Approximation of topology optimization problems using sizing optimization problems

103 Pages·2004·1.26 MB·English

by  Anton Evgrafov

THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
Approximation of
topology optimization problems
using sizing optimization problems
Anton Evgrafov
DepartmentofMathematics
ChalmersUniversityofTechnologyandGöteborgUniversity
Göteborg,Sweden2004
Approximationoftopologyoptimizationproblems
usingsizingoptimizationproblems
AntonEvgrafov
ISBN91–7291–466–1
©AntonEvgrafov,2004
DoktorsavhandlingarvidChalmerstekniskahögskola
Nyserienr2148
ISSN0346–718X
DepartmentofMathematics
ChalmersUniversityofTechnologyandGöteborgUniversity
SE-41296Göteborg
Sweden
Telephone[phone]
PrintedinGöteborg,Sweden2004
Approximationoftopologyoptimizationproblems
using sizing optimizationproblems
AntonEvgrafov
DepartmentofMathematics
ChalmersUniversityofTechnologyandGöteborgUniversity
ABSTRACT
Thepresentworkisdevotedtoapproximationtechniquesforsingularextremalproblems
arisingfromoptimaldesignproblemsinstructuralandfluidmechanics. Thethesiscon-
sists of an introductorypart and four independentpapers, which howeverare united by
thecommonideaofapproximationandtherelatedapplicationareas.
Inthefirsthalfofthethesisweareconcernedwithfindingtheoptimaltopologyoftruss-
likestructures. Thisclassofoptimaldesignproblemsariseswheninordertofindtheop-
timaltrussnotonlyareweallowedtoredistributethematerialamongthestructuralmem-
bers(bars),butalsotocompletelyremovesomepartsalteringtheconnectivity(topology)
ofthestructure.Theotherhalfofthethesisaddressesthequestionoftheoptimaltopolog-
icaldesign offlow domainsforStokesandNavier–Stokesfluids. For flows, optimizing
topology means finding the optimal partition of the given design domain into disjoint
partsoccupiedbythefluidandtheimpenetrablewalls,giventhein-flowandtheout-flow
boundaries.Inparticular,impenetrablewallschangetheshapeandtheconnectivityofthe
flowdomain.
Inthefirstpaperweconstructanexampledemonstratingthesingularbehaviouroftruss
topologyoptimizationproblemsincludingalinearizedglobalbuckling(linearelasticsta-
bility) constraint. This singularity phenomenonhas not been known before and affects
the choice of numerical methods that can be applied to the optimization problem. We
propose a simple approximation strategy and establish the convergenceof globally op-
timalsolutionsto perturbedproblemstowardsgloballyoptimalsolutionsto the original
singularproblem.
Inthesecondpaperweareconcernedwiththeconstructionoffinerapproximatingprob-
lemsthat allowus to reconstructthe localbehaviourof a generalclass of singulartruss
topologyoptimizationproblems,namelytoapproximatestationarypointstothelimiting
problemwithsequencesofstationarypointstotheregularapproximatingproblems. We
dosoontheclassic problemofweightminimizationunderstressconstraintsfortrusses
inunilateralcontactwithrigidobstacles.
Inthethirdpaperwe extenda designparametrizationpreviouslyproposedforthe topo-
logicaldesignofflowdomainsforStokesflowstoalsoincludethelimitingcaseofporous
materials—completelyimpenetrablewalls. Wedemonstratethat,ingeneral,theresulting
design-to-flowmappingisnotclosed,yetundermildassumptionsitispossibletoapprox-
imate globally optimalminimal-power-dissipationdomainsusing porousmaterialswith
diminishingpermeability.
InthefourthandlastpaperweconsidertheoptimaldesignofflowdomainsforNavier–
Stokes flows. We illustrate the discontinuous behaviour of the design-to-flow mapping
caused by the topological changes in the design, and propose “minor” changes to the
designparametrizationandtheequationsthatallowustorigorouslyestablishtheclosed-
ness of the design-to-flow mapping. The existence of optimal solutions as well as the
convergenceofapproximationschemestheneasilyfollowsfromtheclosednessresult.
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LIST OF PUBLICATIONS
Thisthesisconsistsofan introductorypartand thefollowingpapers:
Paper1 A. Evgrafov, On globally stable singular topologies, to appear in
Structuraland MultidisciplinaryOptimization,2004.
Paper2 A. Evgrafov and M. Patriksson, On the convergence of station-
arysequencesintopologyoptimization,submittedtoInternational
JournalforNumericalMethodsinEngineering,2004.
Paper3 A. Evgrafov, On the limits of porous materials in the topology
optimization of Stokes flows, submitted to Applied Mathematics
andOptimization,2003.
Paper4 A. Evgrafov, Topology optimization of slightly compressible flu-
ids, submitted to Zeitschrift für Angewandte Mathematik und
Mechanik,2004.
iv
CONTENTS
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Listofpublications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Introductionandoverview . . . . . . . . . . . . . . . . . . . . . . . . . ix
1. Ongloballystablesingulartopologies . . . . . . . . . . . . . . . . . 1
2. On the convergence of stationary sequences in topology optimiza-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3. On the limits of porous materials in the topology optimization of
Stokesflows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4. Topologyoptimizationofslightlycompressiblefluids . . . . . . . . 55
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ACKNOWLEDGEMENT
DURINGthetimespentonwritingthisdissertation,aswellasonmanyotheroccasions,
Ihavealwaysbeenfeelingtheunflaggingsupportofmanypeople,including
mysupervisor,Prof.MichaelPatriksson;
◦
mywifeElena;
◦
myfriends;
◦
myparents.
◦
Iwilleverbeindebtedtoallofyou.
Many thanks to the School of Mathematical Sciences for being such a welcoming and
friendly place to work, and to the Swedish Research Council for financially supporting
meunderthegrantVR-621-2002-5780.
IdedicatemydissertationtothememoryofJoakimPetersson,whountimelypassedaway
onSeptember30,2002,atthemereageof33. Thisworkistoagreatextentinspiredby
hisresearch.
AntonEvgrafov
Göteborg,May2004
viii