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Page 1: Comparison of the implicit and explicit finite element methods using crystal plasticity

www.elsevier.com/locate/commatsci

Computational Materials Science 39 (2007) 481–494

Comparison of the implicit and explicit finite element methodsusing crystal plasticity

F.J. Harewood a, P.E. McHugh a,b,*

a National Centre for Biomedical Engineering Science, National University of Ireland, Galway, Irelandb Department of Mechanical and Biomedical Engineering, National University of Ireland, Galway, Ireland

Received 28 February 2006; received in revised form 28 April 2006; accepted 1 August 2006

Abstract

The implicit finite element (FE) method can encounter numerical difficulties 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 solutionwithout iteration, thus providing an alternative, more robust method. In this study, a rate-dependent crystal plasticity algorithm wasdeveloped for use with the explicit FE package, ABAQUS/explicit. The subroutine and an equivalent implicit version were used in a seriesof comparative boundary value problem analyses. The suitability of the implicit and explicit solvers to various loading conditions wasassessed 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 efficiency compared to the implicit method foranalyses solved using multiple processors. In conclusion, although the implicit FE method is traditionally favoured when solvingquasi-static problems, it is important to recognise the advantages that the explicit method has in solving certain loading conditions.� 2006 Elsevier B.V. All rights reserved.

Keywords: Implicit finite element method; Explicit finite element method; Crystal plasticity theory; Multiple processor parallelisation

1. Introduction

The finite element method is a popular computationaltool used in engineering research and industrial design. Inthe field of solid mechanics, and specifically non-linearquasi-static problems, finite element equation solutionmethods can generally be classed as either implicit or expli-cit and are typically solved incrementally. In the implicitapproach a solution to the set of finite element equationsinvolves iteration until a convergence criterion is satisfiedfor each increment. The finite element equations in the

0927-0256/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.commatsci.2006.08.002

* Corresponding author. Address: National Centre for BiomedicalEngineering Science, National University of Ireland, Galway, Ireland.Tel.: +353 91 524411x3152.

E-mail address: [email protected] (P.E. McHugh).

explicit approach are reformulated as being dynamic andin this form they can be solved directly to determine thesolution at the end of the increment, without iteration.Several studies have been published comparing the twoand discussing their respective merits [14–16,29,30,35–37].These articles focus on the performance of the two methodsin metal forming analyses. Rebelo et al. [29] found theimplicit method to be preferable in smaller 2D problems,whereas the explicit method is more robust and efficientfor complicated models involving contact. The reason forthis is that the implicit solver can encounter numerical dif-ficulties 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 methodsto various metal deformation analyses when employing auser-written constitutive model.

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482 F.J. Harewood, P.E. McHugh / Computational Materials Science 39 (2007) 481–494

In solid mechanics, crystal plasticity has been used intwo categories of analysis [19]. The first is the investigationof the performance of single metallic crystals under defor-mation. The size scales of these analyses are at the granularmicroscale and each metallic grain is explicitly modelled.They are solved as boundary value problems. The analysisof the macroscale behaviour of polycrystalline aggregatesforms the second category of analysis. Typically duringlarge-scale deformation, randomly oriented crystals experi-ence lattice distortion and rotation which leads to a poly-crystalline texture or orientation pattern. This is achievedusing polycrystalline aggregate constitutive models andresults in material anisotropy (e.g. [4,7,23,27]). The presentstudy focuses on the former category: the implementationof crystal plasticity to study individual grain behaviour.

Boundary value problems employing crystal 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,such as that presented in Peirce et al. [26], the stresses arecalculated based on a non-linear strain hardening responsefunction. Such problems necessitate large deformations ofthe FE mesh and display a highly non-linear mechanicalresponse. For these two reasons, it is of considerable inter-est to assess the effectiveness and efficiency of the explicitsolution method in solving such problems.

Crystal plasticity has been used in many studies to modellarge deformations and strain localisations in metals andmetallic based materials (e.g. [13,19,31–34,39]). 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 updatealgorithm and implement it in an external user-definedmaterial module. Several researchers have developed stressupdate algorithms that define single crystal behaviour foruse with commercial finite element software such as ABA-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 havedeveloped rate-dependent implicit algorithms for use withABAQUS (e.g. [5,8,17,21]).

Although the majority of algorithms developed havecaptured rate-dependent mechanical behaviour, there hasalso been significant effort in developing rate-independentcrystal plasticity formulations. The essential differencebetween these two types 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 difficul-ties in boundary value problems using the rate-independentconstitutive formulation. Several researchers have imple-mented rate-independent algorithms successfully to achievemeaningful results (e.g. [1,22]).

In this paper a new crystal plasticity algorithm is pre-sented. This formulation was developed for use with theFE solver ABAQUS/explicit. The objective of the presentwork is to compare the performance of the implicit andexplicit solution methods using two equivalent crystal

plasticity 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-rithm and is implemented as a user-defined material subrou-tine for both versions of the code.

The implicit and explicit solution procedures are out-lined in Sections 1.1 and 1.2. In Section 2 the crystal plas-ticity formulation of Peirce et al. [26] is presented. For theimplicit analyses the UMAT, developed by Huang [10] forimplementation in ABAQUS/standard, is used. A newsubroutine (VUMAT) was developed by the authors forimplementation in ABAQUS/explicit. The VUMAT is basedon the Huang UMAT [10]. Both user-defined subroutinesare described in Section 3. In Section 4 the methods arecompared in terms of finite deformation analyses in 2Dand 3D. The impact of parallel processing on the compar-ison between the methods is also given in Section 4.Finally, conclusions are drawn in Section 5.

1.1. Implicit solution method

The word ‘implicit’ in this paper refers to the method bywhich the state of a finite element model is updated fromtime t to t + Dt. A fully implicit procedure means thatthe state at t + Dt is determined based on information attime t + Dt, while the explicit method solves for t + Dt

based on information at time t.There are a range of solution procedures used by impli-

cit FE solvers. A form of the Newton–Raphson method isthe most common and is presented here. Vectors andmatrices are denoted as underlined. When solving aquasi-static boundary value problem, a set of non-linearequations is assembled:

GðuÞ ¼Z

vBTrðuÞdV �

ZS

N Tt dS ¼ 0 ð1:1Þ

where G is a set of non-linear equations in u, and u is thevector of nodal displacements. B is the matrix relatingthe strain vector to displacement. The product of BT andthe stress vector, r, is integrated over a volume, V. N

is the matrix of element shape functions and is integratedover a surface, S. The surface traction vector is denotedby t. Eq. (1.1) is usually solved by incremental methods,where loads/displacements are applied in time steps, Dt,up to an ultimate time, t.

The state of the analysis is updated incrementally fromtime t to time t + Dt. An estimation of the roots of Eq.(1.1) is made, such that for the ith iteration:

duiþ1 ¼ utþDtiþ1 � utþDt

i ¼ � oGðutþDti Þ

ou

� ��1

GðutþDti Þ ð1:2Þ

where utþDti is the vector of nodal displacements for the ith

iteration at time t + Dt. The partial derivative on the right-hand side of the equation is the Jacobian matrix of thegoverning equations and can be referred to as the global

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F.J. Harewood, P.E. McHugh / Computational Materials Science 39 (2007) 481–494 483

stiffness matrix, K. Eq. (1.2) is manipulated and inverted toproduce a system of linear equations:

KðutþDti Þduiþ1 ¼ �GðutþDt

i Þ ð1:3Þ

Eq. (1.3) must be solved, for each iteration, for the changein incremental displacements, dui+1. In order to solve fordui+1 the global stiffness matrix, K, must be inverted.Although, this is a computationally expensive operation,iteration ensures that a relatively large time incrementcan be used while maintaining accuracy of solution [9,19].Following iteration i, dui+1 has been determined and abetter approximation of the solution has been made,utþDt

iþ1 , through Eq. (1.2). This in turn is used as the currentapproximation to the solution for the subsequent iteration(i + 1).

The accuracy of the solution is dictated by the conver-gence criterion where the updated value for G must be lessthan a tolerance value. Complications can arise in an anal-ysis that has a highly non-linear stress–strain response orwhere there is contact and sliding between two surfaces.For a complex job it can be difficult to predict how longit will take to solve or even if convergence will occur.

ABAQUS/standard uses a form of the N–R iterativesolution method to solve for the incremental set of equa-tions. Formulating and solving the Jacobian matrix is themost computationally expensive process. Several variationson the N–R method exist to improve the solution time. Themodified Newton method is the most commonly used alter-native and is suitable for non-linear problems. The Jaco-bian is only recalculated occasionally and in cases wherethe Jacobian is unsymmetric it is not necessary to calculatean exact value for it. The modified Newton methodconverges quite well using a symmetric estimate of theJacobian [9].

1.2. Explicit solution method

The explicit method was originally developed, and is pri-marily used, to solve dynamic problems involving deform-able bodies. Accelerations and velocities at a particularpoint in time are assumed to be constant during a timeincrement and are used to solve for the next point in time.ABAQUS/explicit uses a forward Euler integration schemeas follows [9]:

uðiþ1Þ ¼ uðiÞ þ Dtðiþ1Þ _uðiþ12Þ ð1:4Þ

_uðiþ12Þ ¼ _uði�

12Þ þ Dtðiþ1Þ þ DtðiÞ

2€uðiÞ ð1:5Þ

where u is the displacement and the superscripts refer to thetime increment. The term ‘explicit’ refers to the fact thatthe state of the analysis is advanced by assuming constantvalues for the velocities, _u, and the accelerations, €u, acrosshalf time intervals. The accelerations are computed at thestart of the increment by

€uðiÞ ¼ M�1 � ðF ðiÞ � I ðiÞÞ ð1:6Þ

where F is the vector of externally applied forces, I is thevector of internal element forces and M is the lumped massmatrix. As the lumped mass matrix is diagonalised it is atrivial process to invert it, unlike the global stiffness matrixin the implicit solution method. Therefore each time incre-ment is computationally inexpensive to solve.

A stability limit determines the size of the time increment:

Dt 62

xmax

ð1:7Þ

where xmax is the maximum element eigenvalue. A conser-vative and practical method of implementing the aboveinequality is:

Dt ¼ minLe

cd

� �ð1:8Þ

where Le is the characteristic element length and cd is thedilatational wave speed:

cd ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffikþ 2l

q

sð1:9Þ

k and l are the Lame elastic constants and q is the materialdensity. A quasi-static problem that is solved using the ex-plicit method would have much smaller time incrementsthan an equivalent problem solved using the implicit meth-od. Although the incremental solution is easy to obtainusing the explicit method, it is not unusual for an analysisto take 100,000 increments to solve. In order to maintainefficiency of the analyses it is important to ensure thatthe sizes of the elements are as regular as possible. This isso that one small element does not reduce the time incre-ment for the whole model.

It is often impractical to run a quasi-static analysis usingits true time scale as the runtime would be very large. Anumber of methods can be used to artificially reduce theruntime of the simulation. The first involves simply speed-ing up the applied deformation or loading rate and thesecond involves scaling the density of the material in themodel. According to Eqs. (1.8) and (1.9), when the densityis scaled by a factor, f2, the runtime is reduced by a factor f.The latter method is preferable as it does not affect thestrain rate dependent response of viscoplastic/rate-depen-dent materials.

It is important when performing a quasi-static simula-tion that the inertial forces do not affect the mechanicalresponse and provide unrealistic dynamic results. Toreduce the dynamic effects Kutt et al. [16] recommend thatthe ratio of the duration of the load and the fundamentalnatural period of the model be greater than five. It has beenshown that by keeping the ratio of kinetic energy to thetotal internal strain energy at <5% dynamic effects in themodel are negligible [6,14]. This is the criterion for quasi-static behaviour that is employed in this paper.

During implicit analyses where the material gives a non-linear stress–strain response many iterations are usuallyneeded to solve for an increment. This leads to progressively

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484 F.J. Harewood, P.E. McHugh / Computational Materials Science 39 (2007) 481–494

smaller time steps being used and should the code encounterlarge non-linearities convergence may be impossible toachieve in practical terms. As there is no iteration involvedin the explicit method, convergence problems are not anissue.

The solution time of the implicit solver is proportionalto the square of the wavefront size in the global stiffnessmatrix. This has implications when increasing the size ofthe model and when running 3D simulations. In the caseof the explicit solver there is a linear relationship betweenthe size of the model and the solution time, as dictatedby the characteristic element length and the number ofelements in the model [9,14,15,29,35–37].

2. Theory

The finite element analyses performed in this studyincorporate elastic and plastic constitutive laws in thecontext of finite deformation kinematics. The elasticity isconsidered to be isotropic and linear in terms of finitedeformation quantities and can be described using theYoung’s modulus, E, and Poisson’s ratio, m. This is a rea-sonable approach as the elastic strains in the model arevery small in comparison to the plastic strains. Plasticityis described using rate-dependent crystal plasticity theory[26].

Crystal plasticity theory is a physically based plasticitytheory that represents the deformation of a metal at themicroscale. The flow of dislocations in a metallic crystalalong slip systems is represented in a continuum frame-work. Plastic strain is assumed to be solely due to crystallo-graphic dislocation flow. The rate-dependent, viscoplasticsingle crystal theory as presented in Peirce et al. [26] andHuang [10], and employed in several works such asMcHugh and Connolly [20], Savage et al. [31] and McGarryet al. [18], is used. The rate-dependence is implemented inthe formulation through a power-law that relates theresolved shear stress s(a) to the slipping rate _cðaÞ on each slipsystem a,

_cðaÞ ¼ _asgnðsðaÞÞ sðaÞ

gðaÞ

��������n

ð2:1Þ

where _a and n are a reference strain rate and rate sensitivityexponent, respectively, and g(a) is the slip system strainhardness. As n tends to infinity the material reaches therate-independent limit.

The slip system strain hardness, g(a), is determined byintegration of the following evolution equation

_gðaÞ ¼XN

b¼1

habj _cðbÞj ð2:2Þ

where hab is the strain hardening modulus; haa and hab

(a 5 b) are the self and latent hardening moduli, respec-tively, and the summation ranges over the number ofslip systems, N. In this work, Taylor isotropic hardeningis assumed where the self and latent hardening moduli are

considered equal. Quantitatively, slip system strain harden-ing is specified by the following hardness function as definedby Peirce et al. [26]:

gðcaÞ ¼ g0 þ ðg1 � g0Þ tanhh0ca

g1 � g0

�������� ð2:3Þ

From which one can determine the hardening moduli,through differentiation:

haa ¼ hab ¼ hðcaÞ ¼dgðcaÞ

dca

¼ h0sech2 h0ca

g1 � g0

�������� ð2:4Þ

In the above expressions the material hardening parame-ters g0, g1 and h0 are the initial hardness of a slip system,the maximum slip system hardness when og/oc = 0 and thevalue of og/oc when c = 0, respectively. These materialconstants can be derived from the strain hardening partof an experimental tensile stress–strain (r–e) curve for amaterial. The values used in this study are taken from thesingle crystal tensile test data for 316L stainless steel pre-sented by Okamoto et al. [25], and as calibrated in [31]

g0 ¼ 50 MPa; g1 ¼ 330 MPa; h0 ¼ 225 MPa;

_a ¼ 0:001 s�1

A rate sensitivity parameter, n = 20 is used for all of theanalyses except where stated.

A quantity of importance in the above equation is theaccumulated slip, ca. This is a measure of the total crystal-lographic plastic strain at a point and is defined as follows:

ca ¼XN

a¼1

Z t

0

j _cðaÞjdt ð2:5Þ

where t is the time and the summation is over all the slipsystems at a point.

The material chosen, 316L stainless steel, has a face-cen-tred-cubic (FCC) crystalline structure. For this material theslip systems are defined by the following Miller indices:h1 11i{110}.

3. Development of explicit user material subroutine

The crystal plasticity constitutive theory is not providedas standard in any of the commercially available finiteelement analysis software. It is therefore necessary toimplement the theory in the form of a user-defined stressupdate algorithm. This is implemented in the finite elementcode ABAQUS/standard, an implicit solver, by means of aUMAT, as coded in [10]. It was necessary to develop a VUMAT

for use with ABAQUS/explicit. Much of the codinginvolved in the two algorithms is the same but there areseveral key issues that must be addressed to maintain con-sistency of results between the two solvers.

These subroutines, written in Fortran, implement thetheory in the form of a stress update algorithm that iscalled at each integration point for every iteration duringa finite element simulation [9]. Recalling Eq. (1.1), thestress component must be updated during each iteration

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F.J. Harewood, P.E. McHugh / Computational Materials Science 39 (2007) 481–494 485

and this operation is performed by the UMAT/VUMAT. Thestress update is calculated thus:

rðutþDti Þ ¼ rðut þ DuiÞ ¼ rðutÞ þ Dri ð3:1Þ

During every iteration of an analysis the finite elementsolver provides the subroutine with quantities such as thestrain and time increments, Dei and Dt, and the incrementin stress, Dri, is calculated [9].

In this section the development of the crystal plasticityVUMAT for use with ABAQUS/explicit is described. Partic-ular attention is paid to the differences between it andthe crystal plasticity UMAT [10]. A schematic of the VUMAT

layout is presented in the Appendix.

3.1. System equations

The most important difference between the sets of equa-tions used in the two solvers is the lack of presence of a glo-bal stiffness matrix in the explicit solver. When writing aUMAT it is vital that the Jacobian matrix be accuratelyrepresented to get correct and efficient finite strain problemsolutions. It is not necessary to define any such matrix in theVUMAT interface. Although this makes the writing of thesubroutine more straightforward, the choice of elementsavailable to the user is restricted to mainly first order.

3.2. Vectorised interface

The explicit solution process involves a large number ofincrements, each of which is easily solved for. Conse-quently, the explicit finite element calculation procedureis well suited to being split up and solved by a number ofprocessors. With this in mind the VUMAT is constructed witha vectorised interface. At the beginning of each incrementthe stress and state variable data are passed in, in the formof two-dimensional arrays. Each column in an array con-tains the information relating to an integration point ofthe material. When a simulation is performed using multi-ple processors the analysis data can be split up into blocksand solved independently. Thus, vectorisation is preservedin the writing of the subroutine in order that optimalprocessor parallelisation can be achieved.

3.3. Size of time increment

The initial time increment used in ABAQUS/standard ischosen by the user. Subsequent increments are controlledby an automatic incrementation control. However, whenimplementing the UMAT it was necessary to improve the con-trol in order to improve time-stepping accuracy. The varia-tion in the accumulated slip, ca, throughout each iteration ismeasured. If the rate of increase is excessive then the itera-tion is aborted and a smaller time increment size is used.This is found to be an efficient method of incrementationcontrol and prevents premature termination of an analysis.The same procedure is not necessary using ABAQUS/expli-cit as the time increments are sufficiently small.

To determine the size of the initial time increment inABAQUS/explicit a bogus set of tiny strain incrementsare passed in to the VUMAT at the start of the analysis. Fromthe stress response of the material, a conservative value forthe stable time increment is calculated (Eqs. (1.8) and(1.9)). The finite element solver requires that the materialbe elastic for this initial check. Due to the fact that the crys-tal plasticity subroutine is computationally intensive, anelastic stress–strain response must be defined that ensuresa relatively small time increment. During this initial timeincrement calculation stage a material response is definedwith the same elastic properties as are used to describethe elasticity in the body of the crystal plasticity subrou-tine. This ensures that a relatively efficient time incrementis employed. If a smaller increment is required a stiffer elas-tic modulus may be used, although the solution time will belonger. This does not affect the response of the materialduring the analysis; it is purely for the purpose of timeincrement calculation.

3.4. Time integration scheme

The forward gradient time integration scheme thatforms part of the stress update algorithm in [10] involvesthe solution of the following non-linear equation for theincremental slip system shear strains using the Newton–Raphson method:

DcðaÞ ¼ ð1� hÞDt _cðaÞjt þ h _asgnðsaÞsa þ Dsa

ga þ Dga

��������

� n����tþDt

Dt

ð3:2Þ

where h ranges from 0 to 1. This is a non-linear implicitequation as the increments of resolved shear stress, Dsa,and current strength, Dga, are functions of Dc(a). SolvingEq. (3.2) in this way ensures that the convergence rate dur-ing the analysis is high and allows for a relatively large timeincrement to achieve convergence.

The explicit solver does not require the use of iteration.Time rates of change are assumed to be constant through-out each time increment and a value of h = 0 is used in Eq.(3.2) such that an incremental quantity is calculated, in asimple Euler fashion, as the product of the rate quantityand the time increment, for example:

DcðaÞ ¼ _cðaÞDt ð3:3Þ

3.5. Material rotation

During elastic deformation of a crystal, the crystal latticeundergoes rotation and distortion. This effect is captured bythe vectors that define the slip directions, s*(a), and the nor-mals to the slip planes, m*(a), as the deformation continues.These vectors have components s�ðaÞi and m�ðaÞi in the refer-ence Cartesian coordinate system. The incremental valueof s*(a) is calculated thus:

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486 F.J. Harewood, P.E. McHugh / Computational Materials Science 39 (2007) 481–494

DS�ðaÞi ¼ Deij �X

b

lðbÞij DcðbÞ þ xij Dt �X

b

#ðbÞij DcðbÞ

( )S�ðaÞj

ð3:4Þwhere

PblðbÞij DcðbÞ and

Pb#ðbÞij DcðbÞ are the incremental

plastic strain and plastic lattice spin respectively. The quan-tities in the brackets equal the sum of the elastic strain andthe elastic spin increments. The increment of total crystalrotation is denoted by xijDt, where xij is the sum of therotation on each slip system, a, and is calculated by

xðaÞij ¼ asymðS�ðaÞi m�ðaÞj Þ ð3:5Þ

where asym(Aij) denotes the asymmetric part of a tensorAij. The incremental value of the slip normal, Dm*(a), iscalculated similarly to Eq. (3.4).

An important feature of crystal plasticity theory is thatboth the lattice stretch and rotation influence the amountof plasticity:

Lp ¼XN

a¼1

_cðaÞs�ðaÞm�ðaÞ ð3:6Þ

where Lp is plastic velocity gradient in the current state andthe summation ranges over all the slip systems, a, in thecrystal.

The most important issue that the programmer must beaware of is in regard to the way in which stress rates and,consequently, rotations are dealt with. In the case of FEsimulations using solid continuum elements and a user-defined material ABAQUS/standard employs the Jaumannstress rate, whereas ABAQUS/explicit employs the Green–Naghdi stress rate [9,11]. The stress rates are defined,respectively, as:

rrJ ¼ _r� W � rþ r � W ð3:7Þand

rrG ¼ _r� X � rþ r � X ð3:8Þwhere _r is the time derivative of stress, W is a spin ratederived from the velocity gradient

W ¼ asymovox

� �ð3:9Þ

and X is found from the right-hand polar decomposition ofthe total deformation gradient, F;

F ¼ R � U ð3:10Þand

X ¼ _R � RT ð3:11ÞIn the stress update algorithms these quantities must becalculated incrementally.

In ABAQUS/standard the material is treated as beingbased in a fixed global coordinate system. Incrementalrotations are passed in to the UMAT at the start of eachincrement. This array, dR, is the amount by which thestress and strain arrays have been rotated between the

end of the previous increment and the start of the currentone. It is treated thus:

rtþDt ¼ dR � rt � dRT þ dr ð3:12Þwhere rt+Dt is the current Cauchy stress and dr is the stressincrement that has been calculated in the previous incre-ment. The value passed in at the start of each increment isthe Cauchy stress in the model coordinate system. It is ‘ro-tated’, as shown, at the end of each increment. The dR var-iable is calculated using the Hughes–Winget algorithm [9]

dR ¼ ðI � 12DxÞ�1 � ðI þ 1

2DxÞ ð3:13Þ

where I is the identity matrix and Dx is the anti-symmetricpart of the incremental velocity gradient, i.e. the incremen-tal form of W (Eq. (3.9)).

Dx ¼ asymdDu

dxtþDt=2

� �ð3:14Þ

This value is used in the calculations of s*(a) (Eq. (3.4)) andm*(a) and is calculated from the variable dR according toEq. (3.13).

In the case of the VUMAT the material is taken to lie on acorotational coordinate system, i.e. the coordinate systemrotates with the material. As each time increment is sosmall, incrementally, it is assumed that all material rota-tions are rigid body. Hence for each increment the problemcan be viewed as a small strain problem. The rotation usedin ABAQUS/explicit is the Green–Naghdi spin rate. TheHughes–Winget algorithm is also used but takes a slightlydifferent form

dR ¼ ðI � 12DXÞ�1 � ðI þ 1

2DXÞ ð3:15Þ

where DX is found from

DX ¼ D _RRT ð3:16ÞThe rotations used by each code are different. However,these differences are only evident when kinematic hardeningis employed. Johnson and Bamann [11] showed that when akinematic hardening model is used, employing the Jaumannstress rate, a sinusoidal response is exhibited by the shearstress beyond the yield point. This is not physically realistic.When the Green–Naghdi stress rate is employed the shearstress is shown to monotonically increase with the shearstrain. As the crystal plasticity subroutine does not incorpo-rate kinematic hardening, both methods yield the sameresults.

Ultimately, during each time increment of the VUMAT,the increment of stress is calculated thus:

Drij ¼ Dglobalijkl Dekl �

Xa

½vðaÞij � � DcðaÞ � rij � Dedil ð3:17Þ

where

vðaÞij ¼ Dglobalijkl lðaÞkl þ xðaÞik rjk þ xðaÞjk rik ð3:18Þ

Dglobalijkl is the 4th order elastic constitutive tensor in the

global coordinate system and Dedil is the increment of dila-tational strain.

Page 7: Comparison of the implicit and explicit finite element methods using crystal plasticity

Fig. 1. Schematic of the analysis to investigate the rotation of thematerial.

F.J. Harewood, P.E. McHugh / Computational Materials Science 39 (2007) 481–494 487

To investigate the way that material rotation is dealtwith in both user material subroutines a simple analysisis run. In a two-step simulation a square of material istensed uniaxially and in the second step, while maintainingthe axial tension, the material is rotated (Fig. 1). The com-ponents of the stress are monitored in the ABAQUS outputdatabase (odb) and the same components are also writtendirectly from the material subroutines. The analysis is car-ried out for the implicit and explicit solvers.

Two components of stress are plotted in Fig. 2. In thecase of the UMAT the components of stress calculated aredirectly transmitted to the odb. This is not the case withthe VUMAT. As the formulation is based in a corotationalcoordinate system, rotation of the material is not ‘seen’by the VUMAT and there is no change in the orientation ofany tensors. All components are passed into the VUMAT

unrotated and the necessary rotations are performedfollowing the stress update. Therefore when developingthe crystal plasticity VUMAT it was assumed that there iszero incremental rotation of the material and the tensordR (Eq. (3.13)) is the identity matrix. The calculated stresscomponents in the VUMAT and UMAT differ but the odb

Fig. 2. Plot of two components of stress during tension and rotation ofthe material. There are three outputs for each component: from theABAQUS odb, directly from the UMAT, and directly from the VUMAT.

results are identical (Fig. 2). It is interesting to note thatWeber et al. [38] found that the Hughes–Winget algorithmis not absolutely objective. A simultaneous rotation andsimple shearing of material was performed and a non-objective stress response was generated. However, whenthe values of Dt are small the performance of the algorithmis satisfactory.

4. Comparative analyses

A range of 2D and 3D boundary value problems incor-porating crystal plasticity are analysed using the implicitand explicit finite element solvers, ABAQUS/standard andABAQUS/explicit, respectively. The mechanical responseis used as a means of comparing the performance of bothuser material subroutines. As the model has the ability todetect localisations in the material it is also desirable thatthe contour profiles of the accumulated shear strain, c, com-pare well. To ensure a quasi-static analysis in the case of theexplicit method the kinetic energy must be negligible (<5%)compared to the internal energy [6].

As shown in Section 1.2, the system of equations used tosolve for each time increment in the explicit code assumes aconstant acceleration and velocity across time steps. For apractical time increment to be maintained it is necessary toapply displacements that follow an amplitude wave. Asmooth step time–displacement relationship is used toensure that the nodal accelerations and velocities remaincontinuous as the model is being strained.

All computational simulations were performed using anSGI 3800, 500 MHz processor, high performance com-puter, on one processor except in the cases where explicitlystated as being multiple processor. The analyses wereperformed using ABAQUS version 6.3.

4.1. 2D analyses

The 2D geometry used in this study is based on thatpresented in Savage et al. [31]. A 25 lm thick wire, withthe granular microstructure explicitly represented, is con-structed. Each grain is idealised as a hexagon of area92 lm2 (Fig. 3). The material modelled in this study,316L stainless steel, has a face-centred-cubic (FCC) crystalstructure. The orientations of the FCC crystal axes arerandomly generated and assigned to each grain. Orienta-tion mismatch among the grains causes stress localisationsand the coalescence of plastic strain due to favourablegrain orientations determines the site of eventual failure.

Fig. 3. Model of 316L stainless steel wire with a thickness of 25 lm. Thegranular microstructure is idealised as a series of hexagons (highlighted).

Page 8: Comparison of the implicit and explicit finite element methods using crystal plasticity

Fig. 4. Comparison of macroscopic mechanical performance for 2Dtension analyses.

Table 1Computation times for 2D analyses using implicit (UMAT) and explicit(VUMAT) solvers

Analysis Runtime(h, min)

Ratio,explicit/standard

Size of timeincrement,min/max

Tension UMAT 2D 0, 56 2.34 1.86e�4/0.01Tension VUMAT 2D 2, 11 1.37e�5/6.36e�5Bending UMAT 2D 1, 53 2.39 1.85e�4/6.0e�3Bending VUMAT 2D 4, 30 4.86e�6/1.2e�5Contact 1 UMAT 2D 1, 50 1.41 9.25e�5/1.2e�2Contact 1 VUMAT 2D 2, 35 1.86e�5/3.8e�5Contact 2 UMAT 2D 0, 40a 0.875 5.74e�4/3.8e�2Contact 2 VUMAT 2D 0, 35a 2.57e�5/3.81e�5

a The solution time at equivalent points in the two analyses.

488 F.J. Harewood, P.E. McHugh / Computational Materials Science 39 (2007) 481–494

The model is meshed with first order reduced integrationplane strain quadrilateral elements (CPE4R).

4.1.1. Tension

The left-hand edge of the model is constrained in thehorizontal direction and a horizontal displacement isapplied to the right-hand edge. Rigid body motion is pre-vented by constraining the left-hand bottom corner of themodel in all degrees of freedom. The wire is tensed beyondthe failure strain, which is taken as the strain at the point ofultimate tensile stress (UTS). The strain is defined as theratio of the increase in length to the original length,(l1 � l0)/l0, and the stress is the ratio of force applied tothe wire to the original cross-sectional area of the wire,F/A0. This produces engineering stress–engineering strain(reng–eeng) curves and is a convenient method for present-ing tensile data.

The reng–eeng response of the two models shows goodagreement with the failure point particularly well compared(Fig. 4). The distribution of plastic shear strain duringnecking is also very similar for the two analyses (Fig. 5).

The solution time of the implicit method is 56 min ascompared to 2 h 11 min for the explicit method (Table 1).The shorter implicit solution time can be explained by thefact that the size of the incremental global stiffness matrixis relatively small due to the size of the model. As the solu-tion time in implicit analyses is proportional to the squareof the wavefront of the global stiffness matrix in the model,this simulation is computationally easy to solve.

4.1.2. Bending

The left-hand boundary of the model is constrained inall degrees of freedom. The degrees of freedom of the nodesalong the right-hand edge are tied to a reference node and a

Fig. 5. Accumulated shear strain, c, in the 25 lm wire under tensio

rotation, about the z-axis (normal to the page), is appliedto this node. A bending moment is produced that is con-stant along the length of the wire. The average curvaturealong the length of the wire is used as a deformation

n solved using (a) the implicit solver and (b) the explicit solver.

Page 9: Comparison of the implicit and explicit finite element methods using crystal plasticity

Fig. 6. Comparison of macroscopic mechanical performance for 2Dbending analyses. Fig. 8. Comparison of the force–displacement behaviour of the rigid

cylinder for 2D contact analyses.

F.J. Harewood, P.E. McHugh / Computational Materials Science 39 (2007) 481–494 489

measure, analogous to eeng used in the tension analyses.The macroscopic comparison of bending moment–curva-ture curves for the two FE analyses show the same perfor-mance (Fig. 6). The ratio of solution times is similar to the2D tension analyses (Table 1).

4.1.3. Contact

The introduction of contact between surfaces increasesthe computational cost for the implicit solver. Smaller timeincrements are required to achieve convergence and com-putational expense is further increased by the formationand inversion of the stiffness matrix. Due to the nature ofthe explicit FE solver and the small time incrementsinvolved, simple algorithms may be implemented to nego-tiate contact conditions with a minimal loss of computa-tional efficiency [9,15,29,37].

Two types of analyses involving contact are performed.The first analysis simulates a three-point bending of thewire and models contact between a rigid body and thestainless steel. A rigid semi-circular cylinder is brought intocontact with the wire at the midpoint while the left- andright-hand edges provide the reaction forces and are onlyfree to rotate about the z-axis (Fig. 7). A master–slave con-tact algorithm is employed for both analyses. Frictionlesscontact is assumed between the rigid surface and the wire.

It is interesting to note the force–displacement responseof the rigid body in both analyses (Fig. 8). In the explicitcase when the cylinder comes into contact with the wirethere is a vibration that dissipates quickly and subsequentlyboth curves follow the same path. This is a transient effectand does not affect the final state of the model. It is caused

Fig. 7. 2D contact analysis. 25 lm wire is deformed by a rigid cylinder.The left- and right-hand edges are free to rotate.

by the mass scaling factor employed to reduce the solutiontime of the analysis. Solution times are much closer for thisanalysis than for the previous two. The solution times are1 h 50 min and 2 h 35 min for implicit and explicit respec-tively. The presence of contact does not affect the solutionprocess of the explicit code as strongly as the implicit code.

The second contact analysis models the interactionbetween two deformable materials. A block of material(E = 10 GPa, m = 0.3) is modelled as compressing a 25 lm2

of the stainless steel wire. The basis for this analysis is purelyto investigate the ability of the implicit and explicit codes tosolve a more numerically complicated analysis. The geome-try of the model is shown in Fig. 9. The bottom edge of thestainless steel is displaced by 0.01 mm in the vertical direc-tion, while the top edge of the stiff block is held. This ensurescontact and compression of the wire. The same contactalgorithm as in the previous analysis is implemented.

The implicit solver is unable to complete the simulation.A displacement of 7.27 · 10�3 mm is achieved beforenumerical difficulties prevent further analysis. When theimplicit iterative solver encounters a highly non-linearresponse very small time increments must be employed to

Fig. 9. Model of contact simulation involving two deformable materials.

Page 10: Comparison of the implicit and explicit finite element methods using crystal plasticity

Table 2Computation times for 3D analyses using implicit (UMAT) and explicit(VUMAT) solvers

Analysis Runtime(h, min)

Ratio,explicit/standard

Size of timeincrement,min/max

Tension UMAT 3D 27, 30 5.04 4.52e�4/1.24e�2Tension VUMAT 3D 138, 43 1.06e�5/5.01e�5Torsion +

bending UMAT

3D 40, 03 3.78 2.37e�4/1.21e�2

Torsion +bending VUMAT

3D 151, 18 5.06e�7/5.01e�5

Contact UMAT 3D 42, 32 1.46 1.88e�4/3.42e�3Contact VUMAT 3D 62, 05 2.12e�6/8.47e�6

490 F.J. Harewood, P.E. McHugh / Computational Materials Science 39 (2007) 481–494

solve the equilibrium equations. However in this case, thesolver attempts increasingly small time increments withoutachieving equilibrium. The minimum time increment forthe analysis in Table 1 is given as 5.74e�4. However, inthe final time increment, prior to termination of the job,time increments in the order of 1.0e�8 are attempted with-out success. Severe discontinuities at the surfaces in contactare the cause of termination of the analysis. The explicitsolver does not encounter such problems and this analysisis completed. The solution time for the implicit simulationat the point of termination is approximately 40 min. Thetime taken by the explicit solver to solve to the same pointis 35 min while the total solution time is 40 min. Prior tothe premature termination of the implicit analysis theforce–displacement responses and the contour plots ofstress compare very well between both models. It is thusevident that the explicit solver is more suited to this analy-sis in terms of solution time and ability to complete theanalysis.

4.2. 3D analyses

The model used in the 3D analyses is geometricallysimpler in comparison to the 2D analyses, with the grainsbeing represented as cubes. The wire has a 25 lm2 sectionwith a length of 75 lm. Each grain is meshed with first orderreduced integration brick elements (C3D8R) (Fig. 10).Similarly to the 2D models, a random 3D crystallographicorientation is assigned to each grain.

The 3D analyses are carried out in order to investigatewhat effect, if any, a more three-dimensional stress statehas on the solution times. Three types of analyses are car-ried out; tension, contact, and a combination of torsionand bending. The two former loading conditions are imple-mented in the same way as the 2D analyses.

Fig. 10. 3D crystal plasticity model with a cubic crystalline structure. Onegrain is highlighted, there are 3 · 3 grains in the cross-section and ninegrains along the length.

4.2.1. Torsion and bendingA combined loading of rotation around the x-axis (lon-

gitudinal axis of wire) and rotation around the z-axis isapplied to the right-hand end of the wire. The left-handend of the wire is constrained in all degrees of freedom.

The implicit code has a shorter solution time than theexplicit code. This can be expected for a simple deforma-tion case. The ratio of explicit solution time to implicitsolution time is somewhat larger for the 3D displacementdriven analyses compared with the 2D analyses (Table 2).The increase in model size from 2D to 3D has a greatereffect on the explicit solution times. The bandwidth sizein the implicit analyses remains relatively small, while thecombination of the short explicit time increments and thesize of the model significantly increase solution times.

It appears that the implicit solver is the quicker solverfor all three types of 3D model loading but it is importantto comment that in the case of the contact analysis theexplicit solver solves for the initial interaction betweenthe surfaces more quickly. At time, t = 0.4 the solutiontime is 18 h 12 min for the implicit solver while it is 14 h51 min for the explicit solver at the same point in the load-ing history. The contact that follows is not as computation-ally difficult to solve. Therefore, the implicit solverincreases the size of the time increments after the initialcontact. This shows that in an analysis where the contactconditions are more complex, i.e. between two deformablematerials, or are constantly changing, the explicit solverwould prove to be the better option.

4.3. Material strain-rate sensitivity

The material in each of the previous analyses uses a rel-atively rate-dependent value of n = 20 (Eq. (2.1)). Byincreasing the value of n the material becomes more rate-independent. It is known that a material with an increasedrate sensitivity delays the development of shear bands[24,26]. An analysis employing a material with low rate sen-sitivity, i.e. nearly rate-independent, is computationallymore difficult to solve as higher strain gradients must beresolved.

A more rate-independent parameter, n, is assigned to thematerial and the 2D tension case is again simulated. Values

Page 11: Comparison of the implicit and explicit finite element methods using crystal plasticity

Table 3Computation times for the 2D tension analysis using different values forthe rate sensitivity exponent

Tensionanalysis

Rate sensitivityexponent, n

Runtime(h, min)

Size of time increment,min/max

UMAT 20 0, 56 1.89e�3/0.01UMAT 50 1, 24 7.05e�5/2.3e�3UMAT 100 1, 42 5.06e�5/1.0e�3VUMAT 20 2, 11 1.37e�5/6.36e�5VUMAT 50 2, 22 1.24e�5/6.36e�5VUMAT 100 2, 23 1.22e�5/6.36e�5

F.J. Harewood, P.E. McHugh / Computational Materials Science 39 (2007) 481–494 491

of n = 50 and n = 100 are chosen. The resulting runtimesshow that as the material becomes more rate insensitivethe implicit solver has trouble converging to a final solution(Table 3). The explicit solver is affected to a much smallerdegree. Table 3 shows that there is a significant reductionin the size of the time increments using the implicitsolver when the rate sensitivity exponent is increased,whereas, there is little change in the size of the explicit timeincrements.

4.4. Processor parallelisation

As stated in Section 3.2 the VUMAT is constructed in avectorised format, the solver in ABAQUS/explicit is simi-larly constructed. This ensures that a high level of efficiencycan be achieved when multiple processors are used. The 2Dtension case is solved across multiple processors using boththe implicit and explicit solvers. The simulations are solvedusing 2 and 4 processors on the SGI 3800 multiple proces-sor computer.

The results of these analyses are presented in Table 4. Inorder to characterise the efficiency of using multiple proces-sors the following formula was developed:

gprocessor% ¼ffiffiffiffiffiffijCj

p� sgnðCÞ � 100 ð4:1Þ

with

C ¼ t1

tx � v� 1

x

� �� 1� 1

v

� �ð4:2Þ

where t is the solution time and x is the number of proces-sors used. For example, an efficiency of 100% when using

Table 4Computation times and parallelisation efficiency for analyses solved usingmultiple processors

Tensionanalysis

# Processors Runtime(h, min)

Efficiency(%)

UMAT 2D 1 0, 56 –UMAT 2D 2 0, 57 �13.2UMAT 2D 4 1, 02 �20.0UMAT 3D 1 27, 30 –UMAT 3D 2 17, 31 75.5

VUMAT 2D 1 2, 11 –VUMAT 2D 2 1, 13 89.1VUMAT 2D 4 0, 44 81.2VUMAT 3D 1 138, 43 –VUMAT 3D 2 74, 35 92.7

two processors indicates a solution time saving of 50%. Anegative efficiency indicates that a longer solution time isrequired when using multiple processors than when usingone processor.

There is no speedup with the implicit solver when usingmultiple processors for this simple analysis. There is actu-ally a loss in solution time as indicated by the negativeprocessor efficiency. This can be explained by the extracomputation time required to assemble the system equa-tions from a number of processors. The efficiency valuesfor each of the explicit analyses are shown to be quite good(Table 4). When four processors are used to solve the 2Dtension analysis the solution time is 44 min. This comparesfavourably to the quickest solution time of 56 min usingthe implicit solver.

In the 2D analyses a very high speedup efficiency isachieved when the explicit analysis is run across two pro-cessors. The efficiency is not as high when four processorsare used. There is a type of model size dependence associ-ated with the explicit speedup factors. When using thedomain parallelisation setting a larger model can be splitamongst a large number of processors. Each domainrequires a significant amount of processor capability tosolve the analysis. If a large number of processors is usedto solve a relatively small model the computational powerrequired of each processor may be quite low, such that twodomains may be solved by one processor. The speedupover using fewer processors is not simply a product ofthe increase in number of processors.

In order to investigate whether a better speedup can beachieved in a large model the 3D tension case is run acrosstwo processors. The solution time is 17 h 31 min using theimplicit solver. This gives an efficiency of 75.5%. Althoughthis solution time compares favourably to the 3D explicitanalyses it is worth noting the high efficiency in the explicitcase (Table 4). Given the large size of the model, furtherincrease of the number of processors would continue toyield a high efficiency and would likely result in a shortersolution time than the implicit analysis.

5. Conclusions

A crystal plasticity material subroutine has been devel-oped by the authors for use with the explicit finite elementsolver ABAQUS/explicit. The results from a variety of 2Dand 3D loading conditions are shown to be the same as theoriginal UMAT. Either solver is shown to be more efficientwhen solving certain types of simulations. In the simpleranalyses, where deformation is directly applied to the mate-rial, the implicit FE solver is shown to solve more quickly.The ratio of explicit runtime to implicit runtime is approx-imately 2.35 for the simple 2D analyses. The ratio for anal-yses in 3D is approximately double that value. Theinclusion of time increment sizes in Tables 1 and 2 revealsthe influence of different loading conditions on similarmodels. It is noteworthy, when comparing the tension anal-yses to the bending analyses in Table 1, that the bending

Page 12: Comparison of the implicit and explicit finite element methods using crystal plasticity

Appendix

492 F.J. Harewood, P.E. McHugh / Computational Materials Science 39 (2007) 481–494

loading has a far greater effect on the explicit analyses thanthe implicit analyses. A similar trend is clear in the 3Danalyses from Table 2. The opposite is true of the contactanalyses; the introduction of contact has little effect on thesize of the explicit time increments but significantly reducesthe size of the implicit increments.

As regards the implicit solution iteration process, foreach increment, ABAQUS chooses an initial guess, utþDt

0 ,assuming an incrementally linear response for the material,based on the tangent stiffness calculated at the end of theprevious increment ðKðut

finalÞÞ. In the case of a linear elasticmaterial this would provide the correct solution directly.For non-linear problems this should yield a good initialguess for small increments, Dt. Increasingly non-linearproblems require increasingly small time increments tomaintain solution accuracy. Within the constraints ofABAQUS, in the context of the scope of this paper, fewoptions are provided for improving the process. In theproblems of interest here the non-linearities due to contactand material response are of such a severity as to requiretime step sizes far smaller than are practically usable togenerate solutions with the implicit method.

The explicit solver is better suited to deal with complexcontact and sliding conditions, particularly in cases of largeelement deformation. In the contact analyses that include arigid body the runtimes using the implicit code are shorterthan the explicit code. This disguises the fact that the explicitcode actually solves the contact condition more quickly. Thesecond 2D contact analysis involving more complex contactconditions and greater element deformation provides aclearer indication of the benefits of the explicit solver.

It is interesting to note the increase in solution timeswhen a more rate-independent material is used in the 2Dtension analyses. The small time increments used in theexplicit solver ensure that the highly non-linear materialbehaviour is dealt with, with a minimal increase in time.This is not the case with the implicit solver where smallertime increments must be employed to achieve convergence.

When one considers the high multiple processor effi-ciency that is achieved by the explicit solver compared tothe implicit solver for every situation considered in thisstudy, the former option would prove to be the morefavourable when the user has a multiple processor com-puter at his/her disposal. Given the ongoing technologicaladvances, multiple processor computers are becomingmore commonly used. Therefore it is important to recog-nise the link between the speedup efficiency, the numberof processors assigned to solve an analysis, and the sizeof the model. The optimal number of processors shouldbe determined for each analysis so that a high parallelisa-tion efficiency can be maintained.

Acknowledgements

The authors wish to acknowledge funding fromEmbark, Irish Research Council for Science, Engineeringand Technology: Funded by the National DevelopmentPlan. The simulations in this work were performed onthe SGI 3800 high performance computer at NUI, Galway.

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