Aerodynamic Optimization of a Morphing Airfoil Using

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American Institute of Aeronautics and Astronautics

1

Aerodynamic Optimization of a Morphing Airfoil Using

Energy as an Objective

Howoong Namgoong*, William A. Crossley† and Anastasios S. Lyrintzis‡

Purdue University, West Lafayette, Indiana, 47907-2023

Recent advances in materials science and actuation technologies have led to interest in

morphing aircraft. The research discussed in this paper focuses upon the shape design of

morphing airfoil sections. In the efforts herein, the relative strain energy needed to change

from one airfoil shape to another is presented as an additional design objective along with a

drag design objective, while constraints are enforced on lift. Solving the resulting

multiobjective problem generates a range of morphing airfoil designs that represent the best

tradeoffs between aerodynamic performance and morphing energy requirements. From the

multiobjective solutions, a designer can select a set of airfoil shapes with a low relative strain

energy that requires a small actuation cost and with improved aerodynamic performance at

the design conditions.

Nomenclature

C d = coefficient of dragC l = coefficient of lift

C m = coefficient of pitching moment

C f = coefficient of skin frictionC p = coefficient of pressure

F i = objective functions f i = shape functions M = free stream Mach number

Re = Reynolds number

x = x coordinate of airfoil

y = y coordinate of airfoilU ij = relative strain energy of airfoil i and airfoil j

ξ i = design variables

ε i = reference drag coefficient

I. Introduction

IGNIFICANT research has been conducted about how birds fly to provide inspiration for improved man-madeaircraft1. Birds can change their wing shape to attain maneuverability and maximum performance in different

flight conditions. Recently, the concept of a morphing aircraft has been introduced 2,3. A morphing aircraft might

have fully deformable airfoil sections to change shape in flight. Several efforts showed that this type of morphingleads to aerodynamic performance benefits for the aircraft4,5. The Mission Adaptive Wing (MAW) program6 of the

1980s provides an example of aerodynamic performance increase from using a variable camber wing. Despite the

aerodynamic advantages, the MAW’s variable camber design was impractical due to the increased complexity and

weight increase from the actuation system used on the demonstrator aircraft. After the MAW program, most of the

morphing aircraft related research focused on light weight actuator device development to reduce the weight penaltyfor changing the shape of the aircraft. Examples of this device development include the Smart Wing7 and

SAMPSON8 programs; however, many design strategies for morphing aircraft, such as multi-objective optimization

including aerodynamic performance and morphing cost (actuator weight), have not been investigated fully.

*Graduate Student, School of Aeronautics and Astronautics, Grissom Hall, Student Member AIAA.

† Associate Professor, School of Aeronautics and Astronautics, Grissom Hall, Associate Fellow AIAA.‡ Professor, School of Aeronautics and Astronautics, Grissom Hall, Associate Fellow AIAA.

S

44th AIAA Aerospace Sciences Meeting and Exhibit9 - 12 January 2006, Reno, Nevada

AIAA 2006-132

Copyright © 2006 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.




2

The concept of a morphing aircraft1 has generated a new design aspect that must be addressed in aerodynamic

configuration design and optimization. A diverse array of optimization techniques have been applied for traditional

aerodynamic design problems. Gradient-based algorithms such as feasible direction9,10, quasi-Newton11 and adjoint

methods12 have been widely used for airfoil optimization. Recently, stochastic optimization methods (e.g. genetic

algorithms13 and simulated annealing14) have also been applied for aerodynamic optimization 15,16.Generally, in developing the objectives for aerodynamic optimization, a multi-point problem formulation17 is

used for airfoil or wing design. A weighted sum of drag coefficients, computed at various design flight conditions,

serves as the objective, and constraints ensure that the lift coefficient matches specified values at each of the flightconditions. The resulting shape has performance that is essentially a compromise over the flight conditions. In the

case of a morphing aircraft, the wing would be able to change its shape during flight. It should be possible to adjust

the wing shape to the best possible shape for any flight condition encountered by the aircraft; this would suggest that

the morphing airfoil could be designed using a series of single-point problem formulations. However, there is anactuator effort or “cost” associated with these shape changes. Thus, the effort required to effect the morphing must

be included in the optimization process.

The contributions of this paper are the formulation of an energy-objective for morphing airfoil design and the

application of a multi-objective approach to minimize both actuator effort and aerodynamic drag. To investigateenergy as an objective for morphing airfoil shape optimization, typical aerodynamics-only design methods and two

different multi-objective design methods were applied to a morphing airfoil design. These approaches highlight how

strain energy can be included as a measure of morphing airfoil actuation effort. The results demonstrate the trade-off between the morphing energy and the aerodynamic performance.

II. Representative Problem

Some recent studies by the US Air Force Research Laboratory (AFRL) have focused upon a high-altitude, long-

endurance aircraft platform18. A notional representation of this concept appears in Figure 1. This aircraft’s designmission includes a 40+ hour loiter segment, during which

the aircraft will experience a significant weight reduction

as it consumes most of its fuel. If the aircraft is intended to

loiter at a constant altitude and constant airspeed, a fixed

geometry wing would not be operating at its most efficientconditions throughout the mission. However, if the aircraft

utilized a wing with morphing airfoil sections, it would be

possible to change airfoil shape throughout the mission inorder to improve the endurance performance of the

aircraft.Based upon systems studies from AFRL, the required

lift coefficients are known at various times during the longloiter segment. To begin the energy objective

investigations, the flight conditions at three points in time

provide the airfoil shape design conditions. These are

summarized in

Table 1. Near the start of the loiter segment, theaircraft’s weight is high, and the required design lift coefficient is also high. Because of the desire for constant

altitude, constant velocity loiter, the Reynolds number and Mach number for all three conditions are the same. Thelow Mach numbers should not require aerodynamic analyses that include wave drag.

Table 1 Airfoil design conditions.

Condition 1 Condition 2 Condition 3Design lift coefficient 1.52 1.18 0.85Mach number 0.6 0.6 0.6

Reynolds number 1.5×106 1.5×106 1.5×106

III. Aerodynamics Approach

For aerodynamic analysis, the well-known XFOIL19 code is selected as a function evaluator because it provides

rapid calculation of lift and drag coefficient. XFOIL is suitable for high Reynolds number airfoil flows. A linear-

Figure 1 Notional high-altitude, long endurance

aircraft concept.




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vorticity panel method with a Karman-Tsien compressibility correction is used for inviscid calculation. While this is

not generally considered a high-fidelity analysis tool, XFOIL is relatively fast, a requirement for design studies.

XFOIL allows for a good resolution of the airfoil shape and incorporates viscous effects.

To pose the airfoil shape optimization problem when using XFOIL, the design variables describing the airfoil

shape are needed. Several possibilities were investigated in Reference 20. The modified Hicks-Henne21,22 shapefunctions were found to represent an airfoil most accurately and are used here. The design variables are multipliers

that determine the magnitude of the shape function as it is added to the baseline airfoil shape. The y-coordinate positions of the upper and lower surface of the airfoil are then described as functions of the x-coordinate position

using the following equation:

∑ξ+= )()()( AirfoilBase x f x y x y ii(1)

where ξi are the design variables; and f i, the shape functions (i =1, 16 here). Figure 2 provides two plots that depicts

the shape functions and a baseline NACA 0012 airfoil along with the upper and lower limits of the airfoil surfaces.

The left plot shows how the eight shape functions vary as a function of the chordwise position; this plot uses values

ξi=1.0. The right plot shows the available design space using the NACA 0012 base airfoil with the modified Hicks-

Henne shape functions. If all ξi are equal to the upper bound of 0.015, the upper bound airfoil shape is described.

Similary, if all ξi are equal to the lower bound of -0.015, the lower bound shape is found. As a result, all possible

shapes for the airfoil lie between these two bounds.

IV. Strain Energy as an Objective

As mentioned previously, a morphing airfoil could theoretically provide aerodynamically optimal shapes at any

flight condition. However, morphing requires some type of mechanisms to effect shape changes. This work uses the

assumption that the energy needed to change the airfoil shape is proportional to both the actuation system weight

and the power needed by the actuation system. The purpose of this research is to account for the extra cost of morphing devices needed to acquire aerodynamic performance benefits, so an additional objective developed here is

to minimize the strain energy associated with changing the shape of the airfoil.

There are several ways to model the strain energy needed to change the airfoil shape23,24. All of these ways usethe basic idea that the strain energy in a structure is proportional to the square of the change in length of the

structure. In this paper, a simple strain energy model has been considered. Eq.(2) presents the internal linear spring

model concept suggested by Prock, et al23.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

f1 f2 f3 f4 f5 f6 f7 f8

0 0.25 0.5 0.75 1

X

-0.1

-0.05

0

0.05

0.1

0.15

Y

NACA0012Maximum LimitMinimum Limit

Figure 2 Modified Hicks-Henne shape functions (left) and design space available using the NACA 0012 base

airfoil (right)




4

∑∑==

Δ=Δ=n

i

i

i

n

i

ii L L

EA Lk U

1

2

1

2

2

1

2

1 (2)

In this equation, U is the strain energy; k i, is the spring constant, EA is the spring axial stiffness, and Δ Li is the spring

deformation. With no real actuation system envisioned as yet, the spring model strain energy objective does not need

to include the EA terms. This model assumes that springs connect the upper and lower airfoil surfaces; as the airfoil

morphs, the springs deform, which corresponds to an amount of strain energy. Figure 3 presents a simpleillustration of this model.

To formulate the objective function, relative strain energy terms, U ij, representing the strain energy associated

with changing from shape i to shape j. Currently, the objective incorporating strain energy seeks to minimize the

maximum relative strain energy value associated with the

shape changes. For the three conditions used here, theobjective appears as given in Eq.(3).

Minimize [Max(U 12, U 23, U 13)] (3)

This current strain energy model does not directlyaccount for an actuation or mechanization strategy to change

the airfoil shapes. As physically realizable actuation

strategies become available, (like those suggested in Refs. 25and 26) models of these strategies should be used to also

assess strain energy or actuation energy in the same

optimization framework as the simple spring model

presented above.

V. Optimization Algorithm

Since its first descriptions, the Genetic Algorithm (GA)

has been applied to many engineering optimization problems. A genetic algorithm is a computational representation

of natural selection observed in biological populations27. A GA has the ability to search highly multimodal,

discontinuous design spaces and also locates designs at, or near, the global optimum without requiring a good initialdesign point. Because the min-max objective formulation will have discontinuous derivatives and because airfoil

shape design problems appear to frequently have local minima, the GA provides a search method that would not be

hindered by these issues.The population size and mutation rate were selected using empirically derived guidelines 28 for GA using

tournament selection and uniform crossover. Seven bits represent each of the 16 shape function multipliers (8 for

upper surface and 8 for lower surface) in the case of a single airfoil design, for a total chromosome length of 112

bits. In this research, 95% BSA(Bit-String Affinity) stopping criteria29

is applied for all aerodynamic-only designs. Normally, obtaining 95% BSA for this problem requires about 300 GA generations.

Using a GA for design optimization can be computationally expensive. To overcome the computational time

problem, parallelization of the GA is needed. Following the approach of Ref. 30, a manager-worker type

parallelization is applied to convert a serial GA into a parallel program, and a Linux-Cluster machine is used for

computation. Because a relatively small communication time is needed for the parallel GA, the code scales well onthe 52-Processor Linux Cluster used in this effort.

VI. Investigations and Results

Two typical aerodynamics-only design problems and the suggested energy-based multiobjective design areinvestigated. The aerodynamics-only problems provide the extremes of the tradeoff between low drag and low strain

energy.

The design variables are used to describe the airfoil shape as shown in Eq.(1). For the single-point and multi- point aerodynamics-only design, 16 design variables are needed. In the cases using strain energy as an objective, a

total of 48 design variables (16 for each of the three flight conditions) are needed to describe shapes for each design

condition. The NACA0012 airfoil is selected as a base airfoil for all shapes. The same modified Hicks-Henne shape

functions and design variable bounds (ξi=± 0.015) that were described previously are used here. Thus, the same

limits on upper and lower surface shapes shown in Fig. 2 are provided.

Internal springs connecting upper and lower surface

ic L )/(Δ

ic L )/(

Springs contract (or expand) to meet new airfoil shape

Figure 3 Internal linear spring model for strain

energy.




5

In optimal airfoil shape design, the angle of attack also can be, and often is, used as one of the design variables,

so that the solution contains the shape and angle of attack needed to minimize drag and meet the lift constraints.

However, in this research, the angle of attack was not a design variable, because the XFOIL has a built in trimming

subroutine to find the angle of attack that satisfies the design lift constraint. This process might reduce the

complexity of the design space and help find a global optimum design, because the number of design variables issmaller. Also, this trimming can get rid of the possibility that two different airfoil shapes at different angles of

attack have same lift coefficient.

A. Aerodynamics Only Investigation

1. Three single point airfoil designs

With no consideration of the energy needed to change themorphing airfoil’s shape, the airfoil would be able to adjust

so that its performance at any given flight conditions would match the result of a single point optimization at the flight

condition. Each problem results in a single shape that

minimizes C d at each condition. Eq.(4) shows the single-

point problem objectives correspond to each of the flightconditions.

Minimizeid C

Subject toill C C = )3,2,1( =i (4)

In Figure 4, the resulting airfoils from the three single-

point optimization runs are superimposed on each other to

demonstrate the type of shape changes that would berequired for morphing the airfoil to meet the three flight

conditions given in Table 1. These are intended to represent

the best possible aerodynamic shapes for the airfoils.

2. Multi-point optimization

The multi-point approach uses the weighted sum of dragcoefficients as the objective function shown in Eq.(5),

following a typical formulation for this problem type17. Thisapproach finds a single fixed geometry shape that

compromises between all three flight conditions. The lift

coefficient constraints are satisfied by trimming the airfoil.This aerodynamics tradeoff results in an airfoil with higher

drag at each specific flight condition compared to the

corresponding single-point optimized airfoils. Because a

single airfoil shape is acquired by this approach, the airfoil

requires no strain energy.

Minimize321

3

1

3

1

3

1d d d C C C ++

Subject toill C C = )3,2,1( =i (5)

Results of the two methods are compared in Table 2. The single-point design results have lower drag than the

multi-point design results, as expected. This is because the single-point designs have only one objective which is to

minimize drag at one flight condition, but the multi-point design solution is a compromise solution over all threeflight conditions. The multi-point solution is a minimum energy solution, because the result of the multi-point

design is a single airfoil. Thus, the relative strain energies in Table 2 are all zero for the multi-point design. The

0 0.25 0.5 0.75 1

X

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Y

DesignCl=0.85

DesignCl=1.18DesignCl=1.52

Figure 4 Airfoil shapes from single-point

optimizations.

0 0.25 0.5 0.75 1

X

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Y

Multi-PointOptimization

Figure 5 Airfoil shapes from multi-point

optimizations.




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relative energies for the single-point solutions shown in Table 2 are calculated following the energy model described

in Eq.(2). Therefore, the values in Table 2 indicate the range of tradeoff available for a morphing airfoil. The three

single-point shapes are the best possible aerodynamic solution, while the multi-point shape is the best possible

energy solution.

Table 2 Drag comparison of aerodynamics only design

Cd 1 Cd 2 Cd 3 U 12 U 23 U 13

single-point 0.005024 0.007015 0.010249 0.00704 0.00189 0.01241

multi-point 0.007356 0.008178 0.010649 0 0 0

B. Energy-Based Investigation

In the design approach used here, strain energy is proposed as another objective to encourage minimum actuationrequirements for morphing airfoils. Because the aerodynamic performance also needs to be included, the problem

becomes multi-objective.

For the multi-objective optimization, the objective function is a vector function whose components are individual

objectives. In a problem with more than one competing objective, there exist a number of solutions called the

Pareto-optimal set31, instead of a single best solution. This research employs two multi-objective methods (i.e.ε -

constraint and N -branch tournament Genetic Algorithm32) to address the strain energy and drag as objectives.

1. ε -constraint approach

For this problem, there is no guarantee that the multi-objective approach will find an aerodynamically better shape than the multi-point aerodynamics only design. To account for this problem, constraints on the drag

coefficient are enforced via a penalty function in the ε -constraint approach. The reference drag coefficient values

(ε i) are chosen between the values of the single-point

results and multi-point results of aerodynamics only

design. By varying the values of ε i for different runs of the

optimizer, different Pareto optimal solutions are found.

The formulation of the ε -constraint approach is described

in Eq.(6).

Minimize1F

Subject toiF ε ≤2

and, jll C C = (j=1,2,3)

Where

( )[ ]

1

1323121

,,max

C

U U U F = ,

2

31

31

31

2321

C

cccF

d d d ++= ,

C 1=C 2=0.01 (6)

Figure 6 shows the Pareto set found from the ε -

constraint approach. In Fig.6, the “energy-based” points

represent the solutions obtained by including the energyobjective function in the multiobjective problem. The single-point and multi-point results of the aerodynamic-onlydesigns are also plotted for comparison. Figure 6 shows that tradeoff morphing airfoil solutions exist between the

two extreme points represented by the multi-point and single-point solutions.

0.70

0.75

0.80

0.85

0.90

-0.2 0.3 0.8 1.3 1.8

F 1 (Energy Objective)

F 2

( D r a g O

b j e c t i v e )

energy-based

multi-pointsingle-point

Figure 6 Pareto set from ε -constraint approach.




7

Figure 7 shows the set of three shapes associated with

one morphing airfoil tradeoff solution selected from the

Pareto-optimal set. Thus, this airfoil set has better

aerodynamic performance than the multi-point design and

also has smaller strain energy than the single-point design.The morphing airfoil set in Figure 7 has a drag objective of

0.765, which is lower than the multi-point drag objective

of 0.864. Also, the morphing airfoil in Figure 7 has anenergy objective of 0.260, which is lower that the set of

single-point shapes of 1.240. This airfoil set shows little

difference in the lower surface to reduce energy, but has

more variation in the upper surface to reduce drag and maintain required C l values.

2. N-Branch Tournament Genetic Algorithm

Many different versions of modified Genetic

Algorithm have been used for multi-objective

optimization. An appropriately modified Genetic

Algorithm approach can generate a large number of

designs that represent the Pareto set for a multi-objective problem with similar computational effort required to

solve a single objective problem with a genetic algorithm.

In this research, the N -Branch Tournament Genetic Algorithm33 is used as a multi-objective Genetic Algorithm. N -branch tournament differs from non-dominance ranking approaches such as Multi-Objective Genetic Algorithm

(MOGA)34 because it uses the selection operator to perform multi-objective design rather than formulation of a

single fitness function. In N -branch tournament selection, designs compete once on a fitness value associated with

each objective. Eq. (7) is the problem statement for this optimization.

Minimize

( )[ ]

( )⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

++=

2

31

31

31

1

132312

321

,,max

C

ccc

C

U U U

F d d d

G

Subject toil lC C = (i=1,2,3)

Where C 1=C 2=0.01 (7)

Figure 8 shows the Pareto set found from one run (1,000 generations) of the N -branch tournament GA. With less

effort, many more tradeoff solutions were acquired from the N -branch tournament GA than from the ε -constraint

method. Instead of using the BSA stopping criteria like in the aerodynamics-only case, here the Pareto set is

assumed to be converged after 1,000 iterations. Few changes were observed in the Pareto set after about 800

generations.

Figure 9 is one of the tradeoff solutions selected from the Pareto set shown in Figure 8. The morphing airfoil in

Figure 9 has a drag objective of 0.765 and energy objective 0.2. The morphing airfoil shapes in Figure 7 and Figure9 and seem similar, but the energy objective of the airfoil shapes in Figure 9 is slightly less. The airfoil shapes in

Figure 9 are much closer to each other than the single-point generated airfoil shapes in Figure 4, which follows thatthe strain energy of the morphing airfoil in Figure 9 is smaller than that associated with the three single-point

shapes.

0 0.25 0.5 0.75 1

X

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Y

Design Cl=0.85

Design Cl=1.18Design Cl=1.52

Figure 7 Airfoil shapes from energy based design

(ε -constraint approach)




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Figures 10-12 present airfoil shapes and C p distributions associated with the three flight conditions. The left-

hand plot shows the single-point shape designed for the given flight condition, the shape of the multi-point design

problem, a morphing airfoil shape obtained via the ε -constraint method (See Fig. 7), and the morphing airfoil shapefor the given flight condition obtained via the N -branch GA (see Fig. 9)

The airfoil shape plots show that the shapes generated using energy as an objective compromise between energy

and drag when compared with the single-point shapes and the multi-point design, because the multi-objective shapes

lie between the two aerodynamics-only designs.One noticeable trend from the airfoil and C p comparison is that, as the design C l increases, the difference of the

shapes decreases. This might be due to the fact that as the design C l increases, it is more difficult to find airfoils that

satisfy the high C l with small drag.

0.70

0.75

0.80

0.85

0.90

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4


F 2

( D r a g O b j e

c t i v e )

N-branchmulti-point

single-point

Figure 8 Pareto set from N-Branch tournament GA.

0 0.25 0.5 0.75 1

X

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Y

Design Cl=0.85

Design Cl=1.18Design C

l=1.52

Figure 9 Airfoil shapes from energy based design (N-

Branch tournament GA).

0 0.25 0.5 0.75 1X

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Y

single-pointmulti-pointε-constraintN-branch

Design Cl=0.85

M=0.6Re=1.5E6

0 0.25 0.5 0.75 1X

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

C p


DesignCl=0.85

M=0.6Re=1.5E6

Figure 10 Airfoil comparison (left), C p distribution comparison (right) (Design lC =0.85)




9

To see the strain energy reductions more clearly, the strain

energy (see Eq.(2)) distribution along the airfoil is plotted onthe NACA0012, which is not the actual designed airfoil shape.

For example, if the control point associated with a location on

the lower surface trailing edge has a large displacement

between two airfoil shapes, a large bar is placed at thecorresponding control point on the NACA 0012 airfoil section.

In this manner, a visual representation can be made to show

which section of the airfoil is associated with the highest strainenergy. In Fig. 15, the strain energy associated with springs at

the rear upper surface of the single-point airfoil have a

significant amount of strain energy associated with moving

from shape 1 to shape 3 as displayed by the large bars in this

area of the airfoil.The single point and energy-based designs are compared.

Figs. 14-15 show that the strain energy (U 23 , U 13) in the multi-

objective designs has been reduced significantly from that of the single point shapes.

0 0.25 0.5 0.75 1X

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Y


Design Cl=1.18

M=0.6Re=1.5E6

0 0.25 0.5 0.75 1X

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

C p


Design Cl=1.18

M=0.6Re=1.5E6


0 0.25 0.5 0.75 1

X

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Y


Design Cl=1.52

M=0.6Re=1.5E6

0 0.25 0.5 0.75 1

X

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

C p


Design Cl=1.52

M=0.6Re=1.5E6


0

0.2

0.4

0.6

0.8

1

0.10

0.10

0.10

0.1

0

1

2

3

4

5

6

7

x 10−4

X

y

E n e r g y

single−point

ε −constraint

N−branch

Figure 13 Strain energy distributions (U 12)




10

VII. Transonic Morphing Airfoil

As another application of the morphing airfoil design strategy described in the previous sections, a transonic

morphing airfoil design is performed. The main difference from the sensorcraft problem is the flight speed regimeof the morphing aircraft. For this application, the speed regime varies from subsonic to transonic, and the altitude

changes from low to high. This application requires a Navier-Stokes code for flow evaluation to capture the physics

of the shock wave in the transonic regime. The importance of the energy-based optimization is increased here,

because a greater shape change is expected for the optimal shapes compared to the low speed sensorcraft problem.Also, small changes in the airfoil shape could have a large impact in the aerodynamic performance when

considering transonic flow.

A. Problem Definition

The flight conditions considered in this research reflect a notional transonic morphing UAV(Unmanned AerialVehicle). This UAV has a multi-mission capability which includes features of a high altitude reconnaissance UAV

and a high-speed combat UAV.

To reduce the computational time while keeping the aspects of morphing airfoil strategy, only two design

conditions were selected for this study. The selected mission segments are the dash and loiter missions; Table 3

presents these flight conditions. These mission conditions are expected to require large shape change for optimalaerodynamic design. The loiter mission is defined based on the Global Hawk UAV35. In addition, this problem

assumes that the aircraft can significantly reduce its wing area for the dash segment. This type of area change was

presented as a goal of the DARPA morphing aircraft structure program3.

Table 3 Transonic Morphing UAV Mission Profile

Mach Altitude ( ft ) C L Re Wing Area

Dash 0.8 5000 0.20 1.98E+07 25%

Loiter 0.6 60,000 1.00 1.77E+06 100%

B. Objective Function

The objective functions are described in Eq.(8). Only one relative strain energy is needed for the two design

conditions.

0

0.2

0.4

0.6

0.8

1

0.10

0.10

0.10

0.1

0

1

2

3

4

5

6

7

x 10−4

X

y

E n e r g y

single−point

ε −constraint

N−branch


0

0.2

0.4

0.6

0.8

1

0.10

0.10

0.10

0.1

0

1

2

3

4

5

6

7

x 10−4

X

y

E n e r g y

single−point

ε −constraint

N−branch





11

minimize :

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+=

21 2

1

2

1

12

d d C C

U

F G

Subject to: 2.01=lC , M =0.8, Re=1.98E7

0.12=lC , M =0.6, Re=1.77E6 (8)

C. Flow Solver, Design Method and Parameters

The TURNS(Transonic Unsteady Rotor Navier-Stokes)36 code is the flow solver for this problem, because it was

readily adapted to this task. The TURNS code was originally developed to research the unsteady flow around arotor blade, however this code can be used for a 2-D airfoil. In the TURNS code, the inviscid flux is calculated

based on an upwind-biased flux-difference scheme originally suggested by Roe37. To acquire higher order accuracy

the Van Leer MUSCLE(Monoton Upstream-centered scheme for the sonservative laws approach)38 is applied withflux limiters. A 129×30 grid arrangement and resolution was found to be a good compromise between accuracy and

efficiency and applied in this problem (see details in Reference 20). The convergence history of this problem

generally showed that after 1500 iterations, the difference of C l and C d between subsequent iterations was less than5

10−

. From this result, an upper limit of 1500 flow solver iterations are used for each solution here to reduce thecomputational cost without paying a great penalty for accuracy. The same parallel GA described in previously isused for this problem, and the number of design variables is 16 for each airfoil shape (for a total of 32 variables

when including energy as an objective). The RAE2822 airfoil is used as the base airfoil, because it is originally

designed to reduce wave drag in transonic flight conditions. The resolution of the GA is set to 5 bits per design

variable for this problem, which is less than the sensorcraft case (7 bits), in order to control computational cost. TheGA parameters for population size (256) and mutation rate (0.00098) are selected following the guidelines of Ref.

28.

D. Aerodynamics-only Result

As before, an aerodynamics-only optimization isconducted to identify the extremes of the tradeoff between

the drag objective and the energy objective. For thistransonic morphing airfoil problem, the best solutions after 100 GA generations are used as converged solutions. This

stopping criterion also limits computational expense. Even

with these steps to limit to compute times, this requires

76,800 function evaluations and 67 hours of computational

wall-time with 30 CPUs. The single point optimizationresults are shown in Figure 16. As expected, the

aerodynamically best airfoils show a large difference in

shape between the two flight condition, compared to the

difference in shape seen for the subsonic sensor craft problem. Figure 17 presents the resulting shape from the

multi-point design case. 0 0.25 0.5 0.75 1X

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Y

Designcondition1Designcondition2

SingleP ointOptimization

Designcondition1 : M=0.6, Cl=1.0, Re=7.9E6

Designcondition2 : M=0.8, Cl=0.2, Re=2.0E7

Figure 16 Best airfoil shape (Single-point

optimization)




12

E. Energy-based Design Results

For this problem, the N -branch tournament GA is

used to generate a Pareto-set. Because computational

time is high for this problem, the classical multi-

objective methods (weighted sum or ε-constraint) are not

affordable with the available computing power. The

Pareto-set found by the N -branch tournament GA after 300 generations is shown in Figure 18. Figure 18 shows

that the converged Pareto set does not have low drag,

high energy solutions and this appears to be the result of low geometric resolution of the possible airfoil sets.

One Pareto front solution point (F 1=0.199, F 2=1.202)

from Fig. 18 is selected, and the airfoils are drawn in

Figure 19. The shapes are relatively close to each other,

compared with the shapes found by the single pointaerodynamic-only design (Figure 16). Also, the scale

difference of the energy objective value in Fig. 8 and

Fig. 18 reflects that the shape change of the solutionairfoil sets is relatively larger than the subsonic airfoil

solution sets of the sensorcraft application. Because the

larger change in shape provided a broader tradeoff between aerodynamic and actuation energy, the importance of the morphing airfoil optimization strategy suggested

in this paper might be increased compared to the subsonic application and suggests that this approach can identify

shapes that could save significant actuation energy.

VIII. Conclusions

Minimization of actuation energy in designing a morphing aircraft is very important to enhance the advantagesfrom the aerodynamic superiority of a morphing aircraft. In this research, a strain energy objective was formulated

and a multi-objective optimization approach was employed to minimize both actuator energy and aerodynamic drag.

Two multi-objective optimization strategies to design an airfoil set for morphing aircraft are applied to a low-

speed, incompressible flow problem and a transonic flow problem. The relative strain energy needed to change

0 0.25 0.5 0.75 1X

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Y

Multi PointOptimization

Designcondition1 : M=0.6,Cl=1.0,Re=7.9E6

Designcondition2 : M=0.8,Cl=0.2,Re=2.0E7

Figure 17 Best airfoil shape (Multi-point

optimization)

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

-0.5 0 0.5 1 1.5 2 2.5 3


F 2

( D r a g O

b j e c t i v e )

Single-point

Multi-pointN-branch

Figure 18 Pareto-front of transonic morphing wing

case

0 0.25 0.5 0.75 1X

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Y

Design condition1Design condition2

EnergyBasedOptimization

Designcondition1 : M=0.6, Cl=1.0, Re=7.9E6Designcondition2 : M=0.8, C

l=0.2, Re=2.0E7

Figure 19 Airfoil shapes from energy based design

( N -Branch tournament GA)




13

from on airfoil shape to another is presented as a design objective along with the drag objective. From this multi-

objective optimization strategy, we found tradeoff designs of low actuation energy and low drag that lie between the

multi-point shape and the set of single-point shapes. Through these optimization processes, engineers can seek a

solution that maximizes the benefits of the morphing technology, while minimizing the actuation cost.

Acknowledgments

The authors would like to thank Jason Bowman and Brian Sanders of the Air Force Research Laboratory for providing information about the high-altitude, long-endurance aircraft. A significant portion of this work was

supported by the Air Force Research Laboratory award number F33615-00-C-3051. The calculations were performed on a 104-node cluster acquired by a Defense University Research Instrumentation Program (DURIP)

grant.

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