ADAPTIVE COHESIVEVOLUMETRIC FINITE ELEMENT METHODFORDYNAMIC FRACTURESIMULATIONS
BY
MARIUSZ ZACZEK
B.S.,Universityof Illinois at Urbana-Champaign,1999
THESIS
Submittedin partialfulfillment of therequirementsfor thedegreeof Masterof Sciencein AeronauticalandAstronauticalEngineering
in theGraduateCollegeof theUniversityof Illinois atUrbana-Champaign,2001
Urbana,Illinois
c
Copyright by MariuszZaczek,2001
To my mother
iii
ACKNOWLEDGMENTS
First and foremost,I would like to thank the Centerfor Simulationof AdvancedRockets
(CSAR) who hassponsoredmy researchover the pasttwo years. I would also like to express
my appreciationandgratitudeto my advisor, Prof. PhilippeGeubelle.Withouthisadvice,support
andunderstandingI wouldhaveneverbeenableto finishthisthesis.I amespeciallygratefulto him
for spendingcountlesshoursin helpingwith this researchandin takingthetime to answermany
of my questions.Thankyou alsoto DhirendraKubair, SpandanMaiti, JasonKamphausandthe
variousothermembersof theStructuresandSolid MechanicsGroupwho have beengreatfriends
andhavehelpedmetremendouslyovermy time here.In addition,I wouldalsolike to thankProf.
RicardoUribe who hasallowedto expandmy horizonsby working on variousroboticprojectsas
partof theAdvancedDigital SystemsLaboratory.
Lastly, I would like to thankmy motherwho hasalwaysbelievedin meandencouragedmeto
bethebestthatI canandnever forget to smile. Without herI would have never hadtheenergy to
work sohard.
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TABLE OF CONTENTS
CHAPTER PAGE
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 METHODOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1 Review of theCohesive/VolumetricFiniteElementScheme. . . . . . . . . . . . . 6
2.1.1 Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2 Finite ElementImplementation . . . . . . . . . . . . . . . . . . . . . . . 112.1.3 Stability andMeshSize . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 NodalTimeStepSubcycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 DynamicCohesiveNode/ElementInsertion . . . . . . . . . . . . . . . . . . . . . 19
2.3.1 GeometryandDatabaseManagement. . . . . . . . . . . . . . . . . . . . 192.3.1.1 1-D Insertion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.1.2 2-D Insertion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.2 CohesiveElementStability andSystemEquilibrium . . . . . . . . . . . . 262.3.2.1 CohesiveDamping . . . . . . . . . . . . . . . . . . . . . . . . 282.3.2.2 CohesiveElementPre-Stretching. . . . . . . . . . . . . . . . . 30
2.3.3 InsertionRegion Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3.3.1 BoundingBox Approach . . . . . . . . . . . . . . . . . . . . . 332.3.3.2 Stress-basedSelectionApproach . . . . . . . . . . . . . . . . . 35
2.4 ParallelImplementationusingCharm++ . . . . . . . . . . . . . . . . . . . . . . 372.4.1 MeshPartitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.4.2 ComputationalEfficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 402.4.3 Structureof StandardCharm++ FEM Framework . . . . . . . . . . . . . 412.4.4 ParallelStructureof theDynamicInsertionCode . . . . . . . . . . . . . . 44
3 1-D ANALYSIS AND RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.1 ProblemDescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.2 Multi-TimeStepNodalSubcycling Results . . . . . . . . . . . . . . . . . . . . . 493.3 DynamicCohesiveNodeInsertionResults. . . . . . . . . . . . . . . . . . . . . . 54
3.3.1 Blind CohesiveNodeInsertionResults . . . . . . . . . . . . . . . . . . . 553.3.2 Dampingof Blind Insertion . . . . . . . . . . . . . . . . . . . . . . . . . 583.3.3 DynamicInsertionwith Pre-Stretch . . . . . . . . . . . . . . . . . . . . . 60
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3.3.4 CombinedInsertionwith Subcycling . . . . . . . . . . . . . . . . . . . . . 633.4 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4 2-D ANALYSIS AND RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.1 Multi-TimeStepSubcycling Results . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1.1 EqualSubcycledto Non-SubcycledRegionRatioof 1:1 . . . . . . . . . . 694.1.2 UnequalSubcycledto Non-SubcycledRegionRatio . . . . . . . . . . . . 704.1.3 Multi-TimeStepNodalSubcycling Observations . . . . . . . . . . . . . . 73
4.2 DynamicCohesiveElementInsertion . . . . . . . . . . . . . . . . . . . . . . . . 754.2.1 InsertionAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2.1.1 Blind Insertion. . . . . . . . . . . . . . . . . . . . . . . . . . . 784.2.1.2 Dampingof Blind Insertion . . . . . . . . . . . . . . . . . . . . 784.2.1.3 Insertionwith Pre-Stretch. . . . . . . . . . . . . . . . . . . . . 814.2.1.4 InsertionAnalysisObservations. . . . . . . . . . . . . . . . . . 83
4.2.2 InsertionResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.2.2.1 BoundingBox Insertion . . . . . . . . . . . . . . . . . . . . . . 844.2.2.2 StressBasedInsertionin L-Angle Specimen . . . . . . . . . . . 894.2.2.3 Stress-basedInsertionin VerticalInterfaceSpecimen . . . . . . 944.2.2.4 Stress-basedInsertionin AngledInterfaceSpecimen. . . . . . . 964.2.2.5 InsertionInterval Selection . . . . . . . . . . . . . . . . . . . . 1024.2.2.6 DynamicInsertionCombinedwith Subcycling . . . . . . . . . . 105
4.3 ParallelizationUsingCharm++ . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.4 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5 CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . 1125.1 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.2 Recommendationsfor FutureResearch. . . . . . . . . . . . . . . . . . . . . . . . 115
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
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LIST OF FIGURES
Figure Page
1.1 Illustrationof the fractureprocessassociatedwith theTitan IV SRMU graincol-lapseaccident(takenfrom Changetal., (1994)).. . . . . . . . . . . . . . . . . . . 2
2.1 CVFE conceptshowing one 4-nodecohesive elementbetweentwo linear-straintriangularvolumetricelements.The cohesive elementis shown in its deformedconfiguration.In its undeformedconfigurationis hasnothicknessandtheadjacentnodesaresuperposed.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Bilinear cohesive failure law for thepuretensileor modeI (∆t 0, left) andpureshearor modeII (∆n 0, right) cases.An unloadingandreloadingpathis alsoshown in themodeI case.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Coupledcohesive failure modeldescribedby Equation2.4; variationof normal(top)andshear(bottom)cohesivetractionswith respectto normal(∆n) andtangen-tial (∆t) displacementjumps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Timestepdefinedby: (a) elementsizeor (b) elementtype. . . . . . . . . . . . . . 142.5 Subcycling regiondistribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.6 Timestepassignment.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.7 (a)Standard1-D mesh.(b) 1-D meshwith insertedcohesivenode. . . . . . . . . . 202.8 2-D Cohesiveelementrepresentation. . . . . . . . . . . . . . . . . . . . . . . . . 212.9 2-D cohesive elementinsertion: (a) proposededgefor cohesive insertion,(b) in-
sertedcohesiveelement,(c) “criss-crossed”cohesiveelement. . . . . . . . . . . . 212.10 Connectivity updateof nodesandelements. . . . . . . . . . . . . . . . . . . . . . 222.11 Common2-D insertioncases.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.12 Illustrativeexampleof threecohesiveelementinsertionsusingCases#2 and#3 in
Figure2.11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.13 Illustrationof insertionCase#5 in Figure2.11. . . . . . . . . . . . . . . . . . . . 262.14 1-D “blind” insertiontestproblem. . . . . . . . . . . . . . . . . . . . . . . . . . . 272.15 1-D “blind” insertiontest problem: evolution of the displacementjump across
thecohesive element(i.e., betweennodes2 & 4 in Figure2.14) resultingfrom acohesiveelementinsertionat time0, 1000∆t (33 3 s), 2000∆t (66 6 s). . . . . . . . 28
2.16 Schematicrepresentationof adamped1-D cohesiveelement.. . . . . . . . . . . . 28
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2.17 Effect of cohesive damping:evolution of thedisplacementjump acrossthecohe-sive elementfor the simple1-D testproblemshown in Figure2.14andresultingfrom “blind” cohesive elementinsertionwith dampingat time 0, 1000∆t (33 3 s)and2000∆t (66 6 s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.18 1-D cohesiveelementpre-stretchingconcept. . . . . . . . . . . . . . . . . . . . . 302.19 1-D cohesive elementpre-stretchingconcept,with thepre-stretchappliedequally
on thetwo nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.20 2-D separationcontributionsfrom neighboringcohesiveelements. . . . . . . . . . 332.21 1-D test problemseparationoscillationsof nodes2 & 4 (Figure2.14) resulting
from insertionwith pre-stretchingat time0, 1000∆t (33 3 s), 2000∆t (66 6 s). . . . 342.22 Boundingboxmethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.23 Multiple activecohesiveregions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.24 Stress-basedinsertionresultsfor simpleangledcase. . . . . . . . . . . . . . . . . 372.25 SimpleCVFEmesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.26 PartitionedCVFE mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1 Referenceproblemin 1-D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 x-t diagramin 1-D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.3 Analyticalsolutionfor displacementd, velocityv andstressσ in themiddleof the
beamfor the1-D waveproblemdescribedin Figure3.1. . . . . . . . . . . . . . . 483.4 Subcycling testdescribedby Smolinski(1989)by CaseC. . . . . . . . . . . . . . 493.5 Velocityprofileof node#5 with subcycling parameterm 10. . . . . . . . . . . . 503.6 Velocityprofileof node#15with subcycling parameterm 10. . . . . . . . . . . 503.7 Velocityprofileof node#25with subcycling parameterm 10. . . . . . . . . . . 513.8 Subcycling effecton (a)displacementsand(b) velocitiesat node5. . . . . . . . . . 523.9 Testcaseusedto gettiming resultsfor subcycling. . . . . . . . . . . . . . . . . . . 533.10 Nodes12 through20 aremadecohesive. . . . . . . . . . . . . . . . . . . . . . . . 543.11 Velocityprofileof node#15resultingfrom blind insertionat the0th timestep(0 0
s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.12 Velocity profile of node#15resultingfrom blind insertionat the2500th time step
(6 25s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.13 Velocity profile of node#15resultingfrom blind insertionat the5000th time step
(12 5 s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.14 Velocityprofileof node#15resultingfrom blind insertionat the10000th timestep
(25 0 s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.15 Velocity profile of node#15 resultingfrom blind insertionwith dampingat the
5000th timestep(12 5 s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.16 Velocity profile of node#15 resultingfrom blind insertionwith dampingat the
10000th timestep(25 0 s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.17 Velocity profile of node#15 resulting from insertionwith pre-stretchingat the
5000th timestep(12 5 s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
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3.18 Velocity profile of node#15 resulting from insertionwith pre-stretchingat the10000th timestep(25 0 s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.19 Cohesive separationfor node#15 resultingfrom blind insertionat 10000th timestep(25 0 s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.20 Cohesive separationfor node#15 resultingfrom insertionwith pre-stretchingat10000th timestep(25 0 s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.21 Testcasefor dynamiccohesivenodeinsertionwith pre-stretching. . . . . . . . . . 633.22 Testcasefor dynamicinsertionwith subcycling. . . . . . . . . . . . . . . . . . . . 643.23 Velocityprofileof node400of dynamiccohesivenodeinsertionattimestep150000
(3750s) with nodalsubcycling usingm 10. . . . . . . . . . . . . . . . . . . . . 65
4.1 Cohesiveelementdistribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.2 Nodaldisplacementsof a randomnodeaheadof thenotchfor a problemwith an
equalregion ratio (1 : 1) with subcycling parametersof m 1 4 10 16and20. . . 694.3 Nodaldisplacementsof a randomnodefor the2 : 1 region ratio with subcycling
parametersof m 1 4 10 16and20. . . . . . . . . . . . . . . . . . . . . . . . . 724.4 Nodaldisplacementsof a randomnodefor the4 5 : 1 region ratio with subcycling
parametersof m 1 4 10and16. . . . . . . . . . . . . . . . . . . . . . . . . . . 724.5 Percenttimesavingsvs region ratio for varioussubcycling parameters. . . . . . . 744.6 Simple2-D mesheswith threecohesive elementsinsertedalong(a) ”horizontal”
and(b) ”mixed” interfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.7 Normalizedaveragestresslevelsfor thevolumetricelementsof themiddlecohe-
sive element. Vertical lines at the 0th (0 0 s), 2500th (1 0 s) and5000th (2 0 s)timesteprepresentdynamicinsertiontimes. . . . . . . . . . . . . . . . . . . . . 77
4.8 Normalizedseparationof thetrackingnodefor ”horizontal” blind insertionat the0th (0 0 s), 2500th (1 0 s) and5000th (2 0 s) timestep. . . . . . . . . . . . . . . . 79
4.9 Normalizedseparationof thetrackingnodefor ”mixed” blind insertionat the0th(0 0 s), 2500th (1 0 s) and5000th (2 0 s) timestep. . . . . . . . . . . . . . . . . . 79
4.10 Normalizedseparationof the trackingnodefor ”horizontal” blind insertionwithdampingat the 0th (0 0 s), 2500th (1 0 s with η 3 8) and5000th (2 0 s withη 4 4) timestep. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.11 Normalizedseparationof thetrackingnodefor ”mixed”blind insertionwith damp-ing at the0th (0 0 s), 2500th (1 0 s with η 2 4) and5000th (2 0 s with η 4 3)timestep. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.12 Normalizedseparationof the trackingnodefor “horizontal” insertionwith withpre-stretchingat the0th (0 0 s), 2500th (1 0 s) and5000th (2 0 s) timestep. . . . . 82
4.13 Normalizedseparationof the trackingnodefor “mixed” insertionwith with pre-stretchingat the0th (0 0 s), 2500th (1 0 s) and5000th (2 0 s) timestep. . . . . . . 82
4.14 Effect of blind insertionvs pre-stretchingon theamplitudeof thetractionoscilla-tionsfor increasingstressinsertionlevels,for the“horizontal” and“mixed” cases. 83
4.15 Schematicof a modeI crackproblem. . . . . . . . . . . . . . . . . . . . . . . . . 85
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4.16 Numberof cohesive elementspresentin the domainover time for boundingboxinsertion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.17 ModeI casecracktip distanceversustime. . . . . . . . . . . . . . . . . . . . . . 874.18 Mode I referencecasewith cohesive elementspresentfrom the beginning of the
simulation(10x exaggeration). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.19 ModeI boundingbox solution(10x exaggeration)(edgekey: thin = normaledge,
dark= cohesiveelement,dashed= failing cohesiveelement,bold= failedcohesiveelement). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.20 Schematicrepresentationof L-angletestspecimenwith boundaryconditions. . . . 904.21 Numberof cohesive elementspresentin the domainover time for variousstress
insertionlevels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.22 L-anglecasecracktip distanceversustime, for variousstressinsertionlevels. . . . 914.23 L-anglereferencecasewith cohesive elementspresentfrom thebeginningof the
simulation(10x exaggeration). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.24 L-anglecasewith stressbasedcohesive elementinsertionfor a 15% stresslevel
(10xexaggeration)(edgekey: thin = normaledge,dark= cohesiveelement,dashed= failing cohesiveelement,bold= failedcohesiveelement). . . . . . . . . . . . . 92
4.25 L-anglecasewith stressbasedcohesive elementinsertionfor a 30% stresslevel(10xexaggeration). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.26 L-anglecasewith stressbasedcohesive elementinsertionfor a 45% stresslevel(10xexaggeration)(edgekey: thin = normaledge,dark= cohesiveelement,dashed= failing cohesiveelement,bold= failedcohesiveelement). . . . . . . . . . . . . 93
4.27 Schematicrepresentationof interfacetestspecimenwith boundaryconditions. . . 944.28 Deformationafter5000timestepsfor stressinsertionof 45%(10xexaggeration).. 954.29 Deformationafter17500timestepsfor stressinsertionof 45%(10xexaggeration). 954.30 Deformationafter25500timestepsfor stressinsertionof 45%(10xexaggeration)
(edgekey: thin = normaledge,dark= cohesiveelement,dashed= failing cohesiveelement,bold= failedcohesiveelement). . . . . . . . . . . . . . . . . . . . . . . 95
4.31 Schematicrepresentationof interfacetestspecimenwith boundaryconditions. . . 964.32 Cracklengthhistoryfor aweak60 degreeinterface. . . . . . . . . . . . . . . . . 984.33 Crackspeedhistoryfor aweak60 degreeinterface. . . . . . . . . . . . . . . . . . 984.34 ModeI cracktrappedalongtheweakLoctite-384interfacefor a45%stress-based
insertion(no exaggeration)(edgekey: thin = normaledge,dark = cohesive ele-ment,dashed= failing cohesiveelement,bold= failedcohesiveelement). . . . . . 99
4.35 ModeI cracktrappedalongthestrongWeldon-100interfacefor a45%stressbasedinsertion(no exaggeration)(edgekey: thin = normaledge,dark = cohesive ele-ment,dashed= failing cohesiveelement,bold= failedcohesiveelement). . . . . . 100
4.36 Close-upof crackregionalongaweakinterface(noexaggeration). . . . . . . . . 1014.37 Close-upof crackregionalongastronginterface(no exaggeration). . . . . . . . . 1014.38 L-anglereferencecasewith cohesive elementsinsertedevery (a) 100 time steps
(b) 500timestepsat the30%stresslevel (10x exaggeration). . . . . . . . . . . . 103
x
4.39 L-anglereferencecasewith cohesive elementsinsertedevery (a) 1000time steps(b) 10000time stepsat the30%stresslevel (10x exaggeration)(edgekey: thin =normaledge,dark= cohesive element,dashed= failing cohesive element,bold =failedcohesiveelement). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.40 Numberof cohesiveelementspresentover time for insertionintervalsof 100,500,1000,5000and10000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.41 Close-upof thethreeintervalspresentedin Figure4.40. . . . . . . . . . . . . . . 1044.42 Schematicof a modeI crackproblemusingnodalsubcycling anddynamicstress
insertionof 45%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.43 Region ratioover time for thesubcycling solutionswith m 6 10and14. . . . . . 1074.44 Referencesolutionwith cohesive elementspresenteverywherein the domainat
thebeginningof thesimulation.No subcycling is used(10xexaggeration). . . . . 1074.45 (a)Solutionhavingonlydynamicinsertionat45%of thelocalstresswith nosubcy-
cling. (b) Combineddynamicinsertionwith subcycling, m 6 (10xexaggeration).1084.46 (a) Combineddynamicinsertionwith subcycling, m 10 (b) Combineddynamic
insertionwith subcycling, m 14 (10x exaggeration)(edgekey: thin = normaledge,dark = cohesive element,dashed= failing cohesive element,bold = failedcohesiveelement). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.47 Speedupresultsfor L-anglecaseusing1, 2, 4, and6 processors. . . . . . . . . . . 110
5.1 Adaptivecrackpropagation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.2 11thelementbrokeninto 7 pieces. . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.3 Velocityprofile for new node#26afterelementbreakupat timestep10000. . . . . 117
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CHAPTER 1
INTR ODUCTION
Theresearchpresentedin this thesisis sponsoredby ASCI/ASAPCenterfor theSimulationof
AdvancedRockets(CSAR)whosemainobjective is thedetailed,integrated,whole-systemsimu-
lation of solid propellantrocketsunderboth normalandabnormaloperatingconditionsHeathet
al., (2000).Thedevelopmentof numericaltoolsneededto simulatein anadaptive fashionaccident
scenariosinvolving the propagationof one or more cracksin the solid propellant(or grain) or
alongthegrain/caseinterfaceconstitutestheprimaryobjective of the researchwork summarized
hereafter.
Fractureeventstakingplacein thegrainduringtheflight of asolidpropellantrocketoftenhave
detrimentaleffectson theperformanceof therocket. As thecrackpropagatesin thegrainor along
the grain/caseinterface,it createsadditionalburning surfaces,generatingan excessof hot gas,
which, in turn,maystronglyaffect thepressurehistoryin therocket chamberandsometimeslead
to acatastrophicfailure.A classicalexampleof acatastrophicsolidrocket failurethatinvolvedthe
propagationof acrackalongthegrain/caseinterfaceis thatof theTitan IV graincollapseaccident
that took placeon April 1, 1991(Wilson et al., 1990;Changet al. 1994)TheTitan IV accident
scenariois schematicallyillustratedin Figure1.1. Dueto theaerodynamiceffectsassociatedwith
thegrainshapenearaslotandtheinteractionbetweencoreandcrossflows,aregionof lowerpres-
suredevelopedalongthedownstreamportionof thegrain, leadingto its progressive deformation
into therocket chamberandresultingin a dramaticincreasein theheadendpressure.Associated
with thegraindeformation,a crackis believedto have initiatedfrom theaft segmentstressrelief
1
groove,extendedto theadjacentcasebondandpropagatedalongtheinterfaceat speedsexceeding
60m s (Wilsonetal., 1992).Thepropagationof theinterfacecrackaccentuatedthegrainslumping
process,whicheventuallyled to thechokingof thecoreflow andtheexplosionof therocket.
Figure 1.1 Illustrationof the fractureprocessassociatedwith theTitan IV SRMU graincollapseaccident(takenfrom Changet al., (1994)).
Although progresshasbeenmadeover the pastfour decadesin understandingthe complex
physicalphenomenaassociatedwith this classof fractureevents(Kuo andKooker, 1990;Lu and
Kuo, 1994;Smirnov, 1985),no truly predictive numericaltoolsarecurrentlyavailableto capture
adequatelythefailureprocessandits effecton therocketperformance.Thesimulationof dynamic
fractureeventstakingplacewhile thegrainis burningconstitutesamajorcomputationalchallenge
for variousreasons.Firstly, theconstitutiveandfailureresponsesof thesolid propellantarequite
complex andofteninvolvelargedeformationsandratedependence,whichmustbeaccountedfor in
theconstitutive,failureandkinematicdescriptionsof thecontinuum.Secondly, thegeometryof the
problemchangessubstantiallyduringthefractureeventdueto therapidpropagationof thecrack
andthe deformationandprogressive burning of thegrain. Thirdly, this problemis characterized
by a complex fluid/structureinteractiondue to the aeroelasticdeformationsof the grain and to
the pressurizationof the newly createdfracturesurfacesby the reactinggas. This interactionis
2
particularlyhardto modelin thevicinity of theadvancingcrackfront wherethegeometryof the
correspondingfluid domainis especiallycomplex andnew fluid regionsarecontinuouslyadded
dueto the crackmotion. Finally, the problemis highly transient,as the speedof the crackhas
beenshown to besometimesof theorderof tensof meterspersecond,possiblyresultingin failure
eventslastinga fractionof asecond.
As describedin Geubelleet al., (2001),thekey componentof themulti-physicsfluid/structure
codeto beusedin thesimulationof dynamicfailure in ”li ve” solid propellantis anexplicit Arbi-
trary/LagrangianEulerian(ALE) formof theCohesive/VolumetricFiniteElement(CVFE)scheme,
speciallydevelopedfor the simulationof dynamicfractureeventsin structuraldomainswith re-
gressingboundaries.The CVFE schemerelies on a combinationof conventional(volumetric)
elementsandof interfacial(cohesive)elementsto capturetheconstitutiveandfailureresponsesof
the material,respectively. The numericalmethod,which is describedin detail in Chapter2, has
beenshown to bequitesuccessfulin thesimulationof variousdynamicfractureproblemsinvolving
spontaneouscrackinitiation, propagationandarrest. It wasused,for example,by Camachoand
Ortiz (1996)to simulateimpactdamagein ceramicmaterials,andby Needleman(1997)to model
dynamicfailureeventsin brittle materials.SiegmundandNeedleman(1997)haveusedtheCVFE
schemeto studyrate-dependencein thedynamicfailureof elasto-plasticmaterials.Geubelleand
Baylor (1998)have simulatedimpact-induceddelaminationof composites,and,morerecently, Bi
etal., (2001)haveusedtheCVFEschemeto capturedynamicfiberpush-outin modelcomposites.
However, in its currentimplementationderivedfrom thework of GeubelleandBaylor (1998),
the CVFE schemerelieson the initial ”static” introductionof the cohesive elementsin thefinite
elementmesh.In otherwords,theanalystprovidesat theonsetof thesimulationa setof possible
pathsfor thedynamiccrack(s)thatwould resultfrom thedynamicloadingof thestructure.This
approachis particularlyattractive for its simplicity: oncethecohesive/volumetricmeshis created,
thedynamicfracturesimulationproceedswithout theneedto modify thestructuralmodel. How-
ever, it suffersfrom two importantlimitations:firstly, thepresenceof theinterfacialelementsin the
finite elementmeshgreatlyincreasesthenumberof nodes,and,therefore,thenumberof degrees
of freedom.This increasein the problemsizeoftenhassubstantialimpacton thecomputational
3
costof the simulation. Secondly, andperhapsmoreimportantly, the presenceof a large number
of cohesive elementsmayadverselyaffect theprecisionof thenumericalsolution. As shown by
Baylor (1997),theadditionalcomplianceassociatedwith thecohesiveelementsmayleadto under
predictingthestressfieldsin thediscretizedstructure.And sincethefailureprocessis stress-based,
this erroron thestressvaluemayaffect theprecisionof thefractureprediction.
To addressthesetwo issues,we proposeto developandimplementin this projectanadaptive
CVFE scheme,for which the cohesive elementsarenot introducedinitially in the finite element
meshbut areinserteddynamicallyduring thesimulationitself. This approachwill not only sub-
stantiallyreducethenumberof nodaldegreesof freedom,especiallyduring the loadingphaseof
thedynamicproblemduringwhich little failuretakesplace,but alsowill guaranteeamoreprecise
captureof thedynamicstressfield beforeandduringthefractureevent.
The developmentand implementationof the adaptive CVFE schemepresentsvariouschal-
lengesthatneedto beaddressed.Thesechallengesareconcernedwith 1) themanagementof the
databasecontainingtheevolving finite elementdiscretization,2) thecriterion to beusedto insert
cohesive elementsadaptively, 3) themechanicalperturbationcreatedby thedynamicallyinserted
cohesiveelements,and4) theloadimbalanceinherentlypresentin theparallelimplementationof
theadaptiveCVFEscheme.Theapproachadoptedin thepresentstudyto addressthesechallenges
is summarizedin Chapter2.
Thepresentprojectalsoaddressesthe issueof adaptivity of theCVFE schemeat anotherim-
portantlevel. As shown by Baylor (1997), one importantdifficulty associatedwith the CVFE
schemeis the fact that it often requiresthe useof very small time stepstypically representinga
small fraction (3 to 5%) of the Courantlimiting valuecharacterizingexplicit dynamicschemes.
This limitation greatlyimpactsthecomputationaleffort of suchsimulations,astensor hundreds
of thousandsof time stepsareoftenneeded.Theneedto usevery small time stepsfor theentire
meshseemsespeciallywastefulwhenonly a small numberof cohesive elementsareusedin the
analysis,asit is thecasewith theproposedadaptiveCVFE scheme.A naturalway to addressthis
issueis thenodalexplicit subcycling schemeproposedby Smolinski(1989),which allows for the
useof distinctlydifferenttimestepvaluesin variouspartsof thefinite elementdomain.
4
Theapplicationof thenodalsubcycling schemeto theadaptiveCVFEschemeis alsodescribed
in Chapter2, followed, in Chapter3, by a one-dimensional(1-D) study of the adaptive CVFE
scheme,performedbecauseof its simplicity and its ability to provide useful insight on various
stability issues.Finally, wesummarizein Chapter4 variousimplementationissuesassociatedwith
themorecomplex 2-D caseandpresenttheresultsof various2-D simulationsperformedwith the
adaptiveCVFE code.
5
CHAPTER 2
METHODOLOGY
As describedearlier, thebasicgoalof theresearcheffort summarizedin thepresentdocument
is to developandimplementanadaptive Cohesive/VolumetricFinite Element(CVFE) schemeto
simulateefficiently dynamicfractureproblemsinvolving the spontaneousinitiation, propagation
andarrestof oneor morecracks.Theapproachadoptedin this projectrelieson a combinationof
theadaptiveinsertionof cohesiveelementsin thefinite elementmesh,subcycling,meshrefinement
andparallel implementation.Detailson thesevariouscomponentsarepresentedin this chapter,
togetherwith a summaryof theformulationandimplementationof theCVFEscheme.
2.1 Review of the Cohesive/Volumetric Finite ElementScheme
2.1.1 Formulation
As mentionedabove, the backboneof this researchis the CVFE scheme,which is schemati-
cally presentedin Figure2.1. It consistsof a combinationof conventional(volumetric)elements
(representedby 3-nodetrianglesin Figure2.1, althoughmost typesof structuralfinite elements
canbeused)andof interfacial(cohesive)elements(representedby a4-nodeelementin Figure2.1,
althoughhigher-ordercohesive elementsarealsoavailable). The volumetricelementsareused
to characterizethemechanicalresponseof thebulk material,while thecohesive elementsarein-
troducedin thefinite elementmeshto simulatethespontaneousmotionof oneor morecracksin
6
the structure.The captureof the failure processis achieved with the aid of a phenomenological
cohesive failurelaw characterizingtheevolutionof thecohesiveelementresponse.
∆
∆
t
n
VolumetricElement
ElementCohesive
VolumetricElement
Figure 2.1 CVFE conceptshowing one4-nodecohesive elementbetweentwo linear-straintrian-gular volumetricelements.The cohesive elementis shown in its deformedconfiguration. In itsundeformedconfigurationis hasno thicknessandtheadjacentnodesaresuperposed.
Thechoiceof thecohesive failuremodelplaysanimportantrole in thesimulationof thefrac-
ture process.In this study, we usethe bilinear rate-independentintrinsic formulationintroduced
by GeubelleandBaylor (1998),which is presentedin Figure2.2 for thepuremodeI andmodeII
cases.Thecohesive relationconsistsin two distinctportions:a linearly rising part, indicatingan
increasingresistanceof thecohesiveelementto theseparationof theadjacentvolumetricelements,
followedby amonotonicallydecreasingrelationbetweencohesivetractionanddisplacementjump
simulatingtheprogressive failureof thematerial. Themaximumvalueof thenormal(σmax) and
tangential(τmax) cohesivetractionsrespectivelycorrespondto tensileandshearstrengthsof thema-
terial. Oncethedisplacementjump (∆n for thetensilecaseand∆t for theshearcase)hasreached
a critical value(respectively denotedby ∆nc and∆tc for themodeI andII casesin Figure2.2),the
cohesive tractionis assumedto vanish. No moremechanicalinteractionis thenassumedto take
placebetweenthe initially adjacentvolumetricelements,therebycreatinga traction-freesurface
(i.e.,acrack)in thediscretizedsolid domain.
Theareaunderthecohesive traction/separationcurve correspondto theenergy neededto gen-
erateanew fracturesurface,i.e.,thefracturetoughnessof thematerial,denotedby GIc andGI Ic for
themodesI andII, respectively. To accountfor thepossiblecouplingbetweenthefailuremodes,
7
∆
Tt
t
maxτ
∆ tc
GIIc
∆
Tn
n
maxσ
∆ nc
Unloading
Reloading
GIc
Figure2.2Bilinearcohesivefailurelaw for thepuretensileor modeI (∆t 0, left) andpureshearor modeII (∆n 0, right) cases.An unloadingandreloadingpathis alsoshown in themodeI case.
thenormalandtangentialcohesivetractions,Tn andTt , arerelatedto thenormof thedisplacement
jump vector∆ ∆n ∆t throughtheintroductionof theresidualstrengthparameterSdefinedas 1 ∆ (2.1)
where ∆ denotestheEuclideannormof thenon-dimensionaldisplacementjump vector∆
∆ ∆n
∆t ∆n ∆nc
∆t ∆tc (2.2)
To limit thedetrimentaleffect thatthecomplianceof thecohesiveelementsmight have on the
stressfield solution,theresidualstrengthparameterof acohesiveelementis initially givenavalueinitial verycloseto unity. Typically avalueof 0 95to 0 98is used.As theelementfails, thisvalue
progressively decreasesto zero,at which point completefailure is assumedto have occurred.In
orderto maintainamonotonicdecreaseof thisstrengthparameterandtherebypreventthepossible
healingof thecohesive elements,theminimumvalueachievedby
is storedat eachintegration
8
point by using min min max 0 1 ∆ (2.3)
The resultingrate-independentcoupledbilinear cohesive traction-separationlaw can be ex-
pressedas
Tn 1 ∆nσmax Tt
1 ∆tτmax (2.4)
where
∆nc 2GIc
σmax
initial ∆nt 2GI Ic
τmax
initial (2.5)
in which, asindicatedearlier, GIc andGI Ic arethecritical energy releaserates(or fracturetough-
nesses)for modeI andmodeII failures,respectively. The fracturemodecouplingachieved by
Equation2.4canbeseenin Figure2.3.
9
Figure 2.3 Coupledcohesive failuremodeldescribedby Equation2.4; variationof normal(top)andshear(bottom)cohesive tractionswith respectto normal(∆n) andtangential(∆t) displacementjumps.
10
2.1.2 Finite Element Implementation
Theimplementationof theCVFEschemereliesonthefollowing form of theprincipleof virtual
work: Ω
S : δE dΩ internal
Ω
ρa δu dΩ inertial
Γex
Tex δu dΓ external
Γin
T δ∆ dΓ cohesive
0 (2.6)
whereΩ is theundeformeddomain,Γin denotestheinterior “cohesive” boundaryalongwhich the
cohesive tractionsT act, and Γex correspondsto the part of the exterior boundaryalong which
the external tractionsTex areapplied. a andu denotethe accelerationanddisplacementfields,
respectively. S is the secondPiola-Kirchoff stresstensorand E, the Lagrangianstrain tensor,
which is relatedto thedisplacementfield through
E 12 ∇u ∇uT ∇uT∇u (2.7)
Nonlinearkinematicsis usedin this studyto accountfor the possiblelarge rotationspresent
in thestructuredueto the fractureprocess.Theexpression(2.6) of theprinciple of virtual work
is fairly conventional,exceptfor thepresenceof thefourth term,which correspondsto thevirtual
work doneby cohesive tractionT for avirtual separationδ .
Theresultingsemi-discretefinite elementformulationcanbeexpressedin thefollowing matrix
form:
Ma Rin Rco Rex (2.8)
whereM is the lumpedmassmatrix, a is the vectorcontainingthenodalaccelerations,andRin,
Rco andRex respectively denotetheinternal,cohesiveandexternalforcevectors.
The time steppingschemeis basedon the classicalexplicit second-ordercentraldifference
scheme(Belytschko et al., 1976):
dn 1 dn ∆t vn 12
∆t2an (2.9)
an 1 M 1 Rinn 1 Rco
n 1 Rexn 1 (2.10)
11
vn 1 vn 12
∆t an an 1 (2.11)
where∆t is thetimestepanddn denotesthenodaldisplacementvectorat timen∆t.
The expressionof the internal, cohesive and external force vectorscan be found in Baylor
(1997).Whilea varietyof constitutivemodelscanbeusedto characterizetheresponseof thevol-
umetric elements,we use, in this study, a simple linear isotropic relation betweenthe second
Piola-Kirchoff stressesSandtheLagrangianstrainsE:
Si j λEmmδi j 2µEi j (2.12)
whereλ andµ aretheLame’sconstants.
2.1.3 Stability and Mesh Size
Like all explicit time steppingschemes,the centraldifferenceformulation is conditionally
stable(Cooket al., 1989),andthetimestepsizemustsatisfytheCourant(or CFL) condition:
∆t leCD
(2.13)
wherele is thesmallestelementsize,CD, thedilatationalwavespeed,givenby
CD E 1 ν 1 ν 1 2ν ρ (2.14)
in which E, ν andρ denotethematerial’s Young’s modulus,Poisson’s ratio anddensity, respec-
tively.
In theregionwherefailureis takingplace,smallelementshaveto beusedto captureadequately
the stressconcentrationassociatedwith the presenceof the crack front and the failure process.
In particular, a sufficient numberof elements(typically 5 to 10) mustbe usedto discretizethe
cohesive zone,i.e., theregion wherecohesive failure is takingplace.An estimateof thecohesive
12
zonesizecan be obtainedfor the quasi-staticmodeI situationin termsof the constitutive and
failurepropertiesof thematerial,as
R π8
E1 ν2
GIc
σmax2 (2.15)
It is clear, however, thatsmallelementsmustonly beusedin regionswherefailureis takingor
is aboutto takeplace.A coarserdiscretizationcanbeadoptedin therestof thedomain,generating
apossiblylargedisparityin elementsizes,andthereforein timestepsizes.
Furthermore,due to their inherentinstability and to the needto accuratelycapturethe fail-
ure processin the vicinity of the dynamicallypropagatingcrackfront, the presenceof cohesive
elementsin the discretizeddomainfurther reducesthe time stepsize,typically to 1 30th of the
Courantcondition(Baylor, 1997).InanadaptiveCVFE schemewherecohesiveelementsareonly
insertedin critical portionsof thediscretizeddomains,this additionaltime steprequirementalso
suggeststheneedfor timestepsubcycling.
In conclusion,the motivation for the incorporationin the CVFE schemeof the subcycling
algorithmdescribedin the next sectioncanbe schematicallypresentedin Figure2.4: time step
disparitycanbeassociatedwith spatialvariationsin elementsizes(parta)and/orwith thepresence
of cohesivedomains(partb).
2.2 Nodal Time StepSubcycling
Usingauniformtimestepin non-uniformmeshesis averycostlyandinefficient. Theextensive
researchon the topic of time stepoptimizationhasled to the developmentof variousmulti-time
stepsubcycling algorithms.Theinitial work of Belytschko andMullen in 1976led to an“implicit-
explicit” methodfor secondorderequationsusingnodalpartitioning(NealandBelytschko, 1998).
Furtherresearch,by HughesandLui (1978),led to a “implicit-explicit” subcycling methodusing
elementpartitioning. In our researchwe have adoptedan explicit subcycling methodfor second
orderequationspresentedby Smolinski (1989). This algorithmallows the useof different time
13
Large Elements
Large Elements
Small Elements Cohesive
Non−Cohesive
Non−Cohesive
Figure2.4Timestepdefinedby: (a) elementsizeor (b) elementtype.
stepsin different regionsof the sameproblem. As a result,we arenot constrainedby the mi-
nority elements,rathereachregion is given a time stepthat maximizesits efficiency while still
satisfyingthe local Courantcondition. This hasthe addedbenefitof minimizing the numberof
calculationsnecessaryin the larger time stepregionsover thesametime period. Multi-time step
methodspartitiona meshwith differenttime stepsusingeithernodalor elementpartitioning.The
algorithmpresentedby Smolinskiemploys nodalpartitioning. For nodalpartitioning,time steps
aredistributedto nodesor nodalgroups,while in elementpartitioning,the elementsthemselves
aregivendifferenttimesteps.
Although the algorithmpresentedby Smolinski is capableof supportingmultiple time step
regions, we have decided,for simplicity reason,to limit the numberof regions to only two -
designatedasregionsA andB. It shouldbenotedthatthenodesof thetwo regionsarenotrestricted
to begroupedtogether, insteadthey canbedispersedovertheentiremeshasis shown in Figure2.5.
For ouranalysis,wedesignateregionA asthecritical region,having thesmallertimestepequal
to 1∆t. This regionencompassesthesmallervolumetricelementsaswell asany cohesiveelements
in thesystem.Region B is givena time stepof m∆t, wherethe subcycling parameter, m, is thus
theratio of timestepsfrom regionB to regionA. If m 1, all regionsaregivenanequaltimestep
14
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B
AA
A
Figure2.5Subcycling regiondistribution.
andno subcycling is performed.Whensubcycling is used,this time stepratio mustalwaysbe a
positiveevennumber.
Oncethenodalpartitioninghasbeenperformedfor theinitial mesh,thesubcycling algorithm
is directlyappliedto thecentraldifferencetimesteppingloop. As thesolutionis steppedfrom time
t to time t m∆t, thenodesof regionA areupdatedm timesusingthestandarddiscretizationequa-
tions.RegionB, on theotherhand,is thesubcycledregionover thesametimeperiod,to whichan
approximatemethodis applied.Region B is not explicitly updated,ratherthenodalaccelerations
of region B aretoggledafter eachtime step. Sincethe parameterm is even andsincethe nodal
velocity updateis of the form vn 1 vn f an an 1 , alternatingthe sign of the acceleration
hasfor effect to keepthe velocity constantin region B. This toggling of nodalaccelerationsin
region B foregoesthecalculationof the internalandcohesive forces,therebysaving a significant
portion of time that would normally be dedicatedto thesecalculations.The equationsusedfor
directcalculationsin regionA andapproximationsin regionB arethusgivenby
dAn 1 dA
n ∆tvAn 1
2∆t2aA
n (2.16)
dBn 1 dB
n ∆tvBn (2.17)
vAn 1 vA
n 12
∆t aAn aA
n 1 (2.18)
15
vBn 1 vB
n 12
∆t aBn aB
n 1 (2.19)
aAn 1 MA 1 RAin
n 1 RAcon 1 RAex
n 1 (2.20)
aBn 1 aB
n (2.21)
At time t m∆t, we turn to theexplicit updatefor region B. Usinga similar approachpresented
above,region B is explicitly calculatedwhile region A is approximatedthroughthetogglingof its
nodalaccelerations.Accordingto Smolinski(1989),this updateshouldbeperformedtwice using
a time stepof 12m∆t or one-halfthatusedin region A. This approachwasfoundby Smolinskito
correctfor theoscillatorynatureof theaccelerationconstraint(toggling)usedin theexplicit update
of region A. It is alsoa reasonwhy the subcycling parameter, m, mustbe an evennumber. The
resultingequations,which aresolvedtwiceduringthetime t m∆t, aregivenby
dAn 1 dA
n 12
m∆tvAn (2.22)
dBn 1 dB
n 12
m∆tvBn 1
8m2∆t2aB
n (2.23)
vAn 1 vA
n 14
m∆t aAn aA
n 1 (2.24)
vBn 1 vB
n 14
m∆t aBn aB
n 1 (2.25)
aAn 1 aA
n (2.26)
aBn 1 MA 1 RBin
n 1 RBcon 1 RBex
n 1 (2.27)
As implied above, the bulk of the computationalsavings associatedwith the subcycling scheme
is achieved whenwe avoid the calculationsof the internalandcohesive force vectorsfor region
B nodesfrom t to t m∆t. Thesesavings may be quite substantialwhenusingmorecomplex
constitutive modelsfor the volumetric elementsand/orwhen the numberof cohesive elements
(andthereforethesizeof regionA) is small.
16
Sincetheinternalforcecalculationsareperformedby assemblingthelocal internalforcevec-
torsobtainedfor all thevolumetricelements,themostefficient way to take advantageof thesub-
cycling schemeis by flaggingthe volumetricelementsbasedon the type of nodes(A or B) they
contain. This allows to quickly “passby” all theelementscontainingnodesof typeB during the
first m time stepsof thesubcycling loop. In the implementationadoptedin thepresentwork, all
elementsof typeA have a flag of 1, all thoseof typeB have a flag of m, andall thoseof “mixed
type” (i.e.,whichcontainsomenodesof typeA andsomeof typeB) receiveaflagof 1. Thisflag
theninsuresthatinternalforcecalculationsareperformedonly onthoseelementshaving theaflag
of 1 or 1, correspondingto elementshaving at leastonenodefrom regionA. Thecomputational
savingswe achieve areequalto thenumberof elementshaving a flag of m, i.e. elementswhose
internalforcecalculationis skipped.Figure2.6 shows anexampleof applyingtheelementtime
stepflagsto a typicalmesh.
time step =1 t∆ time step = m∆ t
1
−1
m
m
m
m
m
m
−1
−1
−1
−1
−1
1
1
11
1
uniform region
1∆ t or m∆ t
1∆ t
uniform region
mixed region
t∆m
Figure 2.6Timestepassignment.
17
An addedbenefitof thesubcycling algorithmis thatit canbereadily implementedin anadap-
tive fashion.Thetime stepsof theindividualnodes,aswell asof thevolumetricelements,canbe
changedat any time duringthesimulation.This allows themeshto adaptto thechangingcritical
region. Whenusedin conjunctionwith adaptive meshing,thetime stepsarereducedasthe local
meshis madefiner, and increasedas the meshgrows coarser. With dynamiccohesive element
insertion,thetimestepsdecreaseascohesiveelementsareinserted.
Setinitial conditionsfor nodaldisplacements,velocitiesandacceleration.Clearthecounter:c 0Loopover time (conventionalexplicit centraldifferencemethod).
if (c m) PRIMAR Y REGION: A (solveA, approximateB)UpdateA displacementsdA
n 1 usingEquation2.16UpdateB displacementsdB
n 1 usingEquation2.17
CalculateRAinandRco
UpdateA accelerationsaAn 1 usingEquation2.20
ToggleB accelerationsaBn 1 usingEquation2.21
UpdateA velocitiesvAn 1 usingEquation2.18
UpdateB velocitiesvBn 1 usingEquation2.19
Increasecounters:c c 1Increasetime: t t ∆t
if (c ( m) PRIMAR Y REGION: B (solveB, approximateA)UpdateA displacementsdA
n 1 usingEquation2.22UpdateB displacementsdB
n 1 usingEquation2.23
CalculateRBin
ToggleA accelerationsaAn 1 usingEquation2.26
UpdateB accelerationsaBn 1 usingEquation2.27
UpdateA velocitiesvAn 1 usingEquation2.24
UpdateB velocitiesvBn 1 usingEquation2.25
Increasecounters:c c 1If c m 2, thesetc 0
Table 2.1 Codeimplementationof the multi-time stepnodal subcycling algorithm (Smolinski,1989).
18
2.3 Dynamic CohesiveNode/ElementInsertion
As describedearlier, a key componentof the adaptive CVFE schemerelieson the ability to
insertdynamicallycohesiveelementsin afinite elementmesh.Thissectionprovidessomedetails
onthreeissuesassociatedwith thedynamicelementinsertionprocess.Thefirst oneis thecriterion
usedto insertthecohesive elements:in this work, we selecttheregionsfor insertionbasedeither
on a boundingbox approachwhich grows aselementsare inserted,or a stress-basedapproach
wherecohesive elementsareplacedin regionswherestresslevelshave reacheda predetermined
threshold. The secondissueis that of databasemanagement:the insertionalgorithmrequiresa
moredetailedknowledgeof the currentmesh,including connectivity informationfor the nodes,
edgesandelements,aswell asother flagsnot necessarywith standardpre-simulationcohesive
elementinsertions. Finally, we addressthe “mechanics”of the insertionprocess,i.e., how to
insertcohesive elementswith minimal disturbanceof thesolution. Variousapproachesincluding
selectivedampingor pre-stretchingareconsidered.
2.3.1 Geometryand DatabaseManagement
Cohesive elementinsertionpresentschallenginggeometryanddatabasemanagementissues,
asnodesareduplicated,new elementsarecreatedandthe meshconnectivity mustbe appropri-
atelyadjusted.Additionally, theconservationof massandmomentummustbemaintainedon the
nodesof thesystem.This meansthatany nodeduplicationrequirestherecalculationof themass
contributionsfrom connectingelements,aswell astheduplicationof thenodaldisplacements,ve-
locities andaccelerations.Dynamic insertionrequiresmuchmoreinformationpertainingto the
nodes,edges,cohesive andvolumetricelements.Theseincludeflagsdefiningthetypesof nodes,
arraysof edgesconnectedto anode,etc.Thisextra informationis notneededif cohesiveelements
arepresentfrom thevery startof thesimulation. It is requiredonly wheninsertionis performed
dynamicallyto ensurethatthenew cohesiveelementsaswell astheassociatednodesandelements
areadjustedaccordingly.
19
2.3.1.1 1-D Insertion
The 1-D cohesive insertionconceptis illustratedin Figure2.7 for the caseof two adjacent
two-nodevolumetricelements.Cohesive elementsarerepresentedastwo nodehalvesconnected
by a non-linearspringsatisfyingthe chosencohesive traction-separationlaw. In orderto satisfy
the conservation of mass,eachof the nodehalvesreceivesa masscontribution of its connected
volumetricelements.As a result,if theelementon theright of thecohesivenodeis moremassive,
the right half will have a larger massthanthe left. In orderto conserve the linearmomentumof
the system,the newly formedright nodegetsa copy of all pertinentinformation(displacement,
velocity andacceleration)from the left node. Additionally, the volumetricelementson the left
andright sidemusthave their nodalconnectivity informationupdatedto take the new nodeinto
account.Thevariousnodeandelementdataandconnectivity arestoredasarraysin thecodeand
thesearrayswill grow with any duplicationandshouldbe adjustedasnecessarythroughoutthe
simulation.
1 2 3
ProposedCohesive Node
a b
1 32 4
Cohesive Node
a b
Figure2.7 (a)Standard1-D mesh.(b) 1-D meshwith insertedcohesivenode.
2.3.1.2 2-D Insertion
In the 2-D part of the presentstudy, we usethree-nodeconstant-straintriangularvolumetric
elements.Thecohesiveelementsthushave four nodewith thenodesorderedin counter-clockwise
fashionasseenin Figure2.8. Cohesive elementsare insertedin placeof existing or proposed
edgesby duplicatingtheedgeandpossiblyits nodesto form thenew four-nodesystem.In order
to insureproperinsertion,moreinformationis needed.In particular, eachedgemustknow thetwo
nodesconnectedto it aswell asthe volumetricelementsit borders.Anotherlist is alsoneeded,
which contains,for eachnode,thelist of all edgesandvolumetricelementsconnectedto it. This
20
additionalinformationis createdonceatthebeginningof thesimulation,andis updatedascohesive
elementsareinsertedadaptively. Consistency is thekey to asuccessfulcohesiveelementinsertion.
Wealwaysdefinethetopvolumetricelementof anedgeto betheelementpointedto by thenormal
vectorof theedge.Thenormalvectoris itself consistentlycalculatedaspointingto theright of the
segmentfrom thefirst to thesecondnode(Figure2.8).
Bottom Element
Top ElementEdge Normal
2
3
1
4Left Cohesive
Node
Left Node
duplicated edge
Right CohesiveNode
NodeRightoriginal edge
Figure2.82-D Cohesiveelementrepresentation.
Inconsistentnumberingandcalculationscan result in insertionof invalid cohesive elements
which maybe“criss-crossed”asdemonstratedin Figure2.9. Criss-crossedcohesiveelementsare
a resultof incorrectinitiation of top/bottomelementsandleft/right nodes.
4
3
5
1
2 3
2
4ProposedCohesive
Edge
1
4
1
5
2
6
3
21
3
6
45
DuplicatedNodes
DuplicatedEdge
4
1
5
2
21
4
3 6
3
5
DuplicatedEdge
OriginalEdge
6
Figure 2.9 2-D cohesive elementinsertion:(a) proposededgefor cohesive insertion,(b) insertedcohesiveelement,(c) “criss-crossed”cohesiveelement.
21
Onceall the edgeandnodeinformationhasbeendefined,cohesive elementinsertioncanbe
initiated. A selectededgecanonly bemadecohesive if it is not alreadycohesive, or this edgeis
aninternaledgethathasnotbeenflaggedby theuser. Externalboundaryedgesarerestrictedfrom
cohesive insertionbecausecohesive elementsrequiredto besandwichedbetweentwo volumetric
elements.
Thevalid proposededgeis now readyfor cohesive insertion. Standardcohesive insertional-
ways requiresthe duplicationof the proposededge. The nodesof the proposededgeare only
duplicatedif thesenodesarealreadyconnectedto anexisting cohesive or duplicatededge.These
nodescanbeeasilyrecognizedasthey will beflaggedascohesive nodes.Nodesflaggedas“nor-
mal” (i.e., non-cohesive) have no cohesive or duplicatededgesin their lists andasa result they
will not beduplicated.Nodalduplicationschematicallysplits themeshat thenode.Theoriginal
edgesandvolumetricelementsconnectedto this nodearealsosplit betweenthenew nodehalves.
This is similar to the1-D casewherethecohesivenodesystemis composedof two halves,eachof
which is connectedto its own elementscausingtheredistribution of nodalmasses.In 2-D, nodal
massesareobtainedfrom thevolumetricelementsto which thenodeis connected.Whenanodeis
duplicated,themassof theoriginal andduplicatedhalvesmustberecalculatedbasedonly on the
volumetricelementsconnectedto eachhalf. The sumof thesemassesshouldequalthe original
masspresentprior to any insertionaroundtheselectednode.In addition,conservationof momen-
tum mustalsobemaintainedfor eachnodeby duplicatingthenodaldisplacements,velocitiesand
accelerations,aswell asany otherpertinentflagsandmarkers.
Cohesive Edge ProposedCohesive Edge
4
65
3
11
2
New
2 7
Cohesive EdgeCohesive Edge
1
34
52
6
1
Figure 2.10Connectivity updateof nodesandelements.
22
Only five differentcasescanbe encounteredduring cohesive elementinsertion. Thesecases
arepresentedgraphicallyin Figure2.11.Thefirst caseis for anexistingcohesiveelement,which,
for obviousreason,cannothaveany morecohesiveelementsinsertedin its place.Thesecondcase
involvesa proposededgethat hasnever beenmadecohesive and is not connectedto any other
cohesive edges.Insertionfor this caseinvolvesonly the duplicationof the proposededge. The
resultingcohesive elementis considereddormantbecauseit sharesthe left andright nodesets.
Cases3 and4 aremirror imagesof eachother. They eachinvolve a normalproposededgethat
is connectedto oneothercohesive edgeor element. Elementinsertionfor thesecasesrequires
the duplicationof the connectingnodeaswell as the duplicationof the edge. The final caseis
concernedwith anedgethat is borderedby two othercohesive edges.This insertionrequiresthe
duplicationof boththeleft andright nodes.
23
Case #1: Nodes and edge flagged as cohesive or duplicated.
Existing cohesive element.
Edge duplicated
NodeShared
Case #2: Nodes and edge flagged normal.
NodeShared
Edge duplicated
Case #3: Right node is flagged cohesive or duplicated.
NodeShared
DuplicatedRight NodeShared
Node
Edge duplicated
Duplicated
Case #4: Left node is flagged cohesive or duplicated.
Node NodeSharedShared Left Node
Edge duplicated
Duplicated DuplicatedRight Node
Case #5: Both nodes are cohesiveSharedNode
Left NodeNode
Shared
Figure2.11Common2-D insertioncases.
In orderto clarify thevariousinsertioncases,wepresentin Figure2.12anillustrativeexample
of insertionfor asimple2-D mesh.Thefirst proposedcohesiveedgeis anon-cohesiveedgehaving
non-cohesivenodes.Thissituationcorrespondsto Case#2 in Figure2.11,for whichonly theedge
is duplicatedandthetwo nodesareflaggedcohesive. Althoughacohesiveelementis addedto the
generallist of cohesive elements,this elementhasno impacton the structuralsolutionsinceno
displacementjump is possible.Next, we inserttheneighboredgeto theright of thefirst element.
This edgeis alsonon-cohesivebut it containsonecohesivenoderesultingfrom thefirst insertion.
Dependingontheorientationof thenormalvectorof thisedge,thissituationcorrespondsto Cases
24
#3 or #4 in Figure2.11. Finally, the third cohesive elementinsertionis similar in conceptto the
secondone,exceptthattheedgeis notorientedhorizontally.
Edge Duplication Only
Proposed Cohesive Edge
Edge and NodeDuplication
Proposed Cohesive Edge
Edge and NodeDuplication
Proposed Cohesive Edge
Figure 2.12 Illustrative exampleof threecohesive elementinsertionsusingCases#2 and#3 inFigure2.11.
25
In order to illustrate insertionCase#5, we considerthe illustrative exampleshown in Fig-
ure2.13.After theinsertionof thefirst two cohesiveelements,themiddleedgeis still a “normal”
edgealthoughbothof its nodesareflaggedascohesive. Insertionof acohesiveelementalongthis
edgethuscausetheduplicationof boththeleft andright nodes.
Edge Duplications Only
Proposed Cohesive Edges
Edge and Node Duplications
Proposed Cohesive Edge
Figure2.13Illustrationof insertionCase#5 in Figure2.11.
2.3.2 CohesiveElementStability and SystemEquilibrium
Thusfar, we have only discussedthe geometricaspectsof the dynamiccohesive elementin-
sertionalgorithm. It is also importantto investigatethe effect insertionhason the stability and
precisionof thedynamicfinite elementsolution. Typical dynamicinsertionoccursin regionsun-
dergoingdynamicdeformationsandstresses.In theinsertionalgorithmdescribedsofar, we have
not discussedthe effect of existing stressstateon the insertionmethod: our basicapproachin-
volvessimply duplicatingall nodalinformation. Thenew cohesive elementis thusinsertedin an
“unstretched”state,in which theadjacentnodesaresuperposed.
26
However, asillustratedbelow, this “blind” insertionmaycauseoscillationsin theresponseof
thecohesivenodesandotherapproacheshaveto beadopted.To illustratethispoint, let usconsider
thesimple1-D problemshown in Figure2.14. It consistingof two segmentsof equallengthwith
oneendfixed at a wall andthe othersubjectedto a prescribedvelocity V. Eachsegmenthasa
lengthof 1 0 m, a cross-sectionalareaof 1 0 m2, a densityof 1 0 kg m3, anda Young’s modulus
of 1 Pa. Theresultingdilatationalwave speedis thus1 0 m s, andthecritical (Courant)time step
is 1 0 s. The time stepsizechosenhereis 0 033s dueto the inherentinstability of thecohesive
elementto beinserted.A cohesiveelementis insertedat thecenterof thesystem.
For the basesolution,we introducethe cohesive elementinto the meshat the first time step.
Dynamic insertion is investigatedby insertingthe cohesive elementafter 1000∆t (33 3 s) and
2000∆t (66 6 s). We run the simulationfor a total of 3000 time steps(100 s) with a constant
imposedvelocityof 0 01m s.
V31 2 4
Figure2.141-D “blind” insertiontestproblem.
To quantify the nodal oscillationsresultingfrom the cohesive elementinsertion,we plot in
Figure2.15 the evolution of the displacementjump acrossthe cohesive element. As expected,
in the referencecasefor which the cohesive elementis presentthroughoutthe simulation, the
displacementjumpincreaseslinearlywith timedueto theappliedvelocityboundarycondition.No
oscillationsareobserved in that case.On theotherhand,insertionsafter the 1000th and2000th
time stepscreatesubstantiallevels of oscillationsin the cohesive elementresponse.While the
averagedisplacementjump valueremainsthesameasin thereferencecase,theamplitudeof the
oscillationsincreasewith thestresslevel atwhich theinsertionwasperformed.
Thesecohesive nodeoscillationsare likewise presentin 2-D systemsand in order to obtain
satisfactorysolutionswerequirethattheseoscillationsbedampedout.
27
0 10 20 30 40 50 60 70 80 90 100−5
0
5
10
15
20x 10
−3
time (sec)
sepa
ratio
n (m
)
Insertion at 2000th stepInsertion at 1000th stepReference solution
Figure 2.15 1-D “blind” insertiontestproblem: evolution of the displacementjump acrossthecohesive element(i.e., betweennodes2 & 4 in Figure2.14) resultingfrom a cohesive elementinsertionat time0, 1000∆t (33 3 s), 2000∆t (66 6 s).
2.3.2.1 CohesiveDamping
Onepossibleapproachto reducethe potentiallydetrimentaloscillationsassociatedwith the
”blind” insertionof cohesive elementsinvolve the introductionof someform of dampingin the
cohesive response.Schematically, this approachcorrespondsto addinga dash-potin thecohesive
elementdescription(Figure2.16).
Cohesive Node System
Figure2.16Schematicrepresentationof adamped1-D cohesiveelement.
28
In its simplestform, the”damped”cohesive elementresponsecanbecharacterizedby a mul-
tiplicative term 1 ηδ to thecohesive failure law describedin Section2.1,with η denotingthe
dampingcoefficient, δ, thenormof thevelocity jump vector.
To illustratetheeffectof this additionaldampingtermon theresponseof theinsertedcohesive
element,wereconsiderthesimple1-D problemdescribedin Figure2.14.As shown in Figure2.17,
the introductionof a dampingterm in the cohesive elementresponseeliminatesall oscillations
after just a few time steps.However, it wasfound that theamountof damping(i.e., thevalueof
thecoefficient η) is stronglyproblemdependent.For thesimple1-D problemathand,theoptimal
dampingcoefficientsfor the1000th and2000th time stepinsertioncasesareη 20 andη 30,
respectively.
0 10 20 30 40 50 60 70 80 90 1000
0.002
0.004
0.006
0.008
0.01
0.012
time (sec)
sepa
ratio
n (m
)
Insertion at 2000th stepInsertion at 1000th stepReference solution
Figure 2.17Effect of cohesive damping:evolution of thedisplacementjump acrossthecohesiveelementfor thesimple1-D testproblemshown in Figure2.14andresultingfrom “blind” cohesiveelementinsertionwith dampingat time0, 1000∆t (33 3 s) and2000∆t (66 6 s).
29
Furthermore,in 2-D, thepresenceof thedampingtermwasfoundto bemuchlesseffective in
reducingtheoscillations. Theseshortcomingsforcedus to adoptanotherapproachbasedon the
pre-stretchingof thedynamicallyinsertedcohesiveelements.Thisapproachis describednext.
2.3.2.2 CohesiveElementPre-Stretching
As indicatedin thesimple1-D testproblemdescribedabove, theoscillationsassociatedwith
the cohesive elementinsertionarelinked to the amplitudeof the local stressfield presentin the
vicinity of theproposedcohesiveedgeat thetimeof theelementinsertion.Thebasicideapursued
hereafterconsistsin introducingthecohesiveelementin amannerthatwouldminimizethepertur-
bationon the local stressfield. This goalcanbeaccomplishedby insertingthecohesive element
in a pre-deformedstate- eitherstretchedor contracteddependingon the tensileor compressive
natureof the local stressfield. In otherwords,the nodaloscillationswill be minimizedbecause
thepre-deformedcohesiveelementwill bein astateof separationcloseto theexpectedseparation
hadthecohesiveelementbeenpresentfrom thebeginning.
In order to apply the cohesive elementpre-stretchingat the time of insertion,we must de-
terminethe nodaldisplacementjump acrossthe cohesive surfaceto be introduced. A relatively
straightforwardapproachto accomplishthis is to enforcelocal equilibriumon theassemblycom-
posedof thecohesive elementandtheadjacentvolumetricelement.To demonstratethis idea,let
usconsiderthesimple1-D systemshown in Figure2.18.
1 2 3
b
Cohesive NodeProposed
u uu1 2 3
a
1 32 4
a b
Cohesive Node
u u u u1 2 34
Figure 2.181-D cohesiveelementpre-stretchingconcept.
The local equilibriumequationsfor the four nodesinvolvedin thecohesive elementinsertion
canbewritten in theform:
30
)K *,+ D - + R - (2.28)
where,for anaxially loadedbarproblem,thestiffnessmatrix)K * , nodaldisplacementvector + D -
andforcevector + R - aregivenby.//////0 ka ka 0 0 ka ka kc kc 0
0 kc kc kb kb
0 0 kb kb
13222222456666667 6666668
u1
u2
u4
u3
9:666666;666666< .//////0 Ra
internal
0
0
Rbinternal
132222224 (2.29)
in which thestiffnesseska andkb of thebarelementsa andb aregivenby
ka EaAa
lakb EbAb
lb(2.30)
Thecohesivenodestiffnesskc is givenby
kc Sinitial
1 Sinitial
σmax
∆c(2.31)
whereSinitial is theinitial strengthparameter, σmax is thecritical failurestressand∆c is thecritical
separation.
The nodal forces,Rainternal andRb
internal actingon nodes1 and3, respectively, quantify the
existing stressstateon volumetricelementsa andb. Prescribingthenodaldisplacementsat nodes
1 and3 asthosecomputedat thesenodesat thetimeof insertion,wecanreadilysolvetheresulting
2-by-2linearsystemin termsof u2 andu4:.0ka kc kc kc kc kb
14 57 8 u2
u4
9 ;< .0kau1
kbu3
14 (2.32)
31
While this methodis quite simplein 1-D andfor a singlecohesive element,it is quite more
cumbersomein 2-D andwherea largenumberof elementsareinsertedsimultaneously. A simpler
methodinspiredfrom thepre-stretchingapproachconsistsin usingthelocal stressfield directly to
compute,with the aid of the traction-separationlaw, the initial displacementjump to be applied
acrossthecohesivesurface.
Invertingthetraction-separationrelationintroducedearlier, thedisplacementjumpcanbewrit-
tenas
∆ 1 T∆c
σmax(2.33)
Theappliedcohesive traction,T, is simply chosenastheaverageof thenodalinternalforces
appliedonnodes2and4by thevolumetricelementsaandb, respectively. Asshown in Figure2.19,
the separationis thenappliedevenly in both directionsfrom the currentlocationof the original
node(i.e., node#2). This even distribution hasshown to give good resultsalthougha mass-
weightedseparationcanbe usedby which the displacementis greatertowardsthe lighter edge
element.
∆Impose
No Pre−stretch
1 32 4
a b
2/∆ 2/∆
Pre−stretch
1 32 4
a b
Figure 2.191-D cohesive elementpre-stretchingconcept,with thepre-stretchappliedequallyonthetwo nodes.
In two dimensions,thenodalseparationscanreceive contributionsfrom multiple neighboring
cohesive elements,and in both the x and y directions. Figure 2.20 is a schematicexampleof
threeconnectededgeswherecohesive elementsareinserted.We first transformthe normaland
tangentialcohesive separationsinto theseparationsalongtheprincipalx andy axes,resultingin
separationsof ∆x and∆y. Whenapplyingtheseseparationsto thevariousnodeswemustbecareful
notto simplysumthecontributionsfrom eachneighboringcohesiveelement.Insteadwecaneither
32
usethemaximumor minimumnodalseparation,theaverageof all of theneighboringseparations,
or someweighteddistributionbasedon themassof thecurrentnode.After sometestingwe found
thattheoptimalapproachis to usetheaverageof theneighboringcohesiveseparations.Theother
approachesinducedgreateroscillationsfor every testcase.
∆
∆
∆∆
n1
t1n2
t2
Figure 2.202-D separationcontributionsfrom neighboringcohesiveelements.
Figure2.21shows the effect of pre-stretchingon the separationof the cohesive nodefor the
0th (0 s), 1000th (33 3 s) and2000th (66 6 s) timestepinsertioncasesfor thesimple1-D problem
discussedearlier. Theoscillations,while still present,havebeendrasticallyreduced.More testsof
theeffectof adaptivecohesiveelementinsertiononthesolutionarepresentedin Chapters3 and4.
2.3.3 Insertion RegionSelection
Thethird critical componentof theadaptivecohesiveelementschemeis thecriterionto beused
to introducethecohesive elementsin thefinite elementmesh.In this study, two approacheshave
beenconsidered:thefirst onerelieson defininga boundingbox within which all edgesaremade
cohesive andallow this box to grow asnecessary, while thesecondoneconsistsin selectingthe
cohesive edgesbasedon the local stresslevel. Thesetwo approachesaresuccessively discussed
hereafter.
2.3.3.1 Bounding Box Approach
Theboundingbox insertionmethodselectsproposedcohesive edgesby first categorizing the
systemby threemainregions(Figure2.22): a non-cohesive region whereno cohesive elementis
present,a passive cohesive region wherecohesive elementsarepresentbut have not undergone
33
0 10 20 30 40 50 60 70 80 90 1000
0.002
0.004
0.006
0.008
0.01
0.012
time (sec)
sepa
ratio
n (m
)
Insertion at 2000th stepInsertion at 1000th stepReference solution
Figure 2.211-D testproblemseparationoscillationsof nodes2 & 4 (Figure2.14)resultingfrominsertionwith pre-stretchingat time0, 1000∆t (33 3 s), 2000∆t (66 6 s).
any failure,andan active cohesive region wherecohesive elementsarefailing. Failing cohesive
elementaredefinedby the valueof the strengthparameter, S, introducedin Equation2.1,which
decreasesastheelementfails until completefailurefor S 0.
Theboundingbox approachfirst determinestheextentof theactivecohesive region. This box
is thenenlargedby a prescribedamountandnew cohesive elementsareinsertedwithin this new
box. Theregionbetweentheoriginalandnew boundingboxesis thepassivecohesiveregionwhere
failureis expectedto occurin thenearfuture.Theboundingboxwill continueto grow andmoveas
longascohesiveelementsarefailing within thesystem.Theboundingboxmethodis quitecapable
of determiningtheappropriatecohesive regionsalthoughthis methodis not self-starting.Instead,
at leastonecohesive elementmustbepresentin thecritical region, sincetheboundingboxesare
only definedbasedon existing cohesive elements.As a result, we are againlimited to known
problemswherecritical regionsarepredictable,suchasfractureproblemsinvolving apre-existing
notch.
34
AC
PC
NC
AC = Active Cohesive
PC = Passive Cohesive
NC = Non−Cohesive
Figure2.22Boundingbox method.
The seconddrawbackof the boundingbox approachis that the boundingbox may become
very large while the actualactive cohesive region may be quite small. This situationcanoccur
for problemsthathavemultiple critical regionswhich arespacedfar apart,asin a doublenotched
specimenshown in Figure2.23. Whena singleboundingbox approachis used,theextentof the
failing elementsdefinea box which cancrossnon-cohesive regions. The resultingenlargedbox
requiresthat thenon-cohesive region bemadecohesive eventhoughfailure is not expectedthere
until muchlaterin thesimulation,if atall. Thismaycauseinsertionto grow uncontrollably, filling
theentiredomainwithin ashorttime.
2.3.3.2 Stress-basedSelectionApproach
A bettermethodfor selectingsitesfor cohesive elementinsertionis basedon a local stress
criterion. In thismethod,anedgeis selectedfor insertionif thesomemeasureof thestressesin the
neighboringvolumetricelementsreachessomecritical value.This stressmeasuredependson the
typeof failuremodelcharacterizingthematerialof interest.For brittle systems,thestressmeasure
would be basedon the valueof the maximumprincipal stress.For moreductile systems,other
stressmeasuresbased,for example,on thevalueof thevonMisesstress,aremoreadequate.In the
presentstudy, weusethefollowing simpleexpressionof thestresscriterion:
35
PC
Bounding Box
NC
AC
AC
Figure2.23Multiple activecohesiveregions.=σ11
2 σ222 2σ12
2 > X% σmax (2.34)
whereσ11, σ22 and σ12 are the stressescomputedin the adjacentvolumetric elements,X is a
user-definedfractionlevel andσmax denotesthestrengthof thematerial.
With this method,cohesive elementsare insertedinto the systemonly wherever they are
needed,therebysignificantlyminimizing thecomputationaltime andmemorycosts.To illustrate
theconceptof stress-basedcohesiveelementinsertion,Figure2.24presentstheresultof adynamic
fracturesimulationinvolving a simpleL-shapeddomainattachedalongits top edgeandsubjected
to a downwardvelocity alongthe left end. In this particularproblem,cohesive elementswerein-
troducedin thedomainwhenthestresslevel reached45 % of thematerialstrength.No cohesive
elementswerepresentin the domainat the beginning of the simulation. The insertedcohesive
elementsareshown asdarker segmentsandonly representa smallportionof thetotal numberof
edges,therebysignificantlyreducingthe computationalcost. The referencecase,with cohesive
elementpresenteverywherein thesystem,tookapproximately4500s, while thesimulationbased
on thestress-basedinsertiononly requiredabout1000s of CPUtime.
36
Figure2.24Stress-basedinsertionresultsfor simpleangledcase.
2.4 Parallel Implementation usingCharm++
Problemsrequiringa largenumberof computationsor thosehaving longsimulationtimesmay
benefitfrom someform of parallelization.SincetheCVFE schemeis basedon anexplicit central
differencemethod,it lendsitself well to codeparallelization.Distributing a problemacrossmany
processorsallowsusto decreasethesolutiontimeor increasetheproblemsize.
Two well known parallelizationtechniquesareOpenMP andMPI . OpenMP is a collection
of compilerdirectivesand library routineswhich take advantageof sharedmemoryparallelism
of codes(Padua,2000). OpenMP allows us to take a serial codeandapply parallelismto the
individual loops within it. This is a good methodfor simple loops but is not viable for more
complex loopstypical of theCVFE methodor whereraceconditionsmayexist. A racecondition
37
occurswhenmultipleprocessorsattemptto write to thesamememorylocationcausingfutureread
attemptsto potentiallyaccessincorrectdata.
MPI or MessagePassingInterfaceis a morerobust methodby which slave processorscom-
municatedatawith themasterprocessor. Eachprocessorgetsasinglechunkof memoryon which
it performsall necessarycalculations.Thedownsideof MPI is that theserialcodemustbecom-
pletelyrewrittensothatit fits theMPI format.
A betterparallelizationmethodis Charm++ which is basedon the MPI method. The ad-
vantageof this methodis that the communicationoccursdeeperin the background.Using the
Charm++ finite elementframework weareableto write aparallelversionof ourcodethatclosely
resemblestheserialversion,but still takesadvantageof thevariousotherCharm++ featuressuch
as:runtimeloadbalancing,monitoringof performance,etc. (Lawlor, 2000).
2.4.1 MeshPartitioning
Codeparallelizationrequiresthe distribution of dataover several processors.In the CVFE
scheme,wedistributethenodal,volumetricandcohesivedataandconnectivities. Thedataincludes
suchinformationasnodaldisplacements,velocitiesandaccelerations,materialpropertiesof the
elementsaswell asvariousotherflags.Theconnectivity informationincludesthelistsof all nodes
connectedto eachelementof a chunk. A simplerepresentationof a CVFE meshis presentedin
Figure2.25with thecorrespondingconnectivity informationin Table2.2.
Element NodeList
Volumetric- V1 1, 2, 3Volumetric- V2 2, 3, 5Volumetric- V3 4, 6, 8Volumetric- V4 7, 9, 10Volumetric- V5 9, 10,11Cohesive - C1 3, 4, 5, 6Cohesive - C2 6, 7, 8, 9
Table 2.2SimpleCVFE meshconnectivities.
38
V1
V2
V3
V4
V5
2
1
5
6
10
8 9 113 4
C1C2
7
BoundaryLine
Figure 2.25SimpleCVFE mesh.
Partitioning of the simplemeshinto two chunks,A andB, is performedalongthe boundary
line. All of the boundaryelements,volumetricor cohesive, arepartitionedalongtheir edgesso
that they arefully definedin only a singlechunk. Previouspartitioningmethodswould partition
a cohesive meshby splitting the cohesive elementsacrosstwo chunks. Oncepartitioned,each
elementandnodemustberenumberedlocally sothattheloopboundariesin thecodewill nothave
to beadjusted.Theglobalnumberingis alsomaintainedso that themeshcanbereassembledat
any time into its original form. Table2.3shows theresultof partitioningthesimplemeshinto the
two chunks.NotethatnodesA5 andB1 werenode4 of theoriginalmesh- likewisenodesA6 and
B2 werenode6. Thesenodesarenow sharedbetweenthetwo chunks.
ChunkAElement NodeList
Volumetric- AV1 1, 2, 3Volumetric- AV2 2, 3, 4Cohesive- AC1 3, 4, 5, 6
ChunkBElement NodeList
Volumetric- BV1 1, 2, 4Volumetric- BV2 3, 5, 6Volumetric- BV3 5, 6, 7Cohesive - BC1 2, 3, 4, 5
Table2.3PartitionedCVFE meshconnectivities.
39
AV1
AV2
A1
BV1BV3
B4 B5 B7
B6
A3
A4 A5
A6
AC1
B2
BC1
BV2
A2
B1
B3 BoundaryLine
Figure2.26PartitionedCVFEmesh.
2.4.2 Computational Efficiency
A goodrepresentationof theefficiency of theparallelsolutionis to observetheparallelspeedup
of thecode,givenby
speedup T 1T N (2.35)
whereT 1 is the time requiredto run the simulationon one processor, and T N is the time
requiredto run this samesimulationon N processors.With this we cantrackhow well thecode
is ableto performwhendistributedacrossmultiple processors.Ideally, we would preferto have
perfectspeedup,wherethesolutiontime is decreasedin proportionto thenumberof processors.
Unfortunately, thespeedupof mostsimulationsbegins to decreasewith an increasingnumberof
processors.This is becausethecostof theparallelizationbecomesgreaterrelativeto thecostof the
computationsperformedby eachprocessor. Eventually, for many processors,thecommunication
betweenthemdominatesthetotal timeof thesolution.
40
2.4.3 Structure of Standard Charm++ FEM Framework
A standardserialFEM codeis parallelizedwith theCharm++ FEM framework by distributing
it into four main subroutines:init, driver, meshupdateandfinalize. This framework hasbeen
developedespeciallyfor parallelizationof finite elementcodes. Although the original must be
distributedacrossthefour mainsubroutines,it still maintainsthebasicserialform.
The init routinestartsthecodeby readingtheglobalmeshandassociateddata,flagsandcon-
nectivities for thenodesandelements.Theprogramcanhave many differentelementtypes,such
ascohesiveor volumetricelements,but only asinglenodetype.Thenodedataarepackagedusing
thefollowing calls:
call FEM Set Node? #of nodes@ #of datadoublesAB@call FEM Set Node Data r ? nodedataarrayAB@
whereFEM SetNode()setsthenumberof total nodesin thesystemwith #ofnodesandthenum-
ber of datadoublesfor eachnodeusing #ofdatadoubles. A datadoubleis equalto 8 bytesin
32-bit systems,which is equivalentto oneREAL*8 or two INTEGER types. The secondcall,
FemSetNodeData r() sendsthe arrayof nodal information to the driver routine. The sizeof
eachindividual elementof this arrayis equalto thetotal numberof datadoublesspecifiedby the
previouscall. It shouldbenotedthatall databe in full bytes.This is alreadysatisfiedif thedata
elementsareof typeREAL*8 , but if theany datais of typeINTEGER wemusthave INTEGER
pairs.
Theelementdataandconnectivitiesarepassedusingthecalls
call FEM Set Elem? elementtype@ #of elements@ #of datadoubles@ #of conndoublesAB@call FEM Set ElemData r ? elementtype@ dataarrayAB@
call FEM Set ElemConn r ? elementtype@ connectivityarrayAC@whereFEM SetElem()setstheelementtype,thenumberof elementsof this type,thenumberdata
doublesperelementandthenumberof connectivity doublesperelement.Theelementtypeis an
41
integernumberwhich is typically 1 for cohesiveelementsand2 for volumetricelements.Theele-
mentdataandconnectivity arestoredin similar fashionto thenodaldataalthoughthey arebroken
acrosstwo separatearrays.This is becausethepartitioningwill only usetheconnectivity informa-
tion to determinetheproperdistribution acrosschunksandsothedatais not neededat this time.
Theelementdataandconnectivitiesarepassedonto driver usingtheFEM SetElemData R()and
FEM SetElem ConnR()calls,respectively.
Oncefully packaged,the datais senton to Charm++ which usesthe Metis programto par-
tition the meshinto several chunks. Thesechunksarethenpassedon to the processorsusedin
thesimulation. Unlike MPI , which limits only onechunkperprocessor, Charm++ assignssev-
eral chunksper processorwhich enablesit to dynamicallyload balancea simulationby simply
migratingthesmallchunksto lessactiveprocessors,asnecessary.
Thedriver routineis thencalledoneachchunk,whereit performsthevariouscalculationsand
datamanipulations.Thenodeandelementdataandconnectivitiesarereceivedby driver usingthe
following calls,which arethemirror imagesof thedata”send” calls,FEM Set(),initiatedby the
init routine:
call FEM Get Node? #of nodes@ #of datadoublesAB@call FEM Get Node Data r ? nodedataarrayAB@
call FEM Get Elem? elementtype@ #of elements@ #of datadoubles@ #of conndoublesAC@call FEM Get ElemData r ? elementtype@ dataarrayAB@
call FEM Get ElemConn r ? elementtype@ connectivityarrayAB@wheretheparametersarethesameasthosedefinedfor theFEM Set calls.
Duringeachdriver call theCVFEschemeis appliedto thenodesandelementsof theparticular
chunk. Unfortunately, the boundarynodesrequirespecialtreatmentto ensurethat the data is
correctoncethemeshis reassembled.Themassfor eachnodeis thesumof thecontributionsfrom
theneighboringvolumetricelements.If thenodeis a boundarynode,thesevolumetricelements
may be split acrossmultiple chunksso that the local boundarynodesin a given chunk receive
only a contribution from the local volumetricelements.The resultingaccelerationcalculations,
42
whichrely onthenodalmasses,wouldthereforeby incorrectfor all theboundarynodes.TheFEM
framework is ableto accountfor this lackof dataof sharednodesby combiningthedataacrossall
chunks.As a result,all chunkboundarycalculationswill alwaysbeduplicatedin eachchunkbut
this addedcostinsuresthat thesolutionwill beaccurate.Theboundarynodesareall storedin a
field by calling
f ieldid D FEM Create Field ? datatype@ vectorlength @ of f set @ distanceAB@wherefieldid is theID of thecurrentfield. datatypedescribesthetypeof thedatawhich is shared,
eitherFEM BYTE , FEM INT , FEM REAL , or FEM DOUBLE . The vectorlengthdescribes
thenumberof dataitemsassociatedwith eachnode. For example,we storethenodemassesfor
eachdegreeof freedomwherefor 2-D systemsis two - theresultingvectorlengthis 2. Theoffset
is thebyteoffsetform thestartof thenodearrayto theactualdataitemsto beshared.distanceis
thebyteoffsetfrom thefirst nodeto thesecond.During thecalculationswithin driver this field is
updatedby calling
call FEM U pdate Field ? f ieldid @ f ir stnodeAB@wherefieldid specifiedthe ID of thefield definedduring thecreationof thefield. firstnodeis the
locationof thedataarrayfor thesharednodes.
Periodically, wemaywish to outputsomecurrentdatafor theglobalmeshor evenreassemble
themeshinto its original form so thatwe maychangeit andoptionally repartitionit again.This
canall beachievedby acall to themeshupdateroutinevia
call FEM U pdate Mesh? callmeshupdated@ dorepartitionAB@wherecallmeshupdateddeterminesif the meshupdateroutine shouldbe called immediately-
when non-zero. Also, if dorepartition is non-zerothe meshwill be immediatelyreassembled
on the first processor, temporarilysuspendingthe simulation,so that this meshcanbe modified
or tested.Themeshis thenrepartitionedinto several chunksandredistributedto the processors.
Thechunksmight bedifferentfrom thechunksdefinedat thestartof thesimulation.On theother
43
hand,if dorepartition is zero, the call is non-blockingwhich allows the simulationto continue
while meshupdateis calledbecausetheonly actionallowedin this routineis limited to theoutput
of data.
Oncethesimulationis completed,for everychunk,thedatais reassembledonthefirstprocessor
in thefinalizeroutineallowing theuserto performfinal calculationson theserialmeshor simply
outputany necessarydatato thescreenor files.
2.4.4 Parallel Structure of the Dynamic Insertion Code
Sinceour periodic insertionof cohesive elementschangesthe basicmesh,we must remesh
aftereachinsertion.In orderto achievethisusingthecurrentframework, we takeadvantageof the
optionalpartitioningin themeshupdatesubroutine.At specificintervalsduringthegeneralsolu-
tion loop, eachdriver routinesendsall of its datato meshupdate. The datais reassembledinto
a serialmeshwherewe canperformany necessarycohesive elementinsertionsor meshupdates
muchmoreeasilythenwhenthemeshis distributedacrossseveralchunks.Thebiggestdifficulty
with performinginsertionson thedistributedmeshis dueto theboundaryedgesandnodeswhich
may requireinformation from otherchunks. This would requireextensive communicationsbe-
tweenthe chunks,resultingin increasedcomputationaleffort. Instead,we performour insertion
onaserialmeshandthenpartitionit into severalchunkanddistributedit to thevariousprocessors.
Thestructureof our codeis presentedin Table2.4.
44
initreadinput dataandserialmeshcall FEM Set()// senddatato driver
driver//optionallyreadinput dataneededby eachchunkfieldid = FEM CreateField() // createmassandforcefieldsMain UpdateLoop
call FEM Get()// getdatafrom init/meshupdatecall FEM UpdateField(masses)Solutionloop
// calculatedisplacements// calculateforcescall FEM UpdateField(forces)// calculateaccelerations// calculatevelocities
EndSolutionloopcall FEM Set()// senddatato updatemeshcall FEM UpdateMesh(timestep,1)// repartitioncall FEM Get()// getdatabackfrom updatemesh
EndMain UpdateLoop
meshupdatecall FEM Set()// getdatafrom eachdriver
// dynamicinsertionandothermeshmanipulationscall FEM Set()// senddatabackto driver
finalizecloseout theprogram
Table 2.4Parallelstructureof thedynamicinsertioncode.
45
CHAPTER 3
1-D ANALYSIS AND RESULTS
In order to verify andtest the multi-time stepsubcycling anddynamiccohesive elementin-
sertionalgorithmspresentedin the previouschapter, we first apply themto simple1-D systems.
Problemsin 1-D aremuchsimplerto understandandsolve,yet they arestill valid examplesof dy-
namicfractureproblems.Theexperiencegainedin 1-D is very importantin giving usdirectionfor
applyingthesamealgorithmsandany newly developedonesto 2-D problems,while minimizing
our computationaleffort andtime.
In thebeginningof thischapter, wefirst introducethereference1-D problemandits analytical
solution.In thesectionsthatfollow, we takeaparametricapproachin applyingthesubcycling and
dynamiccohesive elementinsertionalgorithmsto various1-D problems.Finally, we discussour
observationsandconclusionsin thelastsection.
3.1 ProblemDescription
Typical 1-D problemsinvolve barsor beamsunderaxial loading. In orderto testour various
algorithmswe definea referenceproblemcomposedof a beamfixedat oneendandfreeandthe
otherwith anappliedaxial loadat thefreeend,asseenin Figure3.1.Weusethisgeneralproblem
for all of our analysesin theupcomingsections.
This beam,of lengthL, is fixedat the right endandhasa compressive axial forceappliedon
the left, correspondingto an axial stressσo DFE 0 G 01 Pa. The beamis homogeneousandhasa
46
x = Lx = 0
F/A
dilatational wave
Figure3.1Referenceproblemin 1-D.
cross-sectionalareaA D 1 G 0 m2, densityρ D 1 G 0 kgH m3 andYoung’s modulusE D 1 G 0 Pa. The
beamwavespeedgivenby
c D Eρ
(3.1)
is thus1 G 0 mH s.
In 1-D theanalyticalsolutionto abeamproblemcanbeobtainedby solvingthewaveequation
∂2u∂x2 D 1
c2
∂2u∂t2 (3.2)
with initial conditions
u ? L @ 0AID 0 ut ? L @ 0AJD 0 (3.3)
andboundaryconditions
u ? L @ t AKD 0 Eux ? 0 @ t AKD F H A DLE σo (3.4)
whereu is thedisplacementandc is thewavespeed.
The solution to this problemis representedby the x-t (displacement-time)diagramin Fig-
ure3.2.Thedisplacement,velocityandstressprofiles,for asamplepointatx D L H 2, areshown in
Figure3.3.
47
v = 0
v = 0
v = 0
v = ο / ρ c
v = −σο / ρ c
x = 0 x = L
t = 4L/c
t = 0
t = L/c
t = 3L/c
t = 5L/c
x = L/2
t = 2L/c
ο
ο
ο+σ
σ = −σ
σ = −2σ
σ = 0
σ = −σ
σ = 0
Figure3.2x-t diagramin 1-D.
t = 0
σ
v
d
L/c 2L/c 3L/c 4L/c 5L/c 6L/c
Incident Wave Reflections
Figure 3.3 Analytical solution for displacementd, velocity v andstressσ in the middle of thebeamfor the1-D waveproblemdescribedin Figure3.1.
48
3.2 Multi-T ime StepNodal SubcyclingResults
Employing themulti-time stepnodalsubcycling algorithmpresentedby Smolinski(1989)and
describedin Chapter2, we apply it to thesimple1-D beamproblemdiscussedabove. In orderto
verify our implementationof thesubcycling algorithmwe compareour resultsto thosepresented
by Smolinski (CaseC) in his paper. The 1-D problemis discretizedinto threeregions of 10
elementseach,aspresentedin Figure3.4. Themiddle region is discretizedinto 0 G 1 m segments,
while theothertwo have1 G 0 m segments.Usingthepropertiesselectedfor thereferenceproblem,
thecritical timestepis 0 G 1s, whichfurtherreducedto 0 G 075s to ensurethatany instabilitiespresent
in thesolutionarea resultof thealgorithmandnotof aninadequatetimestep.
1 5 11 15 21 25 31F
m m1
length = 1.010 elements10 elements
length = 0.1length = 1.010 elements
Figure3.4Subcycling testdescribedby Smolinski(1989)by CaseC.
As in CaseC, we give thetwo non-subcycledboundaryregionsa parameterof m D 10, while
the middle subcycled region retainsthe critical time step. The simulationis run for 1000 time
stepsor 75G 0 s, while we trackthevelocitiesof nodes#5,#15and#25,representingthemiddleof
eachof thethreeregions. Comparingthevelocity profilesin Figures3.5-3.7,we canseethat the
subcycledsolutioncloselymatchesthereferencesolutionfor eachnode.Furthermore,theresults
obtainedareindistinguishablefrom thoseof CaseC presentedby Smolinski,suggestingthatour
implementationis correct.
49
0 10 20 30 40 50 60 70 75−0.015
−0.01
−0.005
0
0.005
0.01
0.015
time (s)
velo
city
(m
/s)
referencem = 10
Figure3.5Velocityprofileof node#5 with subcycling parameterm D 10.
0 10 20 30 40 50 60 70 75−0.015
−0.01
−0.005
0
0.005
0.01
0.015
time (s)
velo
city
(m
/s)
referencem = 10
Figure3.6Velocityprofileof node#15with subcycling parameterm D 10.
50
0 10 20 30 40 50 60 70 75−0.015
−0.01
−0.005
0
0.005
0.01
0.015
time (s)
velo
city
(m
/s)
referencem = 10
Figure3.7Velocityprofileof node#25with subcycling parameterm D 10.
Althoughsubcycling is ableto provide fairly accurateresults,thelevel of accuracy in thenon-
subcycled regionsdecreasesas their time stepgrows closerto the critical valuefor that region.
The advantageof the larger time stepratio is that morecomputationalsavings is achieved since
the non-subcycled regionsareupdatedlessoften. Figure3.8 shows how the increasedtime step
ratioaffectsthedifferencesin displacementsandvelocitiesof node#5 for oursubcycling problem
at node.We cansee,thatasthetime stepratio increases,thedifferenceof thedisplacementsand
velocitiesbetweenthe referenceand the subcycled casesincreases.The differencesat the start
andendof thesimulationarenot displayedbecausethedisplacementsdiffer significantlyrelative
to eachotherbut arequitesmall in respectto therestof thesimulation.As a result,they arenot
plottedsincethey would maskthe effect that subcycling hason the solution. Similarly, for the
velocity profile, themiddle region representsa nearzerovelocity so thatevenminor differences
causelargerelativepercentages.
51
0 10 20 30 40 50 60 70 75−8
−6
−4
−2
0
2
4
time (s)
% d
iffer
ence
in d
ispl
acem
ent
m=2m=4m=6m=8m=10
0 10 20 30 40 50 60 70 75−50
−40
−30
−20
−10
0
10
20
30
40
50
time (s)
% d
iffer
ence
in v
eloc
ity
m=2m=4m=6m=8m=10
Figure3.8Subcycling effect on (a) displacementsand(b) velocitiesat node5.
52
In orderto getsometiming resultsandshow thesavingsgainedusingthesubcycling algorithm,
weselecteda largerproblemfor analysis.Figure3.9showsthe1-D beamwhich is discretizedinto
480equalsegmentseachof length1 G 0 m. The beampropertiesandforcesareselectedto match
theoriginal casepresentedat thebeginningof this chapter. Subcycling is appliedto thefirst and
lastgroupof elementsfor m D 6, 10and14,with atimestep0 G 025s (1/40ththecritical timestep).
The20 innerelementsaregiven thebasetime stepof ∆t, which givesusa ratio of 23 : 1, in the
numberof non-subcycledto subcyclednodes.
length = 1.0m20 elements230 elements
length = 1.0m230 elementslength = 1.0m
m m1
F
Figure 3.9Testcaseusedto gettiming resultsfor subcycling.
Thesimulationsarerun for 200@ 000time stepsor 5000s for eachsubcycling parameter, with
the resultsfor the internalforce vectorandtotal simulationtime presentedin Table3.1. At first
glancetheresultsdo not appearvery favorable.Eventhoughthetime requiredfor calculatingthe
internal force vectorsdecreases,with increasingm, the overall time for the entirecodeis above
thereferencetime. Therearetwo reasonsfor this timing discrepancy. Thefirst is that thecurrent
implementationof the subcycling algorithmmay not be the mostefficient. Whensubcycling is
usedthecomputercodemustmakecopiesof thenodaldisplacements,velocitiesandaccelerations
for every time step. Copying, in itself, is not very expensive but whenusedmultiple times for
eachnodeandfor eachtimestep,this timecostcanaccumulate.Thesecondreasonfor thetiming
discrepancy is that the1-D calculationof the internalforce is very simpleandasa resultis very
fast. In higherdimensions,the complexity of the internalforce vectorcalculationincreasesand
hasa bigger impacton the time of the solution - making it a more ideal testof the subcycling
algorithm.
53
Subroutine ReferenceCase[s], m D 1 m D 6 m D 10 m D 14
Rin 10.19 6.55( 36%) 4.87( 52%) 4.45( 56%)Total 49.66 62.13(-25%) 56.52(-14%) 52.68( -6%)
Table 3.1Timing resultsfor subcycling. TheCPUtimesaving (in %) is givenin parentheses.
3.3 Dynamic CohesiveNodeInsertion Results
As inidcatedearlier, whensolving dynamicfractureproblemsusing the CVFE scheme,the
conservativeapproachdictatesthatcohesiveelementsor nodesbeplacedeverywherein thedomain
at thebeginningof thesimulation.Althoughthis insuresthatall possiblefailureswill becaptured,
thecostof suchanimplementationis extremelylarge. An alternative approachis to dynamically
insertcohesiveelementsatany timein thedomain.In thisway, wecanobtainsometimesavingsby
not performingcomplex cohesive calculations,aswell asmemorysavingssincefewer nodeswill
bepresentin thedomain. In this section,we detail a dynamiccohesive nodeinsertionalgorithm
on simple1-D beamproblems.
Going backto the referenceproblemdescribedin Figure3.10,we insertcohesive nodesinto
the nodes#12 through#20. In orderto avoid cohesive failure, we setthe failing stressesof the
cohesive elementsto be very high. Furthermore,we reducethe critical time stepbasedon the
volumetricelementsby 1H 30th, to avalueof ∆t D 0 G 0025s, whichensuresthatany instabilitiesin
thesystemarea resultof theinsertionalgorithmandnotdueto thecohesiveelementsthemselves.
21 22151110
10 elementslength = 1.0
10 elementslength = 1.0
10 elementslength = 0.1
= normal node = cohesive node
Figure3.10Nodes12 through20 aremadecohesive.
54
Thefirst testis a blind insertionproblemwherenodesareinsertedwithout any thoughtto the
equilibriumof system.As will bepresentedin thenext section,blind insertioncausessomenodal
oscillations,which affect theaccuracy of thesolution. As a result,we attemptto minimize these
oscillationsthroughtheuseof damping.Althoughthis hassomefavorableresults,theimplemen-
tation of dampingis not very efficient, so we insteadpresenta third methodof pre- stretching
cohesive nodesduring insertion. This allows us to minimize the oscillationsmuchmoreeasily
while still maintaininganaccurateresult. Lastly we presentthe resultsfrom combiningdynamic
insertionwith multi-timestepnodalsubcycling to furtherincreaseourcomputationalsavings.
3.3.1 Blind CohesiveNodeInsertion Results
In orderto analyzetheeffect of blind insertion,we dynamicallyinserttheselectednodes(#12
through#20) at time step0 (0 G 0 s), 2500(6 G 25 s), 5000(12G 5 s) or 10000(25G 0 s). Node15 is
monitoredin Figures3.11 through3.14 to determinethe effect of varying the time of insertion.
From thesefigureswe can seethat blind insertionappearsto have an oscillatory effect on the
nodalvelocities.Theamplitudeof theoscillationsincreasesif theinsertionis performedafterthe
dilatationalwave haspassed,or somestresshasbuilt up neartheproposednodes.For thecurrent
case,having a wave speedof 1 G 0 mH s andtime stepof 0 G 0025s, it takesthewave approximately
4000time steps(or 10 s) to reachthefirst cohesive node- node#12. Prior to this time, the local
stressis zeroandsotheblind insertionis stableasseenby Figures3.11and3.12.
55
0 10 20 30 40 50 60 70 75−0.015
−0.01
−0.005
0
0.005
0.01
0.015
time (s)
velo
city
(m
/s)
Figure3.11Velocityprofileof node#15resultingfrom blind insertionat the0th timestep(0 G 0 s).
0 10 20 30 40 50 60 70 75−0.015
−0.01
−0.005
0
0.005
0.01
0.015
time (s)
velo
city
(m
/s)
Figure 3.12 Velocity profile of node#15 resultingfrom blind insertionat the 2500th time step(6 G 25s).
56
0 10 20 30 40 50 60 70 75−0.015
−0.01
−0.005
0
0.005
0.01
0.015
velo
city
(m
/s)
Figure 3.13 Velocity profile of node#15 resultingfrom blind insertionat the 5000th time step(12G 5 s).
0 10 20 30 40 50 60 70 75−0.015
−0.01
−0.005
0
0.005
0.01
0.015
velo
city
(m
/s)
Figure 3.14Velocity profile of node#15 resultingfrom blind insertionat the 10000th time step(25G 0 s).
57
Theseseverevelocity oscillationsarelikely theresultof thesuddenapplicationof volumetric
stresseson thecohesiveelementsduringthedynamicinsertion.Althoughtheoscillatoryeffect in
1-D appearsto belimited to the local node.Theneighbornodesexperienceonly minimal effects
whicharenearlyindistinguishable.This localizationof thedisturbancesis very importantbecause
we areableto insertcohesivenodeswithout fearof affectingtherestof thesystemtoo adversely.
In thenext sectionswewill attemptto completelyremovetheseoscillationsthrougheithertheuse
of dampingor pre-deformationof cohesiveelements.
3.3.2 Damping of Blind Insertion
Using the referenceproblemwe apply dampingduring the blind insertionof cohesive nodes
#12 through#20. From trial anderrorwe find that theoptimaldampingparameter, describedin
Chapter2, is η D 450.Figures3.15and3.16show theeffect thatlineardampinghason node#15
at time steps5000( 12G 5 s ) and10000( 25G 0 s ), respectively. Comparedto thevelocity profiles
dueto blind insertionat thesesametimes(Figures3.13 and3.14), dampingdoesminimize the
previousoscillatoryeffect. Only minoroscillationsarepresentat thetimeof insertion,but arethen
completelydampenedout.
58
0 10 20 30 40 50 60 70 75−0.015
−0.01
−0.005
0
0.005
0.01
0.015
time (s)
velo
city
(m
/s)
Figure3.15Velocityprofileof node#15resultingfrom blind insertionwith dampingat the5000thtimestep(12G 5 s).
0 10 20 30 40 50 60 70 75−0.015
−0.01
−0.005
0
0.005
0.01
0.015
time (s)
velo
city
(m
/s)
Figure 3.16 Velocity profile of node #15 resulting from blind insertion with dampingat the10000th timestep(25G 0 s).
59
Althoughdampingdoesminimizethenodaloscillations,it is quitecumbersometo implement
sincewe usea trial anderrorapproachto determinetheoptimaldampingcoefficients. In orderto
usedampingasefficiently aspossible,werequireananalyticalcorrelationbetweenagivensystem
andtheappropriatedampingparameters.Unfortunately, derivationof suchacorrelationis outside
thescopeof this researchandis left for futureinvestigation.
3.3.3 Dynamic Insertion with Pre-Stretch
To minimize thenodaloscillationsresultingfrom blind insertion,we employ a pre-stretchto
cohesivenodesduringinsertion.With this pre-stretchthecohesivenodeis insertedin astatemost
closelyapproximatingthe stateit would be in hadthe insertionoccurredat the beginningof the
simulation.Usingagainthereferenceproblemdescribedat thebeginningof this chapter, we pre-
stretchnodes#12through#20duringdynamicinsertionat the5000th and10000th timestep.The
velocityprofilesfor node#15arepresentedin in Figures3.17and3.18,for eachof theinsertions.
Comparingthesevelocity profilesto theblind insertionprofilesin Figures3.13and3.14,we
canseethat pre-stretchingis able to minimize the oscillationsquite well. This methodis also
able to avoid the minor stabilizationperiod that is requiredwith damping. Furthermore,Fig-
ures3.19and3.20plot theseparationdistancefor two cohesive halvesof node#15,for dynamic
pre-stretchedinsertionat the 10000th time step. From thesetwo figureswe canclearly seehow
oscillatorytheseparationscohesivenodeare,aswell ashow significantlypre-stretchingminimizes
them.
60
0 10 20 30 40 50 60 70 75−0.015
−0.01
−0.005
0
0.005
0.01
0.015
time (s)
velo
city
(m
/s)
Figure3.17Velocityprofileof node#15resultingfrom insertionwith pre-stretchingat the5000thtimestep(12G 5 s).
0 10 20 30 40 50 60 70 75−0.015
−0.01
−0.005
0
0.005
0.01
0.015
time (s)
velo
city
(m
/s)
Figure3.18Velocityprofileof node#15resultingfrom insertionwith pre-stretchingatthe10000thtimestep(25G 0 s).
61
0 10 20 30 40 50 60 70 75−8
−6
−4
−2
0
2x 10
−5
time (s)
sepa
ratio
n (m
)
Figure 3.19Cohesive separationfor node#15resultingfrom blind insertionat 10000th time step(25G 0 s)
0 10 20 30 40 50 60 70 75−8
−6
−4
−2
0
2x 10
−5
time (s)
sepa
ratio
n (m
)
Figure 3.20 Cohesive separationfor node #15 resulting from insertion with pre-stretchingat10000th timestep(25G 0 s)
62
In order to obtain timing information we have increasedthe size of the referenceproblem
to onehaving 300 segmentsof length1 G 0 m, asseenin Figure3.21,The middle 180 nodesare
madecohesive at the110@ 000th time stepor 2750s for a critical time stepreducedby 1H 40th to
∆t D 0 G 025s Thetotal simulationis run for 200@ 000timesteps(5000s).
length = 1.0m60 elements
length = 1.0m180 elements
length = 1.0m60 elements
F
cohesive region
Figure 3.21Testcasefor dynamiccohesivenodeinsertionwith pre-stretching.
Table3.2givesthetiming resultsfor thecohesiveandinternalforcecalculationsaswell asthe
total simulationtime. We gaina 58 % time savings in the cohesive calculationswhich is on par
with theapproximatetime of insertion,110@ 000th time stepout of 200@ 000,or 55 % throughthe
simulation.Theinternalforcecalculationsareincreasedslightly becausethenumberof nodeshas
increasedandhencethenumberof internalforcecalculationshasalsoincreased.
Subroutine ReferenceCase[s] DynamicCase[s]
Rco 9.17 3.82( 58%)Rin 6.74 6.88( 0%)
Total 60.91 51.34( 16%)
Table3.2Timing resultsfor dynamiccohesivenodeinsertionwith pre-stretching.CPUtimesaving(in %) is givenin parentheses.
3.3.4 Combined Insertion with Subcycling
To further increasethe time savings of our problemwe combinethe dynamiccohesive node
insertionandmulti-time stepsubcycling algorithms.As a testcasewe chosea beamdiscretized
into 800 equalsegmentsof length= 1 G 0 m. Subcycling is appliedto the left andright regions
63
with thesubcycling parameterm D 10,asshown in Figure3.22. Cohesivenodesaredynamically
insertedat the150@ 000th time step(3750s), into themiddle200nodes,with thesimulationbeing
run for 300@ 000timesteps(7500s) usinga ∆t D 0 G 025s.
length = 1.0m200 elements
1m m
300 elementslength = 1.0m
300 elementslength = 1.0m
F
cohesive region
Figure3.22Testcasefor dynamicinsertionwith subcycling.
64
0 1000 2000 3000 4000 5000 6000 7000 7500−0.015
−0.01
−0.005
0
0.005
0.01
0.015
time (s)
velo
city
(m
/s)
referencem = 10
Figure 3.23Velocity profile of node400of dynamiccohesive nodeinsertionat time step150000(3750s) with nodalsubcycling usingm D 10.
FromTable3.3wecanseethatthecombinedmethod,usingdynamicinsertionwith subcycling,
is moreefficient for the individual internalandcohesive force subroutines,althoughthe overall
savings is minimal. This low time savings is dueto thecostly implementationof thesubcycling
algorithmaswell asthe increasednumberof nodalupdatesof the displacements,velocitiesand
accelerationsin themaincode.
Subroutine ReferenceCase[s] CombinedCase,m D 10 [s]
Rco 14.32 9.16( 36%)Rin 27.33 15.93( 42%)
Total 117.80 114.57( 3%)
Table 3.3Timing resultsfor dynamiccohesive nodeinsertionat time step150@ 000(3750s) withnodalsubcycling usingm D 10. CPUtimesaving (in %) is givenin parentheses.
65
3.4 Conclusions
In this chapterwe presentedthe resultsof applyingthedynamiccohesive nodeinsertionand
the multi-time stepnodalsubcycling methodto 1-D systems.Analysisin 1-D allowedus to use
simpleproblemswith verifiablesolutions.Basedon theanalysisof severaldifferentproblems,we
gainedabetterunderstandingof thebestandworstmethodsandwaysto tacklefutureproblemsin
2-D.
Theresultshaveshown thatmulti-timestepnodalsubcycling is agoodapproximatingmethod
which allows for differenttime stepsin differentregionsor a problem. As a result,we areable
to distribute the lower time stepsto morecritical regionswhile larger time stepscanbe usedin
lesscritical ones.Implementingthis algorithmin a standardfinite elementformulationgenerates
significantcomputationalsavings for problemshaving significantly more non-subcycled nodes.
Whentheratio of non-subcycledto subcyclednodesis small, thecostdueto the implementation
of thealgorithmoffsetsany savingsachievedthroughits use.In addition,subcycling is inherently
andapproximationmethodandsosomeinformationis lost which, in turn,decreasestheaccuracy
of thesolution.
We have alsoshown thatdynamicinsertionrequiressomeform of pre-stretchingof cohesive
nodesto minimize the nodaloscillationsassociatedwith blind insertion. The timing resultsfor
this methodhave generatedsignificantcomputationalsavings while the solution wasstill quite
accurate.
66
CHAPTER 4
2-D ANALYSIS AND RESULTS
Having gainedvaluableexperiencefrom ourone-dimensionalanalysisof thepreviouschapter,
we cannow applythevariousoptimizationmethodsto themorecomplex, two-dimensionalprob-
lems. We begin our analysiswith themulti-time stepnodalsubcycling algorithmusinga simple
modeI testproblemwherewe vary the numberof non-subcycled to subcyclednodes.Next, we
analyzethe dynamiccohesive elementinsertionalgorithmon a simplified 2-D problem. Using
two cohesive elementselectioncriteria, the boundingbox methodandthe stress-basedselection
method,weapplytheinsertionalgorithmto aseriesof complex problems,from whichweareable
to gaugetheaccuracy andtiming informationof thesolutions.Finally, weusetheCharm++ code
parallelizationtechniqueto takeadvantageof thebenefitsof multipleprocessorsin solvingagiven
problem.
4.1 Multi-T ime StepSubcyclingResults
In order to verify the 2-D nodalsubcycling algorithmdescriberdin Chapter2, we test it on
a homogeneouslydiscretizedmodeI crackproblempresentedin Figure4.1. The middle region
containscohesive elementswhich defineit as the critical or subcycled region, sincethe overall
domaintime stepmustbereduceddueto thecohesive elements.ThemodeI loadingis dueto an
appliedvelocityof 0 G 25mH s on thetopandbottomof thedomain.
67
Cohesive
Non−Cohesive
Non−Cohesive
Figure4.1Cohesiveelementdistribution.
The entiredomainis composedof PMMA materialwith a Young’s ModulusE D 3 G 24 GPa,
Poisson’s ratio ν D 0 G 35, anddensityρ D 1190kgH m3. Thecohesive elementshave a maximum
stressσmax D 32G 4 MPa, initial strengthparameterSinit D 0 G 995andanormalandtangentialcritical
separationsof ∆nc D ∆tc D 2 G 2 M 10N 5 m. Thetime stepis reducedby thirty to ∆t D 3 G 0 M 10N 9 s,
andthesimulationis run for 60000time steps,or 0 G 00018s, on a PentiumIII, 600MHz, 750Mb
RAMprocessor, runningMandrakeLinux7.2.
In ourprimarily analysis,wewish to seetheeffect thattheratioof non-subcycledto subcycled
nodes- also known as the region ratio - hason the timing and accuracy of the solutions. We
test the algorithm for region ratios of 1 : 1, 2 : 1 and 4 G 5 : 1. For eachof thesecaseswe use
subcycling parametersof m D 4 @ 10@ 16and20in thenon-criticalregionsof thedomain.Optimally,
we shouldbe able to usea maximumof m D 30 sincewe reducethe original time stepby this
amount. Previous resultshave shown that the optimal parametercannotbe achieved dueto the
severeoscillationsthatoccur, asa resultwemaximizeour parameterat m D 20.
Basedonourresultsin 1-D, for low low regionratiosaswell aslow subcycling parameters,the
costof thealgorithmimplementationcanpotentiallyoffsetthesavingsgainedthroughsubcycling.
In increasingbothof theseratioswewill show thatthecomputationalsavingsis increasedalthough
theaccuracy of thesolutionsdecreaseswith thehighersubcycling parameters.
68
4.1.1 Equal Subcycledto Non-SubcycledRegionRatio of 1:1
We first testsubcycling on an equaldistribution of non-subcycled andsubcycled nodes,i.e.
a region ratio of 1 : 1, with 7896nodes,15256edgesand8700volumetricelements.Selectinga
randomnodeaheadof theinitial notch,weplot its nodaldisplacementsin Figure4.2for thevarious
subcycling parameters.Fromthis figurewe canseethatsubcycledsolutionsarevery closeto the
referencesolution,althoughtheiraccuraciesdecreasewith anincreasingsubcycling parameter.
Thetiming resultsfor theinternalandcohesiveforcesubroutines,themainloopandtheoverall
solutionarepresentedin Table4.5. As expected,the time for the internalforce calculationsde-
creasesasthesubcycling parametergrows,althoughfor m D 4, thecostresultingfrom thecohesive
forcesandmainsolutionloopoffsetsthesavingsof theinternalforces- a11%loss.
0 1 2
x 10−4
−2.5
−2
−1.5
−1
−0.5
0
0.5x 10
−5
time (s)
noda
l dis
plac
emen
t (m
)
referencem=4m=10m=16m=20
Figure4.2Nodaldisplacementsof a randomnodeaheadof thenotchfor aproblemwith anequalregion ratio (1 : 1) with subcycling parametersof m D 1 @ 4 @ 10@ 16and20.
69
Subroutine ReferenceCase[s], m= 1 m = 4 m= 10 m = 16 m = 20
Rco 424.13 628.72 503.81 479.24 469.56Rin 733.89 472.52 300.37 256.66 238.05Main 427.29 652.91 499.89 469.59 439.78Total 1628.54 1811.59 1359.70 1259.68 1201.80% Total Savings -11% 17% 23% 26%
Table 4.1 Timing results(in s) for a problemwith an equalregion ratio (1 : 1) with subcyclingparametersof m D 1 @ 4 @ 10@ 16and20.
4.1.2 UnequalSubcycledto Non-SubcycledRegionRatio
Over thenext two problems,we increasetheregion ratio to 2 : 1 ( having 12294nodes,28341
edges,and17381volumetricelements) and4 G 5 : 1 ( having 30889nodes,81148edges,and52020
volumetricelements). We plot thedisplacementprofilesof a randomnodefor eachof theregion
ratiosin Figures4.3 and4.4, respectively. Thecorrespondingtiming resultsarepresentedin Ta-
bles4.2 and4.3. For the4 G 5 : 1 region ratio case,them D 20 resultis not presentedbecausethe
solutionbecameunstablelatein thesimulation,possiblydueto lackof local computermemory.
As we canseefrom the two figures,thedisplacementsfor eachof thesubcycling parameters
arevery closeto thereferencesolution;in fact,thedifferencesarenearlyimperceptibleat certain
pointsin the simulation. In addition,the time savings for bothcasesincreasesasthe subcycling
parameterincreases;with thebiggestincreaseoccurringfor the lower subcycling parametertran-
sitions.For example,for the2 : 1 region ratio results,theincreasefrom m D 4 to m D 10generates
a savingsof 28%,while from m D 10 to m D 16 it is only 4%. The resultsfor the4 G 5 : 1 region
ratio fair slightly betterwith savingsof 32%and6%,for bothsubcycling parametertransitions.
70
Subroutine ReferenceCase[s], m= 1 m = 4 m= 10 m = 16 m = 20
Rco 438.42 634.97 499.39 487.99 473.49Rin 1561.98 893.63 483.02 376.92 339.56Main 704.72 994.00 742.12 736.87 700.87Total 2853.85 2693.81 1894.97 1774.19 1682.16% Total Savings 6% 34% 38% 41%
Table 4.2 Timing results(in s) for a problemwith an equalregion ratio (2 : 1) with subcyclingparametersof m D 1 @ 4 @ 10@ 16and20.
Subroutine ReferenceCase[s], m = 1 m = 4 m = 10 m= 16
Rco 600.59 886.33 703.14 653.98Rin 4215.44 2670.78 1264.39 911.16Main 1119.75 1115.83 1114.14 1116.31Total 7930.09 7577.06 5072.81 4604.52% Total Savings 4% 36% 42%
Table 4.3 Timing results(in s) for a problemwith anequalregion ratio (4 G 5 : 1) with subcyclingparametersof m D 1 @ 4 @ 10and16.
71
0 1 2
x 10−4
−1
0
1
2
3
4
5x 10
−6
time (s)
noda
l dis
plac
emen
t (m
)
referencem=4m=10m=16m=20
Figure 4.3 Nodal displacementsof a randomnodefor the 2 : 1 region ratio with subcycling pa-rametersof m D 1 @ 4 @ 10@ 16and20.
0 1 2
x 10−4
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0x 10
−5
time (s)
noda
l dis
plac
emen
t (m
)
referencem=4m=10m=16
Figure 4.4 Nodal displacementsof a randomnodefor the 4 G 5 : 1 region ratio with subcyclingparametersof m D 1 @ 4 @ 10and16.
72
4.1.3 Multi-T ime StepNodal SubcyclingObservations
Having appliedmulti-time stepnodalsubcycling to variousproblemswe can concludethat
this methodis ableto provide a significanttime savingsat a minimal costto theaccuracy of the
solution.Althoughit is mostoptimalfor problemswherethenumberof calculations(or nodes)in
thenon-subcycledregion is at leasttwicemorethanin thesubcycledregion.
In addition,asthesubcycling parameter, m, is increased,thesavingsalsoincreasesdueto the
decreasednumberof explicit updatesfor eachcycle. In Figure4.5,we presentthecomparisonof
the% savingsobtainedfor thevarioussubcycling parametersasafunctionof theregionratio. The
biggestincreaseoccursasthe region ratio is doubled,with an averageincreaseof 15% for each
subcycling parameter;beyondthis, thesavingsappearsto plateauout.
Althoughsubcycling is ableto achievesignificanttime savings,it comesat thecostdueto the
increasein the sizeof the nodalarraysrequiredto track the displacements,velocitiesandaccel-
erations. Furthermore,as the subcycling parameteris increased,the numberof approximations
alsoincreaseswhich leadsto a lossin accuracy of thesolution. Fromour results,thesubcycling
parametershouldnot bebegreaterthanhalf of thecritical valuefor thesubcyclednodalregions,
to avoid many of theinstabilitiesresultingfrom thenodalapproximations.
73
1 2 4.5−20
−10
0
10
20
30
40
50
Subcycled/Non−subcycled region ratios
% s
avin
gs
m = 4m = 10m = 16m = 20
Figure4.5Percenttime savingsvs region ratio for varioussubcycling parameters.
74
4.2 Dynamic CohesiveElement Insertion
In many dynamicfractureproblemswe arenot ableto easilypredictthecohesive failureloca-
tionsor paths.And so, for conservative CVFE analysis,thedomainis discretizedwith cohesive
elementseverywherefrom the beginning. As a result,theseproblemsincur the greatestcompu-
tationalcosts,especiallyif actualcohesive failuresdo not occuruntil late in the simulation. A
preferredapproachis to placecohesiveelementsanywhereandatany timeduringasimulation.In
thisway, weareableto savemuchcomputationaleffort sincecohesiveelementswon’t beinserted
until they areneeded.We presentthis dynamicinsertionmethodusingblind insertion,damping
andpre-stretchingof cohesive elementson a simple2-D problem. We thenapply thealgorithm,
usingthe two aforementionedselectionmethods:theboundingbox methodandthestressbased
stress-basedapproachto severalproblems.
4.2.1 Insertion Analysis
As a first stepin our insertionanalysiswe selectthetwo simple2-D testproblems,presented
in Figure4.6. Eachdomainis 2 G 5 m tall and1 G 5 m wide,with a fixedbaseandanappliedvelocity
of 0 G 056mH s pulling on thetop. Thematerialpropertiesfor thesystemareselectedsuchthat the
dilatationalwave speedis 10 mH s anda conservative time stepvalueof ∆t D 0 G 0004s is usedto
ensurethatany instabilitiesarea directresultof theinsertionandnot thecohesiveelements.
The two casesdiffer in the orientationof the cohesive elementswith respectto the applied
load. In the first case,all threecohesive elementsareperpendicularto the appliedforce,so that
only normaltractionsandseparationswill bepresenton thesecohesiveelements,with thesecond
having aninclinedcohesiveelementthatwill havebothnormalandsheartractionsandseparations.
In the upcomingsectionswe will presentthe threemain insertionmethods,which include
blind insertion,insertionwith damping,andpre-stretchedinsertion. From our 1-D analysis,we
havedeterminedthatthesolutionfor a dynamicinsertionproblemis dependenton thestresslevel
of the cohesive elementat the time of its insertion. Using the averagestressesfor volumetric
elements#23and#14,thecohesiveelementsareinsertedat timesteps0 (0 G 0 s), 2500∆t (1 G 0 s) or
75
ElementsCohesive
Applied Velocity
TrackingNode
#23
#14ElementsCohesive
Applied Velocity
TrackingNode
#23
#14
Figure 4.6 Simple2-D mesheswith threecohesive elementsinsertedalong(a) ”horizontal” and(b) ”mixed” interfaces.
5000∆t (2 G 0 s) of a 10000(4 G 0 s) time stepsimulation.correspondingto stresslevelsof 0%,23%
and43%,respectively, asseenin Figure4.7. In eachcasewe will follow thenodalseparationof
thetrackingnodepresentedin Figure4.6.
76
0 0.5 1 1.5 2 2.5 3 3.5 4−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time (s)
stre
ss p
erce
ntag
e (σ
xx /
σ max
)
2500
5000
σ11
σ12
σ22
Figure 4.7 Normalizedaveragestresslevels for the volumetricelementsof the middle cohesiveelement. Vertical lines at the 0th (0 G 0 s), 2500th (1 G 0 s) and5000th (2 G 0 s) time steprepresentdynamicinsertiontimes.
77
4.2.1.1 Blind Insertion
Blind insertionis thesimplestinsertionmethod,by whichacohesiveelementis placeddirectly
betweenthevolumetricelements,without thoughtto the local stressesor stability of thesystem.
Figure4.8and4.9presentsthenodalseparationsfor thetrackingnodeof theinsertioncaseusing
the”horizontal” and”mixed” cohesiveelements.
Whenthe cohesive elementsarepresentat the startof the simulation,the nodalseparations
are simply increasingdue to the appliedconstantvelocity. On the other hand,when insertion
occursaftersometime, thenodalseparationsoscillateseverelyaroundthereferencesolution.The
amplitudeof theseoscillationsis directly proportionalto the local stressat the time of insertion,
which increasesover time for a constantvelocity loading,asseenin Figure4.7.
4.2.1.2 Damping of Blind Insertion
Thenodaloscillationsresultingfrom blind insertionmustbeminimizedin orderto obtainan
accuratesolution.Oneknown methodof minimizinggeneraloscillationsis to removesomeenergy
from asystemthroughtheuseof lineardamping,asdiscussedin Chapter2.
Figures4.10and4.11show theresultof applyinglineardampingon thedynamicallyinserted
cohesiveelementsfor the“horizontal” and“mixed” cases.Fromtrial anderrortheoptimaldamp-
ing coefficientsfor insertionat t D 2500∆t and5000∆t areη D 3 G 8 andη D 4 G 4 for the”horizontal”
case,andη D 2 G 4 andη D 4 G 3 for the “mixed” case,respectively. From the figures,we cansee
that linear dampingdoesminimize the oscillationsafter someinitial time, although,our experi-
mentationhasshown thatif thecoefficient is too large,thesolutioncandivergefrom thereference
solution.
78
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.000
0.005
0.010
0.015
0.020
time (s)
∆ / ∆
crit
Insertion at t = 2.0 sInsertion at t = 1.0 sReference Solution
Figure 4.8 Normalizedseparationof thetrackingnodefor ”horizontal” blind insertionat the0th(0 G 0 s), 2500th (1 G 0 s) and5000th (2 G 0 s) timestep.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.000
0.005
0.010
0.015
0.020
time (s)
∆ / ∆
crit
Insertion at t = 2.0 sInsertion at t = 1.0 sReference Solution
Figure 4.9Normalizedseparationof thetrackingnodefor ”mixed” blind insertionat the0th (0 G 0s), 2500th (1 G 0 s) and5000th (2 G 0 s) timestep.
79
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.000
0.005
0.010
0.015
0.020
time (s)
∆ / ∆
crit
Insertion at t = 2.0 sInsertion at t = 1.0 sReference Solution
Figure4.10Normalizedseparationof thetrackingnodefor ”horizontal” blind insertionwith damp-ing at the0th (0 G 0 s), 2500th (1 G 0 swith η D 3 G 8) and5000th (2 G 0 s with η D 4 G 4) timestep.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.000
0.005
0.010
0.015
0.020
time (s)
∆ / ∆
crit
Insertion at t = 2.0 sInsertion at t = 1.0 sReference Solution
Figure4.11Normalizedseparationof thetrackingnodefor ”mixed” blind insertionwith dampingat the0th (0 G 0 s), 2500th (1 G 0 s with η D 2 G 4) and5000th (2 G 0 s with η D 4 G 3) timestep.
80
4.2.1.3 Insertion with Pre-Stretch
Thusfar, we have seenthatblind insertioncausesseverenodaloscillationswhich aredirectly
relatedto the local stressat the insertiontime. Althoughdampingwasableto slightly minimize
theseoscillations,it is not theoptimalmethod.Insteadweapplyapre-stretchto cohesiveelements
during their insertionso that the equilibrium andstability of the local systemis maintainedand
betterminimizationcanbeachieved.
Usingthereferenceproblemwe testthepre-stretchinsertionmethodat the0th (0 G 0 s), 2500th
(1 G 0 s) and5000th (2 G 0 s) timestep.Figures4.12and4.13show theseparationprofilesfor tracking
nodeof the”horizontal” and”mixed” insertioncases.As we canseefrom thesefigures,thesepa-
rationsfor eachinsertionarevery closeto thereferencesolution. We canthereforeconcludethat
thecohesiveelementinsertionwith pre-stretchingmethodis ableto capturethesolutionaccurately
andat any insertiontime (or local stresslevel).
Although greatlyminimized,the oscillationsaremorepronouncedfor the ”mixed” insertion
casewhich hasanangledcohesive elements.This is mostlikely dueto theshearseparationsthat
this angledcohesiveelementexperiences.Underthecurrentinsertionmethod,theseparationsfor
eachcohesivenodeareequallydistributedto theneighboringcohesivenodes.Theneachcohesive
nodeusesan averageof all of the separationscontributed to by the variouscohesive elements.
Although, this hasfound to be the bestmethodthusfar, it is unableto completelyminimize all
of the nodaloscillationssincean averagedseparationis used. Futureresearchin this areamay
provideamoreoptimalmethodfor applyingtheseparations.
81
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.000
0.005
0.010
0.015
0.020
time (s)
∆ / ∆
crit
Insertion at t = 2.0 sInsertion at t = 1.0 sReference Solution
Figure 4.12Normalizedseparationof thetrackingnodefor “horizontal” insertionwith with pre-stretchingat the0th (0 G 0 s), 2500th (1 G 0 s) and5000th (2 G 0 s) timestep.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.000
0.005
0.010
0.015
0.020
time (s)
∆ / ∆
crit
Insertion at t = 2.0 sInsertion at t = 1.0 sReference Solution
Figure 4.13 Normalizedseparationof the tracking node for “mixed” insertionwith with pre-stretchingat the0th (0 G 0 s), 2500th (1 G 0 s) and5000th (2 G 0 s) timestep.
82
4.2.1.4 Insertion Analysis Observations
As in 1-D, we have foundthatblind insertionof cohesive elementsin 2-D systemscausesse-
verenodaloscillations(Figures4.8and4.9),which, in turn,decreasetheaccuracy of thesolution.
Althoughlineardampingdoesminimizetheoscillations(Figures4.10and4.11),theeffect is not
enoughto justify its use. Instead,we have adaptedthe pre-stretchtechniqueusedin 1-D to the
morecomplex 2-D problems.Theresultsof Figures4.12and4.13show thatthenodaloscillations
arenearlycompletelyminimizedfor eachinsertiontime. Theonly drawbackto thepre-stretching
methodis that its effectivenessis diminishedat greaterstresslevelsduring insertion.Figure4.14
shows the amplitudeof the oscillationsasa function of the local stresslevel, at the time of in-
sertion.Thenodalamplitudesarerepresentedby theamplitudeof the tractions,normalizedwith
themaximumcohesivestress.For boththe“horizontal” and“mixed” testcases(presentedearlier)
theoscillationsaresignificantlyminimizedasaresultof pre-stretching,althoughthe“mixed” case
retainsgreateroscillationsafterthepre-stretchingthenthe”horizontal” case.
0.0 0.1 0.2 0.3 0.4 0.50.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
stress insertion level (σins
/ σmax
)
trac
tion
osci
llatio
n am
plitu
de /
σ max
blind insertion (horizontal)blind insertion (mixed)insertion with pre−stretch (horizontal)insertion with pre−stretch (mixed)
Figure4.14Effectof blind insertionvspre-stretchingon theamplitudeof thetractionoscillationsfor increasingstressinsertionlevels,for the“horizontal” and“mixed” cases.
83
4.2.2 Insertion Results
In the analysissectionabove, we have found that blind insertionof cohesive elementsintro-
ducesoscillationsof the local nodes.And, althoughdampingwasableto minimize theseoscil-
lations,it wasnot theoptimalmethodbecauseof its cumbersomeimplementationrequiringtrial
anderror to find thedampingcoefficients. Insteadwe have developeda methodwherethecohe-
siveelementsarepre-stretchedduringtheir dynamicinsertion.This methodis ableto completely
minimizetheoscillationswhile maintainingtheaccuracy of thesolution.
Wenow applythis dynamicinsertionmethodto severalproblemsby usingtwo differentcohe-
sive elementselectioncriteria. Thesecriteriaareusedto determinetheoptimal locationandtime
for insertionof cohesiveelements.Thefirst is basedon a boundingbox approachwhereall edges
within a growing boundingbox aremadecohesive. The secondusesa stresscriteria, basedon
theaveragevolumetricstresses,to determineif a particularedgeshouldhave a cohesive element
inserted. We presentboth of thesemethodsbelow, aswell as the resultsof their applicationto
variousproblems.
4.2.2.1 Bounding Box Insertion
Theboundingboxmethodreliesonachangingboundaryto selectcohesiveedges.At periodic
intervals during the simulation, the boundingbox methodcalculatesthe farthestextentsof all
failing cohesive elementswithin the domain. Wherea failing elementis definedasonewhose
strengthparameteris below the initial value,but hasnot yet reachedzero. The initial bounding
box is theincreasedslightly andall non-cohesiveedgeswithin this new box aremadecohesive.
As a testof theboundingbox method,we have selectedthemodeI crackproblempresented
in Figure 4.15. We apply an appliedvelocity of 0 G 25 mH s along the left and right boundaries.
The bulk materialis PMMA with a Young’s modulusE D 3 G 24 GPa, Poisson’s Ratio ν D 0 G 35,
anddensityρ D 1190kgH m3. Thecohesive elementshave a maximumstressσmax D 32G 4 MPa,
initial strengthparameterSinit D 0 G 995andanormalandtangentialcritical separationsof ∆crit ON D∆crit O T D 2 G 2 M 10N 5 m. Thedomainis meshedinto 4043nodes,11857edges,and7815volumetric
elements.Theresultingcritical timestepfor theproblemis reducedto ∆t D 2 G 8 M 10N 9 s.
84
Velocity Velocity
0.02 m
0.02 m
Figure4.15Schematicof amodeI crackproblem.
Usinga PentiumIII, 600MHz,750MbRAMprocessor, runningMandrake Linux 7.2, thesim-
ulationswererun for 72000time steps- or approximately0 G 0002seconds.For thereferencesim-
ulation,cohesive elementsarepresenteverywherein thedomainfrom time t D 0. Thebounding
boxsimulationrequiresonly a few initial cohesiveelementsin thevicinity of thecracktip, sothat
failurecanbegin. Theselectiontestoccursevery500timestepsatwhichtime theboundingbox is
scaledupby 4 characteristiclengthsin eachof theprincipaldirections.Both theselectioninterval
andscalingfactorscanbe selectedto bestfit a givenproblem. Decreasingthe selectioninterval
increasesthe frequency of the boundingbox selections,which resultsin an increasedcomputa-
tional time aswell asan increasein cohesive insertions.Thesizeof thescalingfactoris directly
proportionalto thenumberof new cohesive elementsthatwill be insertedduring eachbounding
boxselectioncycle. As aresult,if thisscalingfactoris large,thedomaincanbecomequickly filled
with cohesiveelements.
Thetiming resultsfor theboundingbox selectioncasearepresentedin Table4.5 for thecohe-
siveandinternalforcesubroutines,themainsolutioncodeandthetotalsimulation.Fromthistable
we canseethattheboundingbox methodsavesnearly65%of thetime neededto solve themode
I crackproblem.A majorportionof thesavings is dueto thedecreasednumberof internalforce
85
calculations,which savedapproximately42%of the total time. This canbeseenin Figure4.16,
wherethe numberof cohesive elementspresentin the domainwasvery limited for mostof the
simulation.Themajor influx of new cohesive elementsoccurredat approximatelythesametime
asthemaincrackbeganto form. Thesenew cohesiveelementsextendedthecohesive failurezone
allowing for theformationandpropagationof thiscrack.Figure4.17showsthegrowth in thecrack
lengthoccurringat nearlythreequartersof theway throughthesimulation- immediatelyafterthe
cohesive elementinsertions.This figurealsoshows how well theboundingbox selectionmethod
wasableto trackthecracktip distanceover time.
Thefinal solutions,at time t D to P 72000∆t, for boththereferencecaseaswell asthebound-
ing box selectioncasearepresentedin Figures4.18and4.19. In both figures,the non-cohesive
edgesarerepresentedby thin lines, while the cohesive edgesor elementsaredark. Any failing
cohesive elementsarerepresentedby a dashedbold line, andcompletelyfailedonesarebold. In
the boundingbox selectioncase,we canseethat the part of the upperregion is free of cohesive
elements.In addition,thefinal solutionappearsto havemany morefailing cohesiveelements,both
nearthecrackitself, aswell asin fringe regions. This is mostlikely theresultof theoscillations
thatoccurasapartof theinsertion.Sincetheboundingboxmethodselectsnew cohesiveelements
basedon any neighboringcohesive elements,certainregionsmay be underhigh stressalthough
havenoexistingcohesiveelementsin theirvicinity. As aresult,whenacohesiveelementis finally
insertedinto oneof theseregions,evenpre-stretchingis not ableto compensatewell for thehigh
stressesalreadypresent.
Subroutine ReferenceCase[s] BoundingBox Case
Rco 3013.84 619.30Rin 1035.90 794.97Main 1645.03 561.36Total 5749.17 2040.19% Total Savings 65%
Table 4.4 Mode I casetiming results,in seconds,for the referenceandboundingbox insertioncases.
86
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 10−4
0
2000
4000
6000
8000
10000
12000
time (s)
# of
Coh
esiv
e E
lem
ents
referencebounding box
Figure 4.16 Numberof cohesive elementspresentin the domainover time for boundingboxinsertion.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 10−4
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
time (s)
crac
k tip
dis
tanc
e (m
)
referencebounding box
Figure 4.17ModeI casecracktip distanceversustime.
87
Figure 4.18 Mode I referencecasewith cohesive elementspresentfrom the beginning of thesimulation(10xexaggeration).
Figure4.19ModeI boundingboxsolution(10xexaggeration)(edgekey: thin = normaledge,dark= cohesiveelement,dashed= failing cohesiveelement,bold= failedcohesiveelement).
88
4.2.2.2 StressBasedInsertion in L-Angle Specimen
Although, theboundingbox cohesive elementselectionmethodis capableof large computa-
tionalsavings,themethodhassomedrawbacks.Thismethodis notselfstarting,insteadit requires
somecohesive elementsto be presentin regionswherecohesive failure is expected. For fairly
predictableproblems,it canprovide goodresults,but whenthe failure regionsarenot known a
priori , themethodis not optimal. An improvedinsertionmethodusestheaveragestressesof the
neighborvolumetricelementsto determineif theinterfacebetweentheseelementsshouldbemade
cohesive. In effect, this allows us to begin a cohesive elementfree solutionandonly insert the
elementsasthestressesbuild to somepredefinedlevels- definedby Equation2.34.
As a testof the stressbasedinsertionmethod,we usean L-angleproblempresentedin Fig-
ure 4.20. The top andright boundariesarefixed in placeanda vertical velocity of 1 G 25 mH s is
placedin shearalongtheleft boundary. Thebulk materialof thedomainis PMMA with aYoung’s
ModulusE D 3 G 24 GPa, Poisson’s Ratio, ν D 0 G 35, anddensityρ D 1190kgH m3. The cohesive
elementshave a maximumstressσmax D 32G 4 MPa, initial strengthparameterSinit D 0 G 995anda
normalandtangentialcritical separationsof ∆crit ON D ∆crit O T D 2 G 2 M 10N 5 m. Thedomainis dis-
cretizedinto 3263nodes,9531edgesand6269volumetricelements.Takinginto accounttheinsta-
bility of thecohesiveelementsto beinserted,thecritical timestepis reducedto ∆t D 3 G 0 M 10N 9s.
We run four differentsimulationsfor a durationof 60000time stepsor 0 G 00018s, with the
stressinsertionselectionoccurringevery500timesteps.Thefirst representsthereferencesolution
wherecohesive elementsareinsertedeverywherein the domainat the start. The otherthreeuse
thestressbasedinsertionmethodfor stresslevel of 15%,30%and45%,respectively.
In orderto verify theaccuracy of thevarioussolutionsweobservethecracktip distanceversus
time, presentedin Figure4.22. Fromthis figure,we canseethat thecracktip distance,andindi-
rectly thespeed,arevery closeto thereferencesolution.Furthermore,from Figures4.23through
4.26,wecanseethatthecrackprofilesat theendof thesimulationareverysimilar. In addition,to
thecrackprofiles,we canseethat thecohesive elementstendto concentratein thehigh stressre-
gionswith thefewestelementspresentfor the45% stressinsertionlevel. Eventhoughweachieve
thegreatestsavings for the largerstressinsertionlevels, theseresultscontaingreaterinstabilities
89
Velocity
0.02 m
0.02 m
0.01 m
0.01 m
Figure 4.20Schematicrepresentationof L-angletestspecimenwith boundaryconditions.
in the solution. This is visually apparentby the greaternumberof failing cohesive elementson
thefringesof thedomain- representedby dashedlines. For lower stresslevels,aswell asfor the
referencesolution,thefailing cohesive elementstendto belimited to only theimmediatevicinity
of thecrack.
The timing resultsfor thereferenceandstressinsertioncasesarepresentedin Table4.5. The
largestsavings,of 76%,occursfor thestressinsertionof 45%,which usesthe fewestnumberof
cohesiveelementsto obtainthesolution,asseenin Figure4.21.
Subroutine ReferenceCase[s] 15% 30% 45%
Rco 2070.92 988.10 403.28 105.45Rin 773.04 690.75 612.00 536.14Main 1273.18 790.65 465.09 261.64Total 4150.37 2615.30 1585.66 980.68% Total Savings 37% 62% 76%
Table4.5Anglecasetiming results,in seconds,for variousstressinsertionlevels.
90
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
x 10−4
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
time (s)
# of
Coh
esiv
e E
lem
ents
reference15%30%45%
Figure4.21Numberof cohesiveelementspresentin thedomainovertimefor variousstressinser-tion levels.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
x 10−4
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
time (s)
crac
k tip
dis
tanc
e (m
)
reference15%30%45%
Figure 4.22L-anglecasecracktip distanceversustime, for variousstressinsertionlevels.
91
Figure 4.23 L-angle referencecasewith cohesive elementspresentfrom the beginning of thesimulation(10xexaggeration).
Figure 4.24L-anglecasewith stressbasedcohesiveelementinsertionfor a 15%stresslevel (10xexaggeration)(edgekey: thin = normaledge,dark= cohesive element,dashed= failing cohesiveelement,bold= failedcohesiveelement).
92
Figure 4.25L-anglecasewith stressbasedcohesiveelementinsertionfor a 30%stresslevel (10xexaggeration).
Figure 4.26L-anglecasewith stressbasedcohesiveelementinsertionfor a 45%stresslevel (10xexaggeration)(edgekey: thin = normaledge,dark= cohesive element,dashed= failing cohesiveelement,bold= failedcohesiveelement).
93
4.2.2.3 Stress-basedInsertion in Vertical Interface Specimen
In orderto moreclearlyobservetheadaptivecapabilitiesof ourdynamicinsertionalgorithmwe
haveselectedasimplemodeI problemcomposedof two differentmaterialsseparatedby avertical
interfaces,as seenin Figure 4.27. The bulk materialof the pre-notchedregion hasa Young’s
modulusE1 D 3 G 24 M 109Pa andthemaximumnormalandshearstressfor its cohesive elements
is σmaxO 1 D 3 G 24 M 107Pa. Thesecondregion is 100timesstrongerwith E2 D 100E1 andσmaxO 2 D100σmaxO 1, while theinterfacecohesiveelementsareweakenedby 100times.Figures4.28through
4.30 are snapshotsof the solution at varioustime stepsduring the 72000time stepsimulation
usingthe45%stressinsertioncriteria. Fromthesefigureswe canseethebuild-up of thecohesive
elements- andconsequentlythestresses- bothat thecracktip aswell asfarfield alongthevertical
interface.As themaincrackbeginsto propagatethroughthesolution,thebuild-upof stressesnear
theinterfacecausesit to delaminate,prior to thearrival of themaincrack.But oncethemaincrack
finally reachesthe interface,it becomestrappedandgrows in bothdirectionsalongthe interface
till completefailureof thesystemoccurs.
Velocity
Velocity
0.016 m
Inte
rfac
e
0.014 m 0.006 m
E = 3.24e9 Paσmax = E / 100
σ / 100max
100 σmax
Figure4.27Schematicrepresentationof interfacetestspecimenwith boundaryconditions.
94
Figure4.28Deformationafter5000timestepsfor stressinsertionof 45%(10xexaggeration).
Figure4.29Deformationafter17500timestepsfor stressinsertionof 45%(10x exaggeration).
Figure 4.30 Deformationafter 25500time stepsfor stressinsertionof 45% (10x exaggeration)(edgekey: thin = normaledge,dark= cohesive element,dashed= failing cohesive element,bold= failedcohesiveelement).
95
4.2.2.4 Stress-basedInsertion in Angled Interface Specimen
As yet anothertest of the stressbasedinsertionmethod,we attemptto simulatethe effect
of crack propagationanddeflectionat interfacesin homogeneousmaterials. This analysisis a
numericalexampleof recentwork performedby Xu, HuangandRosakis(2001). Their research
hasshown thataninitial crack,undermodeI loading,propagatesatvariousspeedstowardsinclined
interfacesof variousstrengths.Dependingontheinterfacestrengthaswell astheinterfacialangle,
thecrackmaybecometrappedalongtheinterfaceor simply passright throughit.
Forourcomparison,weuseaninterfacialangleof 60degreesandapplyashearvelocityloading
of 1 G 6 mH s alongthe left sidesof thespecimen,aspresentedin Figure4.31. Thebulk materialis
Homalite-100with Young’smodulusE D 3 G 45 GPa, densityρ D 1230kgH m3, andPoisson’s ratio
of ν D 0 G 35. Thecohesiveelementsusedin thebulk materialaregiventhepropertiesof Homalite-
100presentedin Table4.6.
0.2 m
0.12 m
Velocity
Velocity
60 degrees
Homalite−100
Homalite−100
Inte
rface
Figure4.31Schematicrepresentationof interfacetestspecimenwith boundaryconditions.
96
The simulationis run for 70000time steps(∆t D 7 G 0 M 10N 9 s), for both a weakinterfaceof
Loctite-384andastronginterfaceof Weldon-10.Theconstitutivepropertiesof thesematerialsare
alsopresentedin Table4.6. anda stronginterface.
Homalite-100 Loctite-384(weak) Weldon-10(strong)KobayashiandMall (1978) Xu et al., (2001) Xu et al., (2001)
σmax? MPaA 11.0 7.74 6.75τmax? MPaA 25.0 22.0 7.47GIc ? J H m2 A 250.0 199.7 41.9GI Ic ? J H m2 A 568.0 568.0 46.4
Table 4.6 Constitutive cohesive elementpropertiesof thebulk materialandtheweakandstronginterfaces.
Theexperimentalresultsfor boththeweakandstronginterfaces,describedabove,haveshown
that the crackbecomestrappedalongthe interfacefor a shortdurationbeforeturning backinto
thebulk material.Theweakinterfacetrapsthecrackfor a longerdistanceaswe have verifiedin
Figure4.34,while thestrongerinterfacealmostimmediatelyturnsthecrackbackin to thesystem
(seeFigure4.35). Close-upsof thesecracksarepresentedin Figures4.36and4.37for theweak
andstronginterfaces,respectively.
Figures4.32and4.33presentthecracklengthandcrackspeedfor theweakinterfaceproblem.
Fromthesefigures,we canseethat the initial modeI cracktravelsat nearconstantspeeduntil it
reachestheinterface.At this time, it becomesa mixed-modeinterfacialcrackwhosecrackspeed
is increasedwhile travelingalongtheinterface.Comparisonof theseresults,with thosepresented
in Figure13by Xu etal. (2001),showsgoodagreementto thegeneralprofilesof thecracklengths
andspeedcurves.Sincewewereunableto matchall thenecessaryparametersof theexperimental
setup,we usedour bestjudgmentto duplicatetheproblem.Theactualcrackspeedsfoundin our
resultsarenearlytwice asgreatasthosepresentedin theexperimentalresults.This is mostlikely
dueto the applicationof the boundaryconditionsusedto mimic the effect of the impacton the
wedge.
97
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
x 10−4
0
0.02
0.04
0.06
0.08
0.1
0.12
time (s)
crac
k le
ngth
(m
)
mode I incident crackmixed−mode interfacial crack
Figure 4.32Cracklengthhistoryfor aweak60degreeinterface.
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
x 10−4
300
400
500
600
700
800
900
1000
1100
time (s)
crac
k sp
eed
(m/s
)
mode I incident crackmixed−mode interfacial crack
Figure4.33Crackspeedhistoryfor a weak60degreeinterface.
98
Figure 4.34 Mode I crack trappedalong the weakLoctite-384interfacefor a 45% stress-basedinsertion(no exaggeration)(edgekey: thin = normaledge,dark = cohesive element,dashed=failing cohesiveelement,bold= failedcohesiveelement).
99
Figure 4.35ModeI cracktrappedalongthestrongWeldon-100interfacefor a 45%stressbasedinsertion(no exaggeration)(edgekey: thin = normaledge,dark = cohesive element,dashed=failing cohesiveelement,bold= failedcohesiveelement).
100
Figure4.36Close-upof crackregionalongaweakinterface(no exaggeration).
Figure 4.37Close-upof crackregionalonga stronginterface(no exaggeration).
101
4.2.2.5 Insertion Inter val Selection
Thusfar, eachof our simulationsuseda 500time stepinterval betweennew cohesiveelement
insertions. This valuewaschosenarbitrarily andcanbe changedto bestfit the given problem.
Varying this interval effectsboth thesimulationtime andtheaccuracy of thesolution. Table4.7
presentsthesolutiontimesfor theL-anglecase,usingstressinsertionof 30%at intervalsof 100,
500,1000,5000and10000timesteps.As theinterval decreases,thetotal solutiontime increases.
This time increaseis a resultof theincreasednumberof selectionsaswell asa greaternumberof
file outputs,which currentlyoccurafterevery insertion.Furthermore,theaccuracy of thesolution
alsoincreaseswith a smallerinterval sincethethe local stressesarenot ableto vary significantly
betweenthecohesive insertions.We have deducedthis throughobservationof thedistribution of
failing cohesiveelements;therearemany morefailing elements,representinggreaterinstabilities,
as the insertioninterval increases( asseenin Figures4.38 through4.39 ). In fact, in part b of
Figure 4.39, thereare many failing cohesive elementsaroundthe crack but very few cohesive
elementsdirectly aheadof the cracktip. The 10000time stepinsertioninterval doesnot allow
theprogramto insertenoughcohesiveelementsaheadof thecracktip soaccountfor thespeedof
thecrack.As a result,thecrackreachestheendof thecohesive region prior to thenext insertion,
causingit to stopabruptly. As new elementsareinsertedaheadof this cracktip, it is onceagain
ableto continuepropagatingthroughthesystem.Unfortunately, theperiodiccrackarrestingresults
in an inaccuratesolution as presentedin the figure. Overall, the numberof cohesive elements
presentovertime is alsoslightly decreasedfor thelargerinsertionintervals,asseenin Figure4.40.
This possiblyeffectstheaccuracy of thesolutionsincefewer elementsarepresentin thesystem,
althoughthedifferenceareonly about5%.
Insertion Interval 100 500 1000 5000 10000
Total Time [s] 1609.27 1585.66 1300.31 1220.11 1148.40
Table4.7Total simulationtimesfor insertionintervalsof 100,500,1000,5000and10000∆t
102
Figure 4.38L-anglereferencecasewith cohesive elementsinsertedevery (a) 100 time steps(b)500timestepsat the30%stresslevel (10x exaggeration).
Figure 4.39L-anglereferencecasewith cohesive elementsinsertedevery (a) 1000time steps(b)10000timestepsat the30%stresslevel (10xexaggeration)(edgekey: thin = normaledge,dark=cohesiveelement,dashed= failing cohesiveelement,bold= failedcohesiveelement).
103
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
x 10−4
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
time (s)
# of
Coh
esiv
e E
lem
ents
reference1005001000500010000
Figure 4.40Numberof cohesive elementspresentover time for insertionintervals of 100, 500,1000,5000and10000.
1.2 1.4 1.6 1.8
x 10−4
3000
3500
4000
4500
5000
5500
6000
time (s)
# of
Coh
esiv
e E
lem
ents
reference1005001000500010000
Figure 4.41Close-upof thethreeintervalspresentedin Figure4.40.
104
4.2.2.6 Dynamic Insertion Combinedwith Subcycling
The multi-time stepnodal subcycling algorithm,presentedin the beginning of this chapter,
hasshown to generatesignificantcomputationaltime savings,while maintaininga fairly accurate
solution. The solutiondoesbegin to degradethoughfor excessive time stepdifferencesin the
subcycledandnon-subcycledregions,aswell asfor smallregion ratios.In this sectionwepresent
theresultsof combiningboththesubcycling anddynamicinsertionalgorithms.
As atestcasewepresentthemodeI crackspecimenshown in Figure4.42whosebulk material
haspropertiesof PMMA. The critical time stepfor this problemis ∆t D 2 G 8 M 109 s, which is a
1H 30 reductionof theproblem’s Courantcondition. In orderto initiate thesubcycling portionof
thesimulation,wepre-insertcohesiveelementsin asmallregion nearthecracktip. This region is
thereforegivena timestepof 1∆t while thelargerouterregionhasa timestepm∆t.
Thesimulationwasrunfor 60000timestepsfor a referencesolutionhaving cohesiveelements
presenteverywhereandnosubcycling, for asecondreferencesolutionusinga45%stressinsertion
andfor threesolutionsusingbothdynamicinsertionandsubcycling with parametersof m D 4 @ 10
and14. The final crackprofilesfor eachof the above simulationsarepresentedin Figures4.44
through4.46. From thesefigureswe can seethat the combinedsolutionsclosely matchboth
referencesolutions.Only them D 14 combinedsolutionappearsto havea differentfinal solution.
Carefulobservationof this solutionshows that therearelarge instabilitiesnearthe cracktip - as
seenby the large numberof failing cohesive elements.Theseinstabilitiesaswell as the effect
of a large time stepin thenon-subcycledregion have mostlikely compoundedtheerrorspresent
in the solution. In addition,at the endof the simulation,the region ratio is muchlower thanits
initial value- asseenin Figure4.43.This increasein subcyclednodesincreasestheinstabilitiesas
discussedin thesubcycling sectionearlierin this chapter.
Although the solution losessomeaccuracy for the larger subcycling parameters,significant
savings can still be achieved for the lower values- as presentedby Table4.8. The % savings
is presentedfor both the savings with respectto the referencesolutionusingcohesive elements
everywhereat the beginning aswell as the referencesolution of only dynamicinsertionat the
45% stresslevel - the latter savings is presentedin parenthesesin the table. We canseethat the
105
majorsavingsfor thecombinedsolutionsis adirectresultsof thedynamicinsertionandonly small
overall percentageis obtainedthroughsubcycling.
Velocity Velocity
0.02 m
0.02 m
∆1 t
∆tm
Figure 4.42 Schematicof a modeI crack problemusing nodal subcycling and dynamicstressinsertionof 45%.
Subroutine Reference Dynamic,m = 1 m = 6 m = 10 m = 14
Rco 2627.37 76.71 95.23 88.18 79.20Rin 870.83 614.38 247.38 167.45 134.03Main 1425.46 251.39 344.63 291.36 271.16Total 4973.63 1005.79 739.50 617.36 528.33% Total Savings 79%(0%) 85%(26%) 87%(38%) 89%(47%)
Table 4.8 Timing results(in s) for combineddynamicstressinsertionof 45%with subcycling ofm D 6 @ 10 and14. Time savings from thereferencesolutionis givenfirst while thesavings fromthedynamicinsertioncaseis presentedin parentheses.
106
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
x 10−4
0
1
2
3
4
5
6
7
8
time(s)
regi
on r
atio
m = 6m = 10m = 14
Figure4.43Region ratioover time for thesubcycling solutionswith m D 6 @ 10and14.
Figure 4.44Referencesolutionwith cohesive elementspresenteverywherein the domainat thebeginningof thesimulation.No subcycling is used(10xexaggeration).
107
Figure 4.45(a) Solutionhaving only dynamicinsertionat 45%of thelocal stresswith no subcy-cling. (b) Combineddynamicinsertionwith subcycling, m D 6 (10x exaggeration).
Figure 4.46(a) Combineddynamicinsertionwith subcycling, m D 10 (b) Combineddynamicin-sertionwith subcycling,m D 14(10xexaggeration)(edgekey: thin = normaledge,dark= cohesiveelement,dashed= failing cohesiveelement,bold= failedcohesiveelement).
108
4.3 Parallelization UsingCharm++
As problemsgrow computationally, requiringmuchmorememoryresourcesor time to solve
them,we mustturn to usingmultiple processorsto solve them.No only arewe thenableto solve
larger and larger problemsbut we can also take advantageof the multiple processorsto solve
existingproblemsmorequickly.
For ourresearch,wehaveusedtheCharm++ FEM framework in orderto parallelizeourcode.
This framework is ideal for standardCVFE codesand thereforehasbeeneasily integratedinto
our dynamicversion.All of our simulationsarerun on PentiumIII, 600MHzprocessorsrunning
MandrakeLinux.
As a testof theparallelizationwe selectedtheL-angleproblemusingdynamicinsertionat the
30%stresslevel. Thesimulationswererun for 60000time stepson 1, 2, 4 and6 processors.The
dynamicinsertioninterval wasfixedat 500time steps.As expected,thesolutionsfor eachof the
simulationswereidentical,althoughthespeedupsdecreasedwhenmoreprocessorswereusedas
seenin Figure4.47. This decreasein speedupis primarily dueto the serialportion of the code.
Sincemostcodesarenot fully parallel,ratherthey have someserialportions,therelativesolution
times for the serial andparallel portionsof the codesgrow closeras the numberof processors
increases.In our problem,the serialportion correspondsto the initial readingof input files as
well astheperiodiccohesiveelementinsertionsandmeshrepartitioningoccurringevery500time
steps.Thespeedupscanbeincreasedif theinsertioninterval is increasedandif thepre-processing
of input datacanbeimproved.
109
1 2 4 61
2
3
4
5
6
# of processors
spee
dup
ideal speedupparallel
Figure 4.47Speedupresultsfor L-anglecaseusing1, 2, 4, and6 processors.
110
4.4 Conclusions
In this chapterwe presentedtheresultsof applyingthemulti-time stepnodalsubcycling, dy-
namiccohesive elementinsertionandcodeparallelizationmethodsto 2-D problems. From our
subcycling resultswe have foundthatthemostimportantfactorsleadingto anaccurateandstable
solutionarethesubcycling parameterandtheregionratio. Thegreatesttimesavingsis achievedat
highersubcycling parametervaluesalthoughthestability of thesolutiondecreases.Furthermore,
the region ratio mustbe at least2 : 1 to ensurethat thecomputationalsavingsoffsetsthecostof
thesubcycling algorithmimplementation.
Thedynamicinsertionalgorithmhasalsoprovento beextremelysignificant.Usingbothpre-
stretchingcombinedwith stressbasedcohesiveelementinsertionallowsusto generatethegreatest
computationalsavingswhile still maintaininganaccuratesolution.
Finally, weappliedtheCharm++ parallelizationtechniqueto ourcode.Theresultshaveshown
promisingspeedupsalthoughimprovementsin the pre-processingand meshrepartitioningcan
achievebetterresults.
111
CHAPTER 5
CONCLUSIONS AND FUTURE WORK
Thestandardcohesivevolumetricfinite element(CVFE)methodhassuccessfullybeenapplied
to avarietyof dynamicfractureproblemsinvolving theformationandpropagationof cracks.How-
ever, themethodis computationallyinefficient for problemscontainingmany degreesof freedom
aswell many failurepaths.In its currentform, themethodrequiresthatdynamicfractureproblems
beevenlydiscretizedwith smallvolumetricelementsall interconnectedwith theinterfacial(cohe-
sive) elements.Thepresenceof thecohesive elementsrequiresnot only theduplicationof nodes
andthereforeanincreasein thenumberof degreesof freedom,but alsoa reductionin thecritical
time stepof the domain. This time stepreductionis necessaryto ensurethe cohesive element
stabilityandtheaccuracy of thesolution.
The purposeof this thesiswas to develop an adaptive versionof the CVFE schemewhich
addressesits critical issues.Wefirst investigatedtheuseof differenttimestepsin differentregions
throughthemulti-time stepnodalsubcycling algorithm. Next we developeda dynamicinsertion
algorithmto insertcohesive elementsanywherein thedomainandat any time. This allows usto
begin the simulationcohesive elementfree andonly inserttheseelementsasnecessary. Finally,
we implementeda parallelizationtechniquein orderto decreaseour simulationtime or increase
theproblemsize.We initially appliedall of themethodsandalgorithmsto 1-D problemsin order
to gainabetterunderstandingof theresultswhile usingverysimpleproblems.We thenmovedon
to 2-D problemswhereboththeproblemcomplexity andsolutiontimesincreased.Wepresentthe
conclusionsfrom our analysisin thenext section
112
5.1 Conclusions
Thefirst methodappliedto thestandardCVFE schemewasthemulti-time stepnodalsubcy-
cling algorithmpresentedby Smolinski(1989).Thisalgorithmallowsusto usedifferenttimesteps
in differentregionsto solve thetime loopequations.Theprimarysavingsis gainedby approxima-
tionsthatoccurin thelarger time stepregions.Resultsfrom bothour 1-D and2-D analyseshave
shown thatsubcycling is a goodapproximationmethodwhenthetimestepof any regiondoesnot
exceedits local Courantcondition. As theratio of thenon-subcycledto subcycledtime stepsin-
creasestheapproximationseventuallycauseinstabilitiesin thesolutionresultingin its divergence.
Theoptimal time stepratio selectedshouldbeno greaterthanhalf of thecritical time stepof the
largestnon-subcycleddomain.Furthermore,theratioof thenumberof non-subcycledto subcycled
nodes- or region ratio - alsoplaysanimportantrole in thestabilityof thesolutionandin thetime
savings. In orderto maintainthelocal stability, this region ratio shouldbegreaterthan1:1 sothat
many morenon-subcycledthansubcyclednodesexits. This will alsoincreasethecomputational
time savingssincemany morenodeswill beapproximatedduringeachtime cycle. If the ratio is
toosmalltheimplementationof thesubcycling algorithmhasshown to offsetany savingsachieved
throughits use.
Thedynamiccohesive elementinsertionalgorithmthatwe have developedallows usto insert
cohesive elementsanywherein the domainand at any time in the simulation. Eachinsertion
requiresthat nodalandelementdataandconnectivity informationbe adjustedto insurethat all
insertionsarevalid. Furthermore,in orderto maintaintheequilibriumof thesystemwe mustalso
conserveboththemassandmomentumof thesystemby recalculatingthenodalmassdistributions
andduplicationdisplacement,velocityandaccelerationinformationasnecessary.
Initially weusedblind insertionof cohesiveelementswithin thedomain.This insertionsimply
placeda zero-thicknesscohesive elementin betweenthevolumetricelementsandduplicatedthe
nodal information. This insertionresultedin significantnodaloscillationsin both the displace-
ments,velocitiesandaccelerations.Theamplitudeof theseoscillationswasdirectly proportional
to thelocal stressat thetimeof insertion.Weappliedlineardampingto attemptto minimizethese
oscillations.Thoughdampingwasableto minimizetheoscillationsat low stresslevels,it wasless
113
effective at higherlevelsduringwhich mostinsertionwill take place. As a result,we developed
a pre-stretchingtechniquebasedon the traction-separationlaw for thecohesive elements.Using
this techniquewedeformthecohesiveelementsat thetimeof insertionbasedonthelocal tractions
appliedto their interfaces.The elementsaretheninsertedin a statethatmorecloselyresembles
the statethey would be in had they beeninsertedat the start of the simulation. Pre-stretching
hasshown to greatlyminimize thenodaloscillationsat any insertiontime while maintainingthe
accuracy of thefinal solution.
Using the pre-stretchingtechniquewe then investigatedusingdifferentselectioncriteria for
placingthecohesive elementswithin thedomainmostefficiently. First, we useda boundingbox
selectionwherethe cohesive region was encasedin a growing box and all non-cohesive edges
within it wereconvertedto cohesive. This methodwasableto generatesignificanttime savings
but it wasnotveryrobustasit requiredsomestartingcohesiveelementsto definetheinitial bound-
ing box. This limited us to problemswherethe formationof crackscouldbe predicted.Instead,
we thenuseda stress-basedinsertionmethodby which cohesive elementwereinsertedasthe lo-
cal stressreachedsomepredefinedlevel. This methodcanbeusedon any problemanddoesnot
requireany initial cohesive elements.The resultsof applying this methodhave also generated
largecomputationalsavingsfor variousstressinsertionlevels.Althoughasthecritical stresslevel
for insertionis high, thesolutiontendsto bemoreunstablewhile the time savings is not signifi-
cantlymorethanfor smaller, morestableinsertions.A stresslevel of 30%hasshown to provide
significanttimesavingswhile still maintainingtheaccuracy of thesolution.
Finally, we have parallelizedour codeusingtheCharm++ FEM framework. This framework
allows us to easilyparallelizea serial codeusingonly a few directives. We testedour parallel
versiononthesimpleL-angledynamicinsertionproblemusing1, 2, 4 and6 PentiumIII, 600MHz
processors.Althoughwewerenotableto getaperfectspeedup,wedid getverygoodresults.Our
maincostin theimplementationwasdueto thepre-processingof inputfileswhich resultedin less
thenperfectspeedups.
114
5.2 Recommendationsfor Futur eResearch
While someimportantmethodshave beendevelopedandmany encouragingresultspresented
in this research,variousissuesremainto beinvestigated.
Althoughthesubcycling algorithmwasableto generatesometimesavings,its implementation
requiresadditionalmemorystoragerequirementsnot presentin a non-subcycled scheme.Fur-
thermore,the stability of the solution is highly dependenton the time stepratio aswell as the
distribution of the time stepsacrossthe nodal regions. The solutionsmay be morestableif the
subcycledregion is givenabuffer zonewhosethenodesarealsosubcycled,sothatthetransitions
betweenthecritical andnon-criticalregionsarenot sosevere.
Thedynamicinsertionalgorithmcanalsobeimprovedby moreaccuratelyapplyingthenodal
separationsusedin pre-stretching.Underthecurrentimplementation,theseseparationsareaver-
agedfor eachcohesive nodeandequallydistributedacrosseachhalf. Futureresearchmay con-
centrateon applyinga weighteddistribution approachto furtherminimize thenodaloscillations.
Additionally, this methodcanalsobeextendedfor usewith six nodelinearstraintriangles(LST)
in orderto generateevenmoreaccuratesolutions.
As seenin thepreviouschapter, our codeparallelizationusingCharm++ wasnot ableto gen-
eratecompletelyideal speedups,althoughthe resultswerequite promising. Theprimary costof
theparallelizationwasdueto thepre-processingof input dataandserialinsertionportionsof the
code.Parallelizingtheseareascangreatlyimprove theperformanceof thecode.In particular, the
currentserialinsertionrequirestheperiodicassemblyandrepartitioningof theentiremesh.This
interval of this periodicupdatecanbe initially increasedsincemany insertionsarenot required
duringtheinitial stagesof thesimulation,wherethestresseshavenotyet reachedtheirdesiredlev-
els.Furthermore,extendingthedynamicinsertionalgorithmfrom serialbasedinsertionto parallel
canalsoimproveperformancesincenoassemblyandrepartitioningof themeshwouldberequired.
Lastly, wecanalsogeneratesignificantcomputationalsavingsby usingandadaptiveremeshing
technique,asillustratedin Figure5.1. Sincecracksmove throughthesystemsduringthesimula-
tion, theadaptiveremeshingtechniquewouldcontinuallyadjustthecritical region(s)to ensurethat
themostoptimalsolutioncanbeachieved.UsingtheCVFEschemein conjunctionwith theadap-
115
tive remeshingwould requirenot only dynamicinsertionof cohesiveelementsbut alsoa dynamic
removal of failedor unloadedelements.
Figure5.1Adaptivecrackpropagation.
In our initial investigationsof this topic, we have applieda adaptive remeshingtechniqueto a
simple1-D problemshown in Figure5.2 The beamis composedof 21 equalsegmentsof length
1 G 0 m.
Thebeamis similar to thereference1-D beamproblempresentedin Chapter3. We breakthe
11th segmentinto 7 smallerpiecesat time step10000for a 30000time stepsimulation. After
elementbreakup, andcohesive nodeinsertion,the middle of the cohesive region is represented
by node26 for which we plot the velocity in Figure5.3, Comparingthis velocity profile to the
referenceprofile for node15 if thereferencecaseof Figure3.6,we canseethat they matchquite
well, althoughthevelocity for theadaptivemeshingcasearea little oscillatory.
11
11 23 24 25 282726
12
10 11 12
= normal node = cohesive node
12
Figure5.211thelementbrokeninto 7 pieces.
116
0 10 20 30 40 50 60 70 75−0.015
−0.01
−0.005
0
0.005
0.01
0.015
time (s)
velo
city
(m
/s)
Figure5.3Velocityprofile for new node#26afterelementbreakup at timestep10000.
Extendingthisinvestigationinto 2-Drequiresmorecomplex issuessuchaselementconformity,
databasemanagementandtime stepstability. Also, elementremoval presentssimilar, if not more
difficult, issuesbut combinedwith thedynamicinsertionandmulti-time stepnodalsubcycling it
cansignificantlydecreasethecomputationaleffort for solvingdynamicfractureproblems.
117
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