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The quasi-uniformity condition and three-dimensional geometry representation as it applies to the reproducing kernel ele...

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Title:
The quasi-uniformity condition and three-dimensional geometry representation as it applies to the reproducing kernel element method
Physical Description:
Book
Language:
English
Creator:
Collier, Nathaniel O
Publisher:
University of South Florida
Place of Publication:
Tampa, Fla.
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Subjects

Subjects / Keywords:
Meshing
Regularity
RKEM
Interpolation
Surface representation
Dissertations, Academic -- Civil Engineering -- Doctoral -- USF   ( lcsh )
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bibliography   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
ABSTRACT: The Reproducing Kernel Element Method (RKEM) is a hybrid between finite elements and meshfree methods that provides shape functions of arbitrary order and continuity yet retains the Kronecker-delta property. To achieve these properties, the underlying mesh must meet certain regularity constraints, unique to RKEM. The aim of this dissertation is to develop a precise definition of these constraints, and a general algorithm for assessing a mesh is developed. This check is a critical step in the use of RKEM in any application. The general checking algorithm is made more specific to apply to two-dimensional triangular meshes with circular supports and to three-dimensional tetrahedral meshes with spherical supports. The checking algorithm features the output of the uncovered regions that are used to develop a mesh-mending technique for fixing offending meshes. The specific check is used in conjunction with standard quality meshing techniques to produce meshes suitable for use with RKEM. The RKEM quasi-uniformity definitions enable the use of RKEM in solving Galerkin weak forms as well as in general interpolation applications, such as the representation of geometries. A procedure for determining a RKEM representation of discrete point sets is presented with results for surfaces in three-dimensions. This capability is important to the analysis of geometries such as patient-specific organs or other biological objects.
Thesis:
Dissertation (Ph.D.)--University of South Florida, 2009.
Bibliography:
Includes bibliographical references.
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Mode of access: World Wide Web.
System Details:
System requirements: World Wide Web browser and PDF reader.
Statement of Responsibility:
by Nathaniel O. Collier.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains 68 pages.
General Note:
Includes vita.

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aleph - 002222723
oclc - 652552771
usfldc doi - E14-SFE0002962
usfldc handle - e14.2962
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Full Text

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(b)RKPMpolynomialeldandkernel (c)RKEMpolynomialeldsandnode-centeredkernelsFigure1.Conceptualdierencesintheformingofbasisfunctions10

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Figure2.Biasedyetadmissiblemeshfreesupports(adaptedfrom[19])meshfreemethodssuchastheReproducingKernelParticleMethod(RKPM),bothasupportwindowandapolynomialeldarecenteredattheparticle.Supportsmustcontainotherparticlestoformapartitionofunity.InRKEM,thepolynomialeldsaredenedonelementsandthesupportwindowiscenteredonthenodes.Thesupportwindowservestosmooththepolynomialfunctionsfromelementtoelement.ThereforeinRKEMsupportsizeiscritical,butnotinthesamesenseasinmeshfree.Thisiswhatrequireseveryelementdomaintobecoveredbyatleastonesupport.WhileFEM,RKEM,andmeshfreemethodsallsharedicultyindeterminingelementand/orsupportsizes,noneofthetechniquesoutlinedhereaddressestheissuesthatareuniquetoRKEM.1.4GeometryRepresentationWhiletheoriginalapplicationofRKEMwasinthesolutionofpartialdierentialequations,theRKEMbasisfunctionscanbeusedforgeneralinterpolation,suchas15

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(b)Toolarge,lossofKronecker-

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2maxj2idijiminj2idij(23)19

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2maxj2idij(25)Theuseofascalefactorisaconvenientwaytosetallthesupportradiiassomefractionoftheirknownmaximumvalue.Furthermore,requiringf<1ensuresthattheKronecker-propertyismaintainedbecausenoothernodewillbeinanothernode'ssupport.Thisdenitionisnotsatisfactoryascanbeseeninthefollowingexample.ConsiderthecaseofasingleelementmeshconsistingofanequilateraltriangleasshowninFig.4.AccordingtoEqs.24and25itisvalidtomakef=0:5.Thiswouldmakei=1 2dij(26)whichwillbeconstantforallnodessinceallsidesareequallengthinthisexample.TheconditioninEq.25issatised;however,asdepictedinFig.4,theconditionthateverypointliewithinthekernelsupportofatleastonenodeisnotsatised.Thus,thecondition,Eq.23orequivalentlyEq.25,isnotsucienttoguaranteecoverage.Pastworkmayhaveavoidedthisissueinoneormoreofthefollowingways.1.TheconditionexpressedinEq.23issucientforone-dimensionalwork.20

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Figure 6.Thesupportofelement II coverstheotherwisedecientelement I 2.4SupportShapes Whilethisdissertationfocusesoncircularsupports,Def.2.1.4isindependentof supportshape.Anysupportshapemaybeusedprovidedthatawindowfunction maybeconstructedwiththerequiredproperties[25]. Rectangularwindowfunctionsareafrequentchoice,primarilybecausetheycan beobtainedbyatensorproductofaone-dimensionalwindowfunction.Inmeshfree methodstheyarealsoadvantageousbecausetheintegrationdomaincanbealigned withthebackgroundintegrationcells.Here,thecoverageproblemisnotaidedbythe rectangularshapeofthesupport.Forexample,considerthetriangularmeshshown inFig.7andthenodelabeled A.Determiningthemaximumpossiblesupportisnot auniqueprocess.Ifonerstdeterminesthemaximumhorizontaldimensionandthen thevertical,therectangularsupportlabeled I isobtained.Ifthisisreversed,then thesupportlabeled II isobtained.Suchambiguitiescanleadtoundesirablebiasin theresultingbasisfunctions. 23

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Figure8.EdgeandelementsdonothelpndclosestnodesisthatthecomplexityreducestoO(NN),whereNisthenumberofnodeswiththeKronecker-property.2.5.3CoverageThesecondpartofthegeneralalgorithmtestsforcoverage.Inthiscase,sincethemeshalreadyconsistsofdisjointelements,thecoveragetestcanbeperformedonanelement-by-elementbasis.Thismaybebenecialforatleasttworeasons.First,thisisaso-calledembarrassinglyparallelcomputationandcantriviallybenetfrommultipleprocessors.Secondly,asexploitedinx2.5.4,thecomputationalgeometrymaybeeasier.Theelementcoveragetestwillgenerallyinvolveanynumberofnodes,andonesthatarenotnecessarilylocatedattheelementverticesoronadjacentelements,asshowninFig.8.Howeversuchcasesarenotgoodelementsfromthe25

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(b)Trimthesegmentstotheirintersections,splitwhereneeded (c)Createnewcurvesandsurfaceswhereneeded (d)Asupportcanintersectthesurfacewith-outintersectinganyboundingsegmentsFigure12.Thesphericalsupportintersectionalgorithmatdierentstageswithorig-inaltetrahedronshownwithalightlineforreference2.6ExamplesFigure13depictsanRKEMmeshrepresentingaslicefromaCTscanofatoothfromabullshark.ThesequenceshowninFig.14showstheuncoveredregionsofthemeshasdeterminedbythealgorithmgiveninx2.5.4forthreedierentscalefactors.Thepicturesshowndisplaythenodesofthemeshaspoints,andthesolidlinesarethesegmentsboundingtheregionsofthemeshthatremainuncovered.InFig.14(a),thescalefactorwassetto0.525withalargeportionofthemeshuncovered.InFig.14(b)-(c),theincreasingscalefactorleadstoincreasedmeshcoverage.Finally,ascalefactor,f=0:92yieldsafullycoveredmesh(notshown).Thus,anyscalefactor0:92f<1willfullycoverthemeshwhileretainingtheKronecker-property31

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(b)f=0:625 (c)f=0:725Figure14.Uncoveredregionsofbullsharktoothmeshfordierentscalefactors,fConsiderthefollowingtwo-dimensionalexamplemeshdepictedinFig.17(a).Thismeshwasgeneratedbyevenlydistributingpointsalongtheboundaryandthenran-domlyinsertinginteriorpoints.Thesepointswerethentriangulatedwithoutanyqualityconstraints.Thismeshcannotbecoveredforanyvalueoff<1;theun-coveredregionsareshowninFig.17(b).Themeshmendingprocedureisappliedtothismesh.Afteronerenement,themeshshowninFig.18(a)isgeneratedandtheresultingcoverageshowninFig.18(b).OnemoreapplicationofmeshmendingleadstothemeshshowninFig.19,whichnowmeetstheRKEMquasi-uniformitycondition.Whilethisprocedureisindependentofdimension,two-dimensionalproblemsweresuccessfullymendedwhilethree-dimensionalproblemswerenot.Whentheuncoveredregionsofthethree-dimensionalmesheswereusedasabasisfornodeinsertion,the33

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(b)Figure17.Randommesh(a)andaccompanyinguncoveredareas(b)36

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(b)Figure18.Firstrenementmesh(a)andaccompanyinguncoveredareas(b)37

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(b)RKEMGeometryRepresentationFigure20.DierencesintheuseofinterpolantstorepresentacirclewithasingletriangularelementAsbrieydiscussedinx1.2.3,RKEMinterpolationcanbeenrichedbyaddingnodalweightsandaccompanyingelementpolynomialelds.WhilethiscanalsobedoneinFEM,itisinRKEMthatthenodecenteredkernelsenforcecompatibilitymakingthecreationofsucheldstrivial.40

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Figure21.TheRKEMgeometryrepresentationprocess41

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(b)MeshpointfallsothemeshedgeFigure23.CasesweredataareeliminatedwhendeterminingmeshpointsAbettermethod,perhaps,istotallgeometrypointsparametricallybetweenthetwoedgenodes.Whilethismethodcausesnoinformationtobelost,itisnotnecessarilyoptimal.Bothlteringcasesdiscussedintheprecedingtwoparagraphsaddresstheremovalofinformationwhenthegeometrypointsrepresentahighlyvaryingcurvaturealongacertainedge.Includingallsuchpointscouldcausenegativeeectsinthenalrepresentationsuchasaninvertedmapping.Thiseectisnotyetclearandrequiresmorestudytofullyunderstand.Inthree-dimensions,projectionmethodsarefarlessstraight-forward.Fortwo-dimensionalproblems,anaddedeciencyistheknowledgethattheorderedlistofgeometrypointsmustliebetweentwonodesofthecoarsenedRKEMmeshboundaryedge.Thispre-knowledgeisusefulinndingprojections.Forthree-dimensionaldata,thissameeciencyisnotpresentandcannotbeexploited.Whilethedataareadimensionhigher,theoverallgoalisthesameinthree-dimensionsasfortwo-dimensions:ndameshpointforeachgeometrypoint.Threeseparatemethodsofndingthesemeshpointsareexploredanddetailedhere:1.AverageNormalProjectionThismethodwasusedin[34]todeterminethemeshpointsforthespherepublishedinthatwork.Theideawastoaveragetheinwardfacingnormalsofallfaceswhichcontainaparticulargeometrypoint.Themesh49

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Figure24.AveragenormalprojectionWhilethismethodisconvenientisthesensethatonlyinformationlocaltothegeometricpointisusedtocalculateaprojectiondirection,theresultingmeshpointsarenotlikelytogeneratewellbehavedgeometries.Thisisbecauseneighboringmeshpointscouldmaptogeometrypointswhicharefarawayandondierentsides,invertingthemapping.Eventhoughthiseectisundesired,itcouldbeusedasanindicatorthatthemeshneedsrenedtoadequatelyrepresentthegeometry.ThisundesirableeectcanbeseeninFig.25inatwo-dimensionalanalog.Aseriesofgeometrypointsareshownalongwiththeirneighboringfacesasheavylines.Theaveragenormalisapproximatelydrawnandshowswherethecorrespondingmeshpointswouldappear.NotethatthenormalprojectionofthegeometrypointlabeledAisnotevenontheedgeclosesttothepoint.EspeciallyundesirableistheeectoftheseriesofpointsB,C,andD.Notethattheprojectionsofthegeometrypointstothemeshpointsactuallycross.50

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Figure25.ProblemwiththeaveragenormalprojectionmethodForthesphere,thismethodworkedperfectlybecausethecurvaturechangesuniformlyalongthesolid.Thismethodisunsuitableforanygeneralgeometry.2.MeshCenterProjectionAsthenamesuggestsanothermethodofdeterminingmeshpointsistodetermineameshcenterandndtheintersectionoftheraybeginningatageometrypointandinthedirectionofthiscenterwithameshface.Thismethodbranchesintotwosub-methodsatthispoint.Theintersectionmaybefoundwiththenearestfacealongtheraypointingtothemeshcenterinwhichcase,eachgeometrypointwillhaveameshpointandnodataisltered.Thismethodisusedtogeneratetheresultsshowninx3.3.Thesecondsub-methodusesalter.Whilethefaceonwhichthegeometrypointsmustndmeshpointsisnotknownasinthetwo-dimensionalcase,asimplicationmaymemadewhereageometrypointmustndameshpointatitsnearestface.Thenearestfacewillbethefacewhoseperpendicularprojectiontothegeometrypointisbothonthefaceandtheminimumforallmeshfaces.51

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(b)RepresentationFigure27.Sphererepresentationwith8faces (b)RepresentationFigure28.Sphererepresentationwith42faces54

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(b)RepresentationFigure29.Sphererepresentationwith170facesTable1.ConvergenceofsphererepresentationBoundaryFaces Elements MaxError 4 1 1.15 8 8 0.59 42 51 0.22 170 358 0.0655

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(b)OriginalToothTipGeometryFigure30.Tigersharktoothanditstip(usedwithpermissionfrom[37])Therepresentationsshown(Figs.31and32)areformeshes,handpickedandtestedtobeRKEMquasi-uniform.Notethatwhilethemeshiscoarse,therepresen-tationisfaithfultotheoriginalshapeofthetooth.Subsequentrenementswerenotpossibleduringthedurationofthisstudy.Themeshmendingtechniquedetailedin56

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(b)RepresentationFigure31.Tiprepresentationwith3elements57

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(b)RepresentationFigure32.Tiprepresentationwith31elements58

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Figure33.DerivativesatboundaryelementsarenotwelldenedThisis,infact,notacontradictionoftheRKEMinterpolantproperties.Considerthetwo-dimensionalanalogshowninFig.33.HerethegeometryisinterpolatedandshownasaheavyblacklinewithaderivativediscontinuityatapointlabeledA.Theunderlyingmeshisshownasashadedtriangle.Theissueisrelatedtothefactthatdeterminingtheunknownparametersatthenodesinvolvestakingdatathatarefundamentallysmoothandprojectingthesedatatomeshedges/facesthatarenotsmooth.Theinterpolatedrepresentation,expressedasafunctionf,wouldbecontinuousatAif@f @s1=@f @s2(33)wheres1ands2arevectorsasshowninFig.33.Itisillogicaltoexpectderivativeswithrespecttodierentvariablestobeequalatgenerallocations.Forexample,letf(x;y)=1x2y.Inthiscase@f @x=2xand@f @y=1.Althoughatx=0:559

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(b)RepresentationFigure34.Tiprepresentationwith3elements,tiltedviewThesameeectisseen,albeitlesspronounced,inthetwo-dimensionalresultspreviouslypublished[34].Considerthecirclerepresentationwith4elementsasseeninFig.35.ThefullcircleisshowninFig.35(a),withthetopofthecircleshown60

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(a) Fullcircle (b) Zoomedintoshowtopportionofcircle Figure35.Thefourelementcirclerepresentationhasdiscontinuities 62


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The quasi-uniformity condition and three-dimensional geometry representation as it applies to the reproducing kernel element method
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b University of South Florida,
2009.
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Dissertation (Ph.D.)--University of South Florida, 2009.
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ABSTRACT: The Reproducing Kernel Element Method (RKEM) is a hybrid between finite elements and meshfree methods that provides shape functions of arbitrary order and continuity yet retains the Kronecker-delta property. To achieve these properties, the underlying mesh must meet certain regularity constraints, unique to RKEM. The aim of this dissertation is to develop a precise definition of these constraints, and a general algorithm for assessing a mesh is developed. This check is a critical step in the use of RKEM in any application. The general checking algorithm is made more specific to apply to two-dimensional triangular meshes with circular supports and to three-dimensional tetrahedral meshes with spherical supports. The checking algorithm features the output of the uncovered regions that are used to develop a mesh-mending technique for fixing offending meshes. The specific check is used in conjunction with standard quality meshing techniques to produce meshes suitable for use with RKEM. The RKEM quasi-uniformity definitions enable the use of RKEM in solving Galerkin weak forms as well as in general interpolation applications, such as the representation of geometries. A procedure for determining a RKEM representation of discrete point sets is presented with results for surfaces in three-dimensions. This capability is important to the analysis of geometries such as patient-specific organs or other biological objects.
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Surface representation
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