一種基于主應力的雙方向漸進結構拓撲優化方法

Abstract

Thispaperproposesanewtopologyoptimizationmethodforbi-directionalgraduallyevolvingstructuresbasedonprincipalstress.Bytransformingtheoriginalbi-directionalgraduallyevolvingstructureintoalatticestructure,themethodallowsforfindinganoptimizedlatticestructurewithminimalcompliance.Thebasicprinciplebehindthemethodistoidentifythedirectionoftheprincipalstressandapplyregularizediso-densityfilteringtechniqueinthetwoorthogonaldirections.Thismethodhasbeenprovenhighlyeffectiveintopologyoptimizationproblemsforbi-directionalgraduallyevolvingstructures,demonstratingremarkableoptimizationresultsforcomplex3Dstructures.

Introduction

Topologicaloptimizationhasbecomeaninternalpartofmodernengineeringdesign,withthepotentialforsignificantsavingsindesigntime,materialcosts,andimprovedstructuralperformance.Thetopologyoptimizationmethodiswidelyusedintheaerospace,automotive,andmechanicalengineeringindustries.Inrecentyears,thebi-directionalgraduallyevolvingstructureshavereceivedmuchattentionintheresearchcommunityduetotheirspecializedfunctioninthefieldofmicro-electromechanicalsystems(MEMS),aerospaceengineering,andenergyharvesting.However,optimizingthesestructuresisrelativelychallengingcomparedtotraditionaloptimizedstructures.Thetopologyoptimizationmethodcanhelpobtainanoptimizedstructurethatmeetsthedesignobjectivewhilesatisfyingthestringentstructuralconstraints.

Principalstressisanimportantindicatorofthemechanicalpropertiesofmaterials.Calculatingandanalyzingtheprincipalstresses,wecanfindthekeyareaofthestructuralstress,andthenoptimizethestructurebyredistributingthematerialinthekeyarea.Combiningthiswithlatticestructureoptimization,thebi-directionalgraduallyevolvingstructurecanbeeffectivelyoptimized.

ProposedMethod

Theproposedmethodfortopologyoptimizationofbi-directionalgraduallyevolvingstructuresmainlyincludesthreeprocedures.Firstly,wetransformtheoriginalstructureintoalatticestructure.Secondly,we

calculatetheprincipalstressofthestructure.Finally,weapplytheregularizediso-densityfilteringtechniquetoobtaintheoptimaldesign.

Latticestructuretransformation

Theoriginalstructureistransformedintoalatticestructure,andthenthelatticestructureisoptimized.Thetopologyoptimizationisperformedinthelatticegrid,whichcaneffectivelyavoidtheproblemsofmeshormeshdeformationcausedbycomplexstructures.Thelatticestructuretransformationprocessincludesthreesteps:meshing,fixing,andsmoothing.Themeshingprocessdividesthestructureintoauniformgrid.Thefixingprocessfixesthenodesattheboundariesofthestructuretoensurethatthemodeldoesnotdeformduringtheoptimizationprocess.Thesmoothingprocessremovestheverticesthatdonotbelongtothedesignspace,ensuringthatthemodelhasasmoothsurfaceforbetteroptimization.

Calculationofprincipalstress

Afterthelatticestructuretransformation,wecalculatetheprincipalstressofthestructuretoidentifythecriticalareaofthestructure.Weextractthemaximumandminimumprinciplestressfromtheresultsoffiniteelementanalysis.Theprinciplestressdirectionforeachpointinthestructureisfoundandlabelled.Todistinguishdifferentprinciplestressdirections,weusedifferentorientationsforlatticestructuresinthexandydirections.

Regularizediso-densityfilteringtechnique

Theregularizediso-densityfilteringtechniqueisafilteringtechniquewithcontrolledgradient.Itisusedtoadjusttheminimumdensityvaluesandthemaximumdensityvaluesinthelattice.Inthismethod,wefilterthelatticestructuretwice,onceforthexdirectionandoncefortheydirection.Theintermediatevaluesarethedensityvaluesofthecalculatedpointineachdirection.Theminimumandmaximumdensityvaluesofeachdirectionareset,respectively.Theregularizediso-densityfilteringtechniqueiscompletedwhenthelatticestructurewithlowcomplianceisobtained.

Optimizationresultanalysis

Thisproposedtopologyoptimizationmethodisappliedtooptimizethebi-directionalgraduallyevolvingstructure.Thedesignobjectiveistominimizethecomplianceofthestructure,orequivalently,tomaximizethestiffnessofthestructure.TheoptimizationresultofthelatticestructureofaMEMSdeviceisshowninFig.1.

Theoptimizedstructureismorecompactthantheoriginalstructure.Inaddition,theoptimizedstructurehasfewerholes,andthewedge-shapedstructureisremoved.Theoptimizedstructurehasahigherstiffnessthantheoriginalstructure.Theoptimizedstructureeffectivelyavoidsthedifficultycausedbytheexcessivelycomplexstructuralshapeintopologyoptimization.

Conclusion

Thispaperproposesanewtopologyoptimizationmethodbasedonprincipalstressforbi-directionalgraduallyevolvingstructures.Bytransformingtheoriginalstructureintoalatticestructure,themethodallowsforobtaininganoptimizedlatticestructurewithminimalcompliance.TheproposedmethodsuccessfullyoptimizesthestructureofaMEM