Table Of ContentSolid Mechanics and Its Applications
Alan Rothwell
Optimization
Methods in
Structural
Design
Solid Mechanics and Its Applications
Volume 242
Series editors
J.R. Barber, Ann Arbor, USA
Anders Klarbring, Linköping, Sweden
Founding editor
G.M.L. Gladwell, Waterloo, ON, Canada
Aims and Scope of the Series
Thefundamentalquestionsarisinginmechanicsare:Why?,How?,andHowmuch?
The aim of this series is to provide lucid accounts written by authoritative
researchersgivingvisionandinsightinansweringthesequestionsonthesubjectof
mechanics as it relates to solids.
The scope of the series covers the entire spectrum of solid mechanics. Thus it
includes the foundation of mechanics; variational formulations; computational
mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations
of solids and structures; dynamical systems and chaos; the theories of elasticity,
plasticity and viscoelasticity; composite materials; rods, beams, shells and
membranes; structural control and stability; soils, rocks and geomechanics;
fracture; tribology; experimental mechanics; biomechanics and machine design.
Themedianlevelofpresentationistothefirstyeargraduatestudent.Sometexts
aremonographs defining thecurrentstateofthe field; othersareaccessibletofinal
year undergraduates; but essentially the emphasis is on readability and clarity.
More information about this series at http://www.springer.com/series/6557
Alan Rothwell
Optimization Methods
in Structural Design
123
AlanRothwell
Formerly Delft University ofTechnology
Delft
TheNetherlands
Additional material tothis bookcanbedownloaded from http://extras.springer.com.
ISSN 0925-0042 ISSN 2214-7764 (electronic)
Solid MechanicsandIts Applications
ISBN978-3-319-55196-8 ISBN978-3-319-55197-5 (eBook)
DOI 10.1007/978-3-319-55197-5
LibraryofCongressControlNumber:2017933063
©SpringerInternationalPublishingAG2017
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor
for any errors or omissions that may have been made. The publisher remains neutral with regard to
jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
MicrosoftandExcelareregisteredtrademarksoftheMicrosoftCorporation.
ESDUisaregisteredtrademarkofESDUInternationalLimited.
Printedonacid-freepaper
ThisSpringerimprintispublishedbySpringerNature
TheregisteredcompanyisSpringerInternationalPublishingAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
To my wife, Janette, for her love, support and
patience, and to our four children, Katherine,
Sarah, Rachel and Paul.
Preface
The aim of this book is to present numerical optimization methods in structural
designtostudentsinengineeringcoursesatfinalundergraduatelevelorinthefirst
year ofa postgraduate study. Forothers inindustry or elsewhere who may benew
to these highly practical techniques, the book can bridge the gap between familiar
design practice and some of the advanced texts on optimization theory. While the
specificapplicationistostructuraldesign,theprinciplesinvolvedcanbeappliedfar
morewidely.A‘howtodoit’approachisfollowedthroughoutthebook,withless
emphasis at this stage on mathematical derivations. Extensive use is made of the
‘Solver’ optimization tool in Microsoft Excel1, because of its ready availability.
Thisprovidesanidealmeansofillustratingthemethodspresented,howtosetupan
optimizationproblemandtodemonstratetheusefulnessofoptimizationtechniques
ingeneral.WithpracticeintheuseofSolver,useofoptimizationmodulesinmore
extensive computer packages should present little difficulty.
The spreadsheet programs provided with this book are, in the earlier chapters,
principally illustrations of optimization methods. In later chapters, these are of a
morepracticalnature,inparticularforreinforcedshellstructuresandforthedesign
of composite laminates. These topics are chosen to reflect the ever-increasing
demand for lightweight structures in many branches of engineering. Weight
reductionis notonly toreduce operational costs, but also to offset thehigh cost of
many modern, high-performance metallic materials and composites. Detailed
instructions are given for use of the spreadsheets and on the use of Solver.
Exercises, with solutions where appropriate, are provided with each chapter, many
ofthemmakingsome otheruseofSolver orfurtheruseofthespreadsheets.These
areintendedtogivepracticeinsettingupanoptimizationproblemandgenerallyto
explorethecharacteristicsoftheoptimizationprocess.Manyoftheexamplesinthe
book, throughout the text and in the spreadsheets, will be seen to have a distinct
aerospaceflavour,thisbeing simplyareflectionoftheauthor’smainfieldof work
over many years.
1MicrosoftandExcelareregisteredtrademarksoftheMicrosoftCorporation.
vii
viii Preface
Theearlychaptersofthebookshowtherelationshipbetweenformaloptimization
and the traditional methods of design, it not being the intention to replace existing
methods but rather to supplement them with an additional weapon in the armoury
of the designer. Strength-to-weight ratios, limits of feasibility and the concept of
structuralefficiencyarediscussed.Classicaloptimizationisthenintroduced,together
withtheLagrangemultiplier,fundamentaltothediscussionofnumericaloptimiza-
tionmethodsinthefollowingchapters.Numericalmethodsareintroducedinsuffi-
cientdetailtoenablethereadertoappreciatetheprocessestakingplaceinsomeofthe
highlysophisticated‘blackbox’optimizationroutinesinadvancedcomputerpack-
ages.Itisnottheintentiontodescribethesenumericalmethodsinthedetailnecessary
to enable the reader to program them efficiently, this being a task primarily for the
programming specialist. The generalized reduced gradient method and the genetic
algorithm,twoofthemethodsavailableinSolver,aregivendueattention,thelatterin
alaterchapterinthecontextofcompositelaminates.Theremainingchaptersofthe
bookaredevotedtoapplications—reinforcedshellstructures,withthedesignofabox
beam and an aircraft fuselage section, as well as some extended discussion of the
design of composite laminates. For these topics, relevant methods of analysis are
coveredinsufficientdetailbeforeproceedingtospecificoptimizationproblemsand
spreadsheetprogramsfortheirsolution.Compositelaminatesareofparticularinterest
becauseofthespecialproblemintroducedbythediscretenatureoftheindividualplies
of the laminate and because of the freedom to optimize the lay-up to match the
application.Afinalchapterisgiventooptimizationwithfiniteelementanalysis,for
whichsomespecialmethodsarenecessary.
The level of knowledge required to follow the text is no more than in a usual
engineering course. No specific demands are made, and the text should remain
largelyaccessibletothosefromotherdisciplines,sufficientinformationbeinggiven
‘to proceed from this point’. However, it is assumed that the reader already has a
workingknowledgeofMicrosoftExcel,withsomeVisualBasic,andalsoisfamiliar
with matrix notation. With a less mathematical bias, he might in the first place go
rather superficially over Chaps. 4 and 5 and with no experience offinite element
methodsmightbetemptedtomissChap.9.Noattemptismadeatcompletenessin
thebook,butrathertoprovideasoundunderstandingofbasicprinciplesandagood
startforfurtherstudy.Forthis,alistoffurtherreadingisincluded(reflectingperhaps
moretheauthor’spersonalchoice).Specificreferencetoresearchpapersislimitedto
where this is of particular relevance. For a more comprehensive reference list, the
reader should turn to the several excellent, more advanced books on optimization
theoryincluded amongstthereferences attheendof each chapter.
This book is based on lectures given at Delft University of Technology in the
Netherlands,whiletheauthorwasprofessorofaircraftstructures.Hehopesthatthe
reader will enjoy a study of optimization methods as much as he has and will be
able to put them to good use in further study and engineering practice.
Delft, The Netherlands Alan Rothwell
Contents
1 The Conventional Design Process.... .... .... .... .... ..... .... 1
1.1 Fully Stressed Design. ..... .... .... .... .... .... ..... .... 3
1.1.1 Structure Made of Different Materials.... .... ..... .... 7
1.1.2 Structure Under Alternative Loads .. .... .... ..... .... 9
1.2 Strength-to-weight Ratio.... .... .... .... .... .... ..... .... 12
1.2.1 Feasibility.... ..... .... .... .... .... .... ..... .... 15
1.3 Comparison of Layouts .... .... .... .... .... .... ..... .... 16
1.3.1 Classification of Optimization Problems .. .... ..... .... 19
1.4 Spreadsheet Program . ..... .... .... .... .... .... ..... .... 21
1.4.1 ‘Seven-Bar Truss’... .... .... .... .... .... ..... .... 21
1.5 Summary .. .... .... ..... .... .... .... .... .... ..... .... 24
Exercises... .... .... .... ..... .... .... .... .... .... ..... .... 25
References.. .... .... .... ..... .... .... .... .... .... ..... .... 28
2 Optimality Criteria .. .... ..... .... .... .... .... .... ..... .... 29
2.1 Circular Tube in Compression ... .... .... .... .... ..... .... 30
2.1.1 Efficiency Formula .. .... .... .... .... .... ..... .... 33
2.1.2 Material Limitation.. .... .... .... .... .... ..... .... 38
2.2 Criterion for Maximum Stiffness . .... .... .... .... ..... .... 40
2.3 Spreadsheet Programs. ..... .... .... .... .... .... ..... .... 44
2.3.1 ‘Circular and Square Tubes’... .... .... .... ..... .... 44
2.3.2 ‘Truss with Tubular Members’ . .... .... .... ..... .... 48
2.4 Summary .. .... .... ..... .... .... .... .... .... ..... .... 50
Exercises... .... .... .... ..... .... .... .... .... .... ..... .... 51
References.. .... .... .... ..... .... .... .... .... .... ..... .... 53
3 The General Optimization Problem.. .... .... .... .... ..... .... 55
3.1 Box Beam Structure.. ..... .... .... .... .... .... ..... .... 56
3.1.1 General Form of Design Space. .... .... .... ..... .... 58
ix
x Contents
3.2 The Lagrange Multiplier Method . .... .... .... .... ..... .... 61
3.2.1 Interpretation of Lagrange Multipliers.... .... ..... .... 67
3.3 Inequality Constrained Problems . .... .... .... .... ..... .... 70
3.3.1 The Kuhn–Tucker Conditions.. .... .... .... ..... .... 73
3.4 Spreadsheet Program . ..... .... .... .... .... .... ..... .... 73
3.4.1 Eccentrically Loaded Column.. .... .... .... ..... .... 74
3.5 Summary .. .... .... ..... .... .... .... .... .... ..... .... 78
Exercises... .... .... .... ..... .... .... .... .... .... ..... .... 79
References.. .... .... .... ..... .... .... .... .... .... ..... .... 81
4 Numerical Methods for Unconstrained Optimization.... ..... .... 83
4.1 Unconstrained Optimization . .... .... .... .... .... ..... .... 84
4.1.1 Steepest Descent Method . .... .... .... .... ..... .... 85
4.1.2 Fletcher–Reeves Method.. .... .... .... .... ..... .... 90
4.1.3 Quasi–Newton Methods .. .... .... .... .... ..... .... 92
4.2 Line Search Methods . ..... .... .... .... .... .... ..... .... 94
4.2.1 Region Elimination and the Golden Section Method.. .... 95
4.2.2 Polynomial Interpolation.. .... .... .... .... ..... .... 97
4.3 Spreadsheet Program . ..... .... .... .... .... .... ..... .... 100
4.3.1 ‘Hooke and Jeeves Method’ ... .... .... .... ..... .... 100
4.4 Summary .. .... .... ..... .... .... .... .... .... ..... .... 103
Exercises... .... .... .... ..... .... .... .... .... .... ..... .... 105
References.. .... .... .... ..... .... .... .... .... .... ..... .... 106
5 Numerical Methods for Constrained Optimization.. .... ..... .... 107
5.1 Constraint-Following Methods ... .... .... .... .... ..... .... 108
5.1.1 Gradient Projection Method ... .... .... .... ..... .... 109
5.1.2 Generalized Reduced Gradient Method... .... ..... .... 120
5.1.3 Other Methods for Constrained Optimization .. ..... .... 126
5.1.4 Substitution of Variables.. .... .... .... .... ..... .... 129
5.2 Penalty Function Methods .. .... .... .... .... .... ..... .... 129
5.2.1 Interior Penalty Function.. .... .... .... .... ..... .... 130
5.2.2 Exterior Penalty Function . .... .... .... .... ..... .... 133
5.2.3 Augmented Lagrangian Penalty Function . .... ..... .... 135
5.3 Spreadsheet Program . ..... .... .... .... .... .... ..... .... 138
5.3.1 ‘Penalty Function Method’ .... .... .... .... ..... .... 140
5.4 Summary .. .... .... ..... .... .... .... .... .... ..... .... 142
Exercises... .... .... .... ..... .... .... .... .... .... ..... .... 143
References.. .... .... .... ..... .... .... .... .... .... ..... .... 145
6 Optimization of Beams ... ..... .... .... .... .... .... ..... .... 147
6.1 Beam Cross Section.. ..... .... .... .... .... .... ..... .... 148
6.1.1 Thin-Walled Beams.. .... .... .... .... .... ..... .... 150
6.1.2 Geometrically Similar Sections. .... .... .... ..... .... 153
