three-dimensional cst parameterization method applied...

Research ArticleThree-Dimensional CST Parameterization MethodApplied in Aircraft Aeroelastic Analysis

Hua Su Chunlin Gong and Liangxian Gu

Shaanxi Aerospace Flight Vehicle Design Key Laboratory Northwestern Polytechnical University Xirsquoan Shaanxi 710072 China

Correspondence should be addressed to Hua Su sunwpueducn

Received 3 March 2017 Accepted 18 June 2017 Published 4 October 2017

Academic Editor Kenneth M Sobel

Copyright copy 2017 Hua Su et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Classshape transformation (CST) method has advantages of adjustable design variables and powerful parametric geometric shapedesign ability and has been widely used in aerodynamic design and optimization processesThree-dimensional CST is an extensionfor complex aircraft and can generate diverse three-dimensional aircraft and the corresponding mesh automatically and quicklyThis paper proposes a parametric structural modeling method based on gridding feature extraction from the aerodynamic meshgenerated by the three-dimensional CST methodThis novel method can create parametric structural model for fuselage and wingand keep the coordination between the aerodynamicmesh and the structuralmesh Based on the generated aerodynamicmodel andstructural model an automatic process for aeroelastic modeling and solving is presented with the panel method for aerodynamicsolver andNASTRAN for structural solver A reusable launch vehicle (RLV) is used to illustrate the process for aeroelastic modelingand solving The result shows that this method can generate aeroelastic model for diverse complex three-dimensional aircraftautomatically and reduce the difficulty of aeroelastic analysis dramatically It provides an effective approach to make use of theaeroelastic analysis at the conceptual design phase for modern aircraft

1 Introduction

In the wake of requirements for high lift-drag ratio aerody-namic shape and light-weight structure the aeroelastic phe-nomena caused by interaction between fluid and structurehave a growing influence on the integrated performance ofmodern aircraft [1 2] Aeroelastic analysis becomes an impor-tant process for modern aircraft design [3 4] Especially inthe conceptual design phase main performance of an aircraftis determined during this phase and therefore how to carryout the aeroelastic analysis quickly and stably to improvethe integrated performance and the design rationality of theaircraft scheme will help a lot in the following aircraft design

Aeroelastic analysis is given attention by lots ofresearchers [5] Many tools such as ZAERO [6] ENSAERO[7] and NeoCASS [8] are developed to perform aeroelasticanalysis based on the frequency domain analysis methodand time domain analysis method and have been widelyapplied on high aspect ratio wing unmanned aerial vehicleand hypersonic aircraft Although there exist some matureaeroelastic analysis and solving method the aerodynamic

modeling and structural modeling are still a complex andtime-consuming process First the parametric geometricshape and modeling tools should work together closely andautomatically second the aerodynamic solver and structuralsolver should coordinate with each other to ensure that theaeroelastic analysis procedure is executed consistently andaccurately In the conceptual design phase the design schemeof aircraft usually needs constant modification to improveperformance The size parameters and structural topologyof the scheme need to change frequently It is difficult forthe traditional CAD-based geometry modeling method tosatisfy the need of rapid geometry iteration and large rangemodification Aeroelastic analysis applied in the conceptualdesign phase faces the following problems

(1) The aerodynamic model and structural model estab-lished in many aeroelastic literatures were com-plicated the sizing of the aerodynamic shape andstructural layout is difficult and time-consumingwhich makes it hard to meet the demand for rapidmodification at the conceptual design phase

HindawiInternational Journal of Aerospace EngineeringVolume 2017 Article ID 1874729 15 pageshttpsdoiorg10115520171874729

2 International Journal of Aerospace Engineering

(2) Because of the differences between aerodynamicmodeling and structural modeling in mechanismthe aerodynamic mesh and structural mesh are notcompatible The data conversion is used addition-ally to interchange aerodynamic force and structuraldeformation between the aerodynamic model andthe structural model Extra computation overhead isneeded and some errors are introduced inevitably

(3) Quantity and position of the structural inner ele-ments such as beam bulkhead spar and rib arehard to be modified using the traditional CAD-basedgeometry modeling method The structural layoutis vested and cannot obtain the optimal scheme bytopology optimization

These problems hinder the application of aeroelasticanalysis in the conceptual design phase More effective aero-dynamic modeling and structural modeling method shouldbe studied to simplify and improve the aeroelastic modelingand analysis process The CST method is an analyticalmethod developed by Kulfan [9 10] It combines a classfunction representing a specific class of shapes and a shapefunction defining the deviation from the class functionTheseprovide an efficient shape parameterization ability on com-plex geometry using fewer design variables There are somecomparisons with other shape parameterization methods[11 12] showing that the CST method has advantages insmoothness mathematical efficiency fitting performancesand Intuitiveness CST has been widely used to parameterizeand optimize two-dimensional airfoil [13 14] and simple3D aircraft [15 16] Liu [17 18] presents a multiblock CSTmethod tomodel the hypersonic aircraftwith complex shapewhich can join adjacent surfaces smoothly and retain thegood properties of the CST method Leal [19] proposesan aerostructural optimization method for determining ina preliminary manner morphing wing configurations thatprovide benefits during various disparate flight conditionswith CST parameterization

The authors in [20 21] proposed a universal three-dimensional CSTmethod for geometry modeling of complexaircraft It generates complex three-dimensional geometricshape to support various aircraft aerodynamic shape model-ing which gives a simple and effective way to aerodynamicoptimization This novel three-dimensional CST method isextended in aeroelastic analysis of complex aircraft in thisarticle A novel parametric structural modeling method isproposed based on gridding feature extraction from theaerodynamic mesh generated by the three-dimensional CSTmethod An automatic and effective way for aeroelasticmodeling and analysis is also established The structureof this article is as follows First the basic principle ofthe three-dimensional CST method is introduced brieflythen the aircraft characteristic components library is estab-lished including fuselage wing and empennage on thisbasis the structural modeling method is presented in detailand the aeroelastic modeling and analysis process is con-structed finally a static aeroelastic analysis example is usedto verify the proposed aeroelastic modeling and analysisprocess

2 Three-Dimensional CST Method

CST method has an efficient and brief shape description forthe two-dimensional airfoils and simple three-dimensionalgeometriesThis section gives an improvement to expand theoriginal two-dimensional CST method to three dimensionsA universal three-dimensional CST method is proposed toprovide adjustable parameterization ability to complex three-dimensional geometry

21 Basic Cross Section Definition Most of the aircraft geom-etry can be described by innumerable cross sections alongthe axial direction Combining an analytical function (theclass function) with a parametric curve (the shape function)the basic cross section can be defined by the B-spline CSTmethod [22] as follows

120577 (120595)1003816100381610038161003816upplow = 11986211987311198732 (120595) sdot 119878 (120595) + Δ120577119873 (120595) (1)

where 120595 = 119911119871119908 119911 is the lateral coordinate 120595 is thenormalized lateral coordinate 119871119908 is the total length of thebasic cross section in the lateral direction 120577(120595) is the normalheight in the lateral positionΔ120577119873(120595) is the eccentric distancein the lateral position 119862(120595) and 119878(120595) are the class functionand the shape function defined in CST method 1198731 and1198732 are control parameters of the class function For thesymmetrical section 1198731 is equal to 1198732 The class function119862(120595) is defined as follows

11986211987311198732 (120595) = 1205951198731 [1 minus 120595]1198732 (2)

The basic cross section is assembled by the upper 120577(120595)and the inverted lower 120577(120595) If we set Δ120577119873(120595) to 0 ignore theshape function and change the control parameters of the classfunction simultaneously different sections can be generatedVarious cross section shapes can be generated by changingcontrol parameters of the class function

22 B-Spline Shape Function The B-spline function canadjust the range of influence by selecting different orders ofsub-Bernstein polynomials It has better local control abilityand computation efficiency than Bernstein polynomials Soit has been chosen as the shape function of the CST methodin the basic cross section definition The B-spline functionis combined by the massive low-order Bezier curves toapproximate the high-order curve It is defined as follows

119878 (120595) =119899

sum119894=0

119887119894119873119894119896 (120595) (3)

where 119887119894 119894 = 0 1 119899 is the weight factor 119896 + 1 order B-spine basic function is defined as the piecewise polynomialsat node vector 119879 = 1199050 1199051 119905119899+119896+1 119905119894 le 119905119894+1 119873119894119896(120595) isthe B-spline basic function defined at 119905119894119899+119896+1119894=0 which can be

International Journal of Aerospace Engineering 3

obtained by recurrence formula De Boor-Cox

1198731198941 (120595) =1 119905119894 le 120595 le 119905119894+10 otherwise

119873119894119901 (120595) = 120595 minus 119905119894119905119894+119901+1 minus 119905119894119873

119894119901minus1 (120595) +

119905119894+119901+1 minus 120595119905119894+119901+1 minus 119905119894+1119873

119894+1119901minus1 (120595)

00 = 0

(4)

The basic cross section defined with B-spline function isas follows

120577 (120595)1003816100381610038161003816upplow = 11986211987311198732 (120595) sdot119899

sum119894=0

119887119894119873119894119896 (120595) + Δ120577119873 (120595) (5)

23 Three-Dimensional CST Method The three-dimensionalgeometry can be considered as a series of original cross sec-tions arranged parallel along the axial direction By definingproper cross sections using the methodmentioned above wecan get an analytical surface to express geometric shape Inorder to introduce an analytical axial rule the coefficient 119887 isreplaced by the same B-spline CST definition in formula (5)So the two-dimensional CST function can be expanded to athree-dimensional version which is shown as follows

119887119894 = 11986211987211198722 (120578) sdot 119878 (120578) + Δ120577119872 (120578) (6)

119878 (120578) =119898

sum119895=0

119887119895119873119895119896 (120578) (7)

where 120578 = 119909119871 119909 is the axial coordinate 120578 is the normalizedaxial coordinate 119871 is the total length of the generated surfacein the axial direction Bringing formula (6) to formula (5)we have the definition of the three-dimensional analyticalsurface

120577 (120595 120578)1003816100381610038161003816upplow= 11986211987311198732 (120595)11986211987211198722 (120578)

119899

sum119894=0

119898

sum119895=0

119887119894119895119873119894119896 (120595)119873119895119896 (120578)

+ Δ120577119872119873 (120595 120578)

(8)

Formula (8) is the analytical expression of the extensionalthree-dimensional surface where 120577 is the third-dimensionalcoordinate value along the 119910 direction based on the two-dimensional normalized coordinate 120595 and 120578 119862(120595) and 119862(120578)are the CSTrsquos class functions with control parameter119873111987321198721 and 1198722 119873(120595) and 119873(120578) are the basic functions of theB-spline which constitute the CSTrsquos shape function 119887119894119895 isthe discrete control weight factor in the up and low surfacesLower case 119899 and119898 are the orders of B-spline function whichare expressed as the numbers of lateral and axial controlpoints The total points of the geometric surface are (119899 + 1) times(119898 + 1) More control points mean more design parametersand better parametric geometric shape design abilityWhen 119899= 0 and119898 = 0 the discrete control weight factor 119887 is equal to

1 the shape function 119878(120595 120578) is also equal to 1 and then entiregeometric surface is controlled by the control factors119873111987321198721 and1198722 of the class functionΔ120577119872119873(120595 120578) is the eccentricdistance in normal position and the default value is 0

The geometric surface defined in formula (8) can bedescribed as the third-dimensional coordinate value calcu-lated by the two-dimensional mesh points dispersed as thelateral and axial control points 120595 and 120578 are the normalizedcoordinates defined in [0 1] times [0 1] We also need to definethe profile 119885 in the lateral direction with respect to the axialcoordinate 120578 The profile 119885 is expressed the same as the B-spline CST method in formula (5)

119885 (120578) = 11986211987911198792 (120578)119908

sum119905=0

119887119905119873119905119896 (120578) (9)

119887119905 is the discrete control weight factor in the lateral coor-dinate From the above definition we obtain the three-dimensional analytical surface which can be transformedinto global Cartesian coordinate as formula (10)

119883(120578) = 120578

119884 (120595 120578) = 11986211987311198732 (120595)11986211987211198722 (120578)119899

sum119894=0

119898

sum119895=0

119887119894119895119873119894119896 (120595)119873119895119896 (120578)

+ Δ120577119872119873 (120595 120578)1003816100381610038161003816upplow

119885 (120578) = 11986211987911198792 (120578)119908

sum119905=0

119887119905119873119905119896 (120578)

(10)

where 120595 = [0 1] 120578 = [0 1] The outward boundary box isused to control the size of the geometric surface and then weget the full design parameters of the entire geometric surfacewhich includes the following

Axial and lateral length 119871 119871119908Vertical height 1198711198671 1198711198672Sectional section control factor11987311198732Vertical section control factor11987211198722Lateral section control factor 1198791 1198792Surface weight factor 119887119894119895Lateral weight factor 119887119905

119871 119871119908 1198711198671 1198711198672 are the size parameters of the geometricsurface1198731119873211987211198722 1198791 1198792 are the control parametersof the class function 119887119894119895 and 119887119905 are the discrete control weightfactors of the shape function When 119887119894119895 and 119887119905 are equalto 1 the number of total parameters is constant and alsominimumWhen the 119887119894119895 and 119887119905 are expressed as matrix somecontrol points with the same size of the matrix elements arelocated in the geometric surface to improve the complexityof the geometric shape So the size of design parameterscould be adjusted dynamically With these weight factors thethree-dimensional CSTmethod has more flexible parametricgeometric shape design ability


M1 = 00M2 = 00

T1 = 00 T2 = 00

M1 = 05M2 = 00

T1 = 05 T2 = 00

M1 = 05M2 = 05

T1 = 05 T2 = 05

HeadUPP N1 = 05N2 = 05

LOW N1 = 05N2 = 05

TailUPP N1 = 05N2 = 05

LOW N1 = 05N2 = 05

Bupp = 1 Blow = 1

Bupp = 1 Blow = 1


LOW N1 = 05N2 = 05


LOW N1 = 07N2 = 07


LOW N1 = 05N2 = 05


LOW N1 = 07N2 = 07

Bupp = 1 Blow = 1Bupp3lowast9 Bupp3lowast5 Bupp1lowast3

Figure 1 Geometric shapes with different control parameters generated by the three-dimensional CST method

24 Basic Geometric Shape The basic geometric shape isdefined by the upper geometric surface and the inverted lowergeometric surface Figure 1 gives some geometric shapesgenerated by the above three-dimensional CST method withdifferent control parameters 119861upp 119861low and 119861119905 are thematrix generated by 119887119894119895 of the upper surface 119887119894119895 of the lowersurface and 119887119905 of the lateral coordinate The shapes in thesame lines are generated with the same value of 1198731 and1198732 but with different 1198721 1198722 1198791 and 1198792 The shapes inthe same columns are generated with the same values 11987211198722 1198791 and 1198792 but with different 1198731 1198732 and 119861upp Themodified shapes in the third line are generated with differentsizes of control weight factors of the upper surface comparedwith the shapes in the second line

From the various geometric shapes and cross sectionsshown in Figure 1 the following features can be found for thethree-dimensional CST method

(1) The geometric shape can be expressed as an extrudedbody made up of a series of basic cross sections withdifferent control factors along the axial directionThetransformation rule of the body is decided by thecontrol factors 1198731head 1198732head at head and 1198731tail1198732tail at tail

(2) The shape profile at the axial and lateral direction isdecided by the control factors11987211198722 1198791 and 1198792 Itallows continuous control of every single basic crosssectionWhen119872 and119879 are equal to 0 simultaneouslythe corresponding cross section is opened When119872

and 119879 are greater than 0 the corresponding crosssection is closed to a point This feature can be usedto generate the closed head and the opening body

(3) 119861upp 119861low and 119861119905 are matrix with arbitrary sizeThey produce a global correction or a local adjust-ment to the geometric shape The control ability isdecided by the size of the matrix

The shape profile is parameterized by the section controlfactors and the geometric shape can be adjusted globally orlocally by the weight factor matrix These features give thethree-dimensional CST method a comprehensive parametricgeometric shape design ability with adjustable control param-eters Designers could have a more flexible design pattern toparameterize various geometric model

25 Mesh Discretization A generic mesh discretizationmethod is utilized to generate the correspondingmesh whichwill be used as the aerodynamic mesh and structural meshThe procedure for the three-dimensional CST method islikely to generate structured mesh surface along the 120595 and 120578direction With these characteristics the surface is dispersedinto some control points uniformly Then a two-dimensionalmatrix made up of control points is generated which can beexpressed as follows

Nodeupp (119894 119895) = (119883 (120578) 119884 (120595 120578) 119885 (120578)) Nodelow (119894 119895) = (119883 (120578) 119884 (120595 120578) 119885 (120578))

(11)


where 119894 = 1 2 119872 minus 1 119895 = 1 2 119873 minus 1 119872 and 119873are the numbers of total points in the 120595 and 120578 direction Thequadrangle is used as the mesh element consisted with thesediscrete control points

Element (119905) = [Node (119894 119895) Node (119894 119895 + 1) Node (119894 + 1 119895 + 1) Node (119894 + 1 119895)] (12)

The Node(119894 119895) and Element(119905) are used to generate themesh of the analytical geometric shape Then we get thecorresponding quadrilateral mesh

3 Three-Dimensional Parametric GeometryModeling Method

With the above three-dimensional CST method we getthe continuous and smooth geometric shape But singlegeometric shape cannot provide enough degree of freedom togenerate an entire complex aircraftUsually a complex aircraftcan be split into some standard components Sowith the com-ponent combination method a universal three-dimensionalparametric geometry modeling method is proposed basedon the aircraft characteristic components library The typicalaircraft characteristic components include fuselage wingand empennage which are introduced as follows

31 Aircraft Characteristic Components Library

311 Fuselage The fuselage can be divided into head head-body and head-body-tail three types according to the struc-tural cabins or aerodynamic shape features By setting thecontrol factors 1198721 1198722 1198791 and 1198792 to 1 or 0 it is easy togenerate the semiclosed head and tail and the opened bodyThe control factors 1198731 and 1198732 must be kept the same valueat the joint position between two parts to ensure surfacecontinuation The detailed control parameters and the threefuselage types are shown in Figure 2(a)

The control factors of the head include

119873head

= [1198731head upp 1198732head upp 1198731head low 1198732head low] 119872head

= [1198721head upp1198722head upp1198721head low1198722head low] 119879head = [1198791head upp 1198792head upp 1198791head low 1198792head low]

(13)

The weight factors of the head include

119861 = [119861head upp 119861head low] 119861119905 = [119861119905 head upp 119861119905 head low]

(14)

The formulas (13) and (14) are the generic forms of a singlepart of the componentThe others are similar too By varyingthese control parameters various geometric shapes can begenerated The parametric geometric shape design ability isdependent on the size of the matrix 119861 and 119861119905 For complexgeometric shape designers can set 119861 and 119861119905 to some arbitrarymatrix to increase the degree of freedom But one should notethat the number of control parameters also increases as thematrix becomes larger

312 Wing Wing modeling is similar to the fuselage Thecontrol factors11987211198722 1198791 and 1198792 are set to 0 to describethe opening tip and root of the wing The sweepback angleand dihedral angle are also parameterized by mesh offsetoperation as follows

119884119894119895correct = 119884119894119895 + 119883119894119895 lowast tan (120579119910) 119885119894119895correct = 119885119894119895 + 119883119894119895 lowast tan (120579119911)

(15)

120579119910 is the sweepback angle and 120579119911 is the dihedral angle Thesingle wing and double wing are modeled in the componentslibrary The detailed control parameters and the two wingtypes are shown in Figure 2(b) The tip and root of the wingare open so the control factors of the cross section onlyinclude119873in and119873out

313 Empennage Empennage modeling is the same as thewing The single tail and double tail are modeled in thecomponents library and shown in Figure 2(c)

314 Others Some typical components are introducedabove which can be used to create many kinds of commonaircraft There may be other requested components suchas engine and nozzle These can be some simplification ofthe basic three types components library For example thenozzle may be some simplification of the single part fuselagewith opening head and tail For other special componentsdesigners can model it using the three-dimensional CSTmethod andpackage to the aircraft characteristic componentslibrary for reuse

32Three-Dimensional Geometry ModelingThe aircraft char-acteristic components library is introduced in Section 31These components are rotated and moved to the properposition to assemble entire complex aircraft For the discretecontrol points 119875 of component generated in Section 25

119875 =[[[[[[[

1199091 1199101 11991111199092 1199102 1199112 119909119899 119910119899 119911119899

]]]]]]]

(16)

The rotation matrix 119872119903 and transfer matrix 119872119889 are asfollows


119872119903

= [[[[

cos (120579119911) cos (120579119910) sin (120579119911) minus cos (120579119911) sin (120579119910)minus sin (120579119911) cos (120579119910) cos (120579119909) + sin (120579119910) sin (120579119909) cos (120579119911) cos (120579119909) sin (120579119911) sin (120579119910) cos (120579119909) + cos (120579119910) sin (120579119909)sin (120579119911) cos (120579119910) sin (120579119909) + sin (120579119910) cos (120579119909) minus cos (120579119911) sin (120579119909) minus sin (120579119911) sin (120579119910) sin (120579119909) + cos (120579119910) cos (120579119909)

]]]] (17)

119872119889 = [119889119909 119889119910 119889119911] (18)

With these matrices the modified control points 119875neware evaluated in formula (18) Then the modified componentmesh is generated by these points using the mesh discretiza-tion method discussed in Section 25

119875new = 119875119872119903 +119872119889 (19)

4 Universal Structural Modeling Method

Although structural finite element analysis has been widelyused in industrial department and research institution thestructure parameterized modeling is still a difficult phaseThe most common structural parameterization method isbased on parametric CAD model It is complicated to modelcomplex aircraft and also difficult to support the changingof the structural layout These limitations cannot satisfy theneeds of rapid modification and iterations in the conceptualdesign phase of the modern aircraft design This section willprovide a detailed illustration of a novel structural modelingmethod based on the three-dimensional parametric geome-try modeling method mentioned above Structural model isconstructed based on the aircraft characteristic componentslibrary Three typical structural models are elucidated below

41 Fuselage Fuselage is the main body of an aircraft It isusually designed as a thin-walled structure and constitutedby longitudinal stiffeners (like beam and stringer) transversestiffeners (like bulkhead) and surface skin A fuselage con-tains fuel tank payload electronic instrument and otherequipment Also it is as a sole central part connected withwings empennages engines and other components to makeup the entire aircraft The main structural forms of fuselageinclude girder structure longeron structure andmonocoqueshell structure Based on the above assumption all of thesestructural forms can be simplified as three basic elementsbeamstringer bulkhead and skin The detailed modelingmethod of these three elements is shown as follows

411 BeamStringer Beam and stringer are the longitudinalstiffeners of fuselage structure used for undertaking axialload coming from fuselage bending and also used fortransferring the load of outer surface skin to bulkhead Herethe beam and stringer are simplified to one-dimensionalbeam element The cross section shape of the beam and thestringer like I-section H-section circle-section and so oncan be ignored at the modeling phase and will be consideredby the structural solver

Assuming the surface mesh of fuselage is continuous andcompatible among all parts of the component the discretenodes of the upper surface or the lower surface can beexpressed as

NodeFuselage (119894 119895) (20)

where 119894 = 1 2 119872 119895 = 1 2 119873 119872 and 119873 are thenumbers of control points along the120595 and 120578 direction Vector119861 is used to locate the position of the beam and the stringeralong transverse direction

119861 = [1198871 1198872 sdot sdot sdot 119887119899] (21)

where 0 le 1198871 lt 1198872 lt sdot sdot sdot lt 119887119899 le 1 119861 is normalized to[0 1] 119887119896 represents the normalized lateral position of the 119896thbeamstringer For the discrete nodes it can be handled asthe nearest point NodeFuselage(119896 119895) which has the minimumdistance to the 119887119896 position in the 119895th line of the discretesurface consisted of NodeFuselage(119894 119895) So the node set of thebeamstringer can be defined as

Node(119896)119861 = Node (119896 119895) 119895 = 1 2 119873 (22)

And the element set of the beamstringer can be definedas

Element(119896)119861 = [Node (119896 119895) Node (119896 119895 + 1)] 119895 = 1 2 119873 minus 1 (23)

Then the mesh of beamstringer can be extractedfrom the surface mesh of the fuselage Figure 3 shows thebeamstringer structural layout scheme of a hypersonic air-craft

412 Bulkhead Bulkhead can be divided into three typesnormal bulkhead reinforced bulkhead and connected bulk-head It supports the skin of the fuselage to maintain geo-metric shape and also undertakes some concentrated loadThe definition of bulkhead is similar to the beamstringerOne-dimensional beam element is selected to simplify thebulkhead model The cross section shape of the beam also isignored at the modeling phase and will be considered by thestructural solver Vector 119865 is used to locate the position of thebulkhead

119865 = [1198911 1198912 sdot sdot sdot 119891119899] (24)


Size parametershead length head width head heightControl factors

Weight factors Bij Bt

Head

Size parametershead length body length body width body heightControl factors per component N1 N2 M1 M2 T1 T2

N1 N2 M1 M2 T1 T2

Weight factors per component Bij Bt

Head-Body

Size parametershead length body length tail lengthbody width body heightControl factors per component


Head-Body-Tail

N1 N2 M1 M2 T1 T2

(a)

Size parametersaspect ratio span swept-back angletaper ratio thick chord lengthdihedral angleControl factors N1 N2Weight factors Bij

Wing

Size parameters per componentaspect ratio span taper ratio swept-back angle thick chord lengthdihedral angleControl factors per componentN1 N2Weight factors per component Bij Bt

Double-Wing

(b)

Size parametersaspect ratio span swept-back angle taper ratio thick chord lengthControl factors N1 N2Weight factors Bij

Tail

Size parametersaspect ratio span swept-back angletaper ratio thick chord length taildihedral angleControl factors N1 N2Weight factors Bij Bt

Double-Tail

(c)

Figure 2 Basic components in the aircraft characteristic components library

where 0 le 1198911 lt 1198912 lt sdot sdot sdot lt 119891119899 le 1119865 is normalized to [0 1]119891119896represents the normalized axial position of the 119896th bulkheadFor the discrete nodes it can be handled as the nearest pointNodeFuselage(119894 119896) which has the minimum distance to the119891119896 position in the 119894th line of the discrete surface consistingof NodeFuselage(119894 119895) So the node set and the element set of

bulkhead can be defined as

Node(119896)119865 = Node (119894 119896) 119894 = 1 2 119872Element(119896)119865= [NodeFuselage (119894 119896) NodeFuselage (119894 + 1 119896)]

119894 = 1 2 119872 minus 1(25)


(a) Location points of the beamstringer (b) Beamstringer structural layout scheme

Figure 3 The beamstringer structural layout scheme of a hypersonic aircraft

(a) Location points of the bulkhead (b) Bulkhead structural layout scheme

Figure 4 The bulkhead structural layout scheme of a hypersonic aircraft

(a) Aerodynamic model of the fuselage (b) Structural model of the fuselage

Figure 5 The aerodynamic model and structural model of a hypersonic aircraft

Then the mesh of the bulkhead can be extracted from thesurface mesh of the fuselage Figure 4 shows the bulkheadstructural layout scheme of a hypersonic aircraft

413 Fuselage Skin Fuselage skin is used to maintain thegeometric shape of the fuselage It should be continuous andsmooth to support the aerodynamic solver and structuralsolver The discrete surface mesh of fuselage is used forboth the structural mesh and the aerodynamic mesh Ithas the following advantages (a) the structural model andthe aerodynamic model use the same surface mesh coming

from mesh discretization in Section 25 the data conversionbetween these models are no longer needed (b) the mesh ofthe beamstringer and the bulkhead are both extracted fromthe discrete surface mesh they are coordinated with surfaceskin mesh naturally Based on the above definition the wholestructural model of the fuselage can be generated as Figure 5

42 Wing Wing structure consists of skin spar stringerrib and connector The following simplifications are usedto simplify the wing structural model (1) the effects of theconnectors are ignored (2) the effects of the flanges of spar


(a) Spanwise structure (b) Chordwise structure

Figure 6 The spanwise and chordwise structural layout scheme of wing

stringer and rib are imputed to the wing skin Based on theseassumptions the wing structure can be simplified to sometwo-dimensional shell elementsThree typical basic elementsare used to construct the wing structure

421 Spanwise Elements Spanwise elements include sparand stringer Assuming these elements are arranged alongthe chordwise direction rigorously For the discrete nodesNode(119894 119895) of the wing vector119882119904 is used to locate the positionof spanwise elements

119882119904 = [1199041 1199042 sdot sdot sdot 119904119899] (26)

where 0 le 1199041 lt 1199042 lt sdot sdot sdot lt 119904119899 le 1 119882119904 is normalizedto [0 1] 119878119896 represents the normalized spanwise position ofthe 119896th spanwise elements For the discrete nodes it can behandled as the nearest point NodeWing(119896 119895) which has theminimum distance to the 119878119896 position in the 119895th line of thediscrete surface consisting of NodeWing(119894 119895) So the node setof spanwise elements can be defined as

Node(119896)119882119904 = NodeWing (119896 119895) (27)

where 119895 = 1 2 119873119873 is the number of the spanwise points119882119905 is used to define the node distribution in the thicknessdirection

119882119905 = [1199051 1199052 sdot sdot sdot 119905119904] (28)

where 0 le 1199051 lt 1199052 lt sdot sdot sdot lt 119905119904 le 1119882119905 is normalized to [0 1]too The layouts of the upper surface points and the lowersurface points are the same So by arranging some middlepoints between the upper surface points and the lower surfacepoints by some node distribution the total nodes of the wingspanwise section can be generated as

Node(119896)119882119904 = linespace(119896)119882119905 (Node(119896)119882119904uppNode(119896)119882119904low) (29)

linespace(119896)119882119905 means to arrange points in the regularity of dis-tribution119882119905The element set can be generated by connectingthese nodes Figure 6(a) shows the spanwise structural layoutscheme of wing

422 Chordwise Elements Rib is the chordwise elementSimilar to the spanwise elements vector119882119888 is used to definethe position of the rib

119882119888 = [1198881 1198882 sdot sdot sdot 119888119899] (30)where 0 le 1198881 lt 1198882 lt sdot sdot sdot lt 119888119899 le 1119882119888 is normalized to [0 1]The modeling process is the same as the spanwise elementsthe node set can be generated as

Node(119896)119882119888 = Nodewing (119894 119896) (31)

where 119894 = 1 2 119872 119872 is the number of the chordwisepoints119882119905 is also used to define the node distribution in thethickness direction So the total nodes of the wing chordwisesection can be generated as


Figure 6(b) shows the chordwise structural layout schemeof wing

423 Wing Skin Similar to the fuselage skin the discretemesh of the wing is used for both structural mesh andaerodynamic mesh to maintain consistency Based on theabove definition the entire structural model of the wing canbe generated as Figure 7

43 Empennage Structural model of the empennage is thesame as the wing Spanwise elements chordwise elementsand skin are used to construct the empennage structure

44 Others Most of the common components have the simi-lar structural layouts to the fuselage and wing They can bestructured by the method mentioned above

5 Aeroelastic Modeling and Analysis Process

The process of aeroelastic modeling and analysis includesthree steps three-dimensional CST modeling aerodynamicstructural modeling and aeroelastic analysis The process ofaeroelastic modeling and analysis is shown in Figure 8 andthe detailed illustration is as follows


(a) Aerodynamic model (b) Structural model

Figure 7 The aerodynamic model and structural model of wing

Design parameters

Geometry model

Component mesh

Aerodynamic mesh Structural mesh

Aerodynamic model

Structural model

Aerodynamic solver

Structural solver

Aeroelastic result

3D geometry CST modeling

Aerodynamicstructural modeling

Aeroelastic analysis

Force

DispΔMaxDisp lt 1e minus 5

Figure 8 The process of aeroelastic modeling and analysis

51 Three-Dimensional CST Modeling Section 3 lists alldesign parameters of the three common components Thereare four types of design parameters available to use to controlthe geometric shape layout parameter size parameter shape

parameter and local control parameter The layout param-eter and the size parameter are global design parametersThe former changes the position and the posture of thecomponents and the latter changes the shape size of thecomponents They can be used to control the global sizingof aircraft The shape parameter and local control parameterare local design parameters The former changes the sketchof the main cross section and the latter adjusts the detailsof the component They can be used to further control thedetailed shape of the aircraft Local control parameter canbe any size of matrix The larger the matrix the betterthe parametric geometric shape design ability Accordingto the requirement of the geometry modeling complexitythe proper design parameters can be selected as the designvariables others may stay constant or vary with respect to theselected parameters With these design variables geometrymodel and corresponding mesh of the component can begenerated automatically

52 AerodynamicStructural Modeling A complex aircraft isconstituted by several components Applying the first step themesh of these components can be generated respectively tomake up the entire aircraft For each of these componentsthe structural mesh also can be obtained in accordance withthe structural modeling method introduced in Section 4 Butthere are two problems that needed to be solved to generatethe integrated aerodynamicstructural model

The first problem is the redundant mesh caused by themutual nesting between the connected components Thismesh could influence the precision of aeroelastic analysis Inthe worst condition it may cause some error to the aerody-namic solver and the structural solver PINPOLYHEDRONan open-source tool is used to remove this nestedmesh PIN-POLYHEDRON is a C++ code It provides a novel algorithmto test whether points are insideoutsideon a polyhedrondefined by triangular faces and vertices It can be used forvarious complicated models such as nonconvex volumesmultimaterial bodies and so on and there is no assumptionabout orientation of the face normal Above all the algorithmis very efficient especially for large-scale problems In this


research the fuselage is as the main polyhedron If a pointof other components is detected inside the main fuselagethis point and the relevant mesh are both removed fromthe component Looping all points of the component andremoving the nested mesh then the remaining mesh is takenas the valid meshThis valid mesh is used as the aerodynamicmesh to evaluate the aerodynamic characteristics and is usedas the structural mesh to analyze the structural performance

The second problem is the connection of the relevantcomponents For aerodynamic analysis the panel method isused as the aerodynamic solver so the connection betweencomponents is not necessary For structure analysis the con-nectionmust be modeled to guarantee the force transmissionbetween the connected components assuming a structuralconnection exists only between the bulkhead of the fuselageand the beamspar of other components The followingmethod is used to generate these structural connections

(1) For each bulkhead of fuselage evaluate the midpoint119875119898119894 by counting the average coordinate of the pointsset 119878119898119894 on the bulkhead

(2) For each beamspar find the nearest elements to thefuselage mark as 119864119904119895 evaluate the midpoint of 119864119904119895and mark as 119875119904119895

(3) For each beamspar find the nearest bulkhead bymin119894|119875119898119894 minus 119875119904119895| and mark as 119875119898lowast

(4) Sort the points set 119878119898lowast on the bulkhead 119875119898lowast bycounting the distance from 119875119904119895 the sorted points setmark as 119878119898119903lowast

(5) The first119873119903 points of the sorted points 119878119898119903lowast are usedas the connection points set 1198781The connection pointsset of the beam mark 1198782 and119873119903 is the number of theconnection points 1198782

(6) Generate some quadrilaterals by connecting the rele-vant points between the connection points set 1198781 andconnection points set 1198782

(7) Looping all of the beams of the components generatenodes set and elements set of the structural connec-tion to obtain the connection mesh

Figure 9 shows the structural connection between thefuselage and wing generated by the above method The redline is the bulkheads of the fuselage The blue line is thebeamstringer of the fuselage The quadrilaterals with greenedges are the structural connection mesh This method pro-vides a simpleway to generate the structural connectionmeshbetween the main fuselage and the connected components

This structural connection mesh guarantees the gridcontinuity for force transmission However the compellingconnection may reduce the quality of these mesh and influ-ence the connection stiffness between the fuselage and thewing These influences can be corrected by modifying thematerial attribute or adding a spring element In this paperthese impacts are ignored temporarily

53 Aeroelastic Analysis The aerodynamic solver and thestructural solver interplay in the aeroelastic analysis process

Figure 9The structural connection mesh between the fuselage andthe wing

AeroCalc a C++ in-house code with the panel method isused as the aerodynamic solver The modified Newtonianimpact theory is used to evaluate the windward surfaceand the Prandtl-Meyer theory is used to evaluate the lee-ward surface NASTRAN is used as the structural solverThe structural mesh and control data are written in BDFformat as the script file Every part of the components hasits own property section to define thickness cross sectionparameters andmaterial attributesThe outside surfacemeshof aerodynamic model and structural model is consistent sothe force obtained from aerodynamic solver can be appliedto the structural model and the displacement obtainedfrom structural solver can be applied to the aerodynamicmodel without any conversionThese simplify the aeroelasticanalysis process tremendously

The three-dimensional CST modeling method is used asa parametric geometry modeling and mesh generation pro-cessor to automatically generate aerodynamic and structuralmodel for aeroelastic analysis The designer can construct anaircraft geometry shape quickly and automatically obtain thecorresponding aeroelastic model An aeroelastic analysis canbe carried out quickly and effectively to give a preliminaryevaluation of the aeroelastic effect This could be very helpfulfor aircraft preliminary design

The aerodynamic model and structural model use thesame surface mesh The aerodynamic force can be appliedto the structural model and the structural displacement canbe added to aerodynamicmesh to generate new aerodynamicmodel directly This simplification will limit the mesh adap-tation for aerodynamic model and structural model whichmay increase the computational cost and reduce analysis pre-cision But it could save the tedious aeroelastic meshing andmodeling process which is time-consuming especially forcomplex aircraft Also these automatic aeroelastic modelingand analysis process can be used for aeroelastic optimization

6 Static Aeroelastic Analysis Example

A reusable launch vehicle (RLV) is used to illustrate theprocess of aeroelastic analysis Three basic components a



Figure 10 The aerodynamic model and the structural model of the RLV

cp

001

002

004

007

013

025

047

089

170

(a) Pressure coefficient distribution

Y X

Z 129 minus 001 0859 minus 003172 minus 002258 minus 002343 minus 002429 minus 002515 minus 002601 minus 002687 minus 002773 minus 002859 minus 002944 minus 002103 minus 001112 minus 001120 minus 001129 minus 001

(b) Structural displacement distribution

Figure 11 Pressure coefficient distribution and structural displacement distribution at the evaluation points

head-body fuselage a double wing and a tail are utilizedto build up the RLV geometry The main parameters of theRLV are listed in Table 1 and the trimmed aerodynamicmodel is shown in Figure 10(a) Eighteen bulkheads and eightbeams are arranged in the head-body fuselage Five spars andseven ribs are arranged in the double wing Three spars andseven ribs are arranged in the vertical tail Ten structuralconnections are used to connect the fuselage and the wingThree structural connections are used to connect the fuselageand the tail The integrated structural model is shown inFigure 10(b)

The aluminum alloys are used as structural material forthe structural frame and skin with a density of 2700Kgm3and an elastic modulus of 72GPa The total number of theaerodynamic elements is 15506 which is controlled by thenumber of points for mesh discretization in the 120595 and 120578direction of each surface The total number of the structuralelements is 18691 including the skin mesh which is the sameas the aerodynamic mesh except the nested mesh the struc-tural component mesh and the structural connection mesh

The maximum dynamic pressure trajectory point duringreentry is chosen as the aeroelastic evaluation point with119872119886= 36 and 120572 = 65 degreeThe pressure coefficient distribution

evaluated by the panel method at the maximum dynamicpressure point is shown in Figure 11(a) The correspondingaerodynamic force is applied to the structure surface directlywithout interpolation The structural mesh and the displace-ment distribution are shown in Figure 11(b)The relative errorof themaximumdeformation in the119884 direction is used as theconvergence criteria during the aeroelastic iterations and it isset to 1119890 minus 5

The aeroelastic analysis of the RLV converges shortly afterfive iterations The iteration history of the lift coefficient 119862119897the drag coefficient119862119889 and themaximumdeformation in the119884 direction 119889119884 are shown in Figure 12 In the final convergedcondition the 119862119897 is 13 percent lower and the 119862119889 is 037percent lower which lead to a 094 percent decrease in thelift-to-drag ratio than the initial undeformed condition Themaximum structure deformation in the 119884 direction occurs atthe wingtip position The maximum deformation at the firststructure analysis and the last structure analysis are 01287mand 01271mThe wingtip deformation is shown in Figure 13The blue line shows the undeformed wing tip the orangeline shows the deformed wing tip after the first structureanalysis and the red line shows the deformed wing tip atthe convergence The change of the wingtip deformation is


Table 1 Main parameters of the RLV

Components Design parameters Value

Head-body

Head length 50000mmBody length 114310mmBody width 18210mm

Body height upp 12840mmBody height low 4050mm

Double wing

Inner wing span length 6500mmOuter wing span length 26000mm

Inner wing root chord length 88000mmInner wing taper ratio 1923Outer wing taper ratio 27

Inner wing sweep back angle 800 degOuter wing sweep back angle 450 deg

Tail

Tail span length 19250mmTail root chord length 20750mm

Tail taper ratio 1724Tail sweep back angle 450 deg

Cl

Cd

dY

2 3 4 51Iteration history

0

005

01

015

Figure 12 Iteration history of the lift coefficient 119862119897 the dragcoefficient119862119889 and themaximumdeformation in the119884 direction 119889119884

not obvious during the aeroelastic iterations which reaches astable level after the second step

The three-dimensional CST parameterization methodand the aeroelastic analysis process are written in MATLABcode The geometry modeling costs less than one secondand the generation of aerodynamicmesh and structuralmeshcost 55 s The analysis time of aerodynamic analysis andstructure analysis per iteration and the total analysis timeare shown in Table 2 The whole aeroelastic modeling andanalysis process is very fast and efficient To generate an entire

The undeformed wingtip

The last structure analysis 01271 m

The first structure analysis 01287 m

Figure 13 The wingtip deformation

Table 2 The cost time of the aeroelastic iterations

Time per iteration (s) Total time (s)Aerodynamic analysis 51 255Structure analysis 103 504Aeroelastic analysis 165 825

complex RLV and its analysis models only needs several sec-onds This will simplify and shorten the aeroelastic analysisand make it easy to use in the aircraft conceptual designphase

7 Conclusion and Future Work

In this paper a novel aerodynamic and structural modelingmethod based on the three-dimensional CST is developed toprovide a fast and simpleway to carry out an entire aeroelasticanalysis process The aerodynamic model is generated bytrimming the embedded aircraft characteristic componentsThe structural model is generated by arranging proper innerparts from the basic mesh topologyThe consistency between


the aerodynamic model and the structural model is satisfiednaturally Finally an efficient aeroelastic analysis process iscreated and tested by a RLV In conclusion

(1) the three-dimensional parametric geometry model-ing method gives a universal way to generate geom-etry model of common three-dimensional complexaircraft It inherits the advantages of the original CSTmethod and has fast and stable parametric geometricshape design ability An aircraft can be modeledwith a few control parameters in a few minutes anddiscretized to surface mesh More common aircraftcharacteristic components library will be introducedin future work to give a broader support to variouskinds of aircraft

(2) a universal aeroelastic modeling and analysis processis introduced By using the three-dimensional para-metric geometry modeling method the aerodynamicmodel and the structural model can be generatedtogether to keep themesh consistency Fluid structureinteraction (FSI) can be ignored to simplify theaeroelastic analysis process This simplification maylimit themesh adaptation for aerodynamicmodel andstructural model but it gives a fast and simple way tocarry out an aeroelastic analysis process for complexaircraft which is very useful in the conceptual designphase

(3) the structuralmodelingmethod in this article gives anautomatic and fast structural layout parameterizationand generation approach Topology of the structurecan be parameterized from a wide range stably Thiscould be used as parameterized structural model fortopology optimization

(4) the generated mesh by the three-dimensional para-metric geometry modeling method will be of poorquality at the position where the slope of the surfacealong axial direction and lateral direction is too largeThese are caused by the uniform mesh discretizationalong the axial direction and lateral direction Thesenarrow grids may reduce the precision of the aeroe-lastic analysis Nonuniform mesh discretization andmesh repairingmethod will be studied in future workto improve mesh quality

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This researchwas supported by a fund from theNational Nat-ural Science Foundation of China (no 51505385) the Shang-hai Aerospace Science and Technology Innovation Founda-tion (no SAST2015010) and the Defense Basic Research Pro-gram (no JCKY2016204B102 and no JCKY2016208C001)The authors are also thankful to Shaanxi Aerospace FlightVehicle Design Key Laboratory of NPU

References

[1] D P Raymer Aircraft Design A Conceptual Approach AIAAEducation Series 4th edition 2006

[2] S A Brandt R J Stiles J Bertin et al Intorduction to Aero-nautics A Design Perspective vol 2nd ofAIAA Education Series2004

[3] BThuruthimattam P Friedmann K Powell and J McNamaraldquoAeroelasticity of a generic hypersonic vehiclerdquo in Proceedingsof the 43rd AIAAASMEASCEAHSASC Structures StructuralDynamics and Materials Conference Denver Colorado April2002

[4] K K Gupta L S Voelker C Bach T Doyle and E HahnldquoCFD-based aeroelastic analysis of the X-43 hypersonic flightvehiclerdquo in Proceedings of the 39th Aerospace Sciences Meetingand Exhibit Reno Nev USA January 2001

[5] J Heeg P Chwalowski J P Florance C D Wieseman D MSchuster and B Perry Jr ldquoOverview of the aeroelastic predic-tion workshoprdquo in Proceedings of the 51st AIAA Aerospace Sci-ences Meeting including the New Horizons Forum and AerospaceExposition Grapevine Tex USA January 2013

[6] D H Lee and P C Chen ldquoNonlinear aeroelastic studies on afoldingwing configurationwith free-play hinge nonlinearityrdquo inProceedings of the 47th AIAAASMEASCEAHSASC StructuresStructural Dynamics andMaterials Conference American Insti-tute of Aeronautics and Astronautics Newport Rhode IslandMay 2006

[7] D Yeh ldquoPreliminary findings in certification of ENSAEROcodefor rigid and flexible configurationrdquo in Proceedings of the FluidDynamics Conference American Institute of Aeronautics andAstronautics Colorado Springs Colo USA June 1994

[8] C Luca R Sergio and T Lorenzo Neocass An IntegratedTool for Structural Sizing Aeroelastic Analysis and MDO atConceptual Design Level American Institute of Aeronauticsand Astronautics 2010

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

[10] B Kulfan ldquoA universal parametric geometry representationmethodmdashlsquoCSTrsquordquo in Proceedings of the 45th AIAA Aerospace Sci-ences Meeting and Exhibit American Institute of Aeronauticsand Astronautics Inc Reno Nev USA January 2007

[11] V Sripawadkul M Padulo and M Guenov ldquoA compari-son of airfoil shape parameterization techniques for earlydesign optimizationrdquo in Proceedings of the 13th AIAAISSMOMultidisciplinary Analysis and Optimization Conference (MAOrsquo10) American Institute of Aeronautics and Astronautics FortWorth Tex USA September 2010

[12] S Nadarajah P Castonguay and A Mousavi ldquoSurvey ofshape parameterization techniques and its effect on three-dimensional aerodynamic shape optimizationrdquo in Proceedingsof the 18th AIAA Computational Fluid Dynamics ConferenceAmerican Institute of Aeronautics and Astronautics IncMiami Fla USA June 2007

[13] G L Mura and N Qin ldquoLocal class shape transformationparameterization (L-CST) for airfoilsrdquo in Proceedings of the 55thAIAA Aerospace Sciences Meeting Institute of Aeronautics andAstronautics Grapevine Tex USA January 2017

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

[15] E D Olson ldquoThree-dimensional piecewise-continuous class-shape transformation of wingsrdquo in Proceedings of the 16th AIAAISSMOMultidisciplinary Analysis andOptimization Conference


Institute of Aeronautics and Astronautics Dallas Tex USAJune 2015

[16] M H Straathof and M J L Van Tooren ldquoAdjoint optimizationof a wing using the class-shape-refinement-transformationmethodrdquo Journal of Aircraft vol 49 no 4 pp 1091ndash1100 2012

[17] C Liu Y Duan J Cai and J Wang ldquoApplication of the 3Dmulti-block CST method to hypersonic aircraft optimizationrdquoAerospace Science and Technology vol 50 pp 295ndash303 2016

[18] C Liu YDuan J Cai andG Yang ldquoApplications ofmulti-blockCST method for quasi-waverider designrdquo in Proceedings of the52nd Aerospace Sciences Meeting Institute of Aeronautics andAstronautics Maryland Md USA January 2014

[19] P B Leal D J Hartl and C L Bertagne ldquoAero-structuraloptimization of shape memory alloy-based wing morphing viaa classshape transformation approachrdquo in Proceedings of the23nd AIAAAHS Adaptive Structures Conference KissimmeeFla USA January 2015

[20] S Hua G Liangxian and G Chunlin ldquoThe research on geome-try modeling method based on three-dimensional CST param-eterization technologyrdquo in Proceedings of the 16th AIAAISSMOMultidisciplinaryAnalysis andOptimizationConference Ameri-can Institute of Aeronautics andAstronautics Dallas Tex USAJune 2015

[21] H Su C-L Gong and L-X Gu ldquoTwo-level aerodynamicshape optimization strategy based on three-dimensional CSTmodeling methodrdquo Journal of Solid Rocket Technology vol 37no 1 pp 1ndash6 2014

[22] M H Straathof and M J L van Tooren ldquoExtension to theclass-shape-transformation method based on B-splinesrdquo AIAAJournal vol 49 no 4 pp 780ndash790 2011

RoboticsJournal of

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Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

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Electrical and Computer Engineering

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Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



(2) Because of the differences between aerodynamicmodeling and structural modeling in mechanismthe aerodynamic mesh and structural mesh are notcompatible The data conversion is used addition-ally to interchange aerodynamic force and structuraldeformation between the aerodynamic model andthe structural model Extra computation overhead isneeded and some errors are introduced inevitably

(3) Quantity and position of the structural inner ele-ments such as beam bulkhead spar and rib arehard to be modified using the traditional CAD-basedgeometry modeling method The structural layoutis vested and cannot obtain the optimal scheme bytopology optimization

These problems hinder the application of aeroelasticanalysis in the conceptual design phase More effective aero-dynamic modeling and structural modeling method shouldbe studied to simplify and improve the aeroelastic modelingand analysis process The CST method is an analyticalmethod developed by Kulfan [9 10] It combines a classfunction representing a specific class of shapes and a shapefunction defining the deviation from the class functionTheseprovide an efficient shape parameterization ability on com-plex geometry using fewer design variables There are somecomparisons with other shape parameterization methods[11 12] showing that the CST method has advantages insmoothness mathematical efficiency fitting performancesand Intuitiveness CST has been widely used to parameterizeand optimize two-dimensional airfoil [13 14] and simple3D aircraft [15 16] Liu [17 18] presents a multiblock CSTmethod tomodel the hypersonic aircraftwith complex shapewhich can join adjacent surfaces smoothly and retain thegood properties of the CST method Leal [19] proposesan aerostructural optimization method for determining ina preliminary manner morphing wing configurations thatprovide benefits during various disparate flight conditionswith CST parameterization

The authors in [20 21] proposed a universal three-dimensional CSTmethod for geometry modeling of complexaircraft It generates complex three-dimensional geometricshape to support various aircraft aerodynamic shape model-ing which gives a simple and effective way to aerodynamicoptimization This novel three-dimensional CST method isextended in aeroelastic analysis of complex aircraft in thisarticle A novel parametric structural modeling method isproposed based on gridding feature extraction from theaerodynamic mesh generated by the three-dimensional CSTmethod An automatic and effective way for aeroelasticmodeling and analysis is also established The structureof this article is as follows First the basic principle ofthe three-dimensional CST method is introduced brieflythen the aircraft characteristic components library is estab-lished including fuselage wing and empennage on thisbasis the structural modeling method is presented in detailand the aeroelastic modeling and analysis process is con-structed finally a static aeroelastic analysis example is usedto verify the proposed aeroelastic modeling and analysisprocess

2 Three-Dimensional CST Method

CST method has an efficient and brief shape description forthe two-dimensional airfoils and simple three-dimensionalgeometriesThis section gives an improvement to expand theoriginal two-dimensional CST method to three dimensionsA universal three-dimensional CST method is proposed toprovide adjustable parameterization ability to complex three-dimensional geometry

21 Basic Cross Section Definition Most of the aircraft geom-etry can be described by innumerable cross sections alongthe axial direction Combining an analytical function (theclass function) with a parametric curve (the shape function)the basic cross section can be defined by the B-spline CSTmethod [22] as follows

120577 (120595)1003816100381610038161003816upplow = 11986211987311198732 (120595) sdot 119878 (120595) + Δ120577119873 (120595) (1)

where 120595 = 119911119871119908 119911 is the lateral coordinate 120595 is thenormalized lateral coordinate 119871119908 is the total length of thebasic cross section in the lateral direction 120577(120595) is the normalheight in the lateral positionΔ120577119873(120595) is the eccentric distancein the lateral position 119862(120595) and 119878(120595) are the class functionand the shape function defined in CST method 1198731 and1198732 are control parameters of the class function For thesymmetrical section 1198731 is equal to 1198732 The class function119862(120595) is defined as follows

11986211987311198732 (120595) = 1205951198731 [1 minus 120595]1198732 (2)

The basic cross section is assembled by the upper 120577(120595)and the inverted lower 120577(120595) If we set Δ120577119873(120595) to 0 ignore theshape function and change the control parameters of the classfunction simultaneously different sections can be generatedVarious cross section shapes can be generated by changingcontrol parameters of the class function

22 B-Spline Shape Function The B-spline function canadjust the range of influence by selecting different orders ofsub-Bernstein polynomials It has better local control abilityand computation efficiency than Bernstein polynomials Soit has been chosen as the shape function of the CST methodin the basic cross section definition The B-spline functionis combined by the massive low-order Bezier curves toapproximate the high-order curve It is defined as follows

119878 (120595) =119899

sum119894=0

119887119894119873119894119896 (120595) (3)

where 119887119894 119894 = 0 1 119899 is the weight factor 119896 + 1 order B-spine basic function is defined as the piecewise polynomialsat node vector 119879 = 1199050 1199051 119905119899+119896+1 119905119894 le 119905119894+1 119873119894119896(120595) isthe B-spline basic function defined at 119905119894119899+119896+1119894=0 which can be



1198731198941 (120595) =1 119905119894 le 120595 le 119905119894+10 otherwise

119873119894119901 (120595) = 120595 minus 119905119894119905119894+119901+1 minus 119905119894119873

119894119901minus1 (120595) +

119905119894+119901+1 minus 120595119905119894+119901+1 minus 119905119894+1119873

119894+1119901minus1 (120595)

00 = 0

(4)


120577 (120595)1003816100381610038161003816upplow = 11986211987311198732 (120595) sdot119899

sum119894=0

119887119894119873119894119896 (120595) + Δ120577119873 (120595) (5)


119887119894 = 11986211987211198722 (120578) sdot 119878 (120578) + Δ120577119872 (120578) (6)

119878 (120578) =119898

sum119895=0

119887119895119873119895119896 (120578) (7)


120577 (120595 120578)1003816100381610038161003816upplow= 11986211987311198732 (120595)11986211987211198722 (120578)

119899

sum119894=0

119898

sum119895=0

119887119894119895119873119894119896 (120595)119873119895119896 (120578)

+ Δ120577119872119873 (120595 120578)

(8)




119885 (120578) = 11986211987911198792 (120578)119908

sum119905=0

119887119905119873119905119896 (120578) (9)


119883(120578) = 120578

119884 (120595 120578) = 11986211987311198732 (120595)11986211987211198722 (120578)119899

sum119894=0

119898

sum119895=0

119887119894119895119873119894119896 (120595)119873119895119896 (120578)

+ Δ120577119872119873 (120595 120578)1003816100381610038161003816upplow

119885 (120578) = 11986211987911198792 (120578)119908

sum119905=0

119887119905119873119905119896 (120578)

(10)





M1 = 00M2 = 00

T1 = 00 T2 = 00

M1 = 05M2 = 00

T1 = 05 T2 = 00

M1 = 05M2 = 05

T1 = 05 T2 = 05


LOW N1 = 05N2 = 05


LOW N1 = 05N2 = 05

Bupp = 1 Blow = 1

Bupp = 1 Blow = 1


LOW N1 = 05N2 = 05


LOW N1 = 07N2 = 07


LOW N1 = 05N2 = 05


LOW N1 = 07N2 = 07











Nodeupp (119894 119895) = (119883 (120578) 119884 (120595 120578) 119885 (120578)) Nodelow (119894 119895) = (119883 (120578) 119884 (120595 120578) 119885 (120578))

(11)










119873head



(13)



(14)




(15)





119875 =[[[[[[[

1199091 1199101 11991111199092 1199102 1199112 119909119899 119910119899 119911119899

]]]]]]]

(16)



119872119903

= [[[[


]]]] (17)

119872119889 = [119889119909 119889119910 119889119911] (18)


119875new = 119875119872119903 +119872119889 (19)






NodeFuselage (119894 119895) (20)


119861 = [1198871 1198872 sdot sdot sdot 119887119899] (21)


Node(119896)119861 = Node (119896 119895) 119895 = 1 2 119873 (22)





119865 = [1198911 1198912 sdot sdot sdot 119891119899] (24)




Head


N1 N2 M1 M2 T1 T2


Head-Body



Head-Body-Tail

N1 N2 M1 M2 T1 T2

(a)


Wing


Double-Wing

(b)


Tail


Double-Tail

(c)





119894 = 1 2 119872 minus 1(25)

















119882119904 = [1199041 1199042 sdot sdot sdot 119904119899] (26)


Node(119896)119882119904 = NodeWing (119896 119895) (27)


119882119905 = [1199051 1199052 sdot sdot sdot 119905119904] (28)






Node(119896)119882119888 = Nodewing (119894 119896) (31)












Design parameters

Geometry model

Component mesh


Aerodynamic model

Structural model

Aerodynamic solver

Structural solver

Aeroelastic result




Force





























cp

001

002

004

007

013

025

047

089

170


Y X












Head-body



Double wing




Tail



Cl

Cd

dY


0

005

01

015





















Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1198731198941 (120595) =1 119905119894 le 120595 le 119905119894+10 otherwise

119873119894119901 (120595) = 120595 minus 119905119894119905119894+119901+1 minus 119905119894119873

119894119901minus1 (120595) +

119905119894+119901+1 minus 120595119905119894+119901+1 minus 119905119894+1119873

119894+1119901minus1 (120595)

00 = 0

(4)


120577 (120595)1003816100381610038161003816upplow = 11986211987311198732 (120595) sdot119899

sum119894=0

119887119894119873119894119896 (120595) + Δ120577119873 (120595) (5)


119887119894 = 11986211987211198722 (120578) sdot 119878 (120578) + Δ120577119872 (120578) (6)

119878 (120578) =119898

sum119895=0

119887119895119873119895119896 (120578) (7)


120577 (120595 120578)1003816100381610038161003816upplow= 11986211987311198732 (120595)11986211987211198722 (120578)

119899

sum119894=0

119898

sum119895=0

119887119894119895119873119894119896 (120595)119873119895119896 (120578)

+ Δ120577119872119873 (120595 120578)

(8)




119885 (120578) = 11986211987911198792 (120578)119908

sum119905=0

119887119905119873119905119896 (120578) (9)


119883(120578) = 120578

119884 (120595 120578) = 11986211987311198732 (120595)11986211987211198722 (120578)119899

sum119894=0

119898

sum119895=0

119887119894119895119873119894119896 (120595)119873119895119896 (120578)

+ Δ120577119872119873 (120595 120578)1003816100381610038161003816upplow

119885 (120578) = 11986211987911198792 (120578)119908

sum119905=0

119887119905119873119905119896 (120578)

(10)





M1 = 00M2 = 00

T1 = 00 T2 = 00

M1 = 05M2 = 00

T1 = 05 T2 = 00

M1 = 05M2 = 05

T1 = 05 T2 = 05


LOW N1 = 05N2 = 05


LOW N1 = 05N2 = 05

Bupp = 1 Blow = 1

Bupp = 1 Blow = 1


LOW N1 = 05N2 = 05


LOW N1 = 07N2 = 07


LOW N1 = 05N2 = 05


LOW N1 = 07N2 = 07











Nodeupp (119894 119895) = (119883 (120578) 119884 (120595 120578) 119885 (120578)) Nodelow (119894 119895) = (119883 (120578) 119884 (120595 120578) 119885 (120578))

(11)










119873head



(13)



(14)




(15)





119875 =[[[[[[[

1199091 1199101 11991111199092 1199102 1199112 119909119899 119910119899 119911119899

]]]]]]]

(16)



119872119903

= [[[[


]]]] (17)

119872119889 = [119889119909 119889119910 119889119911] (18)


119875new = 119875119872119903 +119872119889 (19)






NodeFuselage (119894 119895) (20)


119861 = [1198871 1198872 sdot sdot sdot 119887119899] (21)


Node(119896)119861 = Node (119896 119895) 119895 = 1 2 119873 (22)





119865 = [1198911 1198912 sdot sdot sdot 119891119899] (24)




Head


N1 N2 M1 M2 T1 T2


Head-Body



Head-Body-Tail

N1 N2 M1 M2 T1 T2

(a)


Wing


Double-Wing

(b)


Tail


Double-Tail

(c)





119894 = 1 2 119872 minus 1(25)

















119882119904 = [1199041 1199042 sdot sdot sdot 119904119899] (26)


Node(119896)119882119904 = NodeWing (119896 119895) (27)


119882119905 = [1199051 1199052 sdot sdot sdot 119905119904] (28)






Node(119896)119882119888 = Nodewing (119894 119896) (31)












Design parameters

Geometry model

Component mesh


Aerodynamic model

Structural model

Aerodynamic solver

Structural solver

Aeroelastic result




Force





























cp

001

002

004

007

013

025

047

089

170


Y X












Head-body



Double wing




Tail



Cl

Cd

dY


0

005

01

015





















Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










M1 = 00M2 = 00

T1 = 00 T2 = 00

M1 = 05M2 = 00

T1 = 05 T2 = 00

M1 = 05M2 = 05

T1 = 05 T2 = 05


LOW N1 = 05N2 = 05


LOW N1 = 05N2 = 05

Bupp = 1 Blow = 1

Bupp = 1 Blow = 1


LOW N1 = 05N2 = 05


LOW N1 = 07N2 = 07


LOW N1 = 05N2 = 05


LOW N1 = 07N2 = 07











Nodeupp (119894 119895) = (119883 (120578) 119884 (120595 120578) 119885 (120578)) Nodelow (119894 119895) = (119883 (120578) 119884 (120595 120578) 119885 (120578))

(11)










119873head



(13)



(14)




(15)





119875 =[[[[[[[

1199091 1199101 11991111199092 1199102 1199112 119909119899 119910119899 119911119899

]]]]]]]

(16)



119872119903

= [[[[


]]]] (17)

119872119889 = [119889119909 119889119910 119889119911] (18)


119875new = 119875119872119903 +119872119889 (19)






NodeFuselage (119894 119895) (20)


119861 = [1198871 1198872 sdot sdot sdot 119887119899] (21)


Node(119896)119861 = Node (119896 119895) 119895 = 1 2 119873 (22)





119865 = [1198911 1198912 sdot sdot sdot 119891119899] (24)




Head


N1 N2 M1 M2 T1 T2


Head-Body



Head-Body-Tail

N1 N2 M1 M2 T1 T2

(a)


Wing


Double-Wing

(b)


Tail


Double-Tail

(c)





119894 = 1 2 119872 minus 1(25)

















119882119904 = [1199041 1199042 sdot sdot sdot 119904119899] (26)


Node(119896)119882119904 = NodeWing (119896 119895) (27)


119882119905 = [1199051 1199052 sdot sdot sdot 119905119904] (28)






Node(119896)119882119888 = Nodewing (119894 119896) (31)












Design parameters

Geometry model

Component mesh


Aerodynamic model

Structural model

Aerodynamic solver

Structural solver

Aeroelastic result




Force





























cp

001

002

004

007

013

025

047

089

170


Y X












Head-body



Double wing




Tail



Cl

Cd

dY


0

005

01

015





















Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















119873head



(13)



(14)




(15)





119875 =[[[[[[[

1199091 1199101 11991111199092 1199102 1199112 119909119899 119910119899 119911119899

]]]]]]]

(16)



119872119903

= [[[[


]]]] (17)

119872119889 = [119889119909 119889119910 119889119911] (18)


119875new = 119875119872119903 +119872119889 (19)






NodeFuselage (119894 119895) (20)


119861 = [1198871 1198872 sdot sdot sdot 119887119899] (21)


Node(119896)119861 = Node (119896 119895) 119895 = 1 2 119873 (22)





119865 = [1198911 1198912 sdot sdot sdot 119891119899] (24)




Head


N1 N2 M1 M2 T1 T2


Head-Body



Head-Body-Tail

N1 N2 M1 M2 T1 T2

(a)


Wing


Double-Wing

(b)


Tail


Double-Tail

(c)





119894 = 1 2 119872 minus 1(25)

















119882119904 = [1199041 1199042 sdot sdot sdot 119904119899] (26)


Node(119896)119882119904 = NodeWing (119896 119895) (27)


119882119905 = [1199051 1199052 sdot sdot sdot 119905119904] (28)






Node(119896)119882119888 = Nodewing (119894 119896) (31)












Design parameters

Geometry model

Component mesh


Aerodynamic model

Structural model

Aerodynamic solver

Structural solver

Aeroelastic result




Force





























cp

001

002

004

007

013

025

047

089

170


Y X












Head-body



Double wing




Tail



Cl

Cd

dY


0

005

01

015





















Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










119872119903

= [[[[


]]]] (17)

119872119889 = [119889119909 119889119910 119889119911] (18)


119875new = 119875119872119903 +119872119889 (19)






NodeFuselage (119894 119895) (20)


119861 = [1198871 1198872 sdot sdot sdot 119887119899] (21)


Node(119896)119861 = Node (119896 119895) 119895 = 1 2 119873 (22)





119865 = [1198911 1198912 sdot sdot sdot 119891119899] (24)




Head


N1 N2 M1 M2 T1 T2


Head-Body



Head-Body-Tail

N1 N2 M1 M2 T1 T2

(a)


Wing


Double-Wing

(b)


Tail


Double-Tail

(c)





119894 = 1 2 119872 minus 1(25)

















119882119904 = [1199041 1199042 sdot sdot sdot 119904119899] (26)


Node(119896)119882119904 = NodeWing (119896 119895) (27)


119882119905 = [1199051 1199052 sdot sdot sdot 119905119904] (28)






Node(119896)119882119888 = Nodewing (119894 119896) (31)












Design parameters

Geometry model

Component mesh


Aerodynamic model

Structural model

Aerodynamic solver

Structural solver

Aeroelastic result




Force





























cp

001

002

004

007

013

025

047

089

170


Y X












Head-body



Double wing




Tail



Cl

Cd

dY


0

005

01

015





















Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Head


N1 N2 M1 M2 T1 T2


Head-Body



Head-Body-Tail

N1 N2 M1 M2 T1 T2

(a)


Wing


Double-Wing

(b)


Tail


Double-Tail

(c)





119894 = 1 2 119872 minus 1(25)

















119882119904 = [1199041 1199042 sdot sdot sdot 119904119899] (26)


Node(119896)119882119904 = NodeWing (119896 119895) (27)


119882119905 = [1199051 1199052 sdot sdot sdot 119905119904] (28)






Node(119896)119882119888 = Nodewing (119894 119896) (31)












Design parameters

Geometry model

Component mesh


Aerodynamic model

Structural model

Aerodynamic solver

Structural solver

Aeroelastic result




Force





























cp

001

002

004

007

013

025

047

089

170


Y X












Head-body



Double wing




Tail



Cl

Cd

dY


0

005

01

015





















Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

























119882119904 = [1199041 1199042 sdot sdot sdot 119904119899] (26)


Node(119896)119882119904 = NodeWing (119896 119895) (27)


119882119905 = [1199051 1199052 sdot sdot sdot 119905119904] (28)






Node(119896)119882119888 = Nodewing (119894 119896) (31)












Design parameters

Geometry model

Component mesh


Aerodynamic model

Structural model

Aerodynamic solver

Structural solver

Aeroelastic result




Force





























cp

001

002

004

007

013

025

047

089

170


Y X












Head-body



Double wing




Tail



Cl

Cd

dY


0

005

01

015





















Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Design parameters

Geometry model

Component mesh


Aerodynamic model

Structural model

Aerodynamic solver

Structural solver

Aeroelastic result




Force





























cp

001

002

004

007

013

025

047

089

170


Y X












Head-body



Double wing




Tail



Cl

Cd

dY


0

005

01

015





















Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation































cp

001

002

004

007

013

025

047

089

170


Y X












Head-body



Double wing




Tail



Cl

Cd

dY


0

005

01

015





















Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Head-body



Double wing




Tail



Cl

Cd

dY


0

005

01

015





















Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









three-dimensional cst parameterization method applied...

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