International Journal of Heat and Mass Transfer 55 (2012) 7926–7932

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Phase-field simulations of crystal growth with adaptive mesh refinement

Yibao Li, Junseok Kim ⇑Department of Mathematics, Korea University, Seoul 136-713, Republic of Korea

a r t i c l e i n f o

Article history:Received 9 January 2012Received in revised form 26 June 2012Accepted 7 August 2012Available online 4 September 2012

Keywords:Crystal growthPhase-field simulationOperator splittingMultigrid methodAdaptive mesh refinement

0017-9310/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.08

⇑ Corresponding author. Tel.: +82 2 3290 3077; faxE-mail addresses: yibaoli@korea.ac.kr (Y. Li), cfdkimURL: http://math.korea.ac.kr/~cfdkim/ (J. Kim).

a b s t r a c t

In this paper, we propose the phase-field simulation of dendritic crystal growth in both two- and three-dimensional spaces with adaptive mesh refinement, which was designed to solve nonlinear parabolicpartial differential equations. The proposed numerical method, based on operator splitting techniques,can use large time step sizes and exhibits excellent stability. In addition, the resulting discrete systemof equations is solved by a fast numerical method such as an adaptive multigrid method. Comparisonsto uniform mesh method, explicit adaptive method, and previous numerical experiments for crystalgrowth simulations are presented to demonstrate the accuracy and robustness of the proposed method.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Crystal growth is a phase transformation from the liquid phaseto the solid phase via heat transfer [1]. Predicting the shape ofgrowing crystals is important for industrial crystallization pro-cesses [2]. Various numerical methods such as boundary integral[3–6], cellular automata [7–10], front-tracking [11–15], level-set[16–19], Monte-Carlo [20,21], and phase-field [22–42] have beendeveloped to simulate crystal growth. Among these methods, thephase-field approach is widely used for modeling solidificationproblems since it avoids the explicit tracking of macroscopicallysharp phase boundaries [29].

A great challenge in the simulation of crystal growth with var-ious supercoolings is the large difference in time and length scales.The adaptive mesh refinement is faster and more efficient than auniform mesh in simulating crystal growth because it allows con-centration of effort and multi-resolution in space and time [27–41]. However most previous adaptive phase-field computationsof dendrite crystal growth suffer from severe time step restrictionssince they use explicit schemes [34–38]. Since the discrete Lapla-cian is used in the explicit scheme, its stability criterion becomesDt � O(h2), where Dt is the time step and h is the mesh size. Thusthe crystal growth simulation with various supercoolings is stillvery difficult.

To use large time steps, Rosam et al. [39,40] proposed a fully im-plicit, fully adaptive time and space discretization method for the

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: +82 2 929 8562.@korea.ac.kr (J. Kim).

crystal growth simulation. Its stability is almost unconditionallystable although it is computationally more expensive than explicitones per time step. To have better stability properties, a multipletime-step algorithm was presented in [41]. A larger time step isused for the flow-field calculations while reserving a finer timestep for the phase-field evolution.

In our previous work [42], we introduced a fast, robust, andaccurate operator splitting method for phase-field simulations ofcrystal growth with uniform mesh size, which allows large timesteps, e.g., Dt � O(h). The main purpose of the present paper is toextend our previous work by incorporating adaptive mesh refine-ment. We will demonstrate stability, robustness, and accuracy ofthe proposed method by a set of representative numericalexperiments.

This paper is organized as follows. In Section 2, we give the gov-erning equations for crystal growth based on the phase-field mod-el. In Section 3, computationally efficient operator splittingalgorithm and adaptive mesh refinement are described. In Sec-tion 4, we present numerical results for solving the crystal growthsimulation both in two and three dimensions. Finally, conclusionsare given in Section 5.

2. The phase-field model

The basic equations of the phase-field model can be derivedfrom a single Lyapounov functional [43]. We model the solidifica-tion in two and three dimensions using a standard form of phase-field equations. The model is given by

Y. Li, J. Kim / International Journal of Heat and Mass Transfer 55 (2012) 7926–7932 7927

�2ð/Þ @/@t¼ r � ð�2ð/Þr/Þ þ ½/� kUð1� /2Þ�ð1� /2Þ

þ jr/j2�ð/Þ @�ð/Þ@/x

� �x

þ jr/j2�ð/Þ @�ð/Þ@/y

!y

þ jr/j2�ð/Þ @�ð/Þ@/z

� �z

; ð1Þ

@U@t¼ DDU þ 1

2@/@t; ð2Þ

where / 2 [�1,1] is the order parameter with / = 1 in the solidphase and / = �1 in the liquid phase. / = 0 is defined as the inter-face of two phases. �(/) is the anisotropic function, k is the dimen-sionless coupling parameter, and U = cp(T � TM)/L is thedimensionless temperature field. Here cp is the specific heat at con-stant pressure, TM is the melting temperature, L is the latent heat offusion, D ¼ as0=�2

0; a is the thermal diffusivity, s0 is the character-istic time, �0 is the characteristic length. k is given as k = D/a2 witha2 = 0.6267 [28,29]. For the fourfold symmetry, anisotropic function�(/) with anisotropy strength �4, is defined as:

�ð/Þ ¼ ð1� 3�4Þ 1þ 4�4

1� 3�4

/4x þ /4

y þ /4z

jr/j4

!:

Fig. 1. Block-structured local refinement with four levels.

3. Numerical solution

In this section, we propose a robust hybrid numerical methodfor the crystal growth simulation. For simplicity of exposition, weshall discretize Eqs. (1) and (2) in two-dimensional space, i.e.,X = (a,b) � (c,d). Three-dimensional discretization is analogouslydefined. Let Nx and Ny be positive even integers, h = (b � a)/Nx bethe uniform mesh size, and Xh = {(xi,yj): xi = (i � 0.5)h, yj =(j � 0.5)h, 1 6 i 6 Nx, 1 6 j 6 Ny} be the set of cell-centers. Let /n

ij

be approximations of /(xi,yj,nDt), where Dt = T/Nt is the time step,T is the final time, and Nt is the total number of time steps. Eqs. (1)and (2) are discretized in a similar manner as in [42]:

�2ð/nÞ/nþ1;1 � /n

Dt¼ 2�ð/nÞ�ð/nÞx/

nx þ 2�ð/nÞy�ð/

nÞ/ny

þ16�ð/nÞ�4/x /2

x/2y � /4

y

� �jrd/j4

0@ 1An

x

þ16�ð/nÞ�4/y /2

x/2y � /4

x

� �jrd/j4

0@ 1An

y

; ð3Þ

/nþ1;2 ¼ /nþ1;1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie� 2Dt�2ð/n Þ þ ð/nþ1;1Þ2 1� e

� 2Dt�2ð/n Þ

� �r ; ð4Þ

�2ð/nÞ/nþ1 � /nþ1;2

Dt¼ �2ð/nÞDd/

nþ1 � 4kUnFð/nþ1;2Þ; ð5Þ

Unþ1 � Un

Dt¼ DDdUnþ1 þ /nþ1 � /n

2Dt; ð6Þ

where the discrete differentiation operator is rd/ij = (/i+1,j � /i�1,j,/i,j+1 � /i,j�1)/(2h) and the discrete Laplacian operator is Dd/ij =(/i+1,j + /i�1,j � 4/ij + /i,j+1 + /i,j�1)/h2. With nine local points, wedescribe the following term

/x�ð/Þ /2x /

2y �/4

y

� �jrd/j4

0@ 1An

x;ij

¼

/x�ð/Þ /2x /2

y�/4yð Þ

jrd/j4

� �n

iþ12;j

� /x�ð/Þ /2x /2

y�/4yð Þ

jrd/j4

� �n

i�12;j

h;

where

/x�ð/Þ /2x /

2y �/4

y

� �jrd/j4

0@ 1An

iþ12;j

¼� /n

iþ1;j

� �þ � /n

ij

� �� �/n

iþ1;j �/nij

� �3/n

iþ1;jþ1 �/niþ1;j�1 þ/n

i;jþ1 �/ni;j�1

� �2=8

h /niþ1;j �/n

ij

� �2þ /n

iþ1;jþ1 �/niþ1;j�1 þ/n

i;jþ1 �/ni;j�1

� �2=4

� �2

þ d

�� /n

ij

� �þ � /n

i�1;j

� �� �/n

iþ1;j �/nij

� �/n

iþ1;jþ1 �/niþ1;j�1 þ/n

i;jþ1 �/ni;j�1

� �4=32

h /niþ1;j �/n

ij

� �2þ /n

iþ1;jþ1 �/niþ1;j�1 þ/n

i;jþ1 �/ni;j�1

� �2=4

� �2

þ d

:

The other terms can be described in a similar manner. Note that weadded a small value d = 1e � 10 in the denominator jrd/j4 to avoidsingularities.

To solve the resulting system of discrete Eqs. (3)–(6) at the im-plicit time level, we use an adaptive mesh refinement [44,45],whose schematic diagram is shown in Fig. 1. In the adaptive ap-proach, we introduce a hierarchy of increasingly finer grids,Xlþ1; . . . ;Xlþl� , restricted to smaller and smaller subdomains, whilethe last hierarchy of global grids are X0,X1, . . . ,Xl. That is, we con-sider a hierarchy of grids, X0;X1; . . . ;Xlþ0;Xlþ1; . . . ;Xlþl� . Here wedenote Xl+0 as level zero, Xl+1 as level one, and so on. For example,in Fig. 1, l⁄ = 3.

The grid is adapted dynamically based on the undivided gradi-ent. First, we tag cells that contain the front, i.e., those in which theundivided gradient of the phase-field is greater than a critical va-lue. Then, the tagged cells are grouped into rectangular patchesby using a clustering algorithm as in Ref. [47]. These rectangularpatches are refined to form the grids at the next level. The processis repeated until a specified maximum level is reached.

Next, we describe the adaptive full approximation storage cycleto solve the discrete system on the hierarchy of increasingly finergrids. First, let us rewrite Eqs. (5) and (6) as

Nð/nþ1;Unþ1Þ ¼ ðun;wnÞ;

where Nð/nþ1;Unþ1Þ ¼ /nþ1

Dt � Dd/nþ1; Unþ1

Dt � DDdUnþ1 � /nþ1

2Dt

� �and the

source term is ðun;wnÞ ¼ /nþ1;2

Dt �4kUnFð/nþ1;2Þ

�2ð/nÞ ; 2Un�/n

2Dt

� �:

Using the above notations on all levels k = 0,1, . . . , l, l +1, . . . , l + l⁄, an adaptive multigrid cycle is formally written as fol-lows [46]:

7928 Y. Li, J. Kim / International Journal of Heat and Mass Transfer 55 (2012) 7926–7932

3.1. Adaptive cycle

We calculate unk ; wn

k on all levels and set the previous time solu-tion as the initial guess, i.e., /0

k ;U0k

� �¼ /n

k ;Unk

� �.

/mþ1k ;Umþ1

k

� �¼ ADAPTIVEcycle k;/m

k ;/mk�1;U

mk ;U

mk�1;Nk;un

k ;wnk ; m

� �:

(1) Presmoothing- Compute ð�/m

k ;Umk Þ by applying m smoothing steps to

/mk ;U

mk

� �on Xk.

Fig. 2

�/mk ;U

mk

� �¼ SMOOTHm /m

k ;Umk ;Nk;un

k ;wnk

� �;

where one SMOOTH relaxation operator step consists of solving Eqs.(9) and (10) given below by a 2 � 2 matrix inversion for each i and j.Rewriting Eqs. (5) and (6), we get

1Dtþ 4

h2

� �/nþ1

ij ¼ unij �

/nþ1iþ1;j þ /nþ1

i�1;j þ /nþ1i;jþ1 þ /nþ1

i;j�1

h2 ; ð7Þ

1Dtþ 4D

h2

� �Unþ1

ij �/nþ1

ij

2Dt¼ wn

ij �D Unþ1

iþ1;j þ Unþ1i�1;j þ Unþ1

i;jþ1 þ Unþ1i;j�1

� �h2 : ð8Þ

Next, we replace /nþ1kl and Unþ1

kl in Eqs. (7) and (8) with �/mkl and Um

kl ifk 6 i and l 6 j; otherwise we replace them with /m

kl and Umkl , i.e.,

1Dtþ 4

h2

� ��/m

ij ¼ unij �

/miþ1;j þ �/m

i�1;j þ /mi;jþ1 þ �/m

i;j�1

h2 ; ð9Þ

1Dtþ 4D

h2

� �Um

ij ��/m

ij

2Dt¼ wn

ij �D Um

iþ1;j þ Umi�1;j þ Um

i;jþ1 þ Umi;j�1

� �h2 : ð10Þ

(2) Coarse-grid correction- Compute

�/mk�1;U

mk�1

� �¼ Ik�1

k�/m

k ;Umk

� �on Xk�1 \Xk;

/mk�1;U

mk�1

� �on Xk�1 �Xk:

(

Fig. 3. A sample adaptive mesh in different views in two d

(a) (b). The stability of crystal growth with different time steps: (a) Dt = 2.24 (256 � 2

- Compute the coarse grid source term

imensio

56 mesh

unk�1;w

nk�1

� �¼

Ik�1k un

k ;wnk

� �� Nk

�/mk ;U

mk

� � ;

þNk�1Ik�1k

�/mk ;U

mk

� �on Xk�1 \Xk;

unk�1;w

nk�1

� �on Xk�1 �Xk:

8><>:

- Compute an approximate solution ð/m

k�1;bUm

k�1Þ of the coarsegrid equation on Xk�1, i.e.,

Nk�1 /mk�1;U

mk�1

� �¼ un

k�1;wnk�1

� �: ð11Þ

If k = 1, we explicitly invert a 2 � 2 matrix to obtain the solution. Ifk > 1, we solve Eq. (11) by using ð�/m

k�1;Umk�1Þ as an initial approxima-

tion to perform an adaptive multigrid k-grid cycle:

/mk�1;

bUmk�1

� �¼ADAPTIVEcycle ðk�1; �/m

k�1;/mk�2;U

mk�1;U

mk�2;Nk�1;un

k�1;wnk�1;mÞ:

- Compute the correction at Xk�1 \Xk. umk�1; vm

k�1

� �¼

/mk�1;

bUmk�1

� �� �/m

k�1;Umk�1

� �.

- Set the solution at the other points of Xk�1 �Xk.

/mþ1k�1 ;U

mþ1k�1

� �¼ /m

k�1;bUm

k�1

� �.

- Interpolate the correction to Xk. umk ; vm

k

� �¼ Ik

k�1 umk�1; vm

k�1

� �:

- Compute the corrected approximation on Xk.

/m; after CGCk ;Um; after CGC

k

� �¼ �/m

k þ umk ;U

mk þ vm

k

� �:

(3) Postsmoothing

/mþ1k ;Umþ1

k

� �¼ SMOOTHm /m; after CGC

k ;Um; after CGCk ;Nk;un

k ;wnk

� �:

This completes the description of an adaptive multigrid cycle.For additional details about the adaptive multigrid cycle, please re-fer to [46].

ns. (a) whole view, (b) and (c) closeup views.

(c)), (b) Dt = 1.17 (512 � 512 mesh), and (c) Dt = 0.59 (1024 � 1024 mesh).

Fig. 4. The evolution for crystal growth in two dimensions.

Y. Li, J. Kim / International Journal of Heat and Mass Transfer 55 (2012) 7926–7932 7929

4. Numerical results

In this section, we perform numerical experiments for two- andthree-dimensional solidification to validate that our proposedscheme is accurate, efficient, and robust. For two-dimensionaltests, unless otherwise specified, we take the initial state as

/ðx; y;0Þ ¼ tanhR0 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

pffiffiffi2p

!;

Uðx; y;0Þ ¼ Dð1� /ðx; y;0ÞÞ2

�0 if / > 0D else:

�

Fig. 5. Snapshots of three-dimensional evolution of crystal growth. (a) crystal shape at timt = 0,240, and 480.

The zero level set (/ = 0) represents a circle of radius R0. From thedimensionless variable definition, the value U = 0 corresponds tothe melting temperature of the pure material, while U = D is the ini-tial undercooling. The capillary length d0 is defined as d0 = a1/k[39,43], where a1 = 0.8839 and k = 3.1913 [39]. And the otherparameters are chosen as �4 = 0.05 and D = 2.

4.1. Stability of our proposed algorithm

As mentioned in Section 1, explicit schemes [34–38] suffer fromtime step restrictions Dt 6 O(h2) for the stability. In order to over-come the restriction of time step, Li et al. [42] proposed a fast, ro-bust, and accurate operator splitting method for phase-fieldsimulations of crystal growth. The authors showed that their meth-od can use large time step sizes Dt � O(h). Here, we consider a setof increasingly finer grids to show the stability of our proposedmethod. The computational domain is X = (�100,100)2 with2n � 2n mesh grids for n = 8, 9, and 10. The numerical solutionsare computed up to T = 128.91, with time steps Dt = 3h. Otherparameters are R0 = 14d0 and D = �0.55. Fig. 2 shows crystal shapeat time T = 128.91 with different time steps. Form these results, wecan observe that our proposed method also allows large time stepsizes.

4.2. Evolution for crystal growth in two- and three-dimensional spaces

In this experiment, we will show the evolution of crystal growthin two and three dimensions with adaptive mesh method. A sam-ple two dimensional adaptive meshes with different views areshown in Fig. 3.

The computational domains are set as X = (�800,800)2 withl⁄ = 4 levels in two dimensions and X = (�200,200)3 with l⁄ = 4 lev-els in three dimensional case. And other parameters are chosen asDt = 0.4, R0 = 14d0, and D = �0.55. The calculations are run up totime T = 8000 in two dimensions and T = 480 in three dimensions.

es t = 0,20,40,160,240, and 480 (from left to right). (b) The bounding boxes at times

Table 1Comparison of CPU time and tip positions calculated by the explicit scheme and ourproposed scheme.

Method Time step Tip position CPU time (h)

Explicit method 0.01 45.94 0.10Proposed method 0.01 45.91 0.60Proposed method 0.20 44.48 0.02

7930 Y. Li, J. Kim / International Journal of Heat and Mass Transfer 55 (2012) 7926–7932

Fig. 4 shows the temporal evolution of crystal interface. Fig. 5(a)shows three-dimensional structures at times t = 0,20,40,160,240,and 480 (from left to right). And in Fig 5(b), we show the boundingboxes at times t = 0,240, and 480 to show the structured localrefinement.

4.3. Comparison between our proposed method and explicit adaptivemethod

In general, an explicit scheme is fast, however, the overall CPUtime for long time integration larger than an implicit scheme,which can use larger time steps. This is true for the adaptivemethod used here. In order to show our proposed method is moreefficient than explicit adaptive methods, we consider CPU timecomparison test in two dimensions. The computational domainis set as X = (�200,200)2 with l⁄ = 3 levels. The minimum elementsize hmin = 0.39, R0 = 15d0, and D = �0.55 are used. Since the expli-cit adaptive method surfers the time step size limitation, here weuse a time step Dt = 0.01. For our proposed method, we use timestep Dt = 0.01 and Dt = 0.2. The calculations are run up to time

(a)

(c)

Fig. 6. The comparison between uniform meshes and adaptive meshes in two and three d(a) crystal shape at t = 1800. (c) y-z plane of crystal shape at t = 240. (b) and (d) are clo

T = 200. We list the tip position of crystal and CPU time in Table 1.Form these results, we can observe that our proposed methodwith Dt = 0.01 needs more CPU time than the explicit method.However, with Dt = 0.2 our proposed method is about 5 times fas-ter than the explicit method. Tip positions with different methodsare similar.

4.4. Comparison with uniform mesh simulation

In this experiment, we will compare the results obtained onuniform meshes and on adaptively refined meshes for two andthree dimensions to show the efficiency and accuracy of our pro-posed method. For two dimensions, the computational domain isset as X = (�400,400)2 with 1024 � 1024 mesh grids for uniformmesh calculation and l⁄ = 3 levels for the adaptive mesh method.And in three dimensions, we use X = (�100,100)3,256 � 256 � 256, and l⁄ = 3. With time step Dt = 0.3, the calcula-tions are run up to time T = 1800 and T = 240 in two and threedimensions, respectively. The comparisons with uniform meshesand adaptively refined meshes in two and three dimensions aredrawn in the first row and in the second row of Fig. 6, respectively.As can be observed, the agreement between the results computedby uniform meshes and adaptive meshes is good. In the two-dimensional calculation, the taken CPU times are 16.15 h and0.82 h for uniform and adaptive meshes, respectively. In thethree-dimensional calculation, the taken CPU times are 29.15 hand 2.71 h for uniform and adaptive meshes, respectively.

(b)

(d)

imensions. First and second rows are two and three dimensional cases, respectively.seup views of (a) and (b), respectively.

Table 2Comparison of dimensionless steady-state tip velocities calculated by the proposedscheme (Vtip = Vd0/D), the results in [42] V LLK

tip

� �, results in [29] VKR

tip

� �, and Green’s

function calculations VGFtip

� �.

D �4 D d0/W0 VtipLLK Vtip

KR VtipGF Vtip

�0.55 0.05 2 0.277 0.01710 0.01680 0.01700 0.01700�0.55 0.05 3 0.185 0.01740 0.01750 0.01700 0.01720�0.55 0.05 4 0.139 0.01720 0.01740 0.01700 0.01710�0.50 0.05 3 0.185 0.01030 0.01005 0.00985 0.00997�0.45 0.05 3 0.185 0.00599 0.00557 0.00545 0.00537

100 101 102 103 104 10510−6

10−5

10−4

10−3

10−2

Time t

V tip

Δ =−0.25[32]Δ =−0.1[32]Present studySolvability theory

Fig. 7. Evolution of tip velocity at D = �0.25 and D = �0.1. In order to compare theresults in [32] and the results computed by solvability theory, we put themtogether.

Table 3Comparison of dimensionless steady-state tip velocities calculated by our proposedmethod and the analytic solution.

Case D = �0.25 D = �0.1

Analytic solution 0.00251 0.000139Numerical solution 0.00252 0.000148

Y. Li, J. Kim / International Journal of Heat and Mass Transfer 55 (2012) 7926–7932 7931

4.5. Comparison of the dimensionless steady-state tip velocities

To demonstrate the accuracy of our proposed method, we com-pare the dimensionless steady-state tip velocities obtained by ourproposed adaptive method with the results in [29,42] and Green’sfunction calculations [29]. This test is performed on the domainX = (�200,200)2 with l⁄ = 3, the minimum element size hmin = 0.39,Dt = 0.15, R0 = 3.462, W0 = 1, and k = D/a2.

To calculate the steady-state velocity, we use a quadratic poly-nomial approximation. We only describe the procedure along they-axis since the crystal is symmetric. Given three points, (xk�1,yk�1), (xk,yk), and (xk+1,yk+1) on the interface, where yk is a maxi-mum value. Let the quadratic polynomial approximation passingthese three points be y = ax2 + bx + c. Then we can find the tip po-sition y⁄ which satisfies the following conditions: y0(x⁄) = 0 andy� ¼ ax2

� þ bx� þ c.The crystal tip velocity is defined as the finite difference of tip

positions from consecutive time steps. From a set of numerical re-sults in Table 2, we can observe that values obtained by our pro-posed scheme are in good agreement with results of previousmethods over the whole range of D, �4, D, and d0/W0 investigatedhere.

4.6. Dendritic growth at low undercooling

For low undercoolings, it requires much longer time to reach asteady-state tip velocity due to the lower growth rate. Furthermorean extremely large domain should be used to avoid the far fieldboundary effect. In this case, the adaptive mesh refinement meth-od is a better choice to overcome it. Here, we consider low under-coolings such as D = �0.25 and D = �0.1. The computationaldomain is X = (�6400,6400)2 with the base mesh grids, 32 � 32.l⁄ = 9 is used and the minimum grid spacing hmin is 0.78. With timestep Dt = 0.4, the numerical solutions for D = �0.25 and D = �0.1are computed up to T = 4000 and T = 40000, respectively. Otherparameters are d0 = 0.403, R0 = 100d0, D = 13, and k = 20.744.Fig. 7 shows the evolution of tip velocity (Vtip) for D = �0.25 andD = �0.1 with our proposed method, the results in [32], and solv-ability theory [49,50]. The numerical results show good agreementwith previous results.

Next, we consider steady-state tip velocities in three dimen-sions. There is the simple relationship which was obtained byIvantsov [50]:

D ¼ �Pe expðPeÞZ 1

Pe

expð�sÞs

ds;

where Pe = RtipVtip/(2D) is the Peclet number and Rtip is the tipradius. The stability constant r ¼ 2Dd0= R2

tipVtip

� �was used in

[49,50]. Here we choose r = 0.02. Thus for given D, D, and d0, wecan compute Vtip. In numerical experiment, we perform the simula-tion on the domain X = (�100,100) � (�100,100) � (0,800) withthe base mesh grids, 8 � 8 � 32. Here l⁄ = 5 and the minimum gridspacing hmin = 0.78 are used. The other parameters are same asthose used in two-dimensional space except for R0 = 50d0.

Comparisons with theoretical solutions are drawn in Table 3. Fromthese results, we can observe that dimensionless steady-state tipvelocities obtained by our proposed scheme are in good agreementwith the analytic solutions at low undercoolings.

5. Conclusion

In this paper, we have proposed the phase-field simulation ofdendritic crystal growth in both two- and three-dimensionalspaces with adaptive mesh refinement, which was designed tosolve nonlinear parabolic partial differential equations. The pro-posed operator splitting numerical method can use large time stepsizes and exhibits excellent stability. The resulting discrete systemof equations is solved by an adaptive multigrid method. Compari-sons to uniform mesh method, explicit adaptive method, and pre-vious numerical experiments for crystal growth simulations werepresented to demonstrate the accuracy and robustness of the pro-posed method. In numerical experiments, the stability of the meth-od was found as Dt 6 3h. Compared to uniform mesh method andexplicit adaptive method, our method achieved the equivalentaccuracy with less computational cost. A set of computations forthe dimensionless steady-state tip velocity showed a good agree-ment with results published in [29,42]. In particular, for lowundercoolings, a good agreement with previous study [32] andanalytic solution [48–50] was found as well. These simulationsconfirm that our proposed method is efficient and accurate.

Acknowledgments

This research was supported by Basic Science Research Programthrough the National Research Foundation of Korea (NRF) fundedby the Ministry of Education, Science and Technology (No. 2011-0023794). The authors also wish to thank the anonymous referee

7932 Y. Li, J. Kim / International Journal of Heat and Mass Transfer 55 (2012) 7926–7932

for the constructive and helpful comments on the revision of thisarticle.

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