research article knowledge-based shape optimization of...

Research ArticleKnowledge-Based Shape Optimization ofMorphing Wing for More Efficient Aircraft

Alessandro De Gaspari and Sergio Ricci

Department of Aerospace Science and Technology Politecnico di Milano Via La Masa 34 20156 Milano Italy

Correspondence should be addressed to Sergio Ricci sergioriccipolimiit

Received 31 March 2015 Revised 22 July 2015 Accepted 31 August 2015

Academic Editor Ning Qin

Copyright copy 2015 A De Gaspari and S Ricci This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

An optimization procedure for the shape design of morphing aircraft is presented The process is coupled with a knowledge-based framework combining parametric geometry representation multidisciplinary modelling and genetic algorithm Theparameterization method exploits the implicit properties of the Bernstein polynomial least squares fitting to allow both local andglobal shape controlThe framework is able to introducemorphing shape changes in a feasible way taking into account the presenceof structural parts such as the wing-box the physical behaviour of the morphing skins and the effects that these modificationshave on the aerodynamic performances It inherits CAD capabilities of generating 3D deformed morphing shapes and it is ableto automatically produce aerodynamic and structural models linked to the same parametric geometry Dedicated crossover andmutation strategies are used to allow the parametric framework to be efficiently incorporated into the genetic algorithm Thisprocedure is applied to the shape design of Reference Aircraft (RA) and to the assessment of the potential benefits that morphingdevices can bring in terms of aircraft performances It is adopted for the design of a variable camber morphing wing to investigatethe effect of conformal leading and trailing edge control surfaces Results concerning four different morphing configurations arereported

1 Introduction

The very challenging targets of new environmental require-ments for transport aircraft force the researchers to lookfor more advanced aircraft configurations based on moreefficient aerodynamics and structures together with moresophisticated flight control systems Focusing on Europeantransport it appears as clear that the pressure will increasefor large scheduled European carriers to reequip their shorthaul fleets with more fuel-efficient types in order to remaincompetitivewith low-cost rivals and to ensure theywill not beunduly penalized when European emissions trading comesinto full force

The Breguet range equation [1 2] combines aerodynamicpropulsion and structural figures ofmerit and suggests actingon both the aircraft empty weight and the lift over dragratio in order to improve the modern aircraft efficiencyMany are the approaches investigated during recent yearstrying to improve all the Breguet equation terms such as

advanced and unconventional configurations able to improveaircraft efficiency new materials and structural conceptsactive controls to peak loads reduction and flight controlimprovements more efficient engines and alternative fuels[3] Certainly morphing technologies offer potential benefitsfor a more efficient aircraft and for this reason literaturereports many and different morphing concepts even if aclear evaluation of their real benefits is not available yet [4ndash7] One of the key challenges for morphing technologiesis represented by the identification of the most suitableactuation concept able to reduce the actuation energy and themechanism weight

While current aircraft are already equipped with systemsable to introduce in-flight geometrical variations such aswingarea change variable camber and retractable landing gearthe morphing of next generation still has challenges andleads to the design of morphing wings based on conformablecontrol surfaces [8] In particular the variable wing cambermorphing considered as the capability to change the airfoil

Hindawi Publishing CorporationInternational Journal of Aerospace EngineeringVolume 2015 Article ID 325724 19 pageshttpdxdoiorg1011552015325724

2 International Journal of Aerospace Engineering

shape without surface discontinuities often based on theadoption of ad hoc designed flexible skins [9] appears asan efficient way to maximize the lift over drag ratio and toreduce the fuel consumption over the entire flight envelopeHowever the design of this kind ofmorphing devices requiresdeveloping specific procedures able to assist the engineersduring both the design [10ndash12] and the benefits evaluationphases Currently this target is commonly addressed bymeans of dedicated multilevel and multiobjective optimiza-tion procedures [13] Multilevel capabilities allow performingthe optimization of morphing shapes and the design ofthe morphing mechanism separately [14] multiobjectivetechniques help to design aircraft able to adapt its shape tooptimize the performances along the cruise or at a wide rangeof different flight conditions

The design of morphing wing devices must combine twoopposite requirements often named kinematic and structuralrequirements respectively a flexible structure so tominimizethe energy necessary to adapt its shape as expected and atthe same time an enough rigid structure able to maintain thenew shape under the aerodynamic loads when the morphingmechanism is not actuated The approach proposed by theauthors [15] is based on the optimization of aerodynamicstiffness and actuation sequentially passing through thedefinition of a family of optimal morphing shapes associatedwith a group of as many flight conditions Hence the designof morphing mechanism depends on the availability of theoptimal shapes that must be achieved when it is actuated andthat are computed before the mechanism is known In thisway the optimal shapes guarantee the aerodynamic perfor-mances while the mechanism can be optimized consideringboth kinematic and structural requirementsThekey problemis that the allowable shape variation laws depend on the typeof mechanism while the mechanism design depends on theshape changes that the mechanism must be able to reachFor this reason in recent years many efforts have been madeaiming at considering energy and actuation requirements[16ndash18] as well as structural constraints [19] directly duringthe aerodynamic shape optimization

The work presented in this paper focuses on the first levelof a wider morphing design framework having the follow-ing capabilities wing shape optimization able to combineaerodynamic performances with optimal deformation of theskins (first level) optimal design of compliant mechanismable to produce once actuated the optimal shape comingout from the first level (second level) integration of themorphing devices into a high-fidelity model representing thecomplete aircraft for final aeroelastic assessment to evaluatethe reliability of the designed morphing solutions [20]

Initially the first level was implemented as a simple2D shape optimization linked to a viscous and subsonic2D aerodynamic solver while the second one represented ageneral code for the synthesis of compliant mechanisms [1521] and two software pieces have been developed separatelyRecently the first one has been extended in order to produce3D wing models starting from the results which continuedto be obtained by 2D shape optimization while the secondone has been improved by adding multiobjective capabilities[22]

The paper describes the novel progress beyond thispoint about the first level optimization which is now basedon a comprehensive knowledge-based engineering (KBE)framework able to define the optimal morphing wing shapein terms of mission profile performances directly in thethree-dimensional space It is based on coupling a paramet-ric geometry representation able to predict the structuralresponse of morphing skin with Computer-Aided Engi-neering (CAE) capabilities Object-Oriented Programming(OOP) genetic algorithm and aerodynamic analyses Theresults obtained applying it to a Reference Aircraft (RA)coming from the FP7 EU NOVEMOR project (Novel AirVehicle Configurations from Fluttering Wings to MorphingFlight) and adopted as a benchmark to evaluate the optimalbenefits that can bring in terms of global performances arereported to validate the proposed procedure and to evaluatethe impact of continuous chordwise and spanwise cambervariation in terms of lift to drag ratio and aerodynamic loaddistribution

2 Shape Design Methodology

The shape design procedure presented in this work consistsof a shape optimization based on genetic algorithm andcoupled with the parametric framework introduced in thefollowing section able to design the optimal morphingshapes from the aerodynamic point of view under skinstructural requirements While genetic algorithms have beenalready coupledwith 2D shape optimization of airfoil [23 24]this paper presents an approach based on a particular use ofthe genetic algorithms able to combine the set of the mostimportant cross sections describing the wing in order todefine the optimal shape of a 3D morphing wing

The set of parameterized cross section shapes is thestarting point of the whole process shown in Figure 1 Theouter optimization loop based on genetic algorithm workson the most important design variables affecting the cambermorphing such as the leading and trailing edge deflectionsAfter the cross sections are combined by PHORMA the codedescribed in Section 31 the 3D model is analytically definedand its shape can be easily controlled and used to estimatethe aerodynamic performances by means of the user-definedfitness evaluation function Different performance indicescan be enclosed in the shape optimization process in orderto improve the aerodynamic efficiency and reduce the fuelconsumption or to guarantee that the morphing aircrafthas optimal aerodynamic characteristics over different flightconditions [25 26] The inner loop based on a dedicateddevelopment of CST method [27] guarantees that the outerloop works only on feasible shapes able to satisfy wing-boxvolume constraints and morphing skin structural require-ments

This parametric shape representation is suitable to beincorporated into genetic algorithm and to be parallelized[28 29] The most innovative aspect of this implementationis that the skin structural behaviour recovered from theadopted geometry representation determines the ruleswhichdrive the reproduction process of the population In this waythe resulting morphing shape comes from a natural selection

International Journal of Aerospace Engineering 3

Initial population

PHORMA

Aerodynamic model

Solver

3D aerodynamic fitness evaluation and score computation

Selection

Crossover

New set ofmorphing shape changes

Morphing skinstructural requirements

Termination criteria

Mutation

End

Parent individuals

Yes

Postprocessing class

CST identification

Chromosome assembly

No

Figure 1 Scheme of the genetic algorithm based on the 3D parametric shape representation

able to take into account the physics related to structuraldeformation of the skin

Each individual making up the population is dividedin as many parts as the wing sections are and each partis composed of the same set of optimization variables Thegeneration of each new population is produced by in-housecrossover and mutation dedicated strategies while a codeparallelization allows decreasing the computational cost Twopossible code parallelization modes can be adopted The firstone computes in parallel the fitness functions in terms ofglobal aerodynamic performanceThe second one divides thewing problem into a number of subproblems equal to thenumber of wing sections that are solved simultaneously

3 The Knowledge-Based Framework

The shape design of morphing aircraft able to smoothlychange their external geometry requires interfacing the aero-dynamics that can be optimized for different flight conditionsof the mission profile and the structure of the skin thatconstitutes the outermost component of the aircraft CAEsystems represent parametric tools including Finite ElementAnalysis (FEA) and Computational Fluid Dynamics (CFD)able to assist the engineering in the shape design but theability to introduce complex changes is limited and the onlyrules that can be embedded are restricted to simple geomet-rical statements KBE systems are designed to allow implicit


PHORMAmain exchange class

CSTv3Parametric geometry representation

for aircraft section shapes

CAD interface3D aerodynamicinterface 3D FEM modeler

3D aerodynamic model

Postprocessing

XML file

PFEM class(in-house FEM code)

Solver

2D CFD meshinterface

(structured grids)

ICEM STEPfile

IGESfile

MSCNastraninput file

ANSYSAPDLmodel

class

Figure 2 The parametric framework

rules physics-based solvers and evolutionary algorithmsto be linked to CAD system A specific knowledge-basedframework able to provide advanced implicit parametriccapabilities and more direct shape control than commonCAE systems is here presented It is a fully integratedmultiphysics environment based on a parametric geometryrepresentation of the aircraft that incorporates the abilityto predict the structural response of the skin to the shapechanges provides couplings with aerodynamic solvers anduses genetic algorithm to tackle the optimization problemrelated to the definition of the best morphing shapes

Theknowledge of themorphing skin structural behaviouris included into a so-called skin structural (KBSS) feedbackthat consists of an inference engine that uses rules to deducenew shapes consistent with the structural response of themorphing skin This knowledge-based technology is strictlyrelated to anObject-Oriented Programming-based geometryparameterization that is the Class Shape Transformation(CST) which is integrated with a Computer-Aided Engineer-ing (CAE) tool The integration with CAD allows enrichingthe parametric capabilities of the CST formulation with theparametric features of the CAD systems for a complete andaccurate 3D shape definition When this knowledge-basedsystem is coupled with genetic algorithm as described inSection 2 it offers very efficient criteria able to drive theevolution of the solution during the whole optimizationprocess In this way it acts not only on the results but alsoon how the evolutionary process works to reach the finalsolution

The architecture of the framework is shown in Figure 2where the implementation sequence of the three main partsand their mutual interactions are representedThe generationof the 3D models corresponding to different morphing

configurations is done in two steps the parametric identi-fication of the aircraft and its shape modification accordingto the KBE strategy Once a 3D geometry is identified in aparametric way different FEM CFD and CAD models canbe linked to a common shape representation In this waythe shape changes can be quickly extended to the physicalmodels

The structural aerodynamic and CAD modelling capa-bilities are implicitly connected and inherit their propertiesfrom a main class able to combine in three dimensions theset of the most important parameterized cross sections of theaircraft model A specific code directly connected to the CSTmethod is also available for the automatic and high-fidelity2D aerodynamic analyses of each cross section

31 Aircraft Shape Parameterization The KBE frameworkrevolves around PHORMA (Parametric sHapes for aerOdy-namic and stRuctural Modelling of Aircraft) that is an objectoriented code composed by a suite of modules conceivedto exchange and handle different shapes corresponding tothe set of the most important cross sections of the aircraftmodel in order to generate the 3D geometry The shapesare parameterized and combined in the 3D space throughCAD-based interpolation techniques so that local shapeschanges can be spread out Resulting geometry is shared bythe aerodynamic structural and CAD models that inheritthe parametric capabilities from the so-called classshapefunction transformation (CST) technique originally pro-posed by Kulfan [27] and extended by PoliMi to morphingwing sections [21] The CST parameterization is based onmerging four terms a shape function a class function andtwo additional terms related to the airfoil leading edge andtrailing edge shapes


120577(120595)

120575LE

LE morphing

ZLERLE

CFS

CRS

c

120573

120575TE morphing skins

120595

ZTE

ΔZTE

120575TE

Structural box

xLE

Figure 3 CSTv3 geometric parameters for airfoil sections

The general mathematical expression representing theairfoil geometry already presented in [15] is here defined as

120577 (120595) = 1198621198731

1198732(120595) a sdot b119879

119899(120595) + 120595120577TE + (1 minus 120595) 120577LE (1)

where 120595 = 119909119888 120577 = 119911119888 are the nondimensional coordinateswith respect to airfoil chord 119888 120577TE = 119885TE119888 and 120577LE =119885LE119888 These parameters in addition to the airfoil leadingedge nose radius 119877LE the trailing edge boat-tail angle 120573the thickness Δ119885TE and the angular thickness 120575 representthe set of ldquoCST parametersrdquo and are shown in Figure 3Starting from these quantities the leading edge deflection isconventionally defined as 120575LE = arctan(119885LE(119862FS minus 119909LE))while the trailing edge equivalent deflection can be definedas 120575TE = arctan(119885TE(119888 minus 119862RS)) where 119862FS and 119862RS are thefront and rear spar position respectively It is noticed that itis different and generally smaller than the boat-tail angle 120573

The first term of (1) is the class function and providescontrol of both the leading and the trailing edge shapeIt includes the exponents 1198731 and 1198732 that are used tomathematically define a variety of basic general shapes [15]The second term of (1) is the shape function that is definedusing a set of (119899+1) Bernstein polynomial components 119861

119894(120595)

(of selected order 119899) included in the function array b119899(120595)

scaled by as many unknown extra coefficients 119860119894included in

the vector aThe 119894th shape function component is defined as the prod-

uct between the 119894th Bernstein polynomial component and the119894th associated binomial coefficient Each Bernstein polyno-mial component is multiplied by the class function definingthe systematic decomposition of the airfoil shapes intocorresponding scalable airfoil components Consequently theconstant coefficient 119860

119894can be used to scale the entire airfoil

function which is a smooth function that eliminates localdiscontinuities

The first and last terms of the vector a are related to theleading and trailing edge boundary conditions and are equalto the shape function evaluated in 119909119888 = 0 and 119909119888 = 1

In particular a(1) = radic2119877LE119888 and a(119899 + 1) = tan120573 + 119885TE119888 minus119885LE119888

The perturbation of one of the extra coefficients propor-tionally changes the value that the CST function assumes atthe chordwise station of the peaks of every airfoil componentThey are equally spaced along the airfoil chord and aredefined as

120595max(119885) =1198731 + 119894

1198731 + 1198732 + 119899for 119894 = 0 119899 (2)

Many other properties of the cross-section shapes such asthe airfoil area the nondimensional arc length nal of theleading edge or the upper and lower surfaces the airfoilcamber or thickness can be computed in a parametric wayThanks to its nature the calculation of the first- and second-order derivatives is very fast and can be used to computegeometrical quantities such as the length 119871(119909) and curvature120581(119909) of the upper and lower sides as well as leading edgesurfaces or a portion of them

PHORMA is implemented by the Object-Oriented Pro-gramming so that different cross-sectional shapes of a com-plete aircraft can be identified in order to reproduce the cor-responding reference shape and to have a 3D mathematicalmodel based on CST parameterization suitable to introducemorphing shape changes In this way the estimation of themorphing skin structural behaviour as well as the generationof the aerodynamic or finite element models of the aircraftinherits all methods coming from its parametric representa-tion

32 Parametric Identification In order to have a parametricmodel suitable to introduce morphing shape modificationsan efficient CST identification of preexisting CAD modelsis available By means of a completely automatic procedurethe shapes corresponding to the set of the most importantcross sections of the aircraft model are parameterized andcorresponding CST models are generated The upper andlower extra coefficients (119860up and119860 low) of each shape function


can be evaluated by a variety of techniques depending onthe particular study such as numerical optimizations In thispaper a very efficient CST identification based on closed-form least squares fitting able to match preexisting CADmodels is presented The procedure is completely automaticand allows catching the arbitrary set of cross sections of theCAD model Equation (1) can be rearranged as follows

[1198621198731

1198732(120595) b119899(120595)] sdot a119879 = 120577 (120595) minus 120595120577TE minus (1 minus 120595) 120577LE (3)

The parameterization process is based on a specific leastsquares fitting where the Bernstein basis described by [30]is replaced by the here defined CST basis

119879119899

= 119905(119899)

119894= 1198621198731

1198732(120595)(

119899

119894) (1 minus 120595)

119899minus119894120595119894 119894 = 0 119899

(4)

The CST basis is the basis of the space of the class-shapefunctions of degree less than or equal to 119899 on the interval[0 1] Equation (3) can be expressed in the CST basis asfollows

t119899(120595) sdot a119879 = 119891 (120595) (5)

where t119899(120595) is the vector containing the CST components in

the CST basis and 119891(120595) is equal to the right terms of (3)If 1205941198951le119895le119897+1

isin (0 1) are 119897 + 1 pairwise distinct realpoints and 119891

1198951le119895le119897+1

isin R computing the (119899 + 1) coefficientsa is equivalent to solve in the least squares sense theoverdetermined system

T119899a119879 = f (6)

whereT119899is the here defined (119897+1)times(119899+1)CST-Vandermonde

matrix for the CST basis 119879119899and f is the (119897 + 1) vector both

evaluated in the nodes 1205941198951le119895le119897+1

Taking into account that T

119899has full rank (119899 + 1) the

problem has a unique solution given by the unique solutionof the linear system

T119879119899T119899a119879 = T119879

119899f (7)

Since T119899is usually an ill-conditioned matrix the 119876119877 factor-

ization can be used to solve the systemThe approximation can be improved extracting the

boundary condition terms from the definition of the CSTbasis and embedding them within the right term of (5) TheCST basis is redefined for 119894 = 1 119899 minus 1 while 119891(120595) isrecomputed as

119891 (120595) = 120577 (120595) minus 120595120577TE minus (1 minus 120595) 120577LE

minus 1198621198731

1198732(120595) [119860

01198610(120595) + 119860

119899119861119899(120595)]

(8)

The terms 1198600and 119860

119899of (8) are defined by the extra coeffi-

cients a(1) and a(119899 + 1) and depend on the CST parameters119877LE 119888 120573119885TE and119885LE that can be directly computed startingfrom the data points

33 Structural Response of Morphing Skin The paramet-ric framework introduced in this section is a dedicatedknowledge-based engineering (KBE) environment able toforce the shape design to move inside a domain where theglobal shape is driven to change in a feasible way First ofall a morphing wing shape is feasible if it is able to take intoaccount the structural behaviour of the skins It depends onhow the actuation is introduced and how the morphing skinresponds to the actuation input together with the externalloads A Knowledge-Based Skin Structural (KBSS) feedbackembedded into the shape parameterization scheme allowstaking into account the skin structural response before themorphing mechanism has been defined

One of the most important obstacles in the wing morph-ing is due to two main aspects

(i) The presence of main bearing structures(ii) The structural contribution of the skin

Indeed even if almost all the proposed approaches formorphing wings are based on a different structural config-uration of the ribs the structural contribution of the skin stillremainsMoreover the skin is directly connected to the struc-tural box which represents an obstacle for a full deformationand restricts outside deformable regions In the approachhere presented the knowledge-based parameterization is ableto take into account all of these aspects in an implicit way Itrepresents a tool that assists the engineers in determining thebest morphing shape to limit the deformation energy of theskin and at the same time the actuator power necessary tocontrol the shape change This approach is general and canbe easily applied to skin made of different type of material

When local perturbations are introduced in a wing shapeone of the main feasible constraints is represented by thewing regions that must be kept undeformed Consideringa conventional wing equipped with active camber thisregion corresponds to the structural box Introducing thisequality constraint in an explicit way for example during anoptimization process is an expensive approach in terms ofcomputational effort A different approach is to exploit theCST basis capabilities described in Section 41 not only tohandle volume constraints but also to ensure that the skin fitsthe fixed parts of the wing In this way the knowledge-basedapproach is able to follow a set of implicit rules governingthe skin structural behaviour in order to prevent the casein which the shape changes violate the wing-box constraintduring the optimization process

The CST formulation offers the possibility to capture asecond structural feedback related to the behavior of the por-tion of morphing skin outside the undeformed regions Theestimation of the skin structural deformation here proposedis directly inherited from the analytical CST formulation andit is included in the KBSS feedback

The structural behavior of the skin has a key role inthe shape changes that the skin can assume during thedeformation It can be obstacle or a help to achieve the desiredshape This link is preserved by the CST formulation whichembeds in the same parameterization technique the abilityto introduce shape changes and at the same time to evaluate


the corresponding stress distribution along the skinThe firstability has been previously described while the second one isbased on geometrical consideration already described in [15]The axial strain 120576axial (or axial stress 120590axial) required to stretchor compress the skin and the bending deformation 120576bend (orbending stress 120590bend) can be directly deduced respectivelyby the length Δ119871(119909) and curvature Δ120581(119909) difference functionbetween the initial and the deformed shape with respectto the same normalized arc length function nal(119909) If theskin structural properties such as Youngrsquos Modulus 119864 andthe skin thickness 119905 are known the comparison betweenthe CAD models representing the initial and the deformedshapes allows estimating the strain distribution along theupper and lower as well as leading edge skin surfaces

34 Multidisciplinary Modelling Once a parametric model isavailable and morphing shape changes are introduced cor-responding CAD FEM or CFD models can be linked to thesame shape geometry While the framework described in thissection allows exporting the most common CAD formatssuch as IGES and STEP as well as CATIA files parametricmeshing strategies able to follow the shape changes basedon the CST formulation are implemented to automaticallygenerate structural and aerodynamic models It interactswith different classes and methods as shown in Figure 2to generate 3D models for different disciplines During theshape optimization described in this paper only aerodynamicanalyses have been performed in order to evaluate theperformances of wings with and without morphing devicesTherefore the generation of structural models based on theOOP-based PFEM (Parametric Finite ElementModels) classas well as the postprocessing class including Fluid-StructureInteraction (FSI) techniques is not described [22]

PHORMA is equipped with a CAD interface to automati-cally interpolate in spanwise direction the set of cross sectionsdescribing the wing Corresponding parametric shapes areautomatically transformed into spline-based curves bymeansof a user-defined number of points extracted from theCST formulation and combined to generate a multisectionNURBS-based surface by sweeping themalong a user-definedspine In order to accurately describe the original geometryone or more guide curves can be defined and slope and cur-vature boundary conditions can be introduced at the ends ofthe surface Comparisons between original wing geometriesand identified shapes are performed to identify the correctcombination Bernstein polynomial ordernumber of sectionsin spanwise direction able to guarantee an approximationerror tolerance

The generation of aerodynamic models is strictly con-nected to the full parametric description of the geometryBoth 2D and 3D models can be produced by different in-house procedure depending on the desired model quality

In the case of 2D aerodynamic analyses a specific codeable to automatically produce a 2D structured mesh suitableto perform Navier-Stokes computations is directly linkedto the CST-based code The generation of structured mesharound a parameterized airfoil is based on a script for AnsysICEMCFD [31] It is represented by a TclTk shell withan extended library consisting of ICEMCFD commands

The script is contained in a ldquorplrdquo file while the completeprocess shown in Figure 4 works by adapting the mesh withrespect to a predefined mesh contained in the ldquoblkrdquo filebuilt around a baseline airfoil shape Once the aerodynamicmesh is produced the CFD computations are performed bythe EDGE code [32] The parametric mesh generation hasbeen applied to the morphing leading and trailing edge of theairfoil shown in Figure 5

The whole process allows adding a set of high-fidelitypolar curves to the set of the most important parameterizedcross sections used by PHORMA to describe the completewing model This could be used to make local corrections toa 3D low-fidelity simulation with the computational cost of areduced number of 2D flow solutions equal to the number ofcross sections [33 34]

In the case of 3D aerodynamic analyses the cross sectionsare combined in order to spread the specific geometry detailsalong the wing via particular interpolation surfaces ableto accurately reproduce the correct wing thickness distri-bution in spanwise direction The implemented procedureallows imposing different transition laws among different2D airfoils as well as modifying their properties such asAngle of Attack dihedral or tow angles Different kind oflow-fidelity and high-fidelity aerodynamic models can begenerated exploiting the parametric methods that PHORMAinherits from the CST formulation

Another way to perform 3D simulations is to directlygenerate full volumemeshes for mediumhigh-fidelity analy-ses PHORMA is linked in batch mode to the software sumo[35 36] in order to generate unstructured surface meshesHeuristics parameters can be tuned to obtain acceptablemesh Triangulations are based on a three-dimensional in-sphere criterion which provide high quality meshes Geo-metric refinement criteria are able to produce a finer mesh inregions of strong curvature such as morphing leading edgegeometries Starting from the surface mesh unstructuredor hybrid volume meshes can be generated by TetGen codeand automatically converted to EDGE-compatible files inorder to perform Euler or Reynolds average Navier-Stokesformulation (RANS) computations

The hybrid mesh generation has been applied to theReference Aircraft The prismatic layer around the fuselageand around a wing section placed near the root equippedwith a morphing leading edge is shown in Figure 6 Once theaerodynamic analysis is finished PHORMA is able to extractthe results in terms of 119862

119901distribution and to apply them

along corresponding parametrized cross section shapes Theaerodynamic results can be extrapolated around one or moresections arbitrarily positioned and oriented so that the set ofwing cross sections can be directly associated with a set ofaerodynamic results coming from 3D analyses

4 Optimization Problem Definition

The skin structural requirements introduced in Section 33are embedded into the implementation of the shape opti-mization problem as described in Sections 41 and 42 sothat the aerodynamic performances are evaluated only onphysically acceptable shapesThe process exploits parametric


AnsysconverterICEM

EDGE(CFD solver)

Morphing ResultsCSTv3 shapechanges

KBSS feedback

Identification

lowastblk lowastrpl lowastexe

lowastmsh lowastcgns

Figure 4 The implemented procedure for 2D CFD analyses embedded into the CST class

Figure 5 Parametric meshing around undeformed airfoil and corresponding morphing leading and trailing edge shapes

(a) (b)

Figure 6 3D hybrid mesh for RANS computation around complete aircraft wo engine (a) and around the morphing leading edge (b)

meshing capabilities described in Section 34 to generatethe aerodynamic models This aerostructural scheme allowscombining the skin structural control and the estimationof global aerodynamic performances in the same shapeparametric representation and it is suitable to describe awide variety of wing shapes using a small number of designvariables

41 Wing-Box Implicit Constraint When a perturbation isintroduced in the CST geometry by means of a subsetof coefficients 119860

119894 the shape changes spread on the entire

domain and the other coefficients must be scaled in orderto keep the shape fixed along specific regions defined by Δ120595intervals If the number of perturbed coefficients is 119898 theother 119899minus119898+1 coefficients can be quickly computed startingfrom the following equation

t(119899minus119898)

(120595) sdot a119879 = 120577 (120595) minus 120595120577TE minus (1 minus 120595) 120577LE

minus 1198621198731

1198732(120595)119898

sum119894=1

119860119894119861119894(120595)

(9)


Given 1205941198951le119895le119897+1

isin (Δ1205951 Δ120595

119896) isin (0 1) pairwise distinct

real points included in a number of fixed subregions equal to119896 computing the (119899 minus 119898 + 1) coefficients a it is equivalentto solve the following reduced system smaller than thatreported in (6)

T119879(119899minus119898)

T(119899minus119898)

a119879 = T119879(119899minus119898)

f (10)

where T(119899minus119898)

is here defined (119897 + 1) times (119899 minus 119898 + 1) reducedCST-Vandermonde matrix for the CST basis 119879

(119899minus119898)of degree

less than or equal to 119899 minus 119898 and f is the vector obtainedevaluating the right term of (10) in the (119897 + 1) nodes In thisway the perturbation affects only the domain outside the fixedregions The code automatically selects the119898 coefficients bythe comparison between the subdomain and the positions ofthe airfoil components peaks defined in (2)

This approach is very efficient for the following mainreasons

(i) It allows introducing local perturbations by means ofa global geometry representation method

(ii) It preserves the 1198622 continuity and the local smooth-ness of the wing geometry which is one of the mostimportant features of the CST formulation

(iii) It reduces the number of external variables from 119899 to119898

(iv) It makes implicit constraints along the regions whichmust be kept undeformed

The results of a simple 2D example are reported in Figure 7 toshow the effects of the approach here presentedThe grey areacorresponds to the region of a considered airfoil that mustbe kept undeformed The blue surface is the CST parametricrepresentation of the upper surface corresponding to Bern-stein Polynomials Order of 8 (BPO8) The green surface wasobtained multiplying the coefficient119860

2by a factor equal to 3

The red surface was obtained in the same way but activatingthe solution of (10) It is possible to see how the perturbationspreads along the entire airfoil domain in the first case whileit is restricted outside the undeformed region in the secondcase

The approach here presented is extended to three dimen-sions by the CAD interpolation capabilities described inSection 34 Figure 8 shows an example of shape pertur-bations introduced outside the wing-box area of a 3Dwing

This method allows implementing implicit wing-boxconstraints into the shape optimization Every time the 119898coefficients of each section are changed and the least squareproblem is solved to constrain the shape in the undeformedregion Since the least square error does not guarantee thatthe shape passes through every surface point the new shape isnot exactly the same compared to the original one within theundeformed region In 2D the differences between the refer-ence and the deformed airfoil rapidly vanish increasing theorder of the Bernstein polynomial following the same trendalready described in [37] In 3D these residual differences

04

02

0

minus02

120577(120595

)middotc

05 1 15 2 250

120595 middot c

Figure 7 Comparison between a generic airfoil (black) its param-eterization (blue) and corresponding shapes obtained introducingthe same perturbation with (red) and without (green) solving thereduced system

Figure 8 Local perturbation and fixed wing-box parametric repre-sentation in three dimensions

also suffer the approximation introduced by the CAD-basedinterpolation technique working in the spanwise directionThe comparison between the actual and the deformed wingshape of the Reference Aircraft showed that using BernsteinPolynomial of Order eight and Bernstein Polynomial ofOrder ten to represent eight cross sections along the spanis a good compromise that allows introducing acceptablespanwise deflection law in the morphing regions and at thesame time obtaining a maximum error within the toleranceof 01mm in the wing-box region CFD analyses showedthat the pressure distribution for a wing shape affected bythis level of error appears to match the actual wing pressuredistribution

Wing-box constraints are very common in the aircraftshape design and several approaches that allow achieving thisgoal can be found in the literature Different parameterizationmethods could be used to improve the approximation in theundeformed region [38] or to increase the efficiency whenthey are used within an optimization process [39] Howeverthe estimation of the morphing skin structural behaviour isbased on the use of Bernstein polynomial as shape functionthat allows the structural quantities of Section 33 to beanalytically computed

42 Morphing Skin Structural Requirements Combining theestimation of the structural behavior of the morphing skinand the introduction of feasible perturbation described inSections 33 and 41 the KBSS feedback represents a specificalgorithm that allows introducing only morphing shapechanges that meet preassigned structural requirements Thedesign variables are composed by a subset of119898 independent


Figure 9 Parametric CAD model of a morphing winglet

coefficients 119860119894 while the general form of the algorithm can

be expressed as follows

Minimize 119908 sdot 120576axial (nal) + (1 minus 119908) sdot 120576bend (nal)

such that T119879(119899minus119898)

T(119899minus119898)

a119879 = T119879(119899minus119898)

f1003816100381610038161003816Δ119871dev

1003816100381610038161003816 le 119896119871 sdot 119871dev119906

|Δ119860| le 119896119860 sdot 119860119906

(11)

where 119908 is a weight coefficient that allows scaling the axialand the bending strain component and represents a degree offreedom to obtain differentmorphing shapesThis expressiondepends on the particular case study and can be replacedwitha more suitable form It represents a sort of shape variationlaw which defines a set of ldquoKBE parametersrdquo that can bechanged to provide a different control of the shape After-wards the first term is the implicit constraint for the solutionof (10) It acts on the structural box region delimited by thefront and rear spar position and described by 119897 + 1 pairwisedistinct coordinates 120594

1198951le119895le119897+1

isin (119862FS 119862RS) isin (119909LE 119888) Thesecond term 119896

119871sets the maximum length variation of the

morphing device skin Δ119871dev with respect to the undeformedone (119871dev119906) It can be related to the skin of a leading edgedevice (Δ119871LEskin) or to the upper and lower skin of a trailingedge device (Δ119871TEUpperSkin and Δ119871TELowerSkin) The inflatableterm 119896

119860that allows limiting the maximum allowable airfoil

internal area variation Δ119860 with respect to the undeformedone (119860

119906) is introduced to avoid the generation of unrealistic

morphing shapes with too much stretched thickness Moreterms such as stress limits can be added It was noticed thatquantities such as length and curvature then strain and stressdepend on the actual size of the shape and are computed withrespect to the variable 119909 instead of the nondimensional 120595

Algorithm which implements (11) can be applied to eachparametric shape included in the set of the most importantsections used to describe the complete 3D aircraft It works asa background process that provides a structural retrofit ableto interactly change the morphing wing shape in a feasibleway

The simplest example to demonstrate how the KBSS feed-back acts is to force the skin axial and bending deformation tobe equal to zero that should be equivalent to a rigid rotation of

a control surface A rigid tab connected to a winglet bymeansof two morphing transition regions is shown in Figure 9In this case the KBSS feedback worked to introduce twoequivalent rotations of 10 deg downward and 15 deg upwardover a deformable region which starts at 80 of the localchord

43 Genetic Coordinates When the shape changes are intro-duced by means of the shape optimization of Section 2the KBE parameters are included between the optimizationvariables in addition to the CST parameters 119877LE 119885LE 120573 120575119885TEΔ119885TE and 119888 defined in Section 31 and together with the119899 minus 1 shape function Bernstein polynomial extra coefficientsfor the upper surface and other 119899 minus 1 for the lower surface

According to the constant cross section length (CCL)concept [15] when the morphing leading or trailing edgechanges its shape the airfoil chord becomes smaller in orderto keep the skin length constant or satisfy the structuralconstraintsThe CST formulation as well as the aerodynamiccomputations works on nondimensional equation but allthe structural quantities such as 120576bend Δ120581(119909) and Δ119871dev aredimensional and depend on the actual size of the morphingskin

Once the wing-box region is detected a subset of CSTparameters can be defined as described in Section 41 Ifa morphing leading edge is considered for example only119885LE is considered a design variable used by the high-level optimizationwhich runs the aerodynamic computationwhile the other CST parameters are fixed by the boundaryconditions On the other side the KBSS feedback works ona subset of 119898 extra coefficients in addition to 119877LE and theairfoil chord 119888 that represent implicit variables not includedin the set of the high-level optimization variables and allowsatisfying the structural constraints included in the low-leveloptimization loop (11)

Both crossover and mutation subroutines execute theKBSS feedback in order to preserve the local feasibility whilededicated subroutine allows mixing only the same type ofgenetic coordinates The crossover randomly selects a pair ofparents who produce a new pair of offsprings by switchinginformation about a user-defined number of CST parametersof the same cross section preserving theKBEparametersTheeffect is to mix CST parameters between different individuals


12271m15732m

Figure 10 Reference Aircraft with morphing wing configuration and corresponding wing planform

05

1015

2025

3035

minus15minus10minus5051015

(m)

minus20246

Figure 11 Parametric identification applied to the Reference Aircraft model

and KBE parameters between different cross sections Themutation can directly change a user-defined number of KBEor CST parameter values in order to modify the local shape

5 Results

The procedure described in the previous sections has beenapplied for the optimization of the so-called ReferenceAircraft (RA) The morphing wing configuration and someinformation about the wing planform such as the total wingspan and the aileron span position are shown in Figure 10The RA is a typical regional jet capable of carrying 113 PAXwhich provides operational flexibility to accomplish differentmissions at the transonic regime It is capable of missions thatencompass 600 nm up to 2300 nm The Maximum TakeoffWeight is 58000 kg and the Maximum Landing Weight53000 kg while the Maximum Payload and the MaximumFuel are respectively 14000 kg and 18000 kg The ReferenceAircraft was designed to achieve an optimum cruise perfor-mance at Mach number equal to 078 an altitude of 37000 ftand an Angle of Attack AoA of 2 deg corresponding to alift coefficient 119862

119871of 047 The landing condition corresponds

to an Angle of Attack AoA of 15 deg while the consideredLanding Mach is 015

Different sections were chosen in order to identify theparametric shape of the Reference Aircraft and to build up

the mathematical model suitable to be used to introduce theshape changes into the leading and trailing edge morphingregions The whole process described in Section 2 has beenapplied to the parametric model of the Reference Aircraftwhich is shown in Figure 11 together with the identified crosssections

In the following subsections some results for morph-ing leading and trailing edge at different flight conditionsare summarized Since different design requirements havebeen provided for each morphing device different shapeoptimization problems have been defined They representfour aerodynamic shape optimizations including structuralconstraints on themorphing skin According to the approachdescribed in Section 2 they allow collecting a family ofoptimal morphing shapes that can be adopted as multipletargets for the design of a morphing mechanism

51 2D Shape Optimization of the Morphing Leading Edge forCruise Condition At the beginning a 2D shape optimizationbased on RANS computations has been applied to the wingsection placed at 9m along the wing span in the referencesystem of Figure 10 to lead the shape design of a morphingleading edge After the corresponding airfoil was identifieda simplified version of the procedure described in Section 2has been combined to the scheme shown in Figure 4 for theevaluation of the 2D aerodynamic coefficients In this way


the process automatically produces the mesh of both cleanand morphing airfoil shown in Figure 5 in order to estimatethe maximum benefit in terms of minimum 119862

119889keeping

the 119862119897as constant as possible In order to guarantee the

trim condition this last constraint forces the lift coefficientto be if not equal at least greater than the lift coefficientof the reference airfoil The evaluation of the aerodynamiccoefficients is performed at the cruise condition of the RAThe optimization problem is formulated as follows

Minimize 119862119889

such that T1198798T8a119879 = T119879

8f

Δ120595wing-box equiv (01 1)

119862119897ge 0566

|Δ119860| le 015 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0

max (120590bend) le 215MPa

(12)

where the first two constraints represent the equation system(10) used to keep undeformed region from the 10 of thechord up to the end of the trailing edgeThemain design rulefrom a structural point of view is keeping the circumferentialskin length 119871LEskin constant to avoid axial strains Theamount of droop of the individual section depends on themaximumbending stress into the skin (120590bend)Themaximumvalue of stress is related to a conventional leading edge skinmade in aluminium alloy (119864 = 72GPa) with a minimumthickness 119905 of 05mm and to the curvature change which isrestricted to 12(1119898)

The parametric airfoil shape is described by BernsteinPolynomials Order 119899 = 10 (BPO10) that involves a totalof 22 extra coefficients 11 for the upper skin and 11 forthe lower one Nevertheless the optimization variables arecomposed of 1 CST parameter 119885LE and 1 KBE parameter 119908The dimension of each genetic array is equal to 2 while theKBSS feedback works on a total of 119898 = 6 implicit variables2 for the upper surface (119860up1 and 119860up2) 2 for the lowersurface (119860 low1 and119860 low2) in addition to119877LE and to the airfoilchord 119888 as described in Section 43 The extra coefficients1198601and 119860

2correspond to the shape component peaks placed

in the region of the morphing leading edge The geneticalgorithm started with an initial population of 15 individualsand allows 50 generations The CFD grid is composed of260452 nodes each individual runs in parallel on 1 dedicatedquad-core CPU with processor base frequency of 293GHzand hyperthreading for a total of 8 threads and requiresabout 20minutes to solve RANS equations On the other sidethe KBSS feedback works for about 2 minutes mainly dueto satisfying the circumferential skin length constraint Theoptimal solution was obtained after 5 days and 25 iterationsThe total computational time could be drastically reducedusing Parallel Genetic Algorithm (PGA)

The vertical deflection which results from the shapeoptimization is equal to 119885LE = minus00260m over an airfoilchord of 28486m corresponding to an equivalent leading

edge deflection 120575LE = 529 deg The circumferential skinlength constraint associated with this morphing deflectionis satisfied considering a numerical tolerance of 1 sdot 119890minus8Corresponding benefits are computed by 2D RANS analysesin terms of aerodynamic coefficients that change from 119862

119897=

0566 119862119889

= 00185 to 119862119897= 0569 119862

119889= 00175

for a total efficiency improvement of 62 These resultsshow that the considered morphing leading edge is able toefficiently change its aerodynamic shape without generatingaxial stresses Figure 12 shows how the shape changes act onthe fore region of the shock wave adapting the aerodynamicfield around the airfoil to increase the lift over drag ratioand to keep the lift coefficient constant or greater than thereference one The optimal deformed airfoil shape is able todecrease the 119862

119889value while the 119862

119901values increase along the

upper skin and decrease along the lower skin so that the 119862119897

chordwise balance is similar to the undeformed one

52 3D Shape Optimization of the Morphing Leading Edgefor Cruise Condition The complete procedure described inSection 2 was applied to the 3D wing of the RA to maximisethe improving of the aerodynamic efficiency 119871119863 of the wingin cruise condition by means of a morphing leading edge Aset of 7parametric sections along the span 4 equally spaced inthe inboard region from the wing root to the wing kink and4 equally spaced in the outboard region from the kink to thewing tip was identified in order to introduce the morphingshape changes Each wing section is represented by BernsteinPolynomials Order 119899 = 10 (BPO10) for a total number ofextra coefficients equal to 154 The optimization variables arecomposed of 1 CST parameter 119885LE and 1 KBE parameter119908 for each section for a total of 14 explicit optimizationvariables that form each genetic array Since the algorithm isapplied to a 10 chordwise configuration 7 KBSS feedbackworks on 6 implicit variables (119860

1and 119860

2for the upper and

lower surfaces 119877LE and 119888) for each parametric section Theoptimization problem can be formulated as follows

Maximize 119871119863

such that T1198798T8a119879 = T119879

8f


|Δ119860| le 012 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0


(13)

TheKBSS feedback acting on each wing section works to keepundeformed local regions between the 10 of the chord andthe end of the trailing edge to limit the airfoil area variationand to avoid the circumferential skin length variation Theoptimization parameters are set up to 119896

119871LE = 1 sdot 119890 minus 6(120590axial = 0) and 119896

119860= 012 The maximum deflection range

is defined by the same curvature constraint used in the 2Doptimization while a 3D constraint was added in order tokeep linear deflection law along the wing span


0 05 1 15 2 25 3

minus15

minus1

minus05

0

05

1

Chord (m)

ReferencemLE2c10-529

Cp

Mac

h

130

129

120

110

101

092

083

074

064

055

046

037

028

018

009

000

X

Y

ZM

ach

130

129

120

110

101

092

083

074

064

055

046

037

028

018

009

000

X

Y

Z

Figure 12 Morphing leading edge in high speed conditions comparison between reference and deformed shape 2D results in terms of 119862119901

and Mach chordwise distributions

The genetic algorithm started with an initial populationof 40 individuals and allows 100 generations The CFD gridis composed of 1 million nodes each individual runs inparallel on 1 dedicated quad-core CPU with hyperthreadingfor a total of 8 threads and requires about 15 minutes ofcomputational time to solve Euler equations Each KBSSfeedback works for 2 minutes on different threads Theoptimal solution was obtained after about 2 weeks and 40iterations

The shape optimization process provided a spanwisedistribution of optimal leading edge deflections 120575LE thatlinearly decreases from 8 deg at the wing root to 72 degat the station placed at the 25 of the outboard region

Beyondmorphing shape variations vanish and no deflectionsare introduced by the optimizer from 25 to 100 of theoutboard region

The results are reported in Figure 13 and show how theleading edge shape changes are able to adapt the aerodynamicfield over the wing in order to increase the wing efficiencyThe black line represents the iso-Mach curve correspondingto a local velocity equal to the speed of sound The resultsshow that in the transonic condition the morphing leadingedge is able to change the chordwise119862

119901distribution inside the

shock wave following the same trend of Section 51 Figure 14shows the chordwise119862

119901distribution around the wing section

placed at 47m along the wing span in the reference system


Cp

050

020

minus010

minus040

minus070

minus100

Figure 13 Comparison between the 119862119901distributions at cruise Mach over the reference wing (right) and over the wing equipped with

morphing leading edge (left)

0 05 1 15 2 25 3 35 4 45

minus15

minus1

minus05

0

05

1

Chord (m)

ReferencemLEc10-3D

Cp

Figure 14 Comparison between reference and morphing shape 3Dresults in terms of 119862

119901chordwise distribution of the wing section

placed at 47m along the wing span

of Figure 10 However since the swept wing reduces thelocal speed it is more difficult to achieve the improvementsobtained in the 2D case and the corresponding increase ofwing efficiency is limited to 34 For this reason the shapeoptimization works to introduce leading edge deflectionsonly in the inboard region where the local aerodynamic flowis still parallel to the asymptotic speed and more similar tothe 2D case

The optimization algorithm acted to decrease the refer-ence lift coefficient from 119862

119871= 049 computed by Euler

method to 119862119871= 0487 This means that the Angle of Attack

should be increased from 2 deg to 203 deg to guarantee thetrim in cruise condition Additional computations showedthat this small variation does not reduce significantly thebenefits related to this morphing shape solution

53 3D Shape Optimization of the Morphing Leading Edgefor Low Speed Condition A second 3D optimization processwas run using an objective function that tries to introducethe maximum droop deflection along the wing span insteadof an aerodynamic performance estimation This may besuitable for a low speed condition corresponding to thelanding condition of the RA A set of 4 identified parametricsections equally spaced along the outboard region wereused in order to introduce the morphing shape changesThe droop deflection to be maximized along the wing spanis the equivalent leading edge deflection 120575LE introduced inSection 31 Each wing section is represented by BernsteinPolynomials Order of 119899 = 10 (BPO10) for a total numberof extra coefficients equal to 88 and a family of possibleshapes has been generated for different values of 119908 Each ofthe 4 KBSS feedback algorithms works on 6 implicit variables(1198601and 119860

2for the upper and lower surfaces 119877LE and 119888)

for each section when the algorithm is applied to the 10chordwise configuration and 8 implicit variables (119860

1 1198602

1198603 119877LE and 119888) when the 20 chordwise configuration is

considered The total number of optimization variables is 4explicit variables and 24 or 32 implicit variables dependingon the type of chordwise morphing configuration Theoptimization problem for the 20 chordwise configurationcan be formulated as follows

Maximize 1

4

4

sum119894=1

120575LE119894

such that T1198797T7a119879 = T119879

7f


|Δ119860| le 02 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0

max (Δ120581 (119909)) le 28 1119898

(14)

The optimization works keeping the KBE parameter 119908 fixedon each section It is considered as an external parameterin addition to the KBSS feedback ones The constraints


include a linear deflection law in spanwise direction alongthe outboard region while the maximum skin curvaturechange is set to avoid strains above 1 according to whatobserved in other works such as that described in [4]Due to the high deflection levels the curvature constraintis expressed in terms of curvature variation that dependsonly on geometrical considerations instead of bending stressthat requires knowing the material properties and the skinthicknessThe first one is not easy to estimate at this stage dueto the specific materials used for the morphing skin the skinthickness could be variable along the leading edge arc lengthThe optimal shapes are chosen according to the followingconsideration the alleviation of the bending stress inside theskin corresponds to the increase of the leading edge radiusthat helps to preserve laminar flow for low speed and high liftconditions

The genetic algorithm started with an initial populationof 20 individuals and allows 80 generations Since no CFDanalysis was performed the optimization process was veryfast and the optimal solution was obtained after less than 1day only to satisfy all the structural constraints along thewing span The shape optimization provided the morphingleading edge shapes shown in Figure 15Themaximumdroopdeflection corresponding to the 20 chordwise configura-tion linearly decreases from 120575LE = 183 deg at the wing kinkto 120575LE = 112 deg at the wing tip Corresponding curvaturedifference functions computed along the morphing leadingedge arc length and along the wing span are reported inFigure 16 and show the values are lower than the maximumcurvature change limit

Different types of 10 and 20 chordwise morphingleading edge shapes for different values of 119908 were analyzedOnce the best shapes were selected they have been imple-mented on the Reference Aircraft 3D model of Figure 11 andpostprocessed adopting RANS computations The analysesperformed after the optimization process showed that forthe subsonic flight regime there is an improvement in thelift over drag ratio between 043 and 10 of the lift coefficient[40] This improvement is about 5 at 119862

119871= 06 for the 10

chordwise configuration and the same value can be achievedat 08 le 119862

119871le 09 for the 20 chordwise configuration

In order to quantify the benefits in terms of fuel saving atypical flightmission was consideredThismission comprises600 nm point to point distance with a holding phase of 45minutes It was noticed that the holding phase is performedat the lift coefficient that falls inside the range in which the10 chordwise configuration can provide benefits in the 119871119863ratio The adoption of this kind of morphing device at thisflight phase would lead to 62Kg of fuel saving

54 3D Shape Optimization of the Morphing Trailing Edgefor Cruise Condition The shape optimization procedure wasapplied to the wing of the Reference Aircraft to improve theperformances by means of a morphing trailing edge opti-mized for the cruise conditionThe fitness function is relatedto the aerodynamic efficiency 119871119863 while the optimizationprocess works without spanwise deflection law constraintsand tries to keep the total 119862

119871not less than the reference

one requested for trimming A set of 7 parametric sections

along the span 4 equally spaced in the inboard region fromthe wing root to the wing kink section and 4 equally spacedin the outboard region from the same kink section to thetransition region before the aileron was identified in order tointroduce the morphing shape changes Each wing section isrepresented by Bernstein Polynomials Order of 119899 = 8 (BPO8)for a total number of extra coefficients equal to 126 Theoptimization variables are composed of 2 CST parameters120573 119885TE and 1 KBE parameter 119908 for each section for a totalof 21 explicit optimization variables that form the geneticarray Since the algorithm is applied to a 28 chordwiseconfiguration each of the 7 KBSS feedback algorithms workson 7 implicit variables (119860

5 1198606 and 119860

7for the upper and

lower surfaces in addition to the airfoil chord 119888) for a totalof 49 implicit variables The optimization problem can beformulated as follows

Maximize 119871119863

such that T1198795T5a119879 = T119879

5f

Δ120595wing-box equiv (0072)

119862119897ge 049

|Δ119860| le 03 sdot 119860119906

10038161003816100381610038161003816Δ119871TEUpperSkin10038161003816100381610038161003816 = 0

1003816100381610038161003816Δ119871TELowerSkin1003816100381610038161003816 le 005 sdot 119888

minus 5 deg le 120575TE le +5 deg

(15)

Different structural requirements were adopted for the trail-ing edge skin the upper skin length constraint is set up to beconstant (119896

119871UpperSkin = 1 sdot 119890 minus 6meaning that the upper skinaxial strain is 120576axialUpperSkin = 0) while the lower skin lengthconstraint has been relaxed in order to obtain an efficientaerodynamic shape This means that the lower skin is free toslide along its contour and corresponds to the introductionof a linear slider to avoid axial strain The constraints relatedto the skin curvature variations are not taken into accountdue to the small deflections which are bounded in the rangeof plusmn5 deg The inflatable term is set up to 119896

119860= 03 and the

trailing edge equivalent deflection 120575TE is bounded betweenminus5∘ and +5∘

The population size and the grid size of the aerodynamicmodel are more or less the same used for the study describedin Section 52 The total computational time was about 10daysThe shape optimization provided the morphing trailingedge shapes shown in Figure 17 Corresponding optimalspanwise deflections increase from minus31 deg at the wing rootto 22 deg at the transition region close to the aileron

The aerodynamic results are reported in Figure 18 andshow how the morphing trailing edge is able to changeits shape in order to adapt the aerodynamic field aroundthe wing Differently from what happens for the morphingleading edge the trailing edge deflection is able to move theshock wave as shown by the black line that represents theiso-Mach curve where the local velocity is equal to the speedof sound The trailing edge spanwise deflection moves the


Figure 15 Parametric CAD model of the 10 and 20 chordwise morphing leading edge for maximum droop angle under skin structuralconstraint

30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)

0 02 04 06 08 1Nondimensional leading edge arc length

minus6minus8minus10minus12minus14

Span

wise (m

)

30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)

Figure 16 Curvature difference functions along the wing span for 10 and 20 chordwise maximum droop angle

shock wave forward in the inboard region and backwards inthe outboard region In order to improve the wing efficiencythe spanwise antisymmetric deflections reduce the local liftcoefficient 119862

119897in the region close to the wing root and

increase it in the region near the aileron This variation canbe partially attributed to the wing load distribution whichbecomes elliptical rather than triangular This effect positivefrom the aerodynamic point of viewmoves the centre ofpressure outboard and leads to increase of the root bendingmoment

The optimization algorithm was able to improve the liftover drag ratio of 3 increasing the lift coefficient of 2The trim condition should be recomputed and correspondingAngle of Attack reduced to obtain the same total lift coeffi-cient required for the cruise For this reason the morphingtrailing edge shapes obtained by the optimization processhave been implemented on the Reference Aircraft and severalhigh-fidelity CFD computations have been performed on thecomplete aircraft [40]The above decreasing deflection law inspanwise direction allowed increasing the119871119863 of 2 for119862

119871=

045 and 585 for 119862119871= 055 This result is an indication that

the potential aerodynamic benefits can be translated into fuelsaving However the observed aerodynamic advantages arejust part of the feasibility evaluation which must involve alarger amount of disciplines so during the structural designof the wing-box the impact on the overall wing weightof increasing the wing bending moment during the cruisecondition should be evaluated

6 Conclusions

Morphing seems a potentially promising technology allowingmatching the new stringent requirements in terms of envi-ronmental impact of next generation aircraft Unfortunatelythe results available in the literature concerning the applica-tion of morphing concepts still combine lights and shadowsin terms or real benefits and open issues like certificationfatigue and so on For sure the lack of design proceduresspecifically dedicated to the optimal design of morphingmechanisms appears clearly This paper introduced in ageneral way the work carried out at Politecnico di Milanoto set up a complete framework for the optimal design


Figure 17 Parametric CAD model of the 28 chordwise morphing trailing edge for high speed condition under skin structural constraint

Cp

050

020

minus010

minus040

minus070

minus100


morphing trailing edge (left)

of morphing mechanisms based on compliant mechanismsIn particular the paper focused on the first level of theproposed multilevel design procedure Indeed the so-calledknowledge-based shape optimization procedure has beenintroduced and described that is able to combine aerody-namic with structural performances mainly related to thebehavior of the skin during the shape variationThe proposedprocedure has been evaluated in the design of morphingwing based on conformable leading and trailing edge surfacesable to adapt the wing camber during the mission profileAn external optimization loop works on the most importantdesign variables that affect the camber morphing and itis dedicated to the aerodynamic performance evaluationA nested loop based on a particular development of CSTmethod guarantees that the outer loopworks only on feasibleshapes able to satisfy wing-box volume constraints andmorphing skin requirements

Some examples are reported concerning the so-calledReference Aircraft a regional-type aircraft developed insideNOVEMOR project and used as a test bench for a finalassessment on morphing applications The results obtainedfor the leading edge device at the subsonic regime and thetrailing edge device at the transonic flight regime have beenselected and the mission analyses have been completed toquantify possible benefits in terms of fuel saving Duringthese studies only the aerodynamic behaviour and the

structural behaviour of the morphing skin were consideredNeither the impact on the maximum takeoff weight of themorphing mechanisms nor the effect of aeroelasticity on thebehavior of morphing devices has been considered A familyof optimal shapes have been produced and verified using bothmedium and high-fidelity tools showing promising results aswell proving the versatility of the proposed approach Duringthe second level of the same procedure all the optimal shapescan be adopted as multiple targets for the optimal design ofthe morphing mechanism according to the multiobjectivedesign strategy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Special thanks go to Alexandre Antunes Felipe Odaguil andGrace Lima from EMBRAER for the CFD and mission anal-ysis performed on Reference Aircraft The research leadingto these results has been partially funded by the EuropeanUnionrsquos Seventh Framework Programme [FP72007ndash2013]under Grant Agreement no 285395


References

[1] L Breguet ldquoAerodynamic efficiency and the reduction of airtransport costsrdquoAeronautical Journal vol 26 pp 307ndash313 1922

[2] M Cavcar ldquoBreguet range equationrdquo Journal of Aircraft vol43 no 5 pp 1542ndash1544 2006

[3] A De Gaspari S Ricci L Riccobene and A Scotti ldquoActiveaeroelastic control over a multisurface wing modeling andwind-tunnel testingrdquoAIAA Journal vol 47 no 9 pp 1995ndash20102009

[4] H P Monner M Kintscher T Lorkowski and S StormldquoDesign of a smart droop nose as leading edge high liftsystem for transportation aircraftsrdquo in Proceedings of the 50thAIAAASMEASCEAHSASC Structures Structural Dynamicsand Materials Conference (SDM rsquo09) AIAA 2009-2128 PalmSprings Calif USA May 2009

[5] J J Spillman ldquoTheuse of variable camber to reduce drag weightand costs of transport aircraftrdquoAeronautical Journal vol 96 no951 pp 1ndash9 1992

[6] A McGowan ldquoOverview morphing activities in the USArdquo inProceedings of the Advanced Course on Morphing Aircraft Mate-rials Mechanisms and Systems Lisbon Portugal November2008

[7] J N Kudva ldquoOverview of the DARPA smart wing projectrdquoJournal of Intelligent Material Systems and Structures vol 15 no4 pp 261ndash267 2004

[8] D Wagg I Bond P Weaver and M Friswell Adaptive Struc-tures Engineering Applications John Wiley amp Sons 2007

[9] C Thill J Etches I Bond K Potter and P Weaver ldquoMorphingskinsrdquoThe Aeronautical Journal vol 38 no 10 2008

[10] D Steenhuizen and M van Tooren ldquoThe implementation of aknowledge-based framework for the aerodynamic optimizationof a morphing wing devicerdquo Advanced Engineering Informaticsvol 26 no 2 pp 207ndash218 2012

[11] G Molinari A F Arrieta and P Ermanni ldquoAero-structuraloptimization of 3-D adaptive wings with embedded smart actu-atorsrdquo in Proceedings of the 54th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials ConferenceBoston Mass USA April 2013

[12] P D Ciampa T Zill T Pfeiffer and B Nagel ldquoA functionalshape parametrization approach for preliminary optimizationof unconventional aircraftrdquo inProceedings of the 3rd CEASAirampSpace Conferencemdash21st AIDAA Congress pp 1513ndash1524 VeniceItaly 2011

[13] S E Gano VM Perez J E Renaud and SM Batill ldquoMultilevelvariable fidelity optimization of a morphing unmanned aerialvehiclerdquo inProceedings of the 45thAIAAASMEASCEAHSASCStructures Structural Dynamics and Materials Conference(AIAA rsquo04) Palm Springs Calif USA April 2004

[14] S Barbarino O Bilgen R M Ajaj M I Friswell and D JInman ldquoA review of morphing aircraftrdquo Journal of IntelligentMaterial Systems and Structures vol 22 no 9 pp 823ndash877 2011

[15] A De Gaspari and S Ricci ldquoA two-level approach for the opti-mal design of morphing wings based on compliant structuresrdquoJournal of IntelligentMaterial Systems and Structures vol 22 no10 pp 1091ndash1111 2011

[16] C O Johnston D A Neal L D Wiggins H H RobertshawW H Mason and D J Inman ldquoA model to compare the flightcontrol energy requirements of morphing and conventionallyactuated wingsrdquo in Proceedings of the 11th AIAAASMEAHSAdaptive Structures Conference (AIAA rsquo03) Norfolk Va USAApril 2003

[17] B C Prock T A Weisshaar and W A Crossley ldquoMorphingairfoil shape change optimization with minimum actuatorenergy as an objectiverdquo in Proceedings of the 9th AIAAISSMOSymposium on Multidisciplinary Analysis and Optimization(AIAA rsquo02) Atlanta Ga USA September 2002

[18] H Namgoong W A Crossley and A S Lyrintzis ldquoAerody-namic optimization of a morphing airfoil using energy as anobjectiverdquo in Proceedings of the 44th AIAA Aerospace SciencesMeeting and Exhibit (AIAA rsquo06) Reno Nev USA January 2006

[19] P Gamboa J Vale F J P Lau andA Suleman ldquoOptimization ofamorphing wing based on coupled aerodynamic and structuralconstraintsrdquo AIAA Journal vol 47 no 9 pp 2087ndash2104 2009

[20] A De Gaspari S Ricci and L Travaglini ldquoAeroelastic analysisof a regional aircraft with active camber morphing devicerdquo inProceedings of the International Forum on Aeroelasticity andStructural Dynamics (IFASD rsquo15) Saint Petersburg Russia June-July 2015

[21] A De Gaspari A two levels approach for the optimal designof morphing wings based on compliant structures [PhD the-sis] Dipartimento di Ingegneria Aerospaziale Politecnico diMilano Milano Italy 2010

[22] A De Gaspari and S Ricci ldquoApplication of the active cambermorphing concept based on compliant structures to a regionalaircraftrdquo in Industrial and Commercial Applications of SmartStructures Technologies vol 9059 of Proceedings of SPIE pp 1ndash21 San Diego Calif USA April 2014

[23] Y He and R K Agarwal ldquoShape optimization of NREL S809airfoil for wind turbine blades using a multiobjective geneticalgorithmrdquo International Journal of Aerospace Engineering vol2014 Article ID 864210 13 pages 2014

[24] A Shahrokhi and A Jahangirian ldquoAirfoil shape parameteriza-tion for optimumNavier-Stokes design with genetic algorithmrdquoAerospace Science and Technology vol 11 no 6 pp 443ndash4502007

[25] T A Weisshaar ldquoMorphing aircraft technologymdashnew shapesfor aircraft designrdquo in Proceedings of the RTO-MP-AVT-141mdashMultifunctional StructuresIntegration of Sensors and Antennaspp 1ndash20 Neuilly-sur-Seine France October 2006

[26] M Secanell A Suleman andPGamboa ldquoDesign of amorphingairfoil using aerodynamic shape optimizationrdquo AIAA Journalvol 44 no 7 pp 1550ndash1562 2006

[27] B M Kulfan ldquoUniversal parametric geometry representationmethodrdquo Journal of Aircraft vol 45 no 1 pp 142ndash158 2008

[28] K Deb ldquoSingle and multi-objective optimization using evolu-tionary computationrdquo in Proceedings of the 6th InternationalConference on Hydro-Informatics pp 14ndash35 Singapore 2004

[29] D Goldberg Genetic Algorithms in Search Optimization andMachine Learning Addison-Wesley Boston Mass USA 1989

[30] A Marco and J-J Martınez ldquoPolynomial least squares fittingin the Bernstein basisrdquo Linear Algebra and its Applications vol433 no 7 pp 1254ndash1264 2010

[31] Icem 121 Programmers Guide ANSYS 2008[32] FOI CFD Flow Solver for Unstructured Grids Division of

Defence and Security Systems and Technology Department ofComputational Physics 2009 httpwwwfoiseedge

[33] S Lebofsky E Ting and N Nguyen ldquoMultidisciplinary dragoptimization of reduced stiffness flexible wing aircraft withvariable camber continuous trailing edge flaprdquo in Proceedings ofthe 56thAIAAASCEAHSASC Structures StructuralDynamicsand Materials Conference pp 1ndash27 AIAA SciTech KissimmeeFla USA January 2015


[34] S Ghasemi A Mosahebi and E Laurendeau ldquoA two-dimensionalinfinite swept wing navier-stokes solverrdquo in Pro-ceedings of the 52nd Aerospace Sciences Meeting pp 1ndash11 AIAASciTech National Harbor Md USA January 2014

[35] M Tomac and D Eller ldquoFrom geometry to CFD gridsmdashan automated approach for conceptual designrdquo Progress inAerospace Sciences vol 47 no 8 pp 589ndash596 2011

[36] M Tomac and D Eller ldquoSteps towards automated robustRANS meshingrdquo in Proceedings of the 4th CEAS Air amp SpaceConference pp 1ndash10 Linkoping Sweden 2013

[37] B M Kulfan ldquolsquoFundamentalrsquo parametric geometry represen-tations for aircraft component shapesrdquo in Proceedings of the11th AIAAISSMO Multidisciplinary Analysis and OptimizationConference Portsmouth Va USA September 2006

[38] F Zhu and N Qin ldquoIntuitive classshape function parameteri-zation for airfoilsrdquo AIAA Journal vol 52 no 1 pp 17ndash25 2014

[39] M H Straathof and M J L van Tooren ldquoExtension tothe classndashshapendashtransformation method based on BndashsplinesrdquoAIAA Journal vol 49 no 4 pp 780ndash790 2011

[40] A De Gaspari S Ricci A Antunes F Odaguil and G LimaldquoApplication of active camber morphing concept to a regionalaircraftrdquo in Proceedings of the 22nd AIAAASMEAHS AdaptiveStructures Conference AIAA 2014-125 pp 1ndash23 National Har-bor Md USA January 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



shape without surface discontinuities often based on theadoption of ad hoc designed flexible skins [9] appears asan efficient way to maximize the lift over drag ratio and toreduce the fuel consumption over the entire flight envelopeHowever the design of this kind ofmorphing devices requiresdeveloping specific procedures able to assist the engineersduring both the design [10ndash12] and the benefits evaluationphases Currently this target is commonly addressed bymeans of dedicated multilevel and multiobjective optimiza-tion procedures [13] Multilevel capabilities allow performingthe optimization of morphing shapes and the design ofthe morphing mechanism separately [14] multiobjectivetechniques help to design aircraft able to adapt its shape tooptimize the performances along the cruise or at a wide rangeof different flight conditions

The design of morphing wing devices must combine twoopposite requirements often named kinematic and structuralrequirements respectively a flexible structure so tominimizethe energy necessary to adapt its shape as expected and atthe same time an enough rigid structure able to maintain thenew shape under the aerodynamic loads when the morphingmechanism is not actuated The approach proposed by theauthors [15] is based on the optimization of aerodynamicstiffness and actuation sequentially passing through thedefinition of a family of optimal morphing shapes associatedwith a group of as many flight conditions Hence the designof morphing mechanism depends on the availability of theoptimal shapes that must be achieved when it is actuated andthat are computed before the mechanism is known In thisway the optimal shapes guarantee the aerodynamic perfor-mances while the mechanism can be optimized consideringboth kinematic and structural requirementsThekey problemis that the allowable shape variation laws depend on the typeof mechanism while the mechanism design depends on theshape changes that the mechanism must be able to reachFor this reason in recent years many efforts have been madeaiming at considering energy and actuation requirements[16ndash18] as well as structural constraints [19] directly duringthe aerodynamic shape optimization

The work presented in this paper focuses on the first levelof a wider morphing design framework having the follow-ing capabilities wing shape optimization able to combineaerodynamic performances with optimal deformation of theskins (first level) optimal design of compliant mechanismable to produce once actuated the optimal shape comingout from the first level (second level) integration of themorphing devices into a high-fidelity model representing thecomplete aircraft for final aeroelastic assessment to evaluatethe reliability of the designed morphing solutions [20]

Initially the first level was implemented as a simple2D shape optimization linked to a viscous and subsonic2D aerodynamic solver while the second one represented ageneral code for the synthesis of compliant mechanisms [1521] and two software pieces have been developed separatelyRecently the first one has been extended in order to produce3D wing models starting from the results which continuedto be obtained by 2D shape optimization while the secondone has been improved by adding multiobjective capabilities[22]

The paper describes the novel progress beyond thispoint about the first level optimization which is now basedon a comprehensive knowledge-based engineering (KBE)framework able to define the optimal morphing wing shapein terms of mission profile performances directly in thethree-dimensional space It is based on coupling a paramet-ric geometry representation able to predict the structuralresponse of morphing skin with Computer-Aided Engi-neering (CAE) capabilities Object-Oriented Programming(OOP) genetic algorithm and aerodynamic analyses Theresults obtained applying it to a Reference Aircraft (RA)coming from the FP7 EU NOVEMOR project (Novel AirVehicle Configurations from Fluttering Wings to MorphingFlight) and adopted as a benchmark to evaluate the optimalbenefits that can bring in terms of global performances arereported to validate the proposed procedure and to evaluatethe impact of continuous chordwise and spanwise cambervariation in terms of lift to drag ratio and aerodynamic loaddistribution

2 Shape Design Methodology

The shape design procedure presented in this work consistsof a shape optimization based on genetic algorithm andcoupled with the parametric framework introduced in thefollowing section able to design the optimal morphingshapes from the aerodynamic point of view under skinstructural requirements While genetic algorithms have beenalready coupledwith 2D shape optimization of airfoil [23 24]this paper presents an approach based on a particular use ofthe genetic algorithms able to combine the set of the mostimportant cross sections describing the wing in order todefine the optimal shape of a 3D morphing wing

The set of parameterized cross section shapes is thestarting point of the whole process shown in Figure 1 Theouter optimization loop based on genetic algorithm workson the most important design variables affecting the cambermorphing such as the leading and trailing edge deflectionsAfter the cross sections are combined by PHORMA the codedescribed in Section 31 the 3D model is analytically definedand its shape can be easily controlled and used to estimatethe aerodynamic performances by means of the user-definedfitness evaluation function Different performance indicescan be enclosed in the shape optimization process in orderto improve the aerodynamic efficiency and reduce the fuelconsumption or to guarantee that the morphing aircrafthas optimal aerodynamic characteristics over different flightconditions [25 26] The inner loop based on a dedicateddevelopment of CST method [27] guarantees that the outerloop works only on feasible shapes able to satisfy wing-boxvolume constraints and morphing skin structural require-ments

This parametric shape representation is suitable to beincorporated into genetic algorithm and to be parallelized[28 29] The most innovative aspect of this implementationis that the skin structural behaviour recovered from theadopted geometry representation determines the ruleswhichdrive the reproduction process of the population In this waythe resulting morphing shape comes from a natural selection


Initial population

PHORMA

Aerodynamic model

Solver


Selection

Crossover




Mutation

End

Parent individuals

Yes


CST identification

Chromosome assembly

No












Postprocessing

XML file


Solver


(structured grids)

ICEM STEPfile

IGESfile


ANSYSAPDLmodel

class









120577(120595)

120575LE

LE morphing

ZLERLE

CFS

CRS

c

120573


120595

ZTE

ΔZTE

120575TE

Structural box

xLE



120577 (120595) = 1198621198731

1198732(120595) a sdot b119879

119899(120595) + 120595120577TE + (1 minus 120595) 120577LE (1)



119894(120595)










120595max(119885) =1198731 + 119894

1198731 + 1198732 + 119899for 119894 = 0 119899 (2)






[1198621198731



119879119899

= 119905(119899)

119894= 1198621198731

1198732(120595)(

119899

119894) (1 minus 120595)

119899minus119894120595119894 119894 = 0 119899

(4)


t119899(120595) sdot a119879 = 119891 (120595) (5)




1198951le119895le119897+1


T119899a119879 = f (6)





119899has full rank (119899 + 1) the


T119879119899T119899a119879 = T119879

119899f (7)





minus 1198621198731

1198732(120595) [119860

01198610(120595) + 119860

119899119861119899(120595)]

(8)



























AnsysconverterICEM

EDGE(CFD solver)


KBSS feedback

Identification





(a) (b)






t(119899minus119898)


minus 1198621198731

1198732(120595)119898

sum119894=1

119860119894119861119894(120595)

(9)


Given 1205941198951le119895le119897+1

isin (Δ1205951 Δ120595



T119879(119899minus119898)

T(119899minus119898)

a119879 = T119879(119899minus119898)

f (10)















04

02

0

minus02

120577(120595

)middotc

05 1 15 2 250

120595 middot c











such that T119879(119899minus119898)

T(119899minus119898)

a119879 = T119879(119899minus119898)

f1003816100381610038161003816Δ119871dev

1003816100381610038161003816 le 119896119871 sdot 119871dev119906

|Δ119860| le 119896119860 sdot 119860119906

(11)


1198951le119895le119897+1
















12271m15732m


05

1015

2025

3035


(m)

minus20246



5 Results










119889keeping



Minimize 119862119889

such that T1198798T8a119879 = T119879

8f


119862119897ge 0566

|Δ119860| le 015 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0


(12)







119897=

0566 119862119889

= 00185 to 119862119897= 0569 119862

119889= 00175







1and 119860

2for the upper and


Maximize 119871119863

such that T1198798T8a119879 = T119879

8f


|Δ119860| le 012 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0


(13)






0 05 1 15 2 25 3

minus15

minus1

minus05

0

05

1

Chord (m)


Cp

Mac

h

130

129

120

110

101

092

083

074

064

055

046

037

028

018

009

000

X

Y

ZM

ach

130

129

120

110

101

092

083

074

064

055

046

037

028

018

009

000

X

Y

Z












Cp

050

020

minus010

minus040

minus070

minus100



0 05 1 15 2 25 3 35 4 45

minus15

minus1

minus05

0

05

1

Chord (m)

ReferencemLEc10-3D

Cp












1 1198602



Maximize 1

4

4

sum119894=1

120575LE119894

such that T1198797T7a119879 = T119879

7f


|Δ119860| le 02 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0

max (Δ120581 (119909)) le 28 1119898

(14)






119871= 06 for the 10








5 1198606 and 119860

7for the upper and


Maximize 119871119863

such that T1198795T5a119879 = T119879

5f


119862119897ge 049

|Δ119860| le 03 sdot 119860119906

10038161003816100381610038161003816Δ119871TEUpperSkin10038161003816100381610038161003816 = 0



(15)



119860= 03 and the






30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)

30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)






119871=



6 Conclusions




Cp

050

020

minus010

minus040

minus070

minus100








Acknowledgments



References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Initial population

PHORMA

Aerodynamic model

Solver


Selection

Crossover




Mutation

End

Parent individuals

Yes


CST identification

Chromosome assembly

No












Postprocessing

XML file


Solver


(structured grids)

ICEM STEPfile

IGESfile


ANSYSAPDLmodel

class









120577(120595)

120575LE

LE morphing

ZLERLE

CFS

CRS

c

120573


120595

ZTE

ΔZTE

120575TE

Structural box

xLE



120577 (120595) = 1198621198731

1198732(120595) a sdot b119879

119899(120595) + 120595120577TE + (1 minus 120595) 120577LE (1)



119894(120595)










120595max(119885) =1198731 + 119894

1198731 + 1198732 + 119899for 119894 = 0 119899 (2)






[1198621198731



119879119899

= 119905(119899)

119894= 1198621198731

1198732(120595)(

119899

119894) (1 minus 120595)

119899minus119894120595119894 119894 = 0 119899

(4)


t119899(120595) sdot a119879 = 119891 (120595) (5)




1198951le119895le119897+1


T119899a119879 = f (6)





119899has full rank (119899 + 1) the


T119879119899T119899a119879 = T119879

119899f (7)





minus 1198621198731

1198732(120595) [119860

01198610(120595) + 119860

119899119861119899(120595)]

(8)



























AnsysconverterICEM

EDGE(CFD solver)


KBSS feedback

Identification





(a) (b)






t(119899minus119898)


minus 1198621198731

1198732(120595)119898

sum119894=1

119860119894119861119894(120595)

(9)


Given 1205941198951le119895le119897+1

isin (Δ1205951 Δ120595



T119879(119899minus119898)

T(119899minus119898)

a119879 = T119879(119899minus119898)

f (10)















04

02

0

minus02

120577(120595

)middotc

05 1 15 2 250

120595 middot c











such that T119879(119899minus119898)

T(119899minus119898)

a119879 = T119879(119899minus119898)

f1003816100381610038161003816Δ119871dev

1003816100381610038161003816 le 119896119871 sdot 119871dev119906

|Δ119860| le 119896119860 sdot 119860119906

(11)


1198951le119895le119897+1
















12271m15732m


05

1015

2025

3035


(m)

minus20246



5 Results










119889keeping



Minimize 119862119889

such that T1198798T8a119879 = T119879

8f


119862119897ge 0566

|Δ119860| le 015 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0


(12)







119897=

0566 119862119889

= 00185 to 119862119897= 0569 119862

119889= 00175







1and 119860

2for the upper and


Maximize 119871119863

such that T1198798T8a119879 = T119879

8f


|Δ119860| le 012 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0


(13)






0 05 1 15 2 25 3

minus15

minus1

minus05

0

05

1

Chord (m)


Cp

Mac

h

130

129

120

110

101

092

083

074

064

055

046

037

028

018

009

000

X

Y

ZM

ach

130

129

120

110

101

092

083

074

064

055

046

037

028

018

009

000

X

Y

Z












Cp

050

020

minus010

minus040

minus070

minus100



0 05 1 15 2 25 3 35 4 45

minus15

minus1

minus05

0

05

1

Chord (m)

ReferencemLEc10-3D

Cp












1 1198602



Maximize 1

4

4

sum119894=1

120575LE119894

such that T1198797T7a119879 = T119879

7f


|Δ119860| le 02 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0

max (Δ120581 (119909)) le 28 1119898

(14)






119871= 06 for the 10








5 1198606 and 119860

7for the upper and


Maximize 119871119863

such that T1198795T5a119879 = T119879

5f


119862119897ge 049

|Δ119860| le 03 sdot 119860119906

10038161003816100381610038161003816Δ119871TEUpperSkin10038161003816100381610038161003816 = 0



(15)



119860= 03 and the






30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)

30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)






119871=



6 Conclusions




Cp

050

020

minus010

minus040

minus070

minus100








Acknowledgments



References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















Postprocessing

XML file


Solver


(structured grids)

ICEM STEPfile

IGESfile


ANSYSAPDLmodel

class









120577(120595)

120575LE

LE morphing

ZLERLE

CFS

CRS

c

120573


120595

ZTE

ΔZTE

120575TE

Structural box

xLE



120577 (120595) = 1198621198731

1198732(120595) a sdot b119879

119899(120595) + 120595120577TE + (1 minus 120595) 120577LE (1)



119894(120595)










120595max(119885) =1198731 + 119894

1198731 + 1198732 + 119899for 119894 = 0 119899 (2)






[1198621198731



119879119899

= 119905(119899)

119894= 1198621198731

1198732(120595)(

119899

119894) (1 minus 120595)

119899minus119894120595119894 119894 = 0 119899

(4)


t119899(120595) sdot a119879 = 119891 (120595) (5)




1198951le119895le119897+1


T119899a119879 = f (6)





119899has full rank (119899 + 1) the


T119879119899T119899a119879 = T119879

119899f (7)





minus 1198621198731

1198732(120595) [119860

01198610(120595) + 119860

119899119861119899(120595)]

(8)



























AnsysconverterICEM

EDGE(CFD solver)


KBSS feedback

Identification





(a) (b)






t(119899minus119898)


minus 1198621198731

1198732(120595)119898

sum119894=1

119860119894119861119894(120595)

(9)


Given 1205941198951le119895le119897+1

isin (Δ1205951 Δ120595



T119879(119899minus119898)

T(119899minus119898)

a119879 = T119879(119899minus119898)

f (10)















04

02

0

minus02

120577(120595

)middotc

05 1 15 2 250

120595 middot c











such that T119879(119899minus119898)

T(119899minus119898)

a119879 = T119879(119899minus119898)

f1003816100381610038161003816Δ119871dev

1003816100381610038161003816 le 119896119871 sdot 119871dev119906

|Δ119860| le 119896119860 sdot 119860119906

(11)


1198951le119895le119897+1
















12271m15732m


05

1015

2025

3035


(m)

minus20246



5 Results










119889keeping



Minimize 119862119889

such that T1198798T8a119879 = T119879

8f


119862119897ge 0566

|Δ119860| le 015 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0


(12)







119897=

0566 119862119889

= 00185 to 119862119897= 0569 119862

119889= 00175







1and 119860

2for the upper and


Maximize 119871119863

such that T1198798T8a119879 = T119879

8f


|Δ119860| le 012 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0


(13)






0 05 1 15 2 25 3

minus15

minus1

minus05

0

05

1

Chord (m)


Cp

Mac

h

130

129

120

110

101

092

083

074

064

055

046

037

028

018

009

000

X

Y

ZM

ach

130

129

120

110

101

092

083

074

064

055

046

037

028

018

009

000

X

Y

Z












Cp

050

020

minus010

minus040

minus070

minus100



0 05 1 15 2 25 3 35 4 45

minus15

minus1

minus05

0

05

1

Chord (m)

ReferencemLEc10-3D

Cp












1 1198602



Maximize 1

4

4

sum119894=1

120575LE119894

such that T1198797T7a119879 = T119879

7f


|Δ119860| le 02 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0

max (Δ120581 (119909)) le 28 1119898

(14)






119871= 06 for the 10








5 1198606 and 119860

7for the upper and


Maximize 119871119863

such that T1198795T5a119879 = T119879

5f


119862119897ge 049

|Δ119860| le 03 sdot 119860119906

10038161003816100381610038161003816Δ119871TEUpperSkin10038161003816100381610038161003816 = 0



(15)



119860= 03 and the






30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)

30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)






119871=



6 Conclusions




Cp

050

020

minus010

minus040

minus070

minus100








Acknowledgments



References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120577(120595)

120575LE

LE morphing

ZLERLE

CFS

CRS

c

120573


120595

ZTE

ΔZTE

120575TE

Structural box

xLE



120577 (120595) = 1198621198731

1198732(120595) a sdot b119879

119899(120595) + 120595120577TE + (1 minus 120595) 120577LE (1)



119894(120595)










120595max(119885) =1198731 + 119894

1198731 + 1198732 + 119899for 119894 = 0 119899 (2)






[1198621198731



119879119899

= 119905(119899)

119894= 1198621198731

1198732(120595)(

119899

119894) (1 minus 120595)

119899minus119894120595119894 119894 = 0 119899

(4)


t119899(120595) sdot a119879 = 119891 (120595) (5)




1198951le119895le119897+1


T119899a119879 = f (6)





119899has full rank (119899 + 1) the


T119879119899T119899a119879 = T119879

119899f (7)





minus 1198621198731

1198732(120595) [119860

01198610(120595) + 119860

119899119861119899(120595)]

(8)



























AnsysconverterICEM

EDGE(CFD solver)


KBSS feedback

Identification





(a) (b)






t(119899minus119898)


minus 1198621198731

1198732(120595)119898

sum119894=1

119860119894119861119894(120595)

(9)


Given 1205941198951le119895le119897+1

isin (Δ1205951 Δ120595



T119879(119899minus119898)

T(119899minus119898)

a119879 = T119879(119899minus119898)

f (10)















04

02

0

minus02

120577(120595

)middotc

05 1 15 2 250

120595 middot c











such that T119879(119899minus119898)

T(119899minus119898)

a119879 = T119879(119899minus119898)

f1003816100381610038161003816Δ119871dev

1003816100381610038161003816 le 119896119871 sdot 119871dev119906

|Δ119860| le 119896119860 sdot 119860119906

(11)


1198951le119895le119897+1
















12271m15732m


05

1015

2025

3035


(m)

minus20246



5 Results










119889keeping



Minimize 119862119889

such that T1198798T8a119879 = T119879

8f


119862119897ge 0566

|Δ119860| le 015 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0


(12)







119897=

0566 119862119889

= 00185 to 119862119897= 0569 119862

119889= 00175







1and 119860

2for the upper and


Maximize 119871119863

such that T1198798T8a119879 = T119879

8f


|Δ119860| le 012 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0


(13)






0 05 1 15 2 25 3

minus15

minus1

minus05

0

05

1

Chord (m)


Cp

Mac

h

130

129

120

110

101

092

083

074

064

055

046

037

028

018

009

000

X

Y

ZM

ach

130

129

120

110

101

092

083

074

064

055

046

037

028

018

009

000

X

Y

Z












Cp

050

020

minus010

minus040

minus070

minus100



0 05 1 15 2 25 3 35 4 45

minus15

minus1

minus05

0

05

1

Chord (m)

ReferencemLEc10-3D

Cp












1 1198602



Maximize 1

4

4

sum119894=1

120575LE119894

such that T1198797T7a119879 = T119879

7f


|Δ119860| le 02 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0

max (Δ120581 (119909)) le 28 1119898

(14)






119871= 06 for the 10








5 1198606 and 119860

7for the upper and


Maximize 119871119863

such that T1198795T5a119879 = T119879

5f


119862119897ge 049

|Δ119860| le 03 sdot 119860119906

10038161003816100381610038161003816Δ119871TEUpperSkin10038161003816100381610038161003816 = 0



(15)



119860= 03 and the






30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)

30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)






119871=



6 Conclusions




Cp

050

020

minus010

minus040

minus070

minus100








Acknowledgments



References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











[1198621198731



119879119899

= 119905(119899)

119894= 1198621198731

1198732(120595)(

119899

119894) (1 minus 120595)

119899minus119894120595119894 119894 = 0 119899

(4)


t119899(120595) sdot a119879 = 119891 (120595) (5)




1198951le119895le119897+1


T119899a119879 = f (6)





119899has full rank (119899 + 1) the


T119879119899T119899a119879 = T119879

119899f (7)





minus 1198621198731

1198732(120595) [119860

01198610(120595) + 119860

119899119861119899(120595)]

(8)



























AnsysconverterICEM

EDGE(CFD solver)


KBSS feedback

Identification





(a) (b)






t(119899minus119898)


minus 1198621198731

1198732(120595)119898

sum119894=1

119860119894119861119894(120595)

(9)


Given 1205941198951le119895le119897+1

isin (Δ1205951 Δ120595



T119879(119899minus119898)

T(119899minus119898)

a119879 = T119879(119899minus119898)

f (10)















04

02

0

minus02

120577(120595

)middotc

05 1 15 2 250

120595 middot c











such that T119879(119899minus119898)

T(119899minus119898)

a119879 = T119879(119899minus119898)

f1003816100381610038161003816Δ119871dev

1003816100381610038161003816 le 119896119871 sdot 119871dev119906

|Δ119860| le 119896119860 sdot 119860119906

(11)


1198951le119895le119897+1
















12271m15732m


05

1015

2025

3035


(m)

minus20246



5 Results










119889keeping



Minimize 119862119889

such that T1198798T8a119879 = T119879

8f


119862119897ge 0566

|Δ119860| le 015 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0


(12)







119897=

0566 119862119889

= 00185 to 119862119897= 0569 119862

119889= 00175







1and 119860

2for the upper and


Maximize 119871119863

such that T1198798T8a119879 = T119879

8f


|Δ119860| le 012 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0


(13)






0 05 1 15 2 25 3

minus15

minus1

minus05

0

05

1

Chord (m)


Cp

Mac

h

130

129

120

110

101

092

083

074

064

055

046

037

028

018

009

000

X

Y

ZM

ach

130

129

120

110

101

092

083

074

064

055

046

037

028

018

009

000

X

Y

Z












Cp

050

020

minus010

minus040

minus070

minus100



0 05 1 15 2 25 3 35 4 45

minus15

minus1

minus05

0

05

1

Chord (m)

ReferencemLEc10-3D

Cp












1 1198602



Maximize 1

4

4

sum119894=1

120575LE119894

such that T1198797T7a119879 = T119879

7f


|Δ119860| le 02 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0

max (Δ120581 (119909)) le 28 1119898

(14)






119871= 06 for the 10








5 1198606 and 119860

7for the upper and


Maximize 119871119863

such that T1198795T5a119879 = T119879

5f


119862119897ge 049

|Δ119860| le 03 sdot 119860119906

10038161003816100381610038161003816Δ119871TEUpperSkin10038161003816100381610038161003816 = 0



(15)



119860= 03 and the






30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)

30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)






119871=



6 Conclusions




Cp

050

020

minus010

minus040

minus070

minus100








Acknowledgments



References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

























AnsysconverterICEM

EDGE(CFD solver)


KBSS feedback

Identification





(a) (b)






t(119899minus119898)


minus 1198621198731

1198732(120595)119898

sum119894=1

119860119894119861119894(120595)

(9)


Given 1205941198951le119895le119897+1

isin (Δ1205951 Δ120595



T119879(119899minus119898)

T(119899minus119898)

a119879 = T119879(119899minus119898)

f (10)















04

02

0

minus02

120577(120595

)middotc

05 1 15 2 250

120595 middot c











such that T119879(119899minus119898)

T(119899minus119898)

a119879 = T119879(119899minus119898)

f1003816100381610038161003816Δ119871dev

1003816100381610038161003816 le 119896119871 sdot 119871dev119906

|Δ119860| le 119896119860 sdot 119860119906

(11)


1198951le119895le119897+1
















12271m15732m


05

1015

2025

3035


(m)

minus20246



5 Results










119889keeping



Minimize 119862119889

such that T1198798T8a119879 = T119879

8f


119862119897ge 0566

|Δ119860| le 015 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0


(12)







119897=

0566 119862119889

= 00185 to 119862119897= 0569 119862

119889= 00175







1and 119860

2for the upper and


Maximize 119871119863

such that T1198798T8a119879 = T119879

8f


|Δ119860| le 012 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0


(13)






0 05 1 15 2 25 3

minus15

minus1

minus05

0

05

1

Chord (m)


Cp

Mac

h

130

129

120

110

101

092

083

074

064

055

046

037

028

018

009

000

X

Y

ZM

ach

130

129

120

110

101

092

083

074

064

055

046

037

028

018

009

000

X

Y

Z












Cp

050

020

minus010

minus040

minus070

minus100



0 05 1 15 2 25 3 35 4 45

minus15

minus1

minus05

0

05

1

Chord (m)

ReferencemLEc10-3D

Cp












1 1198602



Maximize 1

4

4

sum119894=1

120575LE119894

such that T1198797T7a119879 = T119879

7f


|Δ119860| le 02 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0

max (Δ120581 (119909)) le 28 1119898

(14)






119871= 06 for the 10








5 1198606 and 119860

7for the upper and


Maximize 119871119863

such that T1198795T5a119879 = T119879

5f


119862119897ge 049

|Δ119860| le 03 sdot 119860119906

10038161003816100381610038161003816Δ119871TEUpperSkin10038161003816100381610038161003816 = 0



(15)



119860= 03 and the






30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)

30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)






119871=



6 Conclusions




Cp

050

020

minus010

minus040

minus070

minus100








Acknowledgments



References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Given 1205941198951le119895le119897+1

isin (Δ1205951 Δ120595



T119879(119899minus119898)

T(119899minus119898)

a119879 = T119879(119899minus119898)

f (10)















04

02

0

minus02

120577(120595

)middotc

05 1 15 2 250

120595 middot c











such that T119879(119899minus119898)

T(119899minus119898)

a119879 = T119879(119899minus119898)

f1003816100381610038161003816Δ119871dev

1003816100381610038161003816 le 119896119871 sdot 119871dev119906

|Δ119860| le 119896119860 sdot 119860119906

(11)


1198951le119895le119897+1
















12271m15732m


05

1015

2025

3035


(m)

minus20246



5 Results










119889keeping



Minimize 119862119889

such that T1198798T8a119879 = T119879

8f


119862119897ge 0566

|Δ119860| le 015 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0


(12)







119897=

0566 119862119889

= 00185 to 119862119897= 0569 119862

119889= 00175







1and 119860

2for the upper and


Maximize 119871119863

such that T1198798T8a119879 = T119879

8f


|Δ119860| le 012 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0


(13)






0 05 1 15 2 25 3

minus15

minus1

minus05

0

05

1

Chord (m)


Cp

Mac

h

130

129

120

110

101

092

083

074

064

055

046

037

028

018

009

000

X

Y

ZM

ach

130

129

120

110

101

092

083

074

064

055

046

037

028

018

009

000

X

Y

Z












Cp

050

020

minus010

minus040

minus070

minus100



0 05 1 15 2 25 3 35 4 45

minus15

minus1

minus05

0

05

1

Chord (m)

ReferencemLEc10-3D

Cp












1 1198602



Maximize 1

4

4

sum119894=1

120575LE119894

such that T1198797T7a119879 = T119879

7f


|Δ119860| le 02 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0

max (Δ120581 (119909)) le 28 1119898

(14)






119871= 06 for the 10








5 1198606 and 119860

7for the upper and


Maximize 119871119863

such that T1198795T5a119879 = T119879

5f


119862119897ge 049

|Δ119860| le 03 sdot 119860119906

10038161003816100381610038161003816Δ119871TEUpperSkin10038161003816100381610038161003816 = 0



(15)



119860= 03 and the






30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)

30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)






119871=



6 Conclusions




Cp

050

020

minus010

minus040

minus070

minus100








Acknowledgments



References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














such that T119879(119899minus119898)

T(119899minus119898)

a119879 = T119879(119899minus119898)

f1003816100381610038161003816Δ119871dev

1003816100381610038161003816 le 119896119871 sdot 119871dev119906

|Δ119860| le 119896119860 sdot 119860119906

(11)


1198951le119895le119897+1
















12271m15732m


05

1015

2025

3035


(m)

minus20246



5 Results










119889keeping



Minimize 119862119889

such that T1198798T8a119879 = T119879

8f


119862119897ge 0566

|Δ119860| le 015 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0


(12)







119897=

0566 119862119889

= 00185 to 119862119897= 0569 119862

119889= 00175







1and 119860

2for the upper and


Maximize 119871119863

such that T1198798T8a119879 = T119879

8f


|Δ119860| le 012 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0


(13)






0 05 1 15 2 25 3

minus15

minus1

minus05

0

05

1

Chord (m)


Cp

Mac

h

130

129

120

110

101

092

083

074

064

055

046

037

028

018

009

000

X

Y

ZM

ach

130

129

120

110

101

092

083

074

064

055

046

037

028

018

009

000

X

Y

Z












Cp

050

020

minus010

minus040

minus070

minus100



0 05 1 15 2 25 3 35 4 45

minus15

minus1

minus05

0

05

1

Chord (m)

ReferencemLEc10-3D

Cp












1 1198602



Maximize 1

4

4

sum119894=1

120575LE119894

such that T1198797T7a119879 = T119879

7f


|Δ119860| le 02 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0

max (Δ120581 (119909)) le 28 1119898

(14)






119871= 06 for the 10








5 1198606 and 119860

7for the upper and


Maximize 119871119863

such that T1198795T5a119879 = T119879

5f


119862119897ge 049

|Δ119860| le 03 sdot 119860119906

10038161003816100381610038161003816Δ119871TEUpperSkin10038161003816100381610038161003816 = 0



(15)



119860= 03 and the






30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)

30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)






119871=



6 Conclusions




Cp

050

020

minus010

minus040

minus070

minus100








Acknowledgments



References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










12271m15732m


05

1015

2025

3035


(m)

minus20246



5 Results










119889keeping



Minimize 119862119889

such that T1198798T8a119879 = T119879

8f


119862119897ge 0566

|Δ119860| le 015 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0


(12)







119897=

0566 119862119889

= 00185 to 119862119897= 0569 119862

119889= 00175







1and 119860

2for the upper and


Maximize 119871119863

such that T1198798T8a119879 = T119879

8f


|Δ119860| le 012 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0


(13)






0 05 1 15 2 25 3

minus15

minus1

minus05

0

05

1

Chord (m)


Cp

Mac

h

130

129

120

110

101

092

083

074

064

055

046

037

028

018

009

000

X

Y

ZM

ach

130

129

120

110

101

092

083

074

064

055

046

037

028

018

009

000

X

Y

Z












Cp

050

020

minus010

minus040

minus070

minus100



0 05 1 15 2 25 3 35 4 45

minus15

minus1

minus05

0

05

1

Chord (m)

ReferencemLEc10-3D

Cp












1 1198602



Maximize 1

4

4

sum119894=1

120575LE119894

such that T1198797T7a119879 = T119879

7f


|Δ119860| le 02 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0

max (Δ120581 (119909)) le 28 1119898

(14)






119871= 06 for the 10








5 1198606 and 119860

7for the upper and


Maximize 119871119863

such that T1198795T5a119879 = T119879

5f


119862119897ge 049

|Δ119860| le 03 sdot 119860119906

10038161003816100381610038161003816Δ119871TEUpperSkin10038161003816100381610038161003816 = 0



(15)



119860= 03 and the






30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)

30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)






119871=



6 Conclusions




Cp

050

020

minus010

minus040

minus070

minus100








Acknowledgments



References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119889keeping



Minimize 119862119889

such that T1198798T8a119879 = T119879

8f


119862119897ge 0566

|Δ119860| le 015 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0


(12)







119897=

0566 119862119889

= 00185 to 119862119897= 0569 119862

119889= 00175







1and 119860

2for the upper and


Maximize 119871119863

such that T1198798T8a119879 = T119879

8f


|Δ119860| le 012 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0


(13)






0 05 1 15 2 25 3

minus15

minus1

minus05

0

05

1

Chord (m)


Cp

Mac

h

130

129

120

110

101

092

083

074

064

055

046

037

028

018

009

000

X

Y

ZM

ach

130

129

120

110

101

092

083

074

064

055

046

037

028

018

009

000

X

Y

Z












Cp

050

020

minus010

minus040

minus070

minus100



0 05 1 15 2 25 3 35 4 45

minus15

minus1

minus05

0

05

1

Chord (m)

ReferencemLEc10-3D

Cp












1 1198602



Maximize 1

4

4

sum119894=1

120575LE119894

such that T1198797T7a119879 = T119879

7f


|Δ119860| le 02 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0

max (Δ120581 (119909)) le 28 1119898

(14)






119871= 06 for the 10








5 1198606 and 119860

7for the upper and


Maximize 119871119863

such that T1198795T5a119879 = T119879

5f


119862119897ge 049

|Δ119860| le 03 sdot 119860119906

10038161003816100381610038161003816Δ119871TEUpperSkin10038161003816100381610038161003816 = 0



(15)



119860= 03 and the






30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)

30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)






119871=



6 Conclusions




Cp

050

020

minus010

minus040

minus070

minus100








Acknowledgments



References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 05 1 15 2 25 3

minus15

minus1

minus05

0

05

1

Chord (m)


Cp

Mac

h

130

129

120

110

101

092

083

074

064

055

046

037

028

018

009

000

X

Y

ZM

ach

130

129

120

110

101

092

083

074

064

055

046

037

028

018

009

000

X

Y

Z












Cp

050

020

minus010

minus040

minus070

minus100



0 05 1 15 2 25 3 35 4 45

minus15

minus1

minus05

0

05

1

Chord (m)

ReferencemLEc10-3D

Cp












1 1198602



Maximize 1

4

4

sum119894=1

120575LE119894

such that T1198797T7a119879 = T119879

7f


|Δ119860| le 02 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0

max (Δ120581 (119909)) le 28 1119898

(14)






119871= 06 for the 10








5 1198606 and 119860

7for the upper and


Maximize 119871119863

such that T1198795T5a119879 = T119879

5f


119862119897ge 049

|Δ119860| le 03 sdot 119860119906

10038161003816100381610038161003816Δ119871TEUpperSkin10038161003816100381610038161003816 = 0



(15)



119860= 03 and the






30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)

30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)






119871=



6 Conclusions




Cp

050

020

minus010

minus040

minus070

minus100








Acknowledgments



References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Cp

050

020

minus010

minus040

minus070

minus100



0 05 1 15 2 25 3 35 4 45

minus15

minus1

minus05

0

05

1

Chord (m)

ReferencemLEc10-3D

Cp












1 1198602



Maximize 1

4

4

sum119894=1

120575LE119894

such that T1198797T7a119879 = T119879

7f


|Δ119860| le 02 sdot 119860119906

1003816100381610038161003816Δ119871LEskin1003816100381610038161003816 = 0

max (Δ120581 (119909)) le 28 1119898

(14)






119871= 06 for the 10








5 1198606 and 119860

7for the upper and


Maximize 119871119863

such that T1198795T5a119879 = T119879

5f


119862119897ge 049

|Δ119860| le 03 sdot 119860119906

10038161003816100381610038161003816Δ119871TEUpperSkin10038161003816100381610038161003816 = 0



(15)



119860= 03 and the






30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)

30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)






119871=



6 Conclusions




Cp

050

020

minus010

minus040

minus070

minus100








Acknowledgments



References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119871= 06 for the 10








5 1198606 and 119860

7for the upper and


Maximize 119871119863

such that T1198795T5a119879 = T119879

5f


119862119897ge 049

|Δ119860| le 03 sdot 119860119906

10038161003816100381610038161003816Δ119871TEUpperSkin10038161003816100381610038161003816 = 0



(15)



119860= 03 and the






30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)

30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)






119871=



6 Conclusions




Cp

050

020

minus010

minus040

minus070

minus100








Acknowledgments



References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)

30

20

10

0

minus10

minus20

minus30Curv

atur

e diff

eren

ce fu

nctio

n (1

m)



Span

wise (m

)






119871=



6 Conclusions




Cp

050

020

minus010

minus040

minus070

minus100








Acknowledgments



References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Cp

050

020

minus010

minus040

minus070

minus100








Acknowledgments



References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article knowledge-based shape optimization of...

Documents