Table Of ContentAnIntroductiontoStructuralOptimization
SOLIDMECHANICSANDITSAPPLICATIONS
Volume153
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·
Peter W. Christensen Anders Klarbring
An Introduction to Structural
Optimization
PeterW.Christensen,AndersKlarbring
DivisionofMechanics
LinköpingUniversity
SE-58183Linköping
Sweden
peter.christensen@liu.se,anders.klarbring@liu.se
ISBN978-1-4020-8665-6 e-ISBN978-1-4020-8666-3
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Preface
This book has grown out of lectures and courses given at Linköping University,
Sweden, over a period of 15 years. It gives an introductory treatment of problems
andmethodsofstructuraloptimization.Thethreebasicclassesofgeometricalop-
timization problems of mechanical structures, i.e., size, shape and topology opti-
mization,aretreated.Thefocusisonconcretenumericalsolutionmethodsfordis-
crete and (finite element) discretized linear elastic structures. The style is explicit
and practical: mathematical proofs are provided when arguments can be kept ele-
mentary but are otherwise only cited, while implementation details are frequently
provided.Moreover,sincethetexthasanemphasisongeometricaldesignproblems,
wherethedesignisrepresentedbycontinuouslyvarying—frequentlyverymany—
variables,so-calledfirstordermethodsarecentraltothetreatment.Thesemethods
arebasedonsensitivityanalysis,i.e.,onestablishingfirstorderderivativesforob-
jectives and constraints. The classical first order methods that we emphasize are
CONLIN and MMA, which are based on explicit, convex and separable approxi-
mations.Itshouldberemarkedthattheclassicalandfrequentlyusedso-calledopti-
malitycriteriamethodisalsoofthiskind.Itmayalsobenotedinthiscontextthat
zeroordermethodssuchasresponsesurfacemethods,surrogatemodels,neuralnet-
works,geneticalgorithms,etc.,essentiallyapplytodifferenttypesofproblemsthan
the ones treated here and should be presented elsewhere. The numerical solutions
thatarepresentedareallobtainedusingin-houseprograms,someofwhichcanbe
downloadedfromthebook’shomepageatwww.mechanics.iei.liu.se/edu_ug/strop/.
These programs should also be used for solving some of the more extensive exer-
cisesprovided.
Thetextiswrittenforstudentswithabackgroundinsolidandstructuralmechan-
icswithabasicknowledgeofthefiniteelementmethod,althoughinourexperience
such knowledge could be replaced by a certain mathematical maturity. Previous
exposure to basic optimization theory and convex programming is helpful but not
strictlynecessary.
The first three chapters of the book represent an introductory and preparatory
part. In Chap. 1 we introduce the basic idea of mathematical design optimization
andindicateitsplaceinthebroaderframeofproductrealization,aswellasdefine
basicconceptsandterminology.Chapter2isdevotedtoaseriesofsmall-scaleprob-
lemsthat,ontheonehand,givefamiliaritywiththetypeofproblemsencountered
in structural optimization and, on the other hand, are used as model problems in
upcomingchapters.Chapter3reviewsbasicconceptsofconvexanalysis,andexem-
plifiesthesebymeansofconceptsfromstructuralmechanics.Chapter4is,froman
algorithmicpointofview,thecorechapterofthebook.Itintroducesthebasicideaof
sequentialexplicitconvexapproximations,andCONLINandMMAarepresented.
InChap.5thelatterisappliedtostiffnessoptimizationofatruss.Thisisaclassical
v
vi Preface
model problem of structural optimization which we investigate thoroughly. Chap-
ter6concernssensitivityanalysisforfiniteelementdiscretizedstructures.Sensitiv-
ities for shape changes are combined with two-dimensional shape representations
suchasBézierandB-splinesinChap.7,andthisclosesthetreatmentofshapeopti-
mization.Chapter8isessentiallyapreparationfortheformulationoftheproblemof
stiffnesstopologyoptimization.Wereviewsomeclassicalresultsofthecalculusof
variations,andderiveoptimalityconditionsforstiffnessoptimizationofdistributed
parameter systems. In Chap. 9 this problem is slightly extended and discretized,
anditprovidesagatewayintotheproblemoftopologyoptimizationforcontinuous
structures.Wederivetheoptimalitycriteriamethodasaspecialcaseofthegeneral
explicitconvexapproximationmethod, discuss well-posedness and different types
ofregularizationmethods.
Thisbeinganintroductorytreatment,wehavenotmadeanefforttogiveacom-
pletesetofreferences,noranhistoricalaccountofstructuraloptimization.Forthat
we refer to existing monographs such as Haftka and Gürdal [18], Kirsch [22] and
BendsøeandSigmund[4].
As mentioned, this book has its roots in several series of lectures at Linköping
University,wherethefirstonewasgivenbythesecondauthorofthisbookin1992.
Followingthese,intheyear2000,aseparatecourseinstructuraloptimizationwas
established,andJoakimPeterssonwasmaderesponsibleanddefineditsbasiccon-
tents. After having taught the course on two occasions, Joakim very unexpectedly
andsadlypassedawayinSeptember2002,[3].Theauthorsofthisbookthentook
overandsharedresponsibilityforthecourse,initiallyteachingitinawaythatwas
veryclosetothelecturenotesofJoakim.Outofappreciation,wehavecontinuedto
teachthecourse,andeventuallywrittenthisbook,closelyfollowingthespiritand
styleofJoakim,aswerememberandunderstandit.
We like to extend a special thanks to Bo Torstenfelt and Thomas Borrvall for
havingprovidedsomeofthenumericalsolutionspresentedinthebook.Torstenfelt’s
easy-to-usefiniteelementprogramTRINITASmaybedownloadedfromthebook’s
homepage, and should be used for two computer exercises on shape and topology
optimization. A Java applet by Borrvall for performing topology optimization is
also available on the homepage. For the permission to use their programs we are
sincerelygrateful.
Linköping, PeterW.Christensen
July2008 AndersKlarbring
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 TheBasicIdea . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 TheDesignProcess . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 GeneralMathematicalFormofaStructuralOptimizationProblem. 3
1.4 ThreeTypesofStructuralOptimizationProblems . . . . . . . . . 5
1.5 DiscreteandDistributedParameterSystems . . . . . . . . . . . . 7
2 ExamplesofOptimizationofDiscreteParameterSystems . . . . . . 9
2.1 Weight Minimization of a Two-Bar Truss Subject to Stress
Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 WeightMinimizationofaTwo-BarTrussSubjecttoStressand
InstabilityConstraints . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 WeightMinimizationofaTwo-BarTrussSubjecttoStressand
DisplacementConstraints . . . . . . . . . . . . . . . . . . . . . . 14
2.4 WeightMinimizationofaTwo-BeamCantileverSubjecttoa
DisplacementConstraint . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Weight Minimizationof a Three-Bar Truss Subjectto Stress
Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.6 WeightMinimizationofaThree-BarTrussSubjecttoaStiffness
Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 BasicsofConvexProgramming . . . . . . . . . . . . . . . . . . . . . 35
3.1 LocalandGlobalOptima . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 KKTConditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 LagrangianDuality . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4.1 LagrangianDualityforConvexandSeparableProblems . . 47
3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 SequentialExplicit,ConvexApproximations . . . . . . . . . . . . . . 57
4.1 GeneralSolutionProcedureforNestedProblems . . . . . . . . . . 57
vii
viii Contents
4.2 SequentialLinearProgramming(SLP) . . . . . . . . . . . . . . . 58
4.3 SequentialQuadraticProgramming(SQP) . . . . . . . . . . . . . 59
4.4 ConvexLinearization(CONLIN) . . . . . . . . . . . . . . . . . . 59
4.5 TheMethodofMovingAsymptotes(MMA) . . . . . . . . . . . . 66
4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5 SizingStiffnessOptimizationofaTruss . . . . . . . . . . . . . . . . 77
5.1 TheSimultaneousFormulationoftheProblem . . . . . . . . . . . 77
5.2 TheNestedFormulationandSomeofItsProperties . . . . . . . . 84
5.2.1 ConvexityoftheNestedProblem . . . . . . . . . . . . . . 85
5.2.2 FullyStressedDesigns. . . . . . . . . . . . . . . . . . . . 87
5.2.3 MinimizationoftheVolumeUnderaComplianceConstraint 88
5.3 NumericalSolutionoftheNestedProblemUsingMMA . . . . . . 91
6 SensitivityAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.1 NumericalMethods . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.2 AnalyticalMethods . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.2.1 DirectAnalyticalMethod . . . . . . . . . . . . . . . . . . 98
6.2.2 AdjointAnalyticalMethod . . . . . . . . . . . . . . . . . 99
6.3 AnalyticalCalculationofPseudo-loads . . . . . . . . . . . . . . . 100
6.3.1 Bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.3.2 PlaneSheets . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7 Two-DimensionalShapeOptimization . . . . . . . . . . . . . . . . . 117
7.1 ShapeRepresentation . . . . . . . . . . . . . . . . . . . . . . . . 117
7.1.1 BézierSplines . . . . . . . . . . . . . . . . . . . . . . . . 118
7.1.2 B-Splines. . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.2 TreatmentofGeometricalDesignConstraints. . . . . . . . . . . . 127
7.2.1 C1 ContinuityBetweenBézierSplines . . . . . . . . . . . 128
7.2.2 C1 ContinuityataPointonaLineofSymmetry . . . . . . 129
7.2.3 ACompositeCircularArc . . . . . . . . . . . . . . . . . . 131
7.3 MeshGenerationandCalculationofNodalSensitivities . . . . . . 132
7.3.1 B-SplineSurfaceMeshes . . . . . . . . . . . . . . . . . . 133
7.3.2 CoonsSurfaceMeshes. . . . . . . . . . . . . . . . . . . . 134
7.3.3 UnstructuredMeshes . . . . . . . . . . . . . . . . . . . . 137
7.4 SummaryofSensitivityAnalysisforTwo-DimensionalShape
Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8 StiffnessOptimizationofDistributedParameterSystems. . . . . . . 147
8.1 CalculusofVariations . . . . . . . . . . . . . . . . . . . . . . . . 147
8.1.1 OptimalityConditionsandGateauxDerivatives . . . . . . 149
8.1.2 HandlingaConstraint . . . . . . . . . . . . . . . . . . . . 153
8.2 EquilibriumPrinciplesforDistributedParameterSystems . . . . . 156
8.2.1 One-DimensionalElasticity . . . . . . . . . . . . . . . . . 156
Contents ix
8.2.2 BeamProblem . . . . . . . . . . . . . . . . . . . . . . . . 158
8.2.3 Two-DimensionalElasticity . . . . . . . . . . . . . . . . . 159
8.2.4 AbstractEquilibriumPrinciples . . . . . . . . . . . . . . . 162
8.3 TheDesignProblem . . . . . . . . . . . . . . . . . . . . . . . . . 163
8.3.1 OptimalityConditions . . . . . . . . . . . . . . . . . . . . 166
8.3.2 TheStiffestRod . . . . . . . . . . . . . . . . . . . . . . . 168
8.3.3 BeamStiffnessOptimization . . . . . . . . . . . . . . . . 170
8.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
9 TopologyOptimizationofDistributedParameterSystems . . . . . . 179
9.1 TheVariableThicknessSheetProblem . . . . . . . . . . . . . . . 179
9.1.1 ProblemStatementandFE-Discretization. . . . . . . . . . 179
9.1.2 TheOptimalityCriteria(OC)Method . . . . . . . . . . . . 182
9.2 PenalizationofIntermediateThicknessValues . . . . . . . . . . . 188
9.2.1 SolidIsotropicMaterialwithPenalization(SIMP) . . . . . 188
9.2.2 OtherPenalizations . . . . . . . . . . . . . . . . . . . . . 190
9.3 Well-PosednessandPotentialNumericalProblems . . . . . . . . . 190
9.3.1 TheArchetypeProblemandanAnalogy . . . . . . . . . . 190
9.3.2 NumericalInstabilities . . . . . . . . . . . . . . . . . . . . 191
9.4 RestrictionoftheArchetypeProblem . . . . . . . . . . . . . . . . 193
9.4.1 BoundsontheDesignGradient . . . . . . . . . . . . . . . 194
9.4.2 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
9.5 RelaxationoftheArchetypeProblem . . . . . . . . . . . . . . . . 198
9.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
AnswerstoSelectedExercises. . . . . . . . . . . . . . . . . . . . . . . . . 203
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
