The Pseudo-Rigid-Body Model for Fast, Accurate, Non-Linear Elasticity

87 Pages·2016·2.68 MB·English

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Theses and Dissertations
2013-11-22
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Anthony R. Hall
Brigham Young University - Provo
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Hall, Anthony R., "The Pseudo-Rigid-Body Model for Fast, Accurate, Non-Linear Elasticity" (2013). Theses
and Dissertations. 3869.
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ThePseudo-Rigid-BodyModelforFast,Accurate,Non-LinearElasticity
AnthonyR.Hall
Athesissubmittedtothefacultyof
BrighamYoungUniversity
inpartialfulfillmentoftherequirementsforthedegreeof
MasterofScience
MichaelJones,Chair
ParrisEgbert
DanielZappala
DepartmentofComputerScience
BrighamYoungUniversity
November2013
Copyright(cid:13)c 2013AnthonyR.Hall
AllRightsReserved
ABSTRACT
ThePseudo-Rigid-BodyModelforFast,Accurate,Non-LinearElasticity
AnthonyR.Hall
DepartmentofComputerScience,BYU
MasterofScience
Weintroduceto computergraphicsthe Pseudo-Rigid-BodyMechanism (PRBM)andthe
chainalgorithmfrommechanicalengineering,withaunifiedtutorialfromdisparatesourcematerials.
The PRBM has been used successfully to simplify the simulation of non-linearly elastic beams,
usingdeflectionsofananalogousspringandrigid-bodylinkage. Itofferscomputationalefficiency
aswellasanautomaticparameterizationintermsofphysicallymeasurable,intuitiveinputswhich
fit naturally into existing animation work flows for character articulation. The chain algorithm
is a technique for simulating the deflection of complicated elastic bodies in terms of straight
elasticelements,whichhasrecentlybeenextendedtoincorporatePRBMbeam-elementsinthree
dimensions. We present a new, mathematically equivalent optimization of the 3D PRBM chain
algorithm, fromitsformer asymptoticcomplexityof O(n2)inthe numberof elementsn, toO(n).
We also extend an existing PRBM for combined moment-force loads to 3D, where the existing
3D PRBM chain algorithm was limited to 3D PRBM elements for a moment-only load. This
optimizationandextensionarevalidatedbyduplicatingpriorexperimentalresults,butsubstituting
the new optimization and combined-load elements. Finally, a loose road-map is provided with
severalkeyconsiderationsforfutureextensionofthetechniquestodynamicsimulations.
Keywords: simulation,rigid-body,mass-spring,non-linearelasticity
ACKNOWLEDGMENTS
DeanR.WheelerfirstacquaintedmewiththeworkofLarryL.Howell,andpointedmeto
thePRBMasaneffectivemethodformodelingcompliantmechanisms.
MurphyJ.C.Randlewrotepipelinescriptsforinterfacingourresultswithexistingpackages
foranimationandrendering. Hesetupthe lighting,texturing,cameras,rendering,andotherscene
elementsnecessaryfortherenderedimagesfoundinthefigures.
Larry L. Howell generously offered his time and his expertise with PRBM techniques,
and offered early encouragement about the relevance and interest of our research goals. He
independentlyreviewedmythesisproposalforgeneralsoundness,andhisfeedbackregardingthe
PRBMbackgroundmaterialandourproposedextensionshelpedencourageustoproceed.
H.TracyHallre-formulatedmyearliestlong-hand,verboseformulasfortheO(n)-optimized
chainalgorithmintothecompact,elegantrepresentationofsection4.1.2aslinearcombinationsof
basis matrices. The results of chapter 5 came from implementing his formulation; my own later
refinementoftheearliermethodappearsinappendixA.Asamathematicalandtechnicalconsultant,
hehelpedpushtheworkforwardthroughafewkeywhiteboardsessions.
I would like to thank Parris Egbert and Daniel Zappala for willingly serving and giving
theirtime asmembers ofmythesis committee. Dr. Egbert tookadditional timetoreview andgive
importantfeedbackforboththeThesisProposaldocumentandthisThesis.
MyadviserandcommitteeChair,MichaelJones,providedinvaluablesupportthroughout
myentire program. He offeredme thefreedom andtrust totackle aproblem andset oftechniques
inwhichneitherofushadpriorexpertise. Hewasintellectuallygenerous,alwayswillingtoinvest
the time to understand and help me clarify the new concepts and techniques I uncovered in the
literature. Histechnicalinsightswerekeytoanumberofimportantbreakthroughs. Heneverkept
meintellectuallysubordinate,andwelcomedchallengingquestionsandpushback;myconfidence
asa researcherwasable toflourishas aresult. Hishelp withplanning andstrategy was critical in
definingaclear,manageable,andimplementablescopefortheproject;withouthisguidance,the
projectwouldhavebeenmuchmoredifficulttomanageandultimatelyfinish.
TableofContents
ListofFigures viii
ListofTables ix
ListofAlgorithms x
1 Introduction 1
1.1 Immediatecontributionsofthiswork . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Long-termcontributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 RelatedWork 5
2.1 Generalversusspecializedelasticitymethods . . . . . . . . . . . . . . . . . . . . 5
2.2 Finiteelementreductionandcoarsening . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Specialized1Delasticmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Background 10
3.1 Notationandterminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Historyandoverview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2.1 OverviewofthePRBM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2.2 Overviewofthechainalgorithm . . . . . . . . . . . . . . . . . . . . . . . 12
3.2.3 HistoryoftheChainAlgorithmwithPRBMelements . . . . . . . . . . . . 13
3.3 PRBMrecipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3.1 TheplanarPRBMforamomentload . . . . . . . . . . . . . . . . . . . . 15
3.3.2 Theplanar3RPRBMformomentandforceloads . . . . . . . . . . . . . 17
v
3.3.3 The3D1RPRBMforamomentload . . . . . . . . . . . . . . . . . . . . 19
3.4 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4.1 Thechaincalculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4.2 updateElementforthe3D1RPRBM . . . . . . . . . . . . . . . . . . . . . 27
3.4.3 transformElementforthe3D1RPRBM . . . . . . . . . . . . . . . . . . . 30
4 Methods 32
4.1 Linear-timeoptimizationofthechainalgorithm . . . . . . . . . . . . . . . . . . . 34
4.1.1 Thesimplebit: Fi andµi . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Σ Σ
4.1.2 Thenot-so-simplebit: ρi . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Σ
4.1.3 Puttingitalltogether . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1.4 Extensiontobranchinggraphs . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1.5 Acriticalserializationpoint . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2 The3D3RPRBMforforceandmomentloads . . . . . . . . . . . . . . . . . . . 43
4.2.1 updateElementforthe3D3Rrecipe . . . . . . . . . . . . . . . . . . . . . 46
4.2.2 transformElementforthe3D3Rrecipe . . . . . . . . . . . . . . . . . . . 48
5 Results 49
5.1 Re-implementing[Chaseetal.2011]asabaseline . . . . . . . . . . . . . . . . . . 50
5.2 Ourexperimentsetup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.2.1 Definitionoferrormetric . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.3 Experimentresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3.1 Runtimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3.2 ErroroftheO(n)optimization . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3.3 Errorof3Ralgorithmsversus1Ralgorithms . . . . . . . . . . . . . . . . 56
6 ConclusionandFuturework 59
6.1 Extendingthestaticalgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.1.1 Alternative3D3Rconfigurations . . . . . . . . . . . . . . . . . . . . . . 60
vi
6.1.2 Staticconstraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.2 ProposeddynamicPRBMmethods . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.2.1 Reducedcoordinateapproaches . . . . . . . . . . . . . . . . . . . . . . . 61
6.2.2 Numericalintegrationintwokeycases . . . . . . . . . . . . . . . . . . . 62
6.2.3 Approachestoelementintegration(case1) . . . . . . . . . . . . . . . . . 62
6.2.4 Approachestochainintegration(case2) . . . . . . . . . . . . . . . . . . . 63
6.3 Directable,dynamictopologyandmaterialchange . . . . . . . . . . . . . . . . . 65
6.3.1 Automaticstressandfracturedetection . . . . . . . . . . . . . . . . . . . 65
6.3.2 Efficient,onlinematerialchange . . . . . . . . . . . . . . . . . . . . . . . 66
A AlternativeO(n)formulation 68
B Thecrossproductmatrix 72
References 73
vii
ListofFigures
1.1 Volumedeformationmodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3.1 AchainwithPRBMelasticelements . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Thepuremoment-loaded1RPRBM . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 Su’s3RPRBM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4 A3-axisrevolutespringjointfromChase’s3DPRBM . . . . . . . . . . . . . . . 20
3.5 Illustrationofchainalgorithmtotalmomentcomputation . . . . . . . . . . . . . . 26
4.1 Extensionofthe3RPRBMto3D . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.1 Elementtippairingsfortheerrormetricε . . . . . . . . . . . . . . . . . . . . . 52
P
5.2 Graphofruntimesforthefourcasesofthechainalgorithm . . . . . . . . . . . . . 54
5.3 Graphoftheerrorε fromthefourcasesofthechainalgorithm . . . . . . . . . . 55
P
5.4 Rendersofdeflectedchainswith20,25,40,and200elements . . . . . . . . . . . 57
viii
ListofTables
5.1 Runtimesforallfourcasesofthechainalgorithm . . . . . . . . . . . . . . . . . . 54
5.2 Errorε forallfourcasesofthechainalgorithm . . . . . . . . . . . . . . . . . . 55
P
ix