Efficient Structural Optimization of Aircraft Wings Aerospace Engineering

116 Pages·2017·1.75 MB·English

by  Andre C. Marta

Efficient Structural Optimization of Aircraft Wings
Tiago Alexandre Reis Ramalho Moutinho Freire
Thesis to obtain the Master of Science Degree in
Aerospace Engineering
Supervisor: Prof. André Calado Marta
Examination Committee
Chairperson: Prof. Filipe Szolnoky Ramos Pinto Cunha
Supervisor: Prof. André Calado Marta
Member of the Committee: Prof. Aurélio Lima Araújo
February 2017
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Acknowledgments
Iwouldliketothankmysupervisor,ProfessorAndre´ CaladoMarta,forallhissupportandguidance
throughout the development of this thesis. His vast knowledge in engineering and wise advice on how
totackleproblemswereessentialtoproducethiswork.
IwishtoexpressmygratitudetomyfriendsJoa˜oAlmeida,fortakingthetimetoexplainthestructure
of the original framework he developed and for always being there to clarify my doubts, and to Duarte
Ronda˜oforproofreadingthisdissertationandgivinginsightfuladvice.
I would also like to thank the amazing people I met during my studies in Lisbon, Delft and Victoria,
andtoexpressmygratitudeforhavingthemaroundingoodandbadmoments.
Last but not least, I would like to thank my family, specially my mother, for always believing in my
abilitiesandencouragingmetofollowmydreams.
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Resumo
Actualmente,osme´todosdegradienteconstituemumadasferramentasmaisutilizadasnaoptimizac¸a˜o
multidisciplinar de aeronaves. No entanto, estes me´todos necessitam de calcular a sensibilidade das
func¸o˜es de interesse em relac¸a˜o a`s varia´veis de projecto, constituindo uma das etapas de maior
exigeˆncia computacional no processo de optimizac¸a˜o, uma vez que estas sa˜o frequentemente obtidas
por me´todos de aproximac¸a˜o altamente dependentes da dimensa˜o do problema. Assim, este trabalho
tem como objectivo desenvolver uma ferramenta de optimizac¸a˜o eficiente, para uma fase preliminar
do projecto de uma asa, com a utilizac¸a˜o da informac¸a˜o exacta do gradiente. Primeiramente, e´ real-
izado um levantamento dos va´rios me´todos de ana´lise de sensibilidade existentes com exemplos de
aplicac¸a˜oaoprojectodevigasmodeladascomelementosfinitos. Posteriormente,umaferramentapara
arepresentac¸a˜oestruturaldeumaasatridimensionale´ adaptadaemtreˆsblocoscomodesenvolvimento
dosrespectivosmo´dulosparaoca´lculodassensibilidades,utilizandoosme´todosdediferenciac¸a˜oau-
toma´tica, derivac¸a˜o simbo´lica e adjunto. Um estudo parame´trico e´ apresentado com base numa asa
de refereˆncia e as sensibilidades totais sa˜o calculadas com a ferramenta desenvolvida e verificadas
com o me´todo das diferenc¸as finitas. Por fim, sa˜o apresentados exemplos de optimizac¸a˜o estrutu-
ral partindo do caso de refereˆncia. O objectivo de minimizac¸a˜o da massa total da asa e´ alcanc¸ado,
sendonota´veloaumentodaeficieˆncianoprocessodeoptimizac¸a˜ocomautilizac¸a˜odaferramentade-
senvolvida, traduzido pela reduc¸a˜o dos tempos de computac¸a˜o para, aproximadamente, metade e um
terc¸o, quando comparados com os me´todos das diferenc¸as finitas progressivas e centradas, respecti-
vamente.
Palavras-chave:
optimizac¸a˜o estrutural, me´todos de gradiente, ana´lise de sensibilidade,
me´todoadjunto,diferenciac¸a˜oautoma´tica,me´tododoselementosfinitos.
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Abstract
Nowadays,gradient-basedmethodsareoneofthemostwidelyusedtoolsinaircraftMultidisciplinary
DesignOptimization. However,thesemethodsrequirethecomputationofthesensitivitiesoftheinterest
functions with respect to the design variables, representing one of the most computationally expensive
steps in the optimization process, since these are frequently obtained by approximation methods that
are highly dependent on the number of design variables. Therefore, the main objective of this work is
to develop an efficient optimization tool for wing preliminary design, using exact gradient information.
Firstly, a survey on the existent sensitivity analysis methods is conducted, with the application to a
beam design problem modeled with finite elements, providing valuable insight in the implementation
process and advantages of each method. Subsequently, a tool to represent a 3D wing structure is
adapted into three blocks and the correspondent modules for sensitivity computation are developed,
with the application of the automatic differentiation, the symbolic derivative and the adjoint methods. A
parametricstudyispresentedforareferencewingcaseandthetotalsensitivitiesarecomputedwiththe
developedframeworkandverifiedwiththefinitedifferencemethod. Lastly,structuraloptimizationtests,
usingthereferencecaseastheinitialdesignpoint,areperformed. Theobjectiveofminimizingthewing
mass is achieved with a remarkable increase in computational efficiency in the optimization process,
translatedinareductionofthecomputationaltimeto,roughly,halfandonethirdwhencomparedtothe
forwarddifferenceandcentraldifferencemethods,respectively.
Keywords:
structuraloptimization,gradient-basedmethods,sensitivityanalysis,adjointmethod,
automaticdifferentiation,finiteelementmethod.
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Contents
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
ListofTables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
ListofFigures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 AeroelasticToolBackground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 ThesisOutline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 OptimizationMethods 7
2.1 OptimizationBackground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 GradientBasedMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 UnconstrainedOptimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 ConstrainedOptimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Gradient-FreeMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 ChoiceoftheMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 SensitivityAnalysis 13
3.1 SymbolicDifferentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 FiniteDifferenceMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 ComplexStepMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3.1 FiniteDifferenceandComplexMethodsComparison . . . . . . . . . . . . . . . . . 15
3.4 AlternativeAnalyticMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4.1 DirectSensitivityMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4.2 AdjointSensitivityMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.5 AutomaticDifferentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.6 Benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.6.1 ModelDefinition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.6.2 SensitivityAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
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4 StructuralFramework 33
4.1 EquivalentCross-sectionalProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.1.1 Thin-WallAssumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.1.2 ImplementationProcedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.1.3 ApplicationExample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 EquivalentBeamElementProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 FiniteElementModel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3.1 ConstitutiveEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3.2 3DFiniteElementModelDefinition . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3.3 StrongandWeakformulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3.4 DomainDiscretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3.5 ShapeFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3.6 ElementStiffnessMatrixFormulation. . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3.7 ForceVectorDetermination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3.8 BondaryConditionsApplicationandGlobalSystemofEquationsSolution . . . . . 50
5 ParametricStudy 51
5.1 WingBaselineConfiguration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2 FunctionsofInterest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.3 InternalParametersInfluence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3.1 SparsLocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3.2 SparandSkinThickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.4 MaterialParametersInfluence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.5 SummaryofResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6 SensitivityAnalysisFramework 61
6.1 AutomaticDifferentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.2 SymbolicDifferentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.3 AdjointSensitivityModule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.3.1 VerticalTipDisplacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.3.2 TipRotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.4 TotalSensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7 WingStructuralOptimization 75
7.1 Optimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7.2 BoundConstrainedOptimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
7.3 FullyConstrainedOptimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
8 Conclusions 83
8.1 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
8.2 FutureWork. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
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