Approximation of topology optimization problems using sizing optimization problems PDF
Preview Approximation of topology optimization problems using sizing optimization problems
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+46(0)31–7721000 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. ii 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 vi 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
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