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1、MeshSmoothingChallengesintheIndustry SmoothmixedmeshesMovenodesinalldimensions coupled uncoupled Offerenoughuser controlObtainhighermeshqualityTestforinvalidmeshHandlestructured mappedmeshregions WhatisVariationalSmoothing Thisisashellmeshsmoothingtechniquein3Dspacethatcombinesavarietyofconventional
2、smoothingmethodsinanefforttoreapthe bestbenefits andpreventirreversiblemeshdistortions Thevariationalalgorithmsmootheseachnodeaccordingtoaspecificsmoothingtechniquedefinedbyvariationalrules Thesmoothingmethodselectiondependsonthe meshunit connectedtoanode Itisa hybrid and heuristic approach ProblemS
3、tatement Smoothesshellmeshesin1D 2D 3DspaceIsiterative almostasefficientasLaplaceGivesseveralcontrolstotheuserTriestopreservemapped structuredmeshesormeshregionsWorksbetterthanmostsmoothersinconcavedomainsRarelycreatesinvertedelementsImproveselementincludedangles averageelementskewandhencemeshqualit
4、ySmoothedmeshmay maynotbeprojectedbacktosurface VARIATIONALSMOOTHINGMODEL Thegoverningequation NPi Fn C V n C V n 1wherePi Newpositionofnodei Fn Variationalweightfactorforn thelement n Positionalfunctionforn thelementC Connectivitypatternofthenode V Nodalvalency WhatisaMeshUnit MeshUnit VtqV nodalva
5、lencyq noofquadst nooftrianglesAmeshunitisdefinedbyanodeNumberofelementsconvergingatthenodeThetopologyofconnectingelements MeshUnit 303 SMOOTHINGSchemes IncenterSmoothing NPi Pi Wn Pn Pi n 1Pi x y z isthepositionvectorofnodeiPn x y z istheincentervectorofelementnN No ofelementsatnodeiInitialMeshAfte
6、rLaplacianSmoothingAfterIncentersmoothing SMOOTHINGSchemes Isoparametric Laplace 1NPi Wn Pnj Pnl wPnk N 2 w n 1N no ofelementsatnodeiw couplingfactor 0 0 Laplace1 0 Isoparametric0 5 Iso LaplaceLaplaceIsoparametricIsoparametric Laplace SMOOTHINGSchemes Equipotential WinslowSmoothing Thegoverningequat
7、ionforequipotential Winslow smoothingcanwrittenfornodeias Pi 2 Pi Pi 0 where arelogicalvariablesthatareharmonicinnature while areconstantcoefficientsthatdependontheproblem Theweighingfactorsofthe8neighboringnodesaregivenbyW1 2 W2 W3 2 W4 W5 2 W6 W7 2 W8 where xp2 yp2 zp2 xpxq ypyq zpzq xq2 yq2 zq2xp
8、 x2 x6 2 yp y2 y6 2 zp z2 z6 2xq x8 x4 2 yq y8 y4 2 zq z8 z4 2 SMOOTHINGSchemes EquipotentialSmoothing OriginalMeshAfterLaplaciansmoothingAfterWinslowsmoothing SMOOTHINGSchemes EquipotentialSmoothing OriginalmappedmeshMeshaftertangling SMOOTHINGSchemes EquipotentialSmoothing AfterLaplaceSmoothingAft
9、erWinslowSmoothingInitialtangledmesh MeshUnits All Quad MeshUnit 303Isoparametric LaplacesmoothingLength angle weightedLaplaceMeshUnit 404Equi potentialsmoothingMeshUnit 505Isoparametric Laplacesmoothing MeshUnits AllTriangular MeshUnit 660Incenter Angle Zhou Shimada LaplacesmoothingMeshUnit 770Ince
10、nter Laplace Anglesmoothing Zhou Shimada MeshUnit 880Equi potentialsmoothing MeshUnits Mixed MeshUnit 514Incenter Iso LaplaceAnglesmoothingMeshUnit 624Incenter Iso Laplacesmoothing MeshUnit 413Incenter Iso Laplacesmoothing SMOOTHINGSchemes HandlingBivalentnodes SmartSmoothingConstraints Constrainedn
11、odemovement AnglecheckCheckelementincludedanglesduringsmoothingRegionCheckKeepnodeinsidetheboundingboxformedbythebarycentersoftheconnectedelement SmoothingBoundary MorphedOrphanedShellMeshes ShiftthisholeShrinkthishole MeshQualityNoisdefinedasNMQNo Ei 1 Ni 1whereEiistheelementqualitynumberforelement
12、iItmeasureselementskew warp stretch aspectratioandJacobianEiisnon dimensionalandvariesfrom0and1 SmoothingBoundary MorphedOrphanedShellMeshes AfterLaplacesmoothingMeshQualityNo Invalid Unsolvablemesh BadMesh MQN0 4GoodMesh 0 60 5ExcellentMesh MQN 0 6PerfectMesh MQN 1 0AfterVariationalsmoothingMeshQua
13、lityNo 0 504 MorphingAndRemeshingOnLegacyFEM HowVariationalSmoothingHelps Stepstomorphshiftholegougeholeoutstretchendsbendtailaddnewtailcut outsforwiringaccess MorphingAndRemeshingOnLegacyFEM AfterpreliminaryMorphingSteps AfterpreliminarymorphingOriginalmesh MorphingAndRemeshingOnLegacyFEM 3Dsmoothi
14、ngthemorphedmesh AfterLaplacesmoothingMeshQualityNo 0 440 AftervariationalsmoothingMeshQualityNo 0 653 MorphingAndRemeshingOnLegacyFEM Refeaturingsteps B RepisaddedtotherawmorphedmeshNewfeatures cut outsforwiringaccess areadded MorphingAndRemeshingOnLegacyFEM 2Dsmoothingduringremesh Remeshed re smoo
15、thedmorphedlegacyFEMwithnewfeatures AfterLength weightedsmartLaplaciansmoothingMeshQualityNo 0 507 AfterVariationalsmoothingMeshQualityNo 0 759 GlobalSmoothing MeshQualitybeforeglobalsmoothing 3700elements 0 45393elementsfaildifferentelementqualitychecksAfterlength weightedsmartLaplaciansmoothing MeshQualityNo 0 44 379elementsfailAftervariationalsmoothing MeshQualityNo 0 59 210elementsfail Acknowledgements JeanCabelloMichaelHancock
