# Comparison of the implicit and explicit finite element methods using crystal plasticity

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ComparisonoftheimplicitandexplicitniteelementmethodsusingcrystalplasticityF.J.Harewooda,P.E.McHugha,b,*aNationalCentreforBiomedicalEngineeringScience,NationalUniversityofIreland,Galway,IrelandbDepartmentofMechanicalandBiomedicalEngineering,NationalUniversityofIreland,Galway,IrelandReceived28February2006;receivedinrevisedform28April2006;accepted1August2006AbstractThe implicit nite element (FE) method can encounter numerical diculties when solving non-linear quasi-static problems. The iter-ative approach employed may have trouble achieving convergence in analyses with a highly non-linear material behaviour, such as a crys-tal plasticity constitutive model. In the case of the explicit FE method the solver equations can be solved directly to determine the solutionwithoutiteration, thusprovidinganalternative,morerobustmethod. Inthisstudy,arate-dependentcrystalplasticityalgorithmwasdeveloped for use with the explicit FE package, ABAQUS/explicit. The subroutine and an equivalent implicit version were used in a seriesofcomparativeboundaryvalueproblemanalyses.Thesuitabilityoftheimplicitandexplicitsolverstovariousloadingconditionswasassessed and multiple processor speedup rates were also investigated. The results of the study showed that, for simpler loading conditions,the implicit method had a shorter solution time. In the case of loading conditions involving contact, the explicit method proved to be thepreferable choice. The explicit method displayed constantly high levels of parallelisation eciency compared to the implicit method foranalyses solvedusingmultiple processors. Inconclusion, althoughthe implicit FEmethodis traditionallyfavouredwhensolvingquasi-static problems, it is important to recognise the advantages that the explicit method has in solving certain loading conditions.2006ElsevierB.V.Allrightsreserved.Keywords: Implicitniteelementmethod;Explicitniteelementmethod;Crystalplasticitytheory;Multipleprocessorparallelisation1.IntroductionTheniteelement methodisapopularcomputationaltoolusedinengineeringresearchandindustrialdesign.Inthe eld of solid mechanics, and specically non-linearquasi-static problems, nite element equation solutionmethods can generally be classed as either implicit or expli-cit andaretypicallysolvedincrementally. Intheimplicitapproachasolutiontothesetofniteelementequationsinvolvesiterationuntil aconvergencecriterionissatisedfor eachincrement. The nite element equations intheexplicitapproacharereformulatedasbeingdynamicandinthisformtheycanbesolveddirectlytodeterminethesolutionat the endof the increment, without iteration.Several studies have beenpublishedcomparing the twoanddiscussingtheirrespectivemerits[1416,29,30,3537].These articles focus on the performance of the two methodsinmetal forming analyses. Rebeloet al. [29] foundtheimplicitmethodtobepreferableinsmaller2Dproblems,whereas theexplicit methodis morerobust andecientforcomplicatedmodelsinvolvingcontact. Thereasonforthis is that the implicit solver can encounter numerical dif-culties in converging to a correct solution during an anal-ysis involving large element deformation, highly non-linearplasticity or contact between surfaces [9,14,29,35]. Thispaper compares the suitability of the two solution methodstovariousmetal deformationanalyseswhenemployingauser-written constitutive model.0927-0256/$-seefrontmatter 2006ElsevierB.V.Allrightsreserved.doi:10.1016/j.commatsci.2006.08.002*Corresponding author. Address: National Centre for BiomedicalEngineering Science, National University of Ireland, Galway, Ireland.Tel.:+35391524411x3152.E-mailaddress:peter.mchugh@nuigalway.ie(P.E.McHugh).www.elsevier.com/locate/commatsciComputationalMaterialsScience39(2007)481494Insolidmechanics, crystal plasticityhas beenusedintwo categories of analysis [19]. The rst is the investigationoftheperformanceofsinglemetalliccrystalsunderdefor-mation. The size scales of these analyses are at the granularmicroscaleandeachmetallicgrainisexplicitlymodelled.They are solved as boundary value problems. The analysisofthemacroscalebehaviourofpolycrystallineaggregatesforms the secondcategoryof analysis. Typically duringlarge-scale deformation, randomly oriented crystals experi-encelatticedistortionandrotationwhichleadstoapoly-crystallinetextureororientationpattern. Thisisachievedusing polycrystalline aggregate constitutive models andresults in material anisotropy (e.g. [4,7,23,27]). The presentstudyfocusesontheformercategory:theimplementationofcrystalplasticitytostudyindividualgrainbehaviour.Boundaryvalue problems employingcrystal plasticityhave been shown to yield accurate macroscopic predictionsof large strain ductility and the point of mechanical failurein small-scale metallic devices [31]. In a constitutive model,suchasthatpresentedinPeirceetal.[26],thestressesarecalculated based on a non-linear strain hardening responsefunction.SuchproblemsnecessitatelargedeformationsoftheFEmeshanddisplayahighlynon-linearmechanicalresponse. For these two reasons, it is of considerable inter-esttoassesstheeectivenessandeciencyoftheexplicitsolutionmethodinsolvingsuchproblems.Crystal plasticity has been used in many studies to modellargedeformationsandstrainlocalisationsinmetalsandmetallic basedmaterials (e.g. [13,19,3134,39]). Unfortu-nately a crystal plasticity constitutive theory is not providedas standard in the commercially available FE software and,as such, it is generally necessary to develop a stress updatealgorithmand implement it in an external user-denedmaterial module. Several researchers have developed stressupdatealgorithmsthatdenesinglecrystal behaviourforusewith commercialniteelementsoftware suchasABA-QUS and ANSYS [1,3,5,7,8,10,12,17,21,22,28]. In additionto the algorithm by Huang [10], a number of authors havedevelopedrate-dependentimplicitalgorithmsforusewithABAQUS (e.g. [5,8,17,21]).Althoughthe majority of algorithms developedhavecapturedrate-dependent mechanical behaviour, therehasalsobeensignicanteortindevelopingrate-independentcrystal plasticity formulations. The essential dierencebetweenthese twotypes of formulations is the presenceof the relationship between the rate of shear strain on eachslip system and the shear stress, as presented by Asaro andNeedleman [2]. Its absence is a source of numerical dicul-ties in boundary value problems using the rate-independentconstitutive formulation. Several researchers have imple-mented rate-independent algorithms successfully to achievemeaningfulresults(e.g.[1,22]).Inthispaperanewcrystal plasticityalgorithmispre-sented. Thisformulationwas developedfor usewiththeFEsolverABAQUS/explicit.Theobjectiveofthepresentworkis tocomparetheperformanceof theimplicit andexplicit solution methods using two equivalent crystalplasticity algorithms. In the study ABAQUS/standardand ABAQUS/explicit are considered. The rate-dependentcrystal plasticity formulation presented by Peirce et al.[26] is used. It is expressed in terms of a stress update algo-rithmand is implemented as a user-dened material subrou-tine for both versions of the code.Theimplicit andexplicit solutionprocedures areout-linedinSections1.1and1.2.InSection2thecrystalplas-ticityformulation ofPeirceet al.[26] is presented. Fortheimplicit analysesthe UMAT, developedbyHuang[10] forimplementation in ABAQUS/standard, is used. Anewsubroutine (VUMAT) was developed by the authors forimplementationinABAQUS/explicit.TheVUMATisbasedonthe Huang UMAT [10]. Bothuser-denedsubroutinesaredescribedinSection3. InSection4themethodsarecomparedinterms of nite deformationanalyses in2Dand3D.Theimpact of parallelprocessingon the compar-ison between the methods is also given in Section 4.Finally,conclusionsaredrawninSection5.1.1.ImplicitsolutionmethodThe word implicit in this paper refers to the method bywhichthestateofaniteelementmodel isupdatedfromtime t tot + Dt. Afully implicit procedure means thatthestateat t + Dtisdeterminedbasedoninformationattime t + Dt, while the explicit method solves for t + Dtbasedoninformationattimet.There are a range of solution procedures used by impli-cit FE solvers.A formof theNewtonRaphsonmethodisthe most common and is presented here. Vectors andmatrices are denoted as underlined. When solving aquasi-staticboundaryvalueproblem, aset of non-linearequationsisassembled:Gu

vBTru dV

SNTt dS 0 1:1whereGisasetofnon-linearequationsinu,anduisthevector of nodal displacements. Bis the matrix relatingthestrainvectortodisplacement. TheproductofBTandthe stress vector, r, is integratedover a volume, V. Nisthematrixofelementshapefunctionsandisintegratedover asurface, S. Thesurfacetractionvector isdenotedbyt. Eq. (1.1) isusuallysolvedbyincremental methods,where loads/displacements are appliedintime steps, Dt,uptoanultimatetime,t.Thestateoftheanalysisisupdatedincrementallyfromtimet totimet + Dt. Anestimationof theroots of Eq.(1.1)ismade,suchthatfortheithiteration:dui1 utDti1 utDti oGutDtiou 1GutDti 1:2where utDtiis the vectorof nodal displacements for theithiteration at time t + Dt. The partial derivative on the right-handside of the equationis the Jacobianmatrixof thegoverningequationsandcanbereferredtoastheglobal482 F.J.Harewood,P.E.McHugh/ComputationalMaterialsScience39(2007)481494stiness matrix, K. Eq. (1.2) is manipulated and inverted toproduceasystemoflinearequations:KutDtidui1 GutDti 1:3Eq. (1.3) must be solved, for each iteration, for the changeinincremental displacements, dui+1. Inordertosolvefordui+1the global stiness matrix, K, must be inverted.Although, this is acomputationallyexpensiveoperation,iteration ensures that a relatively large time incrementcanbeusedwhilemaintainingaccuracyofsolution[9,19].Following iteration i, dui+1has beendeterminedandabetter approximation of the solution has been made,utDti1,throughEq.(1.2).This inturnis usedasthecurrentapproximation to the solution for the subsequent iteration(i + 1).Theaccuracyofthesolutionisdictatedbytheconver-gence criterion where the updated value forG must be lessthan a tolerance value. Complications

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