 72 percent, depending on the test case.
1 INTRODUCTION
Aerodynamic shape optimization (ASO) involves
the design of aerodynamic components such as
aircraft wings and turbine blades (Leoviriyakit et al.,
2003); (Braembussche, 2008). The state-of-the-art
ASO design methods employ high-fidelity
computational fluid dynamic (CFD) simulations as a
part of efficient numerical optimization algorithms
(Queipo et al., 2005); (Forrester and Keane, 2009);
(Alexandrov et al., 2000); (Robinson et al., 2008).
The accurate CFD simulations typically lead to more
realistic and attainable designs. The downside is that
the high-fidelity CFD analysis is computationally
expensive and design optimization normally requires
a large number of simulations, which leads to a time
consuming design process.
The introduction of surrogate-based optimization
(SBO) methods (Queipo et al., 2005); (Forrester and
Keane, 2009) to ASO permitted reduction of the
overall computational cost, as well as handling noisy
objective functions. Examples of such work can be
found in the literature (see e.g., Alexandrov et al.,
2000); Robinson et al., 2008); (Booker et al., 1999);
(Barrett et al., 2006); (Leifsson and Koziel, 2010);
(Lane and Marshall, 2010).
A computationally efficient design optimization
methodology for the inverse ASO of transonic
airfoils was recently introduced (Leifsson and
Koziel, 2011). The approach replaces the direct
optimization of an accurate, but computationally
expensive, high-fidelity airfoil model by an iterative
re-optimization of a corrected low-fidelity model.
The low-fidelity model is based on the same
governing fluid flow equations as the high-fidelity
one, but uses coarser discretization and relaxed
convergence criteria. The shape-preserving response
prediction technique (Koziel, 2010a) is utilized to
align the pressure distribution of the low-fidelity
model with that of the high-fidelity model. This
alignment process is particularly suitable since a
target pressure distribution is specified in the inverse
design problem.
In this work, we substantially enhance the
optimization methodology introduced in Leifsson
and Koziel (2011). More specifically, the low-
474
Koziel S., Leifsson L. and Ogurtsov S..
TRANSONIC AIRFOIL DESIGN BY THE INVERSE METHOD USING VARIABLE-FIDELITY MODELLING.
DOI: 10.5220/0003646204740482
In Proceedings of 1st International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SDDOM-2011), pages
474-482
ISBN: 978-989-8425-78-2
Copyright
c
2011 SCITEPRESS (Science and Technology Publications, Lda.)
fidelity CFD model is replaced - after a few design
iterations - by its (local) response surface
approximation, which allows us to reduce the overall
design cost and obtain faster convergence when
compared to the original version of the algorithm.
Our approach is demonstrated using several
transonic airfoil design cases.
2 PROBLEM FORMULATION
The airfoil shape optimization can be formulated as
a constrained nonlinear minimization problem. For a
given set of operating conditions, solve
;;,,1,0)(
;,,1 ,0)(s.t.)( min
uxlx
xx
x
Nkh
Mjgf
k
j
(1)
where f(x) is the objective function, x is the design
variable vector, g
j
(x) are the inequality constraints,
M is the number of the inequality constraints, h
k
(x)
are the equality constraints, N is the number of the
equality constraints, and l and u are the design
variables lower and upper bounds, respectively.
There are two main approaches to airfoil design.
One is to directly adjust the geometry parameters of
the airfoil section in order to maximize its
performance. This is called direct design, and the
most common formulations include lift
maximization, drag minimization, and lift-to-drag
ratio maximization (Leifsson and Kozel, 2010).
Another way is to define a priori a specific flow
behavior that is to be attained. The airfoil shape is
then designed to achieve this flow behavior. This is
called inverse design (Dulikravich, 1991).
In inverse design, the role of the designer is to
specify a particular flow feature, which typically is a
target pressure distribution, C
p.t
, on the surface of the
airfoil. The task is then to find the airfoil shape that
can give the target pressure distribution at the
desired flow condition. This can be done by
minimizing the difference between the pressure
distribution of the airfoil C
p
and the target
distribution C
p.t
.
The objective function can be formulated as the L
2
norm of the difference between the airfoil pressure
distribution and the target pressure distribution, or f(x)
= ½ [C
p
(x) - C
p.t
]
2
ds. A minimum thickness is
normally specified so that the optimizer does not
reduce the airfoil to a thin plate. The thickness
constraint can be written as g(x) = A
min
A(x) 0,
where A(x) is the cross-sectional area of the airfoil
and A
min
is the minimum cross-sectional area.
In this paper, we use the NACA airfoil shapes. In
particular, we use the NACA four-digit airfoil
parameterization method, where the airfoil shape is
defined by three parameters m (the maximum
ordinate of the mean camberline as a fraction of
chord), p (the chordwise position of the maximum
ordinate) and t/c (the thickness-to-chord ratio). The
airfoils are denoted by NACA mpxx, where xx
represents the value of t/c. The shapes are
constructed using two polynomials, one for the
thickness distribution and the other for the mean
camber line. The full details of the NACA four-digit
parameterization method are given in Abbott and
Doenhoff (1959). Three example NACA four-digit
airfoils are shown in Fig. 1.
3 CFD MODELLING
A single CFD simulation is, in general, composed of
four steps; the geometry generation (described here
in Section 2), meshing of the solution domain,
numerical solution of the governing fluid flow
equations, and post-processing of the flow results,
which involves, in the case of numerical
optimization, calculating the objectives and
constraints. In this section we present the high- and
low-fidelity CFD models.
3.1 High-fidelity CFD Model
The flow is assumed to be steady, inviscid, and
adiabatic with no body forces. The Euler equations
are taken to be the governing fluid flow equations
(Tannehill et al., 1997). The computational meshes
used in this study are all of structured curvilinear
body-fitted C-topology.
Figure 1: Shown are three different NACA four-digit
airfoil sections; NACA 0012 (m = 0, p = 0, t/c = 0.12)
solid line (-), NACA 2412 (m = 0.02, p = 0.4, t/c = 0.12)
dashed line (--), NACA 4608 (m = 0.04, p = 0.6, t/c =
0.08) dash-dot line (--).
The solution domain boundaries are placed at 24
chord lengths in front of the airfoil, 50 chord lengths
behind it, and 25 chord lengths above and below it.
The meshes are generated with the computer code
ICEM CFD (2006). A fine mesh was developed with
a total of 320 points in the vertical direction, 180
points on the airfoil surface and 160 points in the
0 0.2 0.4 0.6 0.8 1
-0.1
0
0.1
x/
c
z
/
c
NACA 0012
NACA 2412
NACA 4608
TRANSONIC AIRFOIL DESIGN BY THE INVERSE METHOD USING VARIABLE-FIDELITY MODELLING
475
wake behind the airfoil, with a total of 106 thousand
cells. An example computational mesh is shown in
Fig. 2.
The numerical fluid flow simulations are
performed using the computer code FLUENT
(2006). Asymptotic convergence to a steady state
solution is obtained for each case. The iterative
convergence of each solution is examined by
monitoring the overall residual, which is the sum
(over all the cells in the computational domain) of
the L
2
norm of all the governing equations solved in
each cell. In addition to this, the lift and drag forces
(defined in Section 3.3) are monitored for
convergence. The criteria used in this work for the
high-fidelity model is a maximum residual of 10
-6
,
or a maximum number of iterations of 1000.
(a)
(b)
Figure 2: (a) An example computational mesh with a
structured C-topology for a NACA 0012 airfoil; (b) a view
close to the airfoil surface.
3.2 Low-fidelity CFD Model
The low-fidelity CFD model is constructed using the
high-fidelity model, but with a coarser
computational mesh and relaxed convergence
criteria. The parameters of the mesh and the number
of solver iterations were obtained by performing a
parametric study using the NACA 2412 airfoil
section at M
= 0.75 and α = 1 deg.
Initially the fine mesh is solved to full
convergence. The solver needed 216 iterations to
reach a converged solution based on the residuals of
the governing equations. However, the lift and drag
coefficient values reached a converged value after
approximately 50 iterations. Therefore, the number
of iterations limit was set to 100 hundred iterations
in the subsequent steps.
The mesh points were reduced in two steps. First,
the number of mesh points in the z-direction and the
number of mesh points behind the airfoil were
reduced by (approximately) half. This procedure was
repeated and in each step the pressure distribution
was plotted. It was observed that the shock location
moved rearward. After five subsequent steps, the
pressure distribution became highly distorted near
the leading-edge. The previous mesh was then
retained and the number of mesh points on the airfoil
surface was reduced incrementally. Again, the
pressure distribution was plotted for each step. It
was observed that the shock strength reduced in each
step as the mesh got coarser and coarser on the
airfoil surface. After three steps the procedure was
halted. The resulting mesh has 48 points in the z-
direction, 115 points on the airfoil surface, and 20
points in the wake behind the airfoil, with a total of
8295 thousand cells. The overall evaluation time is
reduced to about 34 s, which is approximately 13.5
times faster than the high-fidelity model using the
fine mesh and traditional convergence criteria.
The overall evaluation time of the high-fidelity
model in this parametric study is 471 s with a total
of 216 iterations. In many cases the solver does not
fully converge with respect to the residuals and goes
on up to 1000 iterations. Then the overall evaluation
time goes up to 2500 s, and the low-fidelity model is
approximately 73 times faster. For the sake of
simplicity, we will use a fixed value of 50 in the
numerical computations presented later in the paper.
3.3 Aerodynamic Forces
The non-dimensional force coefficients parallel to
the x- and z-axes, C
x
and C
z
, respectively, are
calculated by integrating the pressure distribution C
p
over the surface of the airfoil as (Tannehill et al.,
1997)
dsCC
px
sin
,
dsCC
pz
cos
(2)
where ds is the surface panel element of length and
is the angle the panel makes relative to the x-axis.
The lift coefficient C
l
and the wave drag coefficient
C
dw
are calculated as
cossin
zxl
CCC
,
sincos
zxdw
CCC
.
(3)
x/c
z/c
-20 0 20 40
-20
-10
0
10
20
x/c
z/c
0 0.2 0.4 0.6 0.8 1
-0.4
-0.2
0
0.2
0.4
SIMULTECH 2011 - 1st International Conference on Simulation and Modeling Methodologies, Technologies and
Applications
476
4 SURROGATE MODELING
In order to use the low-fidelity model as a reliable
prediction tool in the optimization process, it has to
be corrected to become a reliable representation of
the high-fidelity model. The corrected low-fidelity
model is called a surrogate. The surrogate model can
replace the high-fidelity one in the optimization
process and thus reduce the overall optimization
cost.
Here, we adopt a shape-preserving response
prediction (SPRP) methodology introduced in
(Koziel, 2010a) in the context of microwave
engineering, and recently applied for direct airfoil
design (Leifsson and Koziel, 2010). SPRP is easy to
implement, unlike space mapping it does not need
any auxiliary transformations or extractable
parameters (Koziel et al., 2008). Also, it does not
require high-fidelity model derivative information.
By formulation, SPRP works directly with the
model outputs that can be described by certain
number of design-variable-dependent characteristic
features (Koziel, 2010a). In the case of a pressure
distribution, these features may include the location
and strength of the shock, the pressure at the
leading- and trailing-edges, and many others. In
direct airfoil design, the pressure distribution is an
intermediate simulation result, with the figures of
interest, such as lift or wave drag, being derived
from it. In inverse design, the pressure distribution is
the main object of interest, which makes SPRP well
suited for this kind of problem.
We will denote the vector of design variables as
x. The pressure distribution for the high- and low-
fidelity models will be denoted as C
p.f
and C
p.c
,
respectively. The surrogate model is constructed
assuming that the change of C
p.f
due to the
adjustment of the design variables x can be predicted
using the actual changes of C
p.c
. The change of C
p.c
is described by the translation vectors corresponding
to certain (finite) number of its characteristic points.
These translation vectors are subsequently used to
predict the change of C
p.f
, whereas the actual C
p.f
at
the current design, C
p.f
(x
(i)
), is treated as a reference.
Figure 3(a) shows the pressure distribution C
p.c
of the low-fidelity model at x
(i)
= [0.02 0.4 0.12]
T
(NACA 2412 airfoil) for M
= 0.7 and α = 1 deg, as
well as C
p.c
at x = [0.025 0.56 0.122]
T
; x
(i)
will
denote a current design (at the ith iteration of the
optimization algorithm; the initial design will be
denoted as x
(0)
accordingly). Circles denote
characteristic points of C
p.c
(x
(i)
), here, representing,
among others, x/c equal to 0 and 1 (leading and
trailing airfoil edges, respectively), the maxima of
C
p.c
for the lower and upper airfoil surfaces, as well
as the local minimum of C
p.c
for the upper surface.
The last two points are useful to locate the pressure
shock. Squares denote corresponding characteristic
points for C
p.c
(x), while small line segments
represent the translation vectors that determine the
“shift” of the characteristic points of C
p.c
when
changing the design variables from x
(i)
to x.
In order to obtain a reliable prediction, the
number of characteristic points has to be larger than
illustrated in Fig. 3(a). Additional points are inserted
in between initial points either uniformly with
respect to x/c (for those parts of the pressure
distribution that are almost flat) or based on the
relative pressure value with respect to corresponding
initial points (for those parts of the pressure
distribution that are “steep”). Figure 3(b) shows the
full set of characteristic points (initial points are
distinguished using larger markers).
The pressure distribution of the high-fidelity
model at the given design, here, x, can be predicted
using the translation vectors applied to the
corresponding characteristic points of the pressure
distribution of the high-fidelity model at x
(i)
,
C
p.f
(x
(i)
). This is illustrated in Figure 4(a) where only
initial characteristic points and translation vectors
are shown for clarity. Figure 4(b) shows the
predicted pressure distribution of the high-fidelity
model at x as well as the actual C
p.f
(x). The
agreement between both curves is very good.
Rigorous formulation of the SPRP technique can
be found in Koziel (2010a). We omit the details here
for the sake of brevity. It should be mentioned that
the SPRP model assumes that the high- and low-
fidelity model pressure distributions have
corresponding sets of characteristic points. This is
usually the case for the practical ranges of design
variables because the overall shape of the
distributions is similar for both models. In case of a
lack of correspondence, original definitions of
characteristic points are replaced by their closest
counterparts. The typical example would be non-
existence of the local minimum of the pressure
distribution for the upper surface for the high- and/or
low-fidelity model at certain designs. In this case,
the original point (local minimum) is replaced by the
points characterized by the largest curvature.
5 OPTIMIZATION PROCEDURE
5.1 Objective Function
In inverse design, the primary objective is the align-
TRANSONIC AIRFOIL DESIGN BY THE INVERSE METHOD USING VARIABLE-FIDELITY MODELLING
477
ment between the pressure distribution of the airfoil
being designed and the prescribed target. The
alignment can be measured using a norm.
(a)
(b)
Figure 3: (a) Example low-fidelity model pressure
distribution at the design x
(i)
, C
p.c
(x
(i)
) (solid line), the low-
fidelity model pressure distribution at other design x,
C
p.c
(x) (dotted line), characteristic points of C
p.c
(x
(i)
)
(circles) and C
p.c
(x) (squares), and the translation vectors
(short lines); (b) low-fidelity model pressure distributions,
initial characteristic points (large markers) and translation
vectors from Fig. 3(a) as well as additional points (small
markers) inserted in between the initial points either
uniformly with respect to x/c (for those parts of the
pressure distribution that are almost flat) or based on the
relative pressure value with respect to corresponding
initial points (for those parts of the pressure distribution
that are “steep”).
Due to unavoidable misalignment between the
pressure distributions of the high-fidelity model and
its SPRP surrogate, it is not convenient to use some
of the constraints (e.g., a maximum allowable drag)
directly. This is because the design that is feasible
for the surrogate model, may not be feasible for the
high-fidelity model. In particular, the design
obtained as a result of optimizing the surrogate
model C
p.s
(i)
, i.e., x
(i+1)
, will be feasible for C
p.s
(i)
.
However, if x
(i+1)
is not feasible for the high-fidelity
model, it will not be feasible for C
p.s
(i+1)
because we
have C
p.s
(i+1)
(x
(i+1)
) = C
p.f
(x
(i+1)
) by the definition of
the surrogate model. In order to alleviate this
problem, we shall use the penalty function approach
to handle the drag constraint. Of course, this does
not apply to constraints that depend exclusively on
the airfoil geometry, such as the minimum airfoil
cross-sectional area.
(a)
(b)
Figure 4: (a) High-fidelity model pressure distribution at
x
(i)
, C
p.f
(x
(i)
) (solid line) and the predicted high-fidelity
model C
p
at x (dotted line) obtained using SPRP based on
characteristic points of Fig. 3(b); characteristic points of
C
p.f
(x
(i)
) (circles) and the translation vectors (short lines)
were used to find the characteristic points (squares) of the
predicted high-fidelity model pressure distribution (only
initial points are shown for clarity); low-fidelity model
distributions C
p.c
(x
(i)
) and C
p.c
(x) are plotted using thin
solid and dotted line, respectively; (b) high-fidelity model
pressure distribution at x, C
p.f
(x) (solid line), and the
predicted high-fidelity model pressure distribution at x
obtained using SPRP (dotted line).
In this work, all the constraints are handled
through penalty functions so that the objective
function is defined as follows

2
2
.
2
2
.
( ( )) ( ( ))
(()) ()
pdwsp
ls p
HC F C C
CC A





xx
xx
(4)
with F = ||C
p
(x) – C
p.t
||, where C
p.t
is the target
pressure distribution, C
dw.s
= 0 if C
dw.s
C
dw.s.max
and C
dw.s
= C
dw.s
C
dw.s.max
otherwise, C
l.s
= 0 if
C
l.s
C
l.s.min
and C
l.s
= C
l.s
C
l.s.min
otherwise, and
A = 0 if A A
min
and A = AA
min
otherwise. In
our numerical experiments we use
=
=
= 1000.
Here, the pressure distribution for the surrogate
model is C
p
= C
p.s
, and for the high-fidelity model C
p
= C
p.f
. Also, C
l.s
and C
dw.s
denote the lift and wave
drag coefficients (both being functions of the
pressure distribution).
5.2 Basic Optimization Algorithm
The basic optimization algorithm (Leifsson and
0 0.2 0.4 0.6 0.8 1
-1
-0.5
0
0.5
1
1.5
x
/
c
-
C
p
0 0.2 0.4 0.6 0.8 1
-1
-0.5
0
0.5
1
1.5
x
/
c
-
C
p
0 0.2 0.4 0.6 0.8 1
-1
-0.5
0
0.5
1
1.5
x
/
c
-
C
p
0 0.2 0.4 0.6 0.8 1
-1
-0.5
0
0.5
1
1.5
x
/
c
-
C
p
SIMULTECH 2011 - 1st International Conference on Simulation and Modeling Methodologies, Technologies and
Applications
478
Koziel, 2011) exploits the SPRP-based surrogate
model and a trust-region convergence safeguard
(Conn et al., 2000). It can be summarized as follows:
1. Set i = 0; Select
(initial trust region radius);
Evaluate C
p.f
(x
(0)
);
2. Set up SPRP model;
3. Obtain x
(i+1)
= argmin{l x u, ||xx
(i)
||
:
H(C
p.s
(i)
(x))};
4. Evaluate high-fidelity model to get C
p.f
(x
(i)
);
5. If H(C
p.f
(i)
(x
(i+1)
)) < H(C
p.f
(i)
(x
(i)
)) accept x
(i+1)
;
Otherwise x
(i+1)
= x
(i)
;
6. Update
;
7. Set i = i + 1;
8. If termination condition is not satisfied, go to 2.
9. END
The SPRP surrogate model is updated before each
iteration of the optimization algorithm using the
high-fidelity model data at the design obtained in the
previous iteration. The trust-region parameter
is
updated after each iteration, i.e., decreased if the
new design was rejected or if the improvement of
the high-fidelity model objective function was too
small compared to the prediction given by the SPRP
surrogate, or increased otherwise. We use classical
updating rules (Conn et al., 2000; Koziel et al.,
2006). The algorithm is terminated if ||x
(i+1)
x
(i)
|| <
0.001 or
< 0.001.
5.3 Improved Optimization Scheme
In algorithm of Section 5.2, the new solution x
(i+1)
is
obtained by optimizing the SPRP model, which
involves multiple evaluations of the low-fidelity
model. In our case, the low-fidelity model is about
50 times faster than the high-fidelity one, and a
typical surrogate model optimization requires 50 to
100 low-fidelity model calls. Thus, the low-fidelity
model evaluations are responsible for about 50 to 70
percent of the total optimization cost.
In this section, we formulate the improved
optimization scheme that aims at reducing the
aforementioned computational overhead. The idea is
to replace the low-fidelity model—at some point
during the optimization run—by its response surface
approximation model. This replacement is executed
if ||C
p.c
(x
(i–1)
) – C
p.c
(x
(i)
)|| <
max
where
max
is a user-
defined threshold value (here, we use
max
= 0.05).
This allows us to avoid constructing the response
surface model in the entire design space (which
would be too expensive in terms of the number of
necessary data points), but only in the vicinity of the
current solution x
(i)
. In this work, we use kriging
interpolation (Queipo et al., 2005) as the response
surface model. The model is set up using N
kr
= 20
low-fidelity model evaluations allocated—using
Latin Hypercube Sampling (Beachkofski and
Grandhi, 2002)—in the interval [x
(i)
kr
, x
(i)
+
kr
],
where
kr
= [0.001 0.04 0.005]
T
.
The improved algorithm can be summarized as
follows:
1. Set i = 0; Select
(initial trust region radius); Set
the model selector L = 0; Evaluate C
p.f
(x
(0)
);
2. If L = 0 set up SPRP model using the low-
fidelity model; Otherwise, set up SPRP model using
the kriging model of C
p.c
;
3. Obtain x
(i+1)
= argmin{l x u, ||xx
(i)
||
:
H(C
p.s
(i)
(x))};
4. Evaluate high-fidelity model to get C
p.f
(x
(i)
);
5. If H(C
p.f
(i)
(x
(i+1)
)) < H(C
p.f
(i)
(x
(i)
)) accept x
(i+1)
;
Otherwise x
(i+1)
= x
(i)
;
6. Update
;
7. Set i = i + 1;
8. If L = 0 and ||C
p.c
(x
(i–1)
) – C
p.c
(x
(i)
)|| <
max
, set
L = 1 and set up the kriging model of C
p.c
in the
interval [x
(i)
kr
, x
(i)
+
kr
];
9. If termination condition is not satisfied, go to 2.
10. END
In practice (cf. Section 6), the condition ||C
p.c
(x
(i–1)
) –
C
p.c
(x
(i)
)|| <
max
is satisfied after one or two
iterations which allows us to substantially reduce the
number of low-fidelity model evaluations in the
optimization process.
6 NUMERICAL EXAMPLES
6.1 General Setup
The proposed optimization method is applied to the
inverse design optimization of four cases. Designs
are obtained using the basic algorithm in Section
5.2, and the improved algorithm in Section 5.3. The
surrogate model optimization is performed using the
pattern-search algorithm
(Koziel, 2010b). For
comparison purposes, designs obtained through
direct optimization of the high-fidelity model using
the pattern-search algorithm
(Koziel, 2010b) are also
presented.
The design variables are the airfoil shape
parameters in the NACA four-digit parameterization
(Section 2), i.e., x = [m p t]
T
. Note that the chord
length is set to 1. The only inequality constraint is
the minimum cross-sectional area constraint. There
TRANSONIC AIRFOIL DESIGN BY THE INVERSE METHOD USING VARIABLE-FIDELITY MODELLING
479
are no equality constraints. The side constraints are 0
m 0.1, 0.2 p 0.8, and 0.05 t 0.20. The test
cases were chosen only for verification purposes and
they do not represent optimal airfoil designs.
6.2 Results of Case Studies
The numerical results of the case studies are
presented in Table 1. The target pressure distribution
is the same for cases 1, 2, and 3, i.e., x = [0.0163
0.4004 0.1118]
T
with M
= 0.75 and α = 0°. In case
1 the initial design is NACA 2412 and A
min
= 0.078.
For case 1, both the basic and the improved
algorithms hit the cross-sectional area constraint and
match the target pressure distribution only relatively
closely (F = 0.0243 and F = 0.0246, respectively).
However, the improved algorithm requires 5
equivalent high-fidelity model evaluations, while the
basic algorithm needs 18. The direct optimization of
the high-fidelity model required 201 model
evaluations.
In case 2, the constraint value is lowered to
A
min
= 0.075, while keeping other parameters the
same. Now the algorithms are able to match the
target distribution much closer (F = 0.0025 and F =
0.0011, respectively) with the design cost of about
13 equivalent high-fidelity function calls for the
basic algorithm, and 5 for the improved version.
Figure 5 shows the pressure distributions of the
initial and optimized designs. The high-fidelity CFD
model and the pattern-search algorithm required 152
model evaluations.
Table 1: Numerical results of four inverse design case
studies using the proposed optimization methodology. F is
the norm of the difference of pressure distributions for the
optimized and the target designs. N
c
is the number of low-
fidelity model evaluations and N
f
is the number of high-
fidelity model evaluations. All the numerical values are
from the high-fidelity model.
Case 1
M
= 0.75,
= 0°, A
min
= 0.078
Variable Initial Target
Pattern-
Search
#
SPRP
$
Improved
SPRP
M 0.0200 0.0163 0.0153 0.0154 0.0155
P 0.4000 0.4004 0.4089 0.4043 0.3985
T 0.1200 0.1118 0.1159 0.1158 0.1159
C
l
0.4745 0.3832 0.3663 0.3663 0.3671
C
dw
0.0115 0.0049 0.0049 0.0050 0.0052
A 0.0808 0.0753 0.0780 0.0780 0.0781
F N/A N/A 0.0249 0.0243 0.0246
N
c
N/A N/A 0 330 70
N
f
N/A N/A 201 11 3
Total
cost
*
N/A N/A 201 < 18 < 5
Case 2
M
= 0.75,
= 0°, A
min
= 0.075
Variable Initial Target
Pattern-
Search
#
SPRP
$
Improved
SPRP
M 0.0200 0.0163 0.0162 0.0162 0.0163
P 0.4000 0.4004 0.4007 0.4014 0.4011
T 0.1200 0.1118 0.1122 0.1121 0.1119
C
l
0.4745 0.3832 0.3816 0.3822 0.3829
C
dw
0.0115 0.0049 0.0049 0.0048 0.0049
A 0.0808 0.0753 0.0756 0.0755 0.0753
F N/A N/A 0.0025 0.0025 0.0011
N
c
N/A N/A 0 240 70
N
f
N/A N/A 152 8 3
Total
cost
*
N/A N/A 152 < 13 < 5
Case 3
M
= 0.75,
= 0°, A
min
= 0.075
Variable Initial Target
Pattern-
Search
#
SPRP
$
Improved
SPRP
m 0.0000 0.0163 0.0156 0.0163 0.0161
p 0.5000 0.4004 0.4252 0.3978 0.4030
t 0.1000 0.1118 0.1139 0.1117 0.1126
C
l
0.0005 0.3832 0.3776 0.3835 0.3801
C
dw
0.0002 0.0049 0.0045 0.0049 0.0048
A 0.0673 0.0753 0.0767 0.0752 0.0758
F N/A N/A 0.0181 0.0023 0.0048
N
c
N/A N/A 0 468 70
N
f
N/A N/A 201 6 5
Total
cost
*
N/A N/A 201 < 16 < 7
Case 4
M
= 0.75,
= 1°, A
min
= 0.065
Variable Initial Target
Pattern-
Search
#
SPRP
$
Improved
SPRP
m 0.0000 0.0300 0.0287 0.0293 0.0300
p 0.4000 0.2000 0.2067 0.2057 0.2000
t 0.1200 0.1000 0.1083 0.1022 0.1000
C
l
0.2082 0.8035 0.7786 0.7905 0.8034
C
dw
0.0024 0.0410 0.0424 0.0405 0.0410
A 0.0808 0.0675 0.0731 0.0689 0.0675
F N/A N/A 0.0492 0.0182 0.00097
N
c
N/A N/A 0 457 120
N
f
N/A N/A 202 6 9
Total
cost
*
N/A N/A 202 < 16 < 12
#
Design obtained using the high-fidelity model and the grid-
search algorithm (Koziel, 2010b).
$
Design obtained using the basic algorithm of Section 5.2;
surrogate model optimization performed using the grid-search
algorithm (Koziel, 2010b).
%
Design obtained using the improved algorithm proposed in
Section 5.3; surrogate model optimization performed using the
grid-search algorithm (Koziel, 2010b).
*
The total optimization cost is expressed in terms of the
equivalent number of high-fidelity model evaluations. The ratio
of the high-fidelity model evaluation time to the corrected low-
fidelity model evaluation time varies between 13.5 to 73
depending on the design. For the sake of simplicity we use a fixed
value of 50 here.
SIMULTECH 2011 - 1st International Conference on Simulation and Modeling Methodologies, Technologies and
Applications
480
(a)
(b)
Figure 5: The pressure distributions and airfoil shapes for
the initial and optimized designs for case 2.
Case 3 starts with a different initial design,
namely, the NACA 0010. Here, both algorithms
match the target closely. The basic algorithm
requires 16 equivalent high-fidelity function calls,
whereas the improved algorithm 7.
In case 4, the target pressure distribution is the
one of NACA 3210 at M
= 0.75 and α = 1°. The
initial design is NACA 0012 and the minimum
cross-sectional area is A
min
= 0.065. The improved
algorithm is able to match the target closely in less
than 12 equivalent high-fidelity model evaluations.
The basic algorithm and the direct pattern-search are
both unable to match the target closely.
7 CONCLUSIONS
Computationally efficient variable-fidelity design of
transonic airfoils is presented. The algorithm replaces
the direct optimization of a CPU-intensive high-
fidelity CFD model by iterative updating and re-
optimization of its fast surrogate. The surrogate is
constructed using a shape-preserving response
prediction technique with the underlying low-fidelity
CFD model, which is replaced—after a few
iterations—by its local response surface
approximation. The operation and performance of our
algorithm is demonstrated using several transonic
airfoil design cases with the optimized designs
obtained at a low cost corresponding to a few high-
fidelity CFD simulations. Our results indicate that the
algorithm presented here is computationally much
more efficient than its basic version that only exploits
the corrected CFD low-fidelity model but not the
response surface one.
REFERENCES
Abbott, I. H., and Von Doenhoff, A. E., 1959, Theory of
Wing Sections, Dover Publications, 1959.
Alexandrov, N. M., Lewis, R. M., Gumbert, C. R., Green, L.
L., and Newman, P. A., 2000, Optimization with
Variable-Fidelity Models Applied to Wing Design, 38th
Aerospace Sciences Meeting & Exhibit, Reno, NV, AIAA
Paper 2000-0841.
Barrett, T. R., Bressloff, N. W., and Keane, A. J., 2006,
Airfoil Design and Optimization Using Multi-Fidelity
Analysis and Embedded Inverse Design, AIAA Paper
2006-1820, 47th AIAA/ASME/ASCE/AHS/ASC
Structures, Structural Dynamics, and Materials
Conference, Newport, Rhode Island.
Beachkofski B. and Grandhi R., 2002. Improved
distributed hypercube sampling. American Institute of
Aeronautics and Astronautics, paper AIAA 2002-
1274.
Booker, A.J., Dennis Jr., J.E., Frank, P.D., Serafini, D.B.,
Torczon, V., and Trosset, M.W., 1999, A rigorous
framework for optimization of expensive functions by
surrogates, Structural Optimization, vol. 17, no. 1, pp. 1-
13.
Braembussche, R. A., 2008, Numerical Optimization for
Advanced Turbomachinery Design, In Optimization
and Computational Fluid Dynamics, Thevenin, D. and
Janiga, G., editors, Springer, pp. 147-189.
Conn, A. R., Gould, N. I. M., and Toint, P. L., 2000, Trust
Region Methods, MPS-SIAM Series on Optimization.
Dulikravich, G. S., 1991, Aerodynamic Shape Design and
Optimization, 29th AIAA Aerospace Sciences Meeting,
Reno, NV.
FLUENT, ver. 12.1, ANSYS Inc., Southpointe, 275
Technology Drive, Canonsburg, PA 15317, 2006.
Forrester, A. I. J., and Keane, A. J., 2009, Recent advances in
surrogate-based optimization, Progress in Aerospace
Sciences, vol. 45, no. 1-3, pp. 50-79.
ICEM CFD, ver. 12.1, ANSYS Inc., Southpointe, 275
Technology Drive, Canonsburg, PA 15317, 2006.
Koziel, S., Bandler, J. W., and Madsen, K., 2006, Space-
mapping based interpolation for engineering
optimization. IEEE Trans. Microwave Theory and
Tech., 54 (6), pp. 2410–2421.
Koziel, S., Cheng, Q. S., and Bandler, J. W., 2008, Space
mapping, IEEE Microwave Magazine, vol. 9, no. 6,
pp. 105-122.
Koziel, S., 2010a, Shape-preserving response prediction
for microwave design optimization, IEEE Trans.
Microwave Theory and Tech., vol. 58, pp. 2829-2837.
Koziel, S., 2010b, Multi-fidelity multi-grid design
optimization of planar microwave structures with
Sonnet. International Review of Progress in Applied
Computational Electromagnetics, Tampere, Finland,
pp. 719-724.
Labrujere, T. E. and Slooff, J. W., 1993, Computational
Methods for the Aerodynamic Design of Aircraft
Components, Annual Review of Fluid Mechancis, vol. 25,
pp. 183-214.
0 0.2 0.4 0.6 0.8 1
-1
-0.5
0
0.5
1
1.5
x
/
c
-
C
p
Initial
Optimized
0 0.2 0.4 0.6 0.8 1
-0.05
0
0.05
0.1
x
/c
z/c
TRANSONIC AIRFOIL DESIGN BY THE INVERSE METHOD USING VARIABLE-FIDELITY MODELLING
481
Lane, K. A., and Marshall, D. D., 2010, Inverse Airfoil
Design Utilizing CST Parameterization, 48th AIAA
Aerospace Sciences Meeting Including the New Horizons
Forum and Aerospace Exposition, Orlando, Florida.
Leifsson, L., and Koziel, S., 2010, Multi-fidelity design
optimization of transonic airfoils using physics-based
surrogate modeling and shape-preserving response
prediction, Journal of Computational Science, vol. 1, no.
2, pp. 98-106.
Leifsson, L., and Koziel, S., 2011, Inverse Design of
Transonic Airfoils Using Variable-Resolution Modeling
and Pressure Distribution Alignment, International
Conference on Computational Science, Singapore, June
1-3.
Leoviriyakit, K., Kim, S., and Jameson, A., 2003, Viscous
Aerodynamic Shape Optimization of Wings including
Planform Variables, 21st Applied Aerodynamics
Conference, Orlando, Florida, June 23-26.
Robinson, T. D., Eldred, M. S., Willcox, K. E., and Haimes,
R., 2008, Surrogate-Based Optimization Using
Multifidelity Models with Variable Parameterization and
Corrected Space Mapping, AIAA Journal, vol. 46, no. 11.
Tannehill, J. A., Anderson, D. A., and Pletcher, R. H.,
1997, Computational fluid mechanics and heat
transfer, 2
nd
edition, Taylor & Francis.
Queipo, N. V., Haftka, R. T., Shyy, W., Goel, T.,
Vaidynathan, R., and Tucker, P.K., 2005, Surrogate-
Based Analysis and Optimization. Progress in
Aerospace Sciences, 41(1), pp. 1-28.
SIMULTECH 2011 - 1st International Conference on Simulation and Modeling Methodologies, Technologies and
Applications
482