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

Citation
The quasi-uniformity condition and three-dimensional geometry representation as it applies to the reproducing kernel element method

Material Information

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

Subjects

Subjects / Keywords:
Meshing
Regularity
RKEM
Interpolation
Surface representation
Dissertations, Academic -- Civil Engineering -- Doctoral -- USF ( lcsh )
Genre:
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.
System Details:
Mode of access: World Wide Web.
System Details:
System requirements: World Wide Web browser and PDF reader.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains 68 pages.
General Note:
Includes vita.
Statement of Responsibility:
by Nathaniel O. Collier.

Record Information

Source Institution:
University of South Florida Library
Holding Location:
University of South Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
002222723 ( ALEPH )
652552771 ( OCLC )
E14-SFE0002962 ( USFLDC DOI )
e14.2962 ( USFLDC Handle )

Postcard Information

Format:
Book

Downloads

This item has the following downloads:


Full Text
xml version 1.0 encoding UTF-8 standalone no
record xmlns http:www.loc.govMARC21slim xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.loc.govstandardsmarcxmlschemaMARC21slim.xsd
leader nam 2200385Ka 4500
controlfield tag 001 002222723
005 20101007115953.0
007 cr cnu|||uuuuu
008 100804s2009 flua ob 000 0 eng d
datafield ind1 8 ind2 024
subfield code a E14-SFE0002962
035
(OCoLC)652552771
040
FHM
c FHM
049
FHMM
090
TA145 (Online)
1 100
Collier, Nathaniel O.
4 245
The quasi-uniformity condition and three-dimensional geometry representation as it applies to the reproducing kernel element method
h [electronic resource] /
by Nathaniel O. Collier.
260
[Tampa, Fla.] :
b University of South Florida,
2009.
500
Title from PDF of title page.
Document formatted into pages; contains 68 pages.
Includes vita.
502
Dissertation (Ph.D.)--University of South Florida, 2009.
504
Includes bibliographical references.
516
Text (Electronic dissertation) in PDF format.
3 520
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.
538
Mode of access: World Wide Web.
System requirements: World Wide Web browser and PDF reader.
590
Advisor: Daniel C. Simkins, Ph.D.
653
Meshing
Regularity
RKEM
Interpolation
Surface representation
0 690
Dissertations, Academic
z USF
x Civil Engineering
Doctoral.
773
t USF Electronic Theses and Dissertations.
856
u http://digital.lib.usf.edu/?e14.2962



PAGE 19

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

PAGE 24

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

PAGE 27

(b)Toolarge,lossofKronecker-

PAGE 28

2maxj2idijiminj2idij(23)19

PAGE 29

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

PAGE 32

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

PAGE 34

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

PAGE 40

(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

PAGE 42

(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

PAGE 45

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

PAGE 46

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

PAGE 49

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

PAGE 50

Figure21.TheRKEMgeometryrepresentationprocess41

PAGE 58

(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

PAGE 59

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

PAGE 60

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

PAGE 63

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

PAGE 64

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

PAGE 65

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

PAGE 66

(b)RepresentationFigure31.Tiprepresentationwith3elements57

PAGE 67

(b)RepresentationFigure32.Tiprepresentationwith31elements58

PAGE 68

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

PAGE 69

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

PAGE 71

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


printinsert_linkshareget_appmore_horiz

Download Options

close
Choose Size
Choose file type
Cite this item close

APA

Cras ut cursus ante, a fringilla nunc. Mauris lorem nunc, cursus sit amet enim ac, vehicula vestibulum mi. Mauris viverra nisl vel enim faucibus porta. Praesent sit amet ornare diam, non finibus nulla.

MLA

Cras efficitur magna et sapien varius, luctus ullamcorper dolor convallis. Orci varius natoque penatibus et magnis dis parturient montes, nascetur ridiculus mus. Fusce sit amet justo ut erat laoreet congue sed a ante.

CHICAGO

Phasellus ornare in augue eu imperdiet. Donec malesuada sapien ante, at vehicula orci tempor molestie. Proin vitae urna elit. Pellentesque vitae nisi et diam euismod malesuada aliquet non erat.

WIKIPEDIA

Nunc fringilla dolor ut dictum placerat. Proin ac neque rutrum, consectetur ligula id, laoreet ligula. Nulla lorem massa, consectetur vitae consequat in, lobortis at dolor. Nunc sed leo odio.