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1、ALINEARWEIGHTEDLAPLACIANSMOOTHINGFRAMEWORKFORWARPINGTETRAHEDRALMESHES∗§SUZANNEM.SHONTZ†ANDSTEPHENA.VAVASIS‡Abstract.Wepresentanewmeshwarpingframeworkfortetrahedralmeshesbaseduponweightedlaplaciansmoothing.Westartwitha3Ddomainthatisboundedbyatriangulatedsurf
2、acemeshandhasatetrahedralvolumemeshasitsinterior.Wethensupposethatamovementofthesurfacemeshisprescribedanduseanalgorithmwithinourframeworktoupdatethenodesofthevolumemesh.Thefirststepistodetermineasetoflocalweightsforeachinteriornodethatdescribeseachinteriorn
3、odeintermsofthepositionsofitsneighbors.Afteraboundarytransformationisapplied,alinearsystemofequationsbasedupontheweightsissolvedtodeterminethefinalpositionsoftheinteriornodes.Thethreestepscomprisethelinearweightedlaplaciansmoothing(LWLS)framework.Wepresenttw
4、omethodswithinthisframework.Thefirst,LBWARP,usesalog-barrierapproachtocomputetheweights.Thesecond,FEMWARP,isbaseduponthefiniteelementmethod.Westudymeshinvertibilityandproveatheoremgivingsufficientconditionsforameshtoresistinversionbyatransformation.Weproveatheo
5、remforgeneralmappingswithinthecontextofFEMWARP.WeshowthatforLWLSalgorithms,theresultingmeshisthesameastheconvergedmeshobtainedfromthelocalversionofweightedlaplaciansmoothingandfromtheGauss-Seidelalgorithm,whenthelattertwoalgorithmsconverge.Wetesttherobustne
6、ssofouralgorithmsandpresentsomenumericalresults.Finally,weuseFEMWARPtostudythemovementofthecanineheart.Keywords.movingmeshes,adaptation,optimization-basedmeshsmoothing,log-barriermethod,finiteelementmethod,tetrahedralmeshes,unstructuredmeshgeneration,cardiol
7、ogyAMSsubjectclassifications.65N50,65N30,92C101.Introduction.Movingmeshesariseincardiology,computergraphics,animation,andcrashsimulation,amongotherapplicationsinscienceandengineering.Withmovingmeshes,themeshisupdatedateachstepintimeduetoamovingdomainboundary
8、,thusresultinginpotentiallydrasticallyvaryingmeshqualityfromsteptostep.Oneproblemthatcanoccurateachtimestepiselementinversion.Wefocusonmaintaininggood-qualitytetrahedralmeshesthroughoutthemeshwarpingpr
