and Structures Efficiency and effectiveness of implicit

Original Article

Journal of Intelligent Material Systemsand Structures2016, Vol. 27(3) 384–402� The Author(s) 2015Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1045389X15592483jim.sagepub.com

Efficiency and effectiveness of implicitand explicit approaches for the analysisof shape-memory alloy bodies

Giulia Scalet1, Ferdinando Auricchio1 and Darren J Hartl2

AbstractThe increasing number of applications incorporating shape-memory alloy (SMA) components motivates the developmentof three-dimensional constitutive models to enhance their analysis and design. These models only reach their full utility ifthey are then implemented into numerical (e.g. often finite-element-based) frameworks. The present article addresses atopic rarely considered in the myriad of SMA computational analysis works in the literature: the analysis of the time andaccuracy of implementation options. In particular, this work proposes to compare the performance of the implicit andexplicit integration methods for two common three-dimensional phenomenological constitutive models: (i) the modelby Lagoudas et al.; and (ii) the model by Auricchio et al. available in all installations of Abaqus. In doing so, the presentwork develops and implements an explicit algorithm for the model by Lagoudas et al. for the first time. The investigatedmodels are compared in a chosen benchmark boundary value problem analysis considering both thermally inducedactuation and isothermal stress-induced transformation of an SMA beam. The performance of the methods in terms ofanalysis time and parallelization efficiency are also investigated.

KeywordsShape-memory alloys, constitutive modeling, finite element analysis, actuator, explicit integration, implicit integration, effi-ciency, parallelization

Introduction

The number of applications incorporating shape-memory alloy (SMA) components both in developmentand in use is continuing to increase. While many suchapplications incorporate SMA wire forms, a growingnumber are making use of more complex forms, suchas wound springs, beams, and torque tubes (Mammanoand Dragoni, 2014; Sgambitterra et al., 2014; Spaggiariand Dragoni, 2014; Spaggiari et al., 2014). These formsare associated with non-homogeneous fields of solutionvariables (e.g. stress, strain, temperature, etc.), and fur-ther require the consideration of multiple componentsof higher order tensorial variables (i.e. stresses andstrains). Analysis of these components thus precludesthe use of the one-dimensional formulations that workso well for simpler forms (Brinson, 1993; Poorasadionet al., 2014), but rather require more advanced andcomplex modeling tools.

Many three-dimensional models have been proposed(Arghavani et al., 2011; Chemisky et al., 2011;Lagoudas et al., 2012; Liang and Rogers, 1992), andmany works exist which seek to develop, verify, vali-date, and utilize the implementations of these models.

Among the others, several works have proposed theimplementation of SMA models into numerical soft-ware (Arghavani et al., 2011; Auricchio et al., 2014;Bartel et al., 2011; Dunic et al., 2013; Kiefer et al.,2012; Peigney and Seguin, 2013; Stupkiewicz andPetryk, 2012; Zaki, 2012). Three-dimensional analysis,that is, finite element analysis (FEA), has been per-formed on most proposed forms, even considering suchcomplexities as martensitic reorientation (Arghavaniet al., 2010; Marketz and Fischer, 1996; Panico andBrinson, 2007; Thamburaja, 2005), porosity (Entchevand Lagoudas, 2002; Panico and Brinson, 2008), fullthermal coupling (Morin et al., 2011; Nae et al., 2003),plasticity (Hartl and Lagoudas, 2009; Peng et al., 2012;Saint-Sulpice et al., 2009), visco-plasticity (Chemiskyet al., 2014; Hartl et al., 2010), anisotropy (Sedlak

1Dipartimento di Ingegneria Civile e Architettura, Universita di Pavia,

Italy2Department of Aerospace Engineering, Texas A&M University, USA

Corresponding author:

Darren Hartl, Department of Aerospace Engineering, Texas A&M

University, College Station, TX 77843-3409, USA.

Email: [email protected]

at Università degli studi di Pavia on April 27, 2016jim.sagepub.comDownloaded from

http://jim.sagepub.com/

et al., 2012), and finite strain (Panoskaltsis et al., 2014).Motivation for these analysis studies has includedexperimental interpretation (Gautier and Patoor, 1997;Nohouji et al., 2014; Rejzner et al., 2002), analysis-driven iterative design (Bhattacharyya et al., 1995;Chemisky et al., 2009; Godard et al., 2003; Jardineet al., 1997; Seeley and Chattopadhyay, 1993; Strelecet al., 2003; Troisfontaine et al., 1999), and topologyoptimization (Langelaar et al., 2011).

However, one topic is rarely considered in the myr-iad of analysis works in the literature: the discussion ofanalysis time and computational efficiency associatedwith the implementation. These aspects of analysis areof grave importance to the engineering analyst, SMAapplications developer, and practicing research engi-neer. This is especially true in the case of iterativedesign, in which hundreds or thousands of analysesmay be required. Some designers, so dissuaded by thesupposed prohibitive runtimes of the non-linear codespreviously referenced, have even proposed andattempted to apply over-simplified elasticity or thermo-elasticity based models for SMA materials, though suchmodels can be wildly inaccurate (Widdle Jr et al.,2009).

The purpose of the present work is to address insome detail the issue of analysis time, parallelizationefficiency, and, to a lesser extent, accuracy by criticallycomparing the performance of implicit and explicitsolution methods.

The implementation framework is restricted to theAbaqus Unified FEA Suite (Abaqus/Standard implicitand Abaqus/Explicit solvers) (Abaqus, 2012). Two phe-nomenological constitutive models formulated in threedimensions and associated implementations are consid-ered: (i) the model developed and enhanced byLagoudas et al. (2012) (herein referred to as the ‘‘TexasA&M University’’, or ‘‘TAMU’’ model); and (ii) themodel by Auricchio et al. available in all installationsof Abaqus (Auricchio and Taylor, 1997; Auricchioet al., 1997) (referred to as the ‘‘Abaqus Auricchio–Taylor’’, or ‘‘AAT’’ model).

While the implementation of the AAT model isavailable in both explicit and implicit schemes withoutthe need for custom-installed user subroutines (Abaqus,2012), the original implementation of the TAMUmodel(i.e. prior to this work) has only been made available inan implicit formulation scheme, as proposed in Boydand Lagoudas (1994). The present work proposes anddevelops an explicit algorithm for the TAMU model,implemented as a user subroutine (VUMAT) for usewith the finite element (FE) solver Abaqus/Explicit.Such a scheme is conceptually simple.

It is important to remark that explicit schemes forSMA models are still generally lacking in the literatureat the present time, thus limiting the applicability ofthese models. Recently, Stebner and Brinson (2013)presented a novel explicit integration scheme for the

rate-independent model by Panico and Brinson (2007),which is easy to implement and use, and which reducesthe computational expense when compared with theoriginal implementation. The work by Thamburaja andNikabdullah (2009) presents a non-local and thermo-mechanically coupled constitutive model for polycrys-talline SMAs, implemented in Abaqus/Explicit.However, it considers simple uniaxial tests and theanalysis of an SMA sheet. Pan et al. (2007) developedan isotropic-plasticity-based constitutive model for ini-tially martensitic SMAs exhibiting martensitic reorien-tation and the shape-memory effect and implemented itin Abaqus/Explicit.

Explicit implementations of SMA models may bebeneficial, especially since they can be utilized in a glob-ally explicit framework. Further, in the Abaqus FEASuite, the only option for fully coupled thermomechani-cal dynamic analysis is via explicit integration. Such ananalysis option would be important in the analysis ofshock and impact in highly thermally coupled SMAmaterials. Moreover, difficulties have been experiencedwith implicit computational mechanics methods andassociated analytical frameworks, especially regardingglobal numerical convergences. This can be particularlyproblematic when small localized regions in a FEAmesh exhibit unstable responses, be they structural (e.g.local buckling) or numerical. As an example, FE simu-lations employing an implicit time integration schemefailed to converge for a number of analyses associatedwith specific SMA mesh-based self-folding sheet config-urations due to localized elastomeric buckling (Peraza-Hernandez et al., 2013). Moreover, explicit schemeshave significant advantages for FEA models with agreat number of degrees of freedom (DOFs) and espe-cially complex contact conditions. An example case ofthis has been observed in biomedical device analysis,where contact between tools and stent, possible self-contact in the final phase of crimping, and contactbetween stent and balloon each had to be considered(Martin and Boyle, 2013). One of the main advantagesof explicit schemes is that they do not require consider-ation of the global Jacobian matrix, the formulationand solution of which is computationally expensive.However, explicit schemes present the disadvantage ofbeing conditionally stable: if the physical time scale isvery long (e.g. in quasi-static analyses), the analysismay require a prohibitively high number of time incre-ments; the advantages of the implicit method are obvi-ous for such types of analyses.

The models considered in the present article arecompared via full three-dimensional boundary valueproblem analyses. Specifically, determination of thequasi-static response of a perforated SMA beam com-ponent is chosen as a benchmark analysis goal. Boththermally induced actuation (i.e. the shape-memoryeffect) and isothermal stress-induced transformation(i.e. an example of the pseudoelastic effect) are

Scalet et al. 385



investigated. For the purposes of simplicity and focus,this work restricts itself to the consideration of onlytwo boundary value problems (geometric configuration,loads, and kinematic constraints) and does not considersuch fundamental FEA issues as variation in element sizeor basis functions order (i.e. so-called, h- and p-refine-ment). Also, because solvers such as Abaqus/Standardincorporate complex and generally well-constructed load-ing time increment adjustment algorithms, an investiga-tion of larger or smaller loading time increments is notconsidered. Rather, the maximum loading time incre-ment for each implicit analysis is fixed.

The performance of the implicit and explicit solversfor the considered problem in terms of analysis timeand parallelization efficiency is also investigated. Allanalyses herein are performed on a single NehalemXeon (2.66 GHz) node of a high-performance comput-ing cluster with 24 GB of RAM and up to 12 totalcores available. Analyses are performed using from 1to 12 cores; the effects of parallelization are described.Recall, however, that the presented results depend onthe computer platform used, the FEA environment, aswell as on the particular coding of the models.

Despite this, the work is intended as a unique andeven practical guide for the active materials engineeringcommunity. In describing multiple implementation andanalysis options, it is one of the first works to assesscomputational efficiency in a direct manner. It is hopedthat this will have a direct impact with regard toresearchers working in this area. The subject of paralle-lization efficiency in particular, addresses the issue ofshared resource allocation, illustrating clearly thediminishing returns involved in committing an increas-ing number of CPUs and licenses to a given SMA anal-ysis problem. Finally, the work also provides anexample and guidance for the consideration of research-ers developing new SMA constitutive models and theirimplementations, many of whom often choose theeasiest numerical options.

The present article is organized as follows. ‘‘TAMUmodel formulation review’’ covers the main equations ofthe TAMU model, and the following sections present itstime-discrete framework, including the implicit integra-tion scheme and the newly proposed explicit integrationalgorithm, and the considered boundary value problemsand analysis options. Then, ‘‘Thermally induced actua-tion response’’ and ‘‘Pseudoelastic response’’ present theresults of the FEA considering both the pseudoelasticand shape-memory responses, respectively. The impactof parallel processing on the comparison between themethods is also given. Finally, conclusions are given.Details about Abaqus analysis options and the adoptedmeasures to quantify the parallelization effectiveness arereported in a separate Appendix.

TAMU model formulation review

The TAMU model used herein is based on the formula-tion originally presented by Boyd and Lagoudas (1996)and then extended and improved by Lagoudas et al.(2012). The model is able to take into account thedescription of smooth transitions in the thermomecha-nical response at the beginning and the end of phasetransformation, the variation of transformation strainmagnitude with applied stress level, and the variationof the transformation hysteresis area with applied stresslevel due to the stress-dependent critical thermody-namic force for transformation, all of which are com-monly observed experimentally.

The model assumes the stress s and the absolutetemperature T as control variables, and the transforma-tion strain et, the total martensitic volume fraction j,and the transformation hardening energy gt as internalones. The total martensitic volume fraction j isbounded such that 0 � j � 1.

In this work, we will assume that the temperature isknown at all material points at all times. However, themodel has also been implemented in a fully thermome-chanically coupled framework for simulations consider-ing both stress and temperature as unknowns(Lagoudas et al., 2012).

The model assumes small strains and an additivedecomposition of the total strain into elastic, thermal,and transformation contributions, as follows

e=S(j)s+a T � T0ð Þ+ et ð1Þ

where a is the thermal expansion tensor, T0 a referencetemperature, and S(j) = SA+ j (SM 2 SA) the com-pliance tensor evaluated by the rule of mixtures andsuch that S(j) = C21(j), C being the stiffness tensor.

The evolution equation for the transformation strainet is defined as follows

_et = _jLt ð2Þ

where

Lt =Lt

fwd =3

2Hcur(�s)

s

�sif _j.0,

Ltrev =

et�r

jr if _j\0

8><>: ð3Þ

with s=s � 13 tr(s)I and �s=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3=2 s : s

p. In the above

definition, Ltfwd and Lt

rev are the transformation direc-tion tensors during forward and reverse transformation,respectively, while et 2 r and jr are the macroscopictransformation strain and the martensite volume frac-tion at the reverse transformation, respectively. Theuniaxial transformation strain magnitude for full trans-formation, Hcur(�s), is defined as

386 Journal of Intelligent Material Systems and Structures 27(3)



Hcur �sð Þ=Hmin if �s� �scrit,

Hmin + Hsat � Hminð Þ 1� exp�k �s��scritð Þ� �if �s.�scrit

�ð4Þ

Here, Hmin corresponds to the observable uniaxial two-way shape-memory effect; Hsat describes the ultimatetransformation strain given in an uniaxial loading (i.e.the maximum recoverable transformation strain gener-ated such that increases in stress magnitude do notincrease the transformation strain). Additionally, �scrit

denotes the critical Mises equivalent stress below whichHcur = Hmin. The parameter k controls the rate atwhich Hcur exponentially evolves from Hmin to Hsat.

The transformation function ft is denoted as follows

ft =ftfwd =pt

fwd � Y tfwd if _j.0,

ftrev = � pt

rev � Y trev if _j\0

�ð5Þ

where the critical thermodynamic driving force Yt (s) is

Y t sð Þ= Y t0 +Ds : Lt ð6Þ

Here, Y t0 is a positive constant and D an additional pos-

itive model parameter that captures the stress depen-dency of the critical thermodynamic force (Lagoudaset al., 2012). The total thermodynamic force pt conju-gate to j is

pt =s : Lt +1

2s : DSs+ rDs0T � rDu0 � f t jð Þ ð7Þ

where the operator D denotes the difference in a mate-rial constant between pure martensite and pure auste-nite, r is the density, and s0 and u0 denote the specificentropy and internal energy at the reference state,respectively.

The hardening function ft(j), introduced in equation(7), is defined so as to take into account a smooth tran-sition from elastic to transformation response and isgiven as follows

f t jð Þ=f tfwd jð Þ= 1

2a1 1+ jn1 � 1� jð Þn2ð Þ+ a3 if _j.0,

f trev jð Þ= 1

2a2 1+ jn3 � 1� jð Þn4ð Þ � a3 if _j\0

(

ð8Þ

such thatR 1

0f tfwddj +

R 0

1f trevdj = 0. The parameters a1,

a2, a3 are positive material constants and the exponentsn1, n2, n3, and n4 assume real number values in the inter-val (0,1].

The Kuhn–Tucker conditions for the transformationfunctions ft

fwd and ftrev are, respectively

_j � 0, ftfwd� 0, _jft

fwd = 0,_j� 0, ft

rev� 0, _jftrev = 0

�ð9Þ

The calibration of the model parameter set (rDs0,rDu0, a1, a2, a3, Y

t0, and D) in terms of conventional

SMA phase diagram properties (e.g. transformationtemperatures and stress influence coefficients) isdescribed in detail in Section 4 of Lagoudas et al.(2012).

Time discrete framework for the TAMUmodel

As the TAMU model has been custom implemented,and especially as a point of comparison with the novelexplicit implementation given in ‘‘Explicit integrationscheme (b = 0)’’, we briefly present the implicit algo-rithmic treatment of the governing equations of theTAMU model in ‘‘Implicit integration scheme(b = 1)’’. For the sake of notation simplicity, the con-vention establishes subscript n for all the variables eval-uated at the last time tn, and thus n+1 for all thevariables to be computed at the next time tn+1.

In order to discretize equation (2) in time, we can usethe general trapezoidal rule (Quarteroni et al., 2007), asfollows

etn+ 1 = et

n + jn+ 1 � jnð Þ 1� bð ÞLtn +bLt

n+ 1

� �ð10Þ

where b is a continuous algorithmic parameter thatranges from 0 to 1 and is often chosen and then heldconstant during analysis (Boyd and Lagoudas, 1996;Lagoudas et al., 2012; Ortiz and Pinsky, 1981; Qidwaiand Lagoudas, 2000). In particular, the case b = 0leads to a set of explicit equations corresponding to thewell-known forward Euler integration, while b . 0results in a set of implicit equations. For example, thecase b = 1 leads to the well-known backward Eulerintegration. Both the cases b = 1 and b = 0 are pre-sented in the following.

Implicit integration scheme (b = 1)

The implementation of the constitutive model followsthe procedure first described in detail in Qidwai andLagoudas (2000) and then updated in Lagoudas et al.(2012), where backward integration is performed bysetting b = 1 in equation (10), which gives

etn+ 1 = et

n + jn+ 1 � jnð ÞLtn+ 1 ð11Þ

The convex cutting plane algorithm form of thereturn mapping algorithm (Ortiz and Simo, 1986;Qidwai and Lagoudas, 2000) is used for implementa-tion into the current three-dimensional FEA frame-work. This iterative approach to solve the implicitsystem of equation (11) makes the highly enablingassumption that the transformation direction changesonly slightly from loading increment to loading incre-ment (i.e. the loading is relatively proportional). Thus

Scalet et al. 387



equation (11) can be written in the following iterativeform (see Lagoudas et al. (2012))

et(k + 1)n+ 1 = e

t(k)n+ 1 +Dj

(k)n+ 1Lt(k)

n+ 1 ð12Þ

where Dj(k)n+ 1 = j

(k + 1)n+ 1 � j

(k)n+ 1.

An initial thermoelastic prediction assumes that thetransformation strain increment is null. If the predictedthermoelastic state violates the transformation criter-ion, a series of transformation corrections is applied torestore this consistency condition. In particular, the cor-rection in j at the given iteration k yields (Lagoudaset al., 2012)

Djkð Þ

n+ 1 = � ft kð Þn+ 1

Atð13Þ

where

At = ∂jft kð Þn+ 1 � ∂sf

t kð Þn+ 1 : C

t kð Þn+ 1 DSs

kð Þn+ 1 +L

t kð Þn+ 1

� ð14Þ

The iterative scheme continues until ft k + 1ð Þn+ 1 is smaller

than some tolerance. If jk + 1ð Þ

n+ 1 reaches a bound of 0 or1, it is set equal to the violated bound.

In addition to requiring the updated thermomecha-nical state of the material (updated stress and internalvariables), the global solver also requires the tangentmodulus (or tangent stiffness tensor), which defines thecurrent rate of change of stress with a change in totalstrain. Coupled thermal and mechanical analysis (i.e.thermomechanical analysis including heat transfer) alsorequires the thermal tangent modulus, which definesthe current rate of change of stress with a change in

temperature. The reader is referred to Lagoudas et al.(2012) for the derivation of these two tensors.

Table 1 provides a summary of the full implicit algo-rithm (convex cutting plane), needed to integrate theconstitutive relations in an FE framework (Lagoudaset al., 2012). Such a scheme has been implementedwithin an Abaqus/Standard user material subroutine(UMAT) (Abaqus, 2012).

Explicit integration scheme (b = 0)

Forward integration is performed by setting b = 0 inequation (10), which gives

etn+ 1 = et

n + jn+ 1 � jnð ÞLtn ð15Þ

where a purely explicit forward integration depends onthe ability to approximate Lt and calculate jn+1 usingonly data from the solution at the previous time step.

As before, a thermoelastic prediction first assumesthat the transformation strain increment is null. If thepredicted thermoelastic state violates the transforma-tion criterion, jn+1 must be calculated. We derive theneeded relation using the transformation consistencyconditions introduced in equation (9). Enforcing thiscriterion in the current time step at the initial iterationk = 0, we have

ft 0ð Þn+ 1 =ft Tn+ 1,s

0ð Þn+ 1, jn+ 1

� = 0 ð16Þ

In the special case of n1 = n2 = n3 = n4 = 1 (seeequation (8)), it is possible to invert equation (16) andform instead jn+ 1 = j Tn+ 1,s

0ð Þn+ 1

� , that is

Table 1. Implicit backward Euler integration scheme for the TAMU model (Lagoudas et al., 2012) via the convex cutting planealgorithm form of the return mapping algorithm (Ortiz and Simo, 1986; Qidwai and Lagoudas, 2000).

(1) Initialize: Let k = 0, j0ð Þ

n+ 1 = jn, et 0ð Þn+ 1 = et

n, C0ð Þ

n+ 1 =Cn

(2) Elastic prediction:

(a) s0ð Þ

n+ 1 =C0ð Þ

n+ 1 en+ 1 � a Tn+ 1 � T0ð Þ � et 0ð Þn+ 1

h i(b) Evaluate f

t 0ð Þn+ 1

(c) IF ft 0ð Þn+ 1<0 then EXIT (elastic response)

(3) Transformation correction:

(a) ELSE IF ft 0ð Þn+ 1.0 CONTINUE

(b) Find Djkð Þ

n+ 1 via equation (13)

(c) jk+ 1ð Þ

n+ 1 = jkð Þ

n+ 1 +Djkð Þ

n+ 1

(d) etðk+ 1Þn+ 1 = e

t(k)n+ 1 +Dj

(k)n+ 1L

t(k)n+ 1

(e) IF 0<jk+ 1ð Þ

n+ 1 <1 then CONTINUE, ELSE set jn+ 1 to violated bound; recalculate Djkð Þ

n+ 1, jk+ 1ð Þ

n+ 1 , and et k+ 1ð Þn+ 1

(f) Update sk+ 1ð Þ

n+ 1 =Ck+ 1ð Þ

n+ 1 en+ 1 � a Tn+ 1 � T0ð Þ � et k+ 1ð Þn+ 1

h i(g) IF f

t k+ 1ð Þn+ 1 .tol THEN CONTINUE, ELSE EXIT

(h) k k+ 1 and GOTO 3b




jn+ 1 =

1a1½(1� D)s

0ð Þn+ 1 : Lt

n+ 1 +12

s0ð Þ

n+ 1 : DSns0ð Þ

n+ 1

+ rDs0Tn+ 1 � rDu0 � a3 � Y t0� if _j.0,

1a2½(1+D)s

0ð Þn+ 1 : Lt

n+ 1 +12

s0ð Þ

n+ 1 : DSns0ð Þ

n+ 1

+ rDs0Tn+ 1 � rDu0 + a3 + Y t0� if _j\0

8>>><>>>:

ð17Þ

where the transformation strain direction tensor Ltn+ 1

has a known functional form per equation (3), that is,

Ltn+ 1 =Lt s

0ð Þn+ 1

� .

Using the forward Euler explicit methodology andrecalling that Tn+1 is provided by the global solver, wethen assume that for small increments

jn+ 1’j Tn+ 1,s0ð Þ

n+ 1

� . If jn+1 reaches a bound of 0

or 1, it is set equal to the violated bound. Then, we canwrite the explicit forward Euler integration of the trans-formation strain evolution equation as follows

etn+ 1 = et

n + j Tn+ 1,s0ð Þ

n+ 1

� � jn

h iLt

n+ 1 ð18Þ

Equation (18) allows for simplified coding and rapidcalculation, but very small time steps are required toavoid accumulation of error and to achieve significantaccuracy. Further, the explicit method is conditionallystable and the stability limit requires that the maximumtime increment must be less than a critical value. A pos-sible effect of violating this stability limit is that theintegration algorithm may not converge to the correctsolution (Lagoudas et al., 2012; Ortiz and Pinsky, 1981;Qidwai and Lagoudas, 2000). On the other hand, theglobal explicit formulation naturally does not requirethe tangent matrix (local or global).

Table 2 provides a summary of the full explicit algo-rithm needed to integrate the constitutive relations inan FE framework. Such a scheme has been implemen-ted within an Abaqus/Explicit user material subroutine(VUMAT) (Abaqus, 2012) for the current model forthe first time in this work.

Boundary value problem and analysisoptions

Here we describe the benchmark boundary value prob-lem employed for the purposes of comparative analysisin addition to the general analysis options adopted toperform the FEA using both implicit and explicit sol-vers. The problem under investigation considers bothpseudoelastic and shape-memory effects in an SMAbeam, each of which is important to practicing engi-neers and researchers. The reader is referred to theAppendix for further details about Abaqus paralleliza-tion options and the measures adopted to quantify par-allelization effectiveness.

Boundary value problem: Geometric configuration

The present work considers the quasi-static analysis ofa generally rectangular SMA beam; the experimentalcharacterization and preliminary analysis of the beamconfigured in this study have been described in a previ-ous work where a high degree of accuracy in the globalpredictions was demonstrated (Gravatt et al., 2010).The geometric configuration of the beam is shown inFigure 1(a). The beam is 152 mm in total length,19.0 mm in total width, and has a thickness of6.35 mm. Round holes with a diameter of 3.175 mmand spaced 3.175 mm apart pass through the width ofthe beam, vertically centered relative to the beam thick-ness. The consideration of many round holes, each ofwhich require many elements, makes this a suitablylarge analysis problem of many DOFs.

In all analysis cases presented herein, the beam issubjected to centered three-point loading, where verticalsupports are spaced a total of 96.5 mm apart, as shownin Figure 1(a). A downward vertical force is applied atthe beam center, and all reported applied forces andvertical displacement responses (denoted F and u inFigure 1(a), respectively) are taken from this point.1

Table 2. Proposed explicit forward Euler integration scheme for the TAMU model.

(1) Initialize:Let etn+ 1 = et

n, Cn+ 1 = Cn

(2) Elastic prediction:

(a) s0ð Þ

n+ 1 =Cn+ 1 en+ 1 � a Tn+ 1 � T0ð Þ � etn+ 1

� �(b) Evaluate f

t 0ð Þn+ 1

(c) IF ft 0ð Þn+ 1<0 then EXIT (elastic response)

(3) Transformation correction:

(a) ELSE IF ft 0ð Þn+ 1.0 CONTINUE

(b) Find jn+ 1 = j Tn+ 1,s0ð Þ

n+ 1

� via equation (17)

(c) IF 0 <jn+ 1 < 1 then CONTINUE, ELSE set jn+ 1 to violated bound

(d) Update etn+ 1 = et

n + jn+ 1 � jnð ÞLtn+ 1

(e) Update Cn+ 1 = C(jn+ 1)

(f) Update sn+ 1 =Cn+ 1 en+ 1 � a Tn+ 1 � T0ð Þ � etn+ 1

� �(g) EXIT

Scalet et al. 389



For the FEA, symmetry along the beam width isexploited such that the total width of the computa-tional domain is (19.0 mm)/2 = 9.50 mm. However,symmetry about the mid-length is not assumed in thisbenchmarking FEA model to allow for considerationof general asymmetric loading (not considered here).For full three-dimensional analysis, we adopt the meshpresented in Gravatt et al. (2010). The body is discre-tized into 22,188 first-order hexahedral elements withreduced integration (Abaqus designation C3D8R). Thisrefinement was found in that previous work to be suffi-cient. This results in 30,632 nodes and 91,896 totalDOFs. The details of the mesh on the lateral surfacenear the holes are shown in Figure 1(b).

As an alternative technique, implicit plane strainanalysis is also considered. For this reduced-order anal-ysis, the lateral (i.e. perforated) surface of the beamcomprises the entire computational domain, which isdiscretized into 3698 second-order quadrilateral ele-ments with reduced integration (Abaqus designationCPE8R). This resulted in 12,474 nodes and thus 24,948total DOFs. Note that the mesh discretization on thissurface is defined to be identical for both three-dimensional and plane strain analyses.

Boundary value problem: Loading paths

Two thermomechanical loading paths associated withSMAs are applied to the same beam geometry and

considered in the proposed work. First, thermallyinduced actuation (i.e. the shape-memory effect) is con-sidered, which is directly motivated by the experimentalwork previously performed and reported (Gravattet al., 2010). The material properties assumed in thisanalysis are similar to those derived from comparinganalyses to experiments on an alternate (non-perfo-rated) beam composed of Ni60 Ti40 (Gravatt et al.,2010). Also motivated by experimental efforts, theapplied temperature changes are assumed to be uni-form over the entire computational domain. This set ofshape-memory analyses will be addressed in‘‘Thermally induced actuation response’’. Secondly, iso-thermal stress-induced transformation (i.e. the pseu-doelastic effect) is considered. For these analyses, thematerial properties are artificially altered to better rep-resent materials processed for this purpose, as Ni60 Ti40is not an appropriate alloy for pseudoelastic applica-tions. This set of analyses will be addressed in‘‘Pseudoelastic response’’. In each section, the globalstructural response and local solution field distributionswill be provided, and the effects of analysis paralleliza-tion will be addressed.

Analysis options

In the following sections, we investigate the perfor-mance of the implicit and explicit solvers in terms ofmodel response, analysis time, and parallelizationefficiency.

To this end, both the parallelization options pro-vided by the Abaqus/Standard implicit solver are con-sidered: (i) the integration of the constitutive equationsat each material point and the calculation of quantitiesat the element level; and (ii) the solution of the largesystem of coupled linear equations at the global level 2

(Abaqus, 2011). In contrast, in the Abaqus/Explicit sol-ver we adopt the domain-level method together withthe message passing interface (MPI) parallelizationmode. The number of subdomains is set equal to thenumber of processors used for analysis execution.3

As discussed in the introduction, the drawback ofthe explicit method is that it is conditionally stable andrequires very small time increments. Therefore, it isoften not practical to run quasi-static analyses usingtrue time scales as runtimes would be very large. Inparticular, the stable time increment for the investi-gated solid model, evaluated by Abaqus at the begin-ning of the analysis, is 6.28 3 1028 s. To artificiallyreduce the simulation runtime, we employ the methodof mass scaling and we artificially change the density ofthe modeled material4 (Abaqus, 2011). We adopt thesame mass scaling options for both the TAMU andAAT models to compare the results in terms of analysistime.

To quantify the parallelization effectiveness, twomeasures are proposed in the following sections: (i) the

Figure 1. Illustration of the analyzed perforated beam. (a)Beam configuration. (b) Details of the mesh (three-dimensionaland two-dimensional plane strain).




parallelization efficiency; and (ii) the normalized speed-up rate. Both measures are discussed and defined in theAppendix.

Table 3 is provided to summarize the nine types ofanalyses that have been performed in the current work.For each type, various numbers of cores on a singlecluster node (n = 1,2,4,8,12) are utilized to assess par-allelization efficiency. The results from each of theseanalyses will be presented in the following sections.

Thermally induced actuation response

We first focus on the prediction of the perforated beamresponse as a thermally driven actuator. This was thepurpose for which the original perforated beam wasphysically fabricated and experimentally characterized(Gravatt et al., 2010). The perforated actuator was oneof six original flexural actuators of various geometricconfigurations fabricated and tested, each having beencomposed of the same material and exposed to the samethermomechanical processing history. An iterative cali-bration process was used whereby the material proper-ties were tuned such that the experimental response of adifferent (non-perforated) beam was most accuratelypredicted by an FEA model of the same beam (Gravattet al., 2010). The final set of material properties associ-ated with this closest fit was assumed to describe theresponse of the material comprising the perforatedbeam. In the present work, we adopt these same mate-rial properties with the additional assumption thatn1 = n2 = n3 = n4 = 1 to equally compare explicitand implicit methods (cf. equations (8), (16), and (17)).The final properties are provided in Table 4.

The loading is one of three-point bending (seeFigure 1(a)), where two center forces of 738 and1130 N are applied. Regarding the mass scaling optionsdiscussed in ‘‘Analysis options’’, during explicit analysisthe minimum stable increment is set to 0.0005 s, andthe fixed time incrementation to 0.0001 and 0.000025 sduring heating and cooling, respectively. Since theAAT model is not intended for simulation of theshape-memory effect, especially for the complex geome-try and loading conditions considered, only the TAMUmodel as implemented via both the implicit and explicitschemes is considered.

Structural response

Figure 2(a) shows the beam deflection–temperaturecurves calculated via the implicit and explicit analysesfor the two loading levels (738 and 1130 N). The twocurves match well, verifying the newer explicit imple-mentation and validating the analysis parameters cho-sen.5 Only very small local oscillations are visibleduring heating (e.g. at T = 55 �C), which may indicatethe need for a smaller time increment of 0.000025 s.

The width-to-thickness ratio of the beam configura-tion combined with the goal of reduced computationaleffort motivates the consideration of a plane strainmodel for the analysis of the same problem. For thispurpose, the lateral surface mesh from the three-dimensional model (see Figure 1(b)) is used to form atwo-dimensional discretized domain as described in‘‘Boundary value problem: geometric configuration’’,and the implicit analysis is repeated. Comparison ofthe global deflection–temperature responses for thethree- and two-dimensional analyses is shown in Figure2(b). Clearly the plane strain analysis results in a stifferresponse during the initial elastic loading atT = 100 �C; this is further amplified in the recoverableinelastic response. This difference corresponds to amarked decrease in accuracy and clearly indicates theinappropriateness of a plane strain analysis assumptionfor the boundary value problem while justifying ouradoption of a more computationally expensive fullthree-dimensional analysis.

Local predictions of the Mises stress are illustratedin Figure 3 for the free and symmetry surfaces (cf.Figure 1(a)) of the three-dimensional model at the endof cooling to T \ Ms under an applied load of 1130 N.Note especially the change in local Mises stress betweenthe flexure symmetry plane and the free surface,

Table 4. Model parameters associated with analysis ofperforated beam thermally induced actuation (TAMU model)(after Gravatt et al. (2010)).

Parameter(s) Value(s) or functional form(s)

r 6500 kg/m3

(Thermoelastic behavior)EA = EM 48 GPanA = nM 0.33a ;0/ �C

(Phase diagram)Ms, Mf 35 �C, 13 �CAs, Af 30 �C, 73 �CCM =CA� �

�s= 300 MPa13.8 MPa/ �C

(Maximum transformation strain and smooth hardening)

Hcur �sð Þ= 0; �s<69 MPa0:0095 1� exp (� 0:03(�s � 69))½ �; �s.69 MPa

�n1 = n2 = n3 = n4 1

Table 3. Summary of the nine analysis types considered in thepresent work (corresponding to 45 total runs).

Solver Thermally induced Pseudoelastic

Abaqus/Standard

standard_parallel=all

TAMU TAMU, AAT

standard_parallel=solver

TAMU TAMU, AAT

Abaqus/Explicit

TAMU TAMU, AAT

Scalet et al. 391



corresponding to a change in transformation strainmagnitude generated at the end of cooling, and furtherdemonstrating the need for three-dimensional analysis.

Advantages of parallelization

The effect of parallelization can be observed in Figure4 in terms of wallclock time versus number of cores.The total single core runtime for the implicit analysiswas 55 min. In contrast, the total single core runtimefor the explicit analysis was nearly 85 h, clearly animpediment to any type of iterative design study oroptimization, should one be needed. The shorter impli-cit runtime can be explained by the size of the incre-mental global stiffness matrix which is relatively smalldue to the size of the considered beam model. Mostimportantly, it can been seen that full 12-core paralleli-zation6 of the implicit and explicit analyses significantlyreduces the computational time by factor of 5.7 and7.7, respectively. On the other hand, the much lessaccurate implicit plane strain analysis (;24 kDOFs)reduces the total single core runtime to about 5 min,thus reducing computational time. However, althoughthe savings in computational time are substantial, theanalysis of this two-dimensional configuration does notwarrant the utilization of plane strain analysis (seeFigure 2(b)).

To quantify the parallelization effectiveness, we firstconsider the parallelization efficiency hp, given as

hp =

ffiffiffiffiffiffiffiffiffiffiffiffiffit1

n tn� 1

n

1� 1n

sð19Þ

where t1 and tn are the wallclock times required to com-plete an analysis using 1 and n cores, respectively. Note

Figure 2. Thermally induced actuation response of the perforated SMA beam under two applied loads of 738 and 1130 N. (a)Global temperature–deflection curves predicted by the TAMU model in its implicit and explicit formulation (three-dimensionalmodel). (b) Comparison between three- and two-dimensional implicit analyses.

Figure 3. Thermally induced actuation response of theperforated SMA beam: comparison of Mises stress distributionsat the end of cooling to T \ Ms under an applied load of1130 N. See Figure 1(a) for an illustration of free and symmetrysurfaces.




that an efficiency of 100% indicates that as the numberof dedicated cores doubles, the analysis time is halved.As expected from examination of Figure 4, paralleliza-tion of the implicit analysis is less efficient than theexplicit. It is also important to note that, although par-allelization of the solver only provides some reductionin the required wallclock time, it is not nearly as effec-tive as full parallelization. Clearly, the parallelization ofelement operations, including the important calculationof the constitutive response via the UMAT, is essentialto effective parallelization of such analyses. For theseanalyses, hp is plotted as a function of increasing coreutilization in Figure 5(a).

It is observed in all cases that the efficiency decreasesas the number of utilized cores increases, as expected.This raises an important question: at what point isincreasing core utilization no longer sufficiently benefi-cial so as to justify the increased dedication of availableand often shared computational resources? To answerthis, the normalized speed-up rate together with condi-tion (23) (see the Appendix) is graphically illustrated in

Figure 5(b). A user might therefore use this conditionto determine that parallelization at or beyond eightcores is not sufficiently beneficial to warrant theincreased dedication of limited software (i.e. Abaqus)licenses.

Pseudoelastic response

We now focus on the prediction of the pseudoelasticresponse of the perforated SMA beam at a constanttemperature of 27 �C. For the TAMU and AAT mod-els we adopt the material parameters reported inTables 5 and 6, respectively. For the AAT model, weemploy the user-material subroutines already availablein the Abaqus material library, which are invoked bynaming the material in a very specific way(ABQ_SUPER_ELASTIC_1 and ABQ_SUPER_ELASTIC_N3D in Abaqus/Standard and Abaqus/Explicit, respectively) (Abaqus, 2012). The results forthe AAT model are used as a reference for the TAMUmodel results in terms of material response, analysis

Figure 4. Thermally induced actuation response of the perforated SMA beam: effect of increasing number of cores on analysiswallclock time. (a) Implicit analysis times (both parallelization options). (b) Explicit analysis times. (c) Comparison between implicitand explicit times.

Scalet et al. 393



time, and computational efficiency. Accordingly,Figure 6 reports the stress–strain diagram with thecurves obtained with both the implicit and explicit inte-gration schemes for both the models by simulating anuniaxial tensile test with an internal subloop at a con-stant temperature of 27 �C. The curves match well, andthus additional model comparisons will be meaningful.Slightly different responses of the two models areobserved in the internal subloop.

Structural response

A force of 3030 N is applied on the SMA beam. Asregards the mass scaling options discussed in ‘‘Analysisoptions’’, the minimum stable time increment is set to0.0005 s and the time incrementation to 0.0001 s.

Figure 7 shows the analysis predictions for the pseu-doelastic response of the perforated beam in terms ofapplied force and vertical displacement. The curvesgenerally match well, with the exception of the resultscorresponding to the implicit AAT model. Figures 8and 9 report local predictions at the free and symmetrysurfaces of the Mises stress and martensite volume frac-tion, respectively, at the end of full loading to 3030 N.As observed, the martensite volume fraction reaches amaximum value of ;0.75 in all the analyses. This canexplain the differences observed in Figure 7 duringreverse transformation for the two models, similar tothose observed in the internal subloop of Figure 6. Theimplicit and explicit analyses of the AAT model presentslightly different distributions of the martensite fraction

Figure 5. Thermally induced actuation response of the perforated SMA beam: two measures of parallelization effectiveness. (a)The parallelization efficiency (equation (19)) is plotted as a function of increasing core utilization. (b) The normalized speed-up rate isplotted as a function of increasing core utilization. Condition (23) is represented by the horizontal dotted line.

Table 6. Model parameters associated with analysis of aperforated beam experiencing pseudoelastic deformation (AATmodel); see Abaqus (2012) for parameter definitions.

Parameter(s) Value(s)

r 6500 kg/m 3

(Thermoelastic behavior)EA,EM 55 GPa, 46 GPanA = nM 0.33

(Phase diagram)sS

L,sEL 407 MPa, 518 MPa

sSU,s

EU 222 MPa, 148 MPa

ds

dT

�L

=ds

dT

�U

7.4 MPa/ �C

T0 27 �C

(Maximum transformation strain)eL = eL

V 0.056

Table 5. Model parameters associated with analysis of aperforated beam experiencing pseudoelastic deformation(TAMU model).

Parameter(s) Value(s) or functional form(s)

r 6500 kg/m 3

(Thermoelastic behavior)EA,EM 55 GPa, 46 GPanA = nM 0.33a ;0/ �C

(Phase diagram)Ms,Mf 228 �C, 243 �CAs,Af 23 �C, 7 �CCM =CA� �

�s= 300 MPa7.4 MPa/ �C

(Maximum transformation strain and smooth hardening)Hcur �sð Þ=H 0.056n1 = n2 = n3 = n4 1




in the central hole. Note also the change in local Misesstress and martensite fraction between the flexure sym-metry plane and the free surface.

Figures 10(a) and (b) report the number of loadingtime increments and iterations required during theimplicit analyses of the TAMU and AAT models. Thetwo figures also contain light-green regions to highlightwhen transformation is occurring (i.e. _j 6¼ 0). It isobserved that time increments decrease and iterationsincrease during transformation. Moreover, it isobserved that the AAT model requires smaller incre-ments during some portions of both loading andunloading and a larger number of iterations to con-verge during unloading.

Advantages of parallelization

The effect of parallelization can be observed inFigure 11 in terms of wallclock time versus number ofcores. The total single core runtime for the implicit

analysis is about 45 min for both models. The TAMUmodel shows slightly lower wallclock times than theAAT model for the implicit analysis. This could be dueto the fact the AAT model requires smaller increments

Figure 8. Pseudoelastic response of the perforated SMA beam:comparison of Mises stress distributions at the end of fullloading to 3030 N at 27 �C. See Figure 1(a) for an illustrationof free and symmetry surfaces.

Figure 6. Isothermal pseudoelastic uniaxial tensile test with aninternal subloop (27 �C); comparison between implicit andexplicit integration schemes for the TAMU and AAT models.

Figure 7. Pseudoelastic response of the perforated SMA beamat a constant temperature of 27 �C.

Scalet et al. 395



Figure 9. Pseudoelastic response of the perforated SMA beam: comparison of martensite volume fraction distributions at the endof full loading to 3030 N at 27 �C. See Figure 1(a) for an illustration of free and symmetry surfaces.

Figure 10. Pseudoelastic response of the perforated SMA beam: comparison between the implicit analyses of the TAMU and AATmodels, in terms of number of (a) loading time increments and (b) iterations. Recall that a total of 100 time units were used forloading and unloading.




during some portions of both loading and unloadingand a larger number of iterations to converge duringunloading to achieve convergence, as observed in‘‘Structural response’’. In contrast, the total single-coreruntime for the explicit analysis is 22 h for the TAMUmodel and about 20 h for the AAT model. However,the differences between runtimes for the two modelsdecreases with increasing number of cores. Full 12-coreparallelization of the implicit and explicit analyses sig-nificantly reduces computational time by factors ofabout 6 and 10, respectively, for the TAMU model,and of 6.2 and 8, respectively, for the AAT model. It isalso important to note that, once again, although par-allelization of the solver only provides a reduction inthe required wallclock time, it is not nearly as effectiveas full parallelization. The results also show that thewall clock scaling degrades for larger numbers of cores.This is due to the fact that the problem size is getting

too small for the local calculations to dominate thecommunications costs. However, if we consider finermeshes, the performance would be improved.

The parallelization efficiency introduced in equation(19) for these analyses is plotted as a function ofincreasing core utilization in Figure 12. As expected,the explicit formulation clearly has more efficientspeed-up rates with multiple processors, relative to theimplicit formulation. Larger analyses, solved with theexplicit code, are known to show an even better speed-up with multiple processors (Koric et al., 2009).Parallelization efficiency is higher for the TAMUmodel in its explicit formulation. Further, we note thatfor all cases the efficiency decreases as the number ofutilized cores increases, as expected.

The normalized speed-up rate together with condition(23) is graphically illustrated in Figure 12(c).Parallelization at or beyond eight cores once again does

Figure 11. Pseudoelastic response of the perforated SMA beam: effect of increasing number of cores on analysis wallclock time. (a)Implicit analysis times (both parallelization options). (b) Explicit analysis times. (c) Comparison between implicit and explicit times.

Scalet et al. 397



not appear sufficiently beneficial to warrant theincreased dedication of limited Abaqus licenses for thecurrent analysis problem.

Conclusions

Thanks to ongoing technological advances, complexSMA-based devices and multiple processor computersare both becoming much more common. The presentarticle has discussed one topic that is rarely addressedin the myriad of analysis works in the literature but isrelevant for SMA designers, engineers, analysts, andresearchers: the issue of analysis time and accuracy forthe simulation of SMA devices. In particular, this workhas compared the performance of the implicit andexplicit solution methods for two three-dimensionalphenomenological constitutive models: (i) the model by

Auricchio et al. which is actually an industry-standardmodel in the medical community; and (ii) the model byLagoudas et al. which is a leading actuator model. Thepresent work has developed and implemented an expli-cit algorithm for the model by Lagoudas et al. for thefirst time. The considered models have been comparedin quasi-static analyses, involving both thermallyinduced actuation and isothermal stress-induced trans-formation of an SMA beam. The choice of focusing ononly quasi-static analyses is due to the fact that theseallow us to verify the proposed explicit scheme andprovide useful information for the engineering andresearch communities. The model by Lagoudas et al. inboth implicit and explicit formulations has revealed itscapability in predicting the response of SMA bodiesunder the considered geometry and complex loadingconditions. On the other hand, the model by Auricchio

Figure 12. Pseudoelastic response of the perforated SMA beam: measure of parallelization effectiveness. The parallelizationefficiency (equation (19)) is plotted as a function of increasing core utilization for the (a) implicit solver. (b) Comparison betweenimplicit and explicit solvers. (c) The normalized speed-up rate is plotted as a function of increasing core utilization. Condition (23) isrepresented by the horizontal dotted line.




et al. has been able to simulate only the pseudoelasticproperties of the SMA beam. Differences duringreverse transformation and in terms of vertical displa-cement (cf. Figure 7) between model pseudoelastic pre-dictions have been highlighted and are attributed todifferent behaviors during internal subloops. The per-formance of the two models and integration methodsin terms of analysis time and parallelization efficiencyhave been investigated. Such results strongly depend onthe computer platforms used, and the FEA environ-ment, as well as on the particular coding of the models.Particularly, the proposed explicit method has dis-played constantly high levels of parallelization effi-ciency compared to the implicit method for analysessolved using multiple processors. However, for the con-sidered problem and constitutive models, even thoughexplicit local constitutive integration is much faster,global integration/solution remains quite slow. It isexpected that the proposed explicit method wouldremain an inferior option for quasi-static analyses ofFE models of less than ;1 MDOFs. Ongoing work willfocus on testing the proposed explicit algorithm in thedynamic analyses for which it is usually preferred, andit is expected that a wider range of boundary valueproblems will be considered. As suggested by the pre-sented results, the choice of the time discretizationmethod is not an easy task and the results aim at pro-viding useful information according to the purpose ofthe present article. Therefore, designers, researchers,and development engineers need to take into accountthe link between the speed-up efficiency, the number ofassigned processors and the number of availablelicenses (cf. equation (23)), as well as the size of themodel when beginning the analysis and design of com-plex SMA components. Moreover, SMA researchersshould remain active in developing and/or implement-ing custom models, perhaps by working with academicteams to better balance computational efficiency andaccuracy.

Acknowledgement

Numerical analysis was performed via a research license pro-

vided by Simulia.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest withrespect to the research, authorship, and/or publication of thisarticle.

Funding

The author(s) disclosed receipt of the following financial sup-port for the research, authorship, and/or publication of thisarticle: The support of the NSF International Institute ofMaterials for Energy Conversion (award number #0844082)is acknowledged. This work is partially funded by Ministero

dell’Istruzione, dell’Universita e della Ricerca (project num-ber 2010 BFXRHS).

Notes

1. The force is applied as a distributed pressure in Abaqus(keyword *Pressure) on the blue symmetry strip of0.452 3 19.0 mm, depicted in Figure 1(a).

2. One of two options is specified during execution by invok-ing the standard_parallel=all or standard_parallel=sol-ver switch, respectively.

3. In this case the options are specified during execution by

invoking the parallel=domain, mp_mode=mpi,domains=number of processors used switch.

4. All mesh elements are scaled by a single factor so that theminimum stable increment becomes fixed. The mass scal-ing options are labeled by the keywords *Mass Scaling:Semi-Automatic, Whole Model, Fixed Mass Scaling,dt=, type=uniform. Then, the analysis is conducted byfixing the time incrementation, labeled by the keyword*Dynamic, Explicit, direct user control.

5. The reader is referred to Figure 9(a) in Gravatt et al.(2010) to assess model capability in accurately predictingthe experimental deflection–temperature curves for theconsidered case.

6. Recall that the analysis was limited to a single clusternode of 12 CPUs.

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Appendix: Abaqus analysis options

Recall that the purpose of the present work is to assessanalysis efficiency, especially with regard to analysistime. For this purpose, we restrict our discussion towallclock time, which is the total elapsed ‘‘real-world’’time required to complete the analysis. Naturally thisincludes the time to complete computing processes (i.e.CPU time), disk writing (i.e. I/O time), and any inter-core communication delays. To provide a reasonablebase of comparison across parallelization options, espe-cially when dealing with both explicit and implicit FEA,the following Abaqus analysis options are consistentlyutilized:

� The same model (i.e. same input file) is used forall analyses.

� Non-linear geometries are activated in all analy-ses (keyword *nlgeom = YES).

� Each model consists of six analysis steps, each ofwhich is taken to be 100 units of time in length.

� The maximum time increment in implicit analy-sis is taken to be 10 time units.

� The UMAT associated with the TAMU materialmodel includes a modification of the PNEWDT

Scalet et al. 401



variable to cut the maximum time incrementback to two time units when any material pointexperiences transformation. Further, the internalAbaqus time incrementation algorithms areallowed to cause additional cutbacks as needed.

� No ‘‘Field output’’ data and no ‘‘history’’ dataare written during time trials, reasonably limit-ing I/O time, regardless of time increment cut-backs, in order to equally compare explicit andimplicit analyses.

� The double precision option is used for all analy-ses. Such an option is specified during commandline execution by invoking the double=bothswitch.

To quantify the parallelization effectiveness, twomeasures are proposed in the work. First, the metricfor the effectiveness of parallelization is the paralleliza-tion efficiency hp, introduced in equation (19). Then,we introduce the normalized speed-up rate Dt=Dn as ametric for determining the relative utility of furtherparallelization. The normalized CPU usage n representsthe number of utilized CPUs n normalized by the totalnumber of CPUs ntot, available to all users sharing alimited resource, or

n=n

ntotð20Þ

Likewise, the normalized license usage l representsthe number of utilized Abaqus tokens l normalized bythe total number of tokens ltot, available to all users, or

l =l

ltotð21Þ

Finally, the normalized wallclock time t is given simplyas

t=tn

t1ð22Þ

In the absence of any other metric, it is proposed thatincreased parallelization be terminated when the nor-malized speed-up rate Dt=Dn falls below the normalizedCPU usage rate Dn=Dn or the normalized license usagerate Dl=Dn. Given the forms of equations (20) and (21)and further given that for Abaqus Dl = Dn, this condi-tion can finally be written for the current work as

Dt

Dn\max

1

ntot,

1

ltot

� �ð23Þ

As an example, we consider for the current set of analy-ses that the maximum number of shared CPUs is ;200,while the maximum number of shared Abaqus tokens is40. Thus, increased commitment of resources ceases tobe reasonable when Dt=Dn\1=40.




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