Aerodynamic Optimization of a Morphing Airfoil Using ...lyrintzi/AIAA-2006-1324-Nam.pdf ·...

American Institute of Aeronautics and Astronautics

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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 Cd = coefficient of drag Cl = coefficient of lift Cm = coefficient of pitching moment Cf = coefficient of skin friction Cp = coefficient of pressure Fi = objective functions fi = shape functions M = free stream Mach number Re = Reynolds number x = x coordinate of airfoil y = y coordinate of airfoil Uij = relative strain energy of airfoil i and airfoil j ξi = design variables εi = reference drag coefficient

I. Introduction IGNIFICANT research has been conducted about how birds fly to provide inspiration for improved man-made aircraft1. 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 introduced2,3. A morphing aircraft might have fully deformable airfoil sections to change shape in flight. Several efforts showed that this type of morphing leads 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 penalty for 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.

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

AIAA 2006-1324

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


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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 flight conditions. 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 an actuator 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 investigate energy 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 design mission 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 efficient conditions 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 in order 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 long loiter 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, the aircraft’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. The low Mach numbers should not require aerodynamic analyses that include wave drag.

Table 1 Airfoil design conditions.

Condition 1 Condition 2 Condition 3 Design lift coefficient 1.52 1.18 0.85 Mach 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 shape functions 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+= )()()( Airfoil Base xfxyxy ii (1)

where ξi are the design variables; and fi, 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 use the 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.

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

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∑∑==

Δ=Δ=n

ii

i

n

iii L

LEALkU

1

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21 (2)

In this equation, U is the strain energy; ki, is the spring constant, EA is the spring axial stiffness, and ΔLi 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 simple illustration of this model.

To formulate the objective function, relative strain energy terms, Uij, 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, the objective appears as given in Eq.(3).

Minimize [Max(U12, U23, U13)] (3)

This current strain energy model does not directly account for an actuation or mechanization strategy to change the airfoil shapes. As physically realizable actuation strategies become available, (like those suggested in Refs. 25 and 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 initial design 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 guidelines28 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 on the 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 are

investigated. 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

icL )/(Δ

icL )/(

Springs contract (or expand) to meet new airfoil shape Figure 3 Internal linear spring model for strain

energy.


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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 is smaller. 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 the

morphing 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 Cd at each condition. Eq.(4) shows the single-point problem objectives correspond to each of the flight conditions.

Minimize idC

Subject to ill CC = )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 be required 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 drag coefficients as the objective function shown in Eq.(5), following a typical formulation for this problem type17. This approach 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.

Minimize 321 3

131

31

ddd CCC ++

Subject to ill CC = )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 three flight 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

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Figure 4 Airfoil shapes from single-point

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Figure 5 Airfoil shapes from multi-point

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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

Cd1 Cd2 Cd3 U12 U23 U13

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 actuation

requirements 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 (εi) are chosen between the values of the single-point results and multi-point results of aerodynamics only design. By varying the values of εi for different runs of the optimizer, different Pareto optimal solutions are found. The formulation of the ε-constraint approach is described in Eq.(6).

Minimize 1F

Subject to iF ε≤2 and, jll CC = (j=1,2,3)

Where

( )[ ]1

1323121

,,maxC

UUUF = , ( )

2

31

31

31

2321

Cccc

F ddd ++= ,

C1=C2=0.01 (6)

Figure 6 shows the Pareto set found from theε -constraint approach. In Fig.6, the “energy-based” points represent the solutions obtained by including the energy objective function in the multiobjective problem. The single-point and multi-point results of the aerodynamic-only designs 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.

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Figure 6 Pareto set from ε -constraint approach.


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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 an energy 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 Cl 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 ( )[ ]

( )⎪⎪⎭

⎪⎪⎬

⎫

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⎧

++=

2

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,,max

Cccc

CUUU

Fddd

Subject to il lC C= (i=1,2,3)

Where C1=C2=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 Figure 9 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 that the strain energy of the morphing airfoil in Figure 9 is smaller than that associated with the three single-point shapes.

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Figure 7 Airfoil shapes from energy based design

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Figures 10-12 present airfoil shapes and Cp 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 shape for 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 Cp comparison is that, as the design Cl increases, the difference of the shapes decreases. This might be due to the fact that as the design Cl increases, it is more difficult to find airfoils that satisfy the high Cl with small drag.

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Figure 8 Pareto set from N-Branch tournament GA.

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


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To see the strain energy reductions more clearly, the strain energy (see Eq.(2)) distribution along the airfoil is plotted on the 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 the corresponding 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 strain energy. 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 (U23, U13) in the multi-objective designs has been reduced significantly from that of the single point shapes.

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Figure 13 Strain energy distributions (U12)


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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 regime of 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 Aerial

Vehicle). 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 optimal aerodynamic 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) CL 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.

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minimize : ⎪⎭

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+=

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12

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UF

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 a rotor 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 with flux 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 Cl and Cd between subsequent iterations was less than

510− . From this result, an upper limit of 1500 flow solver iterations are used for each solution here to reduce the computational cost without paying a great penalty for accuracy. The same parallel GA described in previously is used 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. The GA 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 is

conducted to identify the extremes of the tradeoff between the drag objective and the energy objective. For this transonic 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 optimization results 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 1

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Design condition 1 : M=0.6, Cl=1.0, Re=7.9E6Design condition 2 : M=0.8, Cl=0.2, Re=2.0E7

Figure 16 Best airfoil shape (Single-point

optimization)


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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 (F1=0.199, F2=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 point aerodynamic-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 solution airfoil 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 advantages

from 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

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Figure 17 Best airfoil shape (Multi-point

optimization)

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Figure 18 Pareto-front of transonic morphing wing

case

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Energy Based Optimization


Figure 19 Airfoil shapes from energy based design

(N-Branch tournament GA)


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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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