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一种用于柔顺机构拓扑优化的改进Sigmoid插值函数.docx

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一种用于柔顺机构拓扑优化的改进Sigmoid插值函数 Introduction： Topologyoptimizationisapowerfultoolforengineeringdesign,especiallyinthefieldofengineeringmechanics.Itoptimizesthematerialdistributionofamechanicalstructuretominimizeitsdeflectionorstressunderexternalloads.Inrecentyears,numerousmethodshavebeenproposedanddevelopedtooptimizethetopologyofmechanicalstructures.OneofthemostwidelyusedmethodsforachievingthisistheSigmoidalfunction. TheSigmoidalfunctionisanon-linearfunctionusedinmathematicsandstatisticsforcurvefitting,predictionandmachinelearningalgorithms.Intopologyoptimization,theSigmoidalfunctionisappliedasasmoothingfunction,asithasasmoothtransitionbetween0and1.However,therearesomedisadvantagesassociatedwithusingthisfunction,suchastheover-smoothingeffectthatcanresultinundesiredgeometricalshapes. Inthispaper,weproposeanimprovedSigmoidalinterpolationfunctionfortopologyoptimization,whichcanachieveabetterbalancebetweensmoothingandmaintainingdetailedfeatures. Background： Intopologyoptimization,itisimportanttoensurethattheoptimizedstructuresatisfiesdifferentsetofconstraints.Oneofthemostcommonconstraintsisthevolumeconstraint,whichlimitstheamountofmaterialusedinthestructure.TheSigmoidalfunctionisusedtosmooththeresultandavoidtheformationofsmalldisconnectedregions.Therefore,itcanbesaidthattheSigmoidalfunctionhelpsbalancingtheshapeandvolumeofthestructure. However,amajordownsideoftheSigmoidalfunctionisthefactthatitover-smoothstheresults.ThisisbecausetheSigmoidalfunctionhasasmallgradientnearthebouncingpoint,leadingtoagradualdecreaseininfluenceasthedistancefromtheboundingpointincreases.Furthermore,theSigmoidalfunctionisonlydefinedbetweenzeroandone,whichlimitsitsflexibilitytogeneratecomplexgeometries. Therefore,itisnecessarytoimprovetheSigmoidalfunctionbyaddressingtheover-smoothingeffectandincreasingitsflexibility. ProposedSolution: TheproposedsolutioninvolvesimprovingtheSigmoidalfunctionbymodifyingtheformulaandaddingadjustableparameters.Thenewfunctioncanbeexpressedasfollows: f(x)=(1+e^-k(x-a))^(-a