Comput. Methods Appl. Mech. Engrg. 249–252 (2012) 52–61

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Comput. Methods Appl. Mech. Engrg.

journal homepage: www.elsevier .com/locate /cma

An unconditionally energy-stable method for the phase field crystal equation

Hector Gomez ⇑, Xesús NogueiraGroup of Numerical Methods in Engineering, University of A Coruña, Department of Mathematical Methods, Campus de Elviña, s/n, 15192 A Coruña, Spain

a r t i c l e i n f o a b s t r a c t

Article history:Available online 14 March 2012

Keywords:Isogeometric AnalysisTime-integrationUnconditionally stablePhase-field crystal

0045-7825/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.cma.2012.03.002

⇑ Corresponding author.E-mail addresses: hgomez@ices.utexas.edu, hgom

com (H. Gomez).

The phase field crystal equation has been recently put forward as a model for microstructure evolution oftwo-phase systems on atomic length and diffusive time scales. The theory is cast in terms of an evolutivenonlinear sixth-order partial differential equation for the interatomic density that locally minimizes anenergy functional with the constraint of mass conservation. Here we propose a new numerical algorithmfor the phase field crystal equation that is second-order time-accurate and unconditionally stable withrespect to the energy functional. We present several numerical examples in two and three dimensionsdealing with crystal growth in a supercooled liquid and crack propagation in a ductile material. Theseexamples show the effectiveness of our new algorithm.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

Material properties at the meso- and macro-scales are to a largeextent controlled by complex microstructures exhibiting topologi-cal defects, such as, for example, vacancies, grain boundaries anddislocations. These defects are the result of complicated non-equi-librium dynamics that takes place on atomic length scales. Themodeling and simulation of the onset and evolution of these fea-tures poses significant challenges over current multiscale tech-niques. Typically, predictive models for these phenomena takethe form of Molecular Dynamics simulations, which describe themwith significant accuracy, but are limited by atomic length scalesand femtosecond time scales. Continuum theories permit simulat-ing much larger systems and longer times, but usually fall short ofincorporating the fundamental physical features that governmicrostructure evolution. Recently, a new theory has been put for-ward under the name of phase field crystal equation [17,18,42].This model describes the microstructure of two-phase systemson atomic length scales, but on a diffusive time scale, leading tosignificant computational savings compared to Molecular Dynam-ics simulations. The phase field crystal equation has been em-ployed to simulate a number of physical phenomena, includingcrystal growth in a supercooled liquid, dendritic and eutectic solid-ification, epitaxial growth, and crack propagation in a ductile mate-rial [17,42]. The phase field crystal equation is derived from anenergy functional that is minimized by periodic density fields, nat-urally incorporating the periodicity of a crystal lattice. The model isthen cast as an evolutive sixth-order partial differential equation

ll rights reserved.

ez@udc.es, hgomezd@gmail.

(PDE) that locally minimizes this energy functional under the con-straint of mass conservation.

The numerical simulation of the phase field crystal equation pre-sents several challenges, such as, for example, the discretization ofnonlinear higher-order partial-differential operators and theapproximation of dynamic interfaces that travel over the computa-tional domain. Previous work on the topic include [11,32–34,49,54].Given the fact that the exact solutions to the phase field crystalequation lead to a time-decreasing energy functional, we feel thata significant goal in the numerical simulation of this model is thedevelopment of algorithms that verify this property at the discretelevel irrespectively of the coarseness of the discretization (in whatfollows, algorithms of this type will be called unconditionally en-ergy stable or thermodynamically consistent). Thermodynamicallyconsistent methods have been previously studied in the context ofsolid [3,39,44,45] and fluid mechanics [30,36,47,48], but remain lessinvestigated for phase-field equations (significant works on this to-pic include [16,20,22,24,27,31,51–53]). Remarkably, a second-orderaccurate, unconditionally uniquely solvable algorithm for the phasefield crystal equation has been proposed in [34,54]. This scheme isunconditionally stable with respect to a discrete energy and weaklystable with respect to the physical energy. Here we introduce a newfully-discrete algorithm for the phase field crystal equation that issecond-order time accurate and unconditionally energy stable.Our time integration algorithm is based on a new quadrature for-mula proposed in [27] that may be thought of as a non-symmetrichigher-order extension of the trapezoidal rule. Our space discretiza-tion is based on a new mixed variational form of the phase field crys-tal equation. The well-posedness of our variational form requiresthe use of globally C1-continuous basis functions that we generateusing Isogeometric Analysis [12,35], a recently proposed general-ization of Finite Element Analysis.

H. Gomez, X. Nogueira / Comput. Methods Appl. Mech. Engrg. 249–252 (2012) 52–61 53

We present several numerical examples in two and three dimen-sions dealing with crystal growth in a supercooled liquid, and crackpropagation in a ductile material. These examples show the effec-tiveness of our approach. The outline of this paper is as follows:In Section 2, we describe the phase field crystal equation. Section3 presents our algorithm for this equation. We present numericalexamples in Section 4. Finally, we draw conclusions in Section 5.

2. The phase field crystal equation

The phase field crystal equation describes the microstructure ofsolid–liquid systems at interatomic length scales, and at a diffusivetime scale [17,18]. The two-phase system is described by a localatomistic density field q, which will be approximately uniform inthe liquid phase, and will inherit the symmetry and periodicityof the crystal lattice in the solid phase. The phase field crystalequation has been shown to correctly model the dynamics of crys-tal growth, including naturally elastic and plastic deformations.Other physical phenomena for which the phase field crystal equa-tion has shown potential as a predictive tool include epitaxialgrowth, material hardness, grain growth, reconstructive phasetransitions, and crack propagation in ductile materials [17]. Thephase field crystal equation has also been recently employed tomodel foams [29] and colloidal solidification [50].

The fundamental quantity for the phase field crystal equation isthe following Lyapunov functional:

FðqÞ ¼Z

XUðqÞ þ D

2ðDqÞ2 � 2k2jrqj2 þ k4q2h i� �

dx; ð1Þ

where k and D are positive numbers, and

UðqÞ ¼ � �2q2 � g

3q3 þ 1

4q4: ð2Þ

In Eq. (2), � and g are positive constants with physical significance.The phase field crystal equation was derived as an evolutive PDEthat preserves mass throughout the entire dynamical process, andachieves free-energy dissipation. These requirements lead us tothe equation

@q@t¼ D

dFdq

� �; ð3Þ

where dF=dq denotes the variational derivative of F with respect toq. Note that Eq. (3) follows the typical structure of conserved phase-field models [2,7–10,14,15,21], and satisfies the aforementionedproperties. Using the expression of the variational derivative of Fand the notationuðqÞ ¼ U0ðqÞ, we get the phase field crystal equation

@q@t¼ D uðqÞ þ Dk4qþ 2Dk2Dqþ DD2q

� �; ð4Þ

which involves sixth-order partial derivatives in space.Multiplying Eq. (3) with dF=dq, integrating over the spatial do-

main X, and integrating by parts, the following expression may beobtained,

dFdt¼ �

ZXr dF

dq

� ���������2

dx; ð5Þ

where F is a real-valued function defined as FðtÞ ¼ Fðqð�; tÞÞ. Eq. (5)shows that the energy functional (1) decreases in time over fields qwhich satisfy the phase field crystal equation.

2.1. Initial/boundary-value problem

We state the following initial/boundary-value problem over thespatial domain X and the time interval ð0; TÞ: given q0 : X # R,find q : X� ½0; T�# R such that

@q@t¼ D uðqÞ þ Dk4qþ 2Dk2Dqþ DD2q

� �in X� ð0; TÞ; ð6Þ

r uðqÞ þ Dk4qþ 2Dk2Dqþ DD2q� �

� n ¼ 0 on C� ½0; T�; ð7Þ

rð2Dk2qþ DDqÞ � n ¼ 0 on C� ½0; T�; ð8Þrq � n ¼ 0 on C� ½0; T�; ð9Þqðx;0Þ ¼ q0ðxÞ in X: ð10Þ

3. Numerical formulation for the phase-field crystal equation

Here we present our numerical formulation for the phase fieldcrystal equation. We first derive a semidiscrete formulation, andthen introduce a time integration scheme which preserves massduring the entire dynamical process, and is unconditionally energystable.

3.1. Semidiscrete formulation

To derive the semidiscrete formulation we propose the follow-ing splitting of the phase field crystal equation,

@q@t¼ Dr; ð11Þ

r ¼ uðqÞ þ Dk4qþ 2Dk2Dqþ DD2q: ð12Þ

Let us define the functional space V � H2, where H2 is the Sobolevspace of square integrable functions with square integrable first andsecond derivatives. We derive a weak form of Eqs. (11) and (12) bymultiplying them with functions w; q 2 V, and integrating by parts.At this point, we assume periodic boundary conditions in all direc-tions. The problem can be stated as: find q;r 2 V such that for allw; q 2 V

w;@q@t

� �þ rw;rrð Þ ¼ 0; ð13Þ

q;rð Þ � q;uðqÞ þ Dk4q� �

þ rq;2Dk2rq� �

� Dq;DDqð Þ ¼ 0: ð14Þ

We derive a semidiscrete formulation by replacing (13) and (14) bya finite-dimensional problem defined over the discrete spaceVh � V. The problem can be stated as follows: find qh;rh 2 Vh suchthat for all wh; qh 2 Vh

wh;@qh

@t

� �þ rwh;rrh

¼0 ð15Þ

qh;rh

� qh;uðqhÞþDk4qh� �

þ rqh;2Dk2rqh� �

� Dqh;DDqh

¼0 ð16Þ

Note that the condition Vh � V requires the discrete space to be H2

conforming. We satisfy this requirement using Isogeometric Analy-sis [12,35], a recently proposed generalization of Finite ElementAnalysis. Isogeometric Analysis is based on developments of Com-puter Aided Design (CAD). The main idea of Isogeometric Analysisis to use the parametrizations that underlie CAD designs to generatethe computational mesh and the basis functions necessary for anal-ysis, following the isoparametric concept. This holds promise tosimplify, or even eliminate altogether, the mesh generation andrefinement process, currently the most time-consuming step ofanalysis. CAD parametrizations are usually defined in terms ofNon-Uniform Rational B-Splines (NURBS), although there are otherpossibilities, such as, for example, T-Splines [4] or PHT-Splines [40].NURBS are generated from B-Splines using projective transforma-tions, while B-Splines are simply piecewise polynomials [41,43].Thus, in NURBS-based Isogeometric Analysis, both the computa-tional domain and the solution field are described using NURBS.This not only leads to simpler interface between CAD and analysis,but has also proven superior accuracy than classical Finite Elementson a per-degree-of-freedom basis [1,23]. Isogeometric Analysis has

54 H. Gomez, X. Nogueira / Comput. Methods Appl. Mech. Engrg. 249–252 (2012) 52–61

been successfully applied to a number of problems in solid[13,19,37,38] and fluid mechanics [5,6] showing significant effi-ciency and robustness. Even more importantly for the present work,the use of NURBS permits generating globally C1-continuous basisfunctions easily, which leads to a simple and efficient discretizationof higher-order operators as shown in [25–28]. In what follows, wewill suppose that Vh ¼ spanfNAg; A ¼ 1; . . . ;nb, where NA is a C1-continuous NURBS function associated to the global degree of free-dom A and nb is the dimension of the discrete space.

3.2. Time integration

This section presents our time integration scheme for the phasefield crystal equation. We divide the time interval ½0; T� into subin-tervals In ¼ ½tn; tnþ1� where 0 ¼ t0 < t1 < � � � < tN ¼ T and½0; T� ¼ [N�1

n¼0 In. We define the time step Dtn ¼ tnþ1 � tn. Let us callqh

n the time discrete approximation to qhðtnÞ. Thus, the problemcan be stated as follows: given qh

n, find qhnþ1 2 Vh such that for all

wh; qh 2 Vh

wh;sqh

nt

Dtn

� �þ rwh;rrh

¼ 0; ð17Þ

qh;rh

� qh;12

uðqhnÞþuðqh

nþ1Þ

� sqhnt

2

12u00ðqh

nÞ !

� qh;Dk4qhnþ1=2

� �þ rqh;2Dk2rqh

nþ1=2

� �� Dqh;DDqh

nþ1=2

� �¼ 0;

ð18Þ

where

sqhnt ¼ qh

nþ1 � qhn and qh

nþ1=2 ¼ ðqhnþ1 þ qh

nÞ=2: ð19Þ

We summarize in the following theorem the most relevant proper-ties of our discrete formulation.

Theorem 1. The fully-discrete variational formulation (17) and (18):

(1) Verifies mass conservation, that is,

ZXqh

ndx ¼Z

Xqh

0dx 8n ¼ 1; . . . ;N:

(2) Verifies the nonlinear stability condition

FðqhnÞ 6 Fðqh

n�1Þ 8n ¼ 1; . . . ;N:

irrespectively of the time step.

(3) Gives rise to a local truncation error s that may be bounded asjsðtnÞj 6 KDt2

n for all tn 2 ½0; T�, where K is a constant indepen-dent of Dtn.

Proof

(1) The result can be proven taking wh ¼ 1 in Eq. (17), applyinginductive logic.

(2) The proof relies on the following quadrature formula: Letf : ½a; b�# R be a sufficiently smooth function. It may be pro-ven that

Z b

af ðxÞdx ¼ b� a

2f ðaÞ þ f ðbÞð Þ � ðb� aÞ3

12f 00ðaÞ

� ðb� aÞ4

24f 000ðnÞ; n 2 ða; bÞ: ð20Þ

A complete derivation of this formula may be found in [27]. If weapply this quadrature formula to the right hand side of the identity

Z qhnþ1

qhn

U0ðtÞdt ¼Z qh

nþ1

qhn

uðtÞdt ð21Þ

we get

suðqhnÞt ¼

sqhnt

2uðqh

nÞ þuðqhnþ1Þ

� sqh

nt2

12u00ðqh

nÞ

� sqhnt

4

24u000ðqh

nþ�Þ; � 2 ð0;1Þ: ð22Þ

Now, let us take wh ¼ rh in (17)and qh ¼ sqhnt in (18). It follows that,

rh;sqh

nt

Dtn

� �þ rrh;rrh

¼ 0; ð23Þ

sqhnt;r

h

� sqhnt;

12

uðqhnÞ þuðqh

nþ1Þ

� sqhnt

2

12u00ðqh

nÞ !

� sqhnt;Dk4qh

nþ1=2

� �þ rsqh

nt;2Dk2rqhnþ1=2

� �� Dsqh

nt;DDqhnþ1=2

� �¼ 0: ð24Þ

Taking into account that,

ðsqhnt;q

hnþ1=2Þ ¼

12

ZX

sðqhnÞ

2tdx; ðrsqh

nt;rqhnþ1=2Þ

¼ 12

ZX

sjrqhnj

2tdx; ð25Þ

ðDsqhnt;Dqh

nþ1=2Þ ¼12

ZX

sðDqhnÞ

2tdx ð26Þ

and making use of (22), we conclude that

sFðqhnÞt ¼ �Dtn rrh;rrh

� 1

24sqh

nt4;u000ðqh

nþnÞ� �

ð27Þ

and the result is proven, because u000ðqÞP 0 8q 2 R.(3) We derive a bound on the local truncation error by compar-

ing our method with the midpoint rule, which is known tobe a second-order time-accurate algorithm. Applying themidpoint rule to the semidiscrete formulation of the phasefield crystal equation (15) and (16), we obtain

wh;sqh

nt

Dtn

� �þ rwh;rrh

¼ 0; ð28Þ

qh;rh �uðqhnþ1=2Þ � Dk4qh

nþ1=2

� �þ rqh;2Dk2rqh

nþ1=2

� �� Dqh;DDqh

nþ1=2

� �¼ 0; ð29Þ

where Eq. (29) defines rh, and qhnþ1 is the time discrete solution

using the midpoint rule. The local truncation error of the midpointrule may be obtained by replacing the time discrete solution qh

n

with the time continuous solution qhðtnÞ in Eqs. (28) and (29).The time continuous solution does not satisfy Eqs. (28) and (29),giving rise to the local truncation error. Proceeding this way, weobtain

wh;sqhðtnÞt

Dtn

� �þ rwh;rrh

s

¼ wh; sðtnÞ

; ð30Þ

qh;rhs �u qhðtnþ1=2Þ

� Dk4qhðtnþ1=2Þ

� �þ rqh;2Dk2rqhðtnþ1=2Þ� �

� Dqh;DDqhðtnþ1=2Þ

¼ 0; ð31Þ

where rhs is defined in Eq. (31), sðtnÞ is the local truncation error of

the midpoint rule. Using Taylor series, it may be proven thatjsðtnÞj 6 KDt2

n , where K is a real constant independent of Dtn. If weproceed analogously with our algorithm, we define the local trunca-tion error of our method by replacing the time continuous solutionin Eqs. (17) and (18), which leads to

Fig. 1. Crystal growth in a supercooled liquid. Snapshots of the numerical approximation to the atomistic density field. The parameters of the phase field crystal equation areD ¼ k ¼ 1; g ¼ 0, and � ¼ 0:25. The computational mesh is composed of 20482 C1 quadratic elements. The time step is Dt ¼ 4.

H. Gomez, X. Nogueira / Comput. Methods Appl. Mech. Engrg. 249–252 (2012) 52–61 55

wh;sqhðtnÞt

Dtn

� �þ rwh;rrh

s

¼ wh; sðtnÞ

; ð32Þ

qh;rhs �

12

uðqhðtnÞÞ þuðqhðtnþ1ÞÞ

� sqhðtnÞt2

12u00ðqhðtnÞÞ

�Dk4qhðtnþ1=2Þ�þ rqh;2Dk2rqhðtnþ1=2Þ� �

� Dqh;DDqhðtnþ1=2Þ

¼ 0: ð33Þ

Taking into account that

12

uðqhðtnÞÞ þuðqhðtnþ1ÞÞ

� sqhðtnÞt2

12u00ðqhðtnÞÞ

¼ uðqhðtnþ1=2ÞÞ þ OðDt2nÞ ð34Þ

it follows that,

wh; sðtnÞ

¼ wh; sðtnÞ

þOðDt2nÞ; ð35Þ

qh;rhs

¼ qh;rh

s

þOðDt2nÞ; ð36Þ

which implies that there exists a real constant K, independent of Dtn

such that jsðtnÞj 6 KDt2n , and the pursued result is proven. h

3.3. Implementation

Let Pn and Sn be the global vectors of degrees of freedom asso-ciated to qh

n and its corresponding rh, respectively. We introducethe following residual vectors

RqðPn;Pnþ1; Snþ1Þ; Rq ¼ fRqAg; A ¼ 1; . . . ;nb; ð37Þ

RrðPn;Pnþ1; Snþ1Þ; Rr ¼ fRrAg; A ¼ 1; . . . ;nb; ð38Þ

where

RqA ¼ NA;

sqhnt

Dt

� �þ rNA;rrh

; ð39Þ

RrA ¼ðNA;rhÞ� NA;

12

uðqhnþ1Þþuðqh

nÞ

�sqhnt

2

12u00ðqh

nÞ !

�ðNA;Dk4qhnþ1=2Þþ rNA;2Dk2rqh

nþ1=2

� �� DNA;DDqh

nþ1=2

� �: ð40Þ

When we equate these residual vectors to zero, we obtain a nonlin-ear system of equations for Pnþ1 and Snþ1 that we solve using New-ton’s method.

Let Pnþ1;ðiÞ and Snþ1;ðiÞ be the ith iteration of Newton’s algorithm.Our iterative procedure is defined as follows: Take Pnþ1;ð0Þ ¼ Pn, andSnþ1;ð0Þ ¼ Sn. Then, for i ¼ 1; . . . ; imax.

(1) Compute the residual vectors using the values Pnþ1;ðiÞ; Snþ1;ðiÞ.These will be denoted as Rq

ðiÞ;RrðiÞ.

(2) Compute the tangent matrix KðiÞ using the ith iterates. Thismatrix has a block structure and may be written as

KðiÞ ¼KqqðiÞ Kqr

ðiÞ

KrqðiÞ Krr

ðiÞ

!:

0 2000 4000 6000 8000 10000 12000 14000 16000−3.85

−3.8

−3.75

−3.7

−3.65

−3.6

−3.55

−3.5

−3.45x 104

Fig. 4. Crack propagation on a square domain. Time evolution of the energyfunctional for the two computations presented in Fig. 3. We observe that the energyis decreasing at all times.

0 200 400 600 800 1000 1200 1400 16001.97

1.98

1.99

2

2.01

2.02

2.03

2.04

2.05

2.06x 104

1200 1250 13001.9728

1.9729

1.9729x 104

Fig. 2. Crystal growth in a supercooled liquid. Time evolution of the free energyfunctional for three different time steps. We observe that the energy decreases at alltimes, which confirms that our algorithm is unconditionally stable, as predicted bythe theory. The inset shows the small differences in the energy evolution for theconsidered time steps.

56 H. Gomez, X. Nogueira / Comput. Methods Appl. Mech. Engrg. 249–252 (2012) 52–61

(3) Solve the linear system

KqqðiÞ Kqr

ðiÞ

KrqðiÞ Krr

ðiÞ

!DPðiþ1Þ

DSðiþ1Þ

� �¼ �

RqðiÞ

RrðiÞ

!

using diagonally-preconditioned GMRES [46].(4) Update the solution as,

Fig. 3.step is Dset a smsquare

Pnþ1;ðiþ1Þ

Snþ1;ðiþ1Þ

� �¼

Pnþ1;ðiÞ

Snþ1;ðiÞ

� �þ

DPðiþ1Þ

DSðiþ1Þ

� �:

ig. 5. Crack propagation on a rectangular domain. The computational domain is¼ ½0;2048p=3� � ½0;512p=3�, and the spatial mesh is composed of 2048� 512 C1

uadratic elements. The time step is Dt ¼ 20. The initial condition, shown on top,rresponds to Configuration 1. At the bottom we present the numerical solution at

me t = 110,000.

The process (1)–(4) needs to be repeated until the norms of bothresidual vectors have been reduced to a given tolerance tol of theirinitial value. Taking tol ¼ 10�4, convergence is typically achieved intwo or three iterations.

Remark 1. An important topic that is, however, out of the scope ofthis paper is the solvability of the nonlinear system of Eqs. (37)–(39). In principle, the unique solvability of this nonlinear system of

Crack propagation in a ductile material. The computational domain is X ¼ ½0;102t ¼ 20. The initial condition is a crystal lattice with stretchings of approximatelyall notch. On the left hand side we show the numerical solution using a circular

notch of side 20p=3.

FXqcoti

4p=3�2, and the spatial mesh is composed of 10242 C1 quadratic elements. The time16% and 15% in the x and y directions, respectively. In the center of the domain, wenotch of radius 20p=3. For the computation on the right hand side we employed a

H. Gomez, X. Nogueira / Comput. Methods Appl. Mech. Engrg. 249–252 (2012) 52–61 57

equations may impose a restriction on the time step size. Ournumerical simulations indicate that this potential restriction, ifexisted, would be very mild, because for all the numericalexamples that we performed, we have been able to take timesteps larger than those reported in the literature. We also remarkthat the algorithm proposed by Hu et al. [34] is second-orderaccurate and unconditionally uniquely solvable. The trade-off forthe sake of unconditional solvability is that Hu’s method is notunconditionally stable with respect to the physical energy, butwith respect to a slightly modified energy.

Fig. 6. Crack propagation on a rectangular domain. The computational domain isX ¼ ½0;2048p=3� � ½0;512p=3�, and the spatial mesh is composed of 2048� 512 C1

quadratic elements. The time step is Dt ¼ 20. The initial condition, shown on topcorresponds to Configuration 2. At the bottom we present the numerical solution atime t ¼ 24000.

0 2 4 6 8 10 12x 104

−4.25

−4.2

−4.15

−4.1

−4.05

−4

−3.95x 104

Fig. 7. Crack propagation on a rectangular domain. Time evolution of the freeenergy for two mechanical configurations. We observe that the free energy isdecreasing at all times.

4. Numerical examples

In this section we present some numerical examples for thephase field crystal equation. The examples are related to severalphysical phenomena, such as, for instance, the growth of a poly-crystal in a supercooled liquid, and the dynamic propagation of acrack in a ductile material. Our calculations provide numerical cor-roboration for the theoretical results presented in the previous sec-tions, and illustrate the accuracy, stability and robustness of ournew algorithm.

4.1. Crystal growth in a supercooled liquid in two dimensions

This example presents the growth of a polycrystal in a super-cooled liquid. We simulate the evolution of three crystallites withdifferent orientations. This leads to a complex dynamical processwhich simultaneously involves the motion of liquid–crystal inter-faces and grain boundaries separating the crystals. Similar numer-ical examples may be found in [18,34].

To define the initial configuration we proceed as follows: first;we set all control variables to a constant value �q, which for thisexample takes the value �q ¼ 0:285; second; we modify this con-stant configuration by setting three perfect crystallites in threesmall square patches of the domain as illustrated in Fig. 1(a). Weuse the following expression to define the crystallites:

qðxl; ylÞ ¼ �qþ C cosqffiffiffi3p yl

� �cos qxlð Þ � 0:5 cos

2qffiffiffi3p yl

� �� ; ð41Þ

where xl and yl define a local system of cartesian coordinates that isoriented with the crystallite lattice. The parameters C and q take thevalues C ¼ 0:446, and q ¼ 0:66. To generate crystallites with differ-ent orientations, we define the local coordinates ðxl; ylÞ using an af-fine transformation of the global coordinates ðx; yÞ, that produces arotation given by an angle a. We generated the three crystallitesusing this strategy. We took a ¼ �p=4;a ¼ 0 and a ¼ p=4.Fig. 1(a), which is the computed solution at an early time, gives aprecise idea of the initial condition we employed.

The computational domain for this example is X ¼ ½0;800�2. Onthis domain, we define an uniform computational mesh composedof 20482 C1 quadratic elements. The time step is Dt ¼ 4. Theparameters of the phase field crystal equation take the valuesD ¼ k ¼ 1; g ¼ 0, and � ¼ 0:25.

Fig. 1 shows snapshots of the numerical solution at severalcomputational times. We observe the growth of the crystallinephase and the motion of well-defined crystal–liquid interfaces.The different alignment of the crystallites causes defects and dislo-cations that are clearly observed in the pictures. The solution pre-sents similar features to those obtained in [18,34].

Fig. 2 analyzes the time evolution of the energy functional. Werecomputed this example using three different time steps, namely,Dt ¼ 1; Dt ¼ 2; Dt ¼ 4, and plotted the free-energy evolution. Theplot shows that the free energy is time decreasing in all cases,and the differences between the three cases are negligible. The insetin Fig. 2 is a zoom of the energy evolution in the late dynamics of the

,t

equation. The inset shows that the larger is the time step, the higheris the energy at a given time, which is consistent with our previousexperience with unconditionally stable methods for phase dynam-ics [27]. Additional insight about the dependence of the energy evo-lution on the time step may be obtained examining Eq. (27).

4.2. Crack propagation on a square domain

Following [17], we utilize the phase field crystal equation tomodel crack propagation in a ductile material. We consider asquare domain X ¼ ½0;1024p=3�2. The computational mesh is com-posed of 10242 C1 quadratic elements, and the time step is Dt ¼ 20.We assume periodic boundary conditions in both directions. Theparameters of the phase field crystal equation are D ¼ k ¼ � ¼ 1,and g ¼ 0. As initial condition, we set a crystal lattice given bythe expression

q0ðxÞ ¼ 0:49þ cos qxxð Þ cosqyffiffiffi

3p y� �

� 12

cos2qyffiffiffi

3p y

� �: ð42Þ

A crystal under no mechanical loads would be in equilibrium whenqx ¼ qy ¼

ffiffiffi3p

=2 in the limit �! 0. We take qx ¼ 0:7265625000000,

Fig. 8. Crystal growth in a supercooled liquid in three dimensions. The images show the evolution and interaction of two crystallites initially surrounded by liquid. Thecomputational times are indicated in the labels. On the left hand side, we show isosurfaces of the solution, while on the right hand side we present a slice of the solutionacross the indicated plane. The computational mesh is composed of 1283 C1 quadratic elements. The time step is Dt ¼ 1.

58 H. Gomez, X. Nogueira / Comput. Methods Appl. Mech. Engrg. 249–252 (2012) 52–61

and qy ¼ 0:7307089344312, which induces stretchings with respectto the equilibrium wavelength of approximately 16% and 15% in thex- and y-directions, respectively. These values of qx and qy also en-sure that the initial condition is periodic, and, thus, compatible withboundary conditions. In the center of the domain, we set a smallnotch in which the density takes an homogeneous value of 0.79.We performed simulations using circular and square notches. Thenumerical approximation to the atomistic density field may be ob-served in Fig. 3. We notice that the solution is extremely dependenton the shape of the notch. This is not surprising due to the high non-linearity of the phase field crystal equation and the fact that differ-ent notch shapes induce different stress concentrations.

Fig. 4 shows the time evolution of the energy functional for thetwo computations presented in Fig. 3. We observe that the energyis decreasing at all times.

4.3. . Crack propagation on a rectangular domain

For this calculation we consider the rectangular domainX ¼ ½0;2048p=3� � ½0;512p=3�. The computational mesh is com-posed of 2048� 512 C1 quadratic elements. We assume periodicboundary conditions in both directions. The parameters of thephase field crystal equation are D ¼ k ¼ � ¼ 1, and g ¼ 0. The timestep is Dt ¼ 20. We remark that similar calculations presented in

Fig. 9. Crystal growth in a supercooled liquid in three dimensions. The images show the evolution and interaction of two crystallites initially surrounded by liquid. Thecomputational times are indicated in the labels. On the left hand side, we show isosurfaces of the solution, while on the right hand side we present a slice of the solutionacross the indicated plane. The computational mesh is composed of 1283 C1 quadratic elements. The time step is Dt ¼ 1.

H. Gomez, X. Nogueira / Comput. Methods Appl. Mech. Engrg. 249–252 (2012) 52–61 59

[17] employed a time step 400 times smaller, which indicates thatour method is significantly more effective.

The initial condition is generated using formula (42), but wetake different stretchings than in the last example. We performtwo computations that correspond to two different mechanicalconfigurations.

4.3.1. Configuration 1We define configuration 1 taking qx ¼ 0:8525390625000, and

qy ¼ 0:7713038752455 which induces stretchings of approxi-mately 1% and 11% in the x and y directions, respectively. In thisinitial density field, we define a rectangular notch in which thedensity takes the constant value 0.79. The initial condition may

be observed in Fig. 5(a). As the computation evolves, the crackpropagates over a straight horizontal line, leading to the situationdepicted in Fig. 5(b).

4.3.2. Configuration 2Configuration 2 is defined taking qx ¼ 0:7705078125000, and

qy ¼ 0:7713038752455 which induces stretchings of approxi-mately 11% in both directions. For this example, we set two rect-angular notches in which the density takes the value 0.79. Theinitial condition may be observed in Fig. 6(a). This configurationleads to more complicated cracks that interact with each other.The numerical solution at time t = 24,000 may be observed inFig. 6(b).

0 50 100 150 200−6

−4

−2

0

2

4

6

8x 104

Fig. 10. Crystal growth in a supercooled liquid in three dimensions. The plot showsthat the energy is decreasing at all times. We have appended to the energy curvesome snapshots of the solution to give an indication of the dynamical process thatcorrespond to the energy decay.

60 H. Gomez, X. Nogueira / Comput. Methods Appl. Mech. Engrg. 249–252 (2012) 52–61

Fig. 7 shows the time evolution of the free energy for the twomechanical configurations. It is observed that the energy decreasesat all times.

4.4. Crystal growth in a supercooled liquid in three dimensions

This example deals with the numerical simulation of crystalgrowth in three dimensions. We simulate the growth and interac-tion of two crystallites that originate from two nucleation sites.The computational domain is X ¼ ½0;100�3, and we assume peri-odic boundary conditions in all directions. The parameters of thephase field crystal equation are D ¼ k ¼ � ¼ 1 and g ¼ 0. For thiscalculation, we employed an uniform mesh composed of 1283 C1-quadratic elements. We utilized the time step Dt ¼ 1. The initialconfiguration, depicted in Fig. 8(a) and (b) was generated as fol-lows: we let a randomly perturbed constant (liquid) state evolveto a periodic lattice (solid) state using the set up and parametersmentioned above. We extracted two pieces of the final state withan hexahedric shape, and superposed them to a constant densityfield as shown in Fig. 8(a) and (b). In Fig. 8, snapshot (a) shows iso-surfaces of the density field, while image (b) presents a slice of thesolution across the indicated plane. The evolution of the two crys-tallites may be observed in the second and third rows of Fig. 8(again, the left hand side shows isosurfaces, while the right handside presents a slice of the solution across a plane). At some pointthe crystallites have grown enough as to start interacting as shownin Fig. 9.

The asymptotic state, which can be observed in the third row ofFig. 9 (snapshots (e) and (f)), corresponds to a solid state repre-sented by a periodic lattice.

Fig. 10 shows the time history of the energy functional (1). Itmay be observed that the energy decreases at all times, which,again, provides numerical evidence for our method being uncondi-tionally stable.

5. Conclusions

The phase field crystal equation is a higher-order nonlinear PDEendowed with an stability property. It describes the microstructureof two-phase systems at interatomic length scales. We introducenew space–time discretizations that inherit the nonlinear stability

relationship of the continuous equation irrespectively of the meshand time step sizes, and that are second-order time-accurate. Weutilize our new algorithm to compute a number of numericalexamples that deal with several physical phenomena, such as, forexample, crystal growth, and dynamic crack propagation. Theseexamples provide numerical corroboration for our theoretical re-sults, and show the accuracy, efficiency and robustness of ournew method.

Acknowledgements

The authors were partially supported by Xunta de Galicia (Grants# 09REM005118PR and 09MDS00718PR), Ministerio de Ciencia eInnovacion (Grant # DPI2009-14546-C02-01) cofinanced withFEDER funds, and Universidad de A Coruña.

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