[American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures,...

Optimization of Aircraft Aeroelastic Response Using

the Spectral Level Set Method

Alexandra A. Gomes∗ and Afzal Suleman†

IDMEC, Instituto Superior Tecnico, 1049-001 Lisbon, Portugal

In this paper, we propose a new approach to the optimization of aircraft aeroelasticresponse that greatly reduces the number of design variables, thus enhancing the perfor-mance of multidisciplinary design tools. The current method is an extension of the LevelSet Methods, which represent an interface as the zero level set of a function. According tothe proposed formulation, the Fourier coefficients of the level set function are the designvariables assigned to describe the interface.

Two structural topology optimization examples and two applications to aircraft struc-tures are presented. The first two examples deal with the optimal configuration of shortand long cantilevered beams for maximum stiffness. In the first application, a system ofactuators provides morphing capability to an airfoil by operating on its camber to increaselift. The problem consists in determining the airfoil profile that minimizes the power con-sumption while improving the airfoil effectiveness. In the second application, the aileronreversal speed is maximized by applying reinforcements to the upper skin of a wing torsionbox. These four problems demonstrate that the proposed methodology is able to modifythe topology of the interface while using a reduced number of design variables. Otheradvantages of this methodology include the partial avoidance of local non-global minima,by providing a mechanism for nucleation of new holes, and avoidance of checkerboard-likedesigns and sucessive remeshing.

I. Introduction

In this paper, we propose a formulation for solving topology optimization problems, the Spectral Level SetMethodology, that greatly reduces the dimensionality of the design space. It is motivated by the need

to develop ”practical topological optimization of realistic aeroservoelastic systems”,1 and is meant to be afirst step for achieving better aeroelastic designs without the drawback of an increased number of designvariables. The interaction between disciplines in aeroservoelasticity is complex and the number of designvariables in topology optimization is, typically, very large.

Topology optimization in aeroservoelasticity may appear under different circumstances. For instance, itmay consider the layout of control surfaces, their number, form and relative positioning that best suit acertain goal. It can be used to decide changes in the number and pattern of internal wing components, suchas spars and ribs, modifications of the form, number and distribution of actuators and sensors.

In general, standard optimization algorithms search within a fixed topological class. Consequently, theoptimal solution is topologically equivalent to the initial design. In contrast, topology optimization allowsthe optimizer to search for the minimizing configuration among topologically distinct classes. In fact, theoptimal topology is often unknown and therefore the topological approach goes further in achieving betterdesigns by removing the limitations of classical size and shape optimization techniques.

In structural topology optimization, the optimizer is able to change the topology of the set that constitutesthe structure. Consequently, holes may appear or disappear from the initial design. Also, breakages andmerges of the set may occur during the search for an optimal structure.

∗Graduate Student, Department of Mechanical Engineering, Instituto Superior Tecnico, Av. Rovisco Pais, 1040-001 Lisbon,Portugal, and AIAA Student Member.

†Associate Professor, Department of Mechanical Engineering, University of Victoria, P.O. Box 3055, Victoria, BC, Canada,AIAA Associate Fellow.

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American Institute of Aeronautics and Astronautics

46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference18 - 21 April 2005, Austin, Texas

AIAA 2005-1892

Copyright © 2005 by Alexandra A. Gomes and Afzal Suleman. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

To illustrate the typical structural topology problem, consider the boundary design of a cantilever sub-jected to a prescribed load distribution, such that its stiffness is maximized with a given amount of material.To solve this benchmark problem, it is necessary to redistribute the initial amount of material, eventuallyadding or removing some of it in order to satisfy the constraint.

The various approaches to structural boundary design can be categorized into four distinct groups: levelset methods, homogenization, evolutionary structural optimization methods and the bubble method. Withinthe first group, the authors of Ref. 2 put together the level set and immersed interface methods to achievefully stressed structures based on the stress distribution of the current design.

The second approach to the design of structural boundaries is homogenization.3–8 The differential equa-tion regulating the behavior of a strongly heterogeneous material has rapidly oscillating coefficients. Thehomogenization theory replaces these coefficients with less oscillating ones, or even constants, by assumingperiodicity and performing asymptotic expansions of functions which determine the response of the com-posite. The resulting coefficients sum up the effective properties of the composite. In the case of structuraltopological design, the repeated base cell is made of an arrangement of solid material and void space. How-ever, as pointed out in Ref. 2, the homogenization theory might yield large load sensitivity or non-realisticdesigns extremely difficult to manufacture.

Evolutionary structural optimization is another approach to structural boundary design, presented byXie and Steven.9 In their work, the structure is bounded by a larger domain, upon which is imposed a fixedfinite element mesh. The optimization process, guided by weight reduction, moves forward by deleting solidelements of the discretization mesh.

Reverse adaptivity, by Reynolds and co-authors,10 is another technique for topology optimization. Theseauthors achieve quasi-fully stressed designs by removing a percentage of the least stressed portions of thestructure. This approach refines the finite element mesh to get a higher resolution near the structuralboundary.

Finally, Eschenauer, Schumacher, Masmoudi, Sokolowski and co-workers use functions of the stresses, thestrains and the displacements to position void regions of prescribed shape along the structure, thus changingits initial topology. This has been called the bubble method or the topological gradient method.11–15

The spectral level set methodology is based on the level set methods, introduced by Osher and Sethian,16

following previous work of Sethian.17–19 Within the level set methods framework, the relevant boundary orinterface is the zero level set of a function, the level set function. During the optimization, this functionevolves and so does the boundary. In this way, a structural boundary can easily sustain topological changes,establishing the level set methodology as an adequate tool in topology optimization problems.

The spectral level set methodology inherits from the level set formulation the idea of embedding theinterface into a level set function. In this sense, both formulations provide an implicit description of theinterface. However, instead of defining the level set function using its nodal values, the spectral level setmethodology uses the Fourier coefficients of its series expansion. These coefficients become the designvariables of the optimization problem.

One major advantage of the proposed methodology arises from the global character of the Fourier co-efficients in the space domain, which provides the ability to describe a boundary using fewer degrees offreedom than classic approaches to structural topology optimization, thereby improving the performance ofmultidisciplinary design tools. This is particularly useful in the preliminary or initial stages of the designprocess for which fine details of the structure are not the primary concern.

Other advantages of the spectral level set methodology are the partial avoidance of local non-globalminima, by providing a mechanism for nucleation of holes, the avoidance of checkerboard-like designs, andof successive mesh generation. Furthermore, the spectral level set methodology can be successfully appliedto any problem involving the optimization of an interface.

Two benchmark problems in structural boundary design are presented under the spectral level set method-ology framework. The problems consist in the design of short and long cantilevers subject to a verticalload opposite to the fixed end. The goal is to maximize the structural stiffness subject to a solid volumeconstraint.2,6, 9, 10,20 The initial designs are rectangular short and long beams, which do not satisfy theconstraint. The zero level set of the spectral level set methodology defines the structural boundary.

This work proposes two applications of the spectral level set methodology to aeroelastic optimizationdesign. Both of them deal with aileron control reversal and aileron effectiveness.21–25 First, the spectrallevel set methodology is used to design a morphing airfoil which enhances the aircraft roll maneuveringperformance. A system of actuators provides morphing capability by operating on the airfoil mean camber to

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produce additional lift. The optimization problem consists in determining the airfoil profile which minimizesthe power consumption while improving the aileron effectiveness. The sign of the spectral level set functionestablishes the set of driving actuators and its Fourier coefficients constitute the design variables.

A second application of the spectral level set formulation to aeroelastic design consists in the maximizationof the aileron reversal speed by applying reinforcements to the upper skin of a wing torsion box, subject toa constraint in the amount of reinforcing material. In this case, the zero level set of the spectral level setfunction defines the interface between reinforced and non-reinforced skin.

This paper is divided into two main parts: the presentation and discussion of the key ideas behindthe spectral level set methodology, and applications to the design of aircraft structures with improved rollmaneuvering performance.

II. Spectral Level Set Methodology

The spectral level set methodology is a new tool to formulate optimization problems involving the evolu-tion of an interface. It is based on the level set methods introduced by Osher and Sethian.16 The concept itinherits from the level set formulation is the embedding of the interface propagation in the evolution of thelevel set function, which, for reasons that will be clear later on, we renamed as spectral level set function.As with the level set methods, the description of the interface is implicit in the description of that func-tion. In particular, the zero level set of the spectral level set function corresponds to the closed interface.This framework allows to easily handle topological changes in the interface, during the optimization process,without the aid of artificial schemes. In the formulation of the spectral level set methodology, we arbitratethe spectral level set function, ψ, is positive in the interior of the interface and negative on the outside.

The level set methods use the nodal values of the level set function in a neighborhood of the propagatinginterface to discretize the function.26 Under the proposed formulation, the discretization of the spectral levelset function uses the coefficients of a finite Fourier series. These coefficients become the design variablesassigned to the description of the interface within the optimization problem.

In the following, we present the key ideas of the spectral level set methodology. A detailed descriptionmay be found in Ref. 27.

Let ψ : C0 (Tn) → R be the spectral level set function. Since C0 (Tn) ⊂ L1 (Tn), the Fourier coefficientsof ψ are

ψ (k) =1

(2π)n

∫

Tn

ψ (θ) e−ik·θdθ. (1)

The Fourier inversion formula is given by

ψ (θ) =∑

k∈Zn

ψ (k) eik·θ (2)

for ψ defined as a function of θ, with θ ∈ [0, 2π]n. In general, we wish to identify [0, 2π]n with the physicalspace defined by the variable x ∈ Rn. Assume x ∈ [a1, b1] × . . . × [an, bn]. Then, each component of θ islinearly related to the corresponding component of x, through

θi =2π

pixi − 2π

piai (3)

in which pi = bi − ai is the period corresponding to coordinate i.

A. Reduction of the Design Space Dimension

The main advantage of the proposed methodology arises from the global character of the Fourier coefficientsin the space domain. This global character provides the ability to describe an interface using fewer degrees offreedom than classic spacial discretizations techniques. Since the Fourier coefficients are the design variablesof the optimization problem, this results in a reduction of the design space dimension.

The reduction in the number of variables used to describe the structural interface is particularly useful inmultidisciplinary optimization design problems. In fact, a desirable feature in these problems is to achievean acceptable preliminary or initial designs, for which the finer details of the interface are not an issue,in an expeditiously fashion. One way to pursue this goal is to describe the structural boundary using asmall number of design variables. In this way, the spectral level set methodology is expected to enhance theperformance of multidisciplinary design tools.

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B. Error Estimates

Consider the set of functions constructed using a finite Fourier series. A key question in the theory of thespectral level set methodology consists in deciding whether it is possible to describe, with arbitrary accuracy,any interface as the zero level set of one of those functions.

To show the space of functions constructed from a finite Fourier series is sufficiently rich to approximateany interface to any desired accuracy, the first decision concerns the concept of approximation or how tomeasure distances between functions and, eventually, between interfaces. For this, we must settle on a normto use. A good candidate is the sup-norm because it encodes pointwise behavior whereas any other type ofLp-norm does not. The convergence criteria associated with the sup-norm is the uniform convergence, inwhich the rate of convergence is estimated uniformly, as opposed to pointwise convergence.

The infinite number of functions with the same zero level set implies the lack of uniqueness of the levelset function in the description of an interface. Consider the infinite set of all the functions whose zero levelset is the relevant interface. Our goal is to choose one of these functions and then see if its zero level set isadequately represented by the zero level set of a finite Fourier expansion.

The following step is realizing that even if two functions are extremely close in the sup-norm, their zerolevel sets are not necessarily close to each other. In particular, their zero level sets might correspond todrastically different interfaces. Figure 1 gives such an example, in which the problem is the small valuesattained by the three level set functions. However, if those values were large away from the interface, theirzero level sets would indeed be close. This means that to show that a given interface can be described witharbitrary accuracy it suffices to prove that we can approximate arbitrarily well an appropriately chosen levelset function. In particular, one which is large away from the interface.

1

Figure 1. L∞-close level set functions withvery different zero level sets.

Obvious candidates are the indicator function, which is +1on the inside of the interface, −1 on the outside and 0 at theinterface, or the signed distance to the interface. When consid-ering our choice we need to take into account the fact that it isnot possible to approximate discontinuous functions uniformlyby Fourier series.28 The signed distance to the interface is Lip-schitz continuous independently of the degree of regularity ofthe interface and the indicator function is not even continuous.Given the regularity of the two candidates and the above men-tioned convergence results for the Fourier series, we take thesigned distance to the interface as a reference in the estimationof the errors associated with the spectral level set methodology. However, further results from the theory ofthe Fourier series,28 determine that we must increase the regularity of the signed distance to the interfaceto guarantee the convergence of its Fourier series.

The smoothing, also designated mollification,29 results in a C∞ function very similar to the originalexcept at sharp corners. At these points, the original function undergoes an averaging operation. Reference30 provides an upper bound for the error committed in the smoothing procedure, which estimates thedifference between the original and the smoothed level set functions as

|ψε (x)− ψ (x) | ≤ Cε, (4)

in which ψ and ψε are the original and the smoothed spectral level set functions and C > 0 is a constantdepending on the Lipschitz constant of ψ. We conclude the smoothing error is controlled linearly in thefactor ε, which is a measure of the n-dimensional ball we need to smooth ψ out, and therefore it depends onthe degree of smoothness of ψ.

Given the smoothed signed distance to the interface, an estimate of the error committed in using a finiteFourier series expansion, with an a priori specified number of Fourier coefficients, instead of the infiniteseries, is also provided in Ref. 30. This error is the truncation error, an upper bound for which is

||ψε (θ)− ψNε (θ) ||∞ ≤ C

εmNm−n, m > n (5)

where m is arbitrarily large. Here N depends on the number of considered Fourier modes.Finally, the total error bound involved in the proposed methodology, which is the sum of the smoothing

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and truncation error bounds, is

||ψ (θ)− ψNε (θ) ||∞ ≤ C

1

Nm−nm+1

, m > n. (6)

Therefore as m → ∞, the upper bound of the total error is proportional to 1/N . The total error expressesthe distance between an appropriately chosen level set function and one user-specified finite Fourier serieswith a convergent infinite counterpart.

The choice of the signed distance function was arbitrary. In fact, there may be other levels set functionswhich represent the same interface and are better approximated by a truncated Fourier series. This meansour method may perform considerably better that the predicted error bounds.

Another important point to address is the relation between the convergence of the optimization algorithmand the accurate description of the interface. In the end, it is up to the optimization algorithm to searchin the functional space of finite Fourier series for the correct approximation, that is, for the set of Fouriercoefficients that yield a function whose zero level set minimizes the cost functional subject to the problemconstraints.

C. Periodicity and Domain Extension

A pertinent feature of the spectral level set methodology is the choice of a periodic level set function. Inprinciple, the region of interest of the domain of this function is a bounded subset of Rn. In fact, the regionof interest corresponds to the zero level set, which given the physical significance of the interface, be it aflame or a structural boundary, is bounded. But the corresponding level set function is not necessarily ofbounded domain. An example is the signed distance to the interface. If we do not assume periodicity of thelevel set function we have to employ the integral Fourier transform, which might lead to convergence issuesof the integral in a worst case scenario. Therefore, the spectral level set methodology considers a periodicextension of the level set function to Rn, by repetition of a fundamental domain. Given this periodicity, theuse of the Fourier series is fully justified.

The complex exponentials in Eq. (2) imply that the spectral level set function, ψ, is a periodic functionin Rn. That is, the domain of ψ is a successive repetition of the fundamental domain.

In principle, the problem domain, that is, the physical domain where the optimization problem is definedcoincides with the fundamental domain. Consequently, when solving the optimization problem, we areinterested in the restriction of ψ to the fundamental domain. However, the periodicity and smoothness ofthis function generate an undesirable, although not restrictive, trait.

The spectral level function is a smooth periodic function in Rn since it is a finite sum of complexexponentials, which belong to C∞ (Cn). Consider the schematics of figure 2. The periodicity of ψ impliesthat the values of the function close to one boundary influence its values close to the opposite boundary. Inparticular, the value at P is approximately the same as the value at O. Therefore, the values of ψ at O andQ are quite close.

fundamental domain=problem domain

repetition of fundamental domain

PQO

Figure 2. Consequences of thesmoothness and periodicity of ψ.

As an example, suppose the solution to a topology optimization prob-lem is a two-bar structure, in which the bars are cantilevered beams joinedat one end and oriented at 45 with respect to the horizontal. The exis-tence of material near the fixed supports implies the existence of materialon the side opposite the wall, which is a consequence of the regularity andperiodicity of the sum of complex exponentials.

To remove this behavior, we extend the fundamental domain beyondthe boundaries of the physical domain of the optimization problem. Wecall this procedure domain extension. The key point is that any materialleakage, as in the case of the two-bar truss mentioned above, stays confinedto the region between the problem and the fundamental domains. In thissense, the extension should provide the necessary amount of space for thespectral level set function to accommodate to the values it attains closeto the boundary of the next period. Nonetheless, the optimization problem is solved on the problem domain,that is, the objective and constraint functions are computed with data gathered from the problem domain.Once the optimization process is over, we restrict the solution to the problem domain.

One relevant issue concerning the domain extension is its dimension. A rule of thumb is to start with10% of the period pi of the problem domain, and then experiment from there. Our experience is that small

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percentages, in the range of 1%−5%, produce in general unsatisfactory results, as they may not be sufficientfor the function to decay, smoothly, from positive to negative values.

Another helpful tip in determining an adequate domain extension is to consider the structure of theoptimization problem. In this way, the designer may make an educated guess of the form of the solutionnear the problem domain boundaries.

D. Avoiding Checkerboard-like Designs

A common numerical problem in structural topology optimization is the development of checkerboard-likepatterns in the optimal design, corresponding to a sequence of alternating solid and void mesh elements.31

The spectral level set approach intrinsically avoids the formation of such patterns.Within the spectral level set methodology, the first Fourier modes are used as design variables. These

modes correspond to the lower frequencies. Therefore, the high frequency modes which are capable ofdescribing the finer details of the interface are not represented in the expansion of the spectral level setfunction. In this way, the checkerboard patterns are automatically deleted from the proposed methodology,unless an extremely large amount of Fourier modes are considered.

Another relevant feature of the proposed methodology is the ability to achieve easily manufacturabledesigns. This arises as a consequence of the use of the first Fourier coefficients as well as of their globalcharacter.

E. Avoiding Successive Mesh Generation

A relevant advantage of the spectral level set methodology is to avert successive mesh generation duringthe optimization algorithm. This property follows from the works of Xie and Steve9 and Sethian andWiegmann.2 Within the level set framework, the latter authors “avoid the mesh generation step by separatingthe representation of the boundary from the uniform computational grid”.2 Following this reasoning, a fixeddiscretization mesh is imposed on the fundamental domain of the spectral level set function.

F. Formation of New Holes

In two-dimensional problems, the level set approach has a major drawback: although it can easily removeholes, it is unable to create new ones in the interior of the interface.2,32 The lack of a nucleation mechanismis a result of the level set function obeying a minimum principle. The failure of the level set methods innucleating holes constitutes a serious handicap in topology design because the optimization algorithm willprobably overlook topological classes. This behavior reduces the chances of escaping local non-global minimain a topology optimization problem.

Under the framework of the spectral level set methodology, the level set function does not comply to aminimum principle, which makes it less likely to settle in a non-global solution. This constitutes a majorimprovement with respect to other topology optimization methods.

G. Relaxation

-10 -5 5 10 Ψ

0.2

0.4

0.6

0.8

1

Χr

Figure 3. Relaxed indicator asa function of ψ for ar = 10.0 andbr = 0.001.

In its simplest form, structural topology optimization considers the dis-tribution of material in space, that is, which points should have materialand which points should remain void in order to best suit a certain goal.This is a discrete value design problem known as the 0-1 problem. Thelack of existence of a solution to this problem is a well known concern.5,31

One way of finding a solution closed to that of the original 0-1 problemis to use a relaxed formulation. Suppose the Young modulus of the mate-rial employed in structural design problems is multiplied by an indicatorfunction of domain 0, 1, which in the case of the proposed methodologyis dependent on the sign of ψ: it is 1 if ψ > 0 and 0 otherwise. In or-der to relaxed the indicator function, its sharp corners are smoothed andthe vertical lines of the steps are transformed into steep but finite rinsingcurves. To do so we write the relaxed indicator function as

χr (ψ) = arctan (arψ) /π + 0.5 + br, (7)

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in which ar > 0 regulates the stiffness of the relaxed indicator function: the largest its value, the closer χr

is to the original indicator function. Here br > 0 is an offset parameter, which in the case of the structuraltopology optimization problems guarantees that the bilinear form of the elasticity problem is strictly positivedefinite, thus avoiding problems in the computation of the displacement field. Figure 3 shows the χr for thecase ar = 10.0 and br = 0.001.

The relaxed approach turned out to be a beneficial tool in the framework of the spectral level setmethodology. To see this, consider any closed interface. A point in the problem domain is either insideor outside it. In this sense, there is a on-off nature inherent to the tracking of an interface that lends itselfto take advantage of the numerical tools developed for the 0-1 problem. In face of this analogy, we decidedto include the relaxation approach in the implementation of the spectral level set methodology. This impliesthat every access to the spectral level set function is done through a filter of the form of Eq. (7).

III. Structural Topology Optimization

In this section, we present two benchmark structural topology optimization problems solved using thespectral level set methodology.

A. Short Cantilever with Single Load

10 m

24 m

100 N

E = 10 Pa

= 0.3

5

n

x

y

(a) Schematics for the shortcantilever.

0 2 4 6 8 10

-10

-5

0

5

10

(b) Optimalstructure.

Figure 4. Short cantilever problem.

Our first problem consists in determining the layout ofmaterial that minimizes the compliance of a structuresubject to a single load. A constraint stating that thesolid volume fraction should be lower or equal to 20%is added as an upper bound on the amount of material.The design problem, boundary conditions and materialproperties are shown in figure 4(a). The initial designconsists of an all-material plate covering the problem do-main, which does not satisfy the solid volume fractionconstraint. This is a benchmark problem in structuralboundary design with a known solution: the optimal de-sign is a two-bar truss, each truss at 45 with the hori-zontal.2,9, 10,20

Figure 4(b) shows the optimal structure obtained forthe above problem. The details of the optimization pro-cedure are found in Ref. 27.

B. Long Cantilever with Single Load

10 m

16 m

100 N

E = 10 Pa

= 0.3

5

n

x

y

(a) Schematics for the long cantilever.

0 2.5 5 7.5 10 12.5 15

-4

-2

0

2

4

(b) Optimal structure.

Figure 5. Long cantilever problem.

The optimization problem for the longcantilever with a single concentrated loadis similar to the problem stated in theprevious section. The only differences arethe dimensions of the beam and the vol-ume constraint, which considers a solidvolume fraction lower or equal to 40%.However, the optimal structure is, in thiscase, more complex which poses an addi-tional challenge in the solving the prob-lem with fewer design variables than clas-sical approaches to structural topologyoptimization.

Figure 5(a) shows the dimensions of the problem domain for the long cantilever optimization, togetherwith boundary conditions and material properties, and figure 5(b) depicts the optimal structural. The detailsof the optimization process are found in Ref. 27.

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IV. Morphing Airfoil

In this section, we apply the spectral level set methodology to determine an optimal design of a morphingairfoil capable of counteracting the loss of aileron effectiveness.

A. Problem Statement

Consider an airfoil equipped with an aileron deflected downwards to provide additional lift. The flow actingon the aileron generates a moment of force which twists the airfoil nose down, thus reducing the angle ofattack and the aileron incidence. Consequently, the net lift generation is decreased and for the same ailerondeflection, the rolling moment for an elastic wing is less than that for a rigid wing. This loss of aileroneffectiveness is a consequence of the elastic properties of the wing and is dependent on the flight condition.If not accounted for, this condition can lead to a reduction of the vehicle’s performance in roll maneuvers.In case the aileron produces no lift, we say the aileron has reversed. The speed at which this occurs in theaileron reversal speed.

Consider the lift coefficient produced by an aileron deflection on the airfoil of a real wing, ∆cel . Also,

consider the lift coefficient produced in a similar way on a rigid wing, ∆crl . A way to measure aileron

effectiveness consists in evaluating the ratio of both coefficients,21 that is,

aileron effectiveness =∆ce

l

∆crl

. (8)

Ideally, this ratio should be one. However, the elastic proprieties of the wing add a negative contribution to∆cr

l . Considering strip theory, ∆cel can be written as

∆cel = ∆cr

l +∂cr

l

∂αθ, (9)

in which θ is the local wing twist, i.e., the airfoil twist, and α is the angle of attack. Here ∂∆crl /∂α > 0 but

θ < 0.We assume the airfoil is equipped with a system of actuators capable of increasing the local lift coefficient,

∆cal , such that the negative twist contribution is counteracted. Also, the designer is satisfied with

∆cal ≥ −∂cr

l

∂αθ. (10)

The actuation system is comprised of a setup of actuators, with two possible states, activated anddeactivated, distributed along the airfoil chord. To generate ∆ca

l , the system changes the airfoil meancamber line.

The specific characteristics of the actuators as well as the actuation mechanism are not of concern inthis study. The requirement is that each actuator is able to increase the camber at its location by a positivevalue hz. For details on integrated actuation in aeronautical systems, we refer the reader to Ref. 33.

Both the twist θ and ∂cal /∂α are dependent on the flight condition. Thus, ∆ca

l , depends also on the flightcondition. As this condition changes, the airfoil mean camber line adapts itself to produce the required ∆ca

l .The statement of the optimization problem is

minFourier coeff.

V (ψ)Vtotal

(11)

subject to ∆cal +

∂crl

∂αθ ≥ 0,

in which V stands for the volume of active actuation and Vtotal stands for the total volume of actuators.In the framework of the spectral level methodology, the sign of the spectral level set function ψ defines the

state of activation of each actuator. This function is defined along the airfoil chord. The Fourier coefficientsof ψ constitute the set of design variables of the optimization problem. Here, the relevant interface definesthe boundary between activated and deactivated actuators.

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c

x

z

z = zc

V

a

Figure 6. Schematic of general thin airfoiltheory nomenclature.

The effect of camber line change on the airfoil lift coefficientis determined using the general thin airfoil theory.34 Given apoint on the airfoil chord, the mean camber line is defined ashalfway between the upper and lower surfaces describing theairfoil geometry at that point. The general thin airfoil theoryassumes the mean camber line stays close to the chord line andthat the maximum airfoil thickness is small compared to thechord length.

Consider an airfoil with a mean camber line given by thegraph of zc, which is a function of the coordinate along the chord direction, x, as depicted in figure 6. Theconvergence of the integral in Eq. (14) requires zc = 0 at x = 0 and x = c.

According to the general thin airfoil theory, the lift coefficient curve slope is 2π for all airfoils. However,the lift coefficient at a zero angle of attack, cl0 is a function of the mean camber line. In fact, neglecting theeffect of the aileron deflection as an additional camber line shape variation, we have

cl0 = 2∫ π

0

dzc

dx(cos τ − 1) dτ, (12)

in which τ is related to x through the relation

x =c

2(1− cos τ) . (13)

Writing Eq. (12) in the x variable yields

cl0 = 2∫ c

0

dzc

dx

[cos τ (x)− 1]c/2 sin τ (x)

dx. (14)

B. Implementation

1. Initial Design

The initial design is a fully activated actuation system, corresponding to an initial spectral level set functionof ψ ≡ 1.0.

2. Relaxation

Let zcnom describe the nominal camber configuration. Each active actuator modifies the camber at itslocation increasing the local zcnom by a positive value hz. The actuation state is regulated by the sign of thespectral level function ψ: at chord locations where ψ is negative, the actuators are not activated, and thecamber coincides with the nominal camber; at positions where ψ is positive, the camber is incremented byhz. Consequently, the zero level set of ψ determines the distribution pattern of active actuators along theairfoil chord. Accordingly,

zc (x) = zcnom (x) + hz

arctan [aactψ (x)]

π+ 0.5

, (15)

in which hz is multiplied by a relaxed indicator function. The larger parameter aact, the fastest are thechanges from zcnom to zcnom + hz and vice-versa.

In this problem, ∆cal , is generated by changing the mean camber line in agreement with Eq. (14) and

Eq. (15). The additional amount of lift is the difference between the lift produced by the nominal camberand the lift provided by the new camber line, that is,

∆cal =

2hz

π

∫ c

0

d

dxarctan [aactψ (x)] [cos τ (x)− 1]

c/2 sin τ (x)dx. (16)

The computation of V in Eq. (11) was also subject to a relaxation of the form of Eq. (7), with br = 0.0.This was done to prevent numerical instabilities in the optimization algorithm. We write ar = aV , whosevalue is discussed later.

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3. Constraints and Domain Extension

Assume the nominal camber line satisfies the restriction zcnom = 0 at x = 0 and x = c imposed by the generalthin airfoil theory. For zc to comply with the same restriction, Eq. (15) implies ψ should be nonpositive atx = 0 and x = c.

In principle, both these constraints on ψ ought to be included in the optimization problem statement.However, within the framework of the spectral level set methodology, ψ is a periodic function in R. Inparticular, this periodicity implies the value of ψ at x = c + δ is the same as its value at x = 0 + δ, in whichδ is a small positive constant. Furthermore, the smoothness of ψ requires its value at x = c − δ to be veryclose to the value at x = c+δ. Therefore, the value of ψ at x = c is close to its value at x = 0. Consequently,we need only to impose either ψ (x = 0) ≤ 0 or ψ (x = c) ≤ 0, but not both.

The domain extension allows the graph of ψ near one part of the boundary to be different from the graphon the opposite part. In the present application, there is no need for an extension because ψ must have thesame value at x = 0 and x = c.

Suppose we adopt inequality ψ (x = 0) ≤ 0. This constraint is enforced by making x = 0 non-designdomain and imposing a negative value of ψ at that point.

According to our optimization algorithm, constraints integrate the random search as penalizations to theobjective function. For problem (11), the constraint is

g1 (ψ) = −2hz

π

∫ c

0

d

dxarctan [aactψ (x)] [cos τ (x)− 1]

c/2 sin τ (x)dx− ∂cr

l

∂αθ, (17)

being satisfied for g1 ≤ 0.0.

4. Domain Discretization

The airfoil chord, corresponding to the domain x ∈ [0, c] was discretized into 5000 elements.

5. Integration Quadratures

The integral in V1, Eq. (11), was computed using a 3-node Newton-Cotes quadrature. The integral inconstraint (17) was computed using the trapezoidal rule.

6. Optimization Parameters

The optimization algorithm is composed of two steps: a quick random search of the design space followedby a local search with cobyla, a derivative-free optimization tool. The setting of the random search isdescribed in the following.

The objective function is f : Rn → R and the m inequality constraints are gj : Rn → R, withj = 1, 2, . . . , m. The constraints are satisfied for gj ≤ 0.0. Define

f (x) = f (x) + penaltymax 0.0, g1 (x) , g2 (x) , . . . , gm (x) , (18)

in which penalty is a positive user-defined constant chosen at the beginning of the random search module.Also chosen is maxtries, which regulates the termination schedule by specifying the maximum number ofcalls to the module calculating f and gj .

The outline of the random search algorithm considers the following steps:

1. Set maxtries, penalty and the initial design point xold ∈ Rn.

2. Determine f (xold) according to Eq. (18).

3. Find new point xtry ∈ Rn using, for each one of its components,

xtryi = xoldi + factor (rand ()− 0.5) , (19)

in which rand() is a fortran function returning a real pseudo-random number from the uniformdistribution in the range [0.0, 1.0[ and i = 1, 2, . . . , n.

4. Evaluate the objective function and the constraints at xtry and calculate f (xtry) according to Eq. (18).

10 of 22


5. Iff (xtry) < f (xold) , (20)

accept xtry as the new design point, make it xold and go to step 1.; if inequality (20) is not verified,discard xtry and go to step 2..

6. Termination occurs after maxtries tentative xtry have been generated.

In Eq. (19), the pseudo-random numbers are chosen from a uniform distribution in the range [−0.5, 0.5[.The constant factor is used to scale the added random number to the order of magnitude of xold. Functionrand() is seeded the first time it is called.

The current xold is the initial design point to the second step of the optimization algorithm.

random parameter valuemaxtries 500penalty 10000.factor 0.1

cobyla parameter valuerhobeg 0.5rhoend 1.E-07maxfun 40000

Table 1. Morphing airfoil: optimization parameters.

Our main optimization tool is cobyla, an acronym for Constrained Optimization BY Linear Approxima-tions, which is a trust region method for constrained minimization without derivatives developed by M.J.D.Powell.35 No a priori conditions are assumed concerning the degree of regularity of f and gj and the typeof relation, linear or nonlinear, between these and x. In practice, the user must set the number of designvariables, the number of inequality constraints, parameters rhobeg and rhoend, defining the size of thetrust region, an integer variable maxfun that limits the number of calls to the module calculating f and gj ,and must also provide the initial vector of design variables. cobyla’s output shows a parameter maxcv,which stands for MAXimum Constraint Violation.

Table 1 shows the values of the parameters for both the random search and the cobyla algorithm usedin the morphing airfoil example.

7. Convergence of the Integral in ∆cal

During the search for a solution to the optimization problem, ψ is likely to achieve positive values at x = 0and x = c, thus violating one of the assumptions on which the general thin airfoil theory stands. To preventthis occurrence, we write ∆ca

l in the following manner:

∆cal =

4hz

πc2

∫ c

0

arctan [aactψ (x)]− arctan [aactψ (c)] sec2 (τ (x) /2)sin τ (x)

dx. (21)

Implementing ∆cal as above also avoids fast transitions along the chord due to the relaxed indicator

function.

8. hz and ∂crl /∂α θ

The value of hz depends both on the actuator system, in particular, on the specifications of each actuator,on the airfoil-flow system and on the coupling between the two. The derivative ∂cr

l /∂α depends on the airfoilgeometry and on flight conditions. Finally, the twist θ depends on the elastic and geometric characteristicsof the wing and, again, on the flight conditions.

Given a value for ∂crl /∂α θ, hz must be such that is able to comply with the inequality (17) and, at

the same time, small enough to obey the requirements of the general thin airfoil theory. In particular, it isassumed that the camber line almost does not depart from the chord line. Typically, in subsonic regime, zc

has a maximum value of 5% of c.34 Suppose that in our case, the airfoil has a zc maximum of 2.5% of c, andassume the increase in camber, hz, is 1.0% of c. Let c = 1.0 m. Then hz = 0.01 m.

Also, let∂cr

l

∂αθ = −0.1,

which for ∂crl /∂α = 2π corresponds, approximately, to a twist angle of 1. These values of hz and ∂cr

l /∂α θwere found to be adequate to satisfy inequality (17).

11 of 22


9. Statement of the Optimization Problem

In the following, we present the results obtained for the morphing airfoil optimization problem

minFourier coeff.

V1 (ψ)Vtotal

(22)

subject to − 4hz

πc2

∫ c

0

arctan [aactψ (x)]− arctan [aactψ (c)] sec2 (τ (x) /2)sin τ (x)

dx

− ∂crl

∂αθ ≤ 0. (23)

solved under the framework of the spectral level set methodology.

C. Results and Discussion

0.2 0.4 0.6 0.8 1xc

-3

-2

-1

1

2

3

Ψ N=3

(a) Optimal ψ for N = 3.

0.2 0.4 0.6 0.8 1xc

0.2

0.4

0.6

0.8

1

Harctan ë ΨLHxL N=3

(b) Optimal χV (ψ) for N = 3.

0.2 0.4 0.6 0.8 1xc

-2

-1

1

2

3

4

ΨHxL N=6

(c) Optimal ψ for N = 6.

0.2 0.4 0.6 0.8 1xc

0.2

0.4

0.6

0.8

1


(d) Optimal χV (ψ) for N = 6.

0.2 0.4 0.6 0.8 1xc

-3

-2

-1

1

2

ΨHxL N=9

(e) Optimal ψ for N = 9.

0.2 0.4 0.6 0.8 1xc

0.2

0.4

0.6

0.8

1


(f) Optimal χV (ψ) for N = 9.

Figure 7. Morphing airfoil: optimal configurations foraV = aact = 10000.0.

The iterative approach in the square radius Nproved to be a good strategy in topology optimiza-tion problems.36 We decided to follow this approachto determine an optimal configuration for the activeactuators. The initial value for the square radiuswas chosen to be N = 3.

The initial design corresponded to an all-activesystem of actuation. For N = 3, we precededcobyla with a random search in the design space.The resulting Fourier coefficients were consecutivelyused to initialize simulations N = 6, 9, without ran-dom search.

The problem was solved for aV = aact = 10000.0and aV = aact = 100000.0. Table 2 and table 3show the values of the optimal objective function J ,Eq. (22), and the maximum constraint violation forthe corresponding optimal configuration, maxcv,obtained for the conducted simulations. Figure 7and figure 8 depict the spectral level set function ψ,as well as the relaxed indicator function in Eq. (15),for each set of parameters.

For each pair (aV , aact), J decreases with in-creasing N , which corresponds to a progressive re-duction of the width of the set of active actuatorsshown particularly in figure 7. In fact, a larger Ncorresponds to a more detailed design space. In thissense, the minimum obtained for the largest N is arefinement of the minimum for the smaller N .

J maxcv

initial 9.99937E-01 9.99990E-01N = 3 2.27479E-01 0.0N = 6 1.88164E-01 6.70552E-08N = 9 1.45582E-01 1.11759E-07

Table 2. Morphing airfoil: quantitative re-sults for aV =aact=10000.0.

J maxcv

initial 9.99941E-01 9.99990E-01N = 3 2.50797E-01 6.70552E-08N = 6 2.22298E-01 0.0N = 9 2.22020E-01 0.0

Table 3. Morphing airfoil: quantitative re-sults for aV =aact=100000.0.

An attentive study of the results in figure 7 and figure 8 shows that the set of active actuators is beingdrawn to x = c. To investigate this behavior further, we look at the integral in the inequality (23). A wayto minimize the actuation while still provide additional lift would be to concentrate all the active actuators

12 of 22


near the trailing edge. In fact, the closer to the trailing edge, the less active actuators are needed. A limitingcase would be to set a spike-type of actuation very close to the trailing edge, since we cannot have actuationat x = c because of constraint ψ (x = c) ≤ 0.0. This means that the interface is evolving according to ourpredictions.

0.2 0.4 0.6 0.8 1xc

-2

-1

1

2

3

4

Ψ N=3

(a) Optimal ψ for N = 3.

0.2 0.4 0.6 0.8 1xc

0.2

0.4

0.6

0.8

1


(b) Optimal χV (ψ) for N = 3.

0.2 0.4 0.6 0.8 1xc

-2

-1

1

2

3

4

Ψ N=6


0.2 0.4 0.6 0.8 1xc

0.2

0.4

0.6

0.8

1



0.2 0.4 0.6 0.8 1xc

-2

-1

1

2

3

4

Ψ N=9


0.2 0.4 0.6 0.8 1xc

0.2

0.4

0.6

0.8

1



Figure 8. Morphing airfoil: optimal configurations foraV = aact = 100000.0.

The next step was to feed the initial con-figuration of figure 9 (a),(b) with N = 3 andaV = aact = 100000.0 to cobyla. This simulationintended to determine whether the spectral levelset formulation was capable of achieving a spike-like function if the initial condition was set closeenough, given a small number of Fourier modes. Fig-ure 9 shows the obtained optimal configuration forN = 3, as well as the optimal configuration withN = 6 considering the solution for N = 3 as theinitial design. In accordance with our predictions,those graphs depict a connected set whose width de-creases from N = 3 to N = 6 and which gets closerto x = c with increasing N . The quantitative resultsare displayed in table 4. These show an optimal ac-tive set of actuators corresponding to 0.5% of thechord.

A final experiment considers the initial config-uration in figure 10 (b), N = 6 and aV = aact =100000.0. The value of the initial objective functionis 0.5, with verified constraints, and the final valuefor the objective function is J = 3.89045E-02, alsowith verified constraints. The spectral level set for-mulation was capable to achieve the one-spike con-figuration depicted in figure 10 (d). In conclusion,for topologically different initial configurations, thespectral level set methodology was able to producea very narrow set of active actuators close to x = cusing a very small number of Fourier modes.

J maxcv

initial 1.52669E-01 0.0N = 3 1.59705E-02 0.0N = 6 5.28274E-03 4.44800E-06

Table 4. Morphing airfoil: quantitative results for figure 9.

V. Reinforced Wing-Box

The determination of the optimal topology of the reinforcement of a structure is a natural extension ofstructural topology optimization design. In this section, we use the spectral level set methodology to deter-mine the reinforcement layout of a wing torsion box for increased aircraft roll maneuverability. In particular,we address the problem of aileron reversal.23 We initialize the optimization problem with configurationsbelonging to different topology classes.

A. Problem Statement

Suppose a straight wing of planform S, span 2l and chord c is subject to an antisymmetrical lift distributionwhich results from differential aileron deflections. Consider the following system of coordinates: the x-axispoints to the aircraft’s rear and is defined along the root of the half wing; the y-axis starts at the root of

13 of 22


0.2 0.4 0.6 0.8 1xc

-6

-4

-2

2

Ψ

(a) Initial ψ.

0.2 0.4 0.6 0.8 1xc

0.2

0.4

0.6

0.8

1Harctan ë ΨLHxL N=3

(b) Initial χV (ψ).

0.2 0.4 0.6 0.8 1xc

-5

-4

-3

-2

-1

Ψ N=3


0.2 0.4 0.6 0.8 1xc

0.2

0.4

0.6

0.8



0.2 0.4 0.6 0.8 1xc

-5

-4

-3

-2

-1

Ψ N=6


0.2 0.4 0.6 0.8 1xc

0.2

0.4

0.6

0.8



Figure 9. Morphing airfoil: generation of a spike-like func-tion.

0.2 0.4 0.6 0.8 1xc

-2

-1

1

2

Ψ

(a) Initial ψ.

0.2 0.4 0.6 0.8 1xc

0.2

0.4

0.6

0.8

1Harctan ë ΨLHxL

(b) Initial χV (ψ).

0.2 0.4 0.6 0.8 1xc

-8

-6

-4

-2

Ψ


0.2 0.4 0.6 0.8 1xc

0.2

0.4

0.6

0.8

1Harctan ë ΨLHxL


Figure 10. Morphing airfoil: generation of a spike-like func-tion from a topological distinct initial configuration.

14 of 22


the half wing, points to its tip and is defined along the wing’s elastic axis. Figure 11 shows this system ofcoordinates and the elastic twist θ. An expression of the aileron reversal dynamic pressure, qR, for this wingbased upon simplifying assumptions is23

q2R = − Clδ

1Sl

∫ l

0a0θ (y) cydy

, (24)

for a unit aileron deflection, in which Clδ is the derivative of the lift coefficient of the rigid airplane withrespect to the aileron deflection δ and a0 is the local two-dimensional slope of the lift coefficient curve. Thedynamic pressure in Eq. (24) corresponds to a speed at which the lift increment due to an aileron deflectionis zero. An increment in the deflection of the aileron would result in a negative lift.

1. Wing-Box Model

V

y

x

z

q

skin

skinfro

nt spar

Figure 11. Schematic of the wing-boxmodel.

One way to improve the performance of the aircraft in roll maneu-vering is to maximize the aileron reversal dynamic pressure. In thisexample, we maximize q2

R, which depends on the wing’s internalstructure through θ. Also, in a preliminary analysis, the resistance toaxial and bending loads is carried by the wing skin and its stringers,whereas torsional and shear loads are supported by the shear stressesdeveloped in the wing skin and spar webs.

A simple idealization of a wing subject to torsional loads forwhich the internal structure is perceived is a wing-box model com-posed of two spars and upper and lower skins between two consec-utive ribs. No stringers or spar flanges were added to the model. Figure 11 shows a schematic of the wingtorsion box model.

To simulate rigid ribs perpendicular to the y axis, we imposed suitable boundary conditions to the wing-box model.37 The four components, two spars and two skins sections, were modelled as four plates fixedat y = 0.0 m. At the opposite end, that is, at y = l, the rotations in turn of the two axes perpendicularto the span direction were set to zero. Also, we defined a rigid body as the set of all the points at y = l.We applied a negative torque in the y direction of magnitude 250000 Nm to this rigid body to simulate theaerodynamic load.

The wing torsion box is 1.0 m along the span, 2.5 m along the x direction and 0.14 m in height. Bothspars have a constant thickness of 0.04 m and the lower skin has a thickness of 0.02 m. The nodes in theupper skin which also belong to the front and aft spar have a constant thickness of 0.04 m and are considerednon-design domain.

2. Statement of the Optimization Problem

Given a prescribed amount of material, the optimization problem consists in maximizing qR in Eq. (24),through the reinforcement of the wing’s upper skin. The zero level set of the spectral level set function, ψ,defines the thickness variation over the upper skin: if the function is positive then the thickness is maximum;if the function is negative, the thickness is minimum. The Fourier coefficients of the spectral level set functionare the design variables of the optimization problem.

In Eq. (24), an aileron-down deflection of δ generates a nose-down twist of the wing, which correspondsto a negative θ according to figure 11. All the other parameters in Eq. (24) are positive. Consequently, theoptimization problem can be set up as

minFourier coeff.

−∫ l

0

θ (y) ydy (25)

subject toVreinf (ψ)

Vtotal≤ 0.3 (26)

elasticity problem,

in which a0 and c in Eq. (24) have been considered constants along the span. Here, Vreinf is the volume ofthe reinforcing material and Vtotal is the volume of the original upper skin.

15 of 22


The twist angle θ distribution along the span is the solution of the elasticity problem defined for thewing-box structure.

In this problem, the design domain is the upper skin of the wing-box. It has a minimum thickness of0.02 m and a maximum thickness of 0.04 m.

In this example, we are particularly interested in the ability of the spectral level set methodology inhandling initial conditions belonging to different topological classes. Given two subsets Ω1 and Ω2 of Q,Ω1 is topological equivalent to Ω2, that is, Ω1 and Ω2 belong to the same topological class, if there exist acontinuous mapping m with continuous inverse from Q to Q, such that, m (Ω1) = Ω2. To this end, we havechosen six initial reinforcements layouts with five different topologies.

B. Implementation

1. Initial Designs

Five of the six initial layouts are depicted in figures 12 (a), 13 (a), 17 (a), 18 (a) and 19 (a). The fifth consistsof an all-reinforced upper skin, which does not satisfy constraint (26). Except for the designs in figures 12 (a)and 17 (a), which are in the same class, all the other configurations belong to different topological classes.

The designs of figures 12 (a) and 13 (a) are educated guesses for the optimal configuration based on thefollowing considerations on the physics and symmetry of the optimization problem: the boundary conditionsimposed on the wing-box structure will result in a twisting distribution which increases with y. This means θachieves a maximum value at the maximum value of y. Accordingly, there are two reasonable configurationsfor the pattern of the reinforcing material. The configuration of figure 12 (a) consists in an evenly distributionof the maximum amount of reinforcement material along the neighborhood of the chord defined by y = `. Inthis way, the reinforced stations become stiffer and therefore the twisting of that part of the wing decreasesin comparison with the non-reinforced wing-box.

The other interesting configuration, depicted in figure 13 (a), is a symmetric distribution of the maximumamount of reinforcement material equally between the leading and the trailing edges. With this type ofreinforcing distribution, the leading and trailing edges twist less and therefore there is a chance the objectivefunction achieves a minimum value.

2. Constraints

According to our optimization algorithm, the constraint of Eq. (26) integrates the random search as apenalization to the objective function. For this problem, we write

g1 (ψ) = Vreinf (ψ)− 0.3Vtotal. (27)

The optimal spectral level set function must satisfy g1 (ψ) ≤ 0.0.

3. Domain Extension

The first initial designs to be experimented with were the configurations in figures 12 (a) and 13 (a). In thecase of figure 13 (a), there is no need for domain extension in either directions since the sign of ψ on oppositepoints of the boundary is the same. For figure 12 (a), a domain extension could be applied along the spancoordinate. However, during testing, we found that the best results obtained for that initial configurationcorresponded to a reinforcement similar to the one of figure 13 (a). For this reason, we decided not to applya domain extension in the case of the reinforced wing torsion box.

4. Relaxation

According to the proposed methodology, the spectral level set function ψ integrates optimization problemsthrough a relaxed indicator function. In this example, the relaxed formulation affects the computation ofthe objective function and the determination of Vreinf in Eq. (27).

To compute the objective function, in particular, the distribution of the twist angle over the wing’s span,we need to provide the elasticity solver with information on the layout of the reinforcements over the upperskin. The thickness at each node of the upper skin discretization mesh is

χt (ψ) = [arctan (atψ) /π + 1.5] bt, (28)

16 of 22


in which at regulates the smoothness of the indicator function and bt is the minimum thickness value.The determination of Vreinf is also subjected to a relaxed approach, thus converting the constraint function

in Eq. (27) into a differentiable function of ψ. The relaxed indicator function, χV , defined as Eq. (7) withar = aV and br = bV = 0.0, constitutes the integrand of the integral used to compute Vreinf.

5. Elasticity Solver and Finite Element Mesh

The elasticity problem solver used for computing the twist angle θ was abaqus/Standard 6.3 c©.38

The wing-box model is composed of four plates: two 2.5 m× 1.0 m plates for the upper and lower skins,and two 1.0 m × 0.14 m for the front and rear spars. The finite element mesh for each skin is composed of250 square 0.1 m× 0.1 m elements. Each spar grid has 40 rectangular 0.1 m× 0.035 m elements.

The plates are discretized using thick shell elements S8R from the abaqus/Standard 6.3 c© library. Thisis a reduced integration 8-node element whose nodal thickness option allows thickness variation over theelement. There are six degrees of freedom at each node: three displacements along the three coordinate axesand three rotations about those axes.

6. Integration Quadratures

The integration quadrature for the objective function is the trapezoidal rule. To compute the volume of thereinforcements, we use a 9-node Newton-Cotes quadrature, in which the value of ψ at the element middlepoint is interpolated from ψ at the other eight nodes.

7. Optimization Parameters

The values of the parameters regulating the random search and cobyla are displayed in table 5.

random parameter valuemaxtries 500penalty 1000.factor 0.1

cobyla parameter valuerhobeg 0.5rhoend 1.E-05maxfun 40000

Table 5. Wing-box: optimization parameters.

8. Computation of θ

At each wing station y, the twist angle θ was computed using the displacement in the z direction of the nodesin the front and aft spars defined by z = 0 m. Given one pair of corresponding spar nodes, the magnitudeof the twist is calculated dividing the average of the absolute value of z by half the station chord. The signof the twist is given accordingly to the sign of the load acting on the wing-box.

C. Results and Discussion

Except for the all-reinforced upper skin initial condition, the optimization algorithm did not include therandom search of the design space. The value of the objective function at the obtained minimizer is denotedby J and the constraint violation at that point is maxcv.

J (E-07 m2) maxcv (m3)initial 424.450 7.73288E-02

at = aV = 100.0 375.835 2.68221E-07at = aV = 1000.0 375.466 4.47035E-07at = aV = 10000.0 375.428 2.08616E-07

Table 6. Wing-box: quantitative results for initially rein-forced tip.

J (E-07 m2) maxcv (m3)initial 378.235 0.0N=3 374.328 2.05636E-06N=2 373.930 8.94070E-08N=1 374.222 8.04663E-07

Table 7. Wing-box: quantitative resultsfor initially reinforced leading and trailingedges; at = aV = 100.0.

17 of 22


Figure 12 (b) depicts the optimal design for N = 1 with the initial configuration of figure 12 (a). Thisconfiguration was obtained with an iterative approach on parameters at and aV , that is, figure 12 (a) wasthe initial design for cobyla with at = aV = 100.0 whose result was introduced in cobyla with at = aV =1000.0, and finally, the resulting coefficients were the initial variables to cobyla and at = aV = 10000.0.Table 6 shows the quantitative results for these simulations.

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

(a) Initial configuration.

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

(b) Optimal configuration.

Figure 12. Wing-box: design of initially reinforcedtip.

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

Figure 13. Wing-box: initial configu-ration of reinforced leading and trailingedges.

Next, we considered the initial design of figure 13. Figures 14, 15 and 16 show the optimal reinforcementconfigurations for N = 1, 2, 3 with at = aV = 100.0, at = aV = 1000.0 and at = aV = 10000.0, respectively.Tables 7, 8 and 8 show the quantitative results for those simulations.

J (E-07 m2) maxcv (m3)initial 378.428 0.0N=3 373.285 6.85453E-07N=2 373.778 2.68221E-07N=1 377.018 2.98023E-08

Table 8. Wing-box: quantitative results for initiallyreinforced leading and trailing edges; at = aV = 1000.0.

J (E-07 m2) maxcv (m3)initial 378.451 0.0N=3 373.706 0.0N=2 373.633 0.0N=1 374.750 0.0

Table 9. Wing-box: quantitative results for initiallyreinforced leading and trailing edges; at = aV =10000.0.


at = aV = 100.0 376.856 0.0at = aV = 1000.0 374.954 0.0at = aV = 10000.0 373.569 0.0

Table 10. Wing-box: quantitative results for initiallyreinforced leading edge.


at = aV = 100.0 373.867 1.16229E-06at = aV = 1000.0 373.561 6.25849E-07at = aV = 10000.0 373.528 1.16229E-06

Table 11. Wing-box: quantitative results for cross-like initial pattern.

Consider the reinforcement layout of figure 17 (a), corresponding to a reinforced leading edge as the initialconfiguration. Figure 17 (b) displays the optimal solution obtained for N = 1 following an iterative approachon parameters at and aV : first, 100.0, then 1000.0 and finally 10000.0. Table 10 shows the quantitative resultsof those simulations.

Suppose the cross-like layout of figure 18 (a) is the initial configuration. Figure 18 (b) depicts the optimalsolution obtained after an iterative procedure in parameters at and aV , running from 100.0 to 10000.0 forN = 1. Table 11 displays the corresponding quantitative results.

Figure 19 (a) displays a 3-diagonal stripe initial layout, and figure 19 (b) shows the corresponding optimalconfiguration, which was obtained for N = 2 and an iterative process in parameters at and aV , running from

18 of 22


0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

(a) N=3.

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

(b) N=2.

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

(c) N=1.

Figure 14. Wing-box: optimal configurations for initially reinforced leading and trailing edges; at = aV = 100.0.

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

(a) N=3.

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

(b) N=2.

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

(c) N=1.

Figure 15. Wing-box: optimal configurations for initially reinforced leading and trailing edges; at = aV = 1000.0.

100.0 to 1000.0, with two consecutive simulations with at = aV = 1000.0. Table 12 provides an account ofthe respective quantitative results.


at = aV = 100.0 385.621 1.22190E-06at = aV = 1000.0 374.308 2.08616E-07

Table 12. Wing-box: quantitative results for diagonal stripes initial pattern.

Finally, we consider the all-reinforced upper skin as an initial configuration. The results for N = 1,at = aV = 1000.0 and factor=1.0 as the random search parameter are displayed in figure 20 and table 13.

In view of the optimal designs obtained from initial configurations belonging to different topologicalclasses, we can conclude that the proposed methodology successfully yielded the same reinforcement config-uration in all cases. Moreover, most of the times, the results were attained with N = 1, that is, with fivedesign variables.

VI. Conclusion

In this paper we proposed the optimization of aircraft aeroelastic response using the spectral level setmethodology. According to this formulation, an interface is described as a level set of a function. As thisfunction evolves, during the optimization process, topological changes of the interface may occur. By usingthe Fourier coefficients of the level set function as design variables, a major reduction of the design space

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0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

(a) N=3.

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

(b) N=2.

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

(c) N=1.

Figure 16. Wing-box: optimal configurations for initially reinforced leading and trailing edges; at = aV =10000.0.

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1


0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1


Figure 17. Wing-box: design for initially reinforcedleading edge.

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1


0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1


Figure 18. Wing-box: design for cross-like initial pat-tern.

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1


0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1


Figure 19. Wing-box: optimal design for diagonalstripes initial pattern.

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

Figure 20. Wing-box: optimal design for all-reinforced upper skin initial configuration.

J (E-07 m2) maxcv (m3)initial 2110.59 -

initial (no penalty included) 361.375 -at = aV = 1000.0 374.141 9.23872E-07

Table 13. Wing-box: quantitative results for all-reinforced upper skin initial configuration.

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dimensionality is achieved. Other advantages of the proposed methodology are the partial avoidance of localnon-global minima, by providing a mechanism for nucleation of holes, the avoidance of checkerboard-likedesigns, and of successive mesh generation.

The first numerical example consisted in designing a morphing airfoil that minimizes the power con-sumption while improving the aileron effectiveness. A system of actuators provided morphing capability byoperating on the airfoil mean camber to produce additional lift. This problem showed the spectral level setmethodology can adequately handle the occurrence of spike-like functions during the optimization processwith a very small number of Fourier coefficients.

The spectral level set methodology was also used to study the optimal design of a reinforced wingstructure for increased aircraft roll maneuverability. To this end, the aileron reversal speed was maximizedby applying a limited amount of reinforcing material to the upper skin of a wing torsion box. The spectrallevel set methodology successfully attained the same topological optimum design for initial configurationsbelonging to different topology classes.

Acknowledgments

Alexandra A. Gomes thanks the financial support of Fundacao para a Ciencia e a Tecnologia, Programapraxis xxi, and of the European Social Fund under Community Support Framework III.

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