Multiobjective Topology Optimization for a Multi-layered Morphing FlapConsidering Multiple Flight Conditions*

Keita KAMBAYASHI,1) Nozomu KOGISO,1)† Takayuki YAMADA,2) Kazuhiro IZUI,2)

Shinji NISHIWAKI,2) and Masato TAMAYAMA3)

1)Department of Aerospace Engineering, Osaka Prefecture University, Sakai, Osaka 599–8531, Japan2)Department of Mechanical Engineering and Science, Kyoto University, Kyoto, Kyoto 615–8520, Japan

3)Aeronautical Technology Directorate, Japan Aerospace Exploration Agency, Mitaka, Tokyo 181–0015, Japan

A morphing wing can deform its geometrical shape seamlessly and continuously to improve aerodynamic perform-ance. In our previous study, a multi-layered compliant mechanism composed of stacked compliant mechanisms was pro-posed as the internal structure of the morphing flap to improve the design flexibility of the deformation shape. Each layerhas an independent structural configuration that can be supported under an independently applied load. By connectinglayers throughout the wing skin, the multi-layered compliant mechanism is worked as a single morphing flap. This studyexpands the design flexibility of the multi-layered compliant flap to achieve different morphing shapes by considering mul-tiple flight conditions. For this purpose, load cases with different actuation forces according to the flight conditions areintroduced and the design problem is formulated using a multiobjective optimization problem. Through numerical exam-ples, several Pareto configurations of the multi-layered flap are demonstrated to facilitate deformation to yield differentdesired morphing shapes to maximize the lift-to-drag ratio or maximum lift coefficient. In addition, the trade-off betweenthe two objective shapes is examined to investigate the effects on the values of these functions and the resulting config-urations.

Key Words: Morphing Wing, Airfoil, Multi-layered Compliant Mechanisms, Multiobjective Topology Optimization

1. Introduction

Amorphing wing1) is a wing that can adaptively deform itsgeometrical shape to improve aerodynamic performance de-pending on the flight conditions, thereby achieving economicand ecological flight. Several morphing technologies areclassified into three categories: planform morphing, airfoilmorphing and out-of-plane morphing. These morphing tech-nologies have been researched in national and internationalprojects in the USA2,3) and Europe.4) In Japan, several similarstudies have been pursued. For example, Kawasaki HeavyIndustries, Ltd. and Nippi corporation have investigatedthe morphing wing as an environment-friendly aircraft tech-nology supported by the Society of Japanese AerospaceCompanies (SJAC).5) Recently, the Japan Aerospace Explo-ration Agency (JAXA) commenced research on morphingwing technology as part of an economical and ecologicaltechnology project (i.e., ECO-wing project).6) Recent studiesconducted for morphing technologies are summarized in arecently published review paper.7) With regard to studieson morphing wings, optimization techniques including mul-tidisciplinary design optimization have been widely utilizedin the fields of aerodynamics, structures, and aeroelasticitydesigns.8)

This study focuses on the deformation of the airfoil shapewith the objective of realizing a morphing flap as a high-lift

device by deforming the trailing-edge, as shown in Fig. 1.Though the authors consider compliant mechanisms to real-ize the morphing deformation, other deformation conceptshave been proposed. For example, Yokozeki et al.,9) pro-posed a corrugated structure composed of CFRP as a morph-ing flap. The feasibility of the corrugated structure wasdemonstrated through wind tunnel tests. A compliant mech-anism10) is adopted in this study as the internal structure of amorphing flap to achieve the desired morphing deforma-tion,3,11) where a compliant mechanism is known as the inte-grated structure that utilizes elastic deformation without us-ing mechanical link or joint components. Several otherstudies on morphing wings using compliant mechanismshave been pursued to realize morphing deformation at theleading- or trailing-edges of an airfoil section.12,13) Most ofthese studies adopted topology optimization based on a

(a) Conventional flap

(b) Morphing flap using compliant mechanism

Fig. 1. Conventional and morphing flaps.

© 2020 The Japan Society for Aeronautical and Space Sciences+Received 11 April 2019; final revision received 10 October 2019;accepted for publication 12 December 2019.†Corresponding author, kogiso@aero.osakafu-u.ac.jp

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ground structure approach, where the cross-sectional area ofthe allocated discrete truss elements is defined as a designvariable.

To improve the design flexibility of the deformationshape, the topology optimization method based on continu-um mechanics is widely utilized to obtain an optimum struc-tural configuration of compliant mechanisms.14,15) The au-thors adopted the level set-based topology optimizationmethod16) to obtain the internal structure of the morphingflap as a compliant mechanism.17) The study concluded thatdesign flexibility is not sufficient to achieve several desiredmorphing shapes. Subsequently, the authors proposed amulti-layered compliant mechanism for the internal structureof the morphing flap to improve the design flexibility of thedeformation shape as shown in Fig. 2.18)

In our investigation of the multi-layered compliant mech-anism, each layer has an independent structural configurationthat can be supported under an independently applied load.However, the compliant mechanism should be operated asa single morphing flap by connection through the wing skin.To perform the optimization with a reasonable computationaleffort, the optimization problem is formulated using a two-dimensional structural model, where the deformation of eachlayer is regularized at the wing skin to coincide with eachother. The deformation flexibility of the multi-layered com-pliant mechanism has been demonstrated using numericalexamples.

In our previous research on the multi-layered compliantmechanism, any optimum configuration achieved can onlybe utilized for a predetermined single morphing shape suchas the lift-to-drag ratio maximization shape or the lift coeffi-cient maximization shape,18) although it was important todemonstrate the efficiency of the multi-layered configuration.However, in actual situations, an optimum configuration cangenerate several types of desired morphing shapes; for exam-ple, both the lift-to-drag ratio maximization and lift coeffi-cient maximization shapes.

As part of the evolution of our research, this study consid-ers the optimum design of a morphing flap that achieves mul-tiple desired morphing shapes using a single configuration.For this purpose, the structural model is changed so thatthe desired morphing shapes can be achieved based on thecorresponding load conditions. The design problem is thenformulated using a multiobjective optimization problem toachieve multiple desired shapes. For computational effi-ciency, the satisficing trade-off method (STOM) is

adopted19,20) to obtain a single Pareto solution, because thisapproach achieves such a solution even if the shape of thePareto set is non-convex in the objective function space.STOM transforms the multiobjective optimization probleminto an equivalent single objective problem by introducingan aspiration level. When the given Pareto solution is not sat-isfactory, the searching process is repeated using a differentaspiration level.

Through numerical examples, the optimum configurationof the multi-layered flap is shown to deform to two differentmorphing shapes under the prescribed load conditions. In ad-dition, a trade-off analysis is performed between the conflict-ing objective function for the optimum structural configura-tions as the structural features depending on each objectivefunction can be clarified through investigating the Pareto op-timal configurations. Finally, the aerodynamic performanceis evaluated for the deformed shapes of the obtained Paretooptimal configurations to confirm the difference betweenthe deformed shapes and the ideal morphing shape. Throughthis comparison, the validity of defining the objective func-tion that considers only the structural deformation is verified.

2. Level Set-based Topology Optimization

Consider a structural design problem to obtain a solid do-main + filled with a solid material within an admissible de-sign domain D, where Dn� is regarded as a void domain.The design optimization problem is formulated as followsusing the respective objective and constraint functionalsFð�ð�ÞÞ and Gð�ð�ÞÞ:

Minimize:�

Fð�ð�ÞÞ ¼Z�

fðxÞ d� ð1Þ

Subject to: Gð�ð�ÞÞ ¼Z�

d�� Vmax � 0 ð2Þ

where, fðxÞ is the integrand function and Vmax is the allow-able volume limit.

This method introduces the level set function �ðxÞ to rep-resent a boundary @� between the material and void domainsin the fixed design domain D. As shown in Fig. 3, the boun-dary is described as an iso-surface of the level set function�ðxÞ as follows:

Fig. 2. Multi-layered morphing flap model.18) Fig. 3. Level set function and design domain.

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0 < �ðxÞ � 1 if 8x2�n@��ðxÞ ¼ 0 if 8x2@��1 � �ðxÞ < 0 if 8x2Dn�

8><>: ð3Þ

The level set function in this method is different from that inthe conventional topology optimization method by introduc-ing the boundary in the level set function as ½�1; 1�.16) Theselimits are used to define a fictitious interface energy based onthe phase field method that is incorporated in the objectivefunctional as discussed below.

Since the formulation in Eq. (1) is an ill-posed problemthat allows discontinuity everywhere in the design domain,the regularization method that introduces a fictitious inter-face energy based on the phase field method was proposed.16)

Adding the fictitious interface energy, the regularized objec-tive functional FR is defined as follows:

Minimize:�

FRð�ð�Þ; �Þ ¼Z�

fðxÞ d�þZD

1

2�jr�j2 d� ð4Þ

where, � > 0 is a regularization parameter that represents theratio of the fictitious interface energy and the objective func-tional. The degree of geometric complexity in the optimalconfiguration can be qualitatively controlled by the valueof ¸. Further details can be found in Yamada et al.16)

The optimization problem is reformulated using Lagrange’smethod of undetermined multipliers. Let the Lagrangian be�FR and the Lagrange multiplier of the volume constraintbe ­. The optimization problem is then formulated as

Minimize:�

�FRð�ð�Þ; �Þ ¼ FR þ �Gð�ð�ÞÞ ð5Þ

The Karush-Kuhn-Tucker (KKT) condition for the precedingoptimization problem is described as follows:

�F0R ¼ 0; �G ¼ 0; � � 0; G � 0 ð6Þ

where, �F0R represents the derivative of the regularized La-

grangian �FR with respect to º. The optimal configuration sat-isfies the aforementioned KKT conditions. However, giventhat a direct calculation is almost impossible, the followingtime-evolutionary equation is introduced.

3. Exploration of Pareto Solutions Using STOM

3.1. Multiobjective optimization problemA multiobjective optimization problem is formulated for

simultaneously optimizing two or more conflicting objectivefunctions as follows:

Minimize : FðdÞ ¼ ½f1ðdÞ; f2ðdÞ; � � � ; fmðdÞ�T ð7Þwhere, F is an objective function vector composed of fiðdÞ,the i-th objective function and d is a design variable vector.

The solutions obtained are called Pareto optimal solutionsthat cannot improve a value of an objective function withoutsacrificing the values of any other objective function. For ex-ample, in the simple two-objective function problem shown

in Fig. 4, black circle points D and E are not Pareto solutionsbecause they are dominated by other solutions with whitecircles. The white circle points A, B and C are Pareto solu-tions because the value of an objective function cannot beimproved without sacrificing the value of the other objectivefunction. This means that these Pareto solutions have a trade-off relationship between the two conflicting objectives. It isimportant to note that in a multiobjective topology optimiza-tion problem, the structural features corresponding to theconflicting objective functions can be investigated.3.2. STOM

This study adopts STOM19,20) to obtain a single Pareto so-lution. STOM converts a multiobjective optimization prob-lem into an equivalent single-objective optimization problemby introducing an aspiration level that corresponds to theuser’s preference for each objective function value.

The multiobjective optimization problem is formulated asthe weighted Tchebyshev norm problem from the originalobjective functions fiðxÞ; ði ¼ 1; � � � ; kÞ as follows:

Minimize: maxi¼1; ...; k

wi fiðxÞ � fAi

� �þ �Xki¼1

wifi ð8Þ

where, the first term is the minimum value among theweighted objective functions. The minimum value is selectedin each iteration of optimization run. The second term is in-troduced as numerical stability by setting the coefficient ¡ asa sufficiently small positive value, which is set as 1:0� 10�6

in this study. The weighting factors wi; ði ¼ 1; � � � ; kÞ aredefined using the aspiration level as follows:

wi ¼1

fAi � fI

i

ði ¼ 1; � � � ; kÞ ð9Þ

where, fIi and fA

i , (i ¼ 1; � � � ; k) correspond to the idealpoint and the aspiration level of each objective function, re-spectively.

It should be noted that the objective function defined inEq. (8) does not satisfy a continuous differentiability condi-tion when the maximum objective function value switches.STOM usually introduces a slack variable to satisfy the con-tinuous differentiability condition.19,20) However, this studydirectly uses Eq. (8) as the objective function because pre-liminary analysis revealed that the adverse effects of ignoringthe continuity condition can be negligible.

The weighting factor wi plays an important role in obtain-

Fig. 4. Conceptual scheme of Pareto optimal solutions (two conflictingobjectives).

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ing the Pareto solution according to the designer’s prefer-ence. As shown in Fig. 5, the Pareto optimal solution is usu-ally located on the line connecting the ideal point and the as-piration level in the objective function space, irrespective ofwhether the aspiration level lies in the feasible region. Itshould be noted that the Pareto solution can be obtained evenfor the concave Pareto surface. This is a superior feature ofthe STOM as compared to the ordinal weighted sum ap-proach. In addition, the Pareto curve is obtained by paramet-rically setting the aspiration level.

4. Optimal Design ofMorphingWing Internal Structure

4.1. Modeling of the multi-layered compliant mechanismThe multi-layered morphing flap shown in Fig. 2 has plural

layers of ribs that are connected at the wing skin. In our multi-layered compliant mechanism study,18) each layer has an in-dependent structural configuration that can be supported foran independently applied load. However, this operates as asingle morphing flap by connecting through the wing skin.

To reduce computational time, the multi-layered compli-ant mechanism is modeled as a two-dimensional structuralmodel.The connection of each layer at the wing skin is de-scribed by defining the displacement as being equal to thatof the other skins. Figure 6 is a schematic representation ofthe multi-layered structure, where the other layer is invertedalong the vertical axis. It should be noted that the differentlayers are located in the same space on the two-dimensionalstructural model. As shown in Fig. 6, the condition for whichthe displacements are specified at the wing skin boundary � I

is described as follows:

u1i ¼ u2i on � I ð10Þwhere, u1i and u2i represent the displacement tensors of layer1 and layer 2, respectively, and i represents the tensor. Thedisplacement of the structure is obtained by coupling the dis-placement field of each layer with the conditions representedin Eq. (10). In the topology optimization process, the config-uration of each layer is then independently updated based onthe displacement field obtained for each layer.

4.2. Formulation of topology optimization for compliantmechanism achieving the desired outer deformation

This study considers the compliant mechanism thatachieves a specified displacement. The compliant mechan-ism requires the following two properties:

1. Stiffness to support the working load2. Stiffness to support the reaction force received from the

workpieceIn the case of the design problem, an artificial spring com-ponent is introduced at the input and output ports based onthe idea of Sigmund.15) Then, the reaction force is consideredin the design formulation instead of the explicit objectivefunction.

This study considers the optimum design of the internalstructure of the morphing flap to achieve the desired outer de-formation at the wing’s skin. The desired outer shape thatcorresponds to an ideal morphing shape will be described lat-er in Section 5. In this study, the desired morphing shape isobtained using the concept based on the shape optimizationmethod for the morphing airfoil proposed by one of the au-thors.21) The objective function is defined to minimize theRMS error between the desired and deformed shapes underapplied loads as follows:

Fð�ð�ÞÞ ¼Z�I

ui � Uik k2 d�� �1=2

ð11Þ

where, �I represents the boundary at the wing skin as theouter shape, and Ui and ui are the desired and deformed dis-placements, respectively.

Considering the volume constraint and governing equa-tions, the optimization problem is defined as follows:

Minimize: Fð�ð�ÞÞ ¼Z� I

u1i � Ui

�� ��2 d�� �1=2

þZ�I

u2i � Ui

�� ��2 d�� �1=2

ð12Þ

Subject to: Gð�nð�ÞÞ ¼Z�n

d�� Vnmax � 0 ð13Þ

� divðCnijklu

nk;lÞ ¼ 0 on �n ð14Þ

uni ¼ �uni in �nu ð15Þ

tni ¼ �t ni þ �nuni in �nt ð16Þ

uni ¼ u ~ni in �n

I ð17Þ

Fig. 5. Pareto solution is located on the line connecting the ideal point andaspiration level in the objective function space using STOM.

Fig. 6. Individual layer and region.

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where, the superscripts n denote the number of the domain,i.e., �n represents the design domain of layer n. Vmax isthe upper limit of the volume constraint, ui is the displace-ment, ti ¼ Cijkluk;lnj is the traction and Cijkl is the elastictensor. �ui and �ti are constant values that represent the givendisplacement and traction, respectively. ¬ is a spring constantof the artificial spring component. The displacement �ui isfixed at the boundary �u, the traction ti is applied at theboundary � t, and the displacement of one layer is prescribedto be equal to that of the other at the boundary �n

I as shown inEq. (17), where ~n represents the other layer.

Then, the topological derivatives22) �F 0 for domain �1 and�2 are derived as follows18):

�F0 ¼ � ~uni;jA

nijklu

nk;l þ �n on �n ð18Þ

where, ~uni and ~t ni are defined to satisfy the adjoint equation asfollows:

�div Cnijkl ~u

nk;l

� �¼ 0 on �n ð19Þ

~uni ¼ 0 in �nu ð20Þ

~t ni ¼ �uni in �nt ð21Þ

~t ni ¼Z�I

uni � Ui

�� ��2 d�� ��1=2

ðuni � UiÞ in �nI ð22Þ

~uni ¼ ~u ~ni in �n

I ð23ÞAijkl is defined as follows:

Aijkl ¼3ð1� ÞE

2ð1þ Þð7� 5Þ

� �ð1� 14þ 152Þð1� 2Þ2 ijkl þ 5 ikjl þ iljk

� �( )

ð24Þwhere, E is Young’s modulus, ¯ is the Poisson ratio and ij isthe Kronecker’s delta function. The details are referred to inOtomori et al.22)

In order to facilitate numerical implementation, Aijkl de-scribed in Eq. (24) is approximated by the elastic tensorCijkl. The approximation accuracy is verified via preliminaryanalysis.4.3. Multiobjective optimization considering multiple

morphing shapesAdopting the STOM formulation, the objective functions

of the multiobjective topology optimization problem aretransformed as follows:

Minimize:�

Fð�ð�ÞÞ ¼ maxm¼1;2

wm fmð�ð�ÞÞ � fAm

� �

þ �X2m¼1

wmfmð�ð�ÞÞ ð25Þ

where, fmð�ð�ÞÞ is the objective functional of the m-th con-dition. For numerical stability, fmð�ð�ÞÞ is normalized asfollows:

fmð�ð�ÞÞ ¼R� I

umi � Umi

�� ��2 d�n o1=2R�I

Umi

�� ��2 d�n o1=2 ðm ¼ 1; 2Þ ð26Þ

where, umi and Umi are the actual deformation and desired

morphing shape, respectively, for the m-th morphing condi-tion.4.4. Numerical implementation

The flowchart of the optimization procedure is shown inFig. 7. Initially, the initial configuration and initial level setfunction are set. The equilibrium equations are then solvedusing the two-dimensional finite element structure model,and the objective and constraint function values are eval-uated. In this case, the optimization process is complete ifthe objective function converges. Otherwise, the sensitivitieswith respect to the objective function are evaluated by solvingthe adjoint equation. The level set function is then updatedafter the step size of the fictitious time is adjusted to stabilizethe optimization process. Then, the next step is followed.

5. Structural Design Conditions for Multiple DesiredMorphing Shapes

To achieve multiple desired morphing shapes using a sin-gle compliant mechanism configuration, the structural modelis changed so that the desired morphing shapes are achievedby adjusting the load conditions. In this study, two types ofdesired target deformations are considered. One is the lift-to-drag ratio (L=D) maximized shape and the other is themaximum lift coefficient (Cmax

L ) maximized shape.Before describing the load conditions for each desired

morphing shape, the modeling of the multi-layered morphingflap is described. Initially, the mother wing adopted in thisstudy is NASA’s Common Research Model (CRM),23) asshown in Fig. 8, where the rear side from the 60.5% chordis used as the morphing flap region.

Figure 9 illustrates the two different morphing shapes for aflap angle of 20°, where only the morphing portion of the

Fig. 7. Flow chart of optimization procedure.

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trailing-edge side of the airfoil is shown. The morphingshapes correspond to L=Dmax maximization and Cmax

L max-imization shapes that are obtained using the optimum morph-ing shape design method proposed by one of the authors.21)

By comparing the two shapes, it was determined that theL=Dmax maximization shape has a relatively uniform bend-ing deformation, while the Cmax

L maximization shape has alarge curvature at approximately 80% chord.

The two-layered morphing flap18) is then described. Theflap is modeled as a two-dimensional structural model, asshown in Fig. 10, for topology optimization. Each layerhas an independent structural configuration that can providesupport under an independently applied load, where the mod-els shown in Fig. 10 have different load points. It should benoted that the flap is operated as a single structure by connec-tion only at the wing skin. The wing skin is set as a fixed de-sign domain with 3-mm thickness to avoid local deformationand the skin structure design simultaneously.

Layer 1 has two loading points A and B in the push direc-tion as shown in Fig. 10(a), while Layer 2 has one loadingpoint C in the pull direction as shown in Fig. 10(b). Thetwo different load points in Layer 1 are used for the two dif-ferent objectives as shown in Table 1. The load points are se-lected by considering the desired morphing shapes and set asa constant for topology optimization. The load magnitude isalso set as a constant with a value of 9:5� 104 N for all po-

sitions in the numerical examples. The load conditions givenwere determined from our preliminary analysis includingtrial and error.

6. Numerical Examples

The optimum inner structure configurations of the morph-ing flap to satisfy the two desired shapes shown in Fig. 9 areobtained by setting several aspiration level values for theSTOM as the multiobjective topology optimization problem.

For the deformation analysis, a linear deformation is as-sumed and the mesh size with triangular elements is set be-tween 0.6 mm and 30mm. The material constants are setwith a Young’s modulus of 70GPa and a Poisson’s ratioof 0.35, cited for aluminum alloy. In addition, the stiffnessof the void domain is set as 1/1000 times the material do-main.The parameters of the topology optimization are setas follows; a regularization parameter ¸, a volume upper limitVmax and an artificial spring constant ¬ as 1:0� 10�5, 40% ofthe fixed design domain and 1:0� 106 N/m, respectively.6.1. Pareto solutions

In order to explore the outline of the Pareto frontier, thePareto solution is obtained for several values of the aspirationlevel. When the ideal point is fI

1 ¼ fI2 ¼ 0, the ratio of the

Pareto solution obtained f1=f2 will ideally correspond tothe ratio of the aspiration level fA

1 =fA2 . The Pareto solutions

are investigated under the condition that the ratios of aspira-tion level are set between 1 : 5 to 5 : 1.

The Pareto curve obtained in the objective function spaceis illustrated in Fig. 11 under several values of the aspirationlevels. The figure indicates that the two objective functionshave a trade-off relationship with each other and the Paretocurve has a convex shape. Since the Pareto solutions ob-

Fig. 8. Wing model based on NASA’s Common Research Model.

(a) L/Dmax maximized shape

(b) CLmax maximized shape

Fig. 9. Two desired morphing shapes.

(a) Layer 1

(b) Layer 2

Fig. 10. Design domain of morphing flap.

Table 1. Load conditions for the two desired shapes.

Morphing condition Layer 1 Layer 2

L=Dmax AC

CLmax B

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tained lie on the straight line that corresponds to the ratio ofthe aspiration level except for fA

1 : fA2 ¼ 5 : 1, the optimiza-

tion works well to converge to the Pareto solutions. In thecase of fA

1 : fA2 ¼ 5 : 1, the solution obtained is close to that

obtained for the case of fA1 : fA

2 ¼ 4 : 1. This is because thedesign is close to the optimum design for f2, the Cmax

L maxi-mized design. When this problem is evaluated as a single ob-jective optimization problem for f2, the optimum solution isf2 ¼ 0:03738. The value is almost equivalent to those underthe cases of fA

1 : fA2 ¼ 4 : 1 and 5 : 1. However, the opti-

mum value of a single objective optimization problem for f1

is f1 ¼ 0:01379. The value is approximately 20% smallerthan that for the case of fA

1 : fA2 ¼ 1 : 5.

Convergence histories of the objective functions for thecases of fA

1 : fA2 ¼ 1 : 4; 1 : 3; 1 : 2; 1 : 1 and 2 : 1 are

shown in Fig. 12. In some cases, large oscillations appearduring the optimization run. However, it is found that the ob-jective functions converge to the Pareto solutions by adjust-ing the step size of the fictitious time described in Section4.4.

Then, the deformation shapes for eight Pareto designs arecompared in Fig. 13, where the black and red curves repre-sent the mother wing shape and deformation shape, respec-tively. The gray regions represent the desired morphingshapes. The left side corresponds to the L=Dmax maximizeddesign, the right side corresponds to Cmax

L maximized designand the numbers at the center correspond to the ratio offA1 : fA

2 . For the left-hand side, the deformed shape of theupper solution approaches the desired morphing shape. How-ever, a lower Pareto solution is better for the right-hand side.

Among these Pareto solutions, we select the Pareto designfor the case of fA

1 : fA2 ¼ 1 : 2 as the design candidate be-

cause the design appears to be well-balanced according toFigs. 11 and 13. As such, the design candidate has a smallerdifference between the desired and deformed morphingshapes for both design conditions of a maximized L=Dmax

and CmaxL .

6.2. Trade-off analysis with structural configurationsHere, we compare the structural features between the

Pareto configurations obtained. The structural configurationswith deformation obtained for five out of the eight Pareto de-signs are shown in Fig. 14, where the black curve represents

the mother wing shape and the gray region represents the de-sired morphing shape. The red and blue areas correspond tothe structural configurations of Layers 1 and 2, respectively.The left-hand side corresponds to the L=Dmax maximized de-sign when the actuation load is applied at the upper point(point A in Fig. 10) in Layer 1. The right-hand side repre-sents the Cmax

L maximized design when the actuation loadis applied at the lower point (point B in Fig. 10) in Layer 1.

According to our preference, the Pareto design forfA1 : fA

2 ¼ 1 : 2 shown in Fig. 14(c) is selected. The designsas shown in Fig. 14(a) and (b) have smaller values for f1 but

(a) f1A f2A 1 4

(b) f1A f2A 1 3

(c) f1A f2A 1 2

(d) f1A f2A 1 1

(e) f1A f2A 2 1

Fig. 12. Convergence history.

Fig. 11. Pareto optimum solution in objective function space.

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larger values for f2 such that the difference between the de-sired and deformed shapes is larger at the trailing-edge forthe right-hand side design, for the Cmax

L maximized morphingshapes. In the case of the designs, as shown in Fig. 14(d)and (e), there are larger values for f1 but smaller valuesfor f2. The difference in deformation is larger at the centralupper surface for the left-hand side design, for the L=Dmax

maximized morphing shape. The deformation shape in thedesign of fA

1 : fA2 ¼ 2 : 1 shown in Fig. 14(e) is almost iden-

tical for both conditions.For all designs, Layer 1 has a hinge part in the vicinity of

the upper center area. When actuation load is applied at theupper point (point A), the flap is almost uniformly bent ex-cept for the case of fA

1 : fA2 ¼ 2 : 1 shown in Fig. 14(e).

However, when the load is applied at the lower point (pointB), Layer 1 has large local bending at the hinge point, andhence, Cmax

L maximized deformation is achieved.In addition, Layer 1 of the designs in Fig. 14(a), (b) and

(c) has a connecting part from the main structure to the lowersurface in the left downward direction. The design inFig. 14(a) has a larger cross-sectional area. The connectingpart transfers the deformation of Layer 2 located in the lowerside to Layer 1, and hence, the uniform bending suitable forthe desired deformation shape of the L=Dmax maximized de-sign is achieved. However, the designs in Fig. 14(d) and (e)

do not have such a connecting part. As such, the suitable lo-cal bending for the desired deformation of the Cmax

L design isachieved.

In addition, the structural configurations are shown inFig. 14 and it is observed that (a) and (e) are largely differentfrom that in the case of Fig. 14(c). The deformations forthese designs are almost identical, even for different loadconditions. In design (a), Layer 1 is connected to the lowersurface at two points, and Layer 2 is connected to the uppersurface near the root. This will cause uniform deformationfor the L=Dmax maximized design. However, for the lowestdesign (e), the local bending deformation for the Cmax

L designis achieved using the hinge parts in Layers 1 and 2.6.3. Aerodynamic performance of Pareto solutions

Aerodynamic performance is evaluated for the deformedshapes of some Pareto optimal solutions to confirm the val-idity of the objective function that is defined as RMS errorbetween the ideal morphing and deformed shapes. Thetwo-dimensional panel method XFOIL24) is used for theaerodynamic analysis, where the Reynolds number is set as3:79� 106 per reference chord with a non-compressive vis-cous flow.

Aerodynamic analysis is performed for three Pareto solu-tions under the cases of fA

1 : fA2 ¼ 1 : 5, 1 : 2 and 2 : 1,

where the deformed shape under 1 : 5 is closer to the idealshape than the other two designs for the L=Dmax maximizeddesign. Here, the angle of attack is defined as the angle be-tween a mainstream direction and a camber line for themother wing. It means the actual angle of attack for the idealmorphing shape is 8:0� different for the L=Dmax design and7:9� different for the Cmax

L design. The difference from thetwo ideal shapes is due to negligible numerical error. ThePareto solutions of which the deformation shape is differentfrom the ideal morphing shape have about 1� difference inthe angle of attack.

TheCL–¡ andCD–¡ curves for the nominal airfoil withoutmorphing, the ideal morphing shape and the deformedshapes of the three Pareto designs of L=Dmax maximized de-sign are compared in Fig. 15. It is found that the deformedshape for the Pareto solutions of fA

1 : fA2 ¼ 1 : 5 has curves

closer to the ideal morphing shape than the other two de-signs. However, the difference becomes larger around theangle of attack where L=D takes the maximum value. Thedifference of L=D is quite large at the maximum value be-cause the difference of CD is quite large. There have beensome problems in the aerodynamic analysis around there.Therefore, the aerodynamic performance values are com-pared in Table 2 for both the angle of attack where L=D

takes maximum value and 0�. The difference of L=D is verysmall when the angle of attack is 0�. Therefore, the Pareto so-lutions obtained that consider only the structural deformationwill have reasonable aerodynamic performance.

Then, the CL–¡ curves for the nominal airfoil, the idealmorphing shape and deformed shapes of the three Pareto de-signs of Cmax

L maximized design are compared in Fig. 16,and the values of Cmax

L for the designs are compared inTable 3. It is found that the differences of Cmax

L between

Fig. 13. Deformed shapes for Pareto solutions: (left) L=Dmax maximizeddesign and (right) Cmax

L maximized design.

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the ideal morphing and deformed shapes of the Pareto solu-tions are sufficiently small.

Because deterioration of the aerodynamic performancecaused by the difference between the ideal morphing shapeand deformed shape for the Pareto solutions is small, the ob-jective functions defined as the RMS error between the idealmorphing shape and deformed shape are considered to bereasonable.

7. Conclusions

This study extends our previous study on a multi-layeredmorphing flap18) to obtain an optimal structural configurationthat can deform to multiple desired morphing shapes corre-sponding to multiple flight conditions as L=Dmax and Cmax

L

maximized designs. For this purpose, different actuation loadpoints are selected to achieve different desired deformationshapes using a single structural configuration of the compli-

ant mechanism as a morphing flap. The design problem isthen formulated as a multiobjective topology optimizationwhere the satisficing trade-off method (STOM)19) and levelset-based topology optimization method16) are utilized.

Based on numerical examples, several Pareto designs ofmulti-layered morphing flaps are obtained for different val-ues of aspiration levels. Using the Pareto surface shape inthe objective function space and the convergence history,the method proposed successfully yielded the Pareto set.Based on a trade-off analysis, the best compromise configu-ration can be selected among the Pareto designs. The struc-tural configurations and deformation mechanism of the Par-eto solutions were then investigated in detail. Finally, theaerodynamic performance is evaluated for the deformedshapes of some Pareto optimal configurations. Based onthe results of aerodynamic analysis, it is considered thatthe objective functions defined as the RMS error betweenthe ideal morphing shape and deformed shape are reasonable.

(a) f1A f2A 1 5

(b) f1A f2A 1 3

(c) f1A f2A 1 2

(d) f1A f2A 1 1

(e) f1A f2A 2 1

Fig. 14. Deformed shapes for Pareto solutions: (left) L=Dmax maximized design and (right) CmaxL maximized design.

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However, these configurations are obtained for given loadconditions determined from our preliminary analysis. The se-lection of optimal load conditions will be investigated in thefuture.

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Fig. 16. CL vs ¡ for the ideal morphing shape of the CmaxL maximized de-

sign and deformed shapes for Pareto optimal solutions.

Table 3. Value of CmaxL for Cmax

L maximized designs.

Airfoil CmaxL ¡ (deg)

Nominal wing 1.97 18.1Ideal shape 2.38 13.9

fA1 : fA2 ¼ 1 : 5 2.33 14.3fA1 : fA2 ¼ 1 : 2 2.35 13.4fA1 : fA2 ¼ 2 : 1 2.37 13.8

Table 2. Aerodynamic performances of L=Dmax maximized designs.

(a) Maximum value in terms of angle of attack

L=Dmax CL CD ¡ (deg)

Nominal wing 111 1.22 0.0110 7.3Ideal shapes 218 1.33 0.00611 ¹8.7

fA1 : fA2 ¼ 1 : 5 119 1.20 0.0101 ¹8.8fA1 : fA2 ¼ 1 : 2 77.9 1.14 0.0146 ¹8.7fA1 : fA2 ¼ 2 : 1 48.9 1.35 0.0275 ¹6.1

(b) � ¼ 0 degree

L=D CL CD

Nominal wing 98.6 0.443 0.00449Ideal shapes 34.7 1.635 0.0472

fA1 : fA2 ¼ 1 : 5 36.6 1.643 0.0448fA1 : fA2 ¼ 1 : 2 35.6 1.668 0.0468fA1 : fA2 ¼ 2 : 1 33.5 1.719 0.0513

(a) CL vs a

(b) CD vs a

Fig. 15. Aerodynamic performance for ideal morphing shape for L=Dmax

maximized design and deformed shapes for Pareto optimal solutions.

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Hiroaki TanakaAssociate Editor

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